Relativistic invariance of canonical/Hamiltonian field quantisation

Question

When I read Mukanov's book in "Quantum effects in gravity", I found the following interesting remark at p. 57.

Remark: Lorentz invariance.To quantize a field theory, we use the Hamiltonian formalism which explicitly separates the time coordinate $t$ from the spatial coordinate $x$. However, if the classical theory is relativistic (Lorentz-invariant), the resulting quantum theory is also relativistic.

This is a statement. I would like to ask: is there any proof, or any simple way to see that it is true? Namely, why by a recipe that is not relativistically invariant (at least it looks so), one can be sure that all the consequences are relativistically variant.

I can't see why it should be true.

PS: This question hits my scar. When I asked exactly that question in my university, I could not get any answer, neither from the lecturer, nor from any book I found. I felt so bad that I even abandoned the quantum field theory course after that.

Answer 1

A good/intuitive discussion of how to take a classical field theory and obtain a quantum field theory in the Heisenberg picture is discussed in Bjorken and Drell [BD] section 11.3. A discussion on how classical symmetries like Lorentz covariance are to be expressed on a quantum level is provided in BD 11.4, directly motivating the above-mentioned material in Srednicki. Discussions of Lorentz covariance of specific quantum field theories are provided in BD 12.1, 13.5, 14.3 and in 15.3. Covariance of e.g. classical Bosonic string theory only goes to the quantum level for D = 26, as one checks analogously to the BD references given, so it's not as straightforward as the quoted passage makes it seem.

The way this is done is to take $U(a,b) \psi_r(x) U^{-1}(a,b) = S^{-1}_{rs}(a) \psi_s(ax+b)$ (motivated in [11.4]) to first order for translations and Lorentz transformations, i.e. $i[\hat{P}^{\mu},\psi_r(x)] = \partial^{\mu} \psi_r(x)$ and $i[\hat{M}^{\mu \nu},\psi_r(x)] = x^{\mu} \partial^{\nu} \psi_r(x) - ...$ and to show that these transformation laws, which are expressions of the classical displacement/Lorentz invariance of the classical theory [11.4], do not upset the canonical commutation relations, or rather, the theory with the commutation relations imposed is still Lorentz invariant, which is shown by showing the commutation relations still hold for transformed fields, as is shown in e.g. [13.5] explicitly.

Answer 2

I would like to refer to the presentation of canonical quantization of scalar fields in chapter 3 of Quantum Field Theory, Srednicki (2007). To clarify this covers canonical quantization in standard Quantum Field Theory (not Quantum Gravity) which I understand is the question.

Srednicki is very careful to use a Lorentz invariant differential so you can see clearly that Lorentz invariance is maintained throughout the quantization procedure (moving from classical variables to quantum operators).

The "proof" then (for the free theory) is to show at the end that once you have created the Fock space of particle states then these all transform as expected under Lorentz transformation. For example for a one-particle state $|k\rangle$ you can prove that

$$U(\Lambda)|k\rangle = |\Lambda k\rangle$$

where $U(\Lambda)$ is the unitary operator associated with the classical Lorentz transformation $\Lambda$.

Problem 3.3 works you through this.