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Question

The way I understand it right now is:

We start off with an algebra that includes $\phi(x)$ and $\pi(x)$ and these obey the canonical commutation relations.

Then we represent these as operators on a Hilbert space. This is what they do in QFT textbooks when they rewrite the field and momentum in terms of ladder operators and then construct Fock space: they are finding a representation for the algebra.

However, the issue is that there are many unitarily inequivalent representations of the algebra.

My questions is, why can't we specify the Hilbert space at the beginning, so that rather than viewing $\phi$ and $\pi$ as elements of an algebra for which a representation needs to be found, we instead specify a particular Hilbert space and treat $\phi$ and $\pi$ as known operators on that space?

Answer 1

You're talking about Haag's theorem.

The issue is not really that inequivalent representations exist but that the standard formulation of perturbative QFT pretends to work in a single representation when that cannot work: The standard way in which QFT textbooks set up the perturbative scattering theory which will be computed via Feynman diagrams is that at some point they have to claim something equivalent to the existence of the interaction picture:

You have the free interaction picture fields $\phi_I(x)$ and the interacting fields $\phi(x)$ from the interacting theory you want to compute. The picture of particles arises by doing the mode expansion of the free fields, and then relating the free fields to the interacting fields by the interaction picture time evolution operator $U_I$, schematically:

$$ \phi(x) = U_I^\dagger(x^0)\phi_I(x)U(x_I^0)$$

But since time evolution $U_I$ is unitary, this is the claim that a unitary equivalence between a free field and an interacting field exists. There's lots of details omitted here, of course, such as what exactly constitutes a "unitary equivalence", but this is the gist of it - for the details see e.g. chapter 4-5 of Streater and Wightman's "PCT, Spin and Statistics, and All That".

This is precisely what is forbidden by Haag's theorem: Not only do inequivalent representations exist, but also every field that is unitarily equivalent to a free field is itself free - meaning that $\phi(x)$ must also be free, in contradiction to the setup that it was supposed to be an interacting field. This is exactly the inverse situation of ordinary QM, where all $x$ and $p$ have the same representation enforced by the Stone-von Neumann theorem regardless of which evolution equations they obey.

So Haag's theorem makes a straightforward rigorous version of the usual setup of QFT at the physics level of rigor impossible. The problem is less in defining the QFT (e.g. the Wightman axioms do a perfectly fine job of defining axiomatic QFTs), but rather in making rigorous the usual approach to QFT computations: If you cannot set up the usual perturbation theory, you cannot claim to have made a rigorous version of QFT.