the operatorsP̂andL̂, an integral of a product involving one creation and one
annihilation operator. The boost operators will be given in momentum space
by
K̂=i∫
R^3ωpa†(p)∇pa(p)d^3 p (44.10)One can check that this gives a Poincar ́e Lie algebra representation on the
multi-particle state space, by evaluating first the commutators for the Lorentz
group Lie algebra, which, together with 44.9, are (recall the Lie bracket relations
40.1 and 40.2)
[−iL̂j,−iK̂k] =jkl(−iK̂l), [−iK̂j,−iK̂k] =−jkl(−iK̂l)The commutators with the momentum and Hamiltonian operators
[−iK̂j,−iP̂j] =−iH,̂ [−iK̂j,−iĤ] =−iP̂jshow that the rest of the non-zero Poincar ́e Lie algebra bracket relations (equa-
tions 42.2) are satisfied. All of these calculations are easily performed using
the expressions 43.22, 44.7, 44.8, and 44.10, forH,̂P̂,̂L,K̂and theorem 25.2
(generalized from a sum to an integral), which reduces the calculation to that
of the commutators of
ωp,p,p×i∇p, iωp∇p
44.3 For further reading
The operators corresponding to various symmetries of scalar quantum fields
described in this chapter are discussed in many quantum field theory books, with
a typical example chapter 4 of [35]. In these books the form of the operators is
typically derived from an invariance of the Lagrangian via Noether’s theorem
rather than by the Hamiltonian moment map methods used here.