A Is irreducibility justified as a Wightman axiom?

[Peter Morgan]

My discussion (copying this from a Facebook post dated April 29th, 2018):

A realization that has been steadily growing crystallized for me this morning. The Wightman axioms require that the vacuum state must be irreducible, or, in more common parlance, it must not be a mixed state. For any subset of all fields in a perfect theory, however, if there is any entanglement between the fields of that subset and the other fields, then the reduced vacuum state over the subalgebra generated by that subset must be a reducible or mixed state (unless the subset generates the whole algebra, in which case the other fields are superfluous).

The irreducibility of the vacuum state cannot, therefore, be required a priori, axiomatically, for the vacuum state over, say, the electromagnetic field considered without all other fields (unless, again, all other fields are superfluous, so we could do all physics using just the electromagnetic field). Whether the vacuum state in any given case is in fact irreducible must be established a posteriori, by experiment. Indeed, insofar as there is dark matter and dark energy unaccounted for, even the vacuum state of the standard model of particle physics cannot be axiomatically required to be irreducible.

This is a big deal because whereas there are relatively few models of the Wightman axioms, just the generalized free fields, there are many, many models of the Wightman axioms minus the irreducibility of the vacuum state. Also, for those who care about such things, my understanding is that the proof of the Reeh-Schlieder theorem requires irreducibility of the vacuum state.

This weakening of the Wightman axioms is not considered in, for example, Section 3.4 of R F Streater, "Outline of axiomatic relativistic quantum field theory", Rep. Prog. Phys. 38 771-846 (1975), where Streater critiques various individual axioms. I would be pleased to know of any discussion that does consider this possibility.

In slightly more detail for PF, irreducibility is only indirectly stated as an axiom in Streater and Wightman's "PCT, spin & statistics, and all that", being a consequence of other axioms, but it is stated separately, as completeness, in Ch. II of Haag's "Local Quantum Physics". Haag also includes completeness in the Haag-Kastler axioms in his Ch. III.
So, question, is the discussion above flawed? Can irreducibility be justified in some other way?

I note that when I say that "there are many, many models of the Wightman axioms minus the irreducibility of the vacuum state", the simplest examples are positive energy Poincaré invariant non-Gaussian states, and it would seem that whether the vacuum state is Gaussian or not, for, say, the electromagnetic field observables of QED, taken as a subalgebra of QED or of the standard model of particle physics as an interacting field, should be empirically verifiable, not asserted a priori.

 

[A. Neumaier]

"irreducibility of the vacuum state"

What do you mean by an irreducible state? I know of irreducible representations, but never heard of such a notion for states.

 

[Peter Morgan]

What do you mean by an irreducible state? I know of irreducible representations, but never heard of such a notion for states.

I jumped to the GNS-representation without saying so. Sorry. Given a state over a *-algebra of operators, use the GNS-construction of a Hilbert space and a representation of the *-algebra acting on that Hilbert space. Streater & Wightman, "PCT, ...", p. 101, then says, "the smeared fields form an irreducible set of operators in Hilbert space" (by "in", I suppose they mean "acting on"). So by an irreducible state I mean a state that results in an irreducible GNS-representation.

 

[A. Neumaier]

So by an irreducible state I mean a state that results in an irreducible GNS-representation.

Doesn't reducibility just imply the presence of multiple phases?

 

[Peter Morgan]

Doesn't reducibility just imply the presence of multiple phases?

I wondered whether I would have to produce the concrete reason for being interested in this. Well, here it is, in brief:

Suppose we present the vacuum state in a generating function form,
$$\rho_{\kappa,m}\left(\mathrm{e}^{\mathrm{i}\lambda_1\hat F_{f_1}}\cdots\mathrm{e}^{\mathrm{i}\lambda_n\hat F_{f_n}}\right)
=\exp\left[-\kappa\left(\hspace{-0.2em}\sum\limits_{1\le i\le n}\hspace{-0.3em}\lambda_i^2(f_i^*,f_i)/2
\ +\hspace{-0.2em}\sum\limits_{1\le i<j\le n}\hspace{-0.4em}\lambda_i\lambda_j(f_i^*,f_j)\ \right)\right],
$$
where $(f,g)$ is required to be a manifestly Poincaré invariant positive semi-definite sesquilinear form on the test function space, restricted to the mass $m$ forward mass shell in wave-number space so that the spectrum condition and microcausality are both satisfied. The useful aspect of this presentation is that the state itself fixes the algebraic structure.
This is a state because it satisfies $\rho_{\kappa,m}(\hat A^\dagger)=\overline{\rho_{\kappa,m}(\hat A)}$, $\rho_{\kappa,m}(1)=1$, and $\rho_{\kappa,m}(\hat A^\dagger\hat A)\ge 0$. Equally, any product of different instances of $\rho_{\kappa,m}(...)$ for different masses $m$ and for different scaling $\kappa$ is a state over a generalized quantized free field, which satisfies the spectrum condition and microcausality. So far almost entirely conventional.
We also have, however, that any convex sum of different instances of $\rho_{\kappa,m}(...)$ for different masses $m$ and for different scaling $\kappa$ is a state, over an algebra that still satisfies the spectrum condition and microcausality, but now it doesn't satisfy canonical commutation relations, and, in general, the representation is not irreducible.
Suppose we use a construction $$\rho\left(\mathrm{e}^{\mathrm{i}\lambda_1\hat F_{f_1}}\cdots\mathrm{e}^{\mathrm{i}\lambda_n\hat F_{f_n}}\right)
=\int \rho_{\kappa,m}\left(\mathrm{e}^{\mathrm{i}\lambda_1\hat F_{f_1}}\cdots\mathrm{e}^{\mathrm{i}\lambda_n\hat F_{f_n}}\right)\mathrm{d}\mu(\kappa);$$
this still restricts to the mass $m$ forward mass shell in wave-number space, and it can be as close to a Gaussian distribution as we like, but we can also engineer it to be rather far from a Gaussian distribution.

Even more specifically, my interest here is that whatever form $\rho_{\kappa,m}\left(\mathrm{e}^{\mathrm{i}\lambda_1\hat F_{f_1}}\cdots\mathrm{e}^{\mathrm{i}\lambda_n\hat F_{f_n}}\right)$ takes, it's supposed to be empirically measurable, and perturbative QFT is supposed to be constructing this object, but instead perturbative QFT can only produce this object as an asymptotic series at asymptotically infinite time-like separation. Hopefully the construction I've just given above is not the only construction we could think of that doesn't satisfy the irreducibility requirement, but even if it's the only such construction it still gives us something more than generalized free fields. If we also allow non-vacuum sector representations, in which case energy will not be bounded below, even more is possible.

My point being that even if this "just implies the presence of multiple phases", there seems to be an empirical fact of the matter whether one construction or another is empirically more useful.