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Gauge Theory and Quantum Gravity
Einstein's 1916 theory of gravity, electromagnetism and the Yang-Mills
theory of the weak and strong sub-nuclear forces are all gauge theories, yet in 4D there is a basic difference between gravity and the rest and that is Einstein's local principle of equivalence. But what is the local principle of equivalence other than the universality of localizing the 4-parameter translation subgroup T4 of the 10-parameter Poincare group (1905 special relativity) for ALL non-gravity quantum fields? EM-weak-strong gauge forces are from the internal U(1) SU(2) flavor and SU(3) color groups with parity violation added so that the left and right handed-chiral projections of the basic qubit spinor fields of leptons and quarks see the nonabelian Yang-Mills fields very differently. Note that Noether's theorem connecting global rigid symmetries to global conserved "charges" including total energy and momentum Pu of a non-gravity field for T4 does not survive localizing T4. This is the basic reason why the purely conformal Weyl classical gravity total energy is nonlocal and why in general one cannot define conserved total energy, linear and angular momentum of any field globally in warped spacetime. The local stress-energy current conservation law Tuv(non-gravity field)^;v = 0 does not automatically define globally conserved integrals over 3D spacelike slices of 4D spacetime in the general case. One has approximately conserved global quantities only for special asymptotically flat solutions of the generally covariant fields with special Killing isometries in the metric field. In other words, Noether's theorem connecting symmetries to conservation laws is restricted to globally flat spacetimes in the absence of curvature and torsion. Note, that the total dark energy of our accelerating expanding universe is not conserved because its w = -1 Ricci energy density (c^4/G)/\zpfg00 is "constant", i.e. in the IR cosmological equations http://www-conf.slac.stanford.edu/ss...ew_Page_05.jpg (c^4/G)/\zpfg00 --- (zero point quantum vacuum fluctuation energy density) ~ a(t)^0 = constant where a(t) is the IR (long wave) FRW cosmological scale factor stretch in the fabric of space. Think of 3D space as a FLAT infinite elastic sheet that is at global cosmic time t in the Hubble flow (lots of Killing isometries in this particular solution) so that at every point on the sheet there is an isotropic stretch a(t) in all directions from that point. That is, let d(P1,P2) be the distance between galaxies 1 & 2 at cosmic time t where t = 0 is the moment of inflation, then the distance between those same galaxies at time t' will be d'(P1,P2) where d'(P1,P2)/d(P1,P2) = a(t')/a(t) Now back to the main point - what is the connection? In GR everyone was thrown off track by thinking that the fundamental connection was at the quadratic 2nd-rank tensor geometrodynamic level instead of the 1st rank tensor vector tetrad field level! The EM & Yang-Mills internal symmetry compensating gauge boson VECTOR fields are all spin 1 quantum fields that are renormalizable. Quantum gravity theorists have been thrown off course by noting that the tensor metric fields are unrenormalizable like the old Fermi beta radioactivity model. However, the tetrad fields are spin 1 vector fields! They are the basic gravity fields. The non-renormalizable tensor fields from unthinkingly quantizing guv is asking the WRONG QUESTION of Nature. Quantize the spin 1 e^a tetrad Lorentz vector fields that are GCT invariants! Then it's a piece of cake. ds^2 = e^aea Obviously you quantize e^a. Also quite obviously e^a = I^a + kA^a where k is a dimensionless coupling constant like alpha = e^2/hc in QED. A^a is the spin 1 Lorentz group VECTOR field, i.e. the fundamental CONNECTION FIELD from localizing the 4-parameter translation subgroup of P10 universally for ALL non-gravity field actions. I^a is the trivial globally flat tetrad when k - 0. I^a's components I^au are curvilinear for non-geodesic observers in globally flat Minkowski spacetime - but so what? From localizing the 10-parameter P10 we also get the spin connections S^a^b = - S^b^a which enable GCT covariant derivatives of the Dirac, Weyl & Majorana spinor fields as well as the tensor fields since we easily get the Einstein Levi-Civita derived connection from products of S^ab with the tetrad components e^au. See Rovelli's book Ch II for details. Therefore there was a lot of wasted time in quantum gravity these past 50 years. I have not even touched above on the emergence of the c-number vacuum condensate parts of the above VECTOR tetrad 1-form fields e^a along with their spin connection 1-forms S^a^b. What are the Yang-Mills equations for the tetrad vector fields. The Yang-Mills field tensor 2-form is obviously F^a = dA^a + S^ac/\A^c This is in analogy with F = dA in U(1) internal symmetry electromagnetism. F^a is related to the torsion field i.e. T^a = De^a = de^a + S^ac/\e^c = torsion field 2-form where e^a = I^a + kA^a The curvature field 2-form is R^a^b = DS^a^b = dS^a^b + S^ac/\S^cb Returning to the Yang-Mills spin 1 Lorentz group vector field equations F^a = DA^a = dA^a + S^ac/\A^c DF^a = 0 D*F^a = *J^a *F^a = Hodge dual of F^a *J^a is a current source density 3-form of the *F^a field. D*J^a = 0 is local source current density In general [A^a, A^b] = K^a^bcA^c Where K^a^bc are the structure constants of a Lie algebra. What is the Lie group G here? Is it T4? Is it O(1,3)? Is it P10? On May 20, 2007, at 1:43 PM, Jack Sarfatti wrote: On May 19, 2007, at 5:07 PM, Jack Sarfatti wrote: Commentary 1 (Draft 2 expanded) This is a very useful little book by Lochlainn O' Raifaeartaigh in Dublin published by Princeton 1997. It has seminal papers by Weyl, Klein, Fock, Schrodinger, London & Pauli in English from the original German. The two great battle-tested principles of basic theoretical physics are 1. The local gauge principle, i.e. "relativity" in the most general sense of no action without reaction, no passive absolute Newtonian arenas, Leibniz's relationism of Bohm's "dialogue" not monologue. 2. The spontaneous breaking (or "hiding" Sidney Coleman, Erice Lectures 1970's) of continuous symmetries in the ground state of real on mass shell quanta and also in the vacuum of virtual zero point quanta. AKA "More is different" (P.W. Anderson) "Emergent complexity." "ODLRO" (Onsager & Oliver Penrose), "Goldstone theorem" "Macroquantum states", "Glauber coherent & squeezed states" et-al. The idea of "hidden symmetry" is that whilst the dynamical classical action in the Feynman path integral alternative to second quantization is invariant under the symmetry group G, the vacuum (ground state) is not. One familiar example is a ferromagnet near absolute zero with a Galilean relativity 3-vector order parameter "magnetization" in a coherent domain. This is three real "Higgs fields" with two continuous 2pi periodic "Goldstone phases" defining the S2 vacuum manifold of minima in the Landau-Ginzburg effective quartic renormalizable interaction Higgs-type field Lagrangian. Non-trivial 2nd order homotopy would give point "monopole" defects not actually observed in real ferromagnetic phases. What is observed are wall domain defects with a S0 vacuum manifold without any continuous Goldstone phase at all corresponding to another broken symmetry from three real Higgs fields to only one effective Higgs field that vanishes on the domain wall. That is S3 broken to S1 - curious. Emergence of "The Nine" My original parsimonious theory explains the origin of gravity & inertia, torsion (i.e. emergent "tetrad" local observer/detector frames and "spinor" connections as macroquantum coherent "surface" world hologram Goldstone phase modulations from an "M-Matrix" of non-closed 1-forms made from two Lorentz 4-vectors of eight 0-form continuous periodic Goldstone phases Theta^b & Phi^a from nine post-inflation real Higgs scalar vacuum ODLRO fields. a,b = 0,1,2,3 are Lorentz group indices M^a^b = Phi^a/\dTheta^b - dPhi^a/\Theta^b dM^a^b = 2dPhi^a/\dTheta^b A^a = M^a^a (diagonal elements of the Matrix) Einstein-Cartan tetrad 1-forms are e^a = I^a + @A^a are spin 1 Yang-Mills type Lorentz group VECTOR renormalizable quantum translation group T4 localized gauge fields. @ = (Lp^2/\zpf)^1/3 dimensionless "world hologram" self-gravity coupling I^a are the global Minkowski tetrad frames that we have when either G - 0 gravity coupling switched off h - 0 quantum action switched off c - infinity, i.e. no causal retardation and/or advanced retro-causation /\zpf - 0 i.e. no gravity when supersymmetry is perfect! /\zpf in the IR limit is Lenny Susskind's cosmic landscape parameter ~ (area of future deSitter horizon of pocket Hubble bubble universe in the "megaverse" of eternal chaotic inflation. Supersymmetry is square root of T4, i.e. anticommutator of supersymmetries is T4. "Spinor" connection 1-forms are (gets dynamical degrees of freedom when total 10-parameter Poincare group P10 = T4*O(1,3) is locally gauged (Utiyama, Kibble et-al 1961) S^a^b = - S^b^a = M^[a,b] Torsion field 2-forms are T^a = de^a + S^ace^c Curvature field 2-forms are R^a^b = dS^a^b + S^acS^cb Einstein-Hilbert (E-H) "classical" Lagrangian density is the 0-form L(E-H) ~ {a,b,c,d}R^a^b/\e^c/\e^d ds^2 = guvdx^udx^v = e^aea = I^aIa + @(I^aAa + A^aIa) + @^2A^aAa Quantum noise corrections in second quantized formalism with macro-quantum vacuum |0 condensate ODLRO is A^a = 0|A^a|0 + &A^a &A^a is the quantum excitation spin 1 tetrad vector field annihilation operator Note that R^a^b is quadratic in A^a and its gradients, therefore L(E-H) has up to quartic terms in the tetrads consistent with renormalization. Note that F^a = dA^a = dM^a^a = dPhi^a/\dTheta^a =/= 0 We have "Maxwell" equations dF^a = 0 d*F^a = *J^a (warped tetrad current density) d*J^a = 0 (local conservation of warped tetrad current density) The Einstein geometrodynamic field ds^2 = e^aea is quadratic in the tetrads, therefore obviously made from entangled Einstein-Rosen-Podolsky (1935) pairs of spin 1 tetrad quanta, hence we have quantum geometrodynamic corrections of spin 2 tensor gravitions, spin 1 vector gravi-photons and spin 0 scalar gravitons. 3x3 = 5 + 3 + 1 i.e. 1 + 1 = 2, 1, 0 for the irreducible spin/angular momentum representations of O(3) subgroup of O(1,3). This does not affect the macro-quantum c-number ODLRO sector of the theory in most cases. Dark energy and dark matter are both very simply virtual quanta inside the total physical vacuum of positive and negative zero point fluctuation energy density respectively. Einstein showed that for an isotropic Ricci source in 3+1 spacetime, the space-time bending power of the source is (G/c^2)(energy density of source)(1 + 3w) Lorentz invariance and general coordinate invariance imply w = -1 for isotropic ZPF distributions of all quantum fields of all spins. Bosons have positive ZPF energy density Fermions have negative ZPF energy density. Non-isotropic boundary conditions e.g. Casimir effect will shift the ZPF w, but as long as w - 1/3 i.e. quintessent field region then positive ZPF energy density - antigravity universal repulsion blue shift (dark energy) negative ZPF energy density - gravity universal attraction red shift (clumping dark matter) - this is indistinguishable from w = 0 conventional CDM for distant observers looking at gravity lensing for example. Note when w - 1 that is "phantom energy" BIG RIP I predict that the LHC will NOT find dark matter particles on mass shell as a matter of basic principle. Looking for dark matter inside the LHC is like looking for the motion of the Earth through the mechanical Victorian aether with a Michelson-Morley interferometer. Null results then and in the future I do declare. All of this is only based on 1 & 2 above. The book is mostly about the early history of 1 and how it affected C.N. Yang as a graduate student. Robert Shaw's little-known important independent work is also discussed. "The role of geometry in physics has always been central. But ... it was passive, providing only the stage on which physics took place ... the theory of special relativity in 1905, when it became clear that space and time were not independent ... But the most profound and astonishing entry of geometry into physics came with Einstein's theory of gravitation in 1916, which showed that gravitation was nothing but the curvature of four-dimensional space. ... George Bernard Shaw ... wrote 'Asked to explain why planets did not move in straight lines and run straight out of the universe, Einstein replied that they do not do so because space is not rectilinear but curvilinear.' ... due almost entirely to the genius of Einstein, geometry graduated from being the stage on which the drama of physics took place to being a major player in the drama. There remained however the electromagnetic and nuclear forces, and the geometrization of gravity raised the question as to whether these other fundamental forces were 'true' forces operating in curved space of gravitational theory or whether they also were part of the geometry. This question has still not been fully answered. ... these forces and gravitation have a common geometrical structure. This is ... the gauge structure." to be continued Jack Sarfatti "If we knew what it was we were doing, it would not be called research, would it?" - Albert Einstein http://www.authorhouse.com/BookStore...x?bookid=23999 http://lifeboat.com/ex/bios.jack.sarfatti http://qedcorp.com/APS/Dec122006.ppt http://video.google.com/videoplay?do...ausation&hl=en http://www.flickr.com/photos/lub/sets/72157594439814784 |
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Gauge Theory and Quantum Gravity
What do you mean by 4d ?
-y |
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Gauge Theory and Quantum Gravity
On May 22, 3:41 am, Y wrote:
What do you mean by 4d ? -y " "4-velocity and 4-acceleration" http://farside.ph.utexas.edu/teachin...es/node15.html Sue... |
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