Transcript Document

June 13, 2011
Euler Symposium
Is there any torsion in your future?
Dmitri Diakonov
Petersburg Nuclear Physics Institute
DD, Alexander Tumanov and Alexey Vladimirov, arXiv:1104.2432
General Relativity in terms of the metric tensor:
2
S 
MP
16
d
Planck mass
4
x
g  2  R 
cosmological term
Christoffel symbol or, better, Levi – Civita connection:
Riemann tensor:
Scalar curvature:

R  R  ,  g

Eistein – Hilbert action
Cartan’s formulation of general relativity (1920’s):
Independent variables, instead of the metric tensor, are
g   e  e ,
A
A
e ,
1) vierbein or frame field

AB
2) spin connection
  
BA
A  1, 2, 3, 4 .
A
Yang – Mills potential of the
Lorentz SO(4) group
SO(4) Yang – Mills field strength or curvature [ historically, first example of “Yang-Mills” ] :
F      
AB
AB
   
AB
   
AC
CB
   
AC
CB
Gravity action:
2
P
1  

S 
 d x   2  det( e )  ò òA B C D FA B e C eD 
16 
4


M
4
Classically, and with no sources, it is equivalent to the Einstein – Hilbert action.
Proof:
The action in quadratic in wm, so saddle point integration in wm is exact.
Saddle-point equation for wm :
D  e  D e   2 T   0,
AB
B
AB
B
A
  
AB
D
AB
 
AB
this combination is called torsion
24 algebraic equation on 24 components of   determine the saddle-point uniquely as
AB
Substituting the saddle-point value back into the action, one recovers identically the
Einstein – Hilbert action written in terms of g  . Torsion appears to be zero dynamically,
even if one allows it, as in Cartan formulation.
There are fermions in Nature. Fermions `gravitate’, and are sources of torsion.
However !
NB: all diffeomorphism-invariant (or general covariant) actions are necessarily linear
in eklmn and hence are not sign-definite !!
A  A  x
'

/ x ,
ò
 
A B  C  D 
dx 
dx
 
ò
A B  C  D ;
4
d x
dx 
dx
 d x .
4
The standard (minimal) way to include fermions into General Relativity:
[ V. Fock, H. Weyl (1929) ]
det( e ) e
D

1
 
ò
6
ò
ABCD
A
B
C
e e  e  ,
e
D
e E   E .
D
the contravariant tetrad is the inverse matrix
This action is invariant under
i)
general coordinate transformations (diffeomorphisms) x
ii)
local Lorentz rotations  ( x )  L ( x )  ( x ),


 x ( x)
L ( x )  SO (4)
SU (2) L  SU (2) R .
just as is the bosonic Einstein – Cartan action.
With fermions switched in, the saddle-point variation with respect to the connection  
gives, generally, a nonzero torsion:
AB
T  ~ (    )
A
bilinear fermion current
Einstein’s gravity with matter, as a quantum theory, is not renormaizable; moreover it is
not 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 s-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  2 / M P2 being the expansion parameter.
The expansion coefficients should be in principle determined from observations.
The leading terms in torsion, curvature, gradients have been systematically written down
by Tumanov, Vladimirov and D.D. [arXiv:1104.2432] . The net result is that in the leading
order there is a local 4-fermion addition to the standard gravity:
S 
const .
2
MP
  V ·V
   A ·A    V ·A   ,
V    A ,
A    A 5  .
This correction is, normally, very small and hardly detectable today. However, there have
been 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
G ( p) 
m   0 (   i n   i p i )
m  (   i n )  p
2
2
2
,
1

 n  2 T  n 
2


.

Matsubara
frequency
m = chemical potential, regulates the difference between # of particles and # of antiparticles
For free (non-interacting) ultra-relativistic fermion gas we obtain
 is the charge density (# of particles - # of antiparticles)
m
is the chemical potential, T is temperature
Corrections to the ideal fermion gas eqs. arise only in the second order in the interaction.
Einstein – Friedman equation for cosmological evolution
R  
1
2
g  R 
16 
M
2
P
 
Main piece in the fermion stress-energy tensor is the Stefan – Boltzmann law:
 00 
7
2
4
T
60
Correction due to torsion:

4  ferm
00


2
M
2
P
4
T .
m is the chemical potential for
conserved 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 in
the Standard Model based on the SU(3) x SU(2) x U(1) gauge group, except gravitation
which is also a gauge theory based on the gauged Lorentz S U (2) L  SU ( 2 ) R group.
color
quarks
+…?
leptons
4 x 2 x 2 x 4 = 64
real dof’s per
fermion generation
+…?
1st generation
2nd
third
fourth?
Standard gravity is not a renormalizable field theory, nor is it even a well-formulated
theory, because the requirement of diffeomorphism-invariance makes the action nonpositive-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!
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,
3
e.g. the  theory
How to define a truly quantum general-covariant theory, such that the path integral formally
converges?
Suggested answer: Use fermionic (grassmannian) variables instead of bosonic ones!
 d d exp(  i Aij j )  det( A )
†
†
 d d exp(  i 
†
†
†
j
A{ ij , kl } k  l )
… (more fermion operators in the exponent)
is also finite, for any sign
The idea is to use composite vielbeins or frame fields, roughly:
E 
A
1
2
  A D  
†
g   E  E
A
A
1
2
( D  )  A
†
D    
1
8
  [ A  B ]
AB
spin connection,
gauge field
metric tensor
If we want the metric to be algebraically unrestricted, the number of dof’s in fermions
A
must match the number of dof’s in E  , equal to d x d. This doesn’t happen in any
number of space-time dimensions.
d
1
The dimension of the two spinor representations of the SO(d=2n) group is d f  2 .
These representations are real for “leap” dimensions d=4n, and complex otherwise.
2
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
Standard Dirac action
S 
d
d
xò
 1 ... d
ò
A1  Ad

E 11  E  dd11   Ad ( D  d  )  ( D  d  ) Ad 
A
A

is in fact the cosmological term in disguise:
S 
d
d
xò
 1 ...  d
ò
A1  Ad
E 11  E  dd11 E  dd 
A
A
A
d
d
x det( E ) 
d
d
x
g
Fermion mass term, e.g.
S 
d
d
xò
 1 ...  d
ò
A1  Ad

E 11  E  dd22  F dd11 dd 
A
A
A
A

is in fact (almost) the Einstein – Hilbert term in disguise:
S 
d
d
xò
 1 ...  d
ò
A1  Ad
E 11  E  dd22 F  dd11 dd 
A
A
A
A
d
d
x
gR
etc.
All such kind of actions can be easily UV regularized by putting them on a lattice.
E   ( x) e
i  a

 ( x  a )   ( x ) ( x )
Om
lives on a lattice
link in the m direction

Om
x
ò
Discretized `cosmological term’ action:
1   d
Tr ( E 1  E  d )
x+a
E
x
ò
Discretized `Einstein – Hilbert’ action:
1   d
Tr ( E 1  E  d  2 F d 1  d )
x
gauge- and diffeomorphisminvariant!
F 
Regularized partition function
Z 
 DO 
Z 1 ( ),
Haar measure
normalized to unity
Z 1 ( ) 
 Z1
 ( x )
 D
0
D exp( S cosm  S E H   )
!
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 mean
A
frame field  E  ( x )  , and then calculate it by integrating explicitly over fermion fields.
A
The equation on the mean field  E  ( x )  will be
a) general covariant,
b) gauge invariant.
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 several
low-dimensional spheres:
x
x
The classical solutions must be such as to break the SO(16) gauge group down to
the Standard Model and Lorentz gauge group:
SO (16)  ( S U (2) L  SU (2) R )  SU ( 3) C  SU (2)W  U (1) Y
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.