Foundations of Physics

, Volume 44, Issue 7, pp 762–780 | Cite as

Quantum Phase Space from Schwinger’s Measurement Algebra



Schwinger’s algebra of microscopic measurement, with the associated complex field of transformation functions, is shown to provide the foundation for a discrete quantum phase space of known type, equipped with a Wigner function and a star product. Discrete position and momentum variables label points in the phase space, each taking \(N\) distinct values, where \(N\) is any chosen prime number. Because of the direct physical interpretation of the measurement symbols, the phase space structure is thereby related to definite experimental configurations.


Schwinger’s measurement algebra Discrete quantum phase space Star product Wigner function 

1 Introduction

Classical physics asserts that disturbances of an observed system by a measuring apparatus can be continuously weakened, without affecting any previously measured values, and enabling precise simultaneous measurements of all physical quantities. In accordance with Heisenberg’s uncertainty principle and its generalizations, an arbitrary level of precision cannot be attained at sub-atomic scales when simultaneous measurement of certain observables is attempted. To describe the laws governing observations at the atomic level, Schwinger conceived a symbolic representation of measurement processes. In a series of papers [1, 2, 3] and two monographs [4, 5], he developed the foundations of quantum kinematics and dynamics from abstract elements characterizing idealized measurements.

Schwinger considered a microscopic measurement to be a process that partitions an ensemble of identically prepared quantum systems into sub-ensembles supporting various physical attributes. His approach captured in an intuitive way the inability of a measurement process to indefinitely compensate for fluctuations, and has attracted much pedagogic interest by providing a self-consistent description of reality at the quantum level. The originality of the approach resides in the deduction of a quantum theory based on an abstract symbolic representation of observations, rather than a pre-supposed mathematical framework to be justified á posteriori by agreement with experiment, as in von Neumann’s Hilbert space formulation, [6] and Dirac’s formulation based on transformation theory [7]). Schwinger’s symbolic representation identifies a vector space, corresponding to the Hilbert space of von Neumann, spanned by measurement operators and scaled by (complex) numerical quantities or transformation functions, which appear as artifacts of consecutive measurements of incompatible physical quantities. His intention was to reproduce transformation theory by allowing measurement operators to preserve their level of abstraction, prior to making the assumption that transformation functions are numerically complex quantities.

Another formulation of quantum mechanics, where observables are represented by functions on a real phase space, has in recent years gained much attention.

For spinless systems with continuous degrees of freedom [8, 9, 10, 11], quantum operators on Hilbert space are represented by functions of the canonical variables \((p,q)\) labeling points in a phase space \(\Gamma \). The non-commutative product of operators is replaced by an equivalent non-commutative star product of functions, while quantum mechanical averages are calculated from a quasi-classical formula in the spirit of classical phase space [12]. Weyl’s correspondence rule (quantization map) [8] and its inverse, the Weyl–Wigner transform, are essential ingredients of the phase space picture, and define a bi-directional mapping between functions on \(\Gamma \) and quantum operators. As a special case, the Wigner function is the inverse map of the density operator associated with the state of a quantum system.

For an \(N\)-dimensional quantum system, a discrete phase space can be modeled as an \(N \times N\) set of phase points on a lattice. A generalized Weyl correspondence rule and its inverse, applicable to systems with finitely many degrees of freedom (such as spin), acts between operators \(\widehat{X}\) on the \(N\)-dimensional vector space of quantum states, and corresponding functions \(X(\alpha ,\beta )\) on the lattice points. Once again, a Wigner function represents the quantum density operator (matrix).

There is by now a substantial literature describing various such quasi-probability representations (QPRs) of quantum systems with finite-dimensional Hilbert spaces, arrived at from different starting points. (See [13, 14, 15] and references therein). In a recent study Ferrie et al. [16] have shown how the mathematical theory of frames provides an over-arching structure to describe all known finite-dimensional QPRs. More recently, this has been extended to infinite-dimensional QPRs as well [17].

Schwinger’s algebra of microscopic measurement has found many applications relating to the foundations of quantum mechanics, for instance in demonstrating the physical equivalence of an ideal measurement and the reduction of states [18], and in defining the framework for a generalization from a complex to a quaternionic measurement algebra [19]. However, although studies of the measurement algebra have progressed in many directions, a generalization to include the phase space formulation of quantum mechanics has not been forthcoming. It is the purpose of this paper to fill this gap, at least in the finite-dimensional case, and to demonstrate how a particular QPR naturally arises from a suitable extension of Schwinger’s algebra of microscopic measurement. We emphasize that the QPR obtained is already known [16, 23]. What is novel in what follows is the derivation on the basis of Schwinger’s algebra.

The paper is organized as follows. A brief overview of a finite-dimensional phase space formulation of quantum mechanics is presented, followed by a summary of the symbolic and algebraic structure of the measurement algebra. Based on this groundwork, an appropriate extension of the measurement-algebra formalism is developed, and a number of symbolic expressions derived that define a primitive structure capable of leading to a discrete phase space picture. In Sect. 5 it is confirmed that with suitable substitutions a discrete Weyl–Wigner representation emerges. The results are discussed in Sect. 6.

2 Finite-Dimensional Phase Space Formulation

A generalized Weyl correspondence rule and its inverse, for systems with \(N\) degrees of freedom, are given by the relations [20, 21]
$$\begin{aligned} \widehat{X}&= \frac{1}{N} \sum _{\alpha \beta } X(\alpha ,\beta ) \widehat{\Delta }(\alpha ,\beta ),\end{aligned}$$
$$\begin{aligned} X(\alpha ,\beta )&= \mathrm{Tr}\left( \widehat{X}\widehat{\Delta }(\alpha ,\beta )\right) \,. \end{aligned}$$
Here \(\widehat{\Delta }(\alpha ,\beta )\) is an \(N \times N\) lattice kernel matrix defined on the lattice points \((\alpha , \beta )\), where \(\alpha ,\beta = 0,1, \ldots , N - 1\), and required to satisfy the properties
$$\begin{aligned} \mathrm{Tr}(\widehat{\Delta }(\alpha ,\beta ))&= 1, \end{aligned}$$
$$\begin{aligned} \frac{1}{N} \sum _{\alpha \beta } \widehat{\Delta }(\alpha ,\beta )&= \widehat{1}, \end{aligned}$$
$$\begin{aligned} \frac{1}{N} \mathrm{Tr}(\widehat{\Delta }(\alpha ,\beta )\widehat{\Delta }(\alpha ^{\prime },\beta ^{\prime }))&= \delta _{\alpha \alpha ^{\prime }}\delta _{\beta \beta ^{\prime }} . \end{aligned}$$
A double application of (1a) to the operator product \(\widehat{X}\widehat{Y}\) leads directly to a discrete star product
$$\begin{aligned} (X \star Y)(\alpha ,\beta ) = \frac{1}{N^2} \sum _{\mathop {\alpha ^{\prime }\, \beta ^{\prime }}\limits _{\alpha ^{\prime \prime }\, \beta ^{\prime \prime }}} \mathrm{Tr}\left( \widehat{\Delta }(\alpha ,\beta ) \widehat{\Delta }(\alpha ^{\prime },\beta ^{\prime }) \widehat{\Delta }(\alpha ^{\prime \prime },\beta ^{\prime \prime })\right) X(\alpha ^{\prime },\beta ^{\prime }) Y(\alpha ^{\prime \prime },\beta ^{\prime \prime }), \end{aligned}$$
of two functions defined on the lattice points.
If the state of an \(N\)-dimensional system is defined by a pure state density matrix \(\widehat{\rho }\) then discrete Wigner functions emerge as coefficients in the operator expansion
$$\begin{aligned} \widehat{\rho } = \frac{1}{N} \sum _{\alpha \beta } W(\alpha ,\beta ) \widehat{\Delta }(\alpha ,\beta ), \end{aligned}$$
or explicitly, using (2c), as \(W(\alpha ,\beta ) = \mathrm{Tr}(\widehat{\rho }\;\widehat{\Delta }(\alpha ,\beta ))\). As with the continuous case, a discrete Wigner function is normalized to unity, namely
$$\begin{aligned} \frac{1}{N} \sum _{\alpha \beta } W(\alpha ,\beta ) = 1, \end{aligned}$$
can be used to calculate the average values of an arbitrary \(\widehat{X}\) according to the pseudo-classical formula
$$\begin{aligned} \langle \widehat{X}\rangle = \frac{1}{N} \sum _{\alpha \beta } X(\alpha ,\beta ) W(\alpha ,\beta ), \end{aligned}$$
but can take negative values and thus cannot be interpreted as a true probability density function [20].

3 The Measurement Algebra

Schwinger [1] supposed that by performing a measurement on a ensemble of identically prepared (and independent) systems, one obtains a collection of sub-ensembles labeled by the allowed values of the observable under consideration. In particular, he considered observables denoted \(A\), \(B\)\(\dots \), each of which admits on measurement a finite number of distinct values \(a\in \{a_1,\,a_2,\,...,\,a_{N_A}\}\), \(b\in \{b_1,\,b_2,\,...,\,b_{N_B}\}\), etc.. He then considered measurements of various types, starting with a selective measurement, denoted by \(\mathrm{M}(a)\) for example, defined as a process that ‘accepts’ systems with a value \(a\) while ‘rejecting’ all others. In this case, from an ensemble of identically prepared systems, one obtains a sub-ensemble in which every system has the value \(a\) and is accepted by a repeat of the measurement \(\mathrm{M}(a)\).

Adapting the measuring apparatus to accept systems with either of the values \(a\) and \(a^{\prime }\), and reject all others, produces a less selective measurement, represented symbolically by the sum
$$\begin{aligned} \mathrm{M}(a) + \mathrm{M}(a^{\prime }). \end{aligned}$$
Continuing this process of accepting more and more of the allowed values of \(A\), a maximal (or non-selective) measurement is reached, that is one that selects, or accepts, all systems in the ensemble. This is written symbolically as
$$\begin{aligned} \sum _a \mathrm{M}(a) = 1, \end{aligned}$$
where the sum is over all allowed values of \(a\), and \(1\) denotes the identity measurement that allows accepts all systems and rejects none. Similarly, \(0\) is introduced to denote the null measurement that rejects all systems and accepts none. The key properties of this simple type of measurement, apart from (8), are then summarized as
$$\begin{aligned} \mathrm{M}(a) \mathrm{M}(a^{\prime }) = \delta (a,a^{\prime }) \mathrm{M}(a^{\prime }), \end{aligned}$$
$$\begin{aligned} \delta (a^{\prime },a) = \left\{ \begin{array}{l@{\quad }l} 0, &{} a^{\prime } \ne a \\ 1, &{} a^{\prime } = a \end{array} \right. \end{aligned}$$
Here the product on the left symbolizes the measurement \(\mathrm{M}(a^{\prime })\) followed by the measurement \(\mathrm{M}(a)\), while on the right,
$$\begin{aligned} 1\,\mathrm{M}(a^{\prime })=\mathrm{M}(a^{\prime })\,,\quad 0\,\mathrm{M}(a^{\prime })=0\,, \end{aligned}$$
symbolize the measurement \(\mathrm{M}(a^{\prime })\) followed by the identity or null measurement, respectively. Similarly,
$$\begin{aligned} \mathrm{M}(a^{\prime })\,1=\mathrm{M}(a^{\prime })\,,\quad \mathrm{M}(a^{\prime })\,0=0\,. \end{aligned}$$
The formalism is extended by building upon these simple measurements to obtain ones whose effect is to produce more complicated changes of state. Thus Schwinger next considered measurements of the type denoted \(M(a,a^{\prime })\), which accepts only systems with value \(a^{\prime }\) and produces, in every accepted case, a system with value \(a\), where \(a\) and \(a^{\prime }\) are possible values of the same observable \(A\). The action of consecutive state-changing measurements of this simple type on a system is represented symbolically by
$$\begin{aligned} \mathrm{M}(a,a^{\prime }) \mathrm{M}(a^{\prime \prime },a^{\prime \prime \prime }) = \delta (a^{\prime },a^{\prime \prime }) \mathrm{M}(a,a^{\prime \prime \prime }). \end{aligned}$$
Reversing the order of the operations results in
$$\begin{aligned} \mathrm{M}(a^{\prime \prime },a^{\prime \prime \prime }) \mathrm{M}(a,a^{\prime }) = \delta (a,a^{\prime \prime \prime }) \mathrm{M}(a^{\prime \prime },a^{\prime })\,, \end{aligned}$$
showing that multiplication of measurement symbols as in (13), (14) is inherently non-commutative, even though values of only one observable \(A\) are involved.
More generally, a maximal (or complete) set \({\mathcal A}\) of compatible observables consists of more than one element like \(A\). It may contain observables \(A\), \(B\), \(\dots \), \(C\), say. The measurement product
$$\begin{aligned} \mathrm{M}(a,\,b,\,\dots ,\,c)=\mathrm{M}(a)\mathrm{M}(b)\dots \mathrm{M}(c) \end{aligned}$$
then represents a complete selective measurement of the set of compatible observables, and provides the maximum attainable knowledge of the ensemble of systems allowed by the measurement apparatus. In this case, the order of the measurements starting with \(\mathrm{M}(c)\) and ending with \(\mathrm{M}(a)\) is inconsequential, and multiplication of the symbols is commutative.

In reality, a system admits more than one complete set of compatible observables, with observables from one set incompatible with those from another. A complete selective measurement associated with one set will then be incompatible with a complete selective measurement associated with another set.

It is convenient to restrict attention to complete sets \(\mathcal {A} = \{A\}, \mathcal {B} = \{B\},\ldots \), each of which contains a single observable taking corresponding values \(a\in \{a_1,\,a_2,\,\dots ,\,a_{N}\}\), \(b\in \{b_1,\,b_2,\,\dots ,\,b_{N}\}\)etc. Note that here \(N_A=N_B= \ldots =N\); this is justified later in Schwinger’s treatment. A general state-changing measurement involving two mutually incompatible observables \(A\) and \(B\) of this type is denoted \(\mathrm{M}(a,b)\), and corresponds to a measurement apparatus that allows only systems with value \(b\) for \(B\) to enter, and only systems with value \(a\) for \(A\) to emerge. It is related to the sequence of selective measurements \(\mathrm{M}(b)\) followed by \(\mathrm{M}(a)\), as in
$$\begin{aligned} \mathrm{M}(a) \mathrm{M}(b) = \langle a|b\rangle \mathrm{M}(a, b)\,. \end{aligned}$$
Here the set of quantities \(\langle a|b\rangle \), which are assumed to be complex numbers, is called a transformation function, and allows for the possibility that in the measurements associated with \(\mathrm{M}(a) \mathrm{M}(b)\) and \(\mathrm{M}(a, b)\), respectively, the ensembles of systems emerging with value \(a\) from a given ensemble may differ in size and in other details. Reversing the sequence on the left in (16) results in
$$\begin{aligned} \mathrm{M}(b) \mathrm{M}(a) = \langle b|a\rangle \mathrm{M}(b, a), \end{aligned}$$
where \(\langle b|a\rangle \) may differ from \(\langle a|b\rangle \).
Measurement symbols of different ‘types’ can be written as a linear combinations of each other, with transformation functions as coefficients, as in
$$\begin{aligned} \mathrm{M}(b, c) = \sum _a \langle a|b\rangle \mathrm{M}(a, c), \quad \mathrm{M}(a, c) = \sum _b \langle b|a\rangle \mathrm{M}(b, c)\,. \end{aligned}$$
This leads to the closure relation
$$\begin{aligned} \mathrm{M}(a, b) \mathrm{M}(c, d) = \langle b|c\rangle \mathrm{M}(a, d)\,. \end{aligned}$$
Schwinger went on to justify consideration of arbitrary complex linear combinations of measurement symbols of the general type \(\mathrm{M}(a, b)\).

The relations (16) and (17) are said to be expressed in a ‘mixed description’. Here a ‘description’ refers to the configuration of the measurement apparatus. Thus the set of \(\mathrm{M}(a,a^{\prime })\) characterizes a simple \(A\)-description, whereas the set of \(\mathrm{M}(a, b)\) characterizes a mixed \(AB\)-description, and the set of \(\mathrm{M}(b,a)\) a mixed \(BA\)-description. For a finite-dimensional system there are \(N\) possible \(a\) values and \(N\) possible \(b\) values, which in combination produce a total of \(N^2\) mixed measurement symbols \(\mathrm{M}(a, b)\). A ‘representation’ expresses a mixed measurement symbol in terms of simple measurement symbols. For example, the measurement symbol \(\mathrm{M}(a, b)\) in the mixed \(AB\)-description, has \(A\)-representation given by the expansion \(\sum _{a^{\prime }} \langle b|a^{\prime }\rangle \mathrm{M}(a, a^{\prime })\).

The transformation function provides the basis for a statistical interpretation of measurement processes. Consider the relation
$$\begin{aligned} \mathrm{M}(b)\mathrm{M}(a)\mathrm{M}(b) = \langle a|b\rangle \langle b|a\rangle \mathrm{M}(b)\,, \end{aligned}$$
which follows from (19). The quantity
$$\begin{aligned} p\left( a,b\right) \equiv \langle a|b\rangle \langle b|a\rangle \end{aligned}$$
is interpreted as the fraction of systems, initially in a state characterized by the value \(b\) of \(B\), which are found in a state characterized by the value \(a\) of \(A\) after measurement of \(A\). Such an interpretation requires that \(p\left( a,b\right) \) is non-negative. If the central component of the measurement (20) is made less selective, and arranged to pass systems with values \(a\) and \(a^{\prime }\), then (20) becomes
$$\begin{aligned} \mathrm{M}(b)\left( \mathrm{M}(a) + \mathrm{M}(a^{\prime })\right) \mathrm{M}(b) = \left( p\left( a,b\right) + p\left( a^{\prime },b\right) \right) \mathrm{M}(b). \end{aligned}$$
Extending to a complete non-selective measurement gives
$$\begin{aligned} \mathrm{M}(b) \left( \sum _a \mathrm{M}(a)\right) \mathrm{M}(b) = \left( \sum _a p\left( a,b\right) \right) \mathrm{M}(b) = \mathrm{M}(b)\,, \end{aligned}$$
using (8) and (9), implying that
$$\begin{aligned} \sum _a p\left( a,b\right) = 1\,, \end{aligned}$$
consistent with the interpretation of \(p\left( a,b\right) \) as the fraction described above. Given (21), it is natural to suppose the complex conjugacy relation \(\overline{\langle a|b\rangle } = \langle b|a\rangle \), so that
$$\begin{aligned} p\left( a,b\right) = p\left( b,a\right) = |\langle a|b\rangle |^2 \ge 0\,. \end{aligned}$$
Then \(p\left( a,b\right) \) can be interpreted as the probability of observing the state \(a\) on measurement of \(A\), given a system initially in a state \(b\), and the \(p\left( a,b\right) \) are the weighting factors used to calculate average values of physical quantities in the usual way.
Schwinger then showed that the observable \(A\) can be represented by the measurement symbol
$$\begin{aligned} \widehat{A}= \sum _a a \mathrm{M}(a)\,, \end{aligned}$$
where the sum runs over the possible values \(a\) of \(A\), and the expectation value of \(A\) for a system in a state with the value \(b\) for \(B\) is given by
$$\begin{aligned} \langle \widehat{A}\rangle = \sum _a p\left( a,b\right) \; a = \mathrm{Tr}(\widehat{A} \mathrm{M}(b))\,. \end{aligned}$$
Here the trace of a measurement symbol is defined first by setting \(\mathrm{Tr}(\mathrm{M}(a,b))=\langle b|a\rangle \) and extending by linearity to arbitrary linear combinations of measurement symbols such as (26).
One can now deduce that an arbitrary measurement symbol \(\widehat{X}\) can be decomposed in a mixed \(AB\)-description as
$$\begin{aligned} \widehat{X} = \sum _{ab} \langle a|\widehat{X}|b\rangle \mathrm{M}(a,b)\,, \end{aligned}$$
where the elements \(\langle a|\widehat{X}|b\rangle \) form an \(N \times N\) matrix. The matrix elements are given explicitly by
$$\begin{aligned} \langle a|\widehat{X}|b\rangle = \mathrm{Tr}(\widehat{X}\mathrm{M}(b,a)). \end{aligned}$$
The next important idea is that a measurement symbolized by \(\mathrm{M}(a,b)\) can be considered as the product of two consecutive processes. The first is the annihilation of each system with the value \(b\) for observable \(B\), and the second is the creation in its place of a system with the value \(a\) for observable \(A\). The first process is symbolized by \(\langle b|\), and the second process by \(|a\rangle \). Thus \(\mathrm{M}(a,b)=|a\rangle \langle b|\).

Schwinger went on to show that the algebra of measurement symbols is isomorphic to the \(N^2\)-dimensional algebra of linear operators acting on a complex \(N\)-dimensional Hilbert space, where the natural involution defined by \(\mathrm{M}(a,b)^{\dagger }=\mathrm{M}(b,a)\) is mapped into hermitian conjugation. The Hilbert space itself is then identified as the vector space consisting of all linear combinations of the symbols \(|a\rangle \), and the extension of the involution operation, by setting \(|a\rangle ^{\dagger }=\langle a |\), maps into the extension of hermitian conjugation from Hilbert space vectors to their duals. The scalar product between two creation symbols or ‘states’ \(|a\rangle \), \(|b\rangle \) is defined as \(\langle b|a\rangle \), so that one can speak of orthogonal states and normalized states in the way familiar in Dirac’s formalism for quantum mechanics.

It is also possible to consider more general annihilation and creation symbols (or ‘null states’ [2]) \(\widehat{\Psi }=|\psi \rangle \), \(\widehat{\Xi }^{\dagger }=\langle \xi |\), leading to ‘dyadic’ measurement symbols more general than \(M(a,b)\), such as \(\widehat{X}=\widehat{\Psi }\widehat{\Xi }^{\dagger }\). Noting that \(\mathrm{M}(a)=\mathrm{M}(a,a)=|a\rangle \langle a|\), and using (8), leads to
$$\begin{aligned} \widehat{\Psi }&= \sum _{a} |a\rangle \langle a| \widehat{\Psi } = \sum _{a} |a\rangle \langle a|\psi \rangle \nonumber \\&= \sum _{a} \psi (a) |a\rangle \,, \end{aligned}$$
where \(\psi (a)\) can be called the wavefunction of the ‘state’ \(\widehat{\Psi }\) in an \(A\) description. Similarly, \(\widehat{\Xi }^{\dagger }= \sum _{b} \langle b|\,\overline{\xi (b)}\), leading to
$$\begin{aligned} \widehat{X}=\widehat{\Psi }\widehat{\Xi }^\dagger = \sum _{a\,b}\psi (a)\,\overline{\xi (b)}\,\mathrm{M}(a,b)\,. \end{aligned}$$
Here \(a\) and \(b\) refer to values of observables \(A\) and \(B\), possibly different and possibly incompatible. For example, \(A\) and \(B\) might be different components of a spin vector or, as in what follows, discrete analogues of position and momentum observables.

The foregoing discussion has outlined the basic elements of the measurement algebra. However, a slight modification to the formalism is needed prior to establishing a transition to a discrete phase space description, and this will be taken up in the next section.

4 A Measurement Symbol Representation

The usual starting point for a discrete (or continuous) phase space formulation of quantum mechanics is the decomposition of quantum operators in terms of some suitably parameterized operator basis with properties similar to those given in (2). In this vein a primitive symbolic expression for a measurement symbol decomposition is given by (28) in which the \(M(a,b)\) assume the role of a basis. However, this is unsuitable in its present form, for reasons that will become apparent, and an appropriate modification has to be applied without altering the general form of (28). This is accomplished with the help of the transformation function by rewriting (28) as
$$\begin{aligned} \widehat{X} = \sum _{ab} \widetilde{X}(a,b) \widehat{\Lambda }(a,b)\,, \end{aligned}$$
where the modified \(AB\) matrix elements of \(\widehat{X}\) are now defined as
$$\begin{aligned} \widetilde{X}(a,b) = \langle a|\widehat{X}|b\rangle \langle b|a\rangle , \end{aligned}$$
and the modified form of the measurement symbol basis is
$$\begin{aligned} \widehat{\Lambda }(a,b) = \mathrm{M}(a,b)/\langle b|a\rangle . \end{aligned}$$
It is assumed here that \({A}\) and \({B}\) are chosen such that \(\langle b|a\rangle \ne 0\), to avoid obvious difficulties in (34).
An immediate consequence of (34) is that it leads to similar properties to those given in (2), namely
$$\begin{aligned} \mathrm{Tr}(\widehat{\Lambda }(a,b))&= 1, \end{aligned}$$
$$\begin{aligned} \sum _{ab} |\langle a|b\rangle |^2 \widehat{\Lambda }(a,b)&= \widehat{1}, \end{aligned}$$
$$\begin{aligned} |\langle a|b\rangle |^2 \mathrm{Tr}(\widehat{\Lambda }(a,b) \widehat{\Lambda }(a^{\prime },b\,^{\prime })^\dagger )&= \delta _{aa^{\prime }}\delta _{bb\,^{\prime }}. \end{aligned}$$
Inversion of (32) follows directly from (35c), giving
$$\begin{aligned} \widetilde{X}(a,b) = |\langle a|b\rangle |^2 \; \mathrm{Tr}(\widehat{X}\widehat{\Lambda }(a,b)^\dagger )\,. \end{aligned}$$
The positive numbers \(|\langle a|b\rangle |^2\), whose values depend on the characteristics of the particular system under investigation, and on the choice of observables \({A}\) and \({B}\), play a crucial role in the formulation that follows.
Altering the configuration of the apparatus produces an alternative \(BA\)-description, leading to the decomposition
$$\begin{aligned} \widehat{X} = \sum _{ab} \check{X}(b,a) \widehat{\Lambda }(b,a), \end{aligned}$$
where the measurement symbol basis is now given by
$$\begin{aligned} \widehat{\Lambda }(b,a)=\widehat{\Lambda }(a,b)^{\dagger } = \mathrm{M}(b,a)/\langle a|b\rangle \,, \end{aligned}$$
satisfying similar properties to (35). Replacing (33) and (36) are
$$\begin{aligned} \check{X}(b,a)=\langle b|\widehat{X}|a\rangle \langle a|b\rangle = |\langle a|b\rangle |^2 \; \mathrm{Tr}(\widehat{X}\widehat{\Lambda }(b,a)^\dagger )\,. \end{aligned}$$
Note that if \(\widehat{X}=\widehat{X}^{\dagger }\), then \(\check{X}(b,a)\) is the complex conjugate of \(\widetilde{X}(a,b)\).
Consider the product of any two elements of the measurement algebra \(\widehat{X}\) and \(\widehat{Y}\). A symbolic representation in the \(AB\)-description is obtained from a double application of (32) giving
$$\begin{aligned} \widehat{X}\widehat{Y} = \sum _{ a^{\prime } b\,^{\prime }} \sum _{a^{\prime \prime } b\,^{\prime \prime }} \widetilde{X}(a^{\prime },b\,^{\prime }) \widetilde{Y}(a^{\prime \prime },b\,^{\prime \prime }) \; \widehat{\Lambda }(a^{\prime },b\,^{\prime })\widehat{\Lambda }(a^{\prime \prime },b\,^{\prime \prime }). \end{aligned}$$
From the closure of the algebra the measurement symbol basis product \(\widehat{\Lambda }(a^{\prime },b\,^{\prime })\widehat{\Lambda }(a^{\prime \prime },b\,^{\prime \prime })\) admits the decomposition
$$\begin{aligned} \widehat{\Lambda }(a^{\prime },b\,^{\prime }) \widehat{\Lambda }(a^{\prime \prime },b\,^{\prime \prime }) = \sum _{ab} |\langle a|b\rangle |^2 \; \mathrm{Tr}(\widehat{\Lambda }(a,b)^\dagger \widehat{\Lambda }(a^{\prime },b\,^{\prime })\widehat{\Lambda }(a^{\prime \prime },b\,^{\prime \prime })) \; \widehat{\Lambda }(a,b). \end{aligned}$$
Substituting (41) into (40) results in
$$\begin{aligned} \widehat{X}\widehat{Y} = \sum _{ab} (\widetilde{X }\star \widetilde{Y})(a,b) \; \widehat{\Lambda }(a,b), \end{aligned}$$
where the measurement algebra star product is defined as
$$\begin{aligned}&(\widetilde{X} \star \widetilde{Y})(a,b)\nonumber \\&\quad = |\langle a|b\rangle |^2 \sum _{ a^{\prime } b\,^{\prime }} \sum _{a^{\prime \prime } b\,^{\prime \prime }} \mathrm{Tr}(\widehat{\Lambda }(a,b)^\dagger \widehat{\Lambda }(a^{\prime },b\,^{\prime })\widehat{\Lambda }(a^{\prime \prime },b\,^{\prime \prime })) \; \widetilde{X}(a^{\prime },b\,^{\prime }) \widetilde{Y}(a^{\prime \prime },b\,^{\prime \prime }). \end{aligned}$$
A similar argument gives the product of two measurement symbols in the \(BA\)-description as
$$\begin{aligned} \widehat{X}\widehat{Y} = \sum _{ab} (\check{X} \star \check{Y})(b,a) \widehat{\Lambda }(b,a), \end{aligned}$$
$$\begin{aligned}&(\check{X} \star \check{Y})(b,a)\nonumber \\&\quad = |\langle a|b\rangle |^2 \sum _{a^{\prime } b^{\prime }} \sum _{a^{\prime \prime } b^{\prime \prime }} \mathrm{Tr}(\widehat{\Lambda }(b,a)^\dagger \widehat{\Lambda }(b\,^{\prime },a^{\prime })\widehat{\Lambda }(b\,^{\prime \prime },a^{\prime \prime })) \; \check{X}(b\,^{\prime },a^{\prime })\check{Y}(b\,^{\prime \prime },a^{\prime \prime }). \end{aligned}$$
Using (43) and (45) it is straightforward to show that each form of the star product is associative but, except in special cases, is non-commutative, for example
$$\begin{aligned} (\widetilde{X} \star (\widetilde{Y} \star \widetilde{Z}))(a,b) = ((\widetilde{X} \star \widetilde{Y}) \star \widetilde{Z})(a,b), \quad (\widetilde{X} \star \widetilde{Y})(a,b) \ne (\widetilde{Y} \star \widetilde{X})(a,b). \end{aligned}$$
Composed of elements of the measurement algebra, the star products (43) and (45) form primitive expressions for star products of the type familiar from the phase space formulation of quantum mechanics, which we consider in the next section.
The trace relations for a product of two measurement symbols follow from the definitions of modified matrix elements, giving
$$\begin{aligned} \mathrm{Tr}(\widehat{X}\widehat{Y}) = \sum _{ab} \frac{1}{|\langle a|b\rangle |^2} \widetilde{X}(a,b) \check{Y}(b,a) = \sum _{ab} \frac{1}{|\langle a|b\rangle |^2} \check{X}(b,a) \widetilde{Y}(a,b)\,. \end{aligned}$$
These trace equations, involving quantities from both \(AB\) and \(BA\) descriptions, can be brought back to a single description by applying to (47) the result
$$\begin{aligned} \widetilde{X}(a,b) = \sum _{a^{\prime }b\,^{\prime }} {\mathcal {M}}(a,b\,;\,a^{\prime },b\,^{\prime }) \check{X}(b\,^{\prime }, a^{\prime })\,. \end{aligned}$$
Here \({\mathcal {M}}\) is the transformation kernel relating the two descriptions and defined in terms of the transformation function by
$$\begin{aligned} {\mathcal {M}}(a,b\,:\,a^{\prime },b\,^{\prime }) = \frac{1}{|\langle a^{\prime }|b\,^{\prime }\rangle |^2} \langle b\,^{\prime }|a^{\prime }\rangle \langle a|b\,^{\prime }\rangle \langle a^{\prime }|b\rangle \langle b|a\rangle . \end{aligned}$$
A similar relationship exists between the operator bases, namely
$$\begin{aligned} \widehat{\Lambda }(b,a) = \sum _{a^{\prime }b\,^{\prime }} {\mathcal {M}}(a,b\,:\,a^{\prime },b\,^{\prime }) \widehat{\Lambda }(a^{\prime },b\,^{\prime }). \end{aligned}$$
The kernel \({\mathcal {M}}\) is not in general symmetric under interchange of \(ab\) with \(a^{\prime }b\,^{\prime }\) since it is not necessarily true that \(|\langle a|b\rangle |^2 = |\langle a^{\prime }|b\,^{\prime }\rangle |^2\), but it does satisfy
$$\begin{aligned} \sum _{a} {\mathcal {M}}(a,b\,:\,a^{\prime },b^{\prime })&= \delta _{bb^{\prime }},\end{aligned}$$
$$\begin{aligned} \sum _{b} {\mathcal {M}}(a,b\,:\,a^{\prime },b^{\prime })&= \delta _{aa^{\prime }},\end{aligned}$$
$$\begin{aligned} \sum _{ab} {\mathcal {M}}(a,b\,:\,a^{\prime },b^{\prime })&= N,\end{aligned}$$
$$\begin{aligned} \sum _{ab} {\mathcal {M}}(a,b\,:\,a^{\prime },b^{\prime })\, \overline{{\mathcal {M}}(a^{\prime \prime },b^{\prime \prime }\,:\,a,b)}&= \delta _{a^{\prime }a^{\prime \prime }}\delta _{b^{\prime }b^{\prime \prime }}. \end{aligned}$$
Use of property (51d) enables (48) and (50) to be inverted, so that
$$\begin{aligned} \check{X}(b,a)&= \sum _{a^{\prime }b\,^{\prime }} \overline{{\mathcal {M}}(a,b\,:\,a^{\prime },b\,^{\prime })} \widetilde{X}(a^{\prime },b\,^{\prime })\,,\quad \widehat{\Lambda }(b,a)\nonumber \\&= \sum _{a^{\prime }b\,^{\prime }} {\mathcal {M}}(a,b\,:\,a^{\prime },b\,^{\prime }) \widehat{\Lambda }(a^{\prime },b\,^{\prime })\,. \end{aligned}$$
Use of (48) and (52) with (47) leads to
$$\begin{aligned} \mathrm{Tr}(\widehat{X}\widehat{Y})&= \sum _{a b}\sum _{a^{\prime } b\,^{\prime }} \frac{1}{|\langle a|b\rangle |^2} \widetilde{X}(a,b) \overline{{\mathcal {M}}(a,b\,:\,a^{\prime },b\,^{\prime })} \widetilde{Y}(a^{\prime },b\,^{\prime }) \nonumber \\&= \sum _{a b} \sum _{a^{\prime } b\,^{\prime }} \frac{1}{|\langle a|b\rangle |^2} \check{X}(b,a) {\mathcal {M}}(a,b\,:\,a^{\prime },b\,^{\prime }) \check{Y}(a^{\prime },b\,^{\prime })\,, \end{aligned}$$
thus expressing the trace using only a single description, but at the expense of introducing the transformation kernel.
Consider the special case of a measurement symbol \(\widehat{\rho }=\widehat{\Psi }\widehat{\Psi }^\dagger \) where, as in (30),
$$\begin{aligned} \widehat{\Psi }=\sum _{a} \psi (a) |a\rangle = \sum _{a b} \psi (a) |b\rangle \langle b|a\rangle =\sum _b\phi (b) |b\rangle \,,\nonumber \\ \phi (b)=\sum _a \langle b|a\rangle \psi (a)\,,\quad \psi (a)=\sum _b \langle a|b\rangle \phi (b)\,,\qquad \end{aligned}$$
Such a measurement symbol \(\widehat{\rho }\) can be regarded as the density matrix for the pure state with wavefunction \(\psi (a)\) in the \(A\) description, and \(\phi (b)\) in the \(B\) description. In this case, the modified matrix elements in the \(AB\) and \(BA\) descriptions follow from (33) and (39) as
$$\begin{aligned} \widetilde{\rho }(a,b)=\psi (a)\overline{\phi (b)}\langle b|a\rangle \,,\quad \check{\rho }(b,a)=\phi (b)\overline{\psi (a)}\langle a|b\rangle \,. \end{aligned}$$
From (55) it follows that
$$\begin{aligned} \sum _b \widetilde{\rho }(a,b)&= \sum _b \check{\rho }(b,a) = |\psi (a)|^2, \nonumber \\ \sum _a \widetilde{\rho }(a,b)&= \sum _a \check{\rho }(b,a) = |\phi (b)|^2, \end{aligned}$$
and hence that
$$\begin{aligned} \mathrm{Tr}(\widehat{\rho }) = \sum _a|\psi (a)|^2 = \sum _b |\phi (b)|^2 \ge 0\,. \end{aligned}$$
Using the the identities
$$\begin{aligned} \Psi ^\dagger \mathrm{M}(b)\Psi = |\phi (b)|^2\,, \quad \Psi ^\dagger \mathrm{M}(a)\Psi =|\psi (a)|^2, \end{aligned}$$
with (55) leads to equivalent relations in the two descriptions, in the form
$$\begin{aligned} |\widetilde{\rho }(a,b)|^2 = |\check{\rho }(b,a)|^2 = |\langle a|b\rangle |^2 \; |\psi (a)|^2 \; |\phi (b)|^2. \end{aligned}$$

5 A Discrete Quantum Phase Space

One of the earliest developments of a discrete analogue of quantum mechanics with continuous degrees of freedom is also due to Schwinger [3]. Two sets of \(N\) orthonormal states are introduced, denoted by \(\{|u_j\rangle \}, \{|v_k\rangle \}\) with \(j,k = 1\,,2\,,\, \dots \,,\, N\). Here \(N\) is a prime number, and
$$\begin{aligned} u_k=\overline{v_k}=e^{2\pi i(k/N)}\,\quad k=1\,,2\,,\dots N\,, \end{aligned}$$
so that \((u_k)^N=(v_k)^N=1\) for each value of \(k\); but the \(u_k\) and \(v_k\) (or the corresponding values \(k/N\)) are to be interpreted as the possible values taken by two incompatible observables \(A\) and \(B\), respectively. Two ‘unitary translations symbols’ \(\widehat{U}\), \(\widehat{V}\) are introduced to act between the two sets of states, and satisfy
$$\begin{aligned} \widehat{U}|u_j\rangle&= u_j|u_j\rangle \,,\quad \widehat{V}|v_k\rangle = v_k|v_k\rangle \,,\quad \widehat{U}^N = \widehat{V}^N = 1\,,\nonumber \\ \widehat{U}|v_1\rangle&= |v_2\rangle \,, \quad \widehat{U}|v_2\rangle = |v_3\rangle \,, \quad \ldots \,, \quad \widehat{U}|v_{N-1}\rangle = |v_N\rangle \,, \quad \widehat{U}|v_N\rangle = |v_1\rangle \,,\qquad \nonumber \\ \widehat{V}|u_N\rangle&= |u_{N-1}\rangle \,, \quad \widehat{V}|u_{N-1}\rangle = |u_{N-2}\rangle \,, \quad \ldots \,, \quad \widehat{V}|u_2\rangle = |u_1\rangle \,, \quad \widehat{V}|u_1\rangle = |u_N\rangle \,. \end{aligned}$$
It follows from (60) that
$$\begin{aligned} \sum _{k=1}^N\,u_k =0\,. \end{aligned}$$
and then from (61) that
$$\begin{aligned} \left( \frac{1}{N}\,\sum _{k=1}^N\,(u_j)^{-k}\,{\widehat{U}}^k\right) \,|u_l\rangle =\delta _{jl}\,|u_j\rangle \,, \end{aligned}$$
implying in turn that
$$\begin{aligned} \frac{1}{N}\,\sum _{k=1}^N\,(u_j)^{-k}\,{\widehat{U}}^k = |u_j\rangle \langle u_j|\,. \end{aligned}$$
Operating on the left of (64) with \(\langle v_N|\) and on the right with \(|v_N\rangle \) then gives
$$\begin{aligned} |\langle u_j|v_N\rangle |^2=1/N\,. \end{aligned}$$
From (61) it also follows that
$$\begin{aligned} |u_j\rangle ={\widehat{V}}^{N+1-j}\,|u_1\rangle \,,\quad |v_k\rangle ={\widehat{U}}^{k-1}\,|v_1\rangle ={\widehat{U}}^k\,|v_N\rangle \,, \end{aligned}$$
and hence that
$$\begin{aligned} \langle u_j|v_k\rangle&=\langle u_j|{\widehat{U}}^{k-1}|v_1\rangle =(u_j)^{k-1}\langle u_j|v_1\rangle \nonumber \\&=(u_j)^{k-1}\langle u_1|{\widehat{V}}^{1-j} |v_1\rangle = (u_j)^{k-1}(v_1)^{1-j}\langle u_1|v_1\rangle \nonumber \\&= e^{2\pi i(jk-1)/N}\,\langle u_1|v_1\rangle \,, \end{aligned}$$
which gives, with (65),
$$\begin{aligned} |\langle u_1|v_1\rangle |^2=1/N\,. \end{aligned}$$
Multiplying every \(|v_k\rangle \) by the same phase factor does not affect the relations (61), and it is clear from (68) that this phase factor can be chosen to make
$$\begin{aligned} \langle u_1 |v_1\rangle =\frac{1}{\sqrt{N}}\,e^{2\pi i/N}\,, \end{aligned}$$
so that (67) becomes
$$\begin{aligned} \langle u_j|v_k\rangle = e^{2\pi i(jk)/N}/\sqrt{N}\,. \end{aligned}$$
It then follows on multiplying (64) on the right by \(|v_N\rangle \) and using (66), that
$$\begin{aligned} |u_j\rangle = \frac{1}{\sqrt{N}}\,\sum _{k=1}^N\,e^{-2\pi i (jk)/N}\,|v_k\rangle \,, \end{aligned}$$
which is in the form of a discrete Fourier transform.
Central to the development of a discrete phase space is the determination of an appropriate operator (or measurement symbol) basis. Schwinger showed that the unitaries \(\widehat{U}\) and \(\widehat{V}\) are generators of a complete orthonormal basis given by
$$\begin{aligned} \widehat{X}^{(m,n)} = \frac{\widehat{U}^m\widehat{V}^n}{\sqrt{N}}\,, \quad m,n = 0\,,1\,,\, \ldots \,, N - 1\,. \end{aligned}$$
From (60) and (61) it follows that
$$\begin{aligned} \widehat{U}^m = \sum _j e^{2\pi i(j\,m/N)} \,|u_j\rangle \langle u_j|, \quad \widehat{V}^n = \sum _k e^{2\pi i(k\,n/N)}\, |v_k\rangle \langle v_k|\,. \end{aligned}$$
However, the basis symbols (72) are unsuitable for generating a discrete phase space owing, in part, to their not possessing all of the properties (35).
A suitable measurement symbol basis is obtained as in (32), identifying the states \(|a\rangle \) with the \(|u_j\rangle \) and the states \(|b\rangle \) with the \(|v_k\rangle \) to get, in a \(U\) representation, the measurement symbol expansion
$$\begin{aligned} \widehat{X} = \sum _{jk } \widetilde{X}(j,k) \widehat{\Lambda }(j,k), \end{aligned}$$
with matrix elements
$$\begin{aligned} \widetilde{X}(j,k) = \frac{1}{N} \sum _{j\,^{\prime }} e^{2\pi i[(j\,^{\prime } - j)k/N]}\, \langle u_j|\widehat{X}|u_{j\,^{\prime }}\rangle \,, \end{aligned}$$
and operator basis
$$\begin{aligned} \widehat{\Lambda }(j,k) = \sum _{k^{\prime }} e^{2\pi i[(k - k^{\prime })j/N]}\, |v_{k^{\prime }}\rangle \langle v_k|\,. \end{aligned}$$
A straightforward calculation reveals that (76) can be expressed in terms of unitaries as
$$\begin{aligned} \widehat{\Lambda }(j,k) = \frac{1}{N} \sum _{m\,n} \; \widehat{U}^m \widehat{V}^n \; e^{-2\pi i(jm/N)} e^{-2\pi i(kn/N)}\,. \end{aligned}$$
In particular, for the special case \(N = 2\), the only even prime, representing a spin-\({\frac{1}{2}}\) system perhaps, (77) reduces to
$$\begin{aligned} \widehat{\Lambda }(j,k)&= {\frac{1}{2}}(\widehat{U}\widehat{V} e^{-i\pi (j + k)} + \widehat{U}^2\widehat{V} e^{-2\pi ij}e^{-i\pi k}\nonumber \\&\quad +\, \widehat{U}\widehat{V}^2 e^{-i\pi j}e^{-2\pi ik} + \widehat{U}^2\widehat{V}^2 e^{-2\pi i(j + k)}). \end{aligned}$$
Identifying \(\widehat{U}\) with \(\sigma _z\), \(\widehat{V}\) with \(\sigma _x\), and hence \(\widehat{U}\widehat{V}\) with \(i\widehat{\sigma }_y\) and \(\widehat{U}^2\), \(\widehat{V}^2\) with \(\widehat{1}\), where the \(\widehat{\sigma }_x\)etc. are the usual Pauli spin operators, leads to
$$\begin{aligned} \widehat{\Lambda }(j,k) = {\frac{1}{2}}((-1)^{j + k}\,i\widehat{\sigma }_y + (-1)^k\,\widehat{\sigma }_x + (-1)^j\,\widehat{\sigma }_z + \widehat{1})\,, \end{aligned}$$
which can be compared with an expression used in the discrete phase space formulation of Wootters [20].
Taking advantage of the property preserving symmetry between \(\widehat{U}\) and \(\widehat{V}\), namely \(\widehat{U}\rightarrow \widehat{V}, \; \widehat{V}\rightarrow \widehat{U}^{-1}\) with \(m \rightarrow n\) and \(n \rightarrow -m\), and applying to (77) the substitution [3]
$$\begin{aligned} \widehat{U}^m\widehat{V}^n \rightarrow \widehat{U}^m\widehat{V}^n e^{i\pi (mn/N)}, \end{aligned}$$
provides a more general basis
$$\begin{aligned} \widehat{\Lambda }(j,k) = \frac{1}{N} \sum _{m\,n} \; \widehat{U}^m \widehat{V}^n \;e^{i\pi (m\,n/N)}\; e^{-2\pi i(jm/N)} e^{-2\pi i(kn/N)}, \end{aligned}$$
which also satisfies properties (35) and has previously been identified [22].
For \(\widehat{X}=\widehat{\rho } = \widehat{\Psi }\widehat{\Psi }^\dagger \), the matrix elements (75) in the \(U\) description become
$$\begin{aligned} \widetilde{\rho }(j,k) = \sum _{j\,^{\prime }} e^{2\pi i(j\,^{\prime } - j)k/N} \psi (u_j)\overline{\psi (u_{j\,^{\prime }})}\,, \end{aligned}$$
with associated marginals given from (56) by
$$\begin{aligned} \sum _k \widetilde{\rho }(j,k) = |\psi (u_j)|^2, \quad \sum _j \widetilde{\rho }(j,k) = |\phi (v_k)|^2, \end{aligned}$$
and the star product (43) in the \(U\) description reduces to
$$\begin{aligned} (\widetilde{X} \star \widetilde{Y})(j,k) = \sum _{j\,^{\prime }k^{\prime }} e^{-2\pi i(j - j\,^{\prime })(k - k^{\prime })/N} \widetilde{X}(j,k^{\prime }) \widetilde{Y}(j\,^{\prime },k). \end{aligned}$$
An analysis in the \(V\) description leads to similar expressions.
A discrete phase space defined on a lattice represents a quantum system with a finite number of orthogonal basis states [20, 21], associated with the eigenvalues of discrete analogues of the canonical position and momentum operators of a system with continuous degrees of freedom. Accordingly, we now introduce a constant \(q_0\) with dimensions of length, set \(p_0=2\pi \hbar /q_0\), and define \(\widehat{q}\) and \(\widehat{p}\) such that
$$\begin{aligned} \widehat{q}\,|u_j\rangle = q_0[j-(N+1)/2]|u_j\rangle \,,\quad \widehat{p}\,|v_j\rangle = p_0[j-(N+1)/2]|v_j\rangle \,, \end{aligned}$$
and relabel the states
$$\begin{aligned} |u_j\rangle \longrightarrow |q\rangle \,,\quad |v_j\rangle \longrightarrow |p\rangle \end{aligned}$$
with each of \(q/q_0\) and \(p/p_0\) running over the values \(j-(N+1)/2\) for \(j=1,\,2,\,\cdots ,\,N\). It follows from (61) and (84) that
$$\begin{aligned} \widehat{U}= e^{i\pi (N+1)/N}\,e^{ i\widehat{q}\,p_0/(N\hbar )}\,,\quad \widehat{V}= e^{i\pi (N+1)/N}\,e^{i\widehat{p}\,q_0/(N\hbar )}\,, \end{aligned}$$
so that (80) becomes
$$\begin{aligned} \widehat{\Lambda }(q,p) = \frac{1}{N} \sum _{mn} \; e^{\frac{i\pi }{N}mn} e^{\frac{2\pi i}{Nq_0}(q - \widehat{q})m} e^{\frac{2\pi i}{Np_0}(p - \widehat{p})n}\,. \end{aligned}$$
A general measurement symbol is now represented by the matrix of its matrix elements, relabeled from (75) using (85), as \(\widetilde{X}(q,p)\). In particular, (81) gives a discrete Wigner function
$$\begin{aligned} W(q,p) = \sum _{q^{\prime }} e^{\frac{2\pi i}{N}(q^{\prime } - q)p} \psi (q)\overline{\psi (q^{\prime })}. \end{aligned}$$
It follows immediately from (88) that the marginal in the \(q\)-representation are given from (82) by
$$\begin{aligned} \sum _p W(q,p) = |\psi (q)|^2\,,\quad \sum _q W(q,p)=|\phi (p)|^2\,, \end{aligned}$$
and the star product (83) becomes
$$\begin{aligned} (\widetilde{X} \star \widetilde{Y})(q,p) = \sum _{q\,^{\prime }p^{\prime }} e^{-2\pi i(q - q\,^{\prime })(p - p^{\prime })/N} \widetilde{X}(q,p^{\prime }) \widetilde{Y}(q\,^{\prime },p). \end{aligned}$$
The expectation value of a general operator \(\widehat{X}\) in the state with density operator \(\widehat{\rho }\) and Wigner function \(W(q,p)\) is calculated in the usual way as
$$\begin{aligned} \langle \widehat{X}\rangle \; = \sum _{qp} \widetilde{X}(q,p) W(q,p). \end{aligned}$$
However, although \(\sum _{qp} W(q,p) = 1\), not every value of the discrete Wigner function is positive in general. Similar expressions for the discrete Wigner function can be obtained in the \(p\)-representation by using the appropriate primitive forms.
Under the above lattice identification, the transformation kernel (49) takes the simple form
$$\begin{aligned} {\mathcal {M}}(q,p\,;\,q^{\prime },p^{\prime }) = \frac{1}{N} e^{2\pi i(q - q^{\prime })(p - p^{\prime })/N}, \end{aligned}$$
and can be compared with an expression discussed in [23]. In addition to satisfying the properties (51), the kernel (92) in this case is also symmetric in its arguments since
$$\begin{aligned} |\langle q|p\rangle |^2 = |\langle q\,^{\prime }|p\,^{\prime }\rangle |^2 = 1/N. \end{aligned}$$
Performing suitable limiting processes as \(N\rightarrow \infty \), with the spacings \(\Delta q =q_0\) between eigenvalues of \(\widehat{q}\), and \(\Delta p=p_0\) between eigenvalues of \(\widehat{p}\), becoming negligible relative to \(N\), and with sums replaced by integrals where \((\Delta q\Delta p)/(2\pi \hbar N)\)\((= (1/N))\) is replaced by \(dq\,dp/(2\pi \hbar )\), the whole apparatus for the phase space formulation of quantum mechanics in the case of continuous \(q\) and \(p\) can be recovered.

6 Concluding Remarks

Schwinger’s algebra of microscopic measurement, which is based on the primitive concept of a measurement, has been extended to define a discrete phase space formulation of quantum mechanics.

An operator basis is an essential quantity in the characterization of a quantum phase space, so defining such an operator basis in terms of the fundamental elements of the measurement algebra was the first step towards achieving our goal. It was found that the simplest of all possible candidates for an operator basis that satisfied the conditions (35) was given by the combinations \(\mathrm{M}(a,b)/\langle b|a\rangle \).

We showed that from this modest operator basis a number of new primitive constructs followed. For example, from a product of two general measurement symbols and an application of the closure property of the measurement algebra we deduced a primitive form of the star product of Groenewold and Moyal, and by relating two different descriptions of the measurement process we obtained the transformation kernel (49), composed entirely of a product of transformation functions, and showed this to be a primitive expression of a known result [23].

From these primitive constructs we demonstrated that a straightforward substitution of Schwinger’s states \(\{|u_j\rangle \}, |\{v_k\rangle \}\) with the states \(\{|p\rangle \}, \{|q\rangle \}\) produced a discrete phase space picture that is formally consistent with those already known [21, 22].

Because the measurement algebra establishes a symbolic interpretation of the basic concepts of the measurement process, and because all the expressions that we have obtained are in terms of elements of the measurement algebra (that is measurement symbols and transformation functions) the discrete phase space thus obtained is related by way of the physical interpretation of the measurement symbols to definite experimental configurations.



The authors wish to thank the referee of an earlier version for constructive comments.


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Centre for Mathematical PhysicsUniversity of QueenslandBrisbaneAustralia

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