LQG - Loop Quantum Gravity

2011-11-29T18:15:00.000-08:00

QFT: The cross-section for graviton scatter, scaled by Feynman’s rules from the weak interaction

Article finished at 2am on 30 Nov. (7.3 MB PDF file, 63 pages, downloadable here.) Also hosted at http://rxiv.org/abs/1111.0111, the General Science Journal, and (with some extra material) as the brief mail-order paperback book, Quantum Gravity and the Standard Model, ISBN 978-1-4709-9745-8, which is being listed on Amazon for those who want to help ensure a warmer global future (by turning trees into paper). High-quality book cover PDF (featuring yours truly on the back) is linked here.

Dr Tommaso Dorigo comments on his post Higgs Expectations:

"... in order to really prove that our understanding of electroweak symmetry breaking is flawed and that there is no Higgs boson we would need a much, much more solid evidence than a mere "95% exclusion". I would not be satisfied with anything less than a 99.9% exclusion (over three sigma) across the full mass range.

But I do not honestly believe that we will ever get into such a situation. I do believe, in fact, that the particle is there, and that it will be found very soon! So stay tuned and place your bets if you haven't already. Time is running short.

We avoid the usual electroweak symmetry breaking problem, by changing electromagnetism from U(1) to a massless SU(2) gauge theory (which works out correctly, yielding Maxwell's equations from the Yang-Mills, because charged massless vector bosons can't propagate asymmetrically), so that SU(2) becomes a complete electroweak theory. (This is fine for the weak bosons, while the apparent discrepancy between weak isospin charges and fractional quark electric charges disappears with a vacuum polarization model, which predicts that 1/3 or 2/3 of the electric charge energy of quarks is present as strong colour charge.) U(1) is not abandoned altogether; it is dark energy, which also predicts gravity. The mass of SU(2) weak bosons is then produced by the Glashow-Weinberg mixing of U(1) gravity with SU(2) electromagnetism. Instead of a electroweak symmetry being broken to yield Nambu-Goldstone "Higgs" bosons, the weak interaction emerges from a simple mixing of SU(2) electromagnetism with U(1) gravity. I'll try to get a briefer paper done, ready to replace the Higgs boson.

Update (7 December 2011):

http://www.science20.com/quantum_diaries_survivor/alejandro_rivero_fermion_mass_coincidences_and_other_fun_ideas-85187?nocache=1

"Then Koide went some steps beyond and considered quarks and leptons with substructure, so that lepton mass quotients could predict the Cabibbo angle too, even if this is a mixing between quarks."

{(sqrt(M_e)+sqrt(M_mu)+sqrt(M_tau))^2} /( M_e + M_mu +M_tau) = 2/3

The key factor of 2/3 in the Koide relationship is the fractional electric charge of the up/charm/truth quarks, which arises from a mixing effect. It's the 2/3 electric charge of up/charm/truth quarks that's so interesting. The -1/3 charge of the down/strange/bottom quarks is very easily predicted by analysis of vacuum polarization for the case of the omega minus baryon (Fig. 31 in http://rxiv.org/pdf/1111.0111v1.pdf). It appears that the square root of the product of two very different masses gives rise to an intermediate mass (see http://nige.wordpress.com/2009/08/26/koide-formula-seen-from-a-different-perspective/ for the simple maths) that the Koide relationship implies a bootstrap model of fundamental particles (akin to the bootstrap concept Geoffrey Chew was trying to develop to explain the S-matrix in the 1960s before quarks were discovered). The square root of the product of the masses of a neutrino and a massive weak boson may give an electron mass, for instance. This seems to be the deeper significance of the Koide formula, from my perspective for what it's worth. All fundamental particles are connected by various offshell field quanta exchanges, so their "charges" are dependent on other charges around them. This means that the ordinary approach of analysis fails, because of the reductionist fallacy. If your mathematical model of rope is the same for 100 one-foot lengths as for a single 100 foot length, it leads to customer complaints when you automatically send a sailor the former, not the latter. It's no good patiently explaining to the sailor that mathematically they are identical, and the universe is mathematical. If the Koide formula is correct, then it points to an extension of the square root nature of the Dirac equation. Dirac made the error of ignoring Maxwell's 1861 paper on magnetic force mechanisms: the chiral handedness of magnetism (the magnetic field curls left-handed around the direction of propagation of an electron) is explained in Maxwell's theory by the spin of "field quanta" (Maxwell had gear cogs, but in QFT it's just the spin angular momentum of field quanta). Maxwell's theory makes EM an SU(2) Yang-Mills theory, throwing a different light on the Dirac's spinor. It just so happens that the Yang-Mills equations automatically reduce to Maxwell's if the field quanta are massless, because of the infinite self-inductance of electrically charged field quanta, so SU(2) Maxwellian electromagnetism in practice looks indistinguishable from Abelian U(1), explaining the delusions in modern physics.

The very interesting results Alejandro Rivero gives are from equation 4 on page 3 of his paper http://www.vixra.org/abs/1111.0062, which solves the Koide formula by writing one mass in terms of the two lepton other generation masses. Koide's formula also implies (my 2009 post):

Me + Mm + Mt = 4 * [(Me * Mm)^1/2 + (Me * Mt)^1/2 + (Mm * Mt)^1/2]

where Me = electron mass, Mm = muon mass, Mt = tauon mass. I.e., the simple sum of lepton masses equals four times the sum of square roots of the products of all combinations of the masses, making it seem that if Koide's formula is physically meaningful, then Geoffrey Chew's bootstrap theory of particle democracy must apply to masses (gravitational charge) in 4-d. At high energy, early in the universe, tauons, muons and electrons were all represented and we only see an excess of electrons today because the other generations have decayed, although some of the other masses may actually exist as dark matter, and thus still undergoes the interaction of graviton exchange, which determines the Koide mass spectrum today (this dark matter is analogous to right-handed neutrinos). The basic physics of the Koide formula seems to be the Chew bootstrap applied to gravitation (Chew applied it to the strong force, pre-QCD):

"By the end of the 1950s, [Geoffrey] Chew was calling this [analytic development of Heisenberg’s empirical scattering or S-matrix] the bootstrap philosophy. Because of analyticity, each particle’s interactions with all others would somehow determine its own basic properties and ... the whole theory would somehow ‘pull itself up by its own bootstraps’.” - Peter Woit, Not Even Wrong, Jonathan Cape, London, 2006, p148. (Emphasis added.)

The S-matrix went out when the SM was developed (although S-matrix results were used to help determine the Feynman rules), but at some stage a Chew-type bootstrap mechanism for Koide's mass formula may be needed to further develop a physical understanding for the underlying theory of mass mixing, leading to a full theory of mixing angles for both gravitation (mass) and weak SU(2) interactions of leptons and quarks.

There are pedagogy and presentation problems: "Darwin's theory of evolution is disproved because Lamarke had an evolution theory before Darwin, which was wrong." (Joseph McCarthy's "guilt by false association", applies to LeSage's gravity mechanism. If one person gets something wrong, nobody else is ever allowed to correct the errors in it. Darwin was only able to proceed by pretending that Lamarke hadn't existed. Science is not a logical system where errors get corrected. It's a political process whereby theories are pre-judged in their incorrect nascent state, then dismissed for ever when found incorrect. If someone else later corrects all the errors, that person is wrong by being "associated", much like friends of people who turned out to have been student communist party members were guilty of being Stalin's friends, in McCarthy's eyes. This story is of course usually turned around to a very different conclusion: the fact that McCarthy was wrong in shooting everyone who had ever heard of Marx was used to try to "defend" Stalin's evil, a kind of one upmanship or reversal of McCarthy's trick. Anyone criticising Stalin was then compared to McCarthy, and their message unheeded. Science is more political than normal politics, because it pretends that there is no political element and uses this deception to "disprove" the need for democratic debate, etc. Science is the worst sort of politics, the sort which pretends it's always justified by good intentions, no matter the consequences, exactly like Stalinism and Nazism, but Godwin help you if you say it.)

2008-06-01T07:08:00.000-07:00

Fig. 1 - Newton's geometric proof that an impulsive pushing graviton mechanism is consistent with Kepler's 3rd law of planetary motion, because equal areas will be swept out in equal times (the three triangles of equal area, SAB, SBC and SBD, all have an equal base of length SB, and they all have altitudes of equal length), together with a diagram we will use for a more modern analysis. Newton's geometric proof of centripetal acceleration, from his book Principia, applies to any elliptical orbit, not just circular orbits as Hooke's easier inverse-square law derivation did. (Newton didn't include the graviton arrow, of course.) By Pythagoras' theorem x² = r² + v²t², hence x = (r² + v²t²)^1/2. Inward motion, y = x - r = (r² + v²t²)^1/2 - r = r[(1 + v²t²/r²)^1/2 - 1], which upon expanding with the binomial theorem to the first two terms, yields: y ~ r[(1 + (1/2)v²t²/r²) - 1] = (1/2)v²t²/r. Since this result is accurate for infidesimally small steps (the first two terms of the binomial become increasingly accurate as the steps get smaller, as does the approximation of treating the triangles as right-angled triangles so Pythagoras' theorem can be used), we can accurately differentiate this result for y with respect to t to give the inward velocity, u = v²t/r. Inward acceleration is the derivative of u with respect to t, giving a = v²/r. This is the centripetal force formula which is required to obtain the inverse square law of gravity from Kepler's third law: Hooke could only derive it for circular orbits, but Newton's geometric derivation (above, using modern notation and algebra) applies to elliptical orbits as well. This was the major selling point for the inverse square law of gravity in Newton's Principia over Hooke's argument.

See Newton's Principia, Book I, The Motion of Bodies, Section II: Determination of Centripetal Forces, Proposition 1, Theorem 1:

'The areas which revolving bodies describe by radii drawn to an immovable centre of force ... are proportional to the times on which they are described. For suppose the time to be divided into equal parts ... suppose that a centripetal [inward directed] force acts at once with a great impulse [like a graviton], and, turning aside the body from the right line ... in equal times, equal areas are described ... Now let the number of those triangles be augmented, and their breadth diminished in infinitum ... QED.'

This result, in combination with Kepler's third law, gives the inverse-square law of gravity, although Newton's argument is using geometry plus hand-waving so it is actually far less rigorous than my rigorous algebraic version above. Newton failed to employ calculus and the binomial theorem to make his proof more rigorous, because he was the inventor of them, and most readers wouldn't be familiar with those methods. (It doesn't do to be so inventive as to both invent a new proof and also invent a new mathematics to use in making that proof, because readers will be completely unable to understand it without a large investment of time and effort; so Newton found that it payed to keep things simple and to use old-fashioned mathematical tools which were widely understood.)

Newton in addition worked out an ingeniously simple proof, again geometrically, to demonstrate that a solid sphere of uniform density (or radially symmetric density) has the same net gravity on the surface and at any distance, for all of its atoms in their three dimensional distribution, as would be the case if all the mass was concentrated in a point in the middle of the Earth. The proof for that is very simple: consider the sphere to be made up of a lot of concentric shells, each of small thickness. For any given shell, the geometry is such as that a person on the surface experiences small gravity effects from small quantities of mass nearby on the shell, while most of the mass of the shell is located at large distances. The inverse square effect, which means that for equal quantities of mass, the most nearby mass creates the strongest gravitational field, is thereby offset by the actual locations of the masses: only small amounts are nearby, and most of the mass of the shell is at a great distance. The overall effect is that the effective location for the entire mass of the shell is in the middle of the shell, which implies that the effective location of the mass of a solid sphere seen from a distance is in the middle of the sphere (if the density of each of the little shells, considered to be parts of the sphere, is uniform).

Feynman discusses the Newton proof in his November 1964 Cornell lecture on 'The Law of Gravitation, an Example of Physical Law', which was filmed for a BBC2 transmission in 1965 and can viewed on google video here (55 minutes). Feynman in his second filmed November 1964 lecture, 'The Relation of Mathematics to Physics', also on google video (55 minutes), stated:

'People are often unsatisfied without a mechanism, and I would like to describe one theory which has been invented of the type you might want, that this is a result of large numbers, and that's why it's mathematical. Suppose in the world everywhere, there are flying through us at very high speed a lot of particles ... we and the sun are practically transparent to them, but not quite transparent, so some hit. ... the number coming [from the sun's direction] towards the earth is less than the number coming from the other sides, because they meet an obstacle, the sun. It is easy to see, after some mental effort, that the farther the sun is away, the less in proportion of the particles are being taken out of the possible directions in which particles can come. So there is therefore an impulse towards the sun on the earth that is inversely as square of the distance, and is the result of large numbers of very simple operations, just hits one after the other. And therefore, the strangeness of the mathematical operation will be very much reduced the fundamental operation is very much simpler; this machine does the calculation, the particles bounce. The only problem is, it doesn't work. .... If the earth is moving it is running into the particles .... so there is a sideways force on the sun would slow the earth up in the orbit and it would not have lasted for the four billions of years it has been going around the sun. So that's the end of that theory. ...

'It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of spacetime is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities.’

The error Feynman makes here is that quantum field theory tells us that there are particles of exchange radiation mediating forces normally, without slowing down the planets: this exchange radiation causes the FitzGerald-Lorentz contraction and inertial resistance to accelerations (gravity has the same mechanism as inertial resistance, by Einstein's equivalence principle in general relativity). So the particles do have an effect, but only as a once-off resistance due to the compressive length change, not continuous drag. Continuous drag requires a net power drain of energy to the surrounding medium, which can't occur with gauge boson exchange radiation unless acceleration is involved, i.e., uniform motion doen't involve acceleration of charges in such a way that there is a continuous loss of energy, so uniform motion doesn't involve continuous drag in the sea of gauge boson exchange radiation which mediates forces! The net energy loss or gain during acceleration occurs due to the acceleration of charges, and in the case of masses (gravitational charges), this effect is experienced by us all the time as inertia and momentum; the resistance to acceleration and to deceleration. The physical manifestation of these energy changes occurs in the FitzGerald-Lorentz transformation; contractions of the matter in the length parallel to the direction of motion, accompanied by related relativistic effects on local time measurements and upon the momentum and thus inertial mass of the matter in motion. This effect is due to the contraction of the earth in the direction of its motion. Feynman misses this entirely. The contraction of the earth's radius by this mechanism of exchange radiation (gravitons) bouncing off the particles, gives rise to the empirically confirmed general relativity law due to conservation of mass-energy for a contracted volume of spacetime, as proved in an earlier post. So it is two for the price of one: the mechanism predicts gravity but also forces you to accept that the Earth's radius shrinks, which forces you to accept general relativity, as well. Additionally, it predicts a lot of empirically confirmed facts about particle masses and cosmology, which are being better confirmed by experiments and observations as more experiments and observations are done.

As pointed out in a previous post giving solid checkable predictions for the strength of quantum gravity and observable cosmological quantities, etc., due to the equivalence of space and time, there are 6 effective dimensions: three expanding time-like dimensions and three contractable material dimensions. Whereas the universe as a whole is continuously expanding in size and age, gravitation contracts matter by a small amount locally, for example the Earth's radius is contracted by the amount 1.5 mm as Feynman emphasized in his famous Lectures on Physics. This physical contraction, due to exchange radiation pressure in the vacuum, is not only a contraction of matter as an effect due to gravity (gravitational mass), but it is also a contraction of moving matter (i.e., inertial mass) in the direction of motion (the Lorentz-FitzGerald contraction).

This contraction necessitates the correction which Einstein and Hilbert discovered in November 1915 to be required for the conservation of mass-energy in the tensor form of the field equation. Hence, the contraction of matter from the physical mechanism of gravity automatically forces the incorporation of the vital correction of subtracting half product of the metric and the trace of the Ricci tensor, from the Ricci tensor of curvature. This correction factor is the difference between Newton's law of gravity merely expressed mathematically as 4 dimensional spacetime curvature with tensors and the full Einstein-Hilbert field equation; as explained on an earlier post, Newton's law of gravitation when merely expressed in terms of 4-dimensional spacetime curvature gives the wrong deflection of starlight and so on. It is absolutely essential to general relativity to have the correction factor for conservation of mass-energy which Newton's law (however expressed in mathematics) ignores. This correction factor doubles the amount of gravitational field curvature experienced by a particle going at light velocity, as compared to the amount of curvature that a low-velocity particle experiences. The amazing thing about the gravitational mechanism is that it yields the full, complete form of general relativity in addition to making checkable predictions about quantum gravity effects and the strength of gravity (the effective gravitational coupling constant, G). It has made falsifiable predictions about cosmology which have been spectacularly confirmed since first published in October 1996. The first major confirmation came in 1998 and this was the lack of long-range gravitational deceleration in the universe. It also resolves the flatness and horizon problems, and predicts observable particle masses and other force strengths, plus unifies gravity with the Standard Model. But perhaps the most amazing thing concerns our understanding of spacetime: the 3 dimensions describing contractable matter are often asymmetric, but the 3 dimensions describing the expanding spacetime universe around us look very symmetrical, i.e. isotropic. This is why the age of the universe as indicated by the Hubble parameter looks the same in all directions: if the expansion rate were different in different directions (i.e., if the expansion of the universe was not isotropic) then the age of the universe would appear different in different directions. This is not so. The expansion does appear isotropic, because those time-like dimensions are all expanding at a similar rate, regardless of the direction in which we look. So the effective number of dimensions is 4, not 6. The three extra time-like dimensions are observed to be identical (the Hubble constant is isotropic), so they can all be most conveniently represented by one 'effective' time dimension.

Only one example of a very minor asymmetry in the graviton pressure from different directions, resulting from tiny asymmetries in the expansion rate and/or effective density of the universe in different directions, has been discovered and is called the Pioneer Anomaly, an otherwise unaccounted-for tiny acceleration in the general direction toward the sun (although the exact direction of the force cannot be precisely determined from the data) of (8.74 ± 1.33) × 10⁻¹⁰ m/s² for long-range space probes, Pioneer-10 and Pioneer-11. However these accelerations are very small, and to a very good approximation, the three time-like dimensions - corresponding to the age of the universe calculated from the Hubble expansion rates in three orthagonal spatial dimensions - are very similar.

Therefore, the full 6-dimensional theory (3 spatial and 3 time dimensions) gives the unification of fundamental forces; Riemann’s suggestion of summing dimensions using the Pythagorean sum ds² = å (dx²) could obviously include time (if we live in a single velocity universe) because the product of velocity, c, and time, t, is a distance, so an additional term d(ct)² can be included with the other dimensions dx², dy², and dz². There is then the question as to whether the term d(ct)² will be added or subtracted from the other dimensions. It is clearly negative, because it is, in the absence of acceleration, a simple resultant, i.e., dx²+ dy² + dz² = d(ct)², which implies that d(ct)² changes sign when passed across the equality sign to the other dimensions: ds² = å (dx²) = dx²+ dy² + dz² – d(ct)² = 0 (for the absence of acceleration, therefore ignoring gravity, and also ignoring the contraction/time-dilation in inertial motion); This formula, ds² = å (dx²) = dx²+ dy² + dz² – d(ct)², is known as the ‘Riemann metric’ of Minkowski spacetime. It is important to note that it is not the correct spacetime metric, which is precisely why Riemann did not discover general relativity back in 1854.

Professor Georg Riemann (1826-66) stated in his 10 June 1854 lecture at Gottingen University, On the hypotheses which lie at the foundations of geometry: ‘If the fixing of the location is referred to determinations of magnitudes, that is, if the location of a point in the n-dimensional manifold be expressed by n variable quantities x₁, x₂, x₃, and so on to x_n, then … ds = Ö [å (dx)²] … I will therefore term flat these manifolds in which the square of the line-element can be reduced to the sum of the squares … A decision upon these questions can be found only by starting from the structure of phenomena that has been approved in experience hitherto, for which Newton laid the foundation, and by modifying this structure gradually under the compulsion of facts which it cannot explain.’

[The algebraic Newtonian-equivalent (for weak fields) approximation in general relativity is the Schwarzschild metric, which, ds² = (1 – 2GM/r)^-1(dx² + dy² + dz²) – (1 – 2GM/r) d(ct)². This only reduces to the special relativity metric for the impossible, unphysical, imaginary, and therefore totally bogus case of M = 0, i.e., the absence of gravitation. However this does not imply that general relativity proves the postulates of special relativity. For example, in general relativity the velocity of light changes as gravity deflects light, but special relativity denies this. Because the deflection in light, and hence velocity change, is an experimentally validated prediction of general relativity, that postulate in special relativity is inconsistent and in error. For this reason, it is misleading to begin teaching physics using special relativity.]

WARNING: I've made a change to the usual tensor notation below and, apart from the conventional notation in the Christoffel symbol and Riemann tensor, I am indicating covariant tensors by positive subscript and contravariant by negative subscript instead of using indices (superscript) notation for contravariant tensors. The reasons for doing this will be explained and are to make this post easier to read for those unfamiliar with tensors but familiar with ordinary indices (it doesn't matter to those who are familiar with tensors, since they will know about covariant and contravariant tensors already).

Professor Gregorio Ricci-Curbastro (1853-1925) took up Riemann’s suggestion and wrote a 23-pages long article in 1892 on ‘absolute differential calculus’, developed to express differentials in such a way that they remain invariant after a change of co-ordinate system. In 1901, Ricci and Tullio Levi-Civita (1873-1941) wrote a 77-pages long paper on this, Methods of the Absolute Differential Calculus and Their Applications, which showed how to represent equations invariantly of any absolute co-ordinate system. This relied upon summations of matrices of differential vectors. Ricci expanded Riemann’s system of notation to allow the Pythagorean dimensions of space to be defined by a line element or ‘Riemann metric’ (named the ‘metric tensor’ by Einstein in 1916):

g = ds² = g_{m n} dx_-mdx_-n. The meaning of such a tensor is revealed by subscript notation, which identify the rank of tensor and its type of variance.

‘The special theory of relativity … does not extend to non-uniform motion … The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion. Along this road we arrive at an extension of the postulate of relativity… The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant). … We call four quantities A_v the components of a covariant four-vector, if for any arbitrary choice of the contravariant four-vector B^v, the sum over v, å A_v B^v = Invariant. The law of transformation of a covariant four-vector follows from this definition.’ – Albert Einstein, ‘The Foundation of the General Theory of Relativity’, Annalen der Physik, v49, 1916.

The rank is denoted simply by the number of letters of subscript notation, so that X_a is a ‘rank 1’ tensor (a vector sum of first-order differentials, like net velocity or gradient over applicable dimensions), and X_ab is a ‘rank 2’ tensor (for second order differential vectors, like acceleration). A ‘rank 0’ tensor would be a scalar (a simple quantity without direction, such as the number of particles you are dealing with). A rank 0 tensor is defined by a single number (scalar), a rank 1 tensor is a vector which is described by four numbers representing components in three orthagonal directions and time, a rank 2 tensor is described by 4 x 4 = 16 numbers, which can be tabulated in a matrix. By definition, a covariant tensor (say, X_a) and a contra-variant tensor of the same variable (say, X_-a) are distinguished by the way they transform when converting from one system of co-ordinates to another; a vector being defined as a rank 1 covariant tensor. Ricci used lower indices (subscript) to denote the matrix expansion of covariant tensors, and denoted a contra-variant tensor by superscript (for example xⁿ). But even when bold print is used, this is still ambiguous with power notation, which of course means something completely different (the tensor xⁿ = x¹+ x² + x³+... xⁿ, whereas for powers or indices xⁿ = x₁ x₂ x₃ ...x_n). [Another step towards ‘beautiful’ gibberish then occurs whenever a contra-variant tensor is raised to a power, resulting in, say (x²)², which a logical mortal (who’s eyes do not catch the bold superscript) immediately ‘sees’ as x⁴,causing confusion.] We avoid the ‘beautiful’ notation by using negative subscript to represent contra-variant notation, thus x_-nis here the contra-variant version of the covariant tensor x_n. Einstein wrote in his original paper on the subject, ‘The Foundation of the General Theory of Relativity’, Annalen der Physik, v49, 1916: ‘Following Ricci and Levi-Civita, we denote the contravariant character by placing the index above, and the covariant by placing it below.’

This was fine for Einstein who had by that time been working with the theory of Ricci and Levi-Civita for five years, but does not have the clarity it could have. (A student who is used to indices from normal algebra finds the use of index notation for contravariant tensors absurd, and it is sensible to be as unambiguous as possible.) If we expand the metric tensor for m and n able to take values representing the four components of space-time (1, 2, 3 and 4 representing the ct, x, y, and z dimensions) we get the awfully long summation of the 16 terms added up like a 4-by-4 matrix (notice that according to Einstein’s summation convention, tensors with indices which appear twice are to be summed over):

g = ds² = g_mn dx_-mdx_-n= å (g_{m n} dx_-m dx_-n )= -(g₁₁ dx_-1 dx_-1 + g₂₁ dx_-2 dx_-1 + g₃₁ dx_-3 dx_-1 + g₄₁ dx_-4 dx_-1) + (-g₁₂ dx_-1 dx_-2 + g₂₂ dx_-2 dx_-2 + g₃₂ dx_-3 dx_-2 + g₄₂ dx_-4 dx_-2) + (-g₁₃ dx_-1 dx_-3 + g₂₃ dx_-2 dx_-3 + g₃₃ dx_-3 dx_-3 + g₄₃ dx_-4 dx_-3) + (-g₁₄ dx_-1 dx_-4 + g₂₄ dx_-2 dx_-4 + g₃₄ dx_-3 dx_-4 + g₄₄ dx_-4 dx_-4)

The first dimension has to be defined as negative since it represents the time component, ct. We can however simplify this result by collecting similar terms together and introducing the defined dimensions in terms of number notation, since the term dx_-1 dx_-1 = d(ct)², while dx_-2 dx_-2 = dx², dx_-3 dx_-3 = dy², and so on. Therefore:

g = ds² = g_ct d(ct)² + g_x dx² + g_y dy² + g_z dz² + (a dozen trivial first order differential terms).

It is often asserted that Albert Einstein (1879-1955) was slow to apply tensors to relativity, resulting in the 10 years long delay between special relativity (1905) and general relativity (1915). In fact, you could more justly blame Ricci and Levi-Civita who wrote the long-winded paper about the invention of tensors (hyped under the name ‘absolute differential calculus’ at that time) and their applications to physical laws to make them invariant of absolute co-ordinate systems. If Ricci and Levi-Civita had been competent geniuses in mathematical physics in 1901, why did they not discover general relativity, instead of merely putting into print some new mathematical tools? Radical innovations on a frontier are difficult enough to impose on the world for psychological reasons, without this being done in a radical manner. So it is rare for a single group of people to have the stamina to both invent a new method, and to apply it successfully to a radically new problem. Sir Isaac Newton used geometry, not his invention of calculus, to describe gravity in his Principia, because an innovation expressed using new methods makes it too difficult for readers to grasp. It is necessary to use familiar language and terminology to explain radical ideas rapidly and successfully. Professor Morris Kline describes the situation after 1911, when Einstein began to search for more sophisticated mathematics to build gravitation into space-time geometry:

‘Up to this time Einstein had used only the simplest mathematical tools and had even been suspicious of the need for "higher mathematics", which he thought was often introduced to dumbfound the reader. However, to make progress on his problem he discussed it in Prague with a colleague, the mathematician Georg Pick, who called his attention to the mathematical theory of Ricci and Levi-Civita. In Zurich Einstein found a friend, Marcel Grossmann (1878-1936), who helped him learn the theory; and with this as a basis, he succeeded in formulating the general theory of relativity.’ (M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1990, vol. 3, p. 1131.)

General relativity equates the mass-energy in space to the curvature of motion (acceleration) of an small test mass, called the geodesic path. Readers who want a good account of the full standard tensor manipulation should see the page by Dr John Baez or a good book by Sean Carroll, Spacetime and Geometry: An Introduction to General Relativity.

Curvature is best illustrated by plotting a graph of distance versus time and when the line curves (as for an accelerating car) that curve is ‘curvature’. It’s the curved line on a space-time graph that marks acceleration, be that acceleration due to a force acting upon gravitational mass or inertial mass (the equivalence principle of general relativity means that gravitational mass = inertial mass).

This point is made very clearly by Professor Lee Smolin on page 42 of the USA edition of his 1996 book, ‘The Trouble with Physics.’ See Figure 1 in the post here. Next, in order to mathematically understand the Riemann curvature tensor, you need to understand the operator (not a tensor) which is denoted by the Christoffel symbol (superscript here indicates contravariance):

G _ab^c = (1/2)g^cd [(dg_da/dx^b) + (dg_db/dx^a) + (dg_ab/dx^d)]

The Riemann curvature tensor is then represented by:

R^a_cbe = ( dG _bc^a /dx^e ) – ( dG _be^a /dx^c ) + (G _te^a G _bc^t ) – (G _tb^a G _ce^t ).

If there is no curvature, spacetime is flat and things don’t accelerate. Notice that if there is any (fictional) ‘cosmological constant’ (a repulsive force between all masses, opposing gravity an increasing with the distance between the masses), it will only cancel out curvature at a particular distance, where gravity is cancelled out (within this distance there is curvature due to gravitation and at greater distances there will be curvature due to the dark energy that is responsible for the cosmological constant). The only way to have a completely flat spacetime is to have totally empty space, which of course doesn’t exist, in the universe we actually know.

To solve the field equation, use is made of the simple concepts of proper lengths and proper times. The proper length in spacetime is equal to cò (- g_mn dx_-m dx_-n)^1/2, while the proper time is ò (g_{m n} dx_-mdx_-n)^1/2.

Notice that the ratio of proper length to proper time is always c. The Ricci tensor is a Riemann tensor contracted in form by summing over a = b, so it is simpler than the Riemann tensor and is composed of 10 second-order differentials. General relativity deals with a change of co-ordinates by using Fitzgerald-Lorentz contraction factor, g = (1 – v²/c²)^1/2. Karl Schwarzschild produced a simple solution to the Einstein field equation in 1916 which shows the effect of gravity on spacetime, which reduces to the line element of special relativity for the impossible, not-in-our-universe, case of zero mass. Einstein at first built a representation of Isaac Newton’s gravity law a = MG/r² (inward acceleration being defined as positive) in the form R_{m n} = 4p GT_{m n} /c², where T_mn is the mass-energy tensor, T_{m n} = r u_m u_n. ( This was incorrect since it did not include conservation of energy.) But if we consider just a single dimension for low velocities (g = 1), and remember E = mc², then T_{m n} = T₀₀ = r u² = r (g c)² = E/(volume). Thus, T_{m n} /c² is the effective density of matter in space (the mass equivalent of the energy of electromagnetic fields). We ignore pressure, momentum, etc., here:

Above: the components of the stress-energy tensor (image credit: Wikipedia).

The scalar term sum or "trace" of the stress-energy tensor is of course the sum of the diagonal terms from the top left to the top right, hence the trace is just the sum of the terms with subscripts of 00, 11, 22, and 33 (i.e., energy-density and pressure terms).

The velocity needed to escape from the gravitational field of a mass (ignoring atmospheric drag), beginning at distance x from the centre of mass, by Newton’s law will be v = (2GM/x)^1/2, so v² = 2GM/x. The situation is symmetrical; ignoring atmospheric drag, the speed that a ball falls back and hits you is equal to the speed with which you threw it upwards (the conservation of energy). Therefore, the energy of mass in a gravitational field at radius x from the centre of mass is equivalent to the energy of an object falling there from an infinite distance, which by symmetry is equal to the energy of a mass travelling with escape velocity v. By Einstein’s principle of equivalence between inertial and gravitational mass, this gravitational acceleration field produces an identical effect to ordinary motion. Therefore, we can place the square of escape velocity (v² = 2GM/x) into the Fitzgerald-Lorentz contraction, giving g = (1 – v²/c²)^1/2 = [1 – 2GM/(xc²)]^1/2.

However, there is an important difference between this gravitational transformation and the usual Fitzgerald-Lorentz transformation, since length is only contracted in one dimension with velocity, whereas length is contracted equally in 3 dimensions (in other words, radially outward in 3 dimensions, not sideways between radial lines!), with spherically symmetric gravity. Using the binomial expansion to the first two terms of each: Fitzgerald-Lorentz contraction effect: g = x/x₀ = t/t₀ = m₀/m = (1 – v²/c²)^1/2 = 1 – ½v²/c² + ... . Gravitational contraction effect: g = x/x₀ = t/t₀ = m₀/m = [1 – 2GM/(xc²)]^1/2 = 1 – GM/(xc²) + ..., where for spherical symmetry ( x = y = z = r), we have the contraction spread over three perpendicular dimensions not just one as is the case for the FitzGerald-Lorentz contraction: x/x₀ + y/y₀ + z/z₀ = 3r/r₀. Hence the radial contraction of space around a mass is r/r₀ = 1 – GM/(xc²) = 1 – GM/[(3rc²]. Therefore, clocks slow down not only when moving at high velocity, but also in gravitational fields, and distance contracts in all directions toward the centre of a static mass. The variation in mass with location within a gravitational field shown in the equation above is due to variations in gravitational potential energy. The contraction of space is by (1/3) GM/c². This physically relates the Schwarzschild solution of general relativity to the special relativity line element of spacetime.

This is the 1.5-mm contraction of earth’s radius Feynman obtains, as if there is pressure in space. An equivalent pressure effect causes the Lorentz-FitzGerald contraction of objects in the direction of their motion in space, similar to the wind pressure when moving in air, but without molecular viscosity (this is due to the Schwinger threshold for pair-production by an electric field: the vacuum only contains fermion-antifermion pairs out to a small distance from charges, and beyond that distance the weaker fields can't cause pair-production - i.e., the energy is below the IR cutoff - so the vacuum contains just bosonic radiation without pair-production loops that can cause viscosity; for this reason the vacuum compresses macroscopic matter without slowing it down by drag). Feynman was unable to proceed with the LeSage gravity and gave up on it in 1965.

More information can be found in the earlier posts here, here, here, here, here and here.

For the record

2007-07-02T12:34:00.000-07:00

"Yes, I have verified that the only talk at "Loops 2007" that was not a meaningless pile of nonsense was a popular talk by Moshe Rozali about the current state of background independence according to string theory." - Harvard Assistant physics Professor Lubos Motl, http://motls.blogspot.com/2007/07/strings-2007-summary.html

From the "fast comments" included in the above-quoted blog post:

Lubos: Your comment about the Loops talks is as stupid as insulting, as you haven't heard any of them. -B.

PS: Yes, Moshe's talk was very interesting and definitly among the better ones.

Bee Homepage 07.02.07 - 9:26 am #

Dear Bee, I have read abstracts plus relevant pieces of bulk of papers of all these talks except for Lee Smolin who hasn't written it down but who spoke about left-right asymmetric octopi after plastic surgery.

I know exactly Moshe's opinions about background independence and he surely made a lot of sense but I insist that his talk was a popular talk. Crackpots at these conferences - such as Loops 2007 - have the problem that they are not able to recognize real science from its popularization.

They are not able to recognize patent stupidities from scientific arguments, and neither can you.

Luboš Motl Homepage 07.02.07 - 9:33 am #

It's tragic to see such brilliant physicists insulting one another like this.

Not Even Wrong book inspired comment on LQG

2006-06-21T02:25:00.000-07:00

Copy of comment in case it is deleted:

http://christinedantas.blogspot.com/2006/06/blog-mania.htm;

nigel said...
Some genuine questions I have about the loops of LQG, inspired by Woit's "Not Even Wrong", pages 189-190.

A loop is a rotational transformation in the vacuum. Is the loop physically the exchange of energy-delivering field radiation from one mass to another, and back to the first mass again?

Like the exchange radiation in Yang-Mills (Standard Model) theories, but with the added restriction of the conservation (looping between masses) of the exchange radiation?

Things accelerated by a gravity field are losing gravitational potential energy and gaining kinetic energy, so the exchange radiation carries energy.

If the LQG spinfoam vacuum does describes a Yang-Mills energy exchange scheme, you can get solid checkable predictions by taking account of the effect of the expansion of the universe on these conserved gravity field mediators.

If silly, please delete.
6/21/2006 11:35:07 AM

(I don't think it is silly, but have to show some humility or it looks awful.)

'In loop quantum gravity, the basic idea is to use the standard methods of quantum theory, but to change the choice of fundamental variables that one is working with. It is well known among mathematicians that an alternative to thinking about geometry in terms of curvature fields at each point in a space is to instead think about the holonomy [whole rule] around loops in space. The idea is that in a curved space, for any path that starts out somewhere and comes back to the same point (a loop), one can imagine moving along the path while carrying a set of vectors, and always keeping the new vectors parallel to older ones as one moves along. When one gets back to where one started and compares the vectors one has been carrying with the ones at the starting point, they will in general be related by a rotational transformation. This rotational transformation is called the holonomy of the loop. It can be calculated for any loop, so the holonomy of a curved space is an assignment of rotations to all loops in the space.'

- P. Woit, Not Even Wrong, Cape, London, 2006, p189.

He goes on, on the same page, to discuss the problem that loop quantum gravity doesn't seem to say anything useful about the Standard Model (Yang-Mills type quantum field theory).

String theory will collapse on 1 April 2006...

2006-03-30T04:47:00.000-08:00

New comment on Dr Motl's blog

2006-03-30T01:22:00.000-08:00

Dr Lubos Motl, esteemed Harvard University string theorist, has a new post up on his blog: http://motls.blogspot.com/2006/03/searching-for-answers.html

There he says that because string theory is in his opinion the 'theory of everything', it may potentially be used by the police to decide what murder suspects to arrest:

'Such a theory might, in principle, also include the answer to the question Why did Mary Winkler kill her husband... The only problem is that our current knowledge of string theory is not sufficient to calculate observables such as the motive of this particular murder. It is also too primitive to be able to save Matthew Winkler's life ...'

For more about the con of string theory, see here and here. I notice that a forthright commentator on Lubos' blog has doubts, too. In case Lubos deletes the comment, it is reproduced here:

Before "theory of everything" - string theory - is used to predict murder suspects, it should be used to predict something about the subject called PHYSICS, you silly ...

Dr Lubos Motl and WMAP

2006-03-17T11:26:00.000-08:00

Dr Motl provides entertainment both in his field of stringy theories and in climatic studies, and there is a funny new webpage about both: http://lubos-motl.blogspot.com/

WMAP results are out. Dr Motl uses it to bash someone sensible.

Dr Danny Ross Lunsford, author of a very interesting abstract unification of electrodynamics and general relativity (see a recent post on my other blog), has written on Peter Woit's blog of his slight disappointment with the uncritical attitude with which the data interpretation of WMAP is being being done:

D R Lunsford Says: March 17th, 2006 at 1:44 am

This is my last comment for this blog, I’m done with all of this. You can read me in forthcoming papers if you’re interested. Otherwise, my last comment is this:

Penrose completely destroyed the idea of inflation using nothing but entropy, yet we are supposed to believe, after years of looking and a long wait, that the octupole anomaly magically cleared itself up (and not a word was said about the ecliptic anomaly, so there was nothing to clear up). In fact, the work of Huterer, Copi, and Starkman was not even mentioned. Instead, we got a dissertation on applied inflation theory.

So, who should I believe? An authentic genius, or a star chamber of people with, shall we say, vested interests?

Science is dead. Nice job y’all.

-drl

For the context of his reference to Penrose, you need to recall Lunsford's earlier comment:

D R Lunsford Says: January 31st, 2006 at 1:04 am

John - even Penrose is ignored. He gives the simplest conceivable argument showing that inflation, to take an example, not only doesn’t fix what it’s supposed to fix, it actually makes it worse*. The alternative for it is to resort to anthropism. Since no one seems to be willing to abandon inflation, we can only assume that anthropism has already taken a seat.
Penrose’s argument can be understood by an undergraduate. So why doesn’t it have a prominent place?

-drl
* http://www.princeton.edu/WebMedia/lectures/
Look for Penrose lecture “Fantasy”

D R Lunsford Says: February 1st, 2006 at 4:37 pm

John,
Penrose’s lecture was fascinating. His bizarre visualization of phase space was sort of unnerving (like all fractals). Although you couldn’t call him a dynamic speaker, his argument was so clear that it held my attention. Well worth attending.

-drl

I've put a comment to reply to the stringy-theory dominated Cosmic Variance blog reports on WMAP findings (it has a good but too hat-raising-to-orthodox-ineptitude review of WMAP): http://cosmicvariance.com/2006/03/16/wmap-results-cosmology-makes-sense/

Science on Mar 17th, 2006 at 10:01 am

This is hyped up to get media attention: the CBR from 300,000 years after BB says nothing of the first few seconds, unless you believe their vague claims that the polarisation tells something about the way the early inflation occurred. That might be true, but it is very indirect.

I do agree with Sean on CV that n = 0.95 may be an important result from this analysis. I’d say it’s the only useful result. But the interpretation of the universe as 4% baryons, 22% dark matter and 74% dark energy is a nice fit to the existing LambdaCDM epicycle theory from 1998. The new results on this are not too different from previous empirical data, but this ‘nice consistency’ is a euphemism for ‘useless’.

WMAP has produced more accurate spectral data of the fluctuations, but that doesn’t prove the ad hoc cosmological interpretation which was force-fitted to the data in 1998. Of course the new data fits the same ad hoc model. Unless there was a significant error in the earlier data, it would do. Ptolemies universe, once fiddled, continued to model things, with only occasional ‘tweaks’, for centuries. This doesn’t mean you should rejoice.

Dark matter, dark energy, and the tiny cosmological constant describing the dark energy, remain massive epicycles in current cosmology. The Standard Model has not been extended to include dark matter and energy. It is not hard science, it’s a very indirect interpretion of the data. I’ve got a correct prediction made without a cosmological constant made and published in ‘96, years before the ad hoc Lambda CDM model. Lunsford’s unification of EM and GR also dismisses the CC. [Cosmological Constant]

Obviously this is too weak a defence of new ideas, but a comment which is too long/strong/on alternative ideas will be edited down or deleted altogether. All I can do is to continue studying Lunsford's system until I understand it well enough to write my own detailed review of it. The problem is that Lunsford is well into the category of Dirac-Woit type innovation, which is heavy going. Abstract mathematical physics can correspond to reality without being presented with a simple physical mechanism. Just because both Lunsford's system and the mechanism I'm working on are both consistent with zero cosmological constant, and 6 dimensions of which 3 describe matter, does not necessarily imply that Lunsford is a fellow-traveller. There is a similar problem with Woit and others. I feel free to continue down my road, borrowing wheels from others as and where necessary.

I'll credit whoever 'loans' a wheel I take. If they want to object to my approach, that's their problem. I'm only interested in determining what the facts of the universe are, not joining someone else's bandwaggon at a price to my freedom. On the topic of the spin foam vacuum, the question is precisely how the spin of gauge/vector boson radiation relates to the QFT -> GR proof given by Dr Lee Smolin. I know for sure is that the general principles involved are more solid than string theory, and the vacuum dynamics behind gravity are essentially important to quantum gravity.

String theorist calls all alternatives "crackpot"

2006-03-06T07:54:00.000-08:00

An egotist has kindly informed us via his blog that Sundance O. Bilson-Thompson, Fotini Markopoulou and Lee Smolin have come up with a theory of everything. The egotist has tried to prevent anyone reporting this by concluding at http://motls.blogspot.com/2006/03/theory-of-everything-from-trinions.html: '... I am ... certain that crackpots will embrace it as a great idea. Sorry to say but this is exactly the way they think.'

The egotist has informed us here that people with alternatives to stringy M-theory are automatically to be dismissed and suppressed as 'completely moronic crackpots' and rants that alternative builders are science-haters. This gets to the bottom of the problem for alternatives to string theory: the mainstream masters a lot of hard stringy maths, and doesn't want to see the reality that they have all been wasting their time, at least insofar as the intended physical application of string theory goes. The maths developed for string theory may have other uses even if the intended physics is a failure.

Anyway, let's forget Motl's rant for a while, and examine http://arxiv.org/abs/hep-th/0603022, 'Quantum gravity and the standard model'.

UPDATE: Dr Dantas has a discussion thread for the paper here.

Dantas discussion forum for Smolin lectures

2006-02-20T12:49:00.000-08:00

Dr Christine Dantas has an interesting discussion forum for Dr Lee Smolin's lectures on quantum gravity here. It has some interesting comments, a few extracts follow:

'You don't take General Relativity as a basis and apply quantization recipes. Because, after all, it is the quantum gravity that is the fundamental theory of spacetime and Gen Rel is merely an APPROXIMATION to it. Instead, [Smolin] says, what is basic is differential forms - that is, fields that you can define without resorting to a METRIC or a setup geometry. So let's find out about form theories - BACKGROUND INDEPENDENT field theories like BF, in other words - and then lets play around ... and then (I guess) he pulls a form theory of gravity out of his hat.' - Dr Who

'I want to think in physical terms on how to understand causality in a discrete, background independent framework.' - Dr Dantas

'In Lecture 1, Smolin introduces the important idea of a HISTORY and he gives a toy model example with unlabeled graphs instead of the spin networks of regular LQG and a HISTORY is simply a series of MOVES (each one with an amplitude, a complex number) and it is a series of moves that get you from the INITIAL graph to the FINAL graph and the AMPLITUDE of any given history is just the PRODUCT of the series of amplitudes of each move, step by step. And the PATH INTEGRAL is simply the sum of all those products - the sum of the amplitudes of each history that gets you from the initial graph to the final graph. So understanding the toy model he gives in Lecture 1 really comes down to getting a feel for MOVES.

'Oh, another thing, a history can be visualized as a foam. The path of a changing network describes a foam-like complex - drag a graph through a time-like dimension and it describes a honey-comb looking cellular complex of 2D cells. In regular Quantum Gravity, spin foams are the HISTORIES of evolving spin networks. So a spin foam is what you get if you apply MOVES to get from an initial spin network to a final one. NO ONE SHOULD HAVE TO UNDERSTAND ALL THIS in one sitting, or in one lecture. So in Lecture 1 what Smolin did was basically make us familiar with the idea of an unlabeled network and the idea of a move.' - Dr Who

'STRING THEORY' FINALLY FINDS A USE: FOR EXCUSING INFIDELITY.

Dr Dantas has LQG reading list...

2006-02-09T05:54:00.000-08:00

http://christinedantas.blogspot.com/2006/02/basic-curriculum-for-quantum-gravity.html

Introduction to Quantum Gravity, Parts 3 and 4

2006-01-27T12:55:00.000-08:00

Parts 3 and 4 of Dr Lee Smolin's Introduction to Quantum Gravity are now available. The previous lectures in the series showed how the path integral is the sum of the spatial topography graphs for all the reactions, and how background independence from special relativity arises when you take the metric out.

Last night's lectures deals with the problem of time, and gauge symmetries. To me, papers by quantum field theorists and general relativists look as if the author has a barely hidden message to the reader: all worthwhile physics is abstract mathematics. I love applied mathematics, I love trying to understand quantitative relationships and empirical equations. This is of course discouraged strongly in theoretical physics, which is about abstract maths, not physical theory.

String 'theory' is of course the extreme example. You need to get as far away from string 'theory' failures as possible. String 'theory' failed because it used unobserved extra dimensions to 'predict' unobservable gravitons and it used undetectable supersymmetric partners to 'predict' a unification at an energy far beyond the capability of a particle accelerator the size of the solar system. That is unhelpful. The maths of string 'theory' is also an inelegant mess. It fails to predict testable particle masses or force strengths, because it is not tied to reality: trash in, trash out. Loop quantum gravity should steer clear of outright speculative time-wasting.

Feynman’s statements in Davies & Brown, ‘Superstrings’ 1988, at pages 194-195:

“… I do feel strongly that this is nonsense! … I think all this superstring stuff is crazy and is in the wrong direction. … I don’t like it that they’re not calculating anything. … why are the masses of the various particles such as quarks what they are? All these numbers … have no explanations in these string theories - absolutely none! … “. (From Tony Smith.)

Thomas Larrson on Woit's blog lists further expert conclusions:

Sheldon “string theory has failed in its primary goal” Glashow
Martinus “string theory is a figment of the theoretical mind” Veltman
Phil “string theory a futile exercise as physics”Anderson
Bob “string theory a 50-year-old woman wearing way too much lipstick” Laughlin
Dan “string theory is a complete scientific failure” Friedan

String theorists made a big mistake: for them SR is more important than the physical fabric theory, GR, because SR comes naturally from strings. So because SR is incompatible with accelerations (twins paradox), unlike GR, string theorists are stuck with no curvature!

"GR has a symmetry principle that extends that of SR, not reduce it, so all constraints of SR remain true in GR" - Motl's false claim (at his blog)

We know how E got GR. He expressed Newton's gravity as a field equation and found that you have to include a contraction. The Newtonian field equation is R = 4.Pi.GT, but in 1915 E found that to correct it you need to put a contraction in to the left hand side (curvature), and correct the right hand side (mass-energy) by doubling it: R - x = 8.Pi.GT.

The physical contraction of earth's radius is by 1/3 MG/c^2 = 1.5 mm.

The physical content of GR is the OPPOSITE of SR:

‘… the source of the gravitational field can be taken to be a perfect fluid…. A fluid is a continuum that ‘flows’... A perfect fluid is defined as one in which all antislipping forces are zero, and the only force between neighboring fluid elements is pressure.’ – Professor Bernard Schutz, General Relativity, Cambridge University Press, 1986, pp. 89-90.

Notice that in SR, there is no mechanism for mass, but the Standard Model says the mass has a physical mechanism: the surrounding Higgs field. When you move a fundamental particle in the Higgs field, and approach light speed, the Higgs field has less and less time to flow out of the way, so it mires the particle more, increasing its mass. You can't move a particle at light speed, because the Higgs field would have ZERO time to flow out of the way (since Higgs bosons are limited to light speed themselves), so inertial mass would be infinite. The increase in mass due to a surrounding fluid is known in hydrodynamics:

‘In this chapter it is proposed to study the very interesting dynamical problem furnished by the motion of one or more solids in a frictionless liquid. The development of this subject is due mainly to Thomson and Tait [Natural Philosophy, Art. 320] and to Kirchhoff [‘Ueber die Bewegung eines Rotationskörpers in einer Flüssigkeit’, Crelle, lxxi. 237 (1869); Mechanik, c. xix]. … it appeared that the whole effect of the fluid might be represented by an addition to the inertia of the solid. The same result will be found to hold in general, provided we use the term ‘inertia’ in a somewhat extended sense.’ – Sir Horace Lamb, Hydrodynamics, Cambridge University Press, 6th ed., 1932, p. 160. (Hence, the gauge boson radiation of the gravitational field causes inertia. This is also explored in the works of Drs Rueda and Haisch: see http://arxiv.org/abs/physics/9802031 http://arxiv.org/abs/gr-qc/0209016 , http://www.calphysics.org/articles/newscientist.html and http://www.eurekalert.org/pub_releases/2005-08/ns-ijv081005.php.)

So the Feynman problem with virtual particles in the spacetime fabric retarding motion does indeed cause the FitzGerald-Lorentz contraction, just as they cause the radial gravitationally produced contraction of distances around any mass (equivalent to the effect of the pressure of space squeezing things and impeding accelerations). What Feynman thought may cause difficulties is really the mechanism of inertia!

In his essay on general relativity in the book ‘It Must Be Beautiful’, Penrose writes: ‘… when there is matter present in the vicinity of the deviating geodesics, the volume reduction is proportional to the total mass that is surrounded by the geodesics. This volume reduction is an average of the geodesic deviation in all directions … Thus, we need an appropriate entity that measures such curvature averages. Indeed, there is such an entity, referred to as the Ricci tensor …’ Feynman discussed this simply as a reduction in radial distance around a mass of (1/3)MG/c2 = 1.5 mm for Earth. It’s such a shame that the physical basics of general relativity are not taught, and the whole thing gets abstruse. The curved space or 4-d spacetime description is needed to avoid Pi varying due to gravitational contraction of radial distances but not circumferences.

The velocity needed to escape from the gravitational field of a mass (ignoring atmospheric drag), beginning at distance x from the centre of mass, by Newton’s law will be v = (2GM/x)1/2, so v2 = 2GM/x. The situation is symmetrical; ignoring atmospheric drag, the speed that a ball falls back and hits you is equal to the speed with which you threw it upwards (the conservation of energy). Therefore, the energy of mass in a gravitational field at radius x from the centre of mass is equivalent to the energy of an object falling there from an infinite distance, which by symmetry is equal to the energy of a mass travelling with escape velocity v.

By Einstein’s principle of equivalence between inertial and gravitational mass, this gravitational acceleration field produces an identical effect to ordinary motion. Therefore, we can place the square of escape velocity (v2 = 2GM/x) into the Fitzgerald-Lorentz contraction, giving g = (1 – v2/c2)1/2 = [1 – 2GM/(xc2)]1/2.

However, there is an important difference between this gravitational transformation and the usual Fitzgerald-Lorentz transformation, since length is only contracted in one dimension with velocity, whereas length is contracted equally in 3 dimensions (in other words, radially outward in 3 dimensions, not sideways between radial lines!), with spherically symmetric gravity. Using the binomial expansion to the first two terms of each:

Fitzgerald-Lorentz contraction effect: g = x/x0 = t/t0 = m0/m = (1 – v2/c2)1/2 = 1 – ½v2/c2 + ...
Gravitational contraction effect: g = x/x0 = t/t0 = m0/m = [1 – 2GM/(xc2)]1/2 = 1 – GM/(xc2) + ...,
where for spherical symmetry ( x = y = z = r), we have the contraction spread over three perpendicular dimensions not just one as is the case for the FitzGerald-Lorentz contraction: x/x0 + y/y0 + z/z0 = 3r/r0. Hence the radial contraction of space around a mass is r/r0 = 1 – GM/(xc2) = 1 – GM/[(3rc2]

Therefore, clocks slow down not only when moving at high velocity, but also in gravitational fields, and distance contracts in all directions toward the centre of a static mass. The variation in mass with location within a gravitational field shown in the equation above is due to variations in gravitational potential energy. The contraction of space is by (1/3) GM/c2.
This is the 1.5-mm contraction of earth’s radius Feynman obtains, as if there is pressure in space. An equivalent pressure effect causes the Lorentz-FitzGerald contraction of objects in the direction of their motion in space, similar to the wind pressure when moving in air, but without viscosity. Feynman was unable to proceed with the LeSage gravity and gave up on it in 1965. However, we have a solution…

‘Recapitulating, we may say that according to the general theory of relativity, space is endowed with physical qualities... According to the general theory of relativity space without ether is unthinkable.’ – Albert Einstein, Leyden University lecture on ‘Ether and Relativity’, 1920. (Einstein, A., Sidelights on Relativity, Dover, New York, 1952, pp. 15, 16, and 23.)

‘The Michelson-Morley experiment has thus failed to detect our motion through the aether, because the effect looked for – the delay of one of the light waves – is exactly compensated by an automatic contraction of the matter forming the apparatus…. The great stumbing-block for a philosophy which denies absolute space is the experimental detection of absolute rotation.’ – Professor A.S. Eddington (who confirmed Einstein’s general theory of relativity in 1919), MA, MSc, FRS, Space Time and Gravitation: An Outline of the General Relativity Theory, Cambridge University Press, Cambridge, 1921, pp. 20, 152.

‘It has been supposed that empty space has no physical properties but only geometrical properties. No such empty space without physical properties has ever been observed, and the assumption that it can exist is without justification. It is convenient to ignore the physical properties of space when discussing its geometrical properties, but this ought not to have resulted in the belief in the possibility of the existence of empty space having only geometrical properties... It has specific inductive capacity and magnetic permeability.’ - Professor H.A. Wilson, FRS, Modern Physics, Blackie & Son Ltd, London, 4th ed., 1959, p. 361.

‘All charges are surrounded by clouds of virtual photons, which spend part of their existence dissociated into fermion-antifermion pairs. The virtual fermions with charges opposite to the bare charge will be, on average, closer to the bare charge than those virtual particles of like sign. Thus, at large distances, we observe a reduced bare charge due to this screening effect.’ – I. Levine, D. Koltick, et al., Physical Review Letters, v.78, 1997, no.3, p.424.

If the electron moves at speed v as a whole in a direction orthogonal (perpendicular) to the plane of the spin, then the c speed of spin will be reduced according to Pythagoras: v2 + x2 = c2 where x is the new spin speed. For v = 0 this gives x = c. What is interesting is that this model gives rise to the Lorentz-FitzGerald transformation naturally, because: x = c(1 - v2 / c2 )1/2 . Since all time is defined by motion, this (1 - v2 / c2 )1/2 factor of reduction of fundamental particle spin speed is therefore the time-dilation factor for the electron when moving at speed v.

Motl's quibbles about the metric of SR is just ignorance. The contraction is a physical effect as shown above, with length contraction in direction of motion, mass increase and time dilation having physical causes. The equivalence principle and the contraction physics of spacetime "curvature" are the advances of GR. GR is a replacement of the false SR which gives wrong answers for all real (curved) motions since it can't deal with acceleration: the TWINS PARADOX.

Strangely, the ‘critics’ are ignoring the consensus on where LQG is a useful approach, and just trying to ridicule it. In a recent post on his blog, for example, Motl states that special relativity should come from LQG. Surely Motl knows that GR deals better with the situation than SR, which is a restricted theory that is not even able to deal with the spacetime fabric (SR implicitly assumes NO spacetime fabric curvature, to avoid acceleration!).

When asked, Motl responds by saying Dirac’s equation in QFT is a unification of SR and QM. What Motl doesn’t grasp is that the ‘SR’ EQUATIONS are the same in GR as in SR, but the background is totally different:

‘The special theory of relativity … does not extend to non-uniform motion … The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion. Along this road we arrive at an extension of the postulate of relativity… The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant). …’ – Albert Einstein, ‘The Foundation of the General Theory of Relativity’, Annalen der Physik, v49, 1916.

More Dr Motl rubbish exposed: http://www.math.columbia.edu/~woit/wordpress/?p=338 (I think Motl/'Bottle' [as in 'hit the bottle' or Dutch courage] is saying below 'The landscape of 10^500 vacua predicted by string theory are better than the 1 vacua dealt with by LQG. All you kids out there, remember: theories so vague that they are compatible with every scheme are best, because they can’t be falsified. Don’t risk your career on a potential dead end… join Bottle’s fan club today!'):

Luboš Motl Says: January 30th, 2006 at 8:46 am

Dear Peter,

your comments about the “landscape problem” have really no relevance for the calculation of the Yukawa couplings in the heterotic model of Ovrut et al.

The topic of your blog is to transform everything to the “landscape problem” - much like a student who has learned the birth of date of William Shakespeare and wants to transform every exam question about history and literature to the number 1564.

But one can’t discuss any concrete physics questions in this way. Sorry.
BestBottle

woit Says: January 30th, 2006 at 9:23 am

This is off-topic for the posting about a parody blog, but then again, maybe not, because Lubos and his blog increasingly seem to be a parody.

The scientific point at issue is about a very recent paper claiming to calculate Yukawa couplings in a specific heterotic string background. When Lubos wrote a post hyping this result, I wrote in a comment pointing that that the authors were ignoring the main problems with doing this: how do deal with the moduli and supersymmetry breaking. These problems are what lead to the landscape, since the only known way of fixing the moduli leads to exponentially large numbers of possibilities.

There was an exchange of comments, with Lubos displaying the usual string theory partisan mixture of insult, ad hominem attack and straw man argument. Faced with having this pointed out to him, he moved on to the next tactic: censorship. Here’s the comment that he evidently had no answer for, so dealt with by deleting it:

“Lubos,
Your tactic is always the same: ignore the point I’m making (one that you know well is quite serious), then make up things I never said in order to use those to criticize me as ignorant. This behavior is stupid, dishonest, and highly scientifically unethical. You should be ashamed of yourself.

Let’s look at how your dishonesty works explicitly:

1.In this case, you are ignoring the problem posed by the moduli, and you are well aware that it is a deadly one. The paper in question notesexplicitly that fermion masses will depend on the moduli.

2.You claim to show that I’m wrong and don’t know what I’m talking about by writing:
“Nope. We are not choosing fluxes “in order to stabilize the moduli”.”

I never wrote that “We are choosing fluxes in order to stabilize the moduli”, I just said that the numbers like 10^500 that one hears for the size of the Landscape come from the number of choices of the fluxes, and these correspond to ways to stabilize the moduli. Do you know of a way of stabilizing all the moduli that doesn’t involve this? If so, lets hear it. If not, acknowledge that you were dishonest to bring this up.

As for supersymmetry breaking, your claim that:

“marginal operators can’t be classically affected by supersymmetry breaking”

is irrelevant. We’re not doing classical physics here.

If you have any honest points to make, I’ll respond to them, but I’m not going to waste any more time dealing with your dishonesty, interspersed with stupid, nasty, personal attacks. When you were a young graduate student, this kind of behavior was a bit amusing. At your age and stage of your career, it’s just pathetic.”

Spin Foam Vacuum dynamics

2006-01-23T04:17:00.000-08:00

Lee Smolin's lectures are actually very good. A lot of the mathematical background can be found in the papers he cites, and some of the basics are dealt with in Penrose's Road to Reality. Penrose of course came up with the spin network.

Loop Quantum Gravity suggests that gravity is mediated by a spin foam vacuum, and this is the main rival to string 'theory'. String 'theory' is an empty frame work to describe an unobserved graviton theory using unobserved 10/11 dimensional spacetime and predicting unobserved SUSY (super symmetry) partners. There is no effort in string theory to describe or model anything observable, and the claim that it predicts gravity is empty, as it just predicts and unobserved gravity mediation scheme, without any dynamics. Loop Quantum Gravity is completely different. His recent two lectures at the Perimeter Institute on 18 Jan, Introduction to Quantum Gravity, are available on line if you have a fast internet connection.

There is very little easily digestable material about Loop Quantum Gravity. Christine Dantas has a blog with useful references: http://christinedantas.blogspot.com/ Lee Smolin's lectures can be found here (you have to ignore the main screen and scroll down to the Introduction to Quantum Gravity at the bottom of the left hand side menu).

He starts off with spatial topology, sets of all graphs possible with or without edges, embeddings in all possible graphs, valent nodes on graphs (by analogy to chemistry, I presume). He defines Hilbert spaces on an orthagonal basis, then Penrose's 'spin networks'. Finally he shows how Feynman's 'sum over histories' approach to quantum mechanics arises in the vacuum: each interaction is a graph and you sum over all the graphs describing interactions in spacetime to arrive at the 'sum over histories'.

Smolin's second lecture dealt with 'background independence' which is the problem with special relativity. I think the recent comments of Smolin make 'background independence' clear: you don't worry about how to write the metric. Motl sees this as key to SR, unaware that the testable results of SR were available from electromagnetism of FitzGerald, Lorentz and Larmor, and even E=mc2 can be obtained from electromagnetic theory.

String theory only connects different bits of unobservable, untestable, speculation:

(1) 10/11 dimensions (Calabi-Yau 6-d manifold, Kaluza-Klein 5-d spacetime),

(2) graviton (spin 2 gauge boson) speculation (falsely called "predicting gravity" by Motl and Witten, when in fact it is unobserved speculation, and string theory can't make any testable prediction of the strength of gravity or anything else)

(3) SUSY super symmetric partners. Again, an unobserved speculation, with no definite quantitative predictions to test.

(4) The "Landscape" of 10^500 vacua, one of which may be real.

String 'theory' doesn't exist or have any program to deal usefully with anything that does exist.
The vacuum does mediate force, it does have gauge bosons in it (we know this for electromagnetism...), so some kind of spin foam vacuum is real.

Lee Smolin disproves his critics politely

2006-01-22T07:13:00.000-08:00

Loop Quantum Gravity researcher Lee Smolin has answered 'critics' (mainly string 'theorists' who believe in unobservables) on Peter Woit's blog, LQG for Skeptics, http://www.math.columbia.edu/~woit/wordpress/?p=330. Lubos (Lumos) Motl of Harvard and some others rushed in with all sorts of 'arguments' trying to simply ridicule the spin foam vacuum approach to understanding quantum gravity (unifying general relativity and Feynman path integrals), when in fact they were just plain ignorant. Smolin then had to politely correct all their miscomprehensions:

Lee Smolin Says: January 20th, 2006 at 6:13 pm

Hi Peter, Thanks for mentioning this. After a quick look I can say that there are some statements they make that I agree with (such as about the difficiulties of relating the spin foam to the Hamiltonian constraint theory) and others with which I disagree, such as their statements about uv finiteness.

In particular, the point you quote, “the need to fix infinitely many couplings in the perturbative approach….” seems to disagree with old, well understood results. (Not to mention they don’t seem to make a detailed argument for their claim.) In fact, there is a well understood and detailed explanation for how the theory is cutoff. I won’t repeat the argument here (see hep-th/0408048) but the key points are that, as a result of the finitess of area and volume, one can show that the Planck length cannot suffer an infinite normalization if the theory is to reproduce the finiteness of black hole entropy,and the appearence of gravitons in perturbations around weave states. Thus, the theory is uv cutoff and the divergences do not have to cancel as they were never there in the first place. The key issue is then how is this finite cutoff length compatible with the symmetry of the ground state, which leads to the expectation that the symmetry is DSR.

But honest criticism based on detailed study is always welcome and I will read the paper carefully before seeing if I have any more substantial reponse to make.
In case anyone is interested, I am teaching a course on LQG, and video’s of lectures will be available as they are given at http://streamer.perimeterinstitute.ca:81/mediasite/viewer/FrontEnd/Front.aspx?&shouldResize=False

Thanks,
Lee

Lee Smolin Says: January 21st, 2006 at 9:51 am

On reading NP I am grateful for the hard work that they put in, but I end up feeling that they still miss the point, because they have prejudices about what a quantum theory of gravity should do coming from old expectations. They appear to evaluate LQG and spin foam models as if they were proposed as a unique theory which was a proposals for a final theory of everything. This is in my view a misunderstanding. One should understand these as a large set of models for studying background and diffeo invariant QFT’s. These are based on quantization of a set of classical field theories which are constrained topological field theories. There are three key claims: 1) these theories exist, rigorously. i.e. there are uv finite diffeo invariant QFT’s based on quantization of constrained TQFT’s. 2) there is a common mathematical and conceptual language and some calculational tools which are useful to study such models and 3) there are some common generic consequences of these models, which are relevant for physics.
Nothing NP say questions these key claims. Unfortunately, they do not mention key papers which support these key claims, such as the uniqueness theorems (gr-qc/0504147, math-ph/0407006) which show the necessity of the quantization LQG uses. And while they mention the non-seperability of the kinematical Hilbert space they fail to mention the seperability of the diffeomorphism invariant Hilbert space, (grqc/ 0403047). It is unfortunate that they omit reference to such key results which resolve issues they mention.

A second misunderstanding concerns uv divergences. NP do not discuss the results on black hole entropy, so they miss the point that the finiteness of the black hole entropy fixes the ratio of the bare and low energy planck length to be a finite number of order one. Calculations on a class of semiclassical states they do not discuss-the weave states-lead to the same conclusion (A. Ashtekar, C. Rovelli, L. Smolin, Weaving a classical metric with quantum threads,” Phys. Rev. Lett. 69 (1992) 237.). So there can be no infinite refinement of spin foams and no infinite renormalization. These theories are uv finite, period. This is one of the generic features I mentioned. Thus, their main claim, that the fact that there are many LQG or spin foam models is the same as the problem of uv divergent is just manifestly untrue. The freedom to specify spin foam amplitudes does not map onto the freedom to specify parameters of a perturbatively non-renormalizable theory. For one thing, few if any spin foam models are likely to have a low energy limit which is Poincare invariant, a property shared by all perturbative QFT’s, renormalizable or not, defined in Minkowski spacetime. In fact, we know from recent results that in 2+1 none do-the low energy limit of 2+1 gravity coupled to arbitrary matter is DSR. So their argument is false. They do get a number of things right. The following are open issues, much discussed in the literature: 1) whether there is any regularization of the Hamiltonian constraint that leads to exchange moves, 2) whether thus there are any links between the spin foam amplitudes and Hamiltonian evolution, 3) whether the sum over spin foam diagrams is convergent or, more likely, Borel resummmable (although they miss that this has been proven for 2+1 models, hep-th/0211026). I don’t agree with all the details of their discussion of these issues, but these certainly are open issues.

NP seem to argue as if one has to prove a QFT rigorously exists in order to do physics with it, by which standard we would believe no prediction from the standard model. They mention that there are no rigorous constructed, semiclassical states, which are exact solutions to the dynamics, but this is the case in most QFT’s. This does not prevent us from writing down and deriving predictions from heuristic semiclassical states (hep-th/ 0501091), or from constructing reduced models to describe black holes or cosmologies and likewise deriving predictions (astro-ph/0411124), Nor does it prevent Rovelli et al from computing the graviton propagator and getting the right answer, showing there are gravitons and Newtonian gravity in the theory (gr-qc/0502036).

But, someone may ask, if LQG is the right general direction, shouldn’t there be a unique theory that is claimed to be the theory of nature? Certainly, but should the program be dismissed because no claim has yet been made that this theory has been found? To narrow in on the right theory there are further considerations, all under study:

-Not every spin foam model is ir finite.-Not every spin foam model is likely to have a good low energy limit.-The right theory should have the standard model of particle physics in it.
In addition it must be stressed that there can in physics be generic consequences of classes of theories, leading to experimental predictions. Here are some historical examples: light bending, weak vector bosons, confinement, principle of inertia, existence of black holes. All of these observable features of nature are predicted by large classes of theories, which can be as a whole confirmed or falsified, even in the absence of knowing which precise theory describes nature, and prior to proving the mathematical consistency of the theory. LQG predicts a number of such generic features: discreteness of quantum geometry, horizon entropy, removal of all spacelike singularities, and I believe will soon predict more including DSR, emergence of matter degrees of freedom.

One reason for this is of course that most of the parameters in such classes of such theories are irrelevant in the RG sense, and do not influence large scale predictions. Since we know the theory is uv finite this does not affect existence. The lack of a uv unique theory does not prevent us from testing predictions of QFT in detail, and it is likely to be the same for quantum gravity. The old idea that consistency would lead to a unique uv theory that would give unique low energy predictions was seductive, but given the landscape, it is an idea that is unsupported by the actual results. Having said all this, I hope that NP will put their hard won expertise to work, and perhaps get their hands dirty and do some research in the area.
Sorry to go on so long,

Thanks, Lee

Lee Smolin Says: January 21st, 2006 at 1:09 pm

To Zorq, I apologise that the one argument I gave is far from the whole story. I agree that to define a theory you have first to define a regulated theory and then study the limit as the regulator is removed and show that the expressions for observables that result have the symmetries and gauge invariances of the classical theory. The physical parameters are parameters of the resulting theory. But this was exactly what was done in the hamiltonian construction of LQG, the observables such as area and volume are constructed through a limit of regulated operators and then the limit is taken. The result is finite, diffeomorphism invariant observables. The parameters of the finite, diffeo inv. theory include the bare Newton’s and cosmological constant-not as coefficients in the dynamics but as coefficients in the algebra of observables. The diffeo invariance is proved restored in the limit. And don’t be confused, the bare planck scale was not the regulator, there was another regulator, that has already been taken to zero. Please study the details of the construction.

So what you ask for has been done-the logic is not circular and the theory is finite. I was addressing the proposal to still take a limit of the parameters vanishing, even after the limit of the regulator to zero is taken. This would be non-standard, and it leads to results in disagreement with the semiclassical theory.

To quantoken: obviously it would be better to have precise predictions than generic ones. But generic ones can still distinguish between classes of theories. If GLAST sees an energy dependent, polarization independent speed of light, thus confirming that the symmetry of the vacuum is DSR, this obviouosly kills theories that predicted either broken or naive Poincare invariance and supports the plausibility of theories that predict-even generically DSR.
I can’t agree with your equivalanece of LQG and string theory just because of the vast imbalence in how much work has been put into each. For each point about string theory there are dozens of papers, so the ins and outs of each issue have been often thoroughly explored. Key facts about LQG still rest, in many cases on one or a few papers. There are many obvious things to do that have not been tried for lack of people. So there is a lot still do to, for example to turn generic predictions into precise predictions.

And I don’t “believe” nature is discrete, we showed this is a generic property of a large class of diffeo invariant QFT’s.

I’ll reply to the rest later,
Lee

Lee Smolin Says: January 22nd, 2006 at 8:58 am

Zorq, the distinction between a finite theory and a uv fixed point is the following: a lattice QFT or condensed matter physics model with a fixed lattice spacing is a finite theory. You can still apply the RG to study the infrared behavior. The point in LQG is that AFTER carrying out the regularization procedure and defining the diffeo invariant states and operators from the limit of the regulator removed, there remains a theory with a fixed, but spatially diffeo invariant cutoff. So it is like a lattice theory in being intrinsically finite, except that all the states are spatially diffeo invariant. So the physical theory has a fixed cutoff. One place this is explained in detail is Rovelli’s book, pages 280-282, another is my review hep-th/0408048. To see how this is done rigorously see Thiemann’s gr-qc/0110034 or Ashtekar-Lewandowski GR-QC 0404018.

Lubos, the finiteness is achieved kinematically, once the procedure just described is done the theory is uv finite WHATEVER the dynanmics, again just as in a lattice QFT with fixed lattice spaceing. But, to continue with my example of 2+1, which you misunderstood, is that the symmetry of the groundstate is not put in, it is determined dynamically. In 2+1 it turns out to be kappa-Poincare. This cannot agree with perturbation theory carried out as an expansion around Minkowski spacetime, which order by order assumes ordinary Poincare invariance.
I agree there is an issue in the Hamiltonian theory with full spacetime diffeo invariance, which is one of several reasons I always discuss dynamics in the spin foam picture.

Thomas and Aaron, this has been discussed before. For QFT the harmonic oscillator kind of quantization leads to Fock space. The inner product of Fock space depends on the background metric, hence this quantization will never arise in a background independent formalism. So we do not avoid it because we are stupid, it is simply not an option. The question is then to find an inner product for states which are functionals of a connection mod spatial diffeos. The only known way to do this is to first construct a kinematical Hilbert spaced that carries an exact non-anomalous rep of the spatial diffeos and then use that unitary rep to mod out by the diffeos .

The uniqueness theorems tell us the result is unique. If you don’t like this please at least acknowledge the argument just described and either accept it or propose an alternative background independent quantization and do the work to show it is consistent.

Anonomous: See my talk at the loops 05 conference, details are in a paper under preparation.

Why LQG spin foam vacuum is better than Strings!

2006-01-19T13:33:00.000-08:00

Spin foam vacuum LQG versus false 'stringy theory'!

2006-01-19T03:28:00.000-08:00

‘String theory has the remarkable property of predicting gravity’: false claim by Edward Witten in the April 1996 issue of Physics Today, repudiated by Roger Penrose on page 896 of his book The Road to Reality, 2004. String theory does not predict anything testable about gravity. - Comment on Peter Woit's blog

'... it is thus perhaps best to view spin foam models ... as a novel way of defining a (regularised) path integral in quantum gravity. Even without a clear-cut link to the canonical spin network quantisation programme, it is conceivable that spin foam models can be constructed which possess a proper semi-classical limit in which the relation to classical gravitational physics becomes clear. For this reason, it has even been suggested that spin foam models may provide a possible ‘way out’ if the difficulties with the conventional Hamiltonian approach should really prove insurmountable.' - http://arxiv.org/abs/hep-th/0601129

The spin foam vacuum of loop quantum gravity is the only consistent mathematical formulation of Feynman's path integrals that seems to have the potential of addressing the physical process by which the vacuum operates. The better known 'quantum gravity' is called string theory, and deals with particles as 10/11 dimensional loops. String theory suggests 10^500 different vacuum states, and although it could provide an empty framework for a (unobserved but speculated) spin-2 gauge boson (graviton speculation), it does not provide any dynamics or testable predictions after 20 years of intense funding and effort!

"An important part of all totalitarian systems is an efficient propaganda machine. ... to protect the 'official opinion' as the only opinion that one is effectively allowed to have." - STRING THEORIST Dr Lubos Motl: http://motls.blogspot.com/2006/01/power-of-propaganda.html

The quotation above shows how "string theorists" have been able to brain-wash many people with propaganda. Loop quantum gravity does not have that propaganda. So let's see a little recent work in Loop Quantum Gravity:

"... the spinfoam model of http://arxiv.org/hep-th/0512113. In that paper Freidel and Livine work out a model of 3D spacetime and matter which has QFT as the zero-gravity limit (as G -> 0) and General Relativity with some quantum corrections as the (semi)classical limit (as h-bar -> 0). In that paper I do not see other constants which can vary to give a large space of solutions. However they still have to extend their results to 4D spacetime and matter." - Woit blog.

http://arxiv.org/abs/hep-th/0601129: Loop and spin foam quantum gravity: a brief guide for beginners, by Hermann Nicolai, Kasper Peeters. This is a very nice brief (18 pages) review of 'loop quantum gravity and spin foam models at an introductory level, with special attention to questions frequently asked by non-specialists':

'In contrast to string theory, which posits that the Einstein-Hilbert action is only an effective low energy approximation to some other, more fundamental, underlying theory, loop and spin foam gravity take Einstein’s theory in four spacetime dimensions as the basic starting point, either with the conventional or with a (constrained) ‘BF-type’ formulation. These approaches are background independent in the sense that they do not presuppose the existence of a given background metric. In comparison to the older geometrodynamics approach (which is also formally background independent) they make use of many new conceptual and technical ingredients.

'A key role is played by the reformulation of gravity in terms of connections and holonomies. A related feature is the use of spin networks in three (for canonical formulations) and four (for spin foams) dimensions. These, in turn, require other mathematical ingredients, such as non-separable (‘polymer’) Hilbert spaces and representations of operators which are not weakly continuous. Undoubtedly, novel concepts and ingredients such as these will be necessary in order to circumvent the problems of perturbatively quantised gravity (that novel ingredients are necessary is, in any case, not just the point of view of LQG but also of most other approaches to quantum gravity). However, it is important not to lose track of the physical questions that one is trying to answer.

'... it is thus perhaps best to view spin foam models as models in their own right, and, in fact, as a novel way of defining a (regularised) path integral in quantum gravity. Even without a clear-cut link to the canonical spin network quantisation programme, it is conceivable that spin foam models can be constructed which possess a proper semi-classical limit in which the relation to classical gravitational physics becomes clear. For this reason, it has even been suggested that spin foam models may provide a possible ‘way out’ if the difficulties with the conventional Hamiltonian approach should really prove insurmountable.'

Christine Dantas points out that these authors are those of http://arxiv.org/abs/hep-th/0501114 which is a paper resubmitted to Classical and Quantum Gravity. There is always the prejudice question!

Another vacuum picture: "Another possible “other approach” might be the emergent vacuum picture advocated by Laughlin and Chapline (popularly described in Laughlin’s book A Different Universe (Reinventing Physics from the Bottom Down)). Here is Susskind’s criticism of Laughlin’s approach:

“… Superfluid helium is an example of a material with special “emergent” properties … In a lot of ways, superfluids are similar to the Higgs fluid that fills space and gives particles their properties. Roughly speaking Laughlin’s view can be summarized by saying that we live in such a space-filling material. He might even say … space IS such an emergent material! Moreover, he believes that gravity is an emergent phenomenon. … There are two serious reasons to doubt that the laws of nature are similar to the laws of emergent materials. … The first … Laughlin himself …[argues]… that black holes (in his theory) cannot have properties, such as Hawking radiation, that practically everyone else believes them to have …[second]… insensitivity to the microscopic starting point is the thing that condensed-matter physicists like best about emergent systems. But the probability that … there should be one … endpoint … with the incredibly fine-tuned properties of our anthropic world is negligible. …”.

"It is sad to me that Susskind’s first criticism of Laughlin assumes that “practically everyone” believes that black holes have Hawking radiation, when, in fact, even Hawking himself has repudiated ( see his Dublin 2004 abstract at http://www.dcu.ie/~nolanb/gr17_plenary.htm#hawking ) information loss by Hawking radiation.

"It is also sad that Susskind’s second criticism of Laughlin assumes that the process of emergence CANNOT produce an “endpoint” with precisely the properties that our experiments observe.Perhaps a spin foam with J3(O) nodes might produce our observed universe,similar to the emergence of superfluid helium from helium atoms.Unless and until the physics community encourages research work along such lines, how will we know the answer?
- Tony Smith http://www.valdostamuseum.org/hamsmith/" - Woit blog.

UPDATE: yet another string hoax exposed to be misleading propaganda: http://www.math.columbia.edu/~woit/wordpress/?p=327

FROM WIKIPEDIA ENTRY ON LOOP QUANTUM GRAVITY (BIASED AGAINST LOOP QUANTUM GRAVITY, SINCE IT IS WRITTEN BY STRING THEORISTS IN PART):

Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. This theory is one of a family of theories called canonical quantum gravity. It was developed in parallel with loop quantization, a rigorous framework for nonperturbative quantization of diffeomorphism-invariant gauge theory. In plain English this is a quantum theory of gravity in which the very space that all other physics occurs in is quantized.

Loop quantum gravity (LQG) is a proposed theory of spacetime which is built from the ground up with the idea of spacetime quantization via the mathematically rigorous theory of loop quantization. It preserves many of the important features of general relativity, such as local Lorentz invariance, while at the same time employing quantization of both space and time at the Planck scale in the tradition of quantum mechanics.

This is not the most popular theory of quantum gravity; many physicists have philosophical problems with it. ...

LQG in itself was initially less ambitious than string theory, purporting only to be a quantum theory of gravity. ...

1) It is a nonperturbative quantization of 3-space geometry, with quantized area and volume operators
2) It includes a calculation of the entropy of black holes
3) It is a viable gravity-only alternative to string theory

However, these claims are not universally accepted. While many of the core results are rigorous mathematical physics, their physical interpretations remain speculative. LQG may or may not be viable as a refinement of either gravity or geometry. For example, entropy calculated in (2) is for a kind of hole which may or may not be a black hole.

The spin foam model proposed by Louis Crane, John Baez, et al, adds a time dimension to loop quantum gravity to accomodate relativistic spacetime.

The incompatibility between quantum mechanics and general relativity

Quantum field theory studied on curved (non-Minkowskian) backgrounds has shown that some of the core assumptions of quantum field theory cannot be carried over. In particular, the vacuum, when it exists, is shown to depend on the path of the observer through space-time (see Unruh effect).

Historically, there have been two reactions to the apparent inconsistency of quantum theories with the necessary background-independence of general relativity. The first is that the geometric interpretation of general relativity is not fundamental, but emergent. The other view is that background-independence is fundamental, and quantum mechanics needs to be generalized to settings where there is no a priori specified time.

Loop quantum gravity is an effort to formulate a background-independent quantum theory. Topological quantum field theory is a background-independent quantum theory, but it lacks causally-propagating local degrees of freedom needed for 3 + 1 dimensional gravity.
[edit]

History of LQG

In 1986, Abhay Ashtekar reformulated Einstein's field equations of general relativity using what have come to be known as Ashtekar variables, a particular flavor of Einstein-Cartan theory with a complex connection. He was able to quantize gravity using gauge field theory. In the Ashtekar formulation, the fundamental objects are a rule for parallel transport (technically, a connection) and a coordinate frame (called a vierbein) at each point. Because the Ashtekar formulation was background-independent, it was possible to use Wilson loops as the basis for a nonperturbative quantization of gravity. Explicit (spatial) diffeomorphism invariance of the vacuum state plays an essential role in the regularization of the Wilson loop states.

Around 1990, Carlo Rovelli and Lee Smolin obtained an explicit basis of states of quantum geometry, which turned out to be labelled by Penrose's spin networks. In this context, spin networks arose as a generalization of Wilson loops necessary to deal with mutually intersecting loops. Mathematically, spin networks are related to group representation theory and can be used to construct knot invariants such as the Jones polynomial.

Being closely related to topological quantum field theory and group representation theory, LQG is mostly established at the level of rigour of mathematical physics.

The ingredients of loop quantum gravity

Loop quantization

At the core of loop quantum gravity is a framework for nonperturbative quantization of diffeomorphism-invariant gauge theories, which one might call loop quantization. While originally developed in order to quantize vacuum general relativity in 3+1 dimensions, the formalism can accommodate arbitrary spacetime dimensionalities, fermions (John Baez and Kirill Krasnov), an arbitrary gauge group (or even quantum group), and supersymmetry (Smolin), and results in a quantization of the kinematics of the corresponding diffeomorphism-invariant gauge theory. Much work remains to be done on the dynamics, the classical limit and the correspondence principle, all of which are necessary in one way or another to make contact with experiment.

In a nutshell, loop quantization is the result of applying C*-algebraic quantization to a non-canonical algebra of gauge-invariant classical observables. Non-canonical means that the basic observables quantized are not generalized coordinates and their conjugate momenta. Instead, the algebra generated by spin network observables (built from holonomies) and field strength fluxes is used.

Loop quantization techniques are particularly successful in dealing with topological quantum field theories, where they give rise to state-sum/spin-foam models such as the Turaev-Viro model of 2+1 dimensional general relativity. A much studied topological quantum field theory is the so-called BF theory in 3+1 dimensions. Since classical general relativity can be formulated as a BF theory with constraints, scientists hope that a consistent quantization of gravity may arise from the perturbation theory of BF spin-foam models.

Lorentz invariance

For detailed discussion see the Lorentz covariance page.
LQG is a quantization of a classical Lagrangian field theory which is equivalent to the usual Einstein-Cartan theory in that it leads to the same equations of motion describing general relativity with torsion. As such, it can be argued that LQG respects local Lorentz invariance. Global Lorentz invariance is broken in LQG just as in general relativity. A positive cosmological constant can be realized in LQG by replacing the Lorentz group with the corresponding quantum group.

Diffeomorphism invariance and background independence

General covariance, also known as diffeomorphism invariance, is the invariance of physical laws under arbitrary coordinate transformations. A good example of this are the equations of general relativity, where this symmetry is one of the defining features of the theory. LQG preserves this symmetry by requiring that the physical states remain invariant under the generators of diffeomorphisms. The interpretation of this condition is well understood for purely spatial diffemorphisms. However, the understanding of diffeomorphisms involving time (the Hamiltonian constraint) is more subtle because it is related to dynamics and the so-called problem of time in general relativity. A generally accepted calculational framework to account for this constraint is yet to be found.

Whether or not Lorentz invariance is broken in the low-energy limit of LQG, the theory is formally background independent. The equations of LQG are not embedded in, or presuppose, space and time, except for its invariant topology. Instead, they are expected to give rise to space and time at distances which are large compared to the Planck length. At present, it remains unproven that LQG's description of spacetime at the Planckian scale has the right continuum limit, described by general relativity with possible quantum corrections.
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Problems

As of December 2005, there is not a single experiment which verifies or refutes any aspect of LQG. This problem plagues most current theories of quantum gravity. LQG is affected especially, because it applies on a small scale to the weakest forces in nature. There is no work around for this problem, as it is the biggest problem any scientific theory can have; theory without experiment is just faith. The second problem is that a crucial free parameter in the theory known as the Immirzi parameter can only be computed by demanding agreement with Bekenstein and Hawking's calculation of the black hole entropy. Loop quantum gravity predicts that the entropy of a black hole is proportional to the area of the event horizon, but does not obtain the Bekenstein-Hawking formula S = A/4 unless the Immirzi parameter is chosen to give this value.

Finally, LQG has gained limited support in the physics community, perhaps because of its limited scope. So far, it seeks to describe a quantum theory including gravity and more or less arbitrary other forces and forms of matter. String theory and M-theory are more ambitious, since they seek a more or less unique theory which predicts not only the behavior of gravity but also the detailed behavior of elementary particles and the forces besides gravity. While they have not succeeded in doing so yet, the general feeling is that these competing theories are more potent. Loop theorists disagree, because they believe that we need a proper theory of quantum gravity as a prerequisite for any theory of everything. Only time and experimentation can decide the matter.

Bibliography
Popular books:
Julian Barbour, The End of Time
Lee Smolin, Three Roads to Quantum Gravity
Magazine articles:
Lee Smolin, "Atoms in Space and Time," Scientific American, January 2004
Easier introductory/expository works:
Abhay Ashtekar, Gravity and the quantum, e-print available as gr-qc/0410054
John C. Baez and Javier Perez de Muniain, Gauge Fields, Knots and Quantum Gravity, World Scientific (1994)
Carlo Rovelli, A Dialog on Quantum Gravity, e-print available as hep-th/0310077
More advanced introductory/expository works:
Abhay Ashtekar, New Perspectives in Canonical Gravity, Bibliopolis (1988).
Abhay Ashtekar, Lectures on Non-Perturbative Canonical Gravity, World Scientific (1991)
Abhay Ashtekar and Jerzy Lewandowski, Background independent quantum gravity: a status report, e-print available as gr-qc/0404018
Rodolfo Gambini and Jorge Pullin, Loops, Knots, Gauge Theories and Quantum Gravity, Cambridge University Press (1996)
Hermann Nicolai, Kasper Peeters, Marija Zamaklar, Loop quantum gravity: an outside view, e-print available as hep-th/0501114
Carlo Rovelli, Loop Quantum Gravity, Living Reviews in Relativity 1, (1998), 1, online article, 2001 15 August version
Carlo Rovelli, che cos'è il tempo, che cos'è lo spazio, Di Renzo Editore, Roma, 2004
Carlo Rovelli, Quantum Gravity, Cambridge University Press (2004); draft available online
Thomas Thiemann, Introduction to modern canonical quantum general relativity, e-print available as gr-qc/0110034
Thomas Thiemann, Lectures on loop quantum gravity, e-print available as gr-qc/0210094
Conference proceedings:
John C. Baez (ed.), Knots and Quantum Gravity
Fundamental research papers:
Abhay Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett., 57, 2244-2247, 1986
Abhay Ashtekar, New Hamiltonian formulation of general relativity, Phys. Rev. D36, 1587-1602, 1987
Roger Penrose, Angular momentum: an approach to combinatorial space-time in Quantum Theory and Beyond, ed. Ted Bastin, Cambridge University Press, 1971
Carlo Rovelli and Lee Smolin, Loop space representation of quantum general relativity, Nuclear Physics B331 (1990) 80-152
Carlo Rovelli and Lee Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys., B442 (1995) 593-622, e-print available as gr-qc/9411005
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External links
Quantum Gravity, Physics, and Philosophy: http://www.qgravity.org/
Resources for LQG and spin foams: http://jdc.math.uwo.ca/spin-foams/
Gamma-ray Large Area Space Telescope: http://glast.gsfc.nasa.gov/
Zeno meets modern science. Article from Acta Physica Polonica B by Z.K. Silagadze.

Loop Quantum Gravity - Spin Foam Vacuum

2006-01-18T15:24:00.000-08:00

... a spin foam is a four-dimensional graph made out of two-dimensional faces that represents one of the configurations that must be summed to obtain Feynman's path integral (functional integration) describing the alternative formulation of quantum gravity known as loop gravity or loop quantum gravity. ...

"In loop quantum gravity there are some results from a possible canonical quantization of general relativity at the Planck scale. Any path integral formulation of the theory can be written in the form of a spin foam model, such as the Barrett-Crane model. Spin network is defined as a diagram (like Feynman diagram) which make a basis of connections between the elements of a differentiable manifold for the Hilbert spaces defined over them. Spin network provides a representation for computations of amplitudes between two different hypersurfaces of the manifold. Any evolution of spin network provides a spin foam over a manifold of one dimensional higher than the dimensions of the corresponding spin network. A spin foam may be viewed as a quantum history. ...

"Spacetime is considered as a quantic superposition of spin foams, which is a generalized Feynman diagram where instead of a graph we use a higher-dimensional complecies. In topology this sort of space is called a 2-dimensional complex." - Wikipedia

PHYSICAL REALITY OF A SPIN FOAM VACUUM (in addition to the maths, not replacing it), http://feynman137.tripod.com/:

Maxwell’s 1873 Treatise on Electricity and Magnetism, Articles 822-3: ‘The ... action of magnetism on polarised light [discovered by Faraday not Maxwell] leads ... to the conclusion that in a medium ... is something belonging to the mathematical class as an angular velocity ... This ... cannot be that of any portion of the medium of sensible dimensions rotating as a whole. We must therefore conceive the rotation to be that of very small portions of the medium, each rotating on its own axis [spin] ... The displacements of the medium, during the propagation of light, will produce a disturbance of the vortices ... We shall therefore assume that the variation of vortices caused by the displacement of the medium is subject to the same conditions which Helmholtz, in his great memoir on Vortex-motion [of 1858; sadly Lord Kelvin in 1867 without a fig leaf of empirical evidence falsely applied this vortex theory to atoms in his paper ‘On Vortex Atoms’, Phil. Mag., v4, creating a mathematical cult of vortex atoms just like the mathematical cult of string theory now; it created a vast amount of prejudice against ‘mere’ experimental evidence of radioactivity and chemistry that Rutherford and Bohr fought], has shewn to regulate the variation of the vortices [spin] of a perfect fluid.’

‘… the source of the gravitational field can be taken to be a perfect fluid…. A fluid is a continuum that ‘flows’... A perfect fluid is defined as one in which all antislipping forces are zero, and the only force between neighboring fluid elements is pressure.’ – Professor Bernard Schutz, General Relativity, Cambridge University Press, 1986, pp. 89-90.

‘In this chapter it is proposed to study the very interesting dynamical problem furnished by the motion of one or more solids in a frictionless liquid. The development of this subject is due mainly to Thomson and Tait [Natural Philosophy, Art. 320] and to Kirchhoff [‘Ueber die Bewegung eines Rotationskörpers in einer Flüssigkeit’, Crelle, lxxi. 237 (1869); Mechanik, c. xix]. … it appeared that the whole effect of the fluid might be represented by an addition to the inertia of the solid. The same result will be found to hold in general, provided we use the term ‘inertia’ in a somewhat extended sense.’ – Sir Horace Lamb, Hydrodynamics, Cambridge University Press, 6th ed., 1932, p. 160. (Hence, the gauge boson radiation of the gravitational field causes inertia. This is also explored in the works of Drs Rueda and Haisch: see http://arxiv.org/abs/physics/9802031 http://arxiv.org/abs/gr-qc/0209016 , http://www.calphysics.org/articles/newscientist.html and http://www.eurekalert.org/pub_releases/2005-08/ns-ijv081005.php .)

So the Feynman problem with virtual particles in the spacetime fabric retarding motion does indeed cause the FitzGerald-Lorentz contraction, just as they cause the radial gravitationally produced contraction of distances around any mass (equivalent to the effect of the pressure of space squeezing things and impeding accelerations). What Feynman thought may cause difficulties is really the mechanism of inertia!

At http://www.valdostamuseum.org/hamsmith/cnfGrHg.html, Tony Smith quotes Feynman:

Richard Feynman’s book Lectures on Gravitation (1962-63 lectures at Caltech), Addison-Wesley 1995, contains a section on Quantum Gravity by Brian Hatfield, who says: ‘... Feynman ... felt ... that ... the fact that a massless spin-2 field can be interpreted as a metric was simply a ‘coincidence’ ... In order to produce a static force and not just scattering, the emission or absorption of a single graviton by either particle [of a pair of particles] must leave both particles in the same internal state ... Therefore the graviton must have integer spin. ... when the exchange particle carries odd integer spin, like charges repel and opposite charges attract ... when the exchanged particle carries even integer spin, the potential is universally attractive ... If we assume that the exchanged particle is spin 0, then we lose the coupling of gravity to the spin-1 photon ... the graviton is massless because gravity is a long ranged force and it is spin 2 in order to be able to couple the energy content of matter with universal attraction ...’.

http://feynman137.tripod.com/:

General relativity, absolute causality

Professor Georg Riemann (1826-66) stated in his 10 June 1854 lecture at Gottingen University, On the hypotheses which lie at the foundations of geometry: ‘If the fixing of the location is referred to determinations of magnitudes, that is, if the location of a point in the n-dimensional manifold be expressed by n variable quantities x₁, x₂, x₃, and so on to x_n, then … ds =

Ö [å (dx)²] … I will therefore term flat these manifolds in which the square of the line-element can be reduced to the sum of the squares … A decision upon these questions can be found only by starting from the structure of phenomena that has been approved in experience hitherto, for which Newton laid the foundation, and by modifying this structure gradually under the compulsion of facts which it cannot explain.’

Riemann’s suggestion of summing dimensions using the Pythagorean sum ds² =

å (dx²) could obviously include time (if we live in a single velocity universe) because the product of velocity, c, and time, t, is a distance, so an additional term d(ct)² can be included with the other dimensions dx², dy², and dz². There is then the question as to whether the term d(ct)² will be added or subtracted from the other dimensions. It is clearly negative, because it is, in the absence of acceleration, a simple resultant, i.e., dx²+ dy² + dz² = d(ct)², which implies that d(ct)² changes sign when passed across the equality sign to the other dimensions: ds² = å (dx²) = dx²+ dy² + dz² – d(ct)² = 0 (for the absence of acceleration, therefore ignoring gravity). This formula, ds² = å (dx²) = dx²+ dy² + dz² – d(ct)², is known as the ‘Riemann metric’. It is important to note that it is not the correct spacetime metric, which is precisely why Riemann did not discover general relativity back in 1854. [The algebraic Newtonian-equivalent (for weak fields) approximation in general relativity is the Schwarzschild metric, which, ds² = (1 – 2GM/r)^-1 (dx²+ dy² + dz² ) – (1 – 2GM/r) d(ct)².]

Professor Gregorio Ricci-Curbastro (1853-1925) took up Riemann’s suggestion and wrote a 23-pages long article in 1892 on ‘absolute differential calculus’, developed to express differentials in such a way that they remain invariant after a change of co-ordinate system. In 1901, Ricci and Tullio Levi-Civita (1873-1941) wrote a 77-pages long paper on this, Methods of the Absolute Differential Calculus and Their Applications, which showed how to represent equations invariantly of any absolute co-ordinate system. This relied upon summations of matrices of differential vectors. Ricci expanded Riemann’s system of notation to allow the Pythagorean dimensions of space to be defined by a dimensionless ‘Riemann metric’ (named the ‘metric tensor’ by Einstein in 1916)..... continued at http://feynman137.tripod.com/

‘The special theory of relativity … does not extend to non-uniform motion … The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion. Along this road we arrive at an extension of the postulate of relativity… The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant). … We call four quantities A_v the components of a covariant four-vector, if for any arbitrary choice of the contravariant four-vector B^v, the sum over v, å A_v B^v = Invariant. The law of transformation of a covariant four-vector follows from this definition.’ – Albert Einstein, ‘The Foundation of the General Theory of Relativity’, Annalen der Physik, v49, 1916.

Professor Morris Kline describes the situation after 1911, when Einstein began to search for more sophisticated mathematics to build gravitation into space-time geometry: ‘Up to this time Einstein had used only the simplest mathematical tools and had even been suspicious of the need for "higher mathematics", which he thought was often introduced to dumbfound the reader. However, to make progress on his problem he discussed it in Prague with a colleague, the mathematician Georg Pick, who called his attention to the mathematical theory of Ricci and Levi-Civita. In Zurich Einstein found a friend, Marcel Grossmann (1878-1936), who helped him learn the theory; and with this as a basis, he succeeded in formulating the general theory of relativity.’ (M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1990, vol. 3, p. 1131.)

Let us examine the developments Einstein introduced to accomplish general relativity, which aims to equate the mass-energy in space to the curvature of motion (acceleration) of an small test mass, called the geodesic path. Readers who want a good account of the full standard tensor manipulation should see the page by

Dr John Baez or Sir Kevin Aylward

We will give perhaps a slightly more practical and physical interpretation of the basics here. Ricci introduced a tensor, the Ricci tensor, which deals with a change of co-ordinates by using Fitzgerald-Lorentz contraction factor,

g = (1 – i>v²/c²)^1/2. Light is accelerated by gravity exactly twice as much as predicted by Newton’s law. General relativity is a mathematical accounting system and this factor of two comes into it from the energy considerations ignored by Newtonian physics, due to the light speed of the gravitational field itself. When gravity deflects an object with rest mass that is moving perpendicularly to the gravitational field lines, it speeds up the object. Because light is already travelling at its maximum velocity, it cannot be speeded up. Therefore, that half of the gravitational potential energy that goes into speeding up an object with rest mass cannot do so in the case of light and must go instead into additional directional change (downward acceleration). This is the mathematical physics why light is deflected twice the amount suggested by Newton’s law.

The contraction of materials only in the direction of their motion through the physical fabric of space, and their contraction due to the space pressure of gravity in the outward (radial) direction from the centre of a mass indicates a physical nature of space consistent with the 377 ohm property of the vacuum in electronics. Feynman’s approach to quantum electrodynamics, showing that interference creates the illusion that light always travels along the shortest route, accords with this model of space. However, Feynman fails to examine radio wave transmission, which cannot be treated by quantum theory as the waves are continuous and of macroscopic size easily examined and experimented with. The emission of radio is due to the accelerations of electrons as the electric field gradient varies in the transmitter aerial. Because electrons are naturally spinning, even still electrons have centripetal acceleration and emit energy continuously. The natural exchange of such energy creates a continuous, non-periodic equilibrium that is only detectable as electromagnetic forces. Photon emission as described by Feynman is periodic emission of energy. Thus in a sheet of glass there are existing energy transfer processes passing energy around at light speed before light enters. The behaviour of light therefore depends on how it is affected by the existing energy flow inside the glass, which depends on its thickness. Feynman explains in his 1985 book QED that ‘When a photon comes down, it interacts with electrons throughout the glass, not just on the surface. The photon and electrons do some kind of dance, the net result of which is the same as if the photon hit only the surface.’ Feynman in the same book concedes that his path-integrals approach to quantum mechanics explains the chaos of the atomic electron as being simply a Bohm-type interference phenomenon: ‘when the space through which a photon moves becomes too small (such as the tiny holes in the screen) … we discover that … there are interferences created by the two holes, and so on. The same situation exists with electrons: when seen on a large scale, they travel like particles, on definite paths. But on a small scale, such as inside an atom, the space is so small that … interference becomes very important.’ Thus Feynman suggests that a single hydrogen atom (one electron orbiting a proton, which can never be seen without an additional particle as part of the detection process) would behave classically, and it is the presence of a third particle (say in the measuring process) which interrupts the electron orbit by interference, creating the 3+ body chaos of the Schroedinger wave electron orbital.

QFT heuristically explained with a classical model of a polarised virtual charge dielectric

‘As I proceeded with the study of Faraday, I perceived that his method of conceiving the phenomena was also a mathematical one, though not exhibited in the conventional form of mathematical symbols. I also found that these methods were capable of being expressed in the ordinary mathematical forms … For instance, Faraday, in his mind’s eye, saw lines of force transversing all space where the mathematicians saw centres of force attracting at a distance: Faraday saw a medium where they saw nothing but distance: Faraday sought the seat of the phenomena in real actions going on in the medium, they were satisfied that they had found it in a power of action at a distance…’ – Dr J. Clerk Maxwell, Preface, A Treatise on Electricity and Magnetism, 1873.

‘In fact, whenever energy is transmitted from one body to another in time, there must be a medium or substance in which the energy exists after it leaves one body and before it reaches the other… I think it ought to occupy a prominent place in our investigations, and that we ought to endeavour to construct a mental representation of all the details of its action…’ – Dr J. Clerk Maxwell, conclusion, A Treatise on Electricity and Magnetism, 1873 edition.

Analogy of the ‘string theory’ to ‘Copenhagen Interpretation’ quantum mechanics math

‘Statistical Uncertainty. This is the kind of uncertainty that pertains to fluctuation phenomena and random variables. It is the uncertainty associated with ‘honest’ gambling devices…

‘Real Uncertainty. This is the uncertainty that arises from the fact that people believe different assumptions…’ – H. Kahn & I. Mann, Techniques of systems analysis, RAND, RM-1829-1, 1957.

Let us deal with the physical interpretation of the periodic table using quantum mechanics very quickly. Niels Bohr in 1913 came up with an orbit quantum number, n, which comes from his theory and takes positive integer values (1 for first or K shell, 2 for second or M shell, etc.). In 1915, Arnold Sommerfeld (of 137-number fame) introduced an elliptical-shape orbit number, l, which can take values of n –1, n – 2, n – 3, … 0. Back in 1896 Pieter Zeeman introduced orbital direction magnetism, which gives a quantum number m with possible values l, l – 1, l – 2, …, 0, … - (l- 2), -(l – 1), -l. Finally, in 1925 George Uhlenbeck and Samuel Goudsmit introduced the electron’s magnetic spin direction effect, s, which can only take values of +1/2 and –1/2. (Back in 1894, Zeeman had observed the phenomenon of spectral lines splitting when the atoms emitting the light are in a strong magnetic field, which was later explained by the fact of the spin of the electron. Other experiments confirm electron spin. The actual spin is in units of h/(2

p ), so the actual amounts of angular spin are + ½ h/(2p ) and – ½ h/(2p

). ) To get the periodic table we simply work out a table of consistent unique sets of quantum numbers. The first shell then has n, l, m, and s values of 1, 0, 0, +1/2 and 1, 0, 0, -1/2. The fact that each electron has a different set of quantum numbers is called the ‘Pauli exclusion principle’ as it prevents electrons duplicating one another. (Proposed by Wolfgang Pauli in 1925; note the exclusion principle only applies to fermions with half-integral spin like the electron, and does not apply to bosons which all have integer spin, like light photons and gravitons. While you use fermi-dirac statistics for fermions, you have to use bose-einstein statistics for bosons, on account of spin. Non-spinning particles, like gas molecules, obey maxwell-boltzmann statistics.) Hence, the first shell can take only 2 electrons before it is full. (It is physically due to a combination of magnetic and electric force effects from the electron, although the mechanism must be officially ignored by order of the Copenhagen Interpretation ‘Witchfinder General’, like the issue of the electron spin speed.)

For the second shell, we find it can take 8 electrons, with l = 0 for the first two (an elliptical subshell is we ignore the chaos effect of wave interactions between multiple electrons), and l = 1 for next other 6.

Experimentally we find that elements with closed full shells of electrons, i.e., a total of 2 or 8 electrons in these shells, are very stable. Hence, helium (2 electrons) and Argon (2 electrons in first shell and 8 electrons filling second shell) will not burn

Let us now examine how fast the electrons go in the atom in their orbits, neglecting spin speed. Assuming simple circular motion to begin with, the inertial ‘outward’ force on the electron is F = ma = mv²/R, which is balanced by electric ‘attractive’ inward force of F = (q_e/R)²/(4

p e ). Hence, v = ½q_e /(p e

Rm)^1/2.

Now for Werner Heisenberg’s ‘uncertainty principle’ of 1927. This is mathematically sound in the sense that the observer always disturbs the signals he observes. If I measure my car tyre pressure, some air leaks out, reducing the pressure. If you have a small charged capacitor and try to measure the voltage of the energy stored in it with an old fashioned analogue volt meter, you will notice that the volt meter itself drains the energy in the capacitor pretty quickly. A digital meter contains an amplifier, so the effect is less pronounced, but it is still there. A geiger counter held in fallout area absorbs some of the gamma radiation it is trying to measure, reducing the reading, as does the presence of the body of the person using it. A blind man searching for a golf ball by swinging a stick around will tend to disturb what he finds. When he feels and hears the click of the impact of his stick hitting the golf ball, he knows the ball is no longer where it was when he detected it. If he prevents this by not moving the stick, he never finds anything. So it is a reality that the observer always tends to disturb the evidence by the very process of observing the evidence. If you even observe a photograph, the light falling on the photograph very slightly fades the colours. With something as tiny as an electron, this effect is pretty severe. But that does not mean that you have to make up metaphysics to stagnate physics for all time, as Bohr and Heisenberg did when they went crazy. Really, Heisenberg’s law has a simple causal meaning to it, as I’ve just explained. If I toss a coin and don’t show you the result, do you assume that the coin is in a limbo, indeterminate state between two parallel universes, in one of which it is heads and in the other of which it landed tails? (If you believe that, then maybe you should have yourself checked into a mental asylum where you can write your filthy equations all over the walls with a crayon held between your big ‘TOEs’ or your ‘theories of everything’.)

For the present, let’s begin right back before QFT, in other words with the classic theory back in 1873:

Fiat Lux:

‘Let there be Light’

Michael Faraday, Thoughts on Ray Vibrations, 1846. Prediction of light without numbers by the son of a blacksmith who became a bookseller’s delivery boy aged 13 and invented electric motor, generator, etc.

James Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field, 1865. Fiddles with numbers.

I notice that the man (J.C. Maxwell) most often attributed with Fiat Lux wrote in his final (1873) edition of his book A Treatise on Electricity and Magnetism, Article 110:

‘... we have made only one step in the theory of the action of the medium. We have supposed it to be in a state of stress, but we have not in any way accounted for this stress, or explained how it is maintained...’

In Article 111, he admits further confusion and ignorance:

‘I have not been able to make the next step, namely, to account by mechanical considerations for these stresses in the dielectric [spacetime fabric]... When induction is transmitted through a dielectric, there is in the first place a displacement of electricity in the direction of the induction...’

First, Maxwell admits he doesn’t know what he’s talking about in the context of ‘displacement current’. Second, he talks more! Now Feynman has something about this in his lectures about light and EM, where he says idler wheels and gear cogs are replaced by equations. So let’s check out Maxwell's equations.

One source is A.F. Chalmers’ article, ‘Maxwell and the Displacement Current’ (Physics Education, vol. 10, 1975, pp. 45-9). Chalmers states that Orwell’s novel 1984 helps to illustrate how the tale was fabricated:

‘… history was constantly rewritten in such a way that it invariably appeared consistent with the reigning ideology.’

Maxwell tried to fix his original calculation deliberately in order to obtain the anticipated value for the speed of light, proven by Part 3 of his paper, On Physical Lines of Force (January 1862), as Chalmers explains:

‘Maxwell’s derivation contains an error, due to a faulty application of elasticity theory. If this error is corrected, we find that Maxwell’s model in fact yields a velocity of propagation in the electromagnetic medium which is a factor of

2 smaller than the velocity of light.’

It took three years for Maxwell to finally force-fit his ‘displacement current’ theory to take the form which allows it to give the already-known speed of light without the 41% error. Chalmers noted: ‘the change was not explicitly acknowledged by Maxwell.’

Weber, not Maxwell, was the first to notice that, by dimensional analysis (which Maxwell popularised), 1/(square root of product of magnetic force permeability and electric force permittivity) = light speed.

Maxwell after a lot of failures (like Keplers trial-and-error road to planetary laws) ended up with a cyclical light model in which a changing electric field creates a magnetic field, which creates an electric field, and so on. Sadly, his picture of a light ray in Article 791, showing in-phase electric and magnetic fields at right angles to one another, has been accused of causing confusion and of being incompatible with his light-wave theory (the illustration is still widely used today!).

GENERAL RELATIVITY’S HEURISTICALLY EXPLAINED PRESSURE-CONTRACTION EFFECT AND INERTIAL ACCELERATION-RESISTANCE CONTRACTION

Penrose’s Perimeter Institute lecture is interesting: ‘Are We Due for a New Revolution in Fundamental Physics?’ Penrose suggests quantum gravity will come from modifying quantum field theory to make it compatible with general relativity…I like the questions at the end where Penrose is asked about the ‘funnel’ spatial pictures of blackholes, and points out they’re misleading illustrations, since you’re really dealing with spacetime not a hole or distortion in 2 dimensions. The funnel picture really shows a 2-d surface distorted into 3 dimensions, where in reality you have a 3-dimensional surface distorted into 4 dimensional spacetime. In his essay on general relativity in the book ‘It Must Be Beautiful’, Penrose writes: ‘… when there is matter present in the vicinity of the deviating geodesics, the volume reduction is proportional to the total mass that is surrounded by the geodesics. This volume reduction is an average of the geodesic deviation in all directions … Thus, we need an appropriate entity that measures such curvature averages. Indeed, there is such an entity, referred to as the Ricci tensor …’ Feynman discussed this simply as a reduction in radial distance around a mass of (1/3)MG/c² = 1.5 mm for Earth. It’s such a shame that the physical basics of general relativity are not taught, and the whole thing gets abstruse. The curved space or 4-d spacetime description is needed to avoid Pi varying due to gravitational contraction of radial distances but not circumferences.

The velocity needed to escape from the gravitational field of a mass (ignoring atmospheric drag), beginning at distance x from the centre of mass, by Newton’s law will be v = (2GM/x)^1/2, so v² = 2GM/x. The situation is symmetrical; ignoring atmospheric drag, the speed that a ball falls back and hits you is equal to the speed with which you threw it upwards (the conservation of energy). Therefore, the energy of mass in a gravitational field at radius x from the centre of mass is equivalent to the energy of an object falling there from an infinite distance, which by symmetry is equal to the energy of a mass travelling with escape velocity v.

By Einstein’s principle of equivalence between inertial and gravitational mass, this gravitational acceleration field produces an identical effect to ordinary motion. Therefore, we can place the square of escape velocity (v² = 2GM/x) into the Fitzgerald-Lorentz contraction, giving

= (1 – v²/c²)^1/2 = [1 – 2GM/(xc²)]^1/2.

However, there is an important difference between this gravitational transformation and the usual Fitzgerald-Lorentz transformation, since length is only contracted in one dimension with velocity, whereas length is contracted equally in 3 dimensions (in other words, radially outward in 3 dimensions, not sideways between radial lines!), with spherically symmetric gravity. Using the binomial expansion to the first two terms of each:

Fitzgerald-Lorentz contraction effect:

g = x/x₀ = t/t₀ = m₀/m = (1 – v²/c²)^1/2 = 1 – ½v²/c² + ...

Gravitational contraction effect:

g = x/x₀ = t/t₀ = m₀/m = [1 – 2GM/(xc²)]^1/2 = 1 – GM/(xc²) + ...,

where for spherical symmetry ( x = y = z = r), we have the contraction spread over three perpendicular dimensions not just one as is the case for the FitzGerald-Lorentz contraction: x/x₀ + y/y₀ + z/z₀ = 3r/r₀. Hence the radial contraction of space around a mass is r/r₀ = 1 – GM/(xc²) = 1 – GM/[(3rc²]

Therefore, clocks slow down not only when moving at high velocity, but also in gravitational fields, and distance contracts in all directions toward the centre of a static mass. The variation in mass with location within a gravitational field shown in the equation above is due to variations in gravitational potential energy. The contraction of space is by (1/3) GM/c².

This is the 1.5-mm contraction of earth’s radius Feynman obtains, as if there is pressure in space. An equivalent pressure effect causes the Lorentz-FitzGerald contraction of objects in the direction of their motion in space, similar to the wind pressure when moving in air, but without viscosity. Feynman was unable to proceed with the LeSage gravity and gave up on it in 1965. However, we have a solution…

The Generic Fundamental Particle

‘I think the important and extremely difficult task of our time is to try to build up a fresh idea of reality.’ – W. Pauli, letter to Fierz, 12 August 1948.

‘… the Heisenberg formulae can be most naturally interpreted as statistical scatter relations, as I proposed [in the 1934 German publication, ‘The Logic of Scientific Discovery’]. … There is, therefore, no reason whatever to accept either Heisenberg’s or Bohr’s subjectivist interpretation of quantum mechanics.’ – Sir Karl R. Popper, Objective Knowledge, Oxford University Press, 1979, p. 303. (Note statistical scatter gives the energy form of Heisenberg’s equation, since the vacuum is full of gauge bosons carrying momentum like light, and exerting vast pressure; this gives the foam vacuum.)

‘... the view of the status of quantum mechanics which Bohr and Heisenberg defended - was, quite simply, that quantum mechanics was the last, the final, the never-to-be-surpassed revolution in physics ... physics has reached the end of the road.’ – Sir Karl Popper, Quantum Theory and the Schism in Physics, Rowman and Littlefield, NJ, 1982, p. 6.

‘To try to stop all attempts to pass beyond the present viewpoint of quantum physics could be very dangerous for the progress of science and would furthermore be contrary to the lessons we may learn from the history of science … Besides, quantum physics … seems to have arrived at a dead end. This situation suggests strongly that an effort to modify the framework of ideas in which quantum physics has voluntarily wrapped itself would be valuable …’ – Professor Louis de Broglie, Foreword to Dr David Bohm’s book, Causality and Chance in Modern Physics, Routledge and Kegan Paul, London, 2^nd ed., 1984, p. xiv.

‘Niels Bohr brain-washed a whole generation of physicists into believing that the problem had been solved fifty years ago.’ – Murray Gell-Mann, in The Nature of the Physical Universe, Wiley, New York, 1979, p. 29.

Before analysing general relativity and quantum field theory, let’s show how the Heaviside mechanism of electricity suggests the spin speed of static charge, giving a model for the electron.

How physically, can you picture this sort of mathematics? Is it not better to deny that it is possible to understand mathematics in terms of simplicity and causality? In the 1960s while at Motorola, Catt (born 1935, B.Eng. 1959) charged up a 1 m length of coaxial cable to 10 volts, and then discharged it, measuring with a Tektronix 661sampling osclloscope with 4S1 and 4S2 (100 picosecond) plug-ins, finding an output of a 2 m long 5 v pulse. In any static charge, the energy is found to be moving at the speed of light for the adjacent insulator; when discharged, the 50% of the energy already moving towards the exit point leaves first, while the remaining 50% first goes in the opposite direction, reflects back off the far edge, and then exits, creating a pulse of half the voltage and twice the duration needed light to transit the length. Considering a capacitor reduced to simply two oppositely charged particles separated by a vacuum, e.g., an atom, we obtain the particle spin speed.

So the electromagnetic energy of charge is trapped at light speed in any ‘static’ charge situation. David Ash, BSc, and Peter Hewitt, MA, in their 1994 book reviewing electron spin ideas, The Vortex (Gateway, Bath, page 33), stated: ‘… E = mc² shows that mass (m) is equivalent to energy (E). The vortex goes further: it shows the precise form of energy in matter. A particle of matter is a swirling ball of energy … Light is a different form of energy, but it is obvious from Einstein’s equation that matter and light share a common movement. In E = mc², it is c, the speed of light, which related matter to energy. From this, we can draw a simple conclusion. It is obvious: the speed of movement in matter must be the speed of light.’ However, Ash and Hewitt don’t tackle the big issue: ‘It had been an audacious idea that particles as small as electrons could have spin and, indeed, quite a lot of it. … the ‘surface of the electron’ would have to move 137 times as fast as the speed of light. Nowadays such objections are simply ignored.’ – Professor Gerard t’Hooft, In Search of the Ultimate Building Blocks, Cambridge University Press, 1997, p27. In addition, quantum mechanical spin, given by Lie’s mathematics, is generally obscure, and different fundamental particles have different spins. Fermions have half-integer spin while bosons have integer spin. Neutrinos and antineutrinos do have a spin around their propagation axis, but the maths of spin for electrons and quarks is obscure. The twisted paper loop, the Mobius strip, illustrates how a particle can have different quantum mechanical spins in a causal way. If you half twist a strip of paper and then glue the ends, forming a loop, the result has only one surface: in the sense that if you draw a continuous line on the looped paper, you find it will cover both sides of the paper! Hence, a Mobius strip must be spun around twice to get back where it began! The same effect would occur in a spinning fundamental particle, where the trapped energy vector rotates while spinning.

Magnetism, in Maxwell’s mechanical theory of spinning virtual particles in space, may be explained akin to vortices, like whirlpools in water. If you have two whirlpools of similar spin (either both being clockwise, or both being anticlockwise), they attract. If the two whirlpools have opposite spins, they repel. In 1927, Samuel Goudsmit and George Uhlenbeck introduced the spin quantum number. But under Bohr’s and Heisenberg’s ‘Machian’ (‘non-observables like atoms and viruses are not real’) paranoid control, it was subsumed into Lie algebra as a mathematical trick, not a physical reality, despite Dirac’s endorsement of the ‘aether’ in predicting antimatter. Apart from the spin issue above that we resolved by the rotation of the Heaviside-Poynting vector like a Mobius strip, there is also the issue that the equator of the classical spherical electron would revolve 137.03597 times faster than light. Taking Ivor Catt’s work, the electron is not a classical sphere at all, but a Heaviside-Poynting energy current trapped gravitationally into a loop, and it goes at light speed, which is the ‘spin’ speed.

If the electron moves at speed v as a whole in a direction orthogonal (perpendicular) to the plane of the spin, then the c speed of spin will be reduced according to Pythagoras: v² + x² = c² where x is the new spin speed. For v = 0 this gives x = c. What is interesting is that this model gives rise to the Lorentz-FitzGerald transformation naturally, because: x = c(1 - v² / c² )^1/2 . Since all time is defined by motion, this (1 - v² / c² )^1/2 factor of reduction of fundamental particle spin speed is therefore the time-dilation factor for the electron when moving at speed v. So there is no metaphysics in such ‘time travel’! Mass increase occurs due to the snowplough effect of the fabric of spacetime ahead of the particle, since it doesn’t have time to flow out of the way when the speed is great.

The light photon has a spin angular momentum is cmr where the effective mass m is of course energy equivalent, m = E/c² (from E = mc² ). Using Planck’s E = hf = hc/

l where f is frequency and l is wavelength (l = 2p r ), we find that the spin angular momentum is cmr = ½ h/p , which is well verified experimentally. Since the unit of atomic angular momentum is ½ h/p

, we find the light boson has a spin or 1 unit, or is a spin-1 boson, obeying Bose-Einstein statistics. The electron, however, has only half this amount of spin, so it is like half a photon (the negative electric field oscillation of a 1.022 MeV gamma ray, to be precise). The electron is called a fermion as it obeys Fermi-Dirac statistics, which applies to half-integer spins. (The spins of two fermions can, of course, under some special conditions ‘add up’ to behave as a boson, hence the ‘Bose-Einstein condensate’ at very low temperatures.)

I corresponded on these topics with Dr Arnold C. Lynch, who later gave the IEE Centenary Lecture on Sir J. J. Thomson’s discovery of the electron in 1997, and he also presented the Catt Anomaly to the IEE in HEE/26 the next year. Lynch had been chosen to give the lecture on the electron because J. J. Thomson (1856-40) told him about it in Cambridge. (J. J. Thomson had always favoured a vortex picture of the fundamental particle, but had been caught up in Kelvin’s mythical ‘vortex atom’ speculations, which were falsely promoted as mathematically beautiful just like string theory today, and so ‘queered the pitch’ for any attempt to develop vortex fundamental particle ideas, just as Maxwell had ‘queered the pitch’ on aether by making errors and false predictions disproved in the Michelson-Morley experiment of 1887.) Sir J. J. Thomson wrote in his 1884 Treatise on the Motion of Vortex Rings: ‘… the vortex theory of matter is of a much more fundamental character than the ordinary solid particle theory.’ Notice that he does not automatically and outrageously jump to the unproved claim that the vortices are atoms, thus he paves the way for his own discovery of 1897 that the atoms are not fundamental particles, but contain fundamental particles. Back in 1875, James Clerk Maxwell had falsely written in his famous article on ‘The Atom’ in Encyclopaedia Britannica: ‘… the vortex ring of Helmholtz … satisfies more of the conditions than any atom hitherto imagined.’ This was just hype, because it was pushing speculation that ignores the chemical facts and that has not a fig leaf of empirical evidence to support it (exactly like string theory claims today). It offered no testable prediction for the masses of different atoms (just as ‘string theory’ today offers absolutely no testable prediction of the masses of different fundamental particles today).

I don’t really want to discuss vortex theory in detail, because like string theory, most of it is speculative and not useful. Lord Kelvin (William Thomson who became Lord Kelvin in 1892, not to be confused with Sir J. J. Thomson) introduced vortex in a lecture to the Royal Society of Edinburgh on 18 February 1867, in which he claimed matter is moving vortices in the aether of space. (‘The Vortex Theory of Ether’, Pro. Roy. Soc. Edin., v6, 1867, pp 94-105; Phil. Mag., v34, 1867, pp 15-24.) He used chemical smoke rings in the air to dramatically produce smoke rings, which were stable and would diffract around knives without being cut, and bounce off walls! The chemical smoke was produced by mixing acid with ammonia in a box with a flexible wall at one end and a circular hole in the other. By hitting the flexible end of the box, a smoke ring-vortex was produced from the hole that propagated outward. Using two smoke boxes with the holes facing one another, vortex collisions could be produced and studied: they bounced and recoiled! Thus, they did not break up when colliding. They just shook like rubber rings, sending off eddy air currents (a bit like light being generated by colliding particles), so they behaved like atoms.

Obviously the problem is that they do eventually dissipate, so smoke vortices, whirlpools, hurricanes, anti-cyclones and tornadoes are not permanent, despite a brief period of stability. Herman von Helmholtz (1821-94) in 1858 showed that in a frictionless or ‘perfect’ fluid, vortices would not dissipate: Ueber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. This added to the vortex crazy of Victorian mathematics.

The only widely known attempt to introduce some kind of causal fluid dynamics into quantum mechanics was by Professor David Bohm and Professor J. P. Vigier in their paper ‘Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid with Irregular Fluctuation’ (Physical Review, v 96, 1954, p 208). This paper showed that the Schroedinger equation of quantum mechanics arises as a statistical description of the effects of Brownian motion impacts on a classically moving particle. However, the whole Bohm approach is wrong in detail, as is the attempt of de Broglie (his ‘non-linear wave mechanics’) to guess a classical potential that mimics quantum mechanics on the small scale and deterministic classical mechanics at the other size regime. The whole error here is due to the Poincaré chaos introduced by the three-body problem, which destroys determinism (but not causality) in classical, Newtonian physics:

‘… the ‘inexorable laws of physics’ … were never really there … Newton could not predict the behaviour of three balls … In retrospect we can see that the determinism of pre-quantum physics kept itself from ideological bankruptcy only by keeping the three balls of the pawnbroker apart.’ – Tim Poston and Ian Stewart, Analog, November 1981.

So it is not quantum physics that is the oddity, but actually classical physics. The normal teaching of Newtonian physics at low levels falsely claims that it allows the positions of the planets to be exactly calculated (determinism) when it does not. Newton’s laws do not contain any exact solution for more than two bodies, and there are more than two bodies in our solar system. So the problem to address is the error in classical, Newtonian physics, which explains why quantum mechanics is the way it is. Bohm’s approach was to try to obtain a classical model of quantum mechanics, which is the wrong approach, since classical physics is the fiddle. What you first have to admit is that Newton only dealt with two bodies, so his laws simply don’t apply to reality.

Henri Poincaré’s work shows that in any atom, you will have chaos whenever you observe it, even in the Newtonian mechanics framework. The simplest atom is hydrogen, with an electron going around a proton. As soon as you try to observe it, you must introduce another particle like a photon or electron, which gives rise to a 3-body situation! Therefore the chaotic, statistical behaviour of the situation gives rise to the statistical Schroedinger wave equation of the atom without any need to introduce explanations based on ‘hidden variables’. The only mediation is the force gauge boson, which is well known in quantum field theory, and is not exactly a ‘hidden variable of the sort Bohm looked for. Newton’s error is restricting his theory to the oversimplified case of only two bodies, when in fact this is a bit like Euclidean geometry, missing a vital ingredient. (Sometimes you do really have to deepen the foundations to build a taller structure.)

In 1890, Poincaré published a 270-pages book, On the problem of Three Bodies and the Equations of Dynamics. He showed that two bodies of similar mass have predictable, deterministic orbital motion because their orbits trace out closed, repeating loops in space. But he found that three bodies of similar mass in orbit trace out irregular, continuously changing unclosed loops and tangles throughout a volume of space, not merely in the flat plane they began in. The average radius of a chaotic orbit that is equal to the classical (deterministic) radius, and the probability of finding the particle beyond average radius diminishes, so giving the basis of the Schroedinger model, where the probability of finding the electron peaks at the classical radius and diminishes gradually elsewhere. Computer programs approximate chaotic motion roughly by breaking up a three body problem, ABC, into steps AB, AC, and BC, and then cyclically calculating motions of each pair of bodies brief period of time while ignoring the other body for that brief period. This is not exact, but is a useful approximation for understanding how chaos occurs and what statistical variations are possible over a period of time. It disproves determinism! Because most of the physicists working in quantum mechanics have not studied the mathematical application of chaos to classical atomic electrodynamics, they have no idea that Newtonian physics is crackpot off the billiard table, and can’t describe the solar system in the way it claims, and that the ‘contradiction’ usually presented as existing between classical and quantum physics is not a real contradiction but is down to the falsehood that classical physics is supposed to be deterministic, when it is not.

LOOP QUANTUM GRAVITY

2006-01-11T13:15:00.000-08:00

Loop quantum gravity is the main alternative to string theory.

Loop quantum gravity: a spin foam vacuum QFT (quantum field theory)

'In 1986, Abhay Ashtekar reformulated Einstein's field equations of general relativity using what have come to be known as Ashtekar variables, a particular flavor of Einstein-Cartan theory with a complex connection. He was able to quantize gravity using gauge field theory. In the Ashtekar formulation, the fundamental objects are a rule for parallel transport (technically, a connection) and a coordinate frame (called a vierbein) at each point. Because the Ashtekar formulation was background-independent, it was possible to use Wilson loops as the basis for a nonperturbative quantization of gravity. Explicit (spatial) diffeomorphism invariance of the vacuum state plays an essential role in the regularization of the Wilson loop states. Around 1990, Carlo Rovelli and Lee Smolin obtained an explicit basis of states of quantum geometry, which turned out to be labelled by Penrose's spin networks.' - Wikipedia.

Why string theory is a fraud

According to string theory (there is actually no string theory, because the theorists haven't discovered any workable dynamics yet so it can't make any predictions), everything is composed of vibrating Planck-sized tiny strings with 6 dimensions curled up into a Calabi-Yau manifold or 'insect'. The other 4 dimensions of space-time or 5 dimensions of Kaluza-Klein space-time (for unification of gravity and electromagnetism), make a total of 10 or 11 dimensions. String theorist Edward Witten, who unified 10 dimensional superstrings with 11 dimensional supergravity in March 1995, claimed in the April 1996 issue of Physics Today that 'String theory has the remarkable property of predicting gravity.' In fact, string theory doesn't predict the strength of gravity (or anything else that is scientific, i.e., potentially checkable). All it does is to allow the possibility of a spin-2 (graviton) mode if a dynamical theory can be discovered which fits the perturbative expansion, as Peter Woit points out.

anon Says:
Sean Carroll now presents a graph of evidence that dark energy varies with time: http://cosmicvariance.com/2006/01/11/evolving-dark-energy/
I love this expert cosmology. The fact that they keep modifying the theory to fit the new observations really shows how expert these guys are. Thanks Peter, for your patience with them!

A more correct (empirically driven, but still predictive and testable) approach to quantum gravity: http://feynman137.tripod.com/