Classical and quantised resolvent algebras for the cylinder

Buchholz and Grundling (Comm. Math. Phys., 272, 699--750, 2007) introduced a C$^\ast$-algebra called the resolvent algebra as a canonical quantisation of a symplectic vector space, and demonstrated that this algebra has several desirable features. We define an analogue of their resolvent algebra on the cotangent bundle $T^*\mathbb{T}^n$ of an $n$-torus by first generalizing the classical analogue of the resolvent algebra defined by the first author of this paper in earlier work (J. Funct. Anal., 277, 2815--2838, 2019), and subsequently applying Weyl quantisation. We prove that this quantisation is almost strict in the sense of Rieffel and show that our resolvent algebra shares many features with the original resolvent algebra. We demonstrate that both our classical and quantised algebras are closed under the time evolutions corresponding to large classes of potentials. Finally, we discuss their relevance to lattice gauge theory.


Introduction
Much of modern physics concerns the search for and examination of quantum versions of known classical theories.Examples include quantum statistical mechanics, quantum field theory, and quantum gravity.Showing that a classical theory is indeed the limit of the quantum theory at hand can be done at various levels of rigour.The most precise way to establish this limit is by strict deformation quantisation, where one 'quantises' a classical (commutative) Poisson algebra into a quantum (noncommutative) C*-algebra [20,15] (cf.[12, p. 5] for an overview of the various definitions in the literature).
Only few pairs of a classical and a quantum C*-algebra are known to connect in this rigorous fashion [2,15,21,22], and each has its merits and drawbacks.In particular, when taking the torus as a configuration space, we found the known examples too limited in certain respects.Hence, in this paper, we define a quantum observable algebra on the torus, i.e. a C*-algebra A ⊆ B(L 2 (T n )), which satisfies the following properties: P1: The algebra A has a classical counterpart A 0 and can be obtained from this commutative algebra through (strict) quantisation.
Lattice gauge theory In the Hamiltonian lattice gauge theory by Kogut and Susskind [14], one approximates a time-slice of spacetime by a finite 'lattice', or more accurately, an oriented graph Λ.The elements of the set of vertices Λ 0 are points in the time slice, while the set of oriented edges Λ 1 are paths between these points.A gauge field corresponding to some connection on a principal fibre bundle over spacetime with structure group some Lie group G is approximated by the parallel transport maps along the edges of Λ.After choosing a trivialisation of the restriction of the principal fibre bundle to Λ 0 , the set of all possible parallel transporters can be identified with G Λ 1 ; this is the configuration space of the Hamiltonian lattice gauge theory, and it carries a natural action of G Λ 0 (endowed with the obvious group structure).This group is the analogue of the set of gauge transformations.
Let us take a brief moment to comment on the choice of the structure group G of the gauge theory.Lattice gauge theory was originally introduced by Wilson [28] to explain the phenomenon of quark confinement in the context of the gauge theory known as quantum chromodynamics (QCD).The underlying structure group of QCD is SU(3), hence the corresponding configuration space is evidently not a torus and therefore lattice QCD is outside of the scope of this article.However, it is worth noting that the structure group of electromagnetism is T, so the results in this paper may be applied to corresponding lattice gauge theories, or perhaps serve as a stepping stone towards an analogous construction that can be applied to lattice QCD.
We now return to the argument.The Hilbert space of the corresponding quantum lattice gauge theory is H = L 2 (G Λ 1 ), where G Λ 1 is endowed with the normalised Haar measure.The field algebra of the system is some C * -algebra A Λ that is represented on H, from which the observable algebra can be obtained by applying a reduction procedure with respect to the gauge group (cf.[13,25]).The observable algebra is accordingly represented on the set of elements of H that are invariant under gauge transformations.
Since the distinction between field and observable algebras is irrelevant with regard to the issue that motivates the present investigation -the embedding maps take the same form in both cases -we will continue to refer to A Λ as the observable algebra in what follows.
In the context of lattice gauge theory, one is interested in constructing an algebra of the continuum system from the above algebras A Λ .This is done by considering direct systems of lattices, and we are naturally led to consider the following situation.Suppose that Λ 1 and Λ 2 are both lattices approximating a time slice, and that Λ 2 is a better approximation than Λ 1 , i.e., Λ 0 1 ⊆ Λ 0 2 , the graph Λ 2 contains more paths than Λ 1 , and each edge in Λ 1 can be written as a concatenation of paths in Λ 2 ; for a precise definition, we refer to [1].We should then be able to find a corresponding embedding map A Λ 1 → A Λ 2 .The embedding map takes a simple form if Λ 2 is obtained from A Λ 1 by only adding edges: in that case, we have ), and the embedding is given by the restriction of the map to A Λ 1 , where 1 denotes the identity on H c 1 , and ⊗ denotes the von Neumann algebraic tensor product.
A first guess for the observable algebras of the two quantum systems could be K(H 1 ) and K(H 2 ), the algebras of compacts.However, except in trivial cases, the Hilbert space H c 1 will be infinite-dimensional, which means that a ⊗ 1 will not be a compact operator.Thus the algebra K(H 2 ) is too small to accommodate these embeddings.This problem was already noticed by Stottmeister and Thiemann in [26].In an earlier paper [1] on Hamiltonian lattice gauge theory coauthored by one of the authors of the present article, the above problem was not encountered since different embedding maps were used.There are nevertheless good reasons to believe that the embedding maps used in [26] are the correct ones, though we will not elaborate on them here, and refer to [24,Chapter 8] instead.The argument presented there is not specific to lattice gauge theory, but can be made for any physical system that is comprised of smaller subsystems.
Another guess for the observable algebra of the composite system could be the one generated by the embedded algebras of all subgraphs, as is done in [11].However, this raises questions about regulator independence of this procedure in situations where one takes limits corresponding to an infinite volume or continuum limit of a collection of systems parametrised by a cutoff.As this problem is beyond the scope of the present article, we will refer the reader to the discussion in [24, section 5.1].The main point is that there is ample reason to try to solve the problem through an appropriate choice of algebras, i.e., algebras that satisfy P3.
The resolvent algebra on R n In the case where the configuration space is R n , there already exists an algebra satisfying P1, P2, P3 and P4, namely the resolvent algebra R(R 2n , σ n ).The resolvent algebra R(X, σ) on a symplectic vector space (X, σ) is a C * -algebra that was originally introduced by Buchholz and Grundling in [8], and subsequently studied in greater detail in [9] and [5] by the same authors.Before we adapt this algebra to the case of T * T n instead of R 2n as its underlying phase space, let us recall the main idea behind the construction of the resolvent algebra.
The resolvent algebra is constructed as the completion of a * -algebra with respect to a certain C * -seminorm [9,Defintion 3.4]; the * -algebra is defined in terms of generators and relations.To each pair (λ, f ) ∈ (R\{0}) × X, a generator R(λ, f ) is associated.Such a generator is thought of as the resolvent (depending on λ) corresponding to some unbounded operator φ(f ) associated to the vector f , where φ denotes a linear map from X to a space of operators on a dense subspace of a Hilbert space on which R(X, σ) can be represented faithfully.
For example, suppose that (X, σ) is R 2 endowed with the standard symplectic form.Then R(X, σ) admits a faithful representation on L 2 (R) such that the unbounded operators corresponding to the vectors (1, 0) and (0, 1) are the standard position and momentum operators respectively (up to a factor of in the latter case), see [9,Corollary 4.4 and Theorem 4.10].Both of these unbounded operators can be defined on the (invariant) dense subspace C ∞ c (R), on which they are essentially self-adjoint.For each f ∈ X, the generator R(λ, f ) is mapped to the bounded operator (iλ1 − φ(f )) −1 ; in particular, taking f = 0, we see that R(X, σ) is unital.The relations defining the * -algebra from which the resolvent algebra is constructed serve to encode the fact that R(λ, f ) behaves like the resolvent of the unbounded operator φ(f ), as well as the linearity of φ.Last but not least, the canonical commutation relations (CCR) are introduced by the defining relations of R(X, σ) in which the symplectic form appears, thereby justifying the term "canonical quantum systems" in the title of [9].
The resolvent algebra is not the only approach to the reformulation of the CCR in a framework based on bounded operators; another is obtained through exponentiation of the unbounded operators of interest, leading to the Weyl form of the CCR and the Weyl algebra.There is a bijection between certain classes of representations of these two algebras [9,Corollary 4.4].In particular, generators of the resolvent algebras can be expressed in terms of generators of the Weyl algebra by means of the Laplace transform, as is done in [8].By changing the representation in that definition to the usual representation on L 2 (R) of the Weyl algebra on R 2 , one obtains the representation mentioned earlier.
Buchholz and Grundling note that their resolvent algebra has some desirable qualities not shared by the Weyl algebra, such as the presence of observables corresponding to bounded functions in regular representations.Furthermore -and this is particularly relevant for this paper -the resolvent algebra associated to R 2 endowed with the standard symplectic form is closed under (quantum) time evolution for a much larger class of Hamiltonians than the Weyl algebra (cf.[9, Proposition 6.1]).The authors explain this as a consequence of the fact that their resolvent algebra contains resolvents of many Hamiltonians.Moreover, Buchholz has shown that the resolvent algebra is stable under dynamics in the context of oscillating lattice systems [6] and nonrelativistic Bose fields [7].
According to [9, Theorem 5.1], for any symplectic vector space (X, σ) and any decomposition S ⊕ S ⊥ of X into subspaces that are nondegenerate with respect to σ (and were S ⊥ denotes the complement of S with respect to σ), the resolvent algebra R(X, σ) naturally contains a copy of R(S, σ) ⊗ R(S ⊥ , σ); with respect to corresponding faithful representations of these three resolvent algebras, the embeddings of R(S, σ) and R(S ⊥ , σ) are given by the analogues of the aforementioned embedding map for lattice gauge theory.Here, ⊗ denotes any C*-norm (nuclearity of the resolvent algebra is shown in [5]), and σ by abuse of notation denotes the symplectic form on X, as well as its restrictions to S and S ⊥ .
We have seen how properties P3 and P2 where shown by Buchholz and Grundling to hold for the resolvent algebra.Now for P1 the question is whether it arises as the strict quantisation of an algebra that can be considered the observable algebra of a classical system in the sense of Landsman, i.e., the C * -algebra generated by the image of a dense Poisson subalgebra of the classical algebra under a quantisation map [15].This question was answered affirmatively by one of the authors of this paper in [19], where it is shown that in the case where (X, σ) is R 2n endowed with the standard symplectic form, there is a corresponding commutative C * -algebra C R (R 2n ), which is the C * -subalgebra of C b (R 2n ) generated by functions of the form where • denotes the standard inner product.Similar to the way in which the algebra C 0 (R 2n ) may be quantised into the compact operators on L 2 (R n ) by considering the dense Poisson subalgebra S(R 2n ) of Schwartz functions and defining Weyl or Berezin quantisation on them, an analogue of the space of Schwartz functions for C R (R n ) is identified as follows.First, for every linear subspace V ⊆ R n , let P V denote the orthogonal projection onto V .Then the space is defined.It is readily seen that this is a dense Poisson subalgebra of C R (R n ) that is closed under the * -operation of complex conjugation.The Weyl quantisation of g • P V is defined using the Fourier transform of g as a function on V [19, section 3.2], but is otherwise equal to the definition of the Weyl quantisation of ordinary Schwartz functions on R n .It is then argued that the Weyl quantisation map admits a (unique) linear extension to S R (R n ).Furthermore, it is shown that the images of S R (R n ) under Weyl and Berezin quantisation are both dense subspaces of R(R 2n , σ).The resulting algebra C R (R 2n ) is accordingly referred to as the classical resolvent algebra on R 2n .As is shown in [19], these definitions are easily extended to spaces of functions whose domain is an inner product space of arbitrary dimension.
In addition to being the classical counterpart of the resolvent algebra as defined by Buchholz and Grundling, the classical resolvent algebra offers an interesting perspective on our earlier discussion on embeddings of observable algebras.In some sense whilst also containing its analogues associated to linear subspaces of R n .This may be formalised as follows.Consider the category whose objects are finite-dimensional real vector spaces, and whose morphisms consist of projections of a vector space onto one of its subspaces.Then there is a contravariant functor C b from this category to the category of C * -algebras that maps an object V to the space C b (V ), and that maps morphisms to their pullbacks between these spaces.It is now consistent with the definition of the classical resolvent algebra to define C R as the smallest subfunctor of C b with the property that the image of every object V contains C 0 (V ).Note that this implies that C R (R n ) is unital, as it contains the embedding of C 0 ({0}).This makes precise in which sense P4 holds for the resolvent algebras on R 2n .
Resolvent algebras on the cylinder In this paper, we define an analogue of the resolvent algebra for the cotangent bundle T * T n ∼ = T n × R n of the n-torus T n .Our approach differs significantly from that of Buchholz and Grundling, in that we do not define it in terms of generators and relations.Rather, we first identify a classical resolvent algebra C R (T * T n ) using ideas from [19], and indicate how this definition may be generalised.We then give a concrete characterisation of C R (T * T n ).Namely, identifying In addition, we identify a dense * -subalgebra S R (T * T n ) ⊆ C R (T * T n ) carrying a natural Poisson structure.The algebra is spanned by functions of the form e k ⊗h, where e k [x] := e 2πik•x , and h is a smooth function that is a product of an element of S R (R n ) and a function of the form x → e iξ•x for some ξ ∈ R n .
To define a quantum counterpart, we apply Weyl quantisation, making P1 integral to the definition of the (quantum) resolvent algebra on T * T n .Our Weyl quantisation map Q W : S R (T * T n ) → B(L 2 (T n )) is defined with an integral formula inspired by [22].When writing C R (T * T n ) as a tensor product as above, Q W can be characterised by the formula where e k ⊗ h ∈ S R (T * T n ), and ψ k is e k viewed as an element of L 2 (R n ) for each k ∈ Z n .The above formula is consistent with the usual Weyl quantisation on (a Poisson *subalgebra of) the smaller classical algebra C 0 (T * T n ), see e.g.[15, section II.3.4], as T n is in particular a Riemannian manifold with its corresponding Levi-Civita connection.Although this consistency already justifies (2) as a reasonable extension of Weyl quantisation, we start section 4 with a systematic way to arrive at (2).Thereafter, we define the (quantum) resolvent algebra on the torus as before remarking that A ∼ = A for all , ∈ (0, ∞).We check P3 by using this explicit description of Q W and the fact that P3 holds for C R (T * T n ), which is readily seen.P4 is satisfied by definition of C R (T * T n ).In addition, we show that an analogue of P2 holds for our algebras, both the classical and the quantum one, in the following very strong sense: our classical resolvent algebra C R (T * T n ) is closed under the classical time evolution associated to the potential V for each V ∈ C 1 (T n ) with Lipschitz continuous derivative.Our quantum resolvent algebra is closed under the quantum time evolution associated to the potential V for each V ∈ C(T n ).(In both cases, the free part of the Hamiltonian is the usual one.)Unlike the analogous result in [9] in which a similar result is established only for R 2n with n = 1, we give proofs of these statements for arbitrary n ∈ N.
The paper is structured as follows.In section 2, we first give a well-motivated definition of the classical resolvent algebra C R (T * T n ).We proceed by analysing its structure, culminating in an alternative, more practical characterisation of C R (T * T n ), namely as the tensor product C(T n ) ⊗ W 0 R (R n ).Furthermore, we identify a dense Poisson *subalgebra that serves the same purpose as S R (R n ) in [19].
Section 3 proves the fact that C R (T * T n ) is closed under the classical time evolution as mentioned above.
In section 4, we adapt Weyl quantisation to functions on T * T n , proving an explicit formula for generators of C R (T * T n ) in the process.This formula is then used to show that Weyl quantisation is almost a strict quantisation in the sense of Landsman.We say 'almost', because we explicitly show that its norm fails to be continuous with respect to for > 0. However, the quantisation map is continuous in a weaker sense.
In section 5, we show that our quantised resolvent algebra is closed under the quantum time evolution.

Definition and basic results
On the phase space R 2n , we already have a commutative C*-algebra that satisfies P2, P3 and P4 mentioned in the introduction and forms the classical part of a strict deformation quantisation, namely the commutative resolvent algebra C R (R 2n ) defined in [19].We begin this section by adapting its definition to T * T n .As mentioned in the introduction, we identify T * T n with T n × R n , and note that the latter space carries a natural left action of R 2n = R n × R n by translation.
the space of orbits of the restriction of the action of R 2n to {(v, w)} ⊥ ⊆ R 2n , and let be the corresponding canonical projection.We then define the commutative resolvent algebra that is, the set of continuous functions invariant under the action of {(v, w)} ⊥ ⊆ R 2n for which the induced map on (T n × R n )/{(v, w)} ⊥ vanishes at infinity.
To establish the link with the definition of C R (R n ) given in [19], note that there is a straightforward generalisation of the above definition to arbitrary topological spaces M carrying a left action of R m for some m ∈ N. Taking M = R n and m = n then yields the definition of C R (R n ).Unfortunately, T * G does not have an appropriate action of R 2n for a nonabelian Lie group G that would enable us to unambiguously generalise this construction.
The definition of the classical resolvent algebra C R (T * T n ) is clearly motivated, but very unwieldy in practice.Our first task is therefore to find an alternative, more elementary characterisation of C R (T * T n ).To this end we will use the following elementary facts about the action of R n on T n .Throughout the rest of the text, we let (1) Exactly one of the following two statements holds true: (2) Suppose now that t → [tv] is periodic, with period T .Furthermore, let π v : T n → T n /H be the quotient map.Then H is a closed subgroup of T n , and is a well-defined isomorphism of topological groups. Proof.
(1) We show that at least one of the two statements is true; we postpone the proof that the two statements are mutually exclusive to the proof of the second part of this lemma.The case n = 1 is trivial, and the case n = 2 is the well-known result that a line in T 2 is dense iff it has irrational slope.We therefore assume that n > 2, and we will reduce the problem to the known two-dimensional case.Suppose that (ii) is false for n > 2, and let U ⊆ T n \H be a non-empty open subset.Without loss of generality we may assume that U = [y] + U for all y ⊥ v.As v = 0 by assumption, we may choose j such that v j = 0. Now suppose k is a different index with v k = 0. Define which is a subgroup of T n isomorphic to T 2 .Note that To see that U ∩ T 2 is non-empty, let us pick [x] ∈ U .Because v j = 0, there exists a y ⊥ v such that [y + x] ∈ T 2 (for instance y = v•x v j δ j − x, where δ j is the j th standard basis vector).Since (2) We first note that the map ϕ is induced (in two steps) by the continuous group homomorphism Since T v ∈ Z n , this map factors through T n , thereby inducing a continuous group homomorphism ϕ 0 : T n → T. It is readily seen that H is a subgroup of T n that is a subset of ker ϕ 0 , so ϕ 0 factors through the quotient T n /H, thereby inducing the continuous group homomorphism ϕ.
Next, we argue that ϕ is in fact a homeomorphism.We prove this by showing that the map is a well-defined inverse; we will tentatively refer to this map as ϕ −1 in what follows.
First we show that ϕ −1 is well-defined.This amounts to showing that which is the case exactly when v/(T v 2 ) is an element of the set on the right-hand side of the inclusion.Note that the components of the vector T v are coprime; otherwise, T /m would be the period of t → [tv] for some natural number m > 1.By the higherdimensional Bézout identity, there exists a tuple a ∈ Z n such that and note that the first and second term on the right-hand side of this equation are contained in Z n and {v} ⊥ , respectively.Thus ϕ −1 is well-defined.It is straightforward to check that ϕ −1 is both a left-and a right-inverse of ϕ, so ϕ −1 is indeed the inverse of ϕ.It is readily seen that ϕ −1 is continuous, so ϕ is both a group isomorphism and a homeomorphism.
To see that ϕ is in fact an isomorphism of topological groups, note that H = ker ϕ 0 by injectivity of ϕ, so H is a closed subgroup of T n , and the quotient T n /H naturally inherits the structure of a topological group from T n ; in particular, the quotient is Hausdorff.This concludes the proof of part (2).
Finishing up the proof of part (1), we note that in case (i), the quotient T n /H is homeomorphic to T, whereas in case (ii), the quotient is an indiscrete space.Thus the two cases are mutually exclusive.Remark 3. In case (i), H is a Lie subgroup of the Lie group T n by the closed-subgroup theorem.In fact, ϕ is an isomorphism of Lie groups from T n /H to T endowed with their respective canonical Lie group structures.See [24,Lemma 5.4] for a Lie-theoretic version of the previous lemma.
Before we characterise C R (T * T n ), we must introduce another algebra, for which it is in turn useful to recall that the algebra of almost periodic functions on R n is the C *subalgebra of C b (R n ) generated by functions of the form x → e iξ•x , where ξ ∈ R n is arbitrary.Almost periodic functions were originally introduced by H. Bohr in [3] for n = 1 using a different definition, whose equivalence with the one mentioned above he proved in [4].This algebra will be denoted by W 0 (R n ).
Next up is the main result of this section, which unveils C R (T * T n ) as a tensor product of two algebras.We regard the algebraic tensor product of two C* , and denote its corresponding completion by A ⊗ B. Since commutative C * -algebras are nuclear (cf.[18,Theorem 6.4.15]), this is equivalent to any other C*-algebraic tensor product.
By part (1) of Lemma 2 we may distinguish between the following three cases.In each of these cases, we obtain the general form of f by first giving a characterisation of the quotient space (T n × R n )/H : is an isomorphism of topological groups.It follows that f is the pullback of a function in C 0 (R • w) along the above map, from which it is readily seen that To handle the remaining two cases in which v = 0, we introduce the map and show that it is a well-defined group isomorphism and a homeomorphism.To see that it is a well-defined continuous group homomorphism, note that it is induced by a continuous group homomorphism which is defined using a similar formula, and whose kernel contains the subgroup H .
To see that θ is a group isomorphism and a homeomorphism, we note that the map is a well-defined countinuous group homomorphism (by a similar argument as for θ) that can be checked to be the inverse of θ.In particular, H = ker θ 0 .As we will see below, θ need not be an isomorphism of topological groups if we require such groups to be Hausdorff spaces.We proceed with the remaining two cases: (ii) v = 0 and H is dense in T n : in this case, the quotient topology on on T n is periodic: then the map θ 0 defined above is a continuous surjective group homomorphism, hence its kernel H is a closed subgroup of T n × R n , and the map θ is an isomorphism of topological groups.Composing θ with the map ϕ from part (2) of Lemma 2, we obtain the isomorphism of topological groups with T as defined in Lemma 2. Then Since the family of exponential functions (e k ) k∈Z generate C(T), and since pullback along the map for arbitrary g ∈ C(T).

This establishes the inclusion
The reverse inclusion is a consequence of the following three observations: • From case (i) in the previous part of this proof, we readily obtain • From case (iii), setting w = 0 and taking v to be a standard basis vector of R n , we obtain • Finally, by considering case (iii) again, but now with v the first standard basis vector and w ∈ R n arbitrary, we see that C R (T * T n ) contains functions of the form where k ∈ Z\{0}, and ξ ∈ R n is arbitrary.The previous point now implies that functions of the form are elements of the resolvent algebra, so We finish this section by defining the analogue of the space of Schwartz functions of C R (T * T n ).This allows us to introduce the notation h U,ξ,g for the generators of W 0 R (R n ), which will be used in section 4.
For each subspace U ⊆ R n , for each ξ ∈ U ⊥ , and for each Schwartz function g ∈ S(U ), let h U,ξ,g : R n → C, p → e iξ•p g(P U (p)) , where P U : R n → U denotes the orthogonal projection onto U .We define the space S R (T * T n ) as the span of functions of the form e k ⊗ h U,ξ,g : (1) The space span{h U,ξ,g : ) that is closed under multiplication and partial differentiation.
(2) The vector space S R (T * T n ) is a subspace of C R (T * T n ) that is closed under multiplication and partial differentiation, and is consequently a Poisson subalgebra of Proof.
(1) Denote B := span{h U,ξ,g } ⊂ W 0 R .For any h U,ξ,g as in Definition 6, hence B is closed under the * -operation.Assume for the moment that B is closed under multiplication.To see that B is invariant under partial differentiation, it suffices to show that partial derivatives of functions of the form h U,ξ,g are elements of B. Any partial derivative can be written as a sum of two directional derivatives; one in a direction lying in U , and one in a direction lying in U ⊥ .It is readily seen that both of these directional derivatives are elements of B, hence so is their sum.
To show that B is closed under multiplication, it suffices to show that the product of two functions h U 1 ,ξ 1 ,g 1 and h U 2 ,ξ 2 ,g 2 as in Definition 6, is an element of B. Let Note that the restrictions of g to U and U ⊥ are Schwartz and constant, respectively.Setting Thus B is a * -subalgebra of W 0 R (R n ).In addition to this fact, the elements of the form h {0},ξ,1 generate W 0 (R n ), while the elements of the form h U,0, (2) For each k ∈ Z n , define e k as in Definition 6.It is a trivial matter to check that the linear span of {e k : k ∈ Z} is a * -subalgebra of C(T n ) that is closed with respect to partial differentiation, and it is a result from Fourier analysis that this linear subspace is dense in C(T n ).Using these facts in conjunction with part (1) of this proposition and Theorem 5, it is readily seen that all of the assertions are true.

Classical time evolution
In this section, we prove that C R (T * T n ) is preserved under the (time) flow induced by the Hamiltonian H(q, p) = 1 2 p 2 + V (q) , for each potential V ∈ C 1 (T n ) R such that ∇V is Lipschitz continuous.This is arguably the most natural assumption on V ; the Picard-Lindelöf theorem then ensures that the Hamilton equations have unique solutions.
Note that the expression on the right-hand side of the first line of equation ( 3) is the Hamiltonian vector field X H corresponding to H evaluated at (q(t), p(t)).For each t ∈ R, the time evolution of the system after time t is the map which is the flow corresponding to X H evaluated at time t; it is well-known to be a homeomorphism.
Note that we have already made the notation of the flow less cumbersome by writing In what follows, we restrict our attention to the case t = 1, further simplifying the notation by defining Φ V := Φ 1 V .The following lemma shows that we may do so without loss of generality: Proof.For any t = 0 (as t = 0 is trivial), we make the following transformation on phase space φ(q, p) := (q, tp) .
Because the momentum part of φ is linear, its pullback preserves the commutative resolvent algebra.Given an integral curve (q(t), p(t)) of the vector field X H corresponding to the potential V , i.e., a solution of equation ( 3), one can easily check that s → φ(q(ts), p(ts)) is an integral curve corresponding to the potential t 2 V .We therefore conclude that We prove our main theorem in three steps: taking V = 0; taking V trigonometric; and finally taking general V .In the second and third step we will need the following consequence of Gronwall's inequality.Let d denote the canonical distance function on T n as well as on T n × R n .(Note that these distance functions are the ones induced by the canonical Riemannian metrics on T n and T * T n ∼ = T n × R n , respectively.)Lemma 9. Let f, g : T n × R n → R 2n be Lipschitz continuous functions, let c be the Lipschitz constant of f , and let y, z : [0, 1] → T n ×R n be curves that satisfy ẏ(t) = f (y(t)) and ż(t) = g(z(t)) Proof.By translation invariance of the metric on T n × R n , we have With the integral version of Gronwall's inequality, this implies the lemma.

Free time evolution
For each pair (q 0 , p 0 ) ∈ T n × R n , we have q(t) = q 0 + tp 0 and p(t) = p 0 , denoting the usual action of R n on T n by +.The latter notation, explicitly written as [x] + p = [x + p] for x, p ∈ R n , will be used in the remainder of this section.We find that Φ 0 (q 0 , p 0 ) = (q 0 + p 0 , p 0 ), and obtain the following preliminary result.Let * denote the pullback.
Lemma 10.Free time evolution preserves the commutative resolvent algebra, i.e., 0 , and since this map is a * -homomorphism, the lemma follows.

Trigonometric potentials
We say that V is a trigonometric potential if it is real-valued and of the form V = k∈N a k e k , for some coefficients a k ∈ C and a finite subset N ⊆ Z n .The main trick used to establish time invariance of the classical resolvent algebra is to use induction on the size of N .The induction basis, N = ∅, corresponds to free time evolution.In order to carry out the induction step we fix a vector k ∈ N , and compare the dynamics corresponding to V with the dynamics corresponding to V − V k , where Similar to the already defined curves q : [0, 1] → T n and p : [0, 1] → R n , the dynamics corresponding to V − V k of the point (q 0 , p 0 ) is incapsulated by the curves q : [0, 1] → T n and p : (q(0), p(0)) = (q 0 , p 0 ) .
We compare the two dynamics in the following proposition.
Proposition 11.Let k ∈ Z n and δ > 0. There exists a D k > 0 such that for each Proof.Note that the statement is vacuously true for any D k > 0 if k = 0. We therefore fix a nonzero k ∈ Z n .Throughout the proof, we use a variation of big O notation, expanding in the variable ∆t := |k • p 0 | −1 , uniformly in q 0 .That is, we write f (q 0 , p 0 ) = O(∆t d ) if there exist N, C > 0 such that for all q 0 , p 0 with |k • p 0 | > N we have |f (q 0 , p 0 )| ≤ C|k • p 0 | −d .Therefore, to prove the proposition, it suffices to show that Assume that ∆t ∈ (0, 1).We divide the time interval [0, 1] into m intervals of length ∆t, where m := 1 ∆t , and a final interval of length 1 − m∆t.For each t ∈ [0, ∆t] and each j ∈ {0, . . ., m} (these will be the assumptions on t and j throughout the rest of the proof), let q j (t) := q(j∆t + t) , p j (t) := p(j∆t + t) , and define the curves qj and pj analogously.Note that (q j , p j ) and (q j , pj ) satisfy the differential equations ( 3) and ( 4) respectively, but with different initial conditions.Furthermore, for every j, we define the curve γ j : [0, ∆t] → T n as the unique solution to the initial value problem (γ j (0), γj (0)) = (q j (0), p j (0)) .
where on the first line, we have emphasised the similarity of this equation with the equations ( 3) and ( 4) by including γj (t).We do not introduce any special notation for γj , however.As depicted in Figure 1, the curve γ j : [0, ∆t] → T n plays a key rôle in comparing q j with qj ; the curve (γ j , γj ) is an integral curve along the same Hamiltonian vector field as (q j , pj ), but with the same initial conditions as (q j , p j ).We now expand our expressions in orders of ∆t.Using equation ( 3) and the fundamental theorem of calculus, we obtain In particular, taking t = ∆t, we get p j+1 (0) − p j (0) = O(∆t), and therefore by induction Figure 1: The position functions q j ,γ j and qj .Sloping lines correspond to V − V k , whereas the horizontal line that depicts q corresponds to V .for every 0 ≤ j ≤ m.Equations ( 7) and ( 8) give us d(q j (t), q j (0 A result similar to (7) exists for γj instead of p j , and hence p j (t) − γj (t) = O(∆t) , (10) which implies Using the definitions of V k and ∆t, we show that the distance between p j (∆t) and γj (∆t) is in fact of order ∆t 2 .We first note that By (11), the first term is O(∆t 3 ).For the second term we can use (9) and the observation that ∆t 0 ∇V k (q j (0) + sp 0 ) ds = 0 .
Hence the second term is O(∆t 2 ).All in all, we obtain the estimate This estimate, together with (11), implies Since γ j and qj satisfy the same differential equation, say with associated Lipschitz constant c, Lemma 9 (with f = g : (q, p) Taking t = ∆t, we by definition have Combining ( 12) and ( 14), we find that .
Because e jc∆t = O(1), repeated use of the above equation gives Let t := 1 − m∆t.Using (13), we find .
The first term is O(∆t 2 ) by ( 11), the second is O(∆t) by (10), and the last term is O(∆t) by ( 15).This implies (5), and thereby the proposition.
Proposition 11 expresses a property of the classical time evolution associated to a trigonometric potential in terms of points in phase space.To translate this result to the world of observables, we fix > 0 and notice that any g ∈ C R (T * T n ) is uniformly continuous.Hence for every k ∈ N we may fix a D k such that sup where We also define the opens Proof.Let S := span R N .We write our phase space as a product of topological spaces and note that On the other hand, regarding our phase space as a coproduct of abelian Lie groups we define φ t as the restriction of Φ t V to T n × S for each t ∈ R. Because ∇V ⊥ S ⊥ , we have ṗ(t) ⊥ S ⊥ , and hence is a well-defined homeomorphism.Moreover, we find the equation because its two sides solve the same differential equation.Using the above relation in a straightforward calculation on generators, one can show that Actually, the same holds for Φ −1 V , which implies that Φ * V is a *-automorphism of the ideal C 0 (T n ×S)⊗W 0 R (S ⊥ ).Now note that U ∞ is of the form K ×S ⊥ for some compact subset K ⊆ T n × S. By Urysohn's lemma, we may choose a function g We then find that and therefore We can finally prove that our classical resolvent algebra is invariant under any time evolution corresponding to a trigonometric potential.
Proposition 13.For every trigonometric potential V : Proof.We use induction on the size of N in V = k∈N a k e k (while assuming that N is chosen minimally).The induction base is precisely Lemma 10.
We now carry out the induction step.The induction hypothesis says that time evolution with respect to Fixing f ∞ as in Lemma 12, we have f i ∈ C R (T * T n ), and equation ( 16) implies that for each i ∈ I = N ∪ {∞}.We now construct a partition of unity {η i } subordinate to the open cover {U i } of T n × R n , to patch together the functions {f i } and obtain a single function in C R (T * T n ).We start by defining nonnegative functions ζ i ∈ C R (T * T n ) that are 1 on W i and 0 outside of U i .Explicitly, for each k ∈ N , we take ζ k := 1 T n ⊗ (g k • P span(k) ) for some bump function g k on span(k), and we take and therefore every function Since ε > 0 was arbitrary and C R (T * T n ) is norm-closed, the assertion follows.

Arbitrary potentials
Having covered the trigonometric case, we now wish to tackle the general case.The following lemma provides the required approximation of a generic potential by trigonometric ones.
Then there exists a sequence (V m ) m of trigonometric polynomials such that (∇V m ) m converges uniformly to ∇V .Furthermore, if V is realvalued, then every V m can be chosen to be real-valued as well.
Proof.We construct the sequence (V m ) by taking the convolution of V with the ndimensional analogues of the family of Fejér kernels.We first recall that for each m ≥ 1, the m-th Fejér kernel is given by where the most right expression in this definition is understood to be equal to m for x = 0.The sequence (F 1,m ) m≥1 is an approximation to the identity, i.e., for every continuous function f on T, the sequence (F 1,m * f ) m≥1 converges uniformly to f , where * denotes the operation of convolution of functions [23, sections 2.4 and 2.5.2].Next, we define the n-dimensional analogues of these functions: Using the corresponding fact for one-dimensional kernels, it is elementary to show that the sequence (F n,m ) m≥1 is an approximation to the identity.We now define for each m ≥ 1.Because every F n,m is trigonometric, and e k * f = f (k)e k for every f ∈ C(T n ) and k ∈ Z n , the sequence (V m ) m≥1 consists of trigonometric polynomials.Moreover, by a general property of convolutions, we have and since (F n,m ) m≥1 is an approximation to the identity, the right-hand side converges uniformly to ∂V ∂q l as m → ∞, for l = 1, . . ., n.It follows that (∇V m ) m≥1 converges uniformly to ∇V .The final assertion is a consequence of the fact that the family of Fejér kernels (as well as its higher-dimensional analogues) consists of real-valued functions.
We now extend Proposition 13 to general V , thereby arriving at our final result.
Theorem 15.Let V ∈ C 1 (T n ) R , and suppose that ∇V is Lipschitz continuous.Then we have we can replace t by −t and note that (Φ −t V ) * is the inverse of (Φ t V ) * to obtain the reverse inclusion.By Lemma 8, we may assume without loss of generality that t = 1.
Let g ∈ C R (T * T n ).By Lemma 14, there exists a sequence of trigonometric potentials (V m ) on T n such that (∇V m ) converges uniformly to ∇V .We show that this implies that (Φ * Vm (g)) converges uniformly to Φ * V (g); since Φ * Vm (g) ∈ C R (T * T n ) by Proposition 13 and since C R (T * T n ) is norm-closed, the theorem will follow from this.

Quantisation of the resolvent algebra
Having shown the nice properties of C R (T * T n ), we now ask whether there exists a quantum version of this algebra.What complicates matters is that, contrary to the resolvent algebra R(R 2n , σ) of Buchholz and Grundling, on the cylinder it is hard, if not impossible, to define an algebra in terms of generators and relations implementing canonical commutation relations.Thus we must take a different approach.We will define our quantisation of the algebra C R (T * T n ) as an algebra represented on L 2 (T n ), using a version of Weyl quantisation similar to the definition of Landsman [15, section II.3.4] for general Riemannian manifolds.By contrast, Rieffel's algebras on cylinders in [22], apart from being quantisations of C u (T * T n ) and subalgebras thereof, are in some sense universal objects from which a physical quantum system is obtained as the image of one of its irreducible representations, and it is not always clear which representation corresponds to the physical system that one wishes to model.These algebras have many inequivalent irreducible representations due to the fact that T is not simply connected, see e.g.[22,Example 10.6] and the discussion in [16, section 7.7].By no means do we intend to discount such universal objects, however; we will return to this point in the outlook of this paper.The main advantage of quantising C R (T * T n ) as an algebra of operators on L 2 (T n ) lies in the explicit formula for the quantisations of the generators of C R (T * T n ) that we are able to derive.
This section is structured as follows.In subsection 4.1, we define the Weyl quantisation map and prove the aforementioned explicit formula.In subsection 4.2, we show that, except for continuity of the map → Q W (f ) at > 0 for fixed f ∈ C R (T * T n ), the quantisation is strict.

Definition of the quantisation map
Let us first recall the basics of Weyl quantisation in R 2n , the quantisation procedure in [27] conceived by Weyl.Given say, a Schwartz function f ∈ S(R 2n ), one associates an operator Q W (f ) ∈ B(L 2 (R n )) to it as follows.First, one expresses f in terms of functions of the form where a, b ∈ R n , by considering the Fourier transform of f .One subsequently substitutes these exponential functions with the operators e i(a•Q+b•P ) , where Q, P are vectors whose components are the essentially self-adjoint operators on S(R n ) ⊆ L 2 (R n ), defined by Q j ψ(x) := x j ψ(x) and P j ψ(x) := −i dψ dx j (x).Thus, the Weyl quantisation of a function f is informally given by the expression 2 e ia•(Q−q) e ib•(P −p) dq dp da db , where we take > 0. To define the above integrals rigorously, we can insert a function ψ ∈ S(R n ) on the right-hand side of the integrand, and check that the resulting expression is well-defined and that it defines a bounded operator on S(R n ) viewed as a subspace of L 2 (R n ).Since S(R n ) is dense in L 2 (R n ), the operator has a unique bounded extension to L 2 (R n ), which we define to be Q W (f ).Using standard identities for Fourier transforms of functions, and performing a number of substitutions, it can be shown that for each ψ ∈ S(R n ) and each x ∈ R n .We now adapt the Weyl quantisation formula to T * T n in such a way that we can quantise elements of C R (T * T n ).We already identified a dense Poisson algebra of C R (T * T n ) in Section 2, namely the space S R (T * T n ) of finite linear combinations of functions of the form e k ⊗ h U,ξ,g ; see Proposition 7.These are the functions that we will quantise.To handle such functions, we take inspiration from Rieffel's work [22], regarding the integrals in the above formula as oscillatory integrals, and regularising the expression by inserting a factor in the integrand in the form of a member of a net of functions that converges pointwise to the constant function 1 R n , as in part (1) of the next proposition.Part (2) of this proposition is the analogue of [22,Proposition 1.11].
(2) Now assume f = e k ⊗ h U,ξ,g is a function as described in Definition 6.Then the expression in equation ( 18) is equal to For each l ∈ Z n , let ψ l be the function and regard it as an element of L 2 (T n ).
(3) In addition to the assumptions in the previous part of the proposition, suppose that ψ = ψ l for some l ∈ Z n .Then the expression in equation ( 18) is equal to and the map defined on span l∈Z n {ψ l } sending ψ to the function on T n that assigns to a point [x] ∈ T n the limit in (18) extends in a unique way to a bounded linear operator on L 2 (T n ) with norm ≤ g ∞ .
Proof.We first show that for functions f of the form e k ⊗ h U,ξ,g , i.e., f as in part (2) of the proposition, the limit in equation ( 18) exists, and is equal to the formula in part (2) of the proposition.Since S R (T * T n ) is by definition the linear span of such functions, part (1) will follow from this.Thus, take such an f , and note that for any δ > 0, we have The inner integral over p can be written as a product of two integrals; one over U and one over U ⊥ : .
Inserting this back into the previous displayed formula, and splitting the outer integral in that formula into an integral over U and an integral over U ⊥ , we obtain where Now note that the family of functions indexed by δ > 0 is an approximation to the identity for functions on U ⊥ .By continuity of ψ, it follows that the functions converge pointwise to the function as δ → 0.Moreover, they are bounded, with h 3,δ ∞ ≤ ψ ∞ for each δ > 0. In addition, by the dominated convergence theorem, the functions h 1,δ converge pointwise to the function as δ → 0. Indeed, the integrands defining these functions are all dominated by the integrable function |g|.Furthermore, note that where ∆ U denotes the standard Laplacian on U , and that for the family of the functions indexed by δ ∈ (0, C], where C is an arbitrary positive real number, there exists a positive function H C ∈ L 1 (U ) dominating the entire family.It follows that for each δ ∈ (0, C] and each y 1 ∈ U , we have The (absolute values of the) functions are therefore dominated by the integrable function so we may again invoke the dominated convergence theorem to find that lim which completes our proof of part (2).
For part (3), we simply take , and apply the formula we just found: which proves the formula in part (3).We thus see that the linear map on span l {ψ l } uniquely determined by maps an orthonormal basis to an orthogonal system of vectors in L 2 (T n ), and the norm of the image of such a vector ψ l is less than or equal to (Note that the suprema defining these sup-norms are taken over U , R n and T n × R n , respectively.)Because of this and the fact that the ψ l 's densely span L 2 (T n ), the map extends in a unique way to a bounded operator on L 2 (T n ) with norm ≤ g ∞ , which proves the final assertion.
The proposition justifies the following definitions: Definition 17.For each > 0 and each f ∈ S R (T * T n ), we define the Weyl quantisation Q W (f ) of f to be the unique bounded linear extension of the operator on span l∈Z n {ψ l } defined by the formula We thus obtain a map, the Weyl quantisation map , for each > 0. We define the quantum resolvent algebra A on Part (3) of Proposition 16 can now be phrased as an explicit formula for the Weyl quantisation of a generator Proposition 18.Let > 0.
(1) The Weyl quantisation map is linear and *-preserving; (2) For each > 0, we have A = A ; (3) The image of (4) Under the canonical embedding induced by the projection at the level of configuration spaces T n+m → T n onto the first n coordinates, the image of the quantum resolvent algebra on T * T n+m is a subalgebra of the quantum resolvent algebra on T * T n .(Here, ⊗ denotes the von Neumann algebraic tensor product.) (5) Let ρ 0 be the group representation of T n on C b (T * T n ) given by and let ρ be the group representation of where L : T n → U (L 2 (T n )) denotes the left regular representation of T n .Then both C R (T * T n ) and S R (T * T n ) are invariant under ρ 0 .Furthermore, the Weyl quantisation map is equivariant with respect to these representations.Proof.
(1) Linearity of Q W is obvious from the definition.Now let e k ⊗ h be a generator of S R (T * T n ), and let be the Fourier transform.Here, •, • denotes the usual inner product on L 2 (T n ).We follow the physicists' convention, taking the inner product to be linear in its second argument.It follows from (19) that and M h 1 denotes the multiplication operator on 2 (Z n ) associated to the function Next, for each l ∈ Z n , we have where h 2 is defined as h 2 (l) := h(π (−k + 2l)).Also note that so by unitarity of the Fourier transform, we have (2) For each > 0, each f ∈ S R (T * T n ) and each ψ ∈ L, we have where we have made the substitution p = p in the formula defining Q W (f )ψ, and absorbed a factor 2 in δ.Next, we observe that S R (T * T n ) is closed under the map (3) Let B be the left-hand side of the displayed formula in the statement.Now let k ∈ Z n , and let g ∈ S(R n ).Using notation from the proof of part (1) of this proposition, we have where g 1 denotes the function This function vanishes at infinity, so its corresponding multiplication operator M g 1 is compact.All of the other operators that we compose to obtain To prove the assertion that Q W (B) is in fact a dense subspace of B 0 (L 2 (T n )), we note that, given a and b in Z n , we can fix a g ∈ S(R n ) such that and from the fact that a, b ∈ Z n were arbitrary and that the family of vectors (ψ l ) l∈Z n is an orthonormal basis of which implies the same inclusion for the respective generated C*-algebras.
(5) Suppose f is of the form e k ⊗ h.Then it is readily seen that for each [x] ∈ T n , from which it follows that both C R (T * T n ) and S R (T * T n ) are invariant subspaces of the representation ρ 0 .Furthermore, for each l ∈ Z n , we have for each [x] and each generator e k ⊗ h of S R (T * T n ).Since these generators span S R (T * T n ), and the quantisation map and the maps ρ 0 [x] and ρ [x] are linear, we may substitute for e k ⊗ h any element of S R (T * T n ) in the above equation.

Proof of strict quantisation
We now show that Weyl quantisation as defined in the previous section yields a strict quantisation of the dense Poisson subalgebra S R (T * T n ) of the classical resolvent algebra C R (T * T n ) on T * T n ∼ = T n × R n , see [15,section II.1.1.1]or Theorem 22 below.Of these properties, the most difficult one to prove is Rieffel's condition, i.e., convergence of the operator norms of Q W (f ) to the sup-norm of f ∈ S R (T * T n ), which we discuss separately before showing that the other conditions hold.To prepare for the proof, we first make the following observation: Then for any f ∈ ∞ (Z n K ), we have Proof.This is readily seen by conjugating the operator Before we give a precise proof of this proposition, it is instructive to first give a sketch of the underlying idea.To relate the norm of Q W (f ) to that of f , we conjugate the quantised function with the Fourier transform to obtain an operator on 2 (Z n ).We visualise Z n as a lattice of points in R n , and divide it into identical boxes.In each of these boxes, we identify a slightly smaller box such that all of the smaller boxes are translates of each other in the same way that the larger boxes that contain them are translates of each other.See Figure 2. The difference between the sizes of the small boxes and the sizes of the larger boxes is determined by the values of the various k j that appear in the function of which we consider the quantisation; specifically, the shift S k j ∈ B( 2 (Z n )) should always map elements on 2 (Z n ) supported on points inside the smaller box to functions supported on points inside the larger box containing the small one.The size of the larger box is determined by a chosen value of ε > 0 and a crude estimate of Q W (f ) .
Given a function φ ∈ 2 (Z n ), we can now estimate the norm of its image under the conjugated quantised function as follows.First, we consider the projection of φ onto the subspace of 2 (Z n ) of elements supported on the set of points inside one of the smaller boxes, and use the fact that its image under the operator consists of elements supported on the set of points inside the larger box.We can then consider a periodic version of the operator, and use the preceding lemma to get an estimate on its norm and relate it to the norm of f .Finally, we sum the contributions of all projections of φ onto the subspaces corresponding to the smaller boxes to obtain an estimate on the difference of the norm of f and that of the conjugated version of its quantisation.To control the difference between φ and its projection onto the space corresponding to the union of all of the smaller boxes, we note that the partition into boxes can always be offset by some element of Z n in such a way that the part of φ supported on the complement of this union is small.Proof.Fix f ∈ S R (T * T n ) and ε > 0. We first prove the following statement: (a) There exists an 1 ∈ (0, ∞) such that for each ∈ (0, 1 ], we have for k j ∈ Z n and h j = h U j ,ξ j ,g j for some U j , ξ j and g j (which are not needed in the proof).Note that by (19), we have a uniform bound on the norms of the operators (Q W (f )) >0 , namely Since the case C = 0 is trivial, we assume that C > 0 (which also implies that m > 0).Now define L := max 1≤j≤m k j ∞ and fix K ∈ N\{0} such that K ≥ 2L and such that Moreover, for j = 1, . . ., m, the function h j is uniformly continuous, hence there exists 1 ∈ (0, ∞) such that for each ∈ (0, 1 ], each a ∈ Z n and each b ∈ Z n with |b l | < K for l = 1, . . ., n, we have (21) |h j (2π a) − h j (π (k j + 2(a + b)))| < ε 4m .
Now fix ∈ (0, 1 ], fix ψ ∈ L 2 (T n ) with ψ = 1, and let φ be the image of ψ under the Fourier transform F : L 2 (T n ) → 2 (Z n ), which we already defined in part (1) of the proof of Proposition 18.Furthermore, we define the set and we define KZ n and Z n K as in the previous lemma.Then, we have is the cardinality of the set X.It follows that there exists a b ∈ Z n with 0 ≤ b l < K for l = 1, . . ., n such that Let P X,b be the orthogonal projection of 2 (Z n ) onto the subspace so that by the above inequality, we have For each a ∈ KZ n , let where the representative a ∈ Z n has been chosen so that 0 ≤ a l < K for l = 1, . . ., n.Furthermore, for each a ∈ Z n , we have a corresponding shift operator and for each a + KZ n ∈ Z n K , we define the shift operator S a+KZ n per as in the previous lemma.Finally, for each a ∈ KZ n , we define ≤ sup Moreover, using our explicit formula (19), we find that where in the third step, we have used the fact that a + k j ∈ {0, . . ., K − 1} n for each a ∈ X and j = 1, . . ., m.On the other hand, we have where we have used equation (21) in the final step.From equations ( 23) and ( 24), we obtain for each a ∈ KZ n .It is straightforward to see that for each φ ∈ 2 (Z n ), we have which together with equation ( 22) implies and since ψ ∈ L 2 (T n ) was an arbitrary vector with norm 1, we obtain for each ∈ (0, 1 ] which proves (a).
We now turn to the reverse inequality: (b) There exists an 2 ∈ (0, ∞) such that for each ∈ (0, 2 ], we have By uniform continuity of f , there exists a δ > 0 such that for each (x , p ) and therefore, by the triangle inequality and our choice of ( Defining X in the same way as we did in the proof part (a), it follows that Next, we note that the function FQ W (f )F −1 P X,0 φ : Z n → C is supported in the set of a ∈ Z n satisfying a l ≤ a l < a l + K for l = 1, . . ., n. Combining this observation with the estimate just obtained and equation ( 24) yields We use this together with equation ( 22) to obtain Since ψ = 1, this establishes (b).
Finishing up the proof, taking 0 := min( 1 , 2 ), we infer that for each ∈ (0, 0 ], we have We are now ready to prove the main result of this subsection.Let Theorem 22.Let I ⊂ [0, ∞) be a subset containing 0 as an accumulation point.Then, except for continuity at > 0, the triple is a strict quantisation of the Poisson algebra S R (T * T n ), i.e., it satisfies (1) Rieffel's condition at = 0: is continuous at 0.
(2) Von Neumann's condition: for each f, g ∈ S R (T * T n ), we have Proof.
(2) First suppose that f j is a generator e k j ⊗ h U j ,ξ j ,g j of S R (T * T n ) for j = 1, 2. As in Proposition 21, we will write h j instead of h U j ,ξ j ,g j .Let k := k 1 + k 2 .Then Applying part (3) of Proposition 16 yields for each a ∈ Z n .On the other hand, we have a, be the scalar in front of ψ k+a on the right-hand side of the last equation.It is not hard to see from this equation that for each > 0. Now note for j = 1, 2, all derivatives of h j ∈ W 0 R (R n ) are bounded, and so h j is Lipschitz continuous.This implies that where big O notation signifies a limit of → 0, uniformly in a, analogous to the notation in §3.2.When plugging the above formulas into the definition of c By bilinearity, this result extends to arbitrary f 1 , f 2 ∈ S R (T * T n ).
(3) As in the previous part of the proof, we prove the statement for f j = e k j ⊗ h j , from which the general case readily follows.We have where k = k 1 + k 2 , as in part (2) of this theorem, and ∇ v h is the directional derivative of h in the direction of v. Applying part (3) of Proposition ( 16) yields while equation ( 27) yields a, ψ k+a , where for each a ∈ Z n and each > 0, we define It is readily seen that a, | .
We claim that the right-hand side of this inequality converges to 0 as ∈ I goes to 0; evidently, this will show that Dirac's condition holds.
Because the second order derivatives of h j are bounded, Taylor's theorem gives This can be used to show that c (2) a, → 0 uniformly in a ∈ Z n as → 0, which proves the claim.
Proposition 24.Let f ∈ S R (T * T n ).Then the map is continuous with respect to the strong operator topology on the codomain.
Proof.By linearity of the quantisation map and the fact that S R (T * T n ) is the linear span of generators of C R (T * T n ), we may assume without loss of generality that there exists a k ∈ Z n and a generator h of W 0 R (R n ) such that f = e k ⊗ h.Then, by our explicit formula (19), we find whenever → 0 in (0, ∞).This convergence also holds when we replace ψ l by a vector in span l {ψ l }.Furthermore, in part (3) of Proposition 16, we have seen that Now let ψ ∈ L 2 (T n ) be arbitrary.Fix > 0. Since span l {ψ l } is dense in L 2 (T n ), there exists ψ ∈ span l {ψ l } such that ψ − ψ < ε/(4( h ∞ + 1)).By the discussion above, there exists δ > 0 such that Q W (f )ψ − Q W 0 (f )ψ < /2 whenever > 0 satisfies | − 0 | < δ.Then for any such , we have which concludes the proof of the proposition.

Quantum time evolution
Our next task is to show that A = C * Q W (S R (T * T n )) is invariant under time evolution for each Hamiltonian with potential V ∈ C(T n ).The general proof strategy resembles that of Buchholz and Grundling in [9, Proposition 6.1].However, the present setting differs from theirs in two important ways, each of which introduces its own technical problems.First of all, our configuration space is T n rather than R n .Secondly, we consider the problem of invariance under time evolution for arbitrary n ∈ N, whereas Buchholz and Grundling only discuss the case n = 1.We start with the simplest type of time evolution: Lemma 25.Let > 0. The algebra A is closed under the quantum time evolution corresponding to the free Hamiltonian H 0 that is the unique self-adjoint extension of the essentially self-adjoint operator − Remark 26.The fact that for any compact Riemannian manifold M the Laplace-Beltrami operator on C ∞ (M ) has a unique self-adjoint extension, is due to Gaffney [10].
Proof.We show that the quantum time evolution corresponding to H 0 maps the set of quantisations of the generators e k ⊗ h U,ξ,g of C R (T * T n ) into itself; since the time evolution consists of a family of automorphisms of C * -algebras, the lemma will follow from this.Let e k ⊗ h U,ξ,g be such a generator.Note that for each a ∈ Z n , we have (29) e − itH 0 ψ a = e −2π 2 it a 2 ψ a .
Remark 27.Comparing the proof of Lemma 25 with the proof of the analogous Lemma 10, we see that (for t = 1) ξ and g are both the same.Indeed, one can easily obtain which is analogous to a known result for Weyl quantisation on R 2n (proved in higher generality in [15, Theorem II.2.5.1]).There is generally no such result for non-free time evolution.
In order to deal with the general quantum time evolution, we recall some basic theory about lattices that we need due to the appearance of the lattice Z n in T n = R n /Z n .A set of linearly independent vectors v 1 , . . ., v l in a lattice Λ is called primitive in Λ if span Z (v 1 , . . ., v l ) = span R (v 1 , . . ., v l ) ∩ Λ.For instance, every Z-basis of a lattice Λ is primitive in Λ.Furthermore, we have the following result: Lemma 28.Let Λ ⊂ R m be a lattice.Every primitive set v 1 , . . ., v l in Λ can be extended to a Z-basis v 1 , . . ., v l , v l+1 , . . ., v m of Λ.
for some compact subset K ⊆ R m .Let K be the image of K under the projection R k → R l onto the first l coordinates, and define the function f 1 : R l → C by )f 0 T −1 (s (1) ⊕ s (2) ) ds (2) .
One easily finds that f 1 ∈ L ∞ (R l ).We are now left with the integral s j v j ds .
We want to relate the above integral to an integral over the first l components in T n .For this purpose, we apply Lemma 28 once more to extend v 1 , . . ., v l to a Z-basis v 1 , . . ., v n of Z n , and let S be the matrix whose columns are the vectors v 1 , . . ., v n .Since S and its inverse are matrices in GL n (Z), we find that det S = ±1.Moreover, S induces the group automorphism [x] → [Sx] of T n , which we can pull back to the unitary map for which it is straightforward to check (on generators of A ) that U where ι denotes the canonical map T l → [0, 1) l .Note that the above sum has only finitely many nonzero terms since f 1 is compactly supported.
In conclusion, we have proved that a = M (e k )U −1 (F ⊗ 1)U , for an integral operator F ∈ L 2 (L 2 (T l )).By part (3) of Proposition 18, any compact operator, like F , is inside the quantum resolvent algebra on T l × R l .By part (4) of Proposition 18, this implies that F ⊗ 1 ∈ A , and hence U −1 (F ⊗ 1)U ∈ A .As M (e k ) is the quantisation of e k ⊗ 1 R n , we find a ∈ A .As we have seen, linearity and continuity of the Dyson series imply that e itH 0 e −itH ∈ A , and this implies the theorem itself.

Remark 19 .
Because of part (2) of this proposition, we will write A for the C * -algebra generated by Q W (S R (T * T n )) for any value of > 0 without specifying .Part (3) is the analogue of the first part of [15, Corollary II.2.5.4] in the present setting, while part (5) is the analogue of [15, Theorem II.2.5.1].
operator of which the corresponding function is the one within absolute value strokes.Proposition 21. (Rieffel's condition) For each f ∈ S R (T * T n ), we have

Figure 2 :
Figure 2: Part of the lattice Z 2 with two larger boxes that are adjacent, each of which contains a smaller box.

A
a,b P a,b P X,b φ = A a,b P a,b P X,b a ∈Z n φ(a )δ a = A a,b a ∈X φ(a + b + a )δ a +KZ n = m j=1 S k j +KZ n per a ∈X h j (2π (a + b))φ(a + b + a )δ a +KZ n .Writing µ a ,j := h j (2π (a + b)) − h j (π (k j + 2(a + b + a ))) ,for j = 1, . . ., m and a ∈ X, we obtain otherwise, and let ψ := F −1 φ ∈ L 2 (T n ).Then ψ = φ = 1, and A a,0 P a,0 φ = m j=1 e 2πi k j •b K h j (2π a)P a,0 φ , with A a,b and P a,b as defined in part (a).Since P a,0 φ = 1, we have

( 1 )
a, and using the fact that h 1 and h 2 are bounded functions, we find that c

2 j
with domain C ∞ (T n ).