3.1. Physical States

The procedure of quantization can be understood as involving two steps. First, one finds a quantum algebra of observables $A$. Second, one chooses a representation (π, $H$) of $A$ on some Hilbert space. The interpretive debate between Algebraic Imperialism and Hilbert Space Conservatism concerns just whether this second step is necessary.Footnote ⁸ According to the Algebraic Imperialist, it is not: a quantum theory (or, at least its kinematics) can be captured by the abstract algebra $A$ in the sense that the physical quantities of a system can be adequately represented by the elements of $A$—or perhaps by weak limits in $A^{S0031824800010497}$—and the physical states can be represented by states on $A$. But, according to the Hilbert Space Conservative, we need to pick a representation (π, $H$). Then we represent the physical quantities of our system by elements of $\pi \left(A\right)$—or perhaps by ultraweak limits—and the physical states by density operators on $H$.

The objection that has been leveled at Algebraic Imperialism that I want to discuss is that the Imperialist allows for “too many” states, in the sense that many states on the abstract algebra $A$ are unphysical (see Arageorgis Reference Arageorgis1995). Taking a Hilbert space representation allows us to consider an appropriate collection of physical states by focusing our attention on only the states that have density operator representatives in our chosen representation. So the fact that Hilbert space methods provide resources to rule out unphysical states is supposed to count in favor of the Hilbert Space Conservative.

Before we can see what these unphysical states are, we need to specify the algebra of observables we are using—after all, different algebras will in general have different state spaces. The algebra of observables that gives rise to the unphysical states at issue is known as the Weyl Algebra. This algebra puts the canonical commutation relations between position and momentum observables in bounded form by considering only exponentiated forms of those observables.Footnote ⁹

The states that one might consider unphysical come in many varieties. For example, the Weyl algebra allows for nonregular states (Beaume et al. Reference Beaume, Manuceau, Pellet and Sirugue1974; Halvorson Reference Halvorson2001, Reference Halvorson2004), which fail to satisfy a continuity condition and in doing so fail to allow one to simultaneously define both position and momentum observables from the Weyl operators. These nonregular states do not have density operator representatives in the usual Hilbert space representation (the Schrödinger representation) of the Weyl algebra. It is a standard move in algebraic quantum theory to restrict attention only to regular states on the Weyl algebra and their representations to rule out these nonregular states.

But there are many other states on the Weyl algebra that one might consider ruling out as unphysical. Arageorgis (Reference Arageorgis1995) mentions a proposal that we ought to restrict attention to locally definite states, which vanish on smaller and smaller regions of space-time. Or perhaps we ought to restrict attention to Hadamard states, which allow for an appropriately well-defined stress-energy observable (see Wald Reference Wald1994). And Halvorson (Reference Halvorson, Butterfield and Earman2006) takes up a suggestion by Doplicher, Haag, and Roberts that we use only what he calls DHR states, which differ only locally from the vacuum. In addition, we might want to restrict attention to states accessible from Fock states, that is, states that allow for an interpretation in terms of particle states with creation and annihilation operators (see Petz Reference Petz1990, chap. 4).

The results that I present below in section 4 are immediately applicable to two of the suggestions just considered: regular states and states accessible from Fock states. The reason, as I will remark later, is that these states form the folium of an irreducible Hilbert space representation of the Weyl algebra, and I will show explicitly that the results of section 4 apply to the folium of any irreducible Hilbert space representation of any algebra. For the purposes of this article, I will not analyze any of the other suggestions in detail because they require substantially more mathematical apparatus. However, the results that I present in the next section are meant to be sufficiently general to apply to any of these suggestions for appropriate physical state spaces, and I hope further work can make these connections explicit for locally definite states, Hadamard states, and DHR states.Footnote ¹⁰

One might already wonder whether the Algebraic Imperialist really needs a response to the “too many states” objection. Given that there is not a clear consensus as to what the physical states of any quantum theory are, one might think that it is not worth putting stock in an objection that relies on the assumption that at least some states are unphysical. Still, the objection is worth considering for two reasons. First, I think there are situations in which the Weyl algebra is not the natural choice of an algebra of observables for the Algebraic Imperialist, and at least part of the motivation for this claim is that there are states on the Weyl algebra that are viewed as pathological and very rarely used (see sec. 3.3). Addressing unphysical states in general can help us understand this particular case. Second, I think that the tools that one can develop in response, if one takes the “too many states” objection seriously, have the potential to lead to progress in our understanding of a wide variety of issues surrounding quantization. It may very well prove useful for current and future physics to have tools for systematically constructing, analyzing, and interpreting algebras and state spaces. So the objection has at least instrumental value in that it leads to new techniques and interesting questions, some of which I will be able to point to at the end of this article.

Furthermore, I should stress that it is not my purpose here to judge whether any of the proposals cited above provide an adequate specification of the states that we ought to deem physical. I mention these concrete proposals only to show that others have expressed an interest in restricting the space of quantum states. My goal in this article is only to show that however one wants to specify the collection of physical states, there is an intimate relationship between this state space and the algebra of observables of a quantum theory.

3.2. Physical Algebras

The fact that the abstract Weyl algebra allows for so many supposedly unphysical states has been taken by some (e.g., Arageorgis Reference Arageorgis1995) as an argument against Algebraic Imperialism. However, I propose that this only gives us reason to use a different algebra. Even if one particular algebra allows for unphysical states, this does not imply that all abstract algebras allow for unphysical states. There are other options for the abstract algebra of observables.

To motivate this approach, I want to note that the Weyl algebra does not have a particularly stable privileged status. To get the Weyl algebra, one starts with the canonical commutation relations between position and momentum observables, then applies these to particular bounded observables (namely, exponentiated position and momentum), and finally completes the resulting algebra in norm. First, the approach of using bounded observables to generate the Weyl algebra is largely for technical convenience. One can approach quantization using unbounded operators (Dubin, Hennings, and Smith Reference Dubin, Hennings and Smith2000), but one then needs to keep track of distinct domains for these unbounded operators. Some (Kay and Wald Reference Kay and Wald1991; Wald Reference Wald1994) have preferred to approach field quantization in terms of a *-algebra rather than a C*-algebra, which corresponds to choosing a collection of possibly unbounded observables. While there are still open questions concerning the relationship between the unbounded approach following Wightman’s axiomatic formulation (see Haag Reference Haag1992) and the bounded approach, these are beyond the scope of this article. All I wish to point out is that proponents of the unbounded approach already reject the Weyl algebra. In what follows in this article, I consider only bounded approaches in terms of C*-algebras because one can demonstrate the kinds of choices one has to make in that context already. It might also be of interest to investigate whether the results of section 4 can be generalized to arbitrary *-algebras.

Second, using the particular bounded observables defined as exponentiated position and momentum and completing the algebra in norm are also choices made largely to simplify technical machinery. Using the Weyl unitaries—those exponentiated position and momentum operators—allows one to put the canonical commutation relations in a particularly simple form so that they fully define the multiplication relation for the algebra. As I show below, there are other bounded operators, namely, the compact operators, that one could also use to implement the canonical commutation relations. Further, one might consider completing an algebra in other topologies apart from that defined by a norm, as Feintzeig (Reference Feintzeig2017b) considers completing a C*-algebra in its weak topology. So the Weyl algebra, which we arrive at through norm completion, is not the only option.

Finally, it is worth pointing out that the canonical commutation relations we use to generate the Weyl algebra are not a priori principles we must hold onto at all costs. Surely, they form some of the core tenets of quantum theory by underlying the uncertainty relations, but in doing so they constitute empirical hypotheses, which are open to revision just like any other principle of quantum theory. Alternative commutation relations will not play any role in the discussion that follows; as we will see, all of the algebras I consider in this article are modifications of the Weyl algebra so that they, in some sense, implement the same commutation relations. I do not think this poses a problem for this article, as my goal is not to justify the commutation relations but instead to take them on as empirical hypotheses made by all quantum theories. I do, however, think this point raises an important concern—in addition to the considerations raised here concerning the choice of algebra, there is also much philosophical work to be done to analyze the canonical commutation relations (in their many forms), perhaps by applying the approach of Alfsen and Shultz (Reference Alfsen and Shultz2001) to explicate the physical content of the algebraic structure in terms of the constraints it places on the state space of an algebra. Such work on the significance of the canonical commutation relations is beyond the scope of this article. But noticing that one might have occasion to use different commutation relations helps motivate the approach of considering alternative algebras of observables.

Some of the choices made in defining an algebra of observables may affect the physical content of a quantum theory, and some may not. I submit that one of the ways (but not the only way) to probe the physical content—or perhaps better, the physical consequences—of using a particular algebraic structure is to analyze the collection of states that algebraic structure allows. If two algebras differ in their state spaces, and if one wants to countenance these differing states as physically significant and employ them for physical purposes, then the choice of algebra becomes significant for physics as well. By using the phrase ‘physical consequences’, I mean to signal that one may not intend in choosing a particular algebra to assert that all of the states on that algebra are to be regarded as physically significant. In practice, working mathematical physicists use the conditions of the previous section to restrict attention to certain states after they have chosen an algebra. But some (e.g., Halvorson Reference Halvorson2004) have started from the premise that we do regard all states on an algebra as physically significant. So it is at least worth investigating to what extent one can hold onto this premise and make a principled choice of algebra that leads to an appropriate collection of physical states.

As an aside: given that one may have worries about whether there are any principled approaches to choosing the collections of physical states and physical quantities, it is worth pointing out that there is hope on the horizon. One can use tools developed recently for taking the classical limit of a quantum theory via deformation quantization with continuous fields of C*-algebras (Landsman Reference Landsman1998, Reference Landsman, Butterfield and Earman2006) to try to provide a principled way of picking out the physical algebra and physical state space. The principled approach I have in mind is to abide by the following guideline: the classical limit of a quantum theory should lead to a physical algebra and physical state space for the classical theory. This provides substantive constraints on the algebras and state spaces we use for our quantum theories. While not providing a complete and ultimate solution, this guideline at least has the advantage of reducing questions about quantum theories to questions about classical theories, with the hopes that we have a better understanding of how to answer these questions in the classical case. I mention this proposal only to point out future directions for research and also potential applications of the tools developed in this article. Further discussion of this principled approach requires additional technical tools and is beyond the scope of this article.

3.3. Example: Regular States and Compact Operators

In the next section, I present general algebraic results that allow us to change the algebra implementing the canonical commutation relations in order to restrict its physical state space. But before I present these results, I want to note that we already have a procedure, at least in the case of a simple system with finitely many degrees of freedom,Footnote ¹¹ for eliminating unphysical states by choosing an appropriate algebra. For these simple systems, it is standard to restrict attention to only regular states, in part because a result known as the Stone–von Neumann Theorem tells us that there is a unique irreducible representation of the Weyl algebra in which all of the regular states (and, it turns out, only the regular states) have density operator representatives.Footnote ¹² This representation is just the usual Schrödinger representation, and in this representation the ultraweak closure of the Weyl algebra is the algebra $B$($H$). So this leads us back to the familiar setting for nonrelativistic quantum mechanics, where every (self-adjoint) element of $B$($H$) is considered as a physically significant observable and every physical state has a density operator representative.Footnote ¹³

However, for these simple systems with finitely many degrees of freedom, there is another path one can take to get the same theory without using Hilbert space methods.Footnote ¹⁴ Instead of choosing the Weyl algebra, one can directly use the algebra of compact operators on a separable Hilbert space (i.e., L ²($R$ⁿ)).Footnote ¹⁵ Then one immediately finds that the state space of this algebra is equivalent to the collection of regular states. One way to see this is to notice that the compact operators have a unique irreducible representation,Footnote ¹⁶ so one has an immediate analog of the Stone–von Neumann Theorem. The bidual of the algebra of compact operators is its ultraweak closure in this representation, the familiar algebra $B$($H$). And there are no nonregular or otherwise unphysical states on the algebra of compact operators that cannot be represented as density operators on this representation.

Thus, one can construct the same quantum theory that we get through the Weyl algebra and the Stone–von Neumann Theorem by instead using the compact operators and forgoing the need to restrict attention to some subspace of states. This procedure provides a route for the Algebraic Imperialist to arrive at the same collection of physical states the Hilbert Space Conservative arrives at. The key to this procedure is an auspicious choice of algebra of observables for our quantum theory. So, I suggest that one might pay closer attention to the choice of algebra in quantization procedures in other cases when constructing quantum theories. The next section shows that if one takes seriously the importance of this choice of algebra, then one can develop powerful algebraic tools generalizing this procedure for representing physical systems with specified state spaces.

4.1. General Algebraic Results

Suppose that we have a C*-algebra $A$ and some preferred subset of its dual space $V\subseteq A^{*}$. This section establishes necessary and sufficient conditions for the existence of an algebra inheriting “the same” algebraic relations but with V as its entire dual space.

First we will need a condition to ensure that V can be dual to a space supporting an appropriate C*-norm. To that end, define ${\Vert \cdot \Vert }_V:A\to ℝ$ as follows for all $A\in A$:

${\Vert A\Vert }_V=\underset{\omega \in V;\Vert \omega \Vert =1}{\sup }\left|\omega \left(A\right)\right|\mathrm{.}$

With this definition, consider the following condition on V:

• 1) For all $A,B\in A$, ${\Vert \mathrm{AB}\Vert }_V\le {\Vert A\Vert }_V{\Vert B\Vert }_V$.

This is a minimal technical condition to ensure V can be the dual space to a C*-algebra: it ensures that V can be dual to an algebra whose multiplication operation is norm continuous.

We need one more piece of background before we can present the main result. The following piece of apparatus allows us to ensure that V has enough structure to support algebraic operations inherited from $A$, as we will make precise in a moment. For any $\omega \in A^{*}$, define a relation ~_ω on $A$ by

$A{~}_{\omega }B\text{ iff for all C}\in A,\omega \left(\mathrm{AC}\right)=\omega \left(\mathrm{BC}\right)\text{ and }\omega \left(\mathrm{CA}\right)=\omega \left(\mathrm{CB}\right)\mathrm{.}$

Two observables A and B satisfy this relation ~_ω just in case ω assigns the same expectation value to A and B, and if whenever we concatenate another measurement represented by the observable C, then ω yields the same expectation value for a measurement of A and C as it does for B and C, at least when the A–C and B–C measurements are performed in the same relative order.

Now define a relation ~_V on $A$ by

$A{~}_{V}B\text{ iff for all }\omega \in V,A{~}_{\omega }B\mathrm{.}$

Two observables A and B satisfy this relation ~_V just in case each state in V assigns the same expectation values to A as to B, and similarly for concatenating another measurement C, as above.

One easily checks that for any $\omega \in A^{*}$, the relation ~_ω is an equivalence relation, and similarly for any $V\subseteq A^{*}$, the relation ~_V is an equivalence relation. Now consider the following condition:

• 2) V is maximal in the sense that for all $\omega \in A^{*}$, $\text{if }A{~}_{\omega }B\text{ whenever }A{~}_{\text{V}}B\text{ for }A,B\in A,\text{ then }\omega \in V$.Footnote ¹⁷

Condition 2 guarantees that V is as big as possible, by containing all states compatible with the states in V.

Here, a state $\omega \in A^{*}$ is compatible with the states in V just in case it judges equivalent any two observables that all of the states in V judge equivalent. Intuitively, picking a collection of physical states tells us which observables are really the same or distinct according to whether those observables could possibly be assigned different expectation values by the physical states. If two observables are always assigned the same expectation value, then one is hard pressed to think of them as distinct observables, but if two observables are at least sometimes assigned different expectation values, then it is easier to see those observables as distinct. Now, there may be a state ω that judges as distinct all the observables the states in V judge distinct and judges equivalent all the observables the states in V judge equivalent. In this case, we might have just as much reason to countenance the state ω as physically significant as we have for the states in V. If one believes this, then one is likely to say that we have just forgotten to include ω in our specification of V. The maximality condition ensures that we do not forget to include any such states.

As we will see, condition 2 is used to ensure that V is “big enough” to support the algebraic operations of $A$. This can be seen as a technical redundancy because we can generate a maximal subspace from any other subspace $V_0\subseteq A^{*}$ by the following proposition.Footnote ¹⁸

Proposition 1. Let $A$ be a C*-algebra and let $V_0\subseteq A^{*}$. Let

$V=\{\omega \in A^{*}:\text{ for all }A,B\in A,\text{ if }A{~}_{V_0}B,\text{ then }A{~}_{\omega }B\}\mathrm{.}$

Then V is maximal in the sense of condition 2 above.

In this case, we will say that the vector space V is generated by V ₀. Given a collection of states in V ₀, V is the smallest maximal collection of states containing V ₀, and it is the collection of all states compatible with those in V ₀ in the sense above. Now we are ready to present our main result, which says that the conditions listed above are necessary and sufficient for reducing the state space of our algebra.

The main idea of the proof (contained in the appendix) is to construct the C*-algebra $B$ by taking the quotient of the original algebra $A$ by the equivalence relation ~_V; in other words, $B=A/{~}_V$. Moreover, we can show that the algebra we get through this construction is, in a certain sense, unique.

These theorems show that one can specify the physical state space however one likes, as long as it satisfies conditions 1 and 2. Whenever these conditions are satisfied, and only when these conditions are satisfied, we can find a new algebra $A/{~}_V$ that inherits the algebraic relations of $A$ through the surjective *-homomorphism f, and this new algebra has precisely the physical state space V as its entire state space.

4.2. Example: $B$($H$)

In this section we deal with the same simple quantum theories of systems with finitely many degrees of freedom considered in section 3.3. Such a theory has as physical states the density operators on a Hilbert space $H$ with observables given by all self-adjoint operators in $B$($H$). But suppose that we have chosen some other algebra of quantum observables. Let $A$ be any C*-subalgebra of $B$($H$) that contains the constants and separates density operator states in the sense that for any two density operators ρ₁, ρ₂ on $H$, there is an $A\in A$ such that $\mathrm{Tr}\left(A{\rho }_1\right)\ne \mathrm{Tr}\left(A{\rho }_2\right)$. The Schrödinger representation of the Weyl algebra mentioned in the previous section is an example of just such an algebra. Here we show that there is a natural way to transform $A$ into all of $B$($H$) using purely algebraic methods.

Why do we need such a procedure, given that we can just take the ultraweak closure of the algebra $A$ in the natural inclusion representation on $H$ to get all of $B$($H$)?Footnote ¹⁹ Of course, we could obtain all of $B$($H$) in this way, but there is something unsatisfying about this approach from the algebraic perspective. Why should we complete $A$ in the ultraweak topology when there may be states on $A$ that are not ultraweakly continuous? There is a sense in which completing $A$ in the ultraweak topology does not respect the algebraic structure (or really the state space) of $A$, because the completion of $A$ in its abstract weak topology (as opposed to the ultraweak topology) to form the bidual $A^{S0031824800010497}$ in fact leads us to a much larger algebra than $B$($H$).Footnote ²⁰

I will show that theorem 1 provides a general procedure to transform $A$ into $B$($H$). To see this, notice that there is a sense in which $A$ is too small and a sense in which it is too large. The algebra $A$ is too small in the sense that it may not contain many elements of $B$($H$) like projections (as is the case with the Weyl algebra). But $A$ is too large in the sense that it may allow for states that are not ultraweakly continuous and so cannot be represented by density operators on $H$. As such, we will first enlarge $A$ to $A^{S0031824800010497}$ to obtain all of the missing operators including the projections on $H$. Then we will restrict attention to a collection of physical states by applying theorem 1.

Let $V_0^Q$ be the collection of bounded linear functionals on $A$ that are ultraweakly continuous on $H$. The following proposition shows that reducing $A^{S0031824800010497}$ by theorem 1 with this preferred collection of states brings us back to the usual setting of $B$($H$).

This shows that one can generate $B$($H$) by first adding to $A$ the weak limit observables in $A^{S0031824800010497}$ and then reducing the algebra by theorem 1 to restrict attention to only states with density operator representatives on $H$. Again, the first step of enlarging the algebra to $A^{S0031824800010497}$ is necessary because there may be many bounded operators outside $A$. In particular, we cannot reduce $A$ directly to the compact operators (recall that $B$($H$) is the bidual and hence ultraweak closure of the compact operators) because there may be compact operators that are not in $A$, as in the case of representations of the Weyl algebra.

This shows that the Algebraic Imperialist always has the resources to reduce the physical state space of her algebra in at least all of the ways the Hilbert Space Conservative can. Where the Hilbert Space Conservative would take an irreducible representation of the algebra and restrict attention to density operator states in that representation, the Algebraic Imperialist can just choose the vector subspace of $A^{*}$ generated by those states (i.e., the density operator states in that representation) as her privileged subspace and directly apply theorem 1 to obtain a new algebra with precisely the right state space.

Moreover, we can immediately apply proposition 2 to two of the candidates for physical state spaces considered in section 3.1. First, consider the Weyl algebra $W\left({ℝ}^{2n}\right)$ over a finite-dimensional and simply connected phase space $R$²ⁿ with the standard symplectic form. Let $V_0^R$ be the vector space of linear combinations of regular states on $W\left({ℝ}^{2n}\right)$, where the linear combinations are taken according to the vector space operations in ${W\left({ℝ}^{2n}\right)}^{*}$ (not according to the Hilbert space operations of some particular representation). Since the Schrödinger representation on the Hilbert space $H$_S is irreducible and faithful, it follows that $W\left({ℝ}^{2n}\right)$ is *-isomorphic to a C*-subalgebra of $B$($H$_S) that separates density operator states. And, since the Schrödinger representation is regular, every regular state on $W\left({ℝ}^{2n}\right)$ is ultraweakly continuous on $H$_S (see Petz Reference Petz1990), which means $V_0^R$ is the collection of bounded linear functionals on $W\left({ℝ}^{2n}\right)$ that are ultraweakly continuous in the Schrödinger representation on $H$_S. In other words, $V_0^R$ is just the physical state space $V_0^Q$ defined above for the particular case of the Weyl algebra $W\left({ℝ}^{2n}\right)$ in the Schrödinger representation on $H$_S. Thus, we have the following immediate corollary of proposition 2.

This shows that theorem 1 can be applied to the Weyl algebra with the collection of physical states as the regular states.

Next, consider the Weyl algebra for a possibly infinite dimensional phase space $M$ (e.g., a field system), which we will denote $W\left(M\right)$. We will say that a state is accessible from a Fock state ω (see Petz Reference Petz1990, chap. 4) if it is a vector state in the Fock representation for ω that is obtained by some linear combination of creation operators acting on the vacuum vector representing ω.Footnote ²¹ Intuitively, a Fock state is one that can be represented as a superposition of particle configurations obtained by creating particles in the vacuum state ω. As is well known, this collection of vectors spans the Fock space, that is, the Hilbert space $H$_F of the Fock representation (Petz Reference Petz1990, 34). Since the Fock representation for ω is faithful and irreducible (Petz Reference Petz1990, theorem 4.7, 34), we know that $W\left(M\right)$ is *-isomorphic to a C*-subalgebra of $B$($H$_F) that separates density operator states.

Now, in order to apply theorem 1, we need to find a closed vector subspace of ${W\left(M\right)}^{*}$ containing the physical states. Let $V_0^F$ be the smallest closed vector subspace of ${W\left(M\right)}^{*}$ containing the states accessible from the Fock state ω. To generate this vector space $V_0^F$, we need to allow for mixtures and arbitrary linear combinations of the states accessible from the Fock state ω and then close the resulting vector subspace in the norm topology on ${W\left(M\right)}^{*}$. Allowing for mixtures (convex combinations) of states accessible from the Fock state ω yields that all finite rank density operator states on the Fock space $H$_F are in $V_0^F$, while allowing for arbitrary linear combinations yields that all finite rank operators on $H$_F are in $V_0^F$. From basic facts in the theory of Hilbert space operators (see Reed and Simon Reference Reed and Simon1980), we know that the finite rank operators are dense in the collection of all trace class operators on $H$_F with the trace norm, which is identical to the standard supremum norm on those operators, considered as linear functionals in ${W\left(M\right)}^{*}$ by the usual trace prescription. Thus, $V_0^F$ is the collection of all trace class operators on $H$_F and hence is the collection of all bounded linear functionals on $W\left(M\right)$ that are ultraweakly continuous on the Fock space $H$_F. In other words, $V_0^F$ is just the physical state space $V_0^Q$ defined above for the particular case of the Weyl algebra $W\left(M\right)$ in the Fock representation on $H$_F. Thus, we have the following immediate corollary of proposition 2.

This shows that theorem 1 can be applied to the Weyl algebra with the collection of physical states as those accessible from a Fock state.

One might worry at this point about the status of the collection of states V we deem physically significant in order to apply theorem 1. The goal of this article was to show that one can find purely algebraic resources for choosing an appropriate algebra of physical quantities. But it seems that the specification of a space of physical states V is an extra resource we need to avail ourselves of. And in the previous examples the state space that we use consists of the folium of a Hilbert space representation, so one might doubt that the methods of this section provide purely algebraic tools.

In response, I want to note that my purpose in this article is only to show that however one specifies the collection of physically significant states, we can find an algebra with that collection of states as its full state space (as long as it satisfies a few constraints). It is a natural question whether one needs a Hilbert space representation to specify V, but that is not a question I can deal with fully in this article. It may be helpful, however, to make a number of remarks concerning the above examples. First, one can define the regular states in a way completely free of reference to any Hilbert space representation by simply looking at the expectation values assigned by states to one-parameter families of Weyl unitaries. It turns out this definition specifies the folium of a particular Hilbert space representation, but this can be understood as a consequence of the continuity assumptions. In the same vein, although we have used a Fock representation to specify the states accessible from a Fock state, this does not imply that one needs a Hilbert space representation to define this collection of states. It is a further substantive question whether one can give an algebraic specification of these particle-like states, especially given that the definition of Fock states on their own does not require the use of any Hilbert space representations. For example, can one understand these particle-like states to constitute the smallest collection of states containing a Fock state and satisfying conditions 1 and 2?

As may be clear by now, I am not interested in specifying a rigid set of rules that the Algebraic Imperialist must abide by when specifying a collection of physical states. I think asking for such a set of rules in some sense misses the point. The point is that the use of Hilbert space methods has been distracting and obscuring in some cases by confusing conditions of technical convenience with conditions that have physical content. The tools developed in this article constitute just one step toward untangling these conditions. The point is not that one has reason for rejecting Hilbert space methods altogether but that too much focus on the use of Hilbert space methods or even debate about whether Hilbert space methods are useful can distract from the application of alternative algebraic methods that have promise for clarifying issues of physical interpretation that have confused researchers working in the Hilbert space approach.