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

• 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

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

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.

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

Plan of This talk

• The Model• Generalized Effective Action• Gap Equations• Ground State and its Symmetries• Fermion Propagator• Gauge Sector• Example• Gravity

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

The Model

Relation to Ising Type Models

Spin variables S(A)= 0, 1 at each site and the partition function:

ABABS

A SiSJS exp Z

The leading term as 0J

produces the 4-Fermi action

Rewrite the Ising Z(J)

)1/2- exp()1(

)1/2- exp()exp1(dJ) (det2

)S(i2

1-exp

d)2(det Z(J)

0ABA

1ABA

A

1/2-

AA1

AB

A

2/1

S

A

BA

BAA

BA

A

Je

Ji

J

J

Each Differentiation yields a factor of J. The leading term is:

0A )2/1exp()( BABAA

J

Takes the form of a sum over all pairings of the sites and can be expressedby an integral over Grassmann variables

BAA BA

B)J(A, dd exp

Basic Quantity:

12N

n

Impose some structures:Assumptions:

• Our Lattice is composed of 4d cells

labeled by integers n.

• Within each cell there are N sites

• Notation : A=(i, n), where i=1,2,…N

and 4Zn

The Action becomes:

Where

in the Grassmann variables!

This is a 4-Fermi model with no bilinears

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:

• Are there any propagating d.o.f?

• Are there any light modes in the spectrum?

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

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

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.

Rewrite the Fermionic Action:Introduce a scalar field

)(n

S does not have any bilinears in

Generlized Effective Action:

Feynmann Rules

Lowest Order graphs:

• There is only one 2-loop graph

• And one 3-loop graph

The Gap Equations:

Hartree-Fock Approximation

• Retain only the 2-loop contribution to

Where, s=n-m, G(n,m)= G(s), etc

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

The couplings are invariantJ(n’-m’)=J(n-m)

And the independent variables transform as scalars

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 underthese global and local transformations

• Assume that the solution of the gap equations defines a homomorphism

Such that

Where

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)

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

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

The Lattice:

• Comprise the points with integer coefficients where the basis vectors

are expressible in the orthonormal basis by

Some Nearby Orbits:

Spinor Representations

Where

Generators

Dirac Matrices in the lattice basis

The vector representation in the lattice basis:

Suppose now that the solution of the gap equations assigns fermions

to the Dirac representation, i.e

The invariance condition becomes:

Where a ranges over the entire group. For the Fourier transform

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.

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

• Recall the gauge transformation of G

• Adopt the Ansatz

Where The path ordered integral is along a straight line joining n and m

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.

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

• 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!

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

Hartree- Fock Approximation:Keep only 2-loop contributions

+ ….

The derivatives at A=0 are

Where the T’s commute with the background G

In Fourier Space:

Where

The Polarization tensor

Use the Gap equation to eliminateDelta

• In the Hartree-Fock approximation

Using this we obtain

Polarization

using

by checked benow can 0Πkidentity Ward The μμ

)()'( identity the 11

1 kupGu

kupGkμ

.uover gintegratin and 1

)()'()2(

write0 of vicinity theIn4 kkk(k,k')Π

k' k

αβμν

b.b g

basis, lattice thein defined is tensor metric thewhere

1

by given is termleading The H.grouppoint the

of tensorsinvariant bemust expansion thisin tscoefficien The

2

σρρσμννσμρμν k)kggg(g

e(k) Π

The coupling constant , 2e is thereby expressed, in the Hartree-Fock Approximation, as an integral over BZ involving )( pG

Example: Replace fermionsby a single complex scalar

G. particular thegenerate equations gap thefor which couplings

ofset a find andinput an as G(s) give toisStrategy Our

matrix.N Na nrather tha

scalar a becomes G(s), function, Green The

In the Hartree-Fock aproximationthe gap equations become

21211

01

11

cos2cos24~

choose and 2d in lattice square heConsider t

0

kk- M(s)eG(k)G

Δ(s)G(s)σδ(s)G

s)G(s)G((s)J(s)Δ

)G(J(s)σ

s

iks-

s,

s

4cos242

cos2cos242

M,mass

of particlea of npropagatio thedescribes thissmallis MIf

21112

12

121

212

2

2

s)ω(ksik

iks

e)k - (Mπ

dk

k k M

e)

π

dk(G(s)

ghost! no is There

.k 0for lly monotonica increases )(k general inbut

thensmall, arek and M If

4cos242

1cos24

2

1ln

11

22

222

Mkω(k)

k)(Mk)(M -(k)

Solution of the Gap Equations

)cos(1,0)(cosk2 (0,0) )(~

)0,1(

)0,1(G-(1,0)

)0,0(

)0,0(G-(0,0)

neighborsnearest its and origin the

at except everywhere 0(s) imply that thenequations gap The

1- )1,0(G)10(G

4 M)0,0(G

neighborsnearest its

and origin at theexcept everywhere 0)(G all ofFirst

21

1-

1-

1-1-

21-

-1

kk

G

G

s

The Couplings:

solution. given theproduces whichJ(s) of choicesdistinct two

there Hence.for solution real twohas equation This

)(~

)(~

)2

()0,1(/)0,0(4)4(

)4()0,0(

for eq. an

yields0kfor above equation the with thisCompare

)0,0(

(0)J~

1

obtain weequations gap theFrom

determine toisleft isWhat

)(~

)(~

)2

()(

~1

-(k)J

~1

22

2

2

pGpGdp

GGM

MG

G

pGpkGdp

k

Gravity

Let us go back to the 4-fermion action

J. of dynamics thegovern to termaction an add toneed thusWe

model.our in objectsother like fielda it to promote toneed We

couplings. ofset givena as m)J(n, regarded have far we So

m)J(n, )m',(n'J'm)J(n,

particular In scalars. like transformS enetring objects theall

provided,S f wheref(n), nunder invariant is action This

)()(),()()( S

mmmnJnn

J-Dynamics

We have not studied the action for the Pure J-field

We shall assume that our background J- values are in turn determined by

minimizing the enlarged action. The J’s are thus effectively considered as given

background fields, somewhat similar to a background metric.

• Because of the general coordinate invariance, gravity should be contained among the light degrees of freedom!

• Our aim is to calculate the Einstein action

and the Newton constant in terms of the

given couplings.

Rewrite S: Introduce a field

mn mnJ

nmmntrmmnnS

, ),(

),(),()(),()(

NN ),( mnji

Solving the field equations and substituting the result in S

reproduces the original 4-fermi action

Integrate over

Det

mnJ

nmmntrS

mneff ln

),(

),(),(

,

This is an approximation to the Generalized effective action which we Constructed before.

The Gap Equations:

),(),(),( 1 mnmnJmn

Let C(n-m) be a translationalInvariant solution of the Gap eq

• Ansatz for the Yang-Mills field

)(exp)(),( n

mAiTmnCmn

The integral is along a straight path from m to n

A is in the subalgebra commuting with C

Substitution in the effective action generates the Y-M Action for A

How do we obtain gravity?

• We need to introduced more structures.• The lattice states should be embedded in a

continuum .• A projection E commuting with should be

introduced• should be regarded as a functional of a set of

vierbeins as well independent connection coefficients

n

)(ne a

x

Then:

xE

nn

lnx

lnlnDet

Anzatz for the light d.o.f

1/2

1

0

a1141/2

e(y) )duTexp(i

)Ke,y(x,iξexp (K)G)2π

dK(e(x)xΓ,....)χ(e,x

A

a

Where : is the geodesic between x and y ( assuming there is only one) and e(x) is the determinant of the vierbein

The plausibility of the Anzatz

• Its structure is dictated by the symmetries of the effective action,

• It produces the correct Yan-Mills action in the absence of gravity,

• It produces the correct (leading order) minimal coupling of the fermions to the gauge fields,

• If we replace fermions by scalar bosons it reproduces the correct( leading order) kinetic terms for the scalars,

• It reproduces the correct cosmological constant

But……

• Although the anzatz satifies the correct Ward identities for the local gauge and general coordinate transformations it does not satisfy those for the local Lorentz transformations, unless we adjust the couplings to kill the non invariant terms!

Conclusion:

The gravity problem still not fully

understood…….

Outstanding Problems• Find a solution of gap equations with spin

• Study the dynamics of the J-fields

( RG equations, an action for J, etc)

• Obtain Gravity ( A better ansatz for

or G) so that the Ward identities for the local Lorentz transformations are satisfied.

• Calculate Newton’s constant….