Let's say I have some Lorentz-violating theory. For concreteness we could imagine a scalar field \begin{equation} S=\int {\rm d}^4 x \left(-\frac{1}{2} (\partial \phi)^2 + V(x)^\mu \partial_\mu \phi\right) \end{equation} where $V(x)^\mu$ is some space-time dependent vector field (ie a non-trivial external current sourcing $\partial_\mu \phi$).
Classically, I can define a Lorentz transformation that acts in the normal way, $x^\mu \rightarrow \Lambda^{\mu}_{\ \ \nu}x^\nu$, $\phi(x) \rightarrow \phi(\Lambda^{-1}x)$. Of course, this transformation won't be a symmetry because the action will not be invariant.
What I want to understand is what the analog of this logic is in quantum field theory. Ie: I would expect that I could define a Lorentz transformation, but I should find there is some condition defining a symmetry that is violated by this transformation.
Now let's say I quantize the theory, and work in the canonical formalism. Surely I can define the action of Lorentz transformations $\Lambda$ on the Hilbert space. After all, given a state $|\Psi\rangle$, I should be able to ask what the state looks like under a boost. Furthermore, I would expect the set of transformations to form a representation on Hilbert space, $R(\Lambda)$, since the Lorentz transformations themselves form a group.
However, it's not clear to me if the transformations can be taken to be unitary. I basically see two possibilities... (1) it is possible to construct a unitary representation of the Lorentz group on the Hilbert space of the theory, but it won't have any physical relevance because the operators controlling the dynamics (4-momentum operator, angular momentum operator) won't transform as tensors under these transformations, or (2) it is not possible to construct a unitary representation.
In favor of possibility (1): in non-relativistic quantum mechanics, I could always define angular momentum operators $L_i = \epsilon_{ijk}x_j p_k$ which obey the algebra for $SU(2)$, even if the Hamiltonian is not Lorentz invariant, so it seems I could always construct a unitary representation of rotation operators by exponentiating these.
My questions are:
- Is constructing $R(\Lambda)$ possible in a Lorentz-violating theory? (If not, what stops you from constructing it?)
- Can we choose a unitary representation for $R(\Lambda)$ (possibility (1)), or not (possibility (2))?