Trivial symmetries in models of gravity

Rabin Banerjee*, 1, Debraj Roy*, 2

1rabin@bose.res.in, 2debraj@bose.res.in

*S. N. Bose National Centre for Basic Sciences, Kolkata, India.

Overview of the problem

Recover Poincare symmetries in models of gravityvia a canonical hamiltonian method.

Find appropriate canonical gauge generators. Canonical methods apparently do not generate Poincare

symmetries. Two independent off-shell symmetries of the same action ! We show: they are canonically equivalent, modulo a trivial

symmetry.

The Poincare gauge construction

Gauge theory of the Poincare group: Poincare Gauge Theory (PGT)[Utiyama, Kibble, Sciama].

Let’s start on a 3D space spanned bybasis vectors (black lines)

Introduce local frames as tangent spaces ateach point, spanned by (coloured lines)

Global Poincare transformations

Lagrangian invariant under this On localization,

become functions of global coords To maintain invariance of , additional fields & covariant

derivative

are the vielbeins and spin-connections Field strengths gravity: the Riemann and Torsion fields

The new Lagrangian is then found to be invariant under the following PGT symmetries of the basic fields

; i i i j kk k jkb b b bò

;x x xm m m nn

m mmx x q e¢= + = +

i i j k i iP jk

i i i j k i iP jk

b b b br rm m m r r m

r rm m m m r r m

d q x x

d w q w q x w x w

=- - ¶ - ¶=- ¶ - - ¶ - ¶

òò

Mielke-Baekler type model with torsion

Equations of Motion:

Conventions:

Latin indices: i, j, k, … = 0, 1, 2 : Local frame indices (an-holonomic)Beginning Greek indices: , … = 1, 2 : Global indices (holonomic)Middle Greek indices: , … = 0, 1, 2 : Global indices (holonomic)

( ) 43 133 3 2

i i j k i i j k ii ijk i ijk iS dx ab R b b b b Tamnr

m nr m n r m n r m n r m nra w w w w wLé ù= - + ¶ + +ê úë ûò ò ò ò

Einstein-Cartan Cosmologic

al termChern-Simons

Torsion

4

3 4

: 0

: 0

j ki i ijki

j ki i ijki

S a R T b bb

S R aT b b

mnrnr nr n r

m

mnrnr nr n r

m

d add a adw

é ù= + - L =ê úë û

é ù= + + =ê úë û

ò ò

ò ò

, i i j kjkb R b ò

.i i i i j k i i ijkR T b bmn m n m n m n mn m n n mw w w w=¶ - ¶ + =Ñ - Ñò

Hamiltonian Constraints

First Class Second Class

Primary

Hamiltonian generator & symmetries

Define structure functions and ;

Gauge generator sum of 1st class constraints where are gauge parameters. Not all of these are independent

Demanding commutation of arbitrary gauge variation with total time derivative: , where and

Using these to eliminate dependent gauge parameters from the set ,

The hamiltonian gauge symmetries turn out to be:i i i j k i j k

H jk jk

i i i j kH jk

b p b b

q bm m m m

m m m

d e e t

d w t e

=Ñ - +=Ñ -

ò òò

A tale of trivial symmetries

To compare the two symmetries map between gauge parameters

The two symmetries can then be compared

Explicitly, the balance symmetry reads:

where,

Here the coefficients are all antisymmetric in the field indices So the action remains invariant without imposition of eqns of

motion

This is NOT a true gauge symmetry and is NOT generated by 1st class constraints

i i i i ibr rr re x t q x w=- =- -

~ equation of motionH Pd d +

0A ABA A B

S S SS qq q qd d dd dd d d

æ ö÷ç ÷ç ÷ç ÷ç ø=

è= L =

( ) ( )

( ) ( )

, ,

, ,

i j i j

i j i j

ij jb b b

ij jb

S Sbb

S Sb

m n m n

m n m n

m wn n

m w w wn n

d ddd dwd ddwd dw

=L + L

=L + L

( ) ( )

( ) ( )

32 2, ,

3 4 3 4

42 2, ,

3 4 3 4

2( ) 2( )

2( ) 2( )

i j i j

i j i j

ij ijb b b

ij ijb

aa a

aa a

m n m n

m n m n

r rmnr mnrw

r rmnr mnrw w w

a h x h xa a a a

ah x h xa a a a

-L = L =- --L = L =- -

ò ò

ò ò

REFERENCES:1. R. Banerjee, H.J. Rothe, K.D. Rothe Phys.Lett. B479 , 429 (2000) and ibid. Phys.Lett. B463, 248(1999) 2. R. Banerjee D. Roy Phys.Rev. D84, 124034 (2011)3. R. Banerjee, S.Gangopadhyay, P. Mukherjee, D.Roy : JHEP 1002:075 (2010 )

Riemann-Cartan manifold