# Quantum Phase Space from Schwinger’s Measurement Algebra

## Abstract

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.

### Keywords

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

## 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 \(\mathrm{M}(a)\).

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.

*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

*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

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 \(\sum _{a^{\prime }} \langle b|a^{\prime }\rangle \mathrm{M}(a, a^{\prime })\).

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.

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

*star product*is defined as

*transformation kernel*relating the two descriptions and defined in terms of the transformation function by

## 5 A Discrete Quantum Phase Space

*etc.*are the usual Pauli spin operators, leads to

## 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.

## Notes

### Acknowledgments

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

### References

- 1.Schwinger, J.: Algebra of microscopic measurement. Proc. Natl. Acad. Sci.
**45**, 1542 (1959)ADSCrossRefMATHMathSciNetGoogle Scholar - 2.Schwinger, J.: Geometry of states. Proc. Natl. Acad. Sci.
**46**, 257 (1960)ADSCrossRefMATHMathSciNetGoogle Scholar - 3.Schwinger, J.: Unitary operator bases. Proc. Natl. Acad. Sci.
**46**, 570 (1960)ADSCrossRefMATHMathSciNetGoogle Scholar - 4.Schwinger, J.: Quantum Kinematics and Dynamics, Advanced Book Program. Addison-Wesley, New York (1991)Google Scholar
- 5.Schwinger, J.: Quantum Mechanics : Symbolism of Atomic Measurement. Springer, Berlin (2001)CrossRefGoogle Scholar
- 6.von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955)MATHGoogle Scholar
- 7.Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, Oxford (1958)MATHGoogle Scholar
- 8.Weyl, H.: The Theory of Groups and Quantum Mechanics. Dover Publications, New York (1950)Google Scholar
- 9.Wigner, E.P.: On the correction for thermodynamic equilibrium. Phys. Rev.
**40**, 749 (1932)ADSCrossRefGoogle Scholar - 10.Groenewold, H.: On the principles of elementary quantum mechanics. Phys. A
**12**, 405 (1946)ADSMATHMathSciNetGoogle Scholar - 11.Moyal, J.E.: Quantum mechanics as a statistical theory. Proc. Camb. Philos. Soc.
**45**, 99 (1949)ADSCrossRefMATHMathSciNetGoogle Scholar - 12.Zachos, Z., Fairlie, D.B., Curtright, T.L.: Quantum Mechanics in Phase Space. World Scientific Publishing, Singapore (2005)MATHGoogle Scholar
- 13.Ferrie, C., Emerson, J.: Framed Hilbert space: hanging the quasi-probability pictures of quantum systems. New J. Phys.
**11**, 063040 (2009)ADSCrossRefGoogle Scholar - 14.Gross, D.: Hudson’s theorem for finite dimensional quantum systems. J. Math. Phys.
**47**, 122107 (2006)ADSCrossRefMathSciNetGoogle Scholar - 15.Vourdas, A.: Quantum systems with finite Hilbert space. Rep. Prog. Phys.
**67**, 267 (2004)ADSCrossRefMathSciNetGoogle Scholar - 16.Ferrie, C., Emerson, J.: Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations. J. Phys. A
**41**, 352001 (2008)CrossRefMathSciNetGoogle Scholar - 17.Ferrie, C., Morris, R., Emerson, J.: Necessity of negativity in quantum theory. Phys. Rev.
**82**, 044103 (2010)ADSCrossRefMathSciNetGoogle Scholar - 18.Maki, Z.: An algebraic approach to the quantum theory of measurements. Prog. Theor. Phys.
**84**, 574 (1990)ADSCrossRefMATHMathSciNetGoogle Scholar - 19.Horwitz, L.P.: Schwinger algebra for quaternionic quantum mechanics. Found. Phys.
**27**, 1011 (1997)ADSCrossRefMathSciNetGoogle Scholar - 20.Wootters, W.K.: A Wigner function formulation of finite state quantum mechanics. Ann. Phys.
**176**, 1 (1987)ADSCrossRefMathSciNetGoogle Scholar - 21.Buot, F.A.: Method for calculating \({\rm TrH}^n\) in solid state theory. Phys. Rev. B
**10**, 3700 (1974)ADSCrossRefGoogle Scholar - 22.Galetti, D., De Toledo Piza, A.F.R.: An extended Weyl–Wigner transformation for special finite spaces. Physica
**149A**, 267 (1988)Google Scholar - 23.Chaturvedi, S., Ercolessi, E., Marmo, G., Morandi, G., Mukunda, N., Simon, R.: Wigner–Weyl correspondence in quantum mechanics for continuous and discrete systems—a Dirac-inspired view. J. Phys. A
**39**, 1405 (2006)ADSCrossRefMATHMathSciNetGoogle Scholar