An operational construction of the sum of two non-commuting observables in quantum theory and related constructions

The existence of a real linear-space structure on the set of observables of a quantum system -- i.e., the requirement that the linear combination of two generally non-commuting observables $A,B$ is an observable as well -- is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of the composed observable $aA+bB$ ($a,b\in \bR$) if such measuring instruments are given for the addends observables $A$ and $B$ when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of $f(aA+bB)$ out of the spectral measures of $A$ and $B$. We present such a construction with a formula which is valid for generally unbounded selfadjoint operators $A$ and $B$, whose spectral measures may not commute, and a wide class of functions $f: \bR \to \bC$. We prove that, in the bounded case the Jordan product of $A$ and $B$ can be constructed with the same procedure out of the spectral measures of $A$ and $B$. The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman-Kac formula.


Introduction
The elementary formulation of Quantum Theory in a complex separable Hilbert space H can be described as a non-Boolean probability theory. There, quantum states µ are probability measures over the orthomodular separable σ-complete lattice L (H) of orthogonal projectors (the order relation being the standard inclusion of projection subspaces) which generalizes the notion of σ-algebra (see, e.g., [Neu32,Mac63,Var07,BeCa81,Lan17,Mor17]).
H depends on the physical system S one intends to study and the elements of L (H) physically represent of elementary propositions of S, also known as YES-NO elementary observables or tests about S. They are therefore assumed to have a definite operational meaning in terms of experimental devices. Commutativity of P, Q ∈ L (H) has the physical meaning of simultaneous measurability of the elementary observables represented by P and Q. In that case P and Q are said to be compatible.
A quantum state is a probability measure over L (H), namely a map µ : L (H) → [0, 1] which satisfies µ(I) = 1 and µ(s-n∈N P n ) = n∈N µ(P n ) where the former sum is defined with respect to the strong operator topology and the projectors P n ∈ L (H) are compatible and mutually exclusive: P n P m = 0 if n = m. Actually, the map µ reduces to a standard probability measure on every Boolean σ-algebra included in L (H), in particular, on every maximal set of commuting projectors.
If, as assumed, the Hilbert space is separable and furthermore dim(H) = 2, the celebrated Gleason theorem [Gle57] establishes that the measures µ are in one-to-one correspondence with statistical operators, i.e., positive, trace-class, unit-trace operators ρ µ : H → H. The correspondence is made explicit by the identity µ(P ) = tr(ρ µ P ) for all P ∈ L (H). (That crucial theoretical achievement has been the object of many generalisations [Dvu92], but we are here content with the original basic formulation). Extremal elements ν of the convex body of probability measures over L (H) are also known as pure states. Under the said hypotheses on H, they are one-to-one with unit vectors ψ ν up to phases and ν(P ) = tr(P |ψ ν ψ ν |) = ||P ψ ν || 2 is the probability that P ∈ L (H) is true if the state is pure and represented by ψ ν .
The issue we want to address in this work concerns the set O S of observables of S. An observable A is primarily a collection of elementary propositions P admits an explicit operational interpretation in terms of the statement "the outcome of the measurement of the observable A belongs to E". We stress that there must be one (or several) measuring instrument(s) associated to the class {P (A) E } E∈B(R) . Sound physical and logical arguments (see, e.g., [Var07,Lan17,Mor17]) impose that the collection {P Summing up, we are committed to think that the set O S of observables of S is made of some PVMs over R taking values in L (H). Among them, compatible observables A, B are by definition those whose associated PVMs are made of pairwise compatible projectors, i.e., P for all E, F ∈ B(R). At this juncture, the spectral machinery establishes that there is a one-to-one correspondence between PVMs over H and (generally unbounded) selfadjoint operators A : D(A) → H (see section 1). As a matter of fact, it holds (1) The use of the same symbol to denote the observable and the uniquely associated selfadjoint operator is now natural and safe. We conclude that O S is made of some selfadjoint operators A : D(A) → H and the natural question emerges about how O S is large in the whole set of selfadjoint operators over H. Selfadjoint operators enter the theory also through another door, which is widely open in complex Hilbert spaces only 1 . We mean the theory of quantum symmetries which plays a fundamental role in defining (elementary) physical systems. Here, upon the choice of a formulation of Wigner's theorem (according to the preferred definition of quantum symmetry [Lan17,Mor17] one adopts) the Lie group G representing the relevant physical transformations on S gives rise to a strongly continuous (generally projective) unitary representation U : G → B(H). In turn, the representation U produces an associated representation of the real Lie algebra g of G on a suitable common dense invariant domain D (the Garding or the Nelson spaces) in H. In this way, the generator a ∈ g of a oneparameter subgroup of G is mapped to an operator iA(a) via Stone's theorem, where A(a) is selfadjoint. A(a) is naturally interpreted as an observable and its physical meaning is usually grasped from the physical interpretation of G itself, including the nature of the instruments devoted to measure P (A(a)) . In all known situations, it is eventually possible to enlarge G to include the time evolution of S. The celebrated correspondence between dynamical symmetries and associated conserved observables shows up here, in terms of a quantum version of the Noether theorem. However, the argument also illustrates a remarkable feature of the subset of observables A(a) associated to the elements a ∈ g: they form a real linear space of operators acting on the said common dense invariant domain D since a real Lie algebra is first of all a linear space and the map g ∋ a → iA(a) preserves the real Lie algebra structure. Furthermore, D is a core for every element in the image of the representation of g so that all operators aA(a) + bA(b) are selfadjoint for every a, b ∈ R and a, b ∈ g.
In the absence of a symmetry group, and referring to the full set O S , a linear structure in the space of observables is technically more difficult to define due to evident issues with domains. It can be nevertheless defined if referring to a certain subset of bounded, thus everywhere defined, selfadjoint operators representing observables. If S and a, b ∈ R, the operator aA + bB ∈ B(H) is well defined and selfadjoint. It remains to establish if it has the physical meaning of an observable in the general situation where, for instance, A and B do not belong to the representation of the Lie algebra of a symmetry group.
As a matter of fact, it is generally assumed as a further postulate that O S is a linear subspace 2 of the real linear space of selfadjoint operators of B(H) This structure is later enriched [Emc72,Stro05,Lan17,Mor17]) by adding in particular the so-called (nonassociative) Jordan product, and assuming suitable (weak) topological features compatible with the spectral machinery. The final construction turns out to be a concrete Jordan W -algebra in B(H) or, more generally, a concrete von Neumann algebra in B(H) when complex combination of observables are permitted. These structures are actually difficult to physically justify a priori. If assumed, they however promote the theory to a very high level of effectiveness in physics a posteriori.
Already sticking to the linear structure of the set of bounded observables, we argue that a long standing (see, e.g., [Gud78,Stre07]) open issue pops out, with a twofold nature, both physical and mathematical. Suppose S and that we know their respective measuring instruments. If we pick out a, b ∈ R, in spite of the postulate about the linear structure of O S , there is no general way to associate a measuring instrument to aA + bB out of those of A and B. This obstruction is valid unless the observables aA and bB are compatible. In that case, the instruments corresponding to the joint PVM of P (A) and P (B) can be exploited.
The mathematical version of the raised issue is that no general formula is known determining the operator f (aA + bB) -and every projector P (aA+bB) E in particular -as a function of P (A) and P (B) when these PVMs do not commute. This work addresses the outlined problem from the mathematical side. We will prove in particular a formula which connects the selfadjoint operator f (aA+bB) to the PVMs of the selfadjoint operators A and B for a suitable class of continuous functions f : R → R. The formula is valid for generally non-commuting P (A) and P (B) , also in the case of unbounded selfadjoint operators A and B under the condition that aA + bB is essentially selfadjoint over D(A) ∩ D(B). The class of functions f is sufficiently large to include all polynomials when A and B are bounded, in particular aA + bB itself.
As a byproduct of the above-mentioned result for the case of A and B bounded, we find also an identity connecting the Jordan product A • B := 1 2 (AB + BA) to the generally non-commuting PVMs P (A) and P (B) .
Remark 1.1 (1) The problem tackled in this work is a consequence of the existence of incompatible observables so that it does not arise in classical physics. Given a classical system S, for instance described in its phase space, there is a minimal set of observables such that all remaining observables are functions of them. Therefore, measuring the first ones exactly amounts to measuring all observables of S.
(2) The assumption of the existence of a real linear space structure over the set of observables is crucial also in much more abstract, and apparently more operational, formulations of the quantum theory [Mac63,Stro05]. In particular the ones leading to the notion of Jordan Banach * -algebra first and that of C * -algebra finally, and discovering the Hilbert space formulation a posteriori through the GNS construction. There, observables are defined in terms of their expectation values and it is postulated that, given a pair of observables A and B and a corresponding couple of reals a, b, there is an observable denoted by aA + bB whose expectation values are the linear combination of the expectation values of A and B with coefficients a and b. Though this observable is proved to be unique (because the states are reasonably supposed to separate the observables), no ideas are supplied to solve the problem we pointed out: how we can measure aA + bB if we know how to measure A and B when they are incompatible.
Structure of the work. The paper is organized as follows. After recalling some general notion and the general notation used throughout, in section 2 we tackle the problem raised above first in the finite-dimensional case and next in the infinite dimensional case. Dealing with a pair of selfadjoint operators A, B in a complex Hilbert space and a continuous bounded function f : R → C of a certain class, our main result consists of a formula where the operator f (aA + bB) is written as the (strong) limit of a sequence of certain operatorvalued integrals whose measure is an alternating product of spectral measures of A and B. As a matter of fact, we prove Eq. (7) in the finite-dimensional case and Eq. (29) in the much harder infinite-dimensional case. Section 3 is devoted to study some applications: how to construct the PVM of aA + bB out of the ones of A and B, how to write down the Jordan product, and the section eventually ends with the discussion of a nice interplay with both the theory of Feynman integration in the phase space as well as the so called Feynman-Kac formula. Section 4 contains some counterexamples which demonstrate how part of the structure introduced in the infinite dimensional case cannot be improved. The final section 5 concerns some attempts to give some physical interpretation of the constructed formalism in the finite dimensional case where no technical issues take place, allowing us to focus directly on physics.
General conventions and notation. We adopt throughout the paper the standard definition of complex measure [Rud85], as a map µ : Σ → C which is unconditionally σ-additive over the σ-algebra Σ. With this definition the total variation ||µ|| := |µ|(Σ) turns out to be finite. B(X) denotes the Borel σ-algebra over a topological space X and L p (X, ν) denotes the standard L p space with respect to a positive Borel measure ν [Rud85] (in case ν is complex, the associated L p spaces are those referred to |ν|). L p (R, dx) (also with dy in place of dx) always refers to the L p space with respect to the Lebesgue measure (viewed as a Borel measure) on R. In case the measure ν is defined on a generic σ-algebra Σ over the set X, we use the more precise notation L p (X, Σ, ν). If E ⊂ X, for any given set X, then the indicator function A Hilbert space is always supposed to be complex. We assume the usual convention concerning standard domains of composition of operators over a Hilbert space H, A : indicates the unital C * -algebra of bounded everywhere defined operators over the Hilbert space H, and L (H) ⊂ B(H) denotes the lattice of orthogonal projectors: P = P * = P P .
The spectrum of an operator A is always denoted by σ(A). According to the spectral theory (details, e.g., in [ReSa75II,Mor17]), if T : D(T ) → H is a generally unbounded selfadjoint operator in H and g : R → C is measurable, the know definitions are valid is the projection-valued measure (PVM) -also known as spectral measure -uniquely associated to T . The right-hand side of (2) can be defined [Rud93] 3 as the unique operator T on D(g(T )) such that φ|T ψ = R g(τ )dµ φ,ψ (τ ) for all φ ∈ H and ψ ∈ D(g(T )). (3) The support of P (T ) coincides to the spectrum σ(T ) and the integration in (2) can be restricted accordingly without affecting the identity. The spectral decomposition of T is nothing but (2) with g : R ∋ τ → τ ∈ R and the spectral theorem just states that this decomposition exists and P (T ) is uniquely determined by T .
2 From C n to an infinite dimensional Hilbert space

The finite-dimensional Hilbert space case
The finite-dimensional case is a comfortable arena where building up our formalism, since no problems related with domains and choices of topologies take place. Let us therefore suppose that H is finite-dimensional. Without loss of generality, we can assume H = C k . From the elementary spectral theory, we define for every function g : R → R and every selfadjoint operator, that is a k × k a Hermitian matrix T : H → H. σ(T ) ⊂ R denotes its set of eigenvalues and P (T ) τ indicates the orthogonal projector onto the eigenspace of τ ∈ σ(T ), so that {P (T ) τ } τ ∈σ(T ) completely determines the PVM P (T ) of T . No technical snags pop out with the sum in (4), since σ(T ) is a non-empty set of k = dim(H) < +∞ elements at most.
We now focus on a couple of selfadjoint operators A : H → H, B : H → H with, in general, AB = BA. We intend to write f (aA+bB) in terms of the PVMs P (A) and P (B) for a, b ∈ R and where f : R → C is every given sufficiently regular function. For technical reasons which will be evident shortly, we henceforth assume that: . Every f ∈ S (R) in particular satisfies the said three hypotheses. Holding (i),(ii),(iii), the pointwise inversion theorem of the Fourier transform is valid: The pivotal tool underpinning our result is the celebrated Trotter formula, whose convergence is here referred to every normed topology on B(H), since that (Banach) space is finite dimensional. Spectrally decomposing the exponentials on both sides into a finite linear combinations of matrices according to (4) and using of Lebesgue's dominate convergence theorem term by term while integrating both sides againstf (t), we have, Exploiting (4), the found identity can be expanded as Since f (x) = Rf (t)e itx dt and taking (4) into account on the left-hand side, we obtain the prototype of our general formula In H = C k , it is not difficult to prove that AB − BA = 0 implies e i t N aA e i t N aB N = e it(aA+bB) . Hence, following our derivation of (6), the limit in the right-hand side of (6) is not necessary because the argument of the limit turns out to be constant in N. Therefore, for commuting A and B, equation (6) reduces to which is nothing but the usual spectral decomposition of the left-hand side with respect to the joint PVM of A and B.
A physically important case of (7) would be obtained by choosing f = 1 [α,β) (the indicator function of the set [α, β)). As a matter of fact, is the spectral projection of [α, β) and this family of projectors, when α < β, includes the full information of the PVM of aA + bB. Unfortunately such f does not satisfy (i) and (iii), so that a further regularization procedure is necessary by means of a family 1 (ǫ) [α,β) ∈ S (R) which suitably converges to 1 [α,β) as ǫ → 0 + . In this way, the following identity can be established µ N (8) We shall prove it later into a much more general context as identity (35).
The requirement f,f ∈ L 2 (R, dx) can be relaxed making stronger the regularity requirement on f as a consequence of the following argument. If f, g satisfy (i),(ii),(ii) and also f (x) = g(x) for x ∈ [− aA − bB , aA + bB ], where the norm is the operatorial one, then the well-known general estimate for selfadjoint operators σ(T ) ⊂ [− T , T ], yields both f (aA + bB) = g(aA + bB) from (4), and It is consequently safe to extend the validity of (6) to every complex valued f ∈ C ∞ (R) simply smoothly changing f to a function of S (R) outside the interval [− aA − bB , aA + bB ].
In particular, f can be chosen as a polynomial of a single real variable. In this way, we also obtain the elementary Jordan-algebra operations on A and B written in terms of the PVMs of A and B as the second identity below.
The former is simply obtained from (6) choosing f (s) = s. The latter arises from a more involved though elementary procedure. The left-hand side of (11) can be written as . Linearity in f of the right-hand side of (6) (also with different constants a, b) immediately gives rise to (11). A more detailed discussion will appear in section 3.
With an argument similar to the one leading to (9) and using the fact that σ(aA + bB) is made of a finite discrete number of reals, we achieve another useful result. If β ∈ σ(aA + bB), identity (12) can be alternatively written where 1 ′ [α,β) ∈ D(R) is a map which attains the constant value 1 over [α, β) and smootly vanishes on σ(aA + bB) \ [α, β).
When trying to extend (6) to the case of a general (separable) Hilbert space H for generally unbounded selfadjoint operators A and B, several evident technical issues take place. First of all, usual domain problems have to be fixed. These problems are tantamount to corresponding domain issues with Trotter's formula which are well known and definitely fixed [ReSa75I, Theorem VIII.31]. A sufficiently general setup consists of assuming that aA + bB is essentially selfadjoint over its natural domain D(A) ∩ D(B) (if ab = 0). A much harder problem is the interpretation of the operator-valued integral appearing in the right-hand side of (6), especially in case the spectrum of either A or B includes a continuous part. Actually problems of a similar nature arise also when A and B have pure point spectra, but H is infinite dimensional. In the following sections we will address those technical problems and other related ones, ending up with a wide generalization of (6)cf. Theorem 2.5.

The unital Banach algebra of complex Borel measures
Before entering the technical details of the core of the paper it is necessary to introduce the space of functions we will use on both sides of the extension of (6).
The linear map is injective (this is a straightforward extension of [Bill12, Theorem 26.2]) so that it defines a linear isomorphism. From the standard properties of Fourier transform, it is not difficult to prove that F includes the maps satisfying (i),(ii),(iii) we used in section 2.1. Furthermore Above, C b (R) is the commutative unital Banach algebra (with norm · ∞ ) of bounded continuous complex-valued functions over R, S (R) and D(R) are respectively the space of Schwartz (complex) functions and the space of the (complex) test functions over R, and S ′ (R) and D ′ (R) denotes the correspondingly associated spaces of distributions. The linear space M (R) turns out to be a commutative unital Banach algebra where (a) the product of two measures ν and ν ′ is their convolution ν * ν ′ , (b) the unit is the delta measure δ 0 concentrated at 0, (c) the norm of a measure ν is defined as its total variation ν = |ν|(R). M (R) also admits a norm-preserving unit-preserving antilinear involution given by the complex conjugation of measure ν * := ν. The linear isomorphism (14) induces a unital commutative Banach algebra structure over F (R) when defining f ν F := ν . More precisely, the map F promoted to an isomorphism of Banach algebras transforms the convolution of measures into the pointwise product of corresponding functions and the unit of B(R) to the constant function 1. The involution of M (R) becomes the normpreserving unit-preserving antilinear involution over F (R), given by f * (x) := f (−x), the bar denoting the complex conjugation. The definition of f F easily implies

Regularized products of non-commuting PVMs of selfadjoint operators
This section and the subsequent one are devoted to extend formula (6) to the case of an infinite-dimensional, though separable, Hilbert space H. We will suppose that the operators A and B are generally unbounded selfadjoint operators with domains D(A) and D(B) respectively, and that their linear combination aA + bB, for suitable a, b ∈ R, is essentially selfadjoint on its standard domain according to section 1.
In other words, we want to construct a functional calculus and the spectral measure of the selfadjoint operator given by the closure aA + bB out of the PVMs P (A) , P (B) of A and B respectively, proving the following suggestive formula if ψ ∈ H and f ∈ F (R): The operator f aA + bB on the left-hand side is independently defined by (2) and its domain is the whole H according to (2) since f is bounded. As a first step, we address the problem of the interpretation of the operators after the symbol of limit on the right-hand side of (17). In fact, we cannot generally interpret the integration appearing in (18) like the one in (7), i.e. as referred to the joint PVM of N copies of P (A) and P (B) (see, e.g., [ReSa75II,Mor17]). This is because we are focusing on the generic situation where P (A) and P (B) do not commute. In the general case, the operators (18) are defined out of a regularized natural quadratic form we are going to introduce with the following crucial technical result. We stress that we will not use the hypothesis of essential selfadjointness of aA + bB at this stage of the construction. (2), and Q := {Q n } n∈N a sequence of orthogonal projectors over respective finite-dimensional subspaces such Q n → I strongly as n → +∞. If φ, ψ ∈ H and N = 1, 2, . . ., then the following facts are valid.

Lemma 2.2 Let A and B be a pair of selfadjoint operators over the separable Hilbert space H as in
(a) For every n ∈ N, there is a unique complex Borel measure ν with I l , J l arbitrary Borel sets in R. The measure satisfies with ν f := F −1 (f ) according to (13)(14).
Proof. See Appendix A.
Analogously to the use of (3) to define the right-hand side in (2), the idea is now to define the operator in (18) in order that the associated quadratic form coincides to the limit of the regularized sequences as on the left-hand side of (21).

Theorem 2.3 Let A and B be a pair of selfadjoint operators over the separable Hilbert
such that that Above, for every given n ∈ N, the Borel complex measure ν for a sequence of finite-dimensional orthogonal projectors Q := {Q n } n∈N with Q n → I strongly as n → +∞. The limit in (23) is however independent of the choice of {Q n } n∈N .
The following further facts are true.
are respectively a Schwartz distribution and a B(H)-valued Schwartz distribution.
Proof. If ν f := F −1 (f ) according to (13)-(14), the map appearing on the right-hand side of (21) is by construction linear in f and ψ, antilinear in φ and satisfies As a consequence, identity (21) permits us to exploit Riesz' lemma defining the wanted operator as required and independently from the regularizing sequence φ,ψ (f ) and (28) prove (i) and (ii). Regarding (iii), we observe that from the continuity properties of the Fourier transform, S (R) ∋ f n → 0 in the S -topology (as n → +∞) implies that the sequence of antitransformsf n vanishes in the same topology, so that f n L 1 (R,dx) → 0 in particular. Since (16), we have that f n F → 0. The linear maps (25) and (26)  ] so that the integral in (23) vanishes as well for every n, producing 0 as limit for n → +∞.
(v) has a direct proof just using the definitions.
Remark 2.4 (1) If H is finite-dimensional, all Q n on the right-hand side of (19) can be removed, simply choosing Q n = I, finding the formulas achieved in section 2.1. For infinite-dimensional H, there are cases where no complex measure satisfying the identity (19) exists if Q n = I for all ncf. 4.1.
(2) The existence of sequences {Q n } n∈N strongly approximating the identity operator with finite-dimensional projection spaces is equivalent to separability of H. Hence, separability hypothesis cannot be relaxed.

Main result
We have reached a position to establish our main theorem when, in addition to the already assumed hypotheses on A and B, we use the requirement that aA + bB is essentially selfadjoint.
Theorem 2.5 Let A, B be self-adjoint operators on a separable Hilbert space such that aA + bB is essentially self-adjoint for some given a, b ∈ R in its standard domain. The following facts are true for f ∈ F (R).
as the unique operator S ∈ B(H) such that is analogously defined. The former operator is already known. Indeed, from (21) and (23), we find Let us pass to focus on the operator in (31). Let P (aA+bB) be the spectral measure of the self-adjoint operator aA + bB and we set µ φ,ψ (E) = ψ|P (aA+bB) (E)ψ , E ∈ B(R), the Borel complex measure associated to φ and ψ. By standard functional calculus, by writing f ∈ F as in (13) and exploiting Fubini's theorem (all measures are finite and the integrand is bounded), we obtain: In other words, Taking advantage of (32) and (33), the thesis of the theorem can be rephrased to (34) Representing the left-hand side as an inner product and expanding it, taking the previous definitions into account, we find that the squared norm above equals Fubini's theorem permits us to re-write the above integral as where dν(t) ⊗ dν(s) is the product measure. By the Trotter product formula [ReSa75I, Theorem VIII.31], which is valid in our hypotheses on aA and bB, both entries of the scalar product vanish as N → +∞. This fact implies that the integral itself vanishes as a consequence of Lebesgue's dominate convergence theorem, since the product measure is finite and the integrand is uniformly bounded in (t, s) as the involved operators are unitary. In summary, (34) is valid and the proof of (a) ends since the last statement is an obvious consequence of aA + bB = bB + aA.
(b) Exploiting the structure of the proof of (a), the thesis is valid if, for every t ∈ R, Let us prove that this identity is in fact true. Since P (aA) and P (bB) commute, referring to their joint PVM P , we have From the general properties of the integral of bounded functions with respect to a given PVM, Taking the strong limit as N → +∞, Trotter's formula yields As an application of Theorem 2.5, we prove that the knowledge of the PVMs P (A) , P (B) of A and B suffices to determine the PVM of aA + bB among all PVMs over R. We take advantage of the well-known result that 4 P (aA+bB) is known once the projectors P (aA+bB) [α,β) are known for all α, β ∈ R with α < β.
Remark 3.2 (1) A statement analogous to that in Theorem 3.1 can be proved for the elements P Together with the injectivity of the Fourier transform for finite measurecf. equation (14) -equation (37) provides another (more indirect) relation between P (A+B) and P (A) , P (B) .
Notice that Equation (37) can be proved without invoking neither Lemma 2.2 nor Theorem 2.5 -although it can be seen as an application of Theorem 2.5 for the case f = 1 R = f δ ∈ F (R), δ denoting the Dirac measure centred at 0. and aA+bB is essentially selfadjoint thereon for every a, b ∈ R, its closure being a generator of the Weyl algebra [Mor17] according to Stone's theorem. As a consequence, the previously developed theory can be applied to the pair A,B.
Taking advantage of Weyl algebra commutation relations, we have that for all φ, ψ ∈ H We argue that equation (29) reproduces the exact result (38) once we choose f = 1. A popular computation shows that, if E ∈ B(R), , for respectively ψ ∈ D(A) or ψ ∈ D(B) and where· denotes the Fourier-Plancherel transform. With these information we wish to compute the limit for N → +∞ of . Notice that we can avoid the regularization Q n by considering ρ as a distribution -there is no integration in t. A direct computation leads to 5 We recall that our convention for the Fourier transform is g(k) := (2π) −1/2 R g(x)e ixk dx, g(x) = (2π) −1/2 R g(k)e −ikx dk for g ∈ S (R)cf. (13).

Relation with Feynman integration
Next application proves the close relation between formula (29) Let us recall that if T is a selfadjoint operator on H with spectral mesure P (T ) and h : R → R is a Borel measurable map, the spectral measure P (h(T )) of the selfadjoint operator h(T ) (defined via functional calculus out of T ) is related to P (T ) by the formula for any Borel-measurable function f : R → C. As a consequence, if f ∈ F (R) and using h : R ∋ r → r 2 , (29) implies If we now choose the function f ∈ F (R) of the form f (λ) = e −itλ , with t ∈ R, for any φ, ψ ∈ H we have the following representation for the matrix elements of the unitary evolution operator Writing explicitly the Schwartz distribution (25) in terms of the wave functions of the vectors φ, ψ ∈ H, iterated applications of Fourier transform and its inverse yield where we recall that the integrals over R 2N are just a formal symbol standing for the limit of the regularizing procedure described in (23), since in this case the distribution (25) is not associated to a complex measure on B(R 2N ). Formula (41) admits an interpretation in terms of a phase space Feynman path integral representation for the time evolution operator U(t), as discussed, e.g., in [AlGuMa02,KuGo11]. Indeed, it is possible to look at the exponent appearing in the integral on the right-hand side of (41), namely the function as the Riemann sum approximation of the classical action functional in the Hamiltonian formulation: where h : R 2 → R is the classical Hamiltonian of the harmonic oscillator, i.e.
A completely similar discussion can be repeated in the case the harmonic oscillator potential is replaced with a more general potential, namely a measurable map V : R → R such that the sum of the associated multiplication operator M V : D(M V ) ⊂ H → H and B = P 2 is essentially selfadjoint. We stress that the limit for N → ∞ on the righthand side of (41) cannot be interpreted in terms of a well defined integral over an infinite dimensional space of paths (q, p) : [0, t] → R 2 , formally written (in Feynman path integral notation) as φ|e −itH ψ = {(q,p):[0,t]→R 2 } e iS[q,p]φ (q(t))ψ(q(0))dqdp since it is impossible to construct a corresponding complex measure, as extensively discussed in [AlMa16] and in section 4.2.

Bounded or semibounded A, B, Jordan product, Feynman-Kac formula
If the selfadjoint operators A and B are bounded then aA + bB ∈ B(H) is defined for every choices of a, b ∈ R and (20) permits us to extend (29) to the case of f ∈ C ∞ (R). A similar result is valid when A and B are bounded below.
Let us start with A, B ∈ B(H). For any choice of a, b ∈ R we then have As a consequence, (21) is still valid withf in place of f on the right-hand side, and the limit does not depend neither on the choice of the regularizing sequence Q := {Q n } n∈N nor on the choice of the saidf : A similar result is found when A and B are (generally unbounded) selfadjoint operators which are bounded below and a, b ≥ 0 are such that aA + bB is essentially selfadjoint. In this case aA + bB is bounded below as well with inf σ(aA + bB) ≥ a inf σ(A) + b inf σ(B) .
With the said A, B, a, b, both (42) and (43) are Identity (43) is sufficient in both cases for giving the following definition based on Riesz' lemma.
Definition 3.4 Assume that one of the following two cases holds (aλ n + bµ n ) dP is defined as the unique operator satisfying (23).
This definition is an evident extension of the definition for f ∈ F (R) appearing in Theorem 2.3 because, iff ∈ S (R) ⊂ F (R) is defined as above with respect to f , then The remaining properties (i)-(iv) of Theorem 2.3 hold as well provided f F is replaced for f F on the right-hand side of (24) in (ii). With this extended definition, Theorem 2.5 is still valid as the reader immediately proves noticing in particular that for the very definition (2)-(3), f (aA + bB) =f (aA + bB) .

Proposition 3.5 If either (a) or (b) in Definition 3.4 is valid, then
If with A, B ∈ B(H), definition 3.4 applies in particular to the operator aA + bB as in (46) below, when referring to the smooth map f : R ∋ x → x ∈ R. That identity shows how the real linear space structure of the algebra of bounded observables can be constructed employing the PVMs of the involved observables only. The whole algebra of bounded observables B(H) has another natural operation that is the (non-associative) Jordan product: The Jordan product of A and B admits an expansion similar to (46) where, again, only the PVMs of A and B is used.
Theorem 3.6 Let A, B ∈ B(H) be selfadjoint operators, with H separable, whose PVMs are respectively denoted by P (A) and P (B) , then the following facts are true.

(b) The Jordan product of A and B satisfies
is the unique operator such that Above the complex measures ν Proof. We only have to prove (b). First of all we have that As discussed above, the limits for n → +∞ of the final three addends exist and do not depend on the used sequence Q. We conclude that the right-hand side of (48) exists and does not depend on Q. More precisely, from (43), wheref : R → R is every given mapf ∈ S (R) such thatf (x) = x 2 for the values At this point Riesz' lemma proves that there is a unique operator satisfying (48). Decomposing ℓ on the right-hand side of (48), and noticing that analogously is an immediate consequence of (45) applied to the three addends on the right-hand sides of the two identies above.
Case (b) In Proposition 3.5 has an interesting application which proves the interplay of formula (29) and the so-called Feynman-Kac formula.
Example 3.7 Let us consider again the setting of section 3.2. Since in this case both operators A and B have positive spectrum, we can apply formula (45) also for f with exponential growth. In particular we can choose f : R → R of the form f (λ) = e −τ λ , with τ > 0, obtaining the following representation for the matrix elements of the semigroup generated by the selfadjoint operator H (see Eq. (39)): If ψ ∈ L 1 (R, dx), the integrals on R 2N +1 appearing on the right-hand side are absolutely convergent thanks to the fast decaying properties of the exponentials appearing in the integrand. In particular, in this case, by applying Fubini Theorem and integrating with respect to the k variables, we get (52) In this case, thanks to the presence of the Gaussian density in the integrand, the limit for N → ∞ of the finite dimensional integrals appearing in the right-hand side of (52) can be interpreted in terms of an integral over the Banach space C τ (R) of continuous paths ω : [0, τ ] → R endowed with the uniform norm, the corresponding Borel σ-algebra and the Wiener Gaussian measure W . By a standard argument (see, e.g., [Sim05]) formula (52) yields the celebrated Feynman-Kac formula: ψ(x + ω(τ ))e − Ω 2 2 τ 0 (ω(s)+x) 2 ds dW (ω)dx.
As remarked at the end of section 3.2, the discussion above generalizes to the case the harmonic oscillator potential is replaced with a Borel measurable non-negative map V : R → R such that the sum of the associated multiplication operator M V : D(M V ) ⊂ H → H and B = P 2 is essentially selfadjoint.

Counterexamples
In this section, we provide a series of counterexamples which make evident the rigidity of the structures involved in the rigorous generalization of formula (6) to infinite dimensional casecf. Theorem 2.5.
In section 2, we introduced a regularization machinery Q n to safely define a family of measures ν (N,n) φ,ψ whose distributional limit n → +∞ permitted us to define the operator integral in equation (22)cf. Theorem 2.3. This may appear artificial at first glance and one may wonder if the regularization Q n can be removed. This is not the casecf. section 4.1.
Moreover, even in the finite dimensional case -where Q n can be chosen to be the identity and ν

The Q n -regularization is generally necessary
Lemma 2.2 introduces the regularized measure ν (N,n) φ,ψ which is built out of the PVMs P (A) , P (B) associated with the operators A, B together with a suitable "regularizing" sequence of projectors Q n . One may wonder whether it is possible or not to avoid the regularization Q n , thus considering a complex measure ν Unfortunately ν (N ) φ,ψ generally does not extend to a complex measure on R 2N as is in particular shown by the following physically relevant example.
is it evident that ν N φ,ψ does not extend to a well-defined complex measure on R 2N because the total variation of such complex measure would be ||ρ φ,ψ || L 1 (R 2N ,d N xd N k) which is not finite.

The sequence of measures ν (N )
φ,ψ generally does not admit (projective) limit Let us consider the elementary case of a finite-dimensional Hilbert space, where the regularization driven by Q is not necessary. In fact, we can simply fix Q n = I.
A natural issue is whether or not it is possible to interpret the limit for N → ∞ on the right-hand side of (6) in terms of a integral on an infinite dimensional space. Indeed, in the particular case where H is finite dimensional, for any φ, ψ ∈ H it is possible to define a family of complex Borel measures {ν that is for all I 1 , ..., It is then meaningful to investigate whether the projective family of complex measures {ν (N ) φ,ψ } N ∈N admits a projective limit. For that, let us consider the infinite dimensional space (R × R) N of all sequences {(λ n , µ n )} n∈N . Moreover, let Σ be the σ-algebra on (R × R) N generated by the cylinder sets π −1 N (E), where N ∈ N, E ∈ B(R 2N ) and π N : (R×R) N → (R×R) N is the canonical projection map. Within this setting, it is meaningful to investigate the existence of a complex bounded measure ν φ,ψ on ((R × R) N , Σ) such that ν (N ) φ,ψ = (π N ) * ν φ,ψ for all n ∈ N (see appendix C for further details). for all n ∈ N (see appendix C for further details). However, as proved in Appendix C, a necessary condition for the existence of ν φ,ψ is a uniform bound on the total variation ν Example 4.2 Let us consider the special case H = C 2 , where A := σ z and B := σ x , σ z , σ x denoting two Pauli matrices. In this case, the Hilbert space is finite dimensional and for any N ≥ 1 and φ, ψ ∈ C 2 , it is possible to define the complex Borel measure ν N φ,ψ on {±1} 2N (the topology being the discrete one) as: a 1 , b 1 , ..., a N where |z + , |z − , resp.|x + , |x − , are the normalized eigenvectors of σ z , resp. σ x . The total variation of ν (N ) φ,ψ can be computed explicitly here. Indeed, by writing the vectors φ, ψ as linear combinations of the vectors of the orthonormal basis {|z + , |z − } and {|x + , |x − }, namely: we get for any (a 1 , b 1 , ..., a N a 1 , b 1 , ..., a N , b N Hence we obtain ν The sequence of mesures ν (N ) φ,ψ cannot admit a projective limit, since sup N ν The illustrated no-go result is strictly related to the non commutativity of the spectral projectors P (A) and P (B) . In fact, in the general case of a separable Hilbert space H, if the spectral measure P (A) and P (B) commute then, for any couple of normalized vectors φ, ψ ∈ H and for any N ≥ 1, the formula defines a complex measure with finite total variation. In particular, since φ, P (A) the following identity holds for any set E ∈ B(R 2N ), where ν (1) J ψ is associated to the joint spectral measure of A and B. By (53) it is simple to obtain the following inequality valid for any N ≥ 1.
In addition, in the particular case where φ = ψ and φ = 1 formula (6) admits a simple probabilistic interpretation. Indeed, in this case the measures {ν (N ) φ,ψ } N form a consistent family of probability measures that can be interpreted as the joint distributions of a sequence {ξ n , η n } n of random variables with values in R 2 which are in fact trivially completely correlated, i.e. (ξ n , η n ) = (ξ 1 , η 1 ) for all n ∈ N, and the distribution of the random vector (ξ 1 , η 1 ) is given by the joint spectral measure of A and B.
In the case the commutativity condition is not fulfilled, then a uniform bound as (54) cannot be obtained as the folloing example shows.  {±1} 2N defined for any (a 1 , b 1 , ..., a N , b N a 1 , b 1 , ..., a N , b N It is simple to verify that φ,ψ || = +∞, unless the spectral measures commute.

commuting PVMs: a counterexample
Let us consider once again the sequence of finite complex measure ν Unfortunately this is not the case as shown in particular by the following example.  L 1 ([0, 1], dx) to some function therein. However this is impossible since ρ n weakly converges to the Dirac delta δ centred at 0 which cannot be represented by a function of L 1 ([0, 1], dx).

Physical interpretation
In the peculiar case where the Hilbert space H is finite dimensional and no technical issues must be addressed, it is interesting to investigate physical interpretations of formula (6).

Noncommutativity and quantum interference terms
We start the analysis with the following elementary observation. Since the identity operator can be decomposed either as I = λ∈σ(A) P Above, the state vector ψ is decomposed into a coherent superposition of generally non mutually orthogonal vector states. These vectors would be mutually orthogonal if the PVMs commuted. The presence in (6) of these non-orthogonal vectors is the source of a quantum interference phenomenon we are going to illustrate. The weak version of formula (6) reads as follows for any φ, ψ ∈ H. In particular when φ = ψ we get As usual, the left-hand side coincides to the expectation value of the observable f (aA+bB) when the state is pure and represented by the unit vector ψ. In particular, if f = 1 [α,β) (in this case (57) should be replaced by (12)), the left-hand side is nothing but the probability of obtaining an outcome in [α, β) when measuring aA + bB over the pure state represented by the unit vector ψ.
(1) The first type admits to some extent (see (1) Remark 5.1 below) a "classical" probabilistic interpretation relying on a recursive use of Born's rule and the standard Lüders-von Neumann's post-measurement state postulate (see, e.g., [BeCa81,Lan17,Mor17]). As is well-known, the former postulate states that, if the initial pure states is represented by the unit vector ψ ∈ H, the probability that the outcome of an elementary YES-NO observable P ∈ L (H) is 1 (YES) is ||P ψ|| 2 . The latter postulate says that, if the outcome resulted to be 1, then the post measurement state is represented by the normalized vector ||P ψ|| −1 P ψ. As a matter of fact, the positive number P can be interpreted as a conditional probability 6 : the probability that, given the initial pure state of the system described by the normalized vector ψ ∈ H, in a sequence of 2N measurements of observables O j , J = 1, ..., 2N, with O j = B for j odd and O j = A for j even, the sequence of outcomes is exactly (µ N , λ N , ..., µ 1 , λ 1 ). Let us illustrate this point in some details. Given a general observable O in our finite dimensional Hilbert space with (necessarily discrete) spectrum σ(O), and a normalized vector φ ∈ H, we shall denote with the symbol P(O = o|φ) the probability that, given that the (pure) state of the system is represented by φ, the outcome of the measurement of O is the real number o ∈ σ(O), namely is the spectral projector associated to the eigenvalue o ∈ σ(O). Let us denote by ψ λ j ,µ j ,λ j+1 ,...,λ N ,µ N and ψ µ j ,λ j+1 ,µ j+1 ,...,λ N ,µ N the post-measurement normalized vectors with j = 1, ..., N. We then have: By induction over N we get the wanted result: It is now evident that the right-hand side has the natural interpretation as the probability that, given the initial pure state of the system described by the normalized vector ψ ∈ H, in a sequence of 2N measurements of observables O j , J = 1, ..., 2N, with O j = B for j odd and O j = A for j even, the sequence of outcomes is exactly (µ N , λ N , ..., µ 1 , λ 1 ).
(2) The remaining terms appearing in the last line of equation (58) of the form µ } µ∈σ(B) commute, it is easy to check that all the interference terms (60) vanish as already remarked. In this sense, their presence has an evident quantum nature.
Remark 5.1 (1) It is worth stressing that also the quantum conditional probability discussed in (1) enjoys different properties than those of the classical conditional probabilty when the PVMs do not commute. This is because we are here dealing with the quantum notion of probability characterized by Gleason's theorem, defined on a orthomodular lattice rather than a σ-algebra. In particular the classical identity P(U|V )P(V ) = P(V |U)P(U) has no corresponding at quantum level when U and V are replaced for quantum incompatible events. This is reflected from the fact that the order in the sequence of measurements P (2) If H is finite dimensional, we have the following convergence in the operator norm We now provide an expansion of f N (A, B) in terms of spectral commutators of A and B which highlights the link between non commutativity of spectral measures and appearance of terms of second kind in the right-hand side of (58). For that let C λ,µ := [P Notice that the product of the terms P (B) µ ℓ ∆ λ ℓ ,λ ℓ+1 provides the contribution arising from the classical convolution formula, namely N ℓ=1 P µ 1 . Products containing a commutator factor C λ,µ represent a "perturbative" quantum correction to the commutative formula. Writing the product of spectral projectors as a sum over insertion of commutators we obtain a "perturbative" expansion of f (A + B) whose first terms are given by

Sum over all possible histories
From now on, we assume a = b = 1 for the sake of simplicity, still sticking to the case of a finite dimensional Hilbert space H.
We shall henceforth set T (O 2j ) ≡ ξ j and T (O 2j−1 ) ≡ η j , j ≥ 1 to shorten the notation. Formally we have for any N ≥ 1 and f 1 , ..., f N , g 1 , ..., where the expectation value on the right-hand side of (61) stands for the integral Π N j=1 f j (λ j )g j (µ j )dν Referring to a measure space (Ω, F , ν), with Ω = (σ(A) × σ(B)) N , F the σ algebra generated by the cylinder sets (see appendix C) and ν a sigma additive measure on F with finite total variation (see section 3.2), the sets {ξ n } n∈N and {η n } n∈N cannot in general be regarded as sequences of random variables (i.e. measurable functions) thereon, where ν extends ν (A,B,...,A,B) φ,ψ . However, for any finite N the integral (62) is well defined and {ξ n } n∈N and {η n } n∈N are called pseudo stochastic process. They find applications, e.g., in the mathematical definition of Feynman path integrals as well as in the construction of Feynman-Kac type formulae for PDEs that do not satisfy maximum principle [AlMa16].
In this context, the construction so far defined describes the spectral measure of A+ B in terms of a particular limiting procedure. Referring to the sequence of pseudo-random variables {ζ n } n∈N defined as ζ n := ξ n +η n , for N ≥ 1 let S N denote the map S N := N n=1 ζ n . Then, according to formula (6), for any function f ∈ F (R) (or also f ∈ C ∞ (R) since here A, B ∈ B(H) automatically because dim H < +∞), it holds which can be interpreted as a weak convergence of the (complex) distribution of the arithmetic mean of the first N pseudorandom variables ζ n to the measure ψ . Since we are dealing with discrete spectra, the formula above can be rephrased to Each factor on the right-hand side (like φ (A) (r 1 )h k 1 (r 1 )) is made of a product of a pair of L 2 functions with the same argument (r 1 ), therefore it is L 1 with respect to the corresponding measure (ν (A) ). Therefore ρ is a σ-additive complex measure with total variation |µ φ,ψ,Q L 1 . Let Π : Γ 2N → R 2N be the surjective map defined by Π(m 1 , λ 1 , . . . , m N , λ N , l 1 , µ 1 , . . . , l N , µ N ) := (λ 1 , . . . , λ N , µ 1 , . . . , µ N ), where m 1 , . . . , m N , l 1 , . . . , l N ∈ N while λ 1 , . . . , λ N , µ 1 , . . . , µ N ∈ R. Clearly Π is measurable because it is continuous with respect to the natural product topologies. We then define the complex measure ν φ,ψ,Q is L 1 with respect to that measure, we conclude that |µ where ν f = F −1 (f ) referring to (14). An elementary inductive argument establishes that S 1 Q n S 2 Q n · · · Q n S N → S 1 S 2 · · · S N strongly as n → +∞, if (1) the orthogonal projectors Q n satisfy Q n → I in the strong sense and (2) S 1 , . . . , S N ∈ B(H) satisfy S j ≤ c < +∞ for j = 1, . . . , N. Taking advantage of this fact (noticing that e isC is unitary if s ∈ R and C is selfadjoint, so that e isC = 1), Lebesgue's dominated convergence theorem and the Fubini theorem entail φ,ψ (f ) = R φ|e i t N aA e i t N bB · · · e i t N aA e i t N bB ψ dν f (t) = R φ| lim n→∞ e i t N aA Q n e i t N bB · · · e i t N aA Q n e i t N bB ψ dν f (t) = R lim n→∞ φ|e i t N aA Q n e i t N bB · · · e i t N aA Q n e i t N bB ψ dν f (t) = lim n→∞ R φ|e i t N aA Q n e i t N bB · · · e i t N aA Q n e i t N bB ψ dν f (t) Due to (64), the limit does not depend on the chosen sequence Q as declared in the hypothesis. ✷

B Spectral representation theorem
We state and prove here a version of the spectral representation theorem for selfadjoint operators we exploited in the proof of Lemma 2.2. This is nothing but a refinement of part of some results which can be found in [ReSa75II,Mor17].
(c) It easily arises from the standard result supp(P (T ) ) = σ(T ), using supp(ν ψ ) ⊂ supp(P (T ) ) and the disjoint decomposition of Γ into open-closed subsets R n .

C Projective limit of complex measures
We state and prove here a particular case of a general result extensively discussed in [Tho01,AlMa16], concerning an extension to complex measures of Kolmogorov theorem for the existence of a so-called consistent family of probability measures (see, e.g., [Pro56,Xia72,Yam85,Yeh73]). Let N be the index set and let us denote with F N the collection of subsets J ⊂ N with a finite number of elements and with |J| the cardinality of J ∈ F N . F N inherits naturally the structure of a directed set, where the partial order relation ≤ is defined as J ≤ K iff J ⊂ K, J, K ∈ F N . Let us associate to any J ∈ F N a measure space R J , naturally isomorphic to R |J| and let Σ J denote its Borel σ-algebra. For any pair J, K ∈ F N with J ≤ K let π K J : R K → R J be the projection map, which is continuous hence Borel measurable. A collections {µ J } J∈F N of complex measures µ J : Σ J → C is said to be consistent or projective if the following compatibility condition is fulfilled for any J, K ∈ F N , J ≤ K: where (π K J ) * µ K denotes the image (pushforward) measure of µ K under the action of the map π K J , namely (π K J ) * µ K (E) := µ K (π K J ) −1 (E) , for all E ∈ Σ J . Let R N be the space of all real-valued sequences and for any J ∈ F N let π J : R N → R J indicate the projection map. It is simple to check that for any J, K ∈ F N , with J ≤ K the following composition property holds: