Trivial symmetries in models of gravity Rabin Banerjee *, 1 , Debraj Roy *, 2 1 [email protected], 2 [email protected] * S. N. Bose National Centre for Basic Sciences, Kolkata, India. Overview of the problem Recover Poincare symmetries in models of gravity via 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 by basis vectors (black lines) Introduce local frames as tangent spaces at each 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 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) Einstein- Cartan Cosmological term Chern- Simons Torsion Hamiltonian Constraints First Class Second Class Primary Hamiltonian generator & symmetries Define structure functions and ; Gauge generator sum of 1 st 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: 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 eq n s of motion This is NOT a true gauge symmetry and is NOT generated by 1 st class constraints 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 ; i i i j k k k jk b b b b ò ; x x x m m m n n m m m x x q e ¢ = + = + i i j k i i P jk i i i j k i i P jk b b b b r r m m m r r m r r m m m m r r m d q x x dw q wq x w x w =- -¶ - ¶ =-¶ - -¶ - ¶ ò ò ( ) 4 3 1 3 3 3 2 i i j k i i j k i i ijk i ijk i S dx ab R bbb bT a mnr m nr m n r m n r m n r m nr a w w www L é ù = - + ¶ + + ê ú ë û ò ò ò ò 4 3 4 : 0 : 0 j k i i ijk i j k i i ijk i S aR T bb b S R aT bb mnr nr nr n r m mnr nr nr n r m d a d d a a dw é ù = + -L = ê ú ë û é ù = + + = ê ú ë û ò ò ò ò , i i j k jk b R b ò . i i i i j k i i i jk R T b b mn m n m n m n mn m n n m w w ww =¶ -¶ + =Ñ -Ñ ò i i i j k i j k H jk jk i i i j k H jk b p b b q b m m m m m m m d e e t dw t e =Ñ - + =Ñ - ò ò ò i i i i i b r r r r e x t q xw =- =- - ~ equation ofm otion H P d d + 0 A AB A A B S S S S q q q q d d d d d d d d æ ö ÷ ç ÷ ç ÷ ç ÷ ç ø = è = L = ( ) ( ) ( ) ( ) , , , , i j i j i j i j i j j b b b i j j b S S b b S S b m n m n m n m n m w n n m w w w n n d d d d dw d d dw d dw =L + L =L + L ( ) ( ) ( ) ( ) 3 2 2 , , 3 4 3 4 4 2 2 , , 3 4 3 4 2( ) 2( ) 2( ) 2( ) i j i j i j i j ij ij b b b ij ij b a a a a a a m n m n m n m n r r mnr mnr w r r mnr mnr w w w a hx hx aa aa a hx hx aa aa - L = L = - - - L = L = - - ò ò ò ò