EFFECTIVE FIELD THEORIES FOR VECTOR PARTICLES AND ...

4.2 the free tensor model 33
The Hamiltonian density reads via Legendre transformation
H 1 = Π 0j Ẇ 0j + Π i j Ẇ i j − L
= ϕ 1 0jz 0j + 1
8a Π i jΠ i j + 2c(W 0j W 0j − W i j W i j )
+ 2b(∂ i W 0i ∂ j W 0j − ∂ k W 0j ∂ k W 0j + ∂ k W i j ∂ k W i j
(4.20)
− ∂ i W ik ∂ j W jk + ∂ i W ki ∂ j W jk + ∂ i W ik ∂ j W k j
− ∂ i W ki ∂ j W k j ) ,
where again three to be determined functions were introduced and the
constraints in equation (4.18) were identified.
The following Poisson brackets are calculated as usual, except for
the fact that an integration by parts is carried out with respect to the
integral of the Hamiltonian function H 1 = ∫ d 3 xH 1 if necessary. From
the conservation in time one obtains
{ϕ 1 0l , H 1} = 4b∂ l ∂ i W 0i − 4b∂ k ∂ k W 0l + 4cW 0l + ∂ n Π nl − ∂ n Π ln
≡ ϕ 2 0l ≈ 0 . (4.21)
At this point the convention in use shall be stressed again, e.g. it holds for
l = 1 that ∑ n