# Relativistic invariance of canonical/Hamiltonian field quantisation

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.

• What is the question exactly? – gented Jul 29 '16 at 11:10
• Just look at the eventual results for that what is physically relevant: Expectation values, S-matrix elements. They all are properly covariant. – ACuriousMind Jul 29 '16 at 11:10
• @ACuriousMind but that is the question: why starting from some recipes that is not covariant, one gets the covariant expectation values, S-matrix elements... Or asking in another way, how to prove that there is no consequence from the recipes that violate relativity (expectation, S-matrix do not cover all consequences, I suppose) – ophelia Jul 29 '16 at 11:16
• @GennaroTedesco please see my comments above, I'll try to edit the question to make it clearer. – ophelia Jul 29 '16 at 11:18
• Well, the "proof" is that everything you can conceive of that is physically meaningful is covariant, even when computed with the Hamiltonian recipes. Are you also bothered by the ADM formalism giving covariant results? If we are free to choose coordinates how we want, then choosing specific ones must be allowed. I'm not sure what exactly bothers you about that. – ACuriousMind Jul 29 '16 at 11:19

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.

• Thanks, I will have a looked at the book. Sounds like a careful formulation. – ophelia Jul 29 '16 at 16:16