Is there any torsion in your future?

Dmitri Diakonov

Petersburg Nuclear Physics Institute

June 13, 2011 Euler Symposium

DD, Alexander Tumanov and Alexey Vladimirov, arXiv:1104.2432

General Relativity in terms of the metric tensor:

2

4 216

PMd RS x g

Planck mass cosmological term Eistein – Hilbert action

Christoffel symbol or, better, Levi – Civita connection:

Riemann tensor:

Scalar curvature:

,R R g

Cartan’s formulation of general relativity (1920’s):

Independent variables, instead of the metric tensor, are

1) vierbein or frame field

2) spin connection Yang – Mills potential of the Lorentz SO(4) group

, , 1, 2,3,4.A A Ag e e Ae AB BA

SO(4) Yang – Mills field strength or curvature [ historically, first example of “Yang-Mills” ] :

AB AB AB AC CB AC CBF

Gravity action:

24 1

2 det( )16 4

AB C DPABCD

Md x e F eS e

ò ò

Classically, and with no sources, it is equivalent to the Einstein – Hilbert action. Proof: The action in quadratic in so saddle point integration in is exact.

Saddle-point equation for

2 0,AB B AB B A AB AB ABe e TD DD

this combination is called torsion

24 algebraic equation on 24 components of determine the saddle-point uniquely asAB

Substituting the saddle-point value back into the action, one recovers identically theEinstein – Hilbert action written in terms of . Torsion appears to be zero dynamically,even if one allows it, as in Cartan formulation.

g

However ! There are fermions in Nature. Fermions `gravitate’, and are sources of torsion.

NB: all diffeomorphism-invariant (or general covariant) actions are necessarily linearin and hence are not sign-definite !!

4 4; .dx dx

A B C D A B C D d x d xdx dx

ò ò' / ,A A x x

The standard (minimal) way to include fermions into General Relativity: [ V. Fock, H. Weyl (1929) ]

1det( ) , .

6D ABCD A B C D D

E Ee e e e e e e ò ò

This action is invariant under

i) general coordinate transformations (diffeomorphisms)

ii) local Lorentz rotations

just as is the bosonic Einstein – Cartan action.

( )x xx

( ) ( ) ( ), ( ) (4) (2) (2) .L Rx L x x L x SO SU SU

With fermions switched in, the saddle-point variation with respect to the connectiongives, generally, a nonzero torsion:

AB

~ ( )AT bilinear fermion current

the contravariant tetrad is the inverse matrix

Einstein’s gravity with matter, as a quantum theory, is not renormaizable; moreover it isnot well-defined as a path integral since the action is not positive-definite. Therefore,at best it should be viewed as an effective low-energy quantum theory, like the effective chiral Lagrangian in Quantum Chromodynamics, or -models for solid states.

The best one can do in the absence of a well-defined microscopic theory, is to treat General Relativity as a gradient expansion, with being the expansion parameter. The expansion coefficients should be in principle determined from observations.

2 2/ PM

The leading terms in torsion, curvature, gradients have been systematically written downby Tumanov, Vladimirov and D.D. [arXiv:1104.2432] . The net result is that in the leadingorder there is a local 4-fermion addition to the standard gravity:

52

.· · · , , .A A

P

constS V V A A V A V A

M

This correction is, normally, very small and hardly detectable today. However, there havebeen speculations that close to the Big Bang when fermion densities were presumably large,the 4-fermion interaction induced by nonzero torsion could be very important and even replace the `inflation’ and solve the `horizon problem’!

We think it is a) doutful, b) in any case beyond the applicability of the gradient expansion.

One has to evaluate the 4-fermion interaction in the hot relativistic Fermi medium.

For ideal Fermi gas

Matsubarafrequency

= chemical potential, regulates the difference between # of particles and # of antiparticles

02 2 2

( ) 1, 2 .

( 2( )

)n

n

i inG p n

m

m iT

i

p

p

For free (non-interacting) ultra-relativistic fermion gas we obtain

is the charge density (# of particles - # of antiparticles)is the chemical potential, T is temperature

Corrections to the ideal fermion gas eqs. arise only in the second order in the interaction.

Main piece in the fermion stress-energy tensor is the Stefan – Boltzmann law:

Einstein – Friedman equation for cosmological evolution

2

1 16

2 P

g RRM

24

00

7

60T

Correction due to torsion:

24 400 2

.ferm

P

TM

is the chemical potential forconserved quantum numbers

One can hardly imagine that this correction ever becomes sizable!

Therefore, it seems that there is no torsion in your future.

However…

Speculations

All known fundamental interactions (color, electromagnetic and weak) are unified inthe Standard Model based on the SU(3) x SU(2) x U(1) gauge group, except gravitationwhich is also a gauge theory based on the gauged Lorentz group.

+…?

+…?

1st generation 2nd third fourth?

quarks

leptons

color

Standard gravity is not a renormalizable field theory, nor is it even a well-formulatedtheory, because the requirement of diffeomorphism-invariance makes the action non-positive-definite:

Cosmological term det(e) can have any sign in the fluctuating quantum world.Einstein – Hilbert action det(e) R can have any sign Quadratic curvature action det(e) R^2 can have any sign,…….

General covariance is a “curse” that makes any diffeomorphism-invariant action bottomless!

4 x 2 x 2 x 4 = 64real dof’s perfermion generation

( )(2) 2L RS SUU

Of course, we live in a world with Minkowski signature with an oscillating weight, exp(iS),however • Euclidean path integral also needs to make sense – e.g. for describing thermal excitations, and for tunneling, like Hawking radiation

• Usually a theory that is pathological in Euclidean space, is also pathological in Minkowski, e.g. the theory 3

How to define a truly quantum general-covariant theory, such that the path integral formallyconverges?

Suggested answer: Use fermionic (grassmannian) variables instead of bosonic ones!

† †exp( ) det( )i ij jd d A A † † †

{ , }exp( )i j ij lkkld d A is also finite, for any sign

… (more fermion operators in the exponent)

The idea is to use composite vielbeins or frame fields, roughly:

1[ ]

8AB

A BD † †1 1

( )2 2

AA ADE D

spin connection,gauge field

A AEg E metric tensor

If we want the metric to be algebraically unrestricted, the number of dof’s in fermionsmust match the number of dof’s in , equal to d x d. This doesn’t happen in anynumber of space-time dimensions.

The dimension of the two spinor representations of the SO(d=2n) group is .These representations are real for “leap” dimensions d=4n, and complex otherwise.

AE

122d

fd

In d=2 and d=16 fermions carry sufficient number of dof’s to compose arbitrary metric!

On the other hand, 256 real fermion dof’s are exactly what is needed to describe four generations of the Standard Model!

[ recent data on neutrino and anti-neutrino oscillations, if confirmed, suggest that there are more than three generations. ]

Renormalizable (maybe finite?) quantum gravity theory

1 1 11

1 1)( ( )... d d d

d d d d d

A A AAdA Ad x E E D DS

ò ò

Standard Dirac action

is in fact the cosmological term in disguise:

1 1 11

1 1

... det( )d d d d

d d

A A A AAd d dd x E E d x gE ES d x

ò ò

Fermion mass term, e.g.

1 1 2 11

1 2 1

... d d d d d

d d d

A A A A AAdS d x E E F

ò ò

is in fact (almost) the Einstein – Hilbert term in disguise:

1 1 2 11

1 2 1

... d d d d d

d d d

A A A A AAd dd x E E F d gRS x

ò ò etc.

All such kind of actions can be easily UV regularized by putting them on a lattice.

( ) ( ) ( ) ( )i ax e x x xE a

x x+a

O

Olives on a lattice link in the direction

Discretized `cosmological term’ action: 1

1( )d

dx

Tr E E

ò

Discretized `Einstein – Hilbert’ action: 1

1 2 1( )

d

d

d dx

Tr E FE

ò

Regularized partition function

E

Fgauge- and diffeomorphism- invariant!

Haar measurenormalized to unity

1 1), ) exp(( ( )cosm EHZ DO Z Z D D S S 1 0(

!)

Z

x

Therefore, it looks like the theory is not only well-defined and renormalizable, but finite !

If the partition function = 1, it does not mean that there are no non-trivial saddle-points.

What is a saddle point in a multi-Grassmannian path integral, is not clear.

Probably, the mean field method in d=16 must work well: Assume there is a meanframe field , and then calculate it by integrating explicitly over fermion fields.The equation on the mean field will be

a) general covariant, b) gauge invariant.

( )AE x ( )AE x

It may happen that the solution breaks spontaneously the rotational SO(16) symmetry,for example, by compactifying the 16d space down to a direct product of severallow-dimensional spheres:

x x

The classical solutions must be such as to break the SO(16) gauge group down tothe Standard Model and Lorentz gauge group:

( (2) (2) ) ((16) 3) (2) (1)L R C W YSSO U SU SU SU U gravity spin connection Standard Model

256 fermion fields needed to describe the 16d metric, go into 4 generations of the SM.

Conclusions

1. When one includes fermions into General Relativity, nonzero torsion is inevitable.

2. However, the ensuing effects are so small that they are probably unobservable.

3. All general covariant theories have bottomless actions. To make quantum gravity well defined, one needs to introduce a composite frame built from fermion fields.

4. This is a new type of quantum theory! Have to develop anew the saddle-point and mean-field methods.

5. The SO(16)-based “Theory of Everything”, unifying Quantum Gravity and the Standard Model, looks as privileged.

6. Then torsion strikes back, becoming the Yang – Mills field of the Standard Model.