- #1
Peter Morgan
Gold Member
- 274
- 77
My discussion (copying this from a Facebook post dated April 29th, 2018):
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 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.