Classical limits of unbounded quantities by strict quantization

Powerful C^*-algebraic tools have been developed in the last few decades for analyzing the classical limits of quantum theories. These tools form the theory of strict quantization^1–6 (in contrast to formal quantization,^7,8 which is used in perturbative quantum field theory). Working with C^*-algebras provides a rich structure in which to construct quantum theories and their classical limits as well as provide physical interpretations. However, it is sometimes said that one cannot use C^*-algebras to model all of the systems of physical interest. In quantum field theory, for example, researchers often employ more flexible types of *-algebras.^9–11 One reason is that C^*-algebras do not allow one to capture unbounded quantities such as the field operators, field momentum operators, or number operators associated with such systems. Yet, recent developments in the theory of unbounded operator algebras have made precise the relationships between algebras of bounded and unbounded quantities.¹² Certain algebras of unbounded quantities can be understood as completions, in a relevant topology, of C^*-algebras. In this paper, we leverage this fact to make some first steps toward understanding the classical limits of unbounded quantities starting from the framework of strict quantization. Our central contribution is to develop tools for taking the classical limits of unbounded quantities and to illustrate these tools in free bosonic quantum field theories by analyzing classical limits of number operators.

Others have analyzed unbounded quantities in C^*-algebraic terms by working in specified Hilbert space representations. This allows one to consider unbounded operators affiliated with a represented C^*-algebra. This technique is useful for many purposes, but its dependence on a Hilbert space representation has some drawbacks for systems whose kinematical C^*-algebras have unitarily inequivalent representations, including quantum field theories.¹³ By contrast, the methods we develop in this paper work directly at the level of the abstract algebras, thus providing a framework that one can use to simultaneously compare even unbounded quantities that appear in inequivalent representations.

The plan of this paper is as follows. In Sec. II, we provide background on strict quantization. In Sec. III, we develop tools for taking the classical limits of unbounded quantities in a system with finitely many degrees of freedom—e.g., an n-particle system. We use this simpler example to outline key features of a strict quantization that allow one to extend it to unbounded quantities. This serves as a jumping off point for the generalization of these methods in Sec. IV to linear bosonic field theories with infinitely many degrees of freedom. In Sec. V, we focus specifically on the Klein–Gordon field and establish the classical limits of number operators associated with inequivalent representations of the kinematical C^*-algebra. In Sec. VI, we apply these methods to the free Maxwell field. We conclude in Sec. VII with some discussion.

Much previous work on the classical limit precedes the approach of the present paper. For example, Hepp¹⁴ provided a semi-classical analysis of bosonic field theories in a particular Fock space representation. Recent work of Ammari et al.¹⁵ provides a different framework from the current investigation that also allows one to discuss coherent states in bosonic field theories, which are known^2,16 to be closely related to the Berezin quantization map employed in the current paper. Falconi¹⁷ extended that previous work in a representation-independent manner, although the Wigner measures used in that work are associated with the alternative to Berezin quantization called Weyl quantization; since Weyl quantization is not positive, the corresponding Wigner measures are, in general, not positive.² We will not provide a comprehensive comparison of different approaches to the classical limit here. We simply note that our work falls in the tradition of strict quantization, which Feintzeig¹⁸ argues provides an appropriate physical interpretation based on uniform approximations of observables. We will also aim to obtain uniform approximations in the classical limit here, even though we rely on pointwise approximations to generate unbounded observables.

A strict quantization provides the mathematical framework for analyzing classical limits of states and quantities within a C^*-algebraic setting by giving one a notion of the limit of a family of C^*-algebras. For background and examples, see the work of Rieffel^4–6 and Landsman.^{1–3,19–22}

A strict quantization determines the structure of a continuous field of C^*-algebras in which all sections of the form $[ℏ↦Qℏ(A)]$ for $A∈P$ are continuous (see Theorem II.1.2.4, p. 111 in Ref. 2). Thus, a strict quantization defines the classical limits of quantities and states as follows. The classical limit of a family of quantities ${Qℏ(A)}ℏ∈[0,1]$ is understood to be the classical quantity $A∈P$⁠. A family of states ${ωℏ∈S(Aℏ)}ℏ∈[0,1]$ is called a continuous field of states when the map $ℏ↦ωℏ(Qℏ(A))$ is continuous for each $A∈P$⁠. The classical limit of a continuous field of states is understood to be the classical state ω₀.

An illustrative example of strict quantization is the quantization of the Weyl algebra, which we review now and use later on. First, we define the Weyl algebra itself. Let E be a vector space of test functions with a symplectic form σ. In Sec. III, we will focus on the case where $E=R2n$ and σ is the standard symplectic form, but in Sec. IV, we will deal with the case where E is infinite-dimensional, so we proceed here with some generality. The Weyl algebra $W(E,ℏσ)$ is generated by elements W_ℏ(F) for each F, G ∈ E with

The elements of the form W_ℏ(F) are linearly independent, and we denote the their linear span as Δ(E, ℏσ). There is a unique C^*-norm on Δ(E, ℏσ) called the minimal regular norm.^23,24 We define $W(E,ℏσ)$ as the completion of Δ(E, ℏσ) with respect to this norm.

The commutative algebra $W(E,0)$ is *-isomorphic to the algebra AP(E′) of σ(E′, E)-continuous almost periodic functions on some topological dual E′ to E, when E is given a vector space topology (see p. 2902 of Ref. 23). Thus, the algebra of classical quantities can be interpreted in a natural way as an algebra of functions on a phase space E′. Furthermore, the *-algebra Δ(E, 0) carries a Poisson bracket defined as the extension of

(see p. 334 of Ref. 25 or Eq. (2.15), p. 11 of Ref. 26) and so can serve as the domain of a quantization map. We note that in the special case where $E=R2n$⁠, the phase space is the dual $E′=R2n$ and so $W(R2n,0)≅AP(R2n)$⁠.

When $E=R2n$⁠, we have further information about the algebras $W(R2n,ℏσ)$ for ℏ > 0. These algebras have a familiar Hilbert space representation on the Hilbert space $L2(Rn)$⁠, which we denote $πℏS:W(E,ℏσ)→B(L2(Rn))$ and call the Schrödinger representation

for all $a,b∈Rn$⁠. Since the families $t↦πℏS(Wℏ(ta,tb))$ are weak operator continuous, Stone’s theorem (see p. 264 of Ref. 27) implies that these one-parameter unitary groups have self-adjoint generators. These generators are unbounded operators, corresponding to the standard position and momentum operators for n particles, and so this Hilbert space representation reproduces the ordinary formulation of quantum mechanics.

To define a strict quantization, we work with the C^*-algebras $Aℏ≔W(E,ℏσ)$ for each ℏ ∈ [0, 1], where $A0=W(E,0)$ contains $P≔Δ(E,0)$ as a dense Poisson subalgebra. We define the Weyl quantization maps $QℏW:Δ(E,0)→W(E,ℏσ)$ as the linear extension of

Binz et al.²⁵ showed that this structure indeed forms a strict quantization. Thus, this structure allows one to analyze classical limits of states and quantities in the Weyl algebra.

If one knows that quantization maps $Qℏ:P→Aℏ$ not only satisfy the conditions (i)–(iii) of a strict quantization but also furthermore are continuous in a locally convex topology, then one can continuously extend these maps to the completions of the respective algebras in that topology. Recent work on algebras of unbounded operators^12,28–31 shows that such a completion of a C^*-algebra $A$⁠, which we will, in general, denote by $Ã$⁠, will be at least a partial *-algebra containing unbounded operators with some discernible structure.

For what follows, we will not need the details of the rich structure theory that has been developed for algebras of unbounded operators.^32–34 Instead, the issue we encounter in applying these ideas in quantization is that quantization maps may fail to be continuous, and continuity is required to guarantee a unique extension of a quantization map to a completion. For example, the Weyl quantization maps defined in Sec. II fail to be continuous in the norm, and hence weak, topologies. This implies that one cannot continuously extend the Weyl quantization maps to the completion of the Weyl algebra. This is unfortunate because the Weyl quantization maps have some nice properties; they can be defined with the minimal algebraic structure of the Weyl algebra even on an infinite dimensional phase space. However, another quantization prescription called Berezin quantization is known to be continuous in the norm, and hence weak, topologies. Berezin quantization is well-defined for systems with finitely many degrees of freedom with phase space $R2n$⁠, but the standard definition involves phase space integrals that are not, in general, well-defined when E is infinite-dimensional. Our goal in this section is thus to put Berezin quantization into a minimal algebraic form so that it can be applied even to systems whose phase space is infinite-dimensional. As we proceed, we will use the simplified example of a system with finitely many degrees of freedom to illustrate the basic concepts of our approach to dealing with classical limits of unbounded operators.

We begin by defining the Berezin quantization maps for a system with phase space $R2n$⁠. We will work with the algebras $Aℏ≔K(L2(Rn))$ of compact operators on $L2(Rn)$ for each ℏ ∈ (0, 1] and the algebra $A0≔C0(R2n)$ of continuous functions vanishing at infinity on $R2n$⁠, the latter of which contains the dense Poisson subalgebra $P≔Cc∞(R2n)$ of smooth, compactly supported functions. The Berezin quantization maps involve integrals over phase space of certain functions of coherent states, but since these integrals are not, in general, meaningful on infinite dimensional phase spaces, we will seek to put the quantization maps in a different form. A coherent state for $(p,q)∈R2n$ is a vector $ψℏ(p,q)∈L2(Rn)$ of the form

Here and in what follows, x² denotes the dot product x ⋅ x for any $x∈Rm$⁠. The Berezin quantization maps $QℏB:Cc∞(R2n)→K(L2(Rn))$ are then defined for each $f∈Cc∞(R2n)$ by

for each $ψ∈L2(Rn)$⁠, where ⟨⋅, ⋅⟩ is the L² inner product.

It is known that this quantization map is positive, which implies that it is continuous in the norm (see Proposition 1.3.7, p. 47 of Ref. 2), and hence weak, topologies. First, this entails that $QℏB$ extends continuously to a map $C0(R2n)→K(L2(Rn))$⁠, which we will denote by the same symbol. Second, this implies that $QℏB$ extends continuously to the weak completions of the domain and range, which we now denote $Q̃ℏB:C0̃(R2n)→K̃(L2(Rn))$⁠. The algebra $C̃0(R2n)$ contains many unbounded and even discontinuous functions (see Example 4.1, p. 371 of Ref. 28), but for our purposes, we note that it at least contains as a subalgebra the algebra of all continuous functions $C(R2n)⊆C̃0(R2n)$ and so contains unbounded functions $Φ0(a,b):R2n→C$ for fixed $a,b∈Rn$ of the form

for each $p,q∈Rn$⁠, where ⋅ denotes the usual dot product. These functions include standard classical position and momentum observables. Similarly, $K̃(L2(Rn))$ contains many unbounded operators (see Example 4.3, p. 372 of Ref. 28), including all operators Φ_ℏ(a, b) on $L2(Rn)$ for fixed $a,b∈Rn$ of the form

acting on the dense domain of vectors $ψ∈Cc∞(Rn)⊂L2(Rn)$⁠. Again, these operators include standard quantum position and momentum observables.

Note that $AP(R2n)⊆C(R2n)⊆C̃0(R2n)$⁠, and so one can directly compare the maps $Q̃ℏB$ and $QℏW$ on the domain $Δ(E,0)⊆AP(R2n)$ [where we freely identify $W(R2n,0)$ with $AP(R2n)$]. The comparison actually follows directly from the known relationship of Weyl quantization with Berezin quantization [see Eq. (2.117), p. 144 of Ref. 2] or from the representation of Berezin quantization in terms of Toeplitz operators on a Segal–Bargmann space (see p. 294 of Ref. 35). Here, we will establish the comparison in the Schrödinger representation of the Weyl algebra on $L2(Rn)$ by direct computation. (See also Ref. 36 for a generalization related to Rieffel’s deformation.)

Define $cℏ(a,b)=e−ℏ4(a2+b2)$ for all $(a,b)∈R2n$⁠. Let $cQℏW:Δ(E,0)→W(R2n,ℏσ)$ be the linear extension of the map defined on the generators $W0(a,b)∈AP(R2n)$ by

The following proposition establishes that $cQℏW$ is equivalent to the extension of $QℏB$⁠:

The scalars c_ℏ form what Honegger and Rieckers³⁷ call quantization factors, satisfying

• $cℏ(a,b)∈R+$ for all ℏ ∈ [0, 1] and $a,b∈Rn$⁠;

• c_ℏ(0, 0) = e⁰ = 1 and c₀(a, b) = e⁰ = 1 for all ℏ ∈ [0, 1] and $a,b∈Rn$⁠; and

• $ℏ↦cℏ(a,b)=e−ℏ4∥(a,b)∥$ is continuous for all $a,b∈Rn$⁠.

This implies (by Theorem 4.4, p. 129 of Ref. 37) that the maps $cQℏW$ likewise define a strict quantization. Thus, we can use the maps $cQℏW$ to provide a definition of the Berezin quantization on the minimal algebraic structure of the Weyl algebra. Since Berezin quantization is positive, and hence continuous, we can extend these maps to unbounded operators defined from the Weyl algebra.

Our goal is to use the quantization maps $cQℏW$ to analyze the classical limits of unbounded operators such as Φ_ℏ(a, b). To do so, we note that these operators can be constructed from the unitary generators W_ℏ(a, b) of the Weyl algebra by the formal relation

This relation holds strictly in the Schrödinger representation when the limit is understood in the weak operator topology on $B(L2(Rn))$⁠. However, the limit does not, in general, converge in the abstract weak topology on $W(R2n,ℏσ)$⁠. In the service of our goal of analyzing quantization in a representation manner, we seek a different abstract algebra with a natural topology in which these limits converge. Then, we will be able to use Eq. (1) as a definition of Φ_ℏ(a, b) solely in terms of abstract algebraic structure.

To construct such an algebra, we will form the quotient algebra by a certain two-sided ideal. Our ultimate goal is to find an algebra that allows only for states whose expectation values of Eq. (1) to converge. It is known^38,39 that one can limit the collection of states of an algebra if one chooses to quotient by an ideal that is the annihilator of the set of states one wants to focus on. More precisely, given a C^*-algebra $B$ and a collection of functionals $V⊆B*$⁠, under certain conditions on V, one can construct a new C^*-algebra $A$ whose dual space contains only the functionals in V, i.e., $A*≅V$⁠. To do so, first let N(V) denote the annihilator of V in $B$⁠. If N(V) is a closed, two-sided ideal in $B$⁠, then setting $A=B/N(V)$ produces a C^*-algebra with the desired dual space.

This is relevant to the current circumstance if we focus on the states on the Weyl algebra for which the expectation values of Eq. (1) converge. To that end, we focus on the so-called regular states and define

However, in this case, N(V_ℏ) is not a closed, two-sided ideal in $W(R2n,ℏσ)$ because the latter algebra is simple. Hence, we move to the bidual $W(R2n,ℏσ)**$ and consider the weak^* closure $V¯ℏ⊆W(R2n,ℏσ)***$ of V_ℏ, understood now as the regular functionals on the bidual. It follows that the annihilator $N(V¯ℏ)$ in $W(R2n,ℏσ)**$ is now a closed, two-sided ideal. Hence, we can complete the construction by defining a quotient C^*-algebra $Aℏ≔W(R2n,ℏσ)**/N(V¯ℏ)$⁠.

It follows that $Aℏ*≅V¯ℏ$⁠. Moreover, the algebra $A0$ is *-isomorphic to the algebra $BR(R2n)$ of bounded universally Radon measurable functions, and the algebras $Aℏ$ are *-isomorphic to $B(L2(Rn))$ for each ℏ > 0.³⁹ Thus, we have canonical projection (quotient) maps $p0:AP(R2n)**→BR(R2n)$ and $pℏ:W(R2n,ℏσ)**→B(L2(Rn)$⁠, and even further, we have $BR(R2n)≅C0(R2n)**$ and $B(L2(Rn))≅K(L2(Rn))**$ so that both algebras are W^*-algebras carrying natural weak^* topologies. The families t ↦ p_ℏ(W_ℏ(ta, tb)) are weak^* continuous in $Aℏ$ for all ℏ ∈ [0, 1], so the limit in Eq. (1) is well-defined in the weak^* topology.

We are now in a position to consider the functions Φ₀(a, b) in the domain of our quantization maps. To do so, we continuously extend $cQℏW$ in the weak topology to a map $cQ̃ℏW:AP(R2n)**→W(R2n,ℏσ)**$⁠. We have the following corollary of Proposition 1.

Suppose now that E is an infinite dimensional vector space with a symplectic form σ. This is the case when E is the test function space for any free Bosonic field theory whose phase space E′ is a linear space. Although the integral formulas defining Berezin quantization in Sec. III A are no longer meaningful in this context, we proceed to construct an analogous positive quantization, which can likewise be extended to unbounded operators.

We start with the Weyl quantization maps $QℏW$⁠, which are well-defined even in the infinite-dimensional setting, and we aim to define quantization factors in the spirit of Sec. III A. We require a norm on E, which may be determined as follows. Suppose that we are given a complex structure J : E → E compatible with σ—that is, a linear map satisfying

• σ(JF, JG) = σ(F, G);

• σ(F, JF) ≥ 0; and

• J² = −I

for all F, G ∈ E. In general, there is no such unique complex structure; we will see concrete examples below. A complex structure can be used to define a complex inner product

for all F, G ∈ E. This inner product α_J allows us to define quantization factors $cℏJ:E→R+$ by $cℏJ(F)≔e−ℏ4αJ(F,F)$⁠. These quantization factors satisfy the same conditions (a)–(c) of Sec. III A. Now, in analogy with Sec. III A, we define new quantization maps $QℏJ:Δ(E,0)→W(E,ℏσ)$ by the linear extension of

For any choice of complex structure J, this defines a strict quantization equivalent to $QℏW$ in the sense that (see Theorem 4.6, p. 131 of Ref. 37)

for all A ∈ Δ(E, 0). It follows that the strict quantizations defined for different choices of complex structure J and J′ are also all equivalent in this same sense as

for all A ∈ Δ(E, 0).

One can show that the quantization maps $QℏJ$ possess some of the same virtues as the Berezin quantization maps of Sec. III A.

The positivity of $QℏJ$ implies its continuity in the norm and weak topologies (see Proposition 1.3.7 of Ref. 2, p. 47), which means that it can be continuously extended to the completions of its domain and range. As in Sec. III A, we want to use these extended quantization maps to analyze field operators of the form

However, these limits again do not converge in the weak topology. So we must perform the construction of Sec. III B to arrive at a new algebra allowing for this definition.

To construct such an algebra, we again quotient out by a certain two-sided ideal given by the annihilator of a desired set of states. We again focus on the states for which the expectation values of Eq. (2) converge by defining the set of regular states as

Just as before, N(V_ℏ) is not a closed, two-sided ideal because $W(E,ℏσ)$ is simple. Instead, we use the strategy of Sec. III B by passing to the bidual $W(E,ℏσ)**$ and letting $V¯ℏ$ be the weak^* closure of V_ℏ in $W(E,ℏσ)***$⁠. Then, $N(V¯ℏ)$ is a closed, two sided ideal, so we can define a C^*-algebra $Aℏ≔W(E,ℏσ)**/N(V¯ℏ)$ exactly as before.

However, since E is now infinite-dimensional and so fails to be locally compact, the structure of these algebras $Aℏ$ is not as tractable, and we have much less information than in Sec. III B. Still, we can show that the algebras $Aℏ$ are W^*-algebras with an appropriate weak^* topology.

This implies that $QℏJ$ extends continuously to a map whose codomain is the weak^* completion $Ãℏ$⁠, which we now denote $Q̃ℏJ:Ã0→Ãℏ$⁠. The field operators are well-defined in these completed algebras via Eq. (2) with the limit now understood in the weak^* topology. Now we can use the maps $Q̃ℏJ$ to analyze the classical limits of unbounded field operators.

First, we note that the familiar facts about the field operators Φ_ℏ(F) follow from what has been said so far. We present proofs here to emphasize the fact that these statements can be both expressed and derived in the bare algebraic setting we have outlined.