There is a beautiful paper by Clifton and Halvorson , which discusses the ontology of particles in quantum field theory using the famous example of Minkowski vs. Rindler quantizations of a free bosonic field. What is especially nice about this paper is that it contains the clearest exposition of the algebraic approach (AQFT), in particular the Gelfand-Naimark-Segal (GNS) construction, I’ve ever encountered. This framework enables them to make the discussion of particles, and physical observables in general, very precise, to wit: that the Minkowski and Rindler vacua induce disjoint GNS representations of the Weyl algebra.
Now, while I prefer to avoid excessive rigor (or rigor as summum bonum), there is truth in Halvorson’s claim  that AQFT “is a particularly apt tool for studying the foundations of QFT.” Hence this post will attempt to summarize (and/or copy verbatim) those essential aspects of the GNS construction, as detailed in section 2 of , which are necessary to address the question of inequivalent field quantizations. A more thorough introduction to AQFT is, alas, an undertaking for another post.
We begin by introducing the Weyl algebra. This is essentially a more formal/rigorous way of formulating the canonical commutation relations. Consider a (for the moment, finite) classical system with degrees of freedom, which has -dimensional phase space . Each point in is described by a pair of vectors , whose components parametrize the position and momentum of the system via the canonical variables
To quantize the system, we elevate the position and momentum variables to operators on some Hilbert space, and impose the canonical commutation relations
where is the -dimensional identity matrix. Of course, the phrase “elevate to operators on Hilbert space” is precisely the sort of cavalier attitude to which mathematical physicists object; and while we theoretical physicists can usually get away with simply dismissing tedious questions about boundedness, representations, and whatnot, in this case we must be (significantly) more precise about what such a procedure entails.
To that end, observe that one can introduce two -parameter families of unitary operators
whereupon one can show that the canonical commutation relations are formally equivalent to
These are known as the Weyl form of the canonical commutation relations. (As noted in , there are some irregular representations in which this equivalence does not rigorously hold; but it’s solid for the standard Schrödinger representation implied above, so this will not concern us).
One nice feature of this language is that one can put position and momentum degrees of freedom on the same footing by introducing the composite Weyl operator
whereupon the Weyl form of the canonical commutation relations may be encapsulated in the multiplication rule
where is none other than the familiar symplectic form on S. For completeness, one further defines
The point is then that any representation of the Weyl operators on a Hilbert space (more on what this means below) gives rise to a representation of the Weyl form of the canonical commutation relations, and vice-versa.
We can now drop the restriction to finite-dimensional spaces, and let the phase space be an arbitrary infinite-dimensional vector space equipped with a symplectic form. Then the family of unitary operators acting in some Hilbert space satisfies the Weyl relations iff
The (self-adjoint) observables of the system are then obtained by taking arbitrarily linear combinations of Weyl operators.
In the above expressions, the subscript denotes the representation. The idea behind a representation is to make an abstract algebra more concrete by representing the elements thereof as matrices; i.e., a representation reduces an abstract algebra to a linear algebra, which is often more practical to work with. For example, the Pauli matrices provide a convenient representation of the Lie group . In the present context, the representation determines the Hilbert space, i.e., the particle–and, arguably, physical–content of the theory itself. We will return to this important point momentarily, but first we must introduce a bit more machinery.
Trigger warning: this is about to become tedious, involving the introduction of various topologies and the like. I promise it’s important.
Let be the set of bounded operators (specifically, linear combinations of Weyl operators) acting on . One then says that a bounded operator can be uniformly approximated by operators in iff
(Note that this essentially imposes the uniform topology, in the sense of uniform convergence of to , hence the name). Now let (not to be confused with non-scripty !) denote the set of all bounded operators on that can be uniformly approximated by elements in . is the -algebra generated by the Weyl operators . It is important to note that this is only a subalgebra of the algebra of all bounded operators on , denoted , which is uniformly closed under adjoints .
Now, suppose that we have two systems of Weyl operators generated by and , which act on and , respectively, and denote the associated -algebras by , . A bijective mapping is called a -isomorphism iff is linear, multiplicative, and commutes with the adjoint operation. This sets the stage for the following important result:
Theorem 1 (Uniqueness theorem)
a -isomorphism such that .
This means that the -algebra generated by any system of Weyl operators is in fact representation-independent, and hence we may refer to this abstract algebra simply as the Weyl algebra . (Implicitly, we mean the Weyl algebra over , denoted , but we may suppress the arguments henceforth without confusion). Thus the problem of defining the Hilbert space amounts to choosing a representation of the Weyl algebra, i.e., the map . We will frequently denote this representation by .
In principle, since the Weyl algebra is representation-independent, one could refuse to choose a representation and instead proceed purely abstractly (e.g., defining states as positive normalized linear functionals on , describing dynamics in terms of a one-parameter group of automorphisms, etc). But representations are more powerful than mere convenience alone would suggest. In particular, the abstract Weyl algebra does not contain unbounded operators, many of which are of physical significance—for example, the total energy, the position & momentum observables in field theory, as well as the total number operator. However, via the introduction of the weak topology below, a representation can be used to extend the observables of the system beyond those contained in the abstract Weyl algebra itself.
Given as above, one says that a bounded operator is weakly approximated by elements of iff
The important thing to note here is that unlike uniform approximation above, weak approximation requires one to select a representation (in order to evaluate the inner product), and hence has no abstract (representation-independent) counterpart. By von Neumann’s double commutant theorem, the set of bounded operators that can be weakly approximated by elements of is , the von Neumann algebra generated by . Note that , since the latter is the weak closure of the former.
So far so good, but still contains only bounded operators. The final step is to associated unbounded observables with via their spectral projections. Namely, one says that an arbitrary (possibly unbounded) self-adjoint operator on is affiliated with iff all of ‘s spectral projections lie in . The reason we had to first extend to the von Neumann algebra instead of doing this with the representation itself is that -algebras do not contain non-trivial projections of their self-adjoint members. In other words, if we want to included unbounded operators, we need to work with the weak closure of the -algebra (hence my promise above that topology would be important).
Now here’s the kicker, which foreshadows the ontological question underlying this post:
Theorem 2 (Non-uniqueness theorem)
There exist representations of for which there is no -isomorphism from to such that .
Thus the price of extending the set of observables to those affiliated with the von Neumann algebra (that is, including unbounded operators) is the loss of uniqueness. In particular, this occurs when and are disjoint representations, which therefore leads to physically inequivalent Hilbert spaces! This is precisely what happens in the Minkowski vs. Rindler vacua.
In discussing the conceptual significance of “physically inequivalent” representations, it is necessary to distinguish various mathematical notions of equivalence. This will enable us to define the notion of disjoint representations, the importance of which should be obvious from the title of this post. First however, we must introduce two related concepts: irreducibility and factoriality.
A representation is irreducible iff no non-trivial subspace of is invariant under the action of all operators in . Since an invariant subspace exists iff the projection onto it commutes with , irreducibility implies . A representation is factorial iff the associated von Neumann algebra is a factor, meaning it has trivial center—that is, the only operators in which commute with all other operators are proportional to the identity. (Incidentally, note that this supports the familiar QFT notion that the only non-trivial operator that commutes with all local operators is the identity). Furthermore, since is a factor, the fact that is irreducible implies that it is also factorial.
We may now proceed to introduce the following sequence of equivalences: unitarily equivalent quasi-equivalent weakly equivalent. Two representations and are unitarily equivalent iff there exists a unitary operator that maps isometrically onto , such that
The slightly weaker notion of quasi-equivalent is most concisely stated as the existence of a -isomorphism from onto such that (cf. Theorem 2). Unitary equivalence is then simply the special case in which is a unitary operator. If both representations are irreducible, then quasi-equivalence also implies unitary equivalence. If two representations are not even quasi-equivalent, they are disjoint.
Before proceeding to weakly equivalent, it is helpful to recast the above in terms of states. Abstractly, a state of a -algebra is simply a positive normalized linear functional . It turns out that some (but not all!) of these abstract states correspond to the familiar density operators from quantum theory; we denote these so-called normalstates by
The subset of normal states is called the folium of the representation , and is denoted ; i.e., iff there exists a density operator acting on such that
The two forms of equivalence introduced above can then be restated more intuitively as follows: and are quasi-equivalent iff , and disjoint iff . That is, two representations are disjoint iff they have no normal states in common (in which case, all normal states in one are orthogonal to those in the other).
Now, given the folium of , one says that an abstract state in can be weak approximated by states in iff
If all states in can be weak approximated by states in (note that this implies the converse), then and are weakly equivalent.
We are now prepared to state the following important results:
Theorem 3 (Stone-von Neumann uniqueness theorem)
When is finite-dimensional, every regular representation of the Weyl algebra is quasi-equivalent to the Schrödinger representation.
A regular representation is one in which the map is weakly continuous, which by Stone’s theorem (not to be confused with the above) guarantees the existence of unbounded self-adjoint operators on with , which are in turn affiliated with . This is the mechanism by which one recovers the usual Schrödinger representation of canonical position and momentum operators on Hilbert space. The Stone-von Neumann theorem is basically the statement that this process is unique: thanks to quasi-equivalence, any classical theory with finitely many degrees of freedom will yield the same quantum mechanical theory. Note the crucial qualifier “finite”: this theorem does not hold in quantum field theory (where is infinite-dimensional). In other words, the failure of the Stone-von Neumann theorem in QFT is what allows one to have disjoint representations, and hence opens the door to the ontological puzzle of inequivalent field quantizations.
However, this isn’t to say that disjoint representations are entirely incompatible, as alluded by the following theorem:
Theorem 4 (Fell’s theorem)
Every abstract state of a -algebra can be weak approximated by states in .
In other words, all representations of are at least weakly equivalent. Finally, we state the GNS theorem as the grand conclusion to this post:
Theorem 5 (Gelfand-Naimark-Segal theorem)
Any abstract state of a -algebra admits a unique (up to unitary equivalence) representation and a vector such that
and the set is dense in . Furthermore, is irreducible iff is pure.
The triple is referred to as the GNS representation of induced by , and is a cyclic vector for this representation. (Note that is therefore both cyclic and separating, and furnishes a representation of the vacuum state). One consequence of this theorem is that, even if , there always exists some representation with ; i.e., every state is normal in some representation.
Thus, the precise statement of inequivalence of the Minkowski and Rindler vacua is that these induce disjoint GNS representations for the Weyl algebra. To see this in detail, let alone address the physical implications, is the subject of another post.