Quantum Gravity

104 HAMILTONIAN FORMULATION OF GENERAL RELATIVITYdifferent formulation can be found that is in accordance with ultralocality. Forthis purpose it is suggestive to omit the term −p b ,b A a in (4.16) because then thewill become zero. This leads to the replacement (Teitelboim 1980)b baBH a → ¯H a ≡H a + p b ,b A a =(A b,a − A a,b )p b . (4.28)What happens with the Poisson-bracket relation (3.92) after this modification?A brief calculation shows{ ¯H a (x), ¯H b (y)} = ¯H b (x)δ ,a (x, y)+ ¯H a (y)δ ,b (x, y) − F ab (x)p c ,c(x)δ(x, y) , (4.29)where F ab = ∂ a A b − ∂ b A a . The new term in (4.29) will only be harmless if itgenerates physically irrelevant transformations (‘gauge transformations’). Thisis the case if the new term actually vanishes as a constraint. Since setting F ab tozero would appear too strong (leaving only the restricted option A a = ∂ a ϕ), itis suggestive to demand that p a ,a ≈ 0. One therefore introduces the constraintG(x) ≡− 1 e pa ,a(x) ≡− 1 e Ea ,a(x) ≡− 1 ∇E(x) .e(4.30)The constraint G≈0isjustGauss’ law of electrodynamics (in the sourcelesscase) with the momentum being equal to the electric field E (the electric charge,e, has been introduced for convenience). As usual, Gauss’ law generates gaugetransformations,∫δA a (x) =∫dy {A a (x), G(y)}ξ(y) = 1 e ∂ aξ(x) , (4.31)δp a (x) = dy {p a (x), G(y)}ξ(y) =0. (4.32)The electric field is of course gauge invariant, and so is the field strength F ab .Inthe modified constraint (4.28), the first term (H a ) generates the usual transformationsfor a vector field, see (4.15), while the second term (p b ,b A a) generatesgauge transformations for the ‘vector potential’ A a (x). Therefore, A a (x) transformsunder ¯H a not like a covariant vector but only like a vector modulo a gaugetransformation. This fact was already encountered in the space–time picture;see Section 2.1. The electric field, however, transforms as a contravariant tensordensity, since the additional term in (4.28) has no effect.The above introduction of the gauge principle can be extended in a straightforwardmanner to the non-Abelian case. Consider instead of a single A a (x)now a set of several fields, A i a (x), i =1,...,N. The simplest generalization ofthe Abelian case consists in the assumptions that A i a (x) should not mix with itsmomentum p a i (x) under a gauge transformation, that the momenta should transformhomogeneously, and that the gauge constraint (the non-Abelian version ofGauss’ law) is local. This then leads to (Teitelboim 1980)G i = − 1 f p i,a a + C ijk A a j p k a ≈ 0 , (4.33)