We show that the Connes–Rovelli thermal time associated with the quantum harmonic oscillator can be described as an (unsharp) observable, that is, as a positive operator valued measure. We furthermore present extensions of this result to the free massless relativistic particle in one dimension and to a hypothetical physical system whose equilibrium state is given by the noncommutative integral.

It was recognised by Bohr in the early days of quantum mechanics that the formulas E = and λ = h/p, with h the Planck constant suggest the uncertainty relations
ΔxΔp2andΔtΔE2.
Whereas position, momentum, and energy could be given a firm mathematical footing in von Neumann’s mathematical foundations of quantum mechanics as observables, that is, selfadjoint operators acting on a suitable Hilbert space (“energy” being replaced by a Hamiltonian operator), this cannot be done for time. In fact, it is a famous observation of Pauli1 that if H (reserving the letter H for Hilbert spaces) is a semi-bounded Hamiltonian operator, there exists no selfadjoint operator T covariant with the unitary group generated by iH, that is, it cannot be true that
eitHTeitH=Tt,tR.
The common view is that “time,” rather than being an observable, should be regarded as a mere bookkeeping index.
The spectral theorem establishes a one-to-one correspondence between selfadjoint operators on the one hand and projection-valued measures on the real line. Observing that the orthogonal projections on a Hilbert space H are precisely the extreme points of the set of all bounded operators T on H satisfying the relation
0TI,
where AB means that (Ah|h) ≤ (Bh|h) for all hH, a natural generalisation of the notion of a projection-valued measure is obtained upon replacing orthogonal projections by operators satisfying the above relation. The mathematical theory of the resulting positive operator-valued measures (POVM) had already been developed in the 1940s by Naimark, and its usefulness in quantum mechanics was first advocated by Ludwig.2 It was quickly realised that covariant POVMs could be associated with various time measurements, such as arrival time, times of occurrence, screen time, and time of flight, and rigorous versions of the time-energy uncertainty relation could be proved for some of them;3–8 see also the overviews in Refs. 9 and 10. For the free falling particle with Hamiltonian
H=P22mmgQ,
with P and Q the position and momentum operators, respectively, it is even possible to specify a selfadjoint operator covariant with H, namely
T=Pmg.
This operator measures time as the increase of momentum. The problem of finding a projection-valued measure (PVM) covariant with the free-particle Hamiltonian remained elusive, however. As Busch5 (pp. 95–96) puts it,
  • “Seemingly obvious candidates of a time operator conjugate to the free-particle Hamiltonian,
    Hfree=P22m,
    are given by suitably symmetrised expressions for the time-of-arrival variable suggested by classical reasoning, for example:
    12m(QP1+P1Q)ormP1/2QP1/2.
    While these expressions formally satisfy the canonical commutation relation, they are not selfadjoint but only symmetric (on suitably defined dense domains on which they actually coincide, see Sec. 10.4), and they do not possess a selfadjoint extensions. Hence this intuitive approach does not lead to a time observable in the usual sense of a selfadjoint operator conjugate to the free Hamiltonian.”

The problem of constructing a POVM for this setting, covariant with time translations, has been addressed, for instance, in Ref. 11 using the time-of-arrival variable 12m(QP1+P1Q). See also the books,12,13 and, in particular, Example 17.5.13 In a similar vein, a description of the four space-time coordinates of an event in terms of a POVM defined on the Minkowski space-time covariant with respect to the Poincaré group has been proposed in Refs. 14 and 15, while “relative” (or “relational”) versions of time covariant POVMs have been studied in Ref. 16.

The aim of the present paper is to address the problem of finding a POVM associated with the Connes–Rovelli thermal time. This notion is introduced in Sec. II. After recalling some mathematical preliminaries in Sec. III, we provide in Sec. IV a positive answer for case of the quantum harmonic oscillator. In Theorem IV.2 we show that the “clock time POVM” associated with the quantum harmonic oscillator as constructed in Refs. 9 and 17 transforms covariantly (up to a scale factor β) with the modular flow canonically associated with the Gibbs state eβN, where N is the number operator. Key to the Proof of Theorem IV.2 is that this Gibbs state is normal. In order to extend the Theorem IV.2 to states that are not necessarily normal, it turns out to be fruitful to relax the definition of thermal time beyond the normal case while retaining the algebraic properties of the modular flow. The resulting notion of time, which we will call modular time, is introduced in Sec. V.

Our second main result, Theorem VI.1, identifies a POVM for the modular time of a free massless relativistic particle in dimension one at equilibrium with a heat bath of inverse temperature β. The main difficulty to overcome is that the equilibrium state associated with the Hamiltonian |D|=Δ fails to be normal.

In Sec. VII we consider the semifinite singular trace ω associated with the noncommutative integral. This trace, also known as the Dixmier trace, is comprehensively covered in the monograph.18 It has deep applications in index theory,19–21 and has fundamental connections with general relativity22 and the noncommutative geometry approach to the standard model.23 The third main result of this paper, Theorem VII.3, shows that this trace is associated with a POVM on an abstract Hardy space naturally associated with (the Weyl calculus associated with) ω.

We conclude the paper with some open problems.

The notion of thermal time was introduced by Rovelli24 as a tentative solution to the “problem of time” in generally covariant theories (such as general relativity). Such theories have no preferred time, but only the “internal times” of objects such as clocks. For this reason it has been proven difficult to give a covariant formulation of thermodynamics, which requires a preferred time to define the notion of equilibrium. Rovelli observed that in classical Hamiltonian mechanics it is possible to reverse this reasoning: the statistical state of the system can be used to determine a preferred internal time variable. The time variable determined in this way is then called thermal time. Rovelli’s thermal time hypothesis is the postulate that this principle is generally applicable and captures the “common sense time flow” experienced as internal time. In a companion paper,25 Rovelli showed that in a Robertson–Walker universe, the thermal time associated with the cosmic background radiation precisely recovers the Robertson–Walker cosmological time.

The notion of thermal time was extended to the quantum domain by Connes and Rovelli26 by interpreting the well-known connection between Kubo–Martin–Schwinger equilibrium states (KMS states) and the Tomita–Takesaki theory: a normalised state ω on a C*-algebra A is a KMS state with respect the Tomita–Takesaki flow in the Gelfand–Naimark–Segal (GNS) representation induced by ω. It is a deep result of Connes27 that this flow is essentially unique, in the sense that the flows corresponding to two different normalised states are always the same up to conjugation by an inner automorphism (a mapping of the form AUAU for some unitary operator U). In view of its connection with KMS states, this makes it natural to interpret the Tomita–Takesaki flow as the thermal time associated with the state ω.

In this section we discuss the mathematical preliminaries needed for our purposes. Notation is standard and follows28 unless otherwise stated.

We start with recalling the definition and some properties of positive operator-valued measures (POVMs). For more information on this topic we refer to Refs. 2, 9, 12, and 2831; Ref. 28 contains complete proofs of all theorems stated below. POVMs arise in a variety of applications, such as the theory of open quantum systems,32 quantum information theory,33 and the theory of symmetric operators.34 

Let H be a complex Hilbert space with inner product (·|·) and norm ‖·‖. We use the mathematician’s convention that inner products are linear in their first argument and conjugate-linear in their second argument. The Banach space of all bounded linear operators on H is denoted by L(H).

An effect is a bounded operator EL(H) satisfying 0 ≤ EI. The set of effects on H will be denoted by E(H). The set of its extreme points equals P(H), the set of all orthogonal projection on H. This suggests generalising the standard definition of a projection-valued measure by replacing the role of orthogonal projections by effects. Accordingly, a positive operator-valued measure (POVM) on a measurable space (Ω,F) is defined to be a mapping E:FE(H) assigning to every set FF an effect EFE(H) with the following properties:

  • EΩ = I;

  • for all xH the mapping F ↦ (EFx|x) defines a measure Ex on (Ω,F).The measure defined by (ii) is denoted by Ex. From
    Ex(Ω)=(EΩx|x)=(x|x)=x2
    we infer that every measure Ex is finite. It is easy to see that a POVM E:FE(H) is a projection-valued measure if and only if for all F,FF we have
    EFEF=EFF.

The convex set of all probability measures on a measurable space (Ω,F) is denoted by M1+(Ω). The quantum analogue of this set is the set S(H) of all positive trace class operators T on H with unit trace; in the physics literature such operators are referred to as density matrices. The formula
ω(S)=tr(ST)
establishes a one-to-one correspondence between the operators TS and the set of all normal states, that is, bounded linear linear functionals ω:L(H)C with the property that n≥1ϕ(Pn) = ϕ(P) whenever (Pn)n1 is a sequence of disjoint orthogonal projections in H and P is their least upper bound.

Theorem III.1.
If Φ:S(H)M1+(Ω) is a convexity preserving mapping, then there exists a unique POVM E:FE(H) such that for all TS(H) we have
(Φ(T))(F)=tr(EFT),FF.

This theorem establishes the operational role of POVMs as “unsharp observables”: in that it transforms normal quantum states to classical states.

If J is an isometry from H into another complex Hilbert space H̃ and FP̃FP(H̃) is a projection-valued measure in H̃, then
FJP̃FJE(H)
is a POVM in H. A celebrated theorem of Naimark asserts that the converse is also true:

Theorem III.2.
Let (Ω,F) be a measurable space and let E:FE(H) be a POVM. There exists a Hilbert space H̃, a projection-valued measure P̃:FP(H̃), and an isometry J:HH̃ such that for all FF we have
EF=JP̃FJ.

For any POVM E:FL(H) there exists a unique linear mapping Ψ:Bb(Ω)L(H) satisfying
Ψ(1F)=QF,FF,
and
Ψ(f)f,fBb(Ω),
where Bb(Ω) is the Banach space of all bounded measurable functions on Ω endowed with the supremum norm ‖f = supω∈Ω|f(ω)|. This mapping Ψ satisfies
Ψ(f)=Ψ(f̄),fBb(Ω).
In contrast to the corresponding functional calculus for PVMs, unless the POVM is a PVM, this calculus fails to be multiplicative. Nevertheless, in the same way as bounded normal (resp. selfadjoint, unitary) operators are in one-to-one correspondence with PVMs on C (resp. on R, T), contractions are in one-to-one correspondence with POVMs on the unit circle T and admits the following representation:

Theorem III.3.
For every contraction TL(H), there exists a unique POVM E on the Borel σ-algebra B(T) such that
Tn=ππeinθdE(θ),nN.
If E is a POVM with the above property, then T is unitary if and only if E is a projection-valued measure.

Intriguingly, the notion of POVM admits a category-theoretic counterpart, in the sense that a Naimark-type theorem holds and the usual notion of POVM is recovered in the category of Hilbert spaces and bounded linear operators.35 

We can only give the briefest of descriptions here; for a full account the reader is referred to Refs. 36–38, where the proofs of all statements can be found.

Let H be a separable complex Hilbert space (in applications it usually is the representation Hilbert space arising from a GNS-like construction), let ML(H) be a von Neumann algebra, and let Ω ∈ H be a separating and cyclic element for M. Recall that this means that whenever AM satisfies AΩ = 0 it follows that A = 0, and that the subspace MΩ={AΩ:AM} is dense in H. The antilinear operator
S0:AΩAΩ,
is well defined as a densely defined operator in H with domain D(S0)=MΩ, and this operator is closable. Its closure will be denoted by S. By polar decomposition, there exist a unique antiunitary operator J on H and a unique positive selfadjoint operator Δ in H such that
S=JΔ1/2
with equal domains. The operator J and Δ are called the modular conjugation and the modular associated with the triple (M,τ,Ω). These operators have a number of interesting properties that are of no concern here. Important for us is that Δ induces a one-parameter family of automorphisms of M by
σt:AΔitAΔit,tR,
the so-called modular flow associated with Ω. The non-trivial point is that the right-hand side belongs to M for all AM. Note that, following Ref. 26, on the right-hand side we have changed the usual sign convention to enhance compatibility with the physical applications.
Let us now suppose that ω is a state on a C-algebra A with unit I, that is, ω:AC is a bounded linear functional on A with the following two properties:
ω(I)=1,ω(AA)0for allAA.
There is some redundancy in this definition, in that boundedness is a consequence of the other assumptions. It is an easy consequence of the Cauchy–Schwarz inequality36 (Lemma 2.3.10) that the set Nω{AA:ω(AA)=0} is a left ideal of A. On the quotient space A/Nω, ω induces an inner product
(qω(A)|qω(B))ωω(BA),
where qω:AA/Nω is the quotient mapping. Let Hω be the Hilbert space obtained by completing the quotient with respect to it, and define πω:AL(Hω) by
πω(A)qω(B)qω(AB),BL(Hω).
Then πω extends to a contractive ⋆-homomorphism from A to Hω, and this mapping is an isometry if and only if the state ω is faithful, that is, ω(AA) = 0 implies A = 0, or equivalently, if Nω={0}. The element
Ωqω(I)
is cyclic for the range of πω; in particular, Ω is cyclic for the von Neumann algebra Mω generated by the range of πω in L(Hω). Note that for all AA we have
ω(A)=(πω(A)Ω|Ω)ω.
The pair (Hω, πω) is known as the Gelfand–Naimark–Segal (GNS) representation associated with ω. If, in addition to being cyclic, Ω is also separating for Mω, this places us in the setting of the Tomita–Takesaki theory, and the associated modular flow on Mω will be called the modular flow associated with the state ω.
The connection with equilibrium thermodynamics is given by a theorem (see Ref. 37, p. 218) that states that if Ω is separating for Mω, the state ω is a KMS state with respect to the modular flow associated with ω. The precise definition of KMS states is unimportant for our purposes (for an exhaustive treatment see Ref. 39). What matters is that this notion extends the notion of a Gibbs state, in that every Gibbs state in KMS; and the notions are equivalent under suitable finiteness conditions. Recall that if H is a Hamiltonian operator with the property that the bounded operator eβH is of trace class, then the Gibbs state at inverse temperature β > 0 associated with H is the state ω on L(H2(D)) given by
ω(A)=tr(TβA),
where TβeβH/tr(eβH).

The noncommutative spaces L2(M,τ), where M is a von Neumann algebra, are usually introduced for faithful semifinite normal traces τ. Since in Sec. VII we will be dealing with general semifinite traces τ, we will briefly go through the steps of defining L2(M,τ) in that more general setting.

Let ML(H) be a von Neumann algebra over a given abstract separable Hilbert space H, and let τ be a trace τ on L(H), that is, τ is a mapping from the positive cone in M to [0, ] such that the following conditions are satisfied:

  • τ(cA) = (A) for all 0AM and scalars c ≥ 0;

  • τ(A + B) = τ(A) + τ(B) for all 0AM and 0BM;

  • τ(AA) = τ(AA) for all AM.

Let
Mτ1,+{AM:A0,τ(A)<},Mτ2{AM:τ(AA)<}.
The linear span Mτ1 of Mτ1,+ is a ⋆-subalgebra of M and the restriction of τ to Mτ1,+ uniquely extends to a positive linear functional on Mτ1. Moreover, Mτ2 is a left ideal in M, and for any two A,BMτ2 one has BAMτ1 and the Cauchy–Schwarz inequality holds:
|τ(BA)|2τ(AA)τ(BB)
[see Ref. 40, Lemma 5.1.2, which applies here since condition (iii) implies that tr(UAU) = tr(A) if U is unitary]. By polarisation [see the proof40 (Proposition 5.2.2)], for all A,BMτ2 we have
τ(AB)=τ(BA).
By the Cauchy–Schwarz inequality, the set
Nτ{AMτ2:τ(AA)=0}
is a vector space. As in Ref. 40 (Lemma 5.1.2), Nτ is a left ideal in Mτ2. On the quotient space Mτ2/Nτ we obtain a well defined inner product by putting
(qτ(A)|qτ(B))ττ(BA),A,BMτ2,
where qτ:Mτ2Mτ2/Nτ is the quotient mapping. The Hilbert space Hτ is defined as the completion of Mτ2/Nτ with respect to this induced inner product.

As is well known, the (one-dimensional) quantum harmonic harmonic oscillator is described by the Hamiltonian L=12I+N, where N is the number operator and 12 is the ground state energy. In view of the trivial identity eitLAeitL = eitNAeitN, from the point of view of finding covariant POVMs it makes no difference whether we work with L or N. Since the number operator N admits a natural presentation on the space H2(D) (which, incidentally, also suggests a natural approach to the problem of time for the free relativistic particle), we will work with N instead of L.

The problem of finding a POVM covariant with the unitary group (eitN)tR has already been solved in the physics literature9 (using ideas that go back to Ref. 17), where this POVM is interpreted as describing “phase” as a (circle-valued) “unsharp observable.” As was already observed in this paper, since L can be interpreted as a Hamiltonian this POVM can alternatively be interpreted as describing “(oscillator clock) time.” The exposition presented here follows the mathematically rigorous treatment of Ref. 28 (Chap. 15).

Let D denote the open unit disc {zC:|z|<1} in the complex plane. The space H2(D) is defined as the Hilbert space whose elements are the holomorphic functions f:DC whose power series f(z)=nNcnzn satisfies
fH2(D)2=nN|cn|2.
We consider the selfadjoint operator N with domain
D(N)=f=nNcnznH2(D):nNn2|cn|2<,
given by
Nzn=nzn,nN,
where zn(z) ≔ zn as a function on D. Then σ(N)=N and the functions zn form an orthonormal basis of eigenvectors for N.
To define phase as a POVM on T we proceed as follows. For Borel sets BT we define the operator EB on H2(D) by
EB=JPBJ,
where J:H2(D)L2(T) is the isometry given by
J:nNcnznnNcnen,
J:L2(T)H2(D) is its adjoint, and PBf1Bf for fL2(T). It is clear that 0 ≤ EBI, and accordingly the assignment E: BEB defines a POVM on T. This construction goes back to Ref. 17, where a selfadjoint operator F on H2(D) is constructed satisfying the Heisenberg commutation relation
NFFN=iI
on a suitable dense domain of function in H2(D). This operator is defined as the Toeplitz operator with symbol arg(z). The problem with this construction, however, is that there the argument is not uniquely defined; one obtained different operators for different choices. As observed in Ref. 41 the Heisenberg relation extends to the natural maximal domain of the commutator FNNF = −iI, but this domain fails to be dense with respect to the graph norm of N. Moreover, as already observed in Ref. 17, the (stronger) Weyl commutation relations
eisNeitF=eisteitFeisN
do not hold. The POVM constructed above solves these problems in a satisfactory way, in that it enjoys the following covariance property with respect to (cf. Ref. 9, Sec. III.5):

Theorem IV.1.
The phase observable E is covariant with respect to the unitary C0-group generated by iN, that is, for all Borel subsets BT we have
eitNEBeitN=EeitB,tR,
where eitB = {eitz: zB} is the rotation of B over t.

The relationship between the POVM E and the operator F is expressed by the relation
F=Targ(z)dE(z).
We refer the reader to Refs. 28 and 41 for a proof and a fuller discussion of these matters.
Reinterpreted as “clock time” (cf. Ref. 17), we show next that the POVM E constructed in the previous subsection can be interpreted as a special instance of the Connes–Rovelli thermal time associated with the Gibbs state
ω(A)tr(AT),AL(H2(D)),
where
T=eβN/tr(eβN).
In this formula, the scalar β is interpreted as the inverse temperature. This involves showing that E transforms covariantly with the modular flow associated with ω by the Tomita–Takesaki theory, which is the content of Theorem IV.2 below. Apart from its own interest, the proof will serve as a template for our definition of modular time in the next section.
Following Ref. 42 (see also Ref. 37, Chap. V), rather than using the GNS representation associated with ω, we may use the unitarily equivalent representation π of L(H2(D)) on the Hilbert space L2(H2(D)) of all Hilbert–Schmidt operators on H2(D) given by
π(A)BAB,AL(H2(D)),BL2(H2(D)).
In what follows we will write A for π(A).
Since T is of trace class, Ω ≔ T1/2 is Hilbert–Schmidt and therefore defines an element of L2(H2(D)), and it is immediate that
ω(A)=(AΩ|Ω),AL(H2(D)),
where (A|B) = tr(BA) is the inner product of L2(H2(D)). As consequence of the unitary equivalence of this representation with the GNS representation, Ω is cyclic for π [and hence for M, the von Neumann algebra M over L2(H2(D)) generated by L(H2(D)), viewing operators in L(H2(D)) as operators acting on L(L2(H2(D))) by left multiplication] and separating for M. Moreover, by Ref. 36 (Theorem 2.4.24), M=L(L2(H2(D))).
On L2(H2(D)) we consider the unitary operators U(t), tR, defined by
U(t)SeiβtNSeiβtN,SL2(H2(D)).
The family (U(t))tR is easily seen to be a unitary C0-group on L2(H2(D)). By Stone’s theorem, the generator of this group is of the from iH with H selfadjoint. By the functional calculus of unbounded selfadjoint operators, the operator ΔeH is injective and selfadjoint (its domain being given by this calculus). By the composition rule of the functional calculus, for tR we obtain
Δit=(eH)it=eitH=U(t),
where in the last step we used the fact that the unitary group generated by iH also arises through exponentiation in the functional calculus. In particular, for all A,BL(H2(D)),
(ΔitAΩ|BΩ)=ω(BeiβtNAeiβtN).
By spectral theory, for hD(Δ1/2) the mapping z ↦ Δizh is continuous on the closed strip {Imz[0,12]} and holomorphic on its interior. This, and standard properties of fractional powers, implies that for all A,BL(H2(D)), both sides of the identity
(ΔitAΩ|BΩ)=ω(BeiβtNAeiβtN)=tr(BeiβtNAeβ(it+1)N)
admit a continuous extension to the strip {Im z ∈ [0, β]} which is analytic on the interior, and given by substituting z for t. By the edge of the wedge theorem (cf. Ref. 39, Proposition 5.3.6) these extensions agree. It follows that for Rez[0,12] we have AΩD(Δ1/2)D(Δiz), and taking z=12i results in the identity
(Δ1/2AΩ|BΩ)=(e12βHAΩ|BΩ)=trBe12βNAe12βN=(ΩA|BΩ).
Since Ω is cyclic for π in L2(H2(D)), this implies
Δ1/2AΩ=ΩA.
Denoting by J the antiunitary operator that sends an operator BL2(H2(D)) to its adjoint BL2(H2(D)), it follows that
(JΔ1/2)(AΩ)=J(ΩA)=AΩ.
By uniqueness, this implies that J and Δ are the modular conjugation and the modular associated with ω in the representation π, respectively.

Denoting the associated modular flow as before by σtA ≔ ΔitAΔit, we have the following result:

Theorem IV.2.
For all tR and Borel sets BT, the POVM E satisfies
σt(EB)=EeiβtB.

Proof.
For all A,BL(H2(D)) we have
ΔitAΔitBΩ=eitHAeitHBΩ=U(t)AU(t)BΩ=(eβitN(A(eiβtNBeiβtN))eiβtN)Ω=eβitNAeiβtNBΩ.
Since Ω is cyclic in L2(H2(D)), and L2(H2(D)) is dense in the weak operator topology of L(H2(D)), it follows that σt(A) = ΔitAΔit = eβitNAeiβtN. By Theorem IV.1, for A = EB with BR a Borel set this may be rewritten in the form
σt(EB)=eiβtNEBeiβtN=EeiβtB.

In conclusion, the thermal time associated with the Gibbs state ω of N recovers the “preferred time” with respect to which it is Gibbs for the unitary group generated by iN.

Scrutinising the above proof, we note that the unitary group (Δit)tR implementing thermal time has two key properties:

  1. It acts on a noncommutative L2 space, namely the space of Hilbert–Schmidt operators L2(H2(D)).

  2. It is associated with the cyclic vector Ω = T1/2, where T is an injective positive operator in L(H), in the sense that
    ΔitA=TitATitAL(H2(D)).
    In the proof, the modularity relation
    JΔ1/2(AΩ)=A*ΩAL(H2(D))
    with J: AA*, is a consequence of the above properties of (Δit)tR.

This motivates the following definition. Let H be a Hilbert space, and let ML(H) be a von Neumann algebra. Recall from Sec. III A that if τ is a trace on M, then the corresponding noncommutative space L2(M,τ) is denoted by Hτ.

Definition V.1.
Let τ be a trace on M and let T be a (possibly unbounded) positive and selfadjoint operator acting in H admitting bounded imaginary powers. The pair (τ, T) is said to define a modular time if the assignment
t[ATitATit]
defines a unitary C0-group of the form (Δit)tR on Hτ, where Δ is an injective positive selfadjoint operator on Hτ.

For the theory of bounded imaginary powers we refer to Ref. 43.

For operators AM satisfying τ(AA) we use the standard abuse of notation of identifying A with its equivalence class modulo Nτ, which is defined as the left ideal of all AM satisfying τ(AA) = 0. Denoting the quotient map by q, the assignment ATitATit in the above definition should more rigorously be interpreted as the assignment
q(A)q(TitATit),
initially defined for AM such that τ(AA) < . We will omit the mapping q in what follows.

The use of the term “modular” comes from the following fact.

Lemma V.2.
Let (τ, T) define a modular time, and let J be the antiunitary operator given by the assignment AA* on Hτ. If for all AHτ the operator AT1/2 is well defined as an element of Hτ, then for all AHτ it belongs to D(Δ1/2) and
JΔ1/2(AT1/2)=A*T1/2.

Note that if T is bounded, then the assumption that AT1/2 be well defined as an element of Hτ is satisfied. In the setting of Sec. VII the operator T is unbounded and the condition will be checked by hand.

Proof.
By definition, for all f,gD(T1/2) we have
(JΔit(AT1/2)f|g)=(T1/2TitA*Titf|g),
for all AHτ and tR. Since Δ and T is are injective positive selfadjoint operators, by the edge of the wedge theorem this identity uniquely extends by continuity and analyticity to
JΔ1/2(AT1/2)=A*T1/2.
The rigorous justification that the elements AT1/2 belongs to D(Δ1/2) can be given on the basis of an element belongs to the domain of a densely defined closed operator if and only if it belongs to its weak domain (see Ref. 28, Proposition 10.20).□

The quantum harmonic oscillator of Sec. IV corresponds to the special case where H=H2(D), M=L(H), τ is the standard trace on M, and the injective positive operator TL(H) with tr(T) = 1 was given as T=eβH/tr(eβH). In that situation one has TM and T1/2Hτ=L2(H2(D)).

In the next sections, we construct modular times and POVM on the Borel subsets of R, that satisfy the covariance relation defined below.

Definition V.3.
Let (τ, T) define a modular time, and E:B(R)E(H) define a POVM. We say that E is covariant with respect to the modular time if
TitEBTit=EB+t
for all Borel sets BR and all tR.

It is important to note that we do not insist that the operators EB belong to Hτ. When they do, the covariance relation can be rewritten as
ΔitEBΔit=EB+t.
Moreover, we do not insist on having the Hilbert space Hτ arise from Tomita–Takesaki modular theory. Whether or not our constructions could be modified to have these two additional features (and thus fully embed into Tomita–Takesaki modular theory) is left as an open problem (see Sec. VIII).

In this section we address the problem of modular time for the free massless relativistic particle in dimension 1. Its Hamiltonian |D|Δ, where Δ is the second derivative operator, is the Fourier multiplier on L2(R) corresponding to the multiplier x ↦ |x|. Its part in the closed subspace of all fL2(R) whose Fourier transform is supported on the nonnegative half line R+ can then be viewed mathematically as the continuous analogue of the number operator N. This simple observation suggests that we may try to redo the identification of thermal time more or less mutatis mutandis.

The reasoning of Sec. IV suggests that it should be possible to carry out an analogous construction in which H=H2(D) is replaced by H, where
U(u,v)R2:uR,v>0.
The space H2(U) is defined as the Hilbert space of all holomorphic functions f:UC for which
fH2(U)supv>0f(+iv)L2(R)
is finite. Under convolution with the Poisson kernel for the upper half-plane, this space is isometric to the closed subspace in L2(R) consisting of all functions whose Fourier–Plancherel transform vanishes on the the negative half-line.
If we think of the number operator on H2(D) as being given by the Fourier multiplier operator on L2(T) with multiplier n on the nonnegative frequencies, it is natural to consider the operator on H2(U) given by the Fourier multiplier operator on L2(R) with multiplier ξ on the nonnegative frequences. A positive selfadjoint operator on L2(R) implementing this property is the Poisson operator
|D|(D2)1/2=(Δ)1/2,
where Δ is the Laplace operator (not to be confused with the modular operator, which is also denoted by this symbol), and D=1id/dx with domain
D(|D|)fH2(U):|D|fL2(R).
It is not hard to prove that the spectrum of this operator equals [0, ). This operator generates a C0-semigroup of contractions (P(t))t≥0 on L2(R), the so-called Poisson semigroup, given explicitly as a convolution semigroup
P(t)f=pt*f,t0,fL2(R),
where
pt(x)=1πtt2+x2,t0,xR,
is the Poisson kernel and the convolution g * h, with gL1(R) and hL2(R), is defined by
(g*h)(x)Rdg(xy)h(y)dy.
(1)
The Fourier multiplier with symbol 1[0,) can be expressed in the Borel functional calculus of the selfadjoint operator D as 1R+(D). Let L+2(R) denote the range of this operator, that is,
L+2(R)=fL2(R):f̂(ξ)=0for almost allξR+.
Here, ff̂ is the Fourier–Plancherel transform, defined for functions fL1(R)L2(R) by
f̂(ξ)12πf(x)expixξdx,ξR,
(2)
and extended to an isometry from L2(R) onto itself by density. With the normalisations chosen in (1) and (2), fL1(Rd) and let gL1(Rd) or gL2(Rd). For almost all ξRd we have
f*ĝ(ξ)=2πf̂(ξ)ĝ(ξ).
(3)
It is well known that, for all fL+2(R), the function F defined by
F(x+iy)P(y)f(x)
belongs to U with ‖FH = ‖f2. In the converse direction, for all FH the limit
flimy0F(+iy)
exists, with convergence in the norm of L2(R), and F is recovered from f as F(x + iy) = P(y)f(x). This correspondence defines a unitary mapping
P:H2(U)L+2(R),P:Ff,
whose inverse is given by fF.
Let E:B(R)E(H2(U)) be uniquely defined by
PEBF1[0,)(D)1B(X)PF
for Borel sets BB(R) and functions FH2(U). Here, 1B(X) denotes the multiplication operator on L2(R) defined by 1B(X)f: x1B(x)f(x).
We will show that BEB is a POVM in H2(U), and that it is covariant with respect to the flow Seit|D|Seit|D| on the Hilbert space Hτ, where τ is the trace on L(H2(U)) defined by
τ(A)tr(Aeβ|D|).

Theorem VI.1.

Under the above assumptions, the following assertions hold:

  1. The assignment BEB defines a POVM in H2(U).

  2. The pair (τ, eβ|D|) defines a modular time, in the sense of Definition V.1.

  3. The POVM E is covariant with respect to this modular time, in the sense of Definition V.3.

Proof.
To prove (1), we first note that
0EBI,
that is, every EB is an effect. To prove this, set fPF. Using that f = 1[0,)(D)f and that 1[0,)(D) and 1B(X) are orthogonal projections, we obtain
(EBF|F)H=(1[0,)(D)1B(X)f|f)L+2(R)=(1B(X)f|1[0,)(D)f)L+2(R)=(1B(X)f|f)L2(R)=(1B(X)f|1B(X)f)L2(R)
and therefore 0 ≤ EB; the inequality EBI then follows from
(EBF|F)H2(U)=1B(X)fL2(R)2fL2(R)2=FH2(U)2=(F|F)H2(U).
Turning to (2), for AL(H2(U))we have that
tr(eit|D|A*Aeit|D|eβ|D|)=tr(A*Aeit|D|eβ|D|eit|D|)=tr(A*Aeβ|D|),
and thus the group defined by (t[Aeit|D|Aeit|D|])tR is unitary on Hτ.
We finally prove (3). For all F,GH2(U), let f=PF and g=PG. We have, using (3),
(EBeit|D|F|G)=(1[0,)(D)1B(X)Peit|D|F|PG)=(1B(X)eit|D|PF|1[0,)(D)PG)=(1B(X)eit|D|f|g)=(1B(X)eitDf|g)=2π0[1B̂*(eiuf̂)](u)ĝ(u)̄du=2π001B̂(uv)eitvf̂(v)ĝ(u)̄dvdu.
At the same time,
(eit|D|EB+tF|G)=(1[0,)(D)1B+t(X)PF|Peit|D|G)=(1[0,)(D)1B+t(X)f|eit|D|g)=(1B+t(X)f|1[0,)(D)eit|D|g)=(1B+t(X)f|eitDg)=2π0eituĝ(u)̄01B+t̂(uv)f̂(v)dvdu=2π0eituĝ(u)̄0f̂(v)Rei(uv)z1B+t(z)dzdvdu=2π0eituĝ(u)̄0f̂(v)Rei(uv)(z+t)1B(z)dzdvdu=2π0ĝ(u)̄0eitvf̂(v)Rei(uv)z1B(z)dzdvdu=2π001B̂(uv)eitvf̂(v)ĝ(u)̄dvdu.

In this section we consider a von Neumann algebra M contained in L(L2(R)) generated by the Weyl calculus aa(Q, P) of a pair of operators Q, P (defined later) for compactly based classical symbols a, endowed with the noncommutative integral τ as a semifinite trace, that is,
τ(a(Q,P))Res(a(X,D)|D|1(Iη(D)))=12R(a0(x,1)+a0(x,1))dx,
where D=1id/dx, X: f ↦ [xxf(x)], ηCc(R) is an arbitrary cut-off function supported on [−1/2, 1/2], and a0 denotes the principal symbol of a(X, D). Note that this definition only depends on the symbol a (in fact on its principal part) and not on the choice of operators Q, P. Here Res denotes the noncommutative residue (which can be seen as a Dixmier trace). See Ref. 18 (Chaps. 10 and 11) for more information on the noncommutative integral, and Ref. 44 for terminology and fundamental results on pseudo-differential operators and the Weyl calculus.
The expression for τ is reminiscent of the states A ↦ tr(AT) considered in previous sections, but requires the use of |D|−1 instead of expβ|D|. Consequently, the translation group generated by iD will be replaced by the C0 group of imaginary powers (|D|it)tR, i.e., the group generated by iP where
Pln|D|.

We show that the pair (τ, |D|−1) defines a modular time (Δit)tR in the sense of Definition V.1. We then construct a POVM that is covariant with respect to this modular time in the sense of Definition V.3. This will be constructed using generalised pseudo-differential operators of the form a(Q, P) (in the sense of the Weyl calculus defined in Ref. 45), where Q is a suitable replacement for the standard position operator, adapted to our modified momentum operator P.

Definition VII.1.
Let (S(t))tR be the unitary C0-group on L2(R) defined by
S(t):get/2F1(ξĝ(etξ)),
for all tR. We denote its generator by Q.

Lemma VII.2.
The group (S(t))t≥0 satisfies the following Weyl relations:
eisPS(t)=eistS(t)eisP,s,tR.
(4)

Proof.
For fL2(R) and ξR we have
eisPS(t)f̂(ξ)=|ξ|isS(t)f̂(ξ)=et/2|ξ|isf̂(etξ)
as well as
S(t)eisPf̂(ξ)=et/2|etξ|isf̂(etξ)=eistet/2|ξ|isf̂(etξ).

We will use the group functional calculi for P and Q, defined by
f(P)12πRf̂(ξ)expiξPdξ,g(Q)12πRĝ(ξ)expiξQdξ,
for f̂,ĝL1(R), as well as the Weyl functional calculus of Ref. 45, defined for aS0(R2) by
a(Q,P)=12πR2â(u,v)expi2uvexpiuQexpivPdudv.

Theorem VII.3.
For any Borel set BB(R) define EBA by
EB1R+(P)1B(Q)1R+(P),
the following assertions hold:
  1. The assignment BEB defines a POVM on the range space R(1R+(P)).

  2. The pair (τ, |D|−1) defines a modular time, in the sense of Definition V.1.

  3. The POVM E is covariant with respect to this modular time, in the sense of Definition V.3.

Observe that R(1R+(P)) naturally takes over the role of the Hardy spaces used in the Secs. IV and VI.

Proof.
To prove (1), we first note that
1B(Q)2=12πRR1B̂(η)1B̂(ξ)exp(i(ξ+η)Q)dξdη=12πR1B̂1B̂(ξ)expiξQdξ=1B(Q).
Moreover, for all ξR and f,gL2(R) we have
(S(ξ)f|g)=(f|S(ξ)g),
by a change of variables. and thus (1B(Q)f|g) = (f|1B(Q)g), and
(EBf|f)=(1B(Q)1R+(P)f|1R+(P)f)=(1B(Q)21R+(P)f|1R+(P)f)=1B(Q)1R+(P)f20.
Moreover 1B(Q)1R+(P)f2=(1B(Q)1R+(P)f|1R+(P)f)1B(Q)1R+(P)ff, which implies EB ≥ 1. For Borel sets (Bn)nN, we also have
(1nNBn(Q)f|f)=12πRnN1Bn̂(ξ)S(ξ)dξ=nN(1Bn(Q)f|f),
and (1R(Q)f|f)=(S(0)f|f)=(f|f). This completes the proof of (1).
For (2), we consider a(Q, P) ∈ Hτ. Using (4), we first note that, for tR,
|D|ita(Q,P)|D|it=12πR2â(u,v)ei2uvexpitPexpiuQexp(i(vt)P)dudv=12πR2â(u,v)eituei2uvexpiuQexpivPdudv=at(Q,P),
where at: (x, ξ) ↦ a(x + t, ξ). This implies that t ↦ |D|ita(Q, P)|D|it is a group, and that
Tita(Q,P)TitHτ2=12R(a02(x+t,1)+a02(x+t,1))dx=a(Q,P)Hτ2.
To verify the abstract condition in Lemma V.2 that AT1/2Hτ be well defined and an element of Hτ, we argue as follows. Noting that a(Q, P) and a(Q, P)(Iη(exp(P)) are both representatives of the same equivalence class in Hτ, we have that a(Q,P)exp12P agrees on a dense set with a(Q,P)(Iη(exp(P))exp12PHτ.
We argue similarly for (3), using the commutation relation (4) to obtain that, for all tR and BB(R),
|D|itEB|D|it=12π1R+(P)R1B̂(ξ)eitPeiξQeitP1R+(P)dξ=12π1R+(P)R1B̂(ξ)eitξeiξQ1R+(P)dξ=EB+t.

We believe that the following problems are worth investigating:

  • Do the results of Sec. VI extend to the free relativistic particle with mass m?

    The obstruction to overcome is that the algebra underlying our computations no longer works for the Hamiltonian m2Δ with Δ (the Laplace operator). In all likelihood, techniques involving Dirac operators have to be used to handle this case.

  • Embed Theorems VI.1 and VII.3 into the Tomita–Takesaki framework.

    As was already pointed out, there are two obstructions to be overcome: (i) both theorems involve a non-normal semifinite trace; (ii) the operators EB in the POVM’s constructed in these theorems generally do not belong to the von Neumann algebra M used in these theorems.

  • Identify a physical system whose equilibrium state is described by the noncommutative integral.

    This question is speculative; we have included the treatment of the noncommutative integral simply because it invited itself on mathematical grounds.

  • Interpret the “observation” of time given in Ref. 46 as a POVM associated with a modular/thermal time.The paper46 provides a completely different approach to the thermodynamical origin of time compared to the Connes–Rovelli theory. It identifies an information geometric relationship between time and temperature (derived in a simple model from classical mechanics), and suggests using this relationship (stating that a curvature is constant equal to −1) as a way to “observe” time.

  • Interpret the time POVMs given here in the Page–Wootters formalism (introduced in Ref. 47 and corrected in Ref. 48), i.e., as internal quantum degrees of freedom).This could well use the relativisation procedure from Ref. 16.

Most of this paper was written while the first-name author visited the Mathematical Science Institute at ANU (Canberra) through the MSRVP programme. He would like to thank the staff at ANU for the kind and generous hospitality.

The authors have no conflicts to disclose.

Jan van Neerven: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Pierre Portal: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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