Lorentz generators on Fock space

Question

Consider a free massive relativistic scalar field in $d+1$ dimensions. Its Hilbert space can be taken to be the bosonic Fock space on $\mathfrak h = L^2(\mathbb R^d)$: $$\mathcal F = \bigoplus_{n=0}^{+\infty} S^n\mathfrak h.$$ On this space we have creation-anihilation operators corresponding to each $1$-particle state. This is usually written in terms of the $3$-momentum eigenstates $|\vec p\rangle$ so we get operators $a_{\vec p}$ satisfying: \begin{align} [a_\vec{p},a_{\vec q}] &= [{a_\vec p}^\dagger,{a_\vec q}^\dagger] = 0 \\ [a_\vec p,{a_\vec q}^\dagger] &= (2\pi)^d \delta^{(d)}(\vec p -\vec q). \end{align} The (normal ordered) Hamiltonian is then: $$H = \int \frac{\text d^d p}{(2\pi)^d} (\vec p ^2 +m^2)^{1/2}{a_\vec p}^\dagger a_\vec p.$$ Likewise, we get the total $3$-momentum operators: $$P^i = \int \frac{\text d^d p}{(2\pi)^d}\vec p^i{a_\vec p}^\dagger a_\vec p.$$ Together with $P^0 = H$, the are the generators of space-time translation. My question is the following: What are the expressions of the generators $M_{\mu\nu}$ of Lorentz transformations in this formalism ?

I haven't found these in textbooks. In this question, we have the expression of the commutator of the boost generators $K_i$ with a creation operator, $$[K_i,\alpha_p^\dagger]=-iE_p\frac{\partial}{\partial p_i}\alpha^\dagger_p$$ but this does not tell us the action of $K_i$ on a state of the Fock space.

Answer 1

Look at (3-54) of the basic popular text of Itzykson & Zuber. Unfortunately, they use an oscillator normalization different than yours, which appears to follow P&S, so I cannot vouch for a single sign or measure in my crude translation below: $$ \Large M_{0j}= i\int\! \frac{\text d^3 p}{(2\pi)^d}~~ {a_\vec p}^\dagger E_p\partial_{p^j} a_\vec p \\ \Large M_{jl}= i\int\! \frac{\text d^3 p}{(2\pi)^d}~~ {a_\vec p}^\dagger ( p_j\partial_{p^l}- p_l\partial_{p^j}) a_\vec p ~. $$ You may have to fix/fuss signs and prefactors to your satisfaction yourself; but it should be evident how the commutators of the above Jordan map structure yield what you wish, in terms of the Lie algebras and the action on oscillators, hence states, such as you wrote down.

(The more tasteful exercise is to work out $M^{\mu\nu}\sim \int d^3x (x^\mu T^{\nu 0}-x^\nu T^{\mu 0})$ in terms of field operators, and then convert to the above. Cf (2-63) of Paul Roman's Introduction to QFT book, 1969.)