# What is the explicit form of the generators of Lorentz boosts in the Hamiltonian formulation of Quantum Field Theory (QFT)?

Is it possible to express the generators of Lorentz boosts in QFT in terms of the corresponding quantum fields? More specifically, I am interested in an expression in terms of time independent fields i.e. in the Hamiltonian formulation.

For example, in free bosonic QFT (the Klein-Gordon model) both the Hamiltonian $$H$$ and the momentum operator $$P_i$$, which are the generators of time and space translations, respectively, can be expressed in terms of the bosonic field $$\phi(x)$$ and the canonically conjugate field $$\pi(x)$$: $$H = \int d^{3} x \; \left[\frac{1}{2} \pi^{2}+\frac{1}{2}(\nabla \phi)^{2}+\frac{1}{2} m^{2} \phi^{2}\right]$$ $$P_i = -\int d^{3} x \; \pi \partial_i \phi$$ These relations can be derived from the Lagrangian of the model either directly or through the stress-energy tensor. On the other hand, I'm not aware of a general way to derive the generators of Lorentz boosts $$K_i$$ in terms of the $$\phi(x)$$ and $$\pi(x)$$ fields, even though they should be related to the above two operators through the Poincare algebra. In the (1+1)D case, for example, this is: \begin{align} [H,P]&=0\\ [K,P]&=\mathrm{i}H\\ [K,H]&=\mathrm{i}P \end{align} from which, however, it doesn't seem easy to determine $$K$$ from $$H$$ and $$P$$.

Searching the literature, I came across one of Dirac's famous papers: Forms of Relativistic Dynamics, P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949) which explains the ambiguities related to the definition of the Hamiltonian of a system of relativistic particles and the non-triviality of determining these operators in the case of interacting particles, but I'm not sure if these points are relevant to the present problem, at least in the free case, and what they mean in the context of field theory.

Consider the KG scalar field. The boost depends on time parametrically. Its form in the Hamiltonian formulation reads $$K^j(t) = \int_{{\mathbb R}^3}t T^{0j} - x^j T^{00} d^3x$$ where you have to expand the canonical stress energy tensor appearing above in terms of canonical variables. (There could be an overall sign which depends on the use of different conventions.) In free QFT you just have to use the normal ordering procedure to get the corresponding operators.