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Microscopic Origins for Gauge and Space Time Symmetries

Microscopic Origins for Gauge and Space Time Symmetries. S.Randjbar-Daemi, Greece June 2005. References:. S. Randjbar-Daemi and J.A. Strathdee Phys. Lett B337, ( 1994) 309 and Int. Journal of Modern Physics A 10 (1995) 4651 S.R. D Work in Progress. Introduction.

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Microscopic Origins for Gauge and Space Time Symmetries

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  1. Microscopic Origins for Gauge and Space Time Symmetries S.Randjbar-Daemi, Greece June 2005

  2. References: • S. Randjbar-Daemi and J.A. Strathdee Phys. Lett B337, ( 1994) 309 and Int. Journal of Modern Physics A 10 (1995) 4651 • S.R. D Work in Progress

  3. Introduction • Local Quantum Field Theories are inevitable in low energy physics: They reconcile quantum mechanics with special relativity • Renormalizable theories are probably the only consistent ones: They are insensitive to short distance physics. • To each QFT there may correspond a large class of models with the same Low E sector

  4. On the other hand….. • Relativistic Invariance may not be Fundamental but merely an approximate feature of the Low Energy Sector! • General Relativity is a very successful theory of large scale gravitational phenomena, BUT • it is not renormalizable and hence is sensitive to the short distance effects • There is no experimental guide to a Microscopic Theory of Gravity

  5. We have no guide to lead us to a microscopic theory of gravity • Only recently corrections to sub- millimeter gravity have been examined ( Brane World Scenarios) • Any Quantum Theory of Gravity will be SPECULATIVE • Superstring theory is probably the best ( and the most ambitious) speculative theory of gravity and certainly the most elegant! • But there may be others……(less ambitious!) and less elegant! The may help us to appreciate the elegance of string theory.

  6. Our Idea • The microscopic d.o.f of gravity may be different from what we see at long distances, i.e. the metric tensor of space and time • In this sense gravity is an emergent effect which manifests itself in terms of different variables at different scales

  7. Plan of This talk • The Model • Generalized Effective Action • Gap Equations • Ground State and its Symmetries • Fermion Propagator • Gauge Sector • Example • Gravity

  8. Our Model consists of an array of sites with a Grassman variable attached to each site This is invariant under the permutation group provided we also transform the couplings J The total action may contain an extra term, S(J), invariant under the permutation group

  9. The ModelRelation to Ising Type ModelsSpin variables S(A)= 0, 1 at each site and the partition function: The leading term as produces the 4-Fermi action

  10. Rewrite the Ising Z(J)

  11. Each Differentiation yields a factor of J. The leading term is: Takes the form of a sum over all pairings of the sites and can be expressed by an integral over Grassmann variables

  12. Basic Quantity:

  13. Impose some structures:Assumptions: Our Lattice is composed of4d cells labeled by integersn. Within each cell there areNsites Notation :A=(i, n),wherei=1,2,…N and

  14. The Action becomes: This is a 4-Fermi model with no bilinears in the Grassmann variables!

  15. Our Aim: • Obtain information about the structure of the ground state of this model and the spectrum of long-wavelength oscillations • Since there are no bilinears in the Grassmann variables we should ask:

  16. Are there any propagating d.o.f? • Are there any light modes in the spectrum?

  17. Methodology: • Set up a set of Generalized Gap equations for the Propagator, • These Equations are difficult to solve, even in simplest cases. But • We can study the property of the system about an assumed solution by making some simplifying assumptions

  18. Generalized Effective Action: • Ref: C. De Dominicis J. Math. Phys. 3( 1962)983 • R. Jakiw and K. Johnson Phys.Rev. D8 (1973) 2386 • J.M. Cornwall and R.E. Norton Phys. Rev D8 (1973) 3338

  19. Generalized Action: • Depends on the 2 point functions as well as the fields • The equation for the 2 point functions is obtained as an extremal condition • The Generalized action functional is represented by a loop expansion of 2-particle-irreducible vacuum graphs in which the lines are associated with the unknown functions.

  20. Rewrite the Fermionic Action:Introduce a scalar field S does not have any bilinears in

  21. Generlized Effective Action:

  22. Feynmann Rules

  23. Lowest Order graphs: • There is only one 2-loop graph

  24. And one 3-loop graph

  25. The Gap Equations:

  26. Hartree-Fock Approximation • Retain only the 2-loop contribution to Where, s=n-m, G(n,m)= G(s), etc

  27. Ground State Symmetries1. Global symmetries: • The Generalized Effective Action is invariant w.r.t a space group that acts on the lattice: Where : h belongs to the point group

  28. The couplings are invariantJ(n’-m’)=J(n-m) And the independent variables transform as scalars

  29. 2. Local Symmetries: There is a local GL(N,C) Invariance w.r. to which are invariant but: The gap equations which determine the ground state are covariant under these global and local transformations

  30. Assume that the solution of the gap equations defines a homomorphism Such that Where

  31. If such solution could be found:1. Fermions propagate 2. They belong to a non trivial representation of the ground state symmetry group, ie. They carry SPIN • For a reasonable Fermion Spectrum: • Ground state symmetry group should have a factor which is a subgroup of Spin(4)

  32. Require that H is contained in O(4) • Our Lattice is a 4-dimensional Crystal • There are 227 four dimensional Crystals • We chose the largest one. Its point group has 1152 elements and it coincides with the Weyl group of the weight lattice of the exceptional Lie Algebra

  33. The action of point group divides the lattice into orbits • The number of points on each orbit can be as large as the numer of elements in H, but it can be smaller if there is a subgroup of H leaving some points fixed • The couplings J(s) are equal for all points on the same orbit and the Green function G(s) is fully specified if it is known at one point on each orbit

  34. The Lattice: • Comprise the points with integer coefficients where the basis vectors are expressible in the orthonormal basis by

  35. Some Nearby Orbits:

  36. Spinor Representations Where

  37. Generators

  38. Dirac Matrices in the lattice basis The vector representation in the lattice basis:

  39. Suppose now that the solution of the gap equations assigns fermions to the Dirac representation, i.e Where a ranges over the entire group. For the Fourier transform The invariance condition becomes:

  40. Long wavelength modes • The invaraince condition implies that near • K=0 the fermion Green function becomes • With Z and M numbers to be determined by • solving the gap equations. • The leading terms are O(4) invariant. • Higher order terms will break it.

  41. Gauge Sector • The Green Function G(n,m) is a dynamical variable. • In the long wavelength regiem its dynamics should be expressible in terms of scalars, vectors, tensors, etc. • Among the light degrees of freedom we should expect to find a Yang Mills field • Our aim is to isolate this term

  42. Recall the gauge transformation of G Adopt the Ansatz The path ordered integral is along a straight line joining n and m Where

  43. Under local gauge transformations The unbroken local symmetries must be associated with a set of Yang –Mills vectors and these must contribute a factor as indicated above, in the long wave length limit. Of course G(n,m) must represent other d.o.f. but, for simplicity, we shall assume that they are all heavy and therefore irrelevant.

  44. Calculating the gauge coupling constant • The lowest order gauge dynamics should be governed by the standard Yang-Mills action with a calculable gauge coupling. • We must substitute our ansatz in the generalized effective action, • Use the gap equations to eliminate This is obviously independent of A

  45. This depends on A ! • After these eliminations the gauge invariant functional in the lowest order in derivative expansion must have the form With a calculable e The O(4) invariance is a consequence of the invariance of w.r.t the point group. It is true only in the leading order in k!

  46. Some Steps in the calculation of the gauge coupling • Basically we need to calculated the quantum induced vacuum polarization and • Show that it satisfies the Ward identity The basic quantity is To be evaluated at A=0 and fixed at the extremum

  47. Hartree- Fock Approximation:Keep only 2-loop contributions + …. The derivatives atA=0are Where the T’s commute with the background G

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