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Matrix Wigner Function and SU(1,1)

Peter Morrison [email protected] University of Technology, Sydney
Abstract

This paper contains a brief sketch of some methods that can be used to obtain the Wigner function for a number of systems. We give an overview of the technique as it is applied to some simple differential systems related to diffusion problems in one dimension. We compute the Wigner function for the harmonic oscillator, the x​pxp interaction, and a hyperbolic oscillator. These systems are shown to share several properties in common related to the Whittaker function and various formulae for the Laguerre polynomials. To contrast with the techniques that are applicable to problems involving continuous states, we then show that by expanding the solution space to the hyperbolic plane and utilising some results from matrix calculus, we are able to recover a number of interesting identities for SU(1,1) and the pseudosphere. We close with a discussion of some more advanced topics in the theory of the Wigner function.

Abstract

This paper contains a brief sketch of some methods that can be used to obtain the Wigner function for a number of systems. We give an overview of the technique as it is applied to some simple differential systems related to diffusion problems in one dimension. We compute the Wigner function for the harmonic oscillator, the x​pxp interaction, and a hyperbolic oscillator. These systems are shown to share several properties in common related to the Whittaker function and various formulae for the Laguerre polynomials. To contrast with the techniques that are applicable to problems involving continuous states, we then show that by expanding the solution space to the hyperbolic plane and utilising some results from matrix calculus, we are able to recover a number of interesting identities for SU(1,1) and the pseudosphere. We close with a discussion of some more advanced topics in the theory of the Wigner function.

keywords:
;Special Functions; Wigner Function; Quasiprobability; XP Oscillator; Matrix Analysis;
\subjectindex

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1 Introduction

The quasiprobability function of Wigner wigner1997quantum is an interesting counterpart to the familiar probability distributions we associate with the fundamental solutions of diffusion equations. This function reduces, in many cases, to an analysis of the autocorrelation of a stochastic variable as expressed through the eigenfunctions of the solution. By using the Wigner function, we are able to extract all expectations values in a similar way to the use of a transition probability density. However, this function fails to be completely positive, and obeys the laws of quasiprobabilities as opposed to probabilities as a result. This paper addresses several deeper questions that might be asked regarding the theory of probability densities in one dimension, being the question of when it is possible to find a positive definite, bounded solution to a differential system which defines a density. As we shall show using this important, well-known counterexample, it is possible to extract all measurable information from this quasiprobability, and indeed, as we shall show in the following calculation, we can learn valuable properties about the dynamics of diffusion problems by relaxing the assumption of positivity.

In parallel with this investigation, we shall show that another assumption related to the dimensionality of the system may be relaxed in a different way, by utilising some relationships from the study of the quantum brachistochrone morrison2012time . The analysis proceeds in a similar fashion, with the added complication of non-commutative variables. We shall demonstrate that it is possible to use an identical methodology as in the first part to obtain a number of interesting formulae by using the theory of matrix representations.

2 Review

We shall briefly go over some papers and results that are used in the following. As an entry point for the reader who is unfamiliar with the Wigner function, the reference text of Tannor tannor2007introduction has an excellent discussion of the topic suitable for this level of expertise. We refer other readers familiar with the basic structures to the papers of frank2000wigner , who has calculated the Wigner function for the Morse oscillator. This system is very similar to those we are to analyse in the following. For other references, especially with regard to SU(1,1), the paper of Seyfarth et. al seyfarth2020wigner contains an outline of a calculation related to the Wigner function on the hyperbolic disk, by using a coherent state methodology. The papers of Lenz lenz2016mehler ; lenz2017mehler ; lenz2017positive is also of use in understanding the interaction between positive definite kernels and SU(1,1). This paper is closely associated with some results explored in this work, and although we shall not delve deeply into the use of coherent states and their formalism, at many points the results shall be complementary. We shall point this out where appropriate.

3 Properties of the Wigner Transform

We shall now briefly review the necessary formal axioms of Wigner function theory and the projection theories relevant to this calculation. In particular, we shall need the formulae for the Wigner function, which is essentially the characteristic function of the autocorrelated operator. This paper shall use the known relationship between the phase-space version of quantum mechanics and matrix projection operators. For references, the interested reader is directed to the historical works of Wigner wigner1997quantum , the canonical reference hillery1984distribution and description of the application of Poisson brackets and phase space quantum mechanics by Groenewold wigner1997quantum ; groenewold1946principles and Moyal moyal1949quantum . More recent resources may be found in Tannor’s treatise on time dependent quantum mechanics tannor2007introduction and the papers of Curtright et. al curtright1998features ; curtright2001generating , the last of which contains an explicit outline of the method of the associative star product.

3.1 Axioms of the Wigner Function

We shall run over the basic properties of the Wigner transform, and how it relates to the intersection between classical and quantum mechanics. The basic definition of the Wigner function, see e.g. hillery1984distribution ; tannor2007introduction for an modern introduction to the topic, can be written as the Fourier transform of the autocorrelated operator expectation:

Definition 3.1.
𝒜W​(p,q)=∫−∞+∞ei​p​s/ℏ​⟨q−s2|A^|q+s2⟩​𝑑s\mathcal{A}_{W}(p,q)=\int_{-\infty}^{+\infty}e^{ips/\hbar}\left\langle q-\dfrac{s}{2}\right|\hat{A}\left|q+\dfrac{s}{2}\right\rangle ds (1)

The basic concept is to develop a way to move between operators and continuous, differentiable functions. Wigner wigner1997quantum developed a series of axioms for his analysis of quasiprobability densities by applying this functional to the density matrix of the system:

Definition 3.2.
ρW​(p,q)=∫−∞+∞ei​p​(x−x′)/ℏ​⟨x′|ϱ^|x⟩​𝑑s\mathcal{\rho}_{W}(p,q)=\int_{-\infty}^{+\infty}e^{ip(x-x^{\prime})/\hbar}\left\langle x^{\prime}\right|\hat{\varrho}\left|x\right\rangle ds (2)
Definition 3.3.

The basic assumptions on the density matrix are that it be representated as a pure state:

ϱ^=|Ψ⟩​⟨Ψ|\hat{\varrho}=\left|\Psi\right\rangle\left\langle\Psi\right| (3)
Definition 3.4.

We assume that the set of probabilities is exhaustive, so we may write:

𝐓𝐫​(ϱ^)=⟨Ψ|Ψ⟩=1\mathbf{Tr}\left(\hat{\varrho}\right)=\left\langle\Psi\right|\left.\Psi\right\rangle=1 (4)
Definition 3.5.

We can relate expectation values of an operator to a trace over the Wigner function

⟨A^⟩=𝐓𝐫​(ϱ^​A^)=12​π​ℏ​∫−∞+∞∫−∞+∞ρW​(p,q)​𝒜W​(p,q).d​p.d​q\left\langle\hat{A}\right\rangle=\mathbf{Tr}\left(\hat{\varrho}\hat{A}\right)=\dfrac{1}{2\pi\hbar}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\mathcal{\rho}_{W}(p,q)\mathcal{A}_{W}(p,q).dp.dq (5)

This property is representative of a broader class of operators which are tracial in that the trace is preserved as an integral over the phase space:

Definition 3.6.
𝐓𝐫​(A^​B^)=∫−∞+∞∫−∞+∞𝒜W​(p,q)​ℬW​(p,q).d​p.d​q\mathbf{Tr}\left(\hat{A}\hat{B}\right)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\mathcal{A}_{W}(p,q)\mathcal{B}_{W}(p,q).dp.dq (6)

Further, the Wigner quasiprobability function is defined by the normalised quasidistribution:

Definition 3.7.
fW​(p,q)=12​π​ℏ​ρW​(p,q)f_{W}(p,q)=\dfrac{1}{2\pi\hbar}\mathcal{\rho}_{W}(p,q) (7)

Finally, as any probability density is a real number, this is associated to the Hermitian nature of the Wigner transform.

Definition 3.8.
f​(x,p)=⟨Ψ|M^​(x,p)|Ψ⟩f(x,p)=\left\langle\Psi\left|\hat{M}(x,p)\right|\Psi\right\rangle (8)
M^†=M^\hat{M}^{\dagger}=\hat{M} (9)
f∗=ff^{*}=f (10)

and therefore the Wigner function is real. Marginal distributions are produced in the standard way

Lemma 3.9.
∫−∞+∞f​(x,p)​𝑑p=|ψ​(x)|2=⟨x|ρ^|x⟩\int_{-\infty}^{+\infty}f(x,p)dp=|\psi(x)|^{2}=\left\langle x\left|\hat{\rho}\right|x\right\rangle (11)
∫−∞+∞f​(x,p)​𝑑x=|ψ​(p)|2=⟨p|ρ^|p⟩\int_{-\infty}^{+\infty}f(x,p)dx=|\psi(p)|^{2}=\left\langle p\left|\hat{\rho}\right|p\right\rangle (12)

We assume also that the total probability sums to one, in the same way as a classical probability density. The integral in this case is taken over the whole phase space:

Definition 3.10.
∫−∞+∞∫−∞+∞f​(x,p)​𝑑x​𝑑p=1\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}f(x,p)dxdp=1 (13)

The next ingredient is Galilean invariance principles. The basis propositions state that the Wigner function should transform appropriately under translations in space and time. For space translations, we have that if the state is shifted in space, the Wigner function shifts accordingly via:

Ψ​(x)→Ψ​(x+a)f​(x,p)→f​(x+a,p)\begin{array}[]{cc}\Psi(x)\rightarrow&\Psi(x+a)\\ f(x,p)\rightarrow&f(x+a,p)\end{array} (14)

and for momentum, we have that if the wavefunction is rotated through a phase, the momentum component is translated according to:

Ψ​(x)→ei​p′​x/ℏ​Ψ​(x)f​(x,p)→f​(x,p−p′)\begin{array}[]{cc}\Psi(x)\rightarrow&e^{ip^{\prime}x/\hbar}\Psi(x)\\ f(x,p)\rightarrow&f(x,p-p^{\prime})\end{array} (15)

The final ingredient is parity invariance. If we invert the space coordinate by mirror reflection, the Wigner function will change via:

Ψ​(x)→Ψ​(−x)f​(x,p)→f​(−x,−p)\begin{array}[]{cc}\Psi(x)\rightarrow&\Psi(-x)\\ f(x,p)\rightarrow&f(-x,-p)\end{array} (16)

In a complementary way, taking the complex conjugate of the wave function, then the momentum coordinate is altered to yield

Ψ​(x)→Ψ∗​(x)f​(x,p)→f​(x,−p)\begin{array}[]{cc}\Psi(x)\rightarrow&\Psi^{*}(x)\\ f(x,p)\rightarrow&f(x,-p)\end{array} (17)

These are all the ingredients that we need to define the Wigner (c. 1932) distribution. Note that we have not assumed that the density is positive. This is why it is called a quasiprobability. Many authors have discussed the non-positivity of this function so we shall not discuss it here.

3.2 Poisson Bracket

We shall now review some basic properties of the Poisson bracket and elementary phase space mechanics.

Definition 3.11.

The Poisson bracket gives the Liouville evolution for the state space:

∂ρ∂t={ℋ​(q,p),ρ}\dfrac{\partial\rho}{\partial t}=\{\mathcal{H}(q,p),\rho\} (18)

The Poisson bracket is antisymmetric:

{a,b}=−{b,a}\{a,b\}=-\{b,a\} (19)

and obeys the Jacobi identity:

{a,{b,c}}+{b,{c,a}}+{c,{a,b}}=0\{a,\{b,c\}\}+\{b,\{c,a\}\}+\{c,\{a,b\}\}=0 (20)
Theorem 3.12.

The basic system dynamics is described through the action of the 2-form:

ω=d​pμ∧d​qμ\omega=dp_{\mu}\wedge dq^{\mu} (21)

The generator of the Lie vector field is defined through the action of the Poisson bracket:

X^f=∂f∂pμ​∂∂qμ−∂f∂qμ​∂∂pμ\hat{X}_{f}=\dfrac{\partial f}{\partial p_{\mu}}\dfrac{\partial}{\partial q^{\mu}}-\dfrac{\partial f}{\partial q^{\mu}}\dfrac{\partial}{\partial p_{\mu}} (22)

Hamilton’s equations of classical dynamics can then be written as:

f=ℋ​(qμ,pμ)f=\mathcal{H}(q^{\mu},p_{\mu}) (23)
q˙μ=∂ℋ∂pμ\dot{q}^{\mu}=\dfrac{\partial\mathcal{H}}{\partial p_{\mu}} (24)
p˙μ=−∂ℋ∂qμ\dot{p}_{\mu}=-\dfrac{\partial\mathcal{H}}{\partial q^{\mu}} (25)
X^ℋ=q˙μ​∂∂qμ+p˙μ​∂∂pμ=∂∂t\hat{X}_{\mathcal{H}}=\dot{q}^{\mu}\dfrac{\partial}{\partial q^{\mu}}+\dot{p}_{\mu}\dfrac{\partial}{\partial p_{\mu}}=\dfrac{\partial}{\partial t} (26)

We shall now describe the basic equivalence from the phase space picture to quantum dynamics. From quantum mechanics, we know the equivalent of the Liouville equation in statistical theory is the von Neumann equation:

i​∂ρ^∂t=[H^,ρ^]i\dfrac{\partial\hat{\rho}}{\partial t}=[\hat{H},\hat{\rho}] (27)

where under the bracket we have non-commutative matrix operations via:

[A^,B^]=A^​B^−B^​A^[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A} (28)

The dynamic operator is Hermitian:

H^†=H^\hat{H}^{\dagger}=\hat{H} (29)

We assume the standard properties of a pure state density matrix:

Tr​ρ^=1,ρ^2=ρ^\mathrm{Tr}\hat{\rho}=1,\hat{\rho}^{2}=\hat{\rho} (30)

This is the basic setup of the properties of the Wigner function. There are various situations more general than this, in particular it is clear that this system is Galilei-invariant and not Lorentz invariant, which limits the domain of application of this theory i.e. it is not relativistic. In the case where velocities are low this is an acceptable situation, and we can expect there to be a systematic relationship between the behaviour of the quantum and classical theories.

3.3 Moyal Bracket

Groenewold (1946), J.E. Moyal (1949) groenewold1946principles ; moyal1949quantum proposed to investigate the nature of phase-space quantum mechanics. Their insight was to try and understand the exact relationship between the classical and quantum systems by using the Poisson bracket. Independently, they came to similar answers regarding the question as to what replaces the Poisson bracket in quantum mechanics. The solution to this question was to employ the associative star product, defined through:

Definition 3.13.
⋆′′=exp(i​ℏ2[∂←x∂→p−∂←p∂→x])=exp(−i​ℏ2Λ^){}^{\prime}\star^{\prime}=\exp\left(\dfrac{i\hbar}{2}\left[\overleftarrow{\partial}_{x}\overrightarrow{\partial}_{p}-\overleftarrow{\partial}_{p}\overrightarrow{\partial}_{x}\right]\right)=\exp\left(-\dfrac{i\hbar}{2}\hat{\Lambda}\right) (31)
Notation 1.

Overhead arrows in this context indicate if the operator acts to the left or the right:

f​Λ^​g=−(∂f∂x​∂g∂p−∂f∂p​∂g∂x)f\hat{\Lambda}g=-\left(\dfrac{\partial f}{\partial x}\dfrac{\partial g}{\partial p}-\dfrac{\partial f}{\partial p}\dfrac{\partial g}{\partial x}\right) (32)

The great insight of Moyal was to employ this algebra to derive the quantum equivalent of phase space mechanics. Indeed, by using this method we can solve for the dynamics of the Wigner distribution defined through the autocorrelation transform. The Moyal equation may be written:

WA^​B^=WA^​exp⁡(−i​ℏ2​Λ^)​WB^W_{\hat{A}\hat{B}}=W_{\hat{A}}\exp\left(-\dfrac{i\hbar}{2}\hat{\Lambda}\right)W_{\hat{B}} (33)

where WA^W_{\hat{A}} represents the Wigner transform of the operator. The quantum equivalent of the Liouville equation in phase space mechanics goes over into Moyal’s equation:

∂∂t​Wρ^=−2ℏ​WH^​sin⁡(ℏ2​Λ^)​Wρ^\dfrac{\partial}{\partial t}W_{\hat{\rho}}=-\dfrac{2}{\hbar}W_{\hat{H}}\sin\left(\dfrac{\hbar}{2}\hat{\Lambda}\right)W_{\hat{\rho}} (34)

The classical limit of this equation can be evaluated which gives

∂∂t​Wρ^=−WH^​Λ^​Wρ^={WH^,Wρ^}\dfrac{\partial}{\partial t}W_{\hat{\rho}}=-W_{\hat{H}}\hat{\Lambda}W_{\hat{\rho}}=\{W_{\hat{H}},W_{\hat{\rho}}\} (35)

The normal state of affairs in quantum mechanics is the eigenvalue problem:

ℋ​ψ=E​ψ\mathcal{H}\psi=E\psi (36)

In terms of the star-eigenvalue problem, it is simple to show that there is a complementary equation:

ℋ⋆f=E​f\mathcal{H}\star f=Ef (37)

The action of the associative star product is shown to be:

f⋆g=f​(x+i​ℏ2​∂→p,p−i​ℏ2​∂→x)​g​(x,p)f\star g=f(x+\dfrac{i\hbar}{2}\overrightarrow{\partial}_{p},p-\dfrac{i\hbar}{2}\overrightarrow{\partial}_{x})g(x,p) (38)

and the star-eigenvalue problem becomes the differential equation:

Theorem 3.14.
ℋ​(x+i​ℏ2​∂→p,p−i​ℏ2​∂→x)​f​(x,p)=E​f​(x,p)\mathcal{H}(x+\dfrac{i\hbar}{2}\overrightarrow{\partial}_{p},p-\dfrac{i\hbar}{2}\overrightarrow{\partial}_{x})f(x,p)=Ef(x,p) (39)

This is actually two simultaneous differential equations, in that the solution to this expression solves a complex valued PDE which we can separate into real and imaginary parts. For description of this technique in a modern context, consult curtright1998features ; curtright2001generating .

4 Wigner Distribution for Hyperbolic Systems

The star eigenvalue equations guarantee ready access to the formulae for the Wigner distribution, if we can solve them. As we shall show, it is possible to use the Wigner transform to find relationships between the eigenstates of different systems. We shall also encounter the spectral theory of the Wigner distribution, and demonstrate that we can use this to find kernel representations for the various different systems. At its core, the simplified example of a hyperbolic geometry in phase space gives us access to several fundamental examples which we can use to probe the technique. Future works to appear have shown that the Wigner-Weyl transform can be used to find Wigner distributions in systems which have completely continuous eigenvalues, where the basic theory of Curtright curtright1998features ; curtright2001generating suffers from complications which do not present themselves for the discrete states we shall consider in the following.

4.1 Simple Harmonic Oscillator

Proposition 4.1.

It is possible to use the star-eigenvalue equation to define Wigner functions consistent with the results of Groenwold and Moyal, (Curtright).

Proposition 4.2.

The Wigner function in this case is given by a Laguerre function of the distance function in phase space.

Proof.

There are very few results available for calculated Wigner functions. We shall begin with the simplest non-trivial example, which is given by the harmonic oscillator. This has the advantage of having a known solution. The basic differential operator we shall take is specified by:

Example 4.3.
H^=p^22​m+12​m​ω2​x2∼12​(p^2+x^2)\hat{H}=\dfrac{\hat{p}^{2}}{2m}+\dfrac{1}{2}m\omega^{2}x^{2}\sim\dfrac{1}{2}(\hat{p}^{2}+\hat{x}^{2}) (40)

where we are using natural units to simplify the argument. The eigenfunctions are given by the normalised Hermite polynomials:

Ψm​(u)=π−1/42m​m!​Hm​(u)​e−u2/2\Psi_{m}(u)=\dfrac{\pi^{-1/4}}{\sqrt{2^{m}m!}}H_{m}(u)e^{-u^{2}/2} (41)

The star-eigenvalue equation in this situation can be written:

Lemma 4.4.
12​[(p−i2​∂→x)2+(x+i2​∂→p)2]​f​(x,p)=E​f​(x,p)\dfrac{1}{2}[(p-\dfrac{i}{2}\overrightarrow{\partial}_{x})^{2}+(x+\dfrac{i}{2}\overrightarrow{\partial}_{p})^{2}]f(x,p)=Ef(x,p) (42)

We use normalised variables:

ω=ℏ=m=c=1\omega=\hbar=m=c=1 (43)

Taking real and imaginary parts, we find the differential equations:

x​∂f∂p−p​∂f∂x=0x\dfrac{\partial f}{\partial p}-p\dfrac{\partial f}{\partial x}=0 (44)
[(p2+x2)−14​(∂2∂p2+∂2∂x2)−2​E]​f​(x,p)=0\left[(p^{2}+x^{2})-\dfrac{1}{4}(\dfrac{\partial^{2}}{\partial p^{2}}+\dfrac{\partial^{2}}{\partial x^{2}})-2E\right]f(x,p)=0 (45)

The two differential equations are solved using the following.

f​(x,p)=F​(λ​[x2+p2])f(x,p)=F(\lambda[x^{2}+p^{2}]) (46)
f​(x,p)=F​(z),z=2​(x2+p2)f(x,p)=F(z),z=2(x^{2}+p^{2}) (47)

Substituting this into the second differential equation and using the chain rule, we find:

(z4−∂∂z−z​∂2∂z2−E)​F​(z)=0\left(\dfrac{z}{4}-\dfrac{\partial}{\partial z}-z\dfrac{\partial^{2}}{\partial z^{2}}-E\right)F(z)=0 (48)

Using the trial solution F​(z)=e−z/2​L​(z)F(z)=e^{-z/2}L(z) this reduces to the Laguerre differential equation

(E−12)​L​(z)+(1−z)​∂L∂z+z​∂2L∂z2=0(E-\dfrac{1}{2})L(z)+(1-z)\dfrac{\partial L}{\partial z}+z\dfrac{\partial^{2}L}{\partial z^{2}}=0 (49)

The system is solved using the set of eigenstates n=E−12=0,1,2,…n=E-\dfrac{1}{2}=0,1,2,..., then we have L​(z)=Ln​(z)L(z)=L_{n}(z), and the solution for the Wigner distribution is then given by:

fn​(x,p)=Cn​F​(z)=Cn​e−z/2​Ln​(z)=Cn​e−2​ℋ​Ln​(4​ℋ)f_{n}(x,p)=C_{n}F(z)=C_{n}e^{-z/2}L_{n}(z)=C_{n}e^{-2\mathcal{H}}L_{n}(4\mathcal{H}) (50)

or in the original coordinates:

Theorem 4.5.
fn​(x,p)=Cn​e−(x2+p2)​Ln​[2​(x2+p2)]f_{n}(x,p)=C_{n}e^{-(x^{2}+p^{2})}L_{n}[2(x^{2}+p^{2})] (51)

To resolve the value of the constant, we must use other means. The most straightforward way in which to determine this is to use the integral normalisation of the quasidistribution. The integral formula for the Wigner distribution can be written out as:

fm​(x,p)=∫−∞+∞e2​i​p​y​Ψm∗​(x+y)​Ψm​(x−y)​𝑑yf_{m}(x,p)=\int_{-\infty}^{+\infty}e^{2ipy}\Psi_{m}^{*}(x+y)\Psi_{m}(x-y)dy (52)

Using the expression for the eigenstates of the harmonic oscillator, and doing some simple calculations involving Hermite polynomials yield:

fm​(x,p)=12m​m!​π​∫−∞+∞e2​i​p​y​Hm​(x+y)​Hm​(x−y)​e−(x2+y2)​𝑑yf_{m}(x,p)=\dfrac{1}{2^{m}m!\sqrt{\pi}}\int_{-\infty}^{+\infty}e^{2ipy}H_{m}(x+y)H_{m}(x-y)e^{-(x^{2}+y^{2})}dy (53)

This equation will be further simplified by using an integral formula. Extracting factors which do not enter into the integration, the formula may be manipulated into the format:

fm​(x,p)=(−1)m​e−(x2+p2)2m​m!​π​∫−∞+∞e−(y−i​p)2​Hm​(y+x)​Hm​(y−x)​𝑑yf_{m}(x,p)=\dfrac{(-1)^{m}e^{-(x^{2}+p^{2})}}{2^{m}m!\sqrt{\pi}}\int_{-\infty}^{+\infty}e^{-(y-ip)^{2}}H_{m}(y+x)H_{m}(y-x)dy (54)

Using formula from Gradshteyn and Ryzik (eq. 7.377 pp 853)gradshteyn2014table :

Lemma 4.6.
Lm​(−2​ξ1​ξ2)=12m​Γ​(m)​π​∫−∞+∞e−η2​Hm​(η+ξ1)​Hm​(η+ξ2)​𝑑ηL_{m}(-2\xi_{1}\xi_{2})=\dfrac{1}{2^{m}\Gamma(m)\sqrt{\pi}}\int_{-\infty}^{+\infty}e^{-\eta^{2}}H_{m}(\eta+\xi_{1})H_{m}(\eta+\xi_{2})d\eta (55)
ξ1=i​p+x,ξ2=i​p−x,ξ1​ξ2=−(x2+p2)\xi_{1}=ip+x,\xi_{2}=ip-x,\xi_{1}\xi_{2}=-(x^{2}+p^{2}) (56)

The solution for the Wigner distribution from this perspective is then:

fm​(x,p)=(−1)m​e−(x2+p2)π​Lm​[2​(x2+p2)]f_{m}(x,p)=\dfrac{(-1)^{m}e^{-(x^{2}+p^{2})}}{\pi}L_{m}[2(x^{2}+p^{2})] (57)

The only major difference between the results is the normalisation. This can be evaluated using the statement of total probability ∫−∞+∞∫−∞+∞fn​(x,p)​𝑑x​𝑑p=1\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}f_{n}(x,p)dxdp=1. Using this on the un-normalised form of the Wigner distribution, we find:

Cn​∫−∞+∞∫−∞+∞e−(x2+p2)​Ln​[2​(x2+p2)]​𝑑x​𝑑p=1C_{n}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}e^{-(x^{2}+p^{2})}L_{n}[2(x^{2}+p^{2})]dxdp=1 (58)

In the space represented by the co-ordinate z=2​(x2+p2)z=2(x^{2}+p^{2}), the integral over all of phase space is transformed via

∫−∞+∞∫−∞+∞𝑑x​𝑑p→12​∫−∞+∞𝑑z\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}dxdp\rightarrow\dfrac{1}{2}\int_{-\infty}^{+\infty}dz (59)

Normalisation of the quasidistribution is then:

Cn2​∫0+∞𝑑z​e−z/2​Ln​(z)=1\dfrac{C_{n}}{2}\int_{0}^{+\infty}dze^{-z/2}L_{n}(z)=1 (60)

Using Gradshetyn and Ryzik (eq. 7.414.6 pp809) gradshteyn2014table :

∫0+∞e−b​x​Ln​(x)​𝑑x=(b−1)n​b−n−1\int_{0}^{+\infty}e^{-bx}L_{n}(x)dx=(b-1)^{n}b^{-n-1} (61)

we can resolve the normalised Wigner function to be given by:

Theorem 4.7.
fn​(x,p)=(−1)n​e−(x2+p2)​Ln​[2​(x2+p2)]f_{n}(x,p)=(-1)^{n}e^{-(x^{2}+p^{2})}L_{n}[2(x^{2}+p^{2})] (62)

in agreement with the other methods.

∎

Remark 1.

The missing numerical factor of π\pi comes from our original definition of the distribution itself. This solution dates to moyal1949quantum . Obviously the class of systems which are deformable into a simple harmonic oscillator and the myriad of variations forms a large part of quantum mechanics. Consequently, this representation of the Wigner function performs an analogous role to the kernel function in the two point correlation function.

4.2 XP oscillator

We are led naturally to think of other examples related to the harmonic oscillator. There are several directions one may take, the simplest being the change from spherical symmetry, as expressed through the simple harmonic oscillator, to a hyperbolic geometry.

Proposition 4.8.

The XP oscillator is a hyperbolic variant of the simple harmonic oscillator.

Proposition 4.9.

The Wigner function in this case can be evaluated by similar means, with the distance metric modified to a suitable form for the hyperbolic space, and the Laguerre polynomial replaced by a Whittaker function.

Proof.

The following two systems are indicative of various ways in which this change can be achieved satisfactorily, the hyperbolic plane in multiplicative form:

Definition 4.10.
ℋ=12​(x^​p^+p^​x^)=−i​(x​∂∂x+12)\mathcal{H}=\dfrac{1}{2}(\hat{x}\hat{p}+\hat{p}\hat{x})=-i\left(x\dfrac{\partial}{\partial x}+\dfrac{1}{2}\right) (63)

and the hyperbolic oscillator which we can write as:

Definition 4.11.
ℋ=12​(p2−x2)=−12​[∂2∂x2+x2]\mathcal{H}=\dfrac{1}{2}(p^{2}-x^{2})=-\dfrac{1}{2}\left[\dfrac{\partial^{2}}{\partial x^{2}}+x^{2}\right] (64)

We assume the standard form of the momentum operator p^=−i​∂x\hat{p}=-i\partial_{x} familiar from quantum mechanics. The three fundamental operators defined by these different Hamiltonians + the SHO are related to the fundamental invariants that can be formed between the space and momentum. We have a method for evaluating the Wigner function in a unitary space, but the formulae for these Hamiltonian operators is only pseudounitary, hence our basic method requires modification to function correctly in this case. For the situation where the dynamics is specified through:

ℋ​f=−i​(x​∂∂x+12)​f\mathcal{H}f=-i\left(x\dfrac{\partial}{\partial x}+\dfrac{1}{2}\right)f (65)

we can carry out most of the analysis in exactly the same way as for the harmonic oscillator. In this case, the star-eigenvalue equations are given by:

ℋ​(x+i​ℏ2​∂→p,p−i​ℏ2​∂→x)​f​(x,p)=E​f​(x,p)\mathcal{H}(x+\dfrac{i\hbar}{2}\overrightarrow{\partial}_{p},p-\dfrac{i\hbar}{2}\overrightarrow{\partial}_{x})f(x,p)=Ef(x,p) (66)

and can be written:

Theorem 4.12.
x​p​f+14​∂2f∂p​∂x+i2​(p​∂f∂p−x​∂f∂x)=E​fxpf+\dfrac{1}{4}\dfrac{\partial^{2}f}{\partial p\partial x}+\dfrac{i}{2}\left(p\dfrac{\partial f}{\partial p}-x\dfrac{\partial f}{\partial x}\right)=Ef (67)

Taking real and imaginary parts, we obtain a solvable set of equations. Writing out the paired equations explicitly, we find:

Theorem 4.13.
(x​p+14​∂2∂p​∂x)​f=E​f\left(xp+\dfrac{1}{4}\dfrac{\partial^{2}}{\partial p\partial x}\right)f=Ef (68)
p​∂f∂p−x​∂f∂x=0p\dfrac{\partial f}{\partial p}-x\dfrac{\partial f}{\partial x}=0 (69)

Similar considerations to the harmonic oscillator calculation allow us to derive the following:

F​(z)=C1​Wi​E,1/2​(4​i​z)+C2​Mi​E,1/2​(4​i​z)F(z)=C_{1}W_{iE,1/2}(4iz)+C_{2}M_{iE,1/2}(4iz) (70)

where the variable changes are given by f​(x,p)=F​(x​p)=F​(z)f(x,p)=F(xp)=F(z) . To analyse which of these solutions has the correct behaviour, we must examine the structure of the Wigner distribution. Proceeding in a similar way to the SHO calculation, we can write the integral form of the Wigner distribution as:

f​(x,p)=∫ei​p​s​Ψm∗​(x−s/2)​Ψm​(x+s/2)​𝑑sf(x,p)=\int e^{ips}\Psi_{m}^{*}(x-s/2)\Psi_{m}(x+s/2)ds
=|dm|2​∫0∞ei​p​s​(x+s2)−i​Em​(x−s2)i​Em​𝑑s=|d_{m}|^{2}\int_{0}^{\infty}e^{ips}(x+\dfrac{s}{2})^{-iE_{m}}(x-\dfrac{s}{2})^{iE_{m}}ds (71)

We have assumed the eigenfunction for the pseudo-Hamiltonian operator can be specified by Ψm​(x)=dm​(x)​xi​Em\Psi_{m}(x)=d_{m}(x)x^{iE_{m}}. Evaluating the integral can be done using Whittaker functions:

Theorem 4.14.
Wi​Em,1/2​(4​i​x​p)=e−2​i​x​pΓ​(1−i​Em)​∫0∞e−4​i​x​p​u​u−i​Em​(1+u)i​Em​𝑑uW_{iE_{m},1/2}(4ixp)=\dfrac{e^{-2ixp}}{\Gamma(1-iE_{m})}\int_{0}^{\infty}e^{-4ixpu}u^{-iE_{m}}(1+u)^{iE_{m}}du (72)

after some lengthy algebra. This form of the confluent hypergeometric function is new and is not found in buchholz2013confluent , but follows naturally from the geometry of the problem. The result found for the Wigner function in this instance is given by the un-normalised formula:

fm​(x,p)=Cm​(−1)−i​Em​Wi​Em,1/2​(4​i​x​p)f_{m}(x,p)=C_{m}(-1)^{-iE_{m}}W_{iE_{m},1/2}(4ixp) (73)

whence upon using the statement of total probability we conclude that the normalisation is given by:

∫A​Wi​Em,1/2​(4​i​x​p)​𝑑x​𝑑p=1\int AW_{iE_{m},1/2}(4ixp)dxdp=1 (74)

A formula from Gradshetyn and Ryzik (eq 7.622.11) gradshteyn2014table enables resolution of the constant:

∫0∞e−x/2​xν−1​Wκ,1/2​(x)​𝑑x=Γ​(ν)​Γ​(ν+1)Γ​(ν−κ+1)\int_{0}^{\infty}e^{-x/2}x^{\nu-1}W_{\kappa,1/2}(x)dx=\dfrac{\Gamma(\nu)\Gamma(\nu+1)}{\Gamma(\nu-\kappa+1)} (75)

where the integration measure in this case goes over ∫∫𝑑x​𝑑p=∫0∞𝑑z\int\int dxdp=\int_{0}^{\infty}dz. Putting all the ingredients together, we find the normalised formula for the Wigner distribution on the half-plane as given by:

Theorem 4.15.
fm​(x,p)=(−1)−i​Em​e−2​i​x​p(4​i​x​p)​Γ​(1−i​Em)​Wi​Em,1/2​(4​i​x​p)f_{m}(x,p)=(-1)^{-iE_{m}}\dfrac{e^{-2ixp}}{(4ixp)}\Gamma(1-iE_{m})W_{iE_{m},1/2}(4ixp) (76)

∎

Remark 2.

This is a new result. Note the appearance of the “distance” function in the multiplicative hyperbolic system in the arguments of the different functions. We shall now show that the other example can be addressed in a similar fashion.

4.3 Hyperbolic Oscillator

Proposition 4.16.

A third variant with an analytic Wigner function is given by the hyperbolic oscillator.

Corollary 4.17.

This can be viewed as the extension of the harmonic oscillator to complex momentum or space.

Proposition 4.18.

The Wigner function may be found in a similar fashion as with the previous two examples. In this case, the distance metric is replaced by a hyperbolic distance in phase space, and the Wigner function is given by a special type of Laguerre function.

Proof.

The hyperbolic oscillator equation is defined by the Hamiltonian:

Definition 4.19.
ℋ=12​(p2−x2)\mathcal{H}=\dfrac{1}{2}(p^{2}-x^{2}) (77)

We can think of the three systems as being either spherical, pseudospherical, or hyperspherical. We can see directly here that this system will not be positive definite. There are deeper connections to differential geometry here which may be seen as deformations of the Hamiltonian into a curved space. See e.g. grosche1988path ; grosche1990path for an outline of the path integral kernel on the hyperbolic plane/pseudosphere. Calculating the Wigner function here is much harder, as instead of unitary, or pseudounitary, we have to deal with a different type of symmetry relation. Writing out the star-eigenvalue equations in full, we have:

Theorem 4.20.
ℋ⋆f=(p2−x2)​f+14​(∂2f∂p2−∂2f∂x2)−i​(p​∂f∂x+x​∂f∂p)=E​f\mathcal{H}\star f=(p^{2}-x^{2})f+\dfrac{1}{4}\left(\dfrac{\partial^{2}f}{\partial p^{2}}-\dfrac{\partial^{2}f}{\partial x^{2}}\right)-i\left(p\dfrac{\partial f}{\partial x}+x\dfrac{\partial f}{\partial p}\right)=Ef (78)

The fundamental differential equation for the eigenstate is given by:

ℋ​Ψm​(x)=Em​Ψm​(x)\mathcal{H}\Psi_{m}(x)=E_{m}\Psi_{m}(x) (79)
ℋ=−12​[∂2∂x2+x2]\mathcal{H}=-\dfrac{1}{2}\left[\dfrac{\partial^{2}}{\partial x^{2}}+x^{2}\right] (80)

Now, the star-eigenvalue equations can be solved to find:

Lemma 4.21.
f​(x,p)=F​(p2−x2)=F​(z)f(x,p)=F(p^{2}-x^{2})=F(z) (81)
∂2F∂z2+(1−Ez)​F​(z)=0\dfrac{\partial^{2}F}{\partial z^{2}}+(1-\dfrac{E}{z})F(z)=0 (82)
F​(z)=C1​Mi​E/2,1/2​(2​i​z)+C2​Wi​E/2,1/2​(2​i​z)F(z)=C_{1}M_{iE/2,1/2}(2iz)+C_{2}W_{iE/2,1/2}(2iz) (83)

There are two cases which can be distinguished here, which could take the place of the Hermitian symmetry. The eigenvalue might be a complex number, in which case:

−i2​[∂2Ψm∂x2+x2​Ψm]=Em​Ψm-\dfrac{i}{2}\left[\dfrac{\partial^{2}\Psi_{m}}{\partial x^{2}}+x^{2}\Psi_{m}\right]=E_{m}\Psi_{m} (84)

with adjoint

i2​[∂2Ψ¯m∂x2+x2​Ψ¯m]=Em​Ψ¯m\dfrac{i}{2}\left[\dfrac{\partial^{2}\bar{\Psi}{}_{m}}{\partial x^{2}}+x^{2}\bar{\Psi}_{m}\right]=E_{m}\bar{\Psi}_{m} (85)

The solutions are easily found:

Ψm=1x​[cm+​M−E/2,1/4​(i​x2)+cm−​W−E/2,1/4​(i​x2)]\Psi_{m}=\dfrac{1}{\sqrt{x}}\left[c_{m}^{+}M_{-E/2,1/4}(ix^{2})+c_{m}^{-}W_{-E/2,1/4}(ix^{2})\right] (86)
Ψ¯m=1x​[cm+​M+E/2,1/4​(i​x2)+cm−​W+E/2,1/4​(i​x2)]\bar{\Psi}_{m}=\dfrac{1}{\sqrt{x}}\left[c_{m}^{+}M_{+E/2,1/4}(ix^{2})+c_{m}^{-}W_{+E/2,1/4}(ix^{2})\right] (87)

The other alternative is that the eigenvalue is a real number:

ℋ​Ψm=−12​[∂2Ψm∂x2+x2​Ψm]=Em​Ψm\mathcal{H}\Psi_{m}=-\dfrac{1}{2}\left[\dfrac{\partial^{2}\Psi_{m}}{\partial x^{2}}+x^{2}\Psi_{m}\right]=E_{m}\Psi_{m} (88)

and the adjoint is skew Hermitian.

Ψ¯m​ℋ†=+12​[∂2Ψ¯m∂x2+x2​Ψ¯m]=Ψ¯m​Em\bar{\Psi}_{m}\mathcal{H}^{\dagger}=+\dfrac{1}{2}\left[\dfrac{\partial^{2}\bar{\Psi}_{m}}{\partial x^{2}}+x^{2}\bar{\Psi}_{m}\right]=\bar{\Psi}_{m}E_{m} (89)

The solutions in this case can also be written in terms of Whittaker functions:

Lemma 4.22.
Ψm=1x​[cm+​Mi​Em/2,1/4​(i​x2)+cm−​Wi​Em/2,1/4​(i​x2)]=Ψm​(x,E)\Psi_{m}=\dfrac{1}{\sqrt{x}}\left[c_{m}^{+}M_{iE_{m}/2,1/4}(ix^{2})+c_{m}^{-}W_{iE_{m}/2,1/4}(ix^{2})\right]=\Psi_{m}(x,E) (90)

We have a symmetry relation Ψ¯m=Ψm​(x,−E)\bar{\Psi}_{m}=\Psi_{m}(x,-E) for the adjoint. This represents the correct solution. The correct part of the solution taking into account boundary conditions is:

Ψm​(x)=cm​Wi​Em/2,1/4​(i​x2)x\Psi_{m}(x)=\dfrac{c_{m}W_{iE_{m}/2,1/4}(ix^{2})}{\sqrt{x}} (91)

The adjoint in this case is:

Ψ¯m​(x)=cm​W−i​Em/2,1/4​(i​x2)x\bar{\Psi}_{m}(x)=\dfrac{c_{m}W_{-iE_{m}/2,1/4}(ix^{2})}{\sqrt{x}} (92)

Evaluating the Wigner function in this situation seems to be a daunting task, however we can access the results by using the theory of special functions and Hermite polynomials. The following known conversion formulae are available for Whittaker functions, Laguerre functions and the Hermite polynomials. For the even states, we may write:

Theorem 4.23.
Wn+1/4,−1/4​(z)z=(−1)n​Γ​(n−1/2)​z−1/4​Ln−1/2(−1/2)​(z)​e−z/2\dfrac{W_{n+1/4,-1/4}(z)}{\sqrt{z}}=(-1)^{n}\Gamma(n-1/2)z^{-1/4}L_{n-1/2}^{(-1/2)}(z)e^{-z/2}
=z1/4​2−2​n​H2​n​(z)​e−z/2=z^{1/4}2^{-2n}H_{2n}(\sqrt{z})e^{-z/2} (93)

and the complementary formulae for the odd states are given by:

Wn+3/4,1/4​(z)z=(−1)n​Γ​(n)​z1/4​Ln(1/2)​(z)\dfrac{W_{n+3/4,1/4}(z)}{\sqrt{z}}=(-1)^{n}\Gamma(n)z^{1/4}L_{n}^{(1/2)}(z)
=z1/4​2−2​n​H2​n+1​(z)​e−z/2=z^{1/4}2^{-2n}H_{2n+1}(\sqrt{z})e^{-z/2} (94)

It is important to realise that this is a physical model, and as such we must respect the difference in parity between these two sets of complementary eigenstates which form the basis of the system. The other key ingredient here is the order switching formula for Laguerre polynomials:

(−x)kk!​Ln(k−n)​(x)=(−x)nn!​Ln−(k−n)​(x)\dfrac{(-x)^{k}}{k!}L_{n}^{(k-n)}(x)=\dfrac{(-x)^{n}}{n!}L_{n}^{-(k-n)}(x) (95)

Another valuable formula can be found in Gradshetyn and Ryzik 7.377 gradshteyn2014table :

∫−∞+∞e−u2​Hm​(u+y)​Hn​(u+z)​𝑑u=2n​π​Γ​(m)​zn−m​Lm(n−m)​(−2​y​z)\int_{-\infty}^{+\infty}e^{-u^{2}}H_{m}(u+y)H_{n}(u+z)du=2^{n}\sqrt{\pi}\Gamma(m)z^{n-m}L_{m}^{(n-m)}(-2yz) (96)

We are now in a position to evaluate the Wigner distribution for this system:

fm​(x,p)=∫−∞+∞e2​i​p​y​Ψ¯m​(x−i​y)​Ψm​(x+i​y)​𝑑yf_{m}(x,p)=\int_{-\infty}^{+\infty}e^{2ipy}\bar{\Psi}_{m}(x-iy)\Psi_{m}(x+iy)dy (97)

where we note that the formula for the Wigner distribution in this system is not the same due to the imaginary number in the argument of the eigenstates. This is the analytical continuation of the autocorrelation. Using the properties of Hermite polynomials, the solution is shown to be:

Theorem 4.24.
fn​(x,p)=i​A​eζ​(−1)2​n​22​n+1​π​Γ​(2​n)​L2​n(1)​[−2​ζ]f_{n}(x,p)=iAe^{\zeta}(-1)^{2n}2^{2n+1}\sqrt{\pi}\Gamma(2n)L_{2n}^{(1)}[-2\zeta] (98)
ζ=p2−x2\zeta=p^{2}-x^{2} (99)

Comparing this with the solution from the star-eigenvalue equation, we have:

fm​(x,p)=C​Wi​Em/2,1/2​(2​i​z)f_{m}(x,p)=CW_{iE_{m}/2,1/2}(2iz) (100)

There is a way to relate these two seemingly different representations. The conversion formula gives:

Wm+1,1/2​(z)=(−1)m​Γ​(m)​e−z/2​z​Lm(1)​(z)W_{m+1,1/2}(z)=(-1)^{m}\Gamma(m)e^{-z/2}zL_{m}^{(1)}(z) (101)

Q.E.D. ∎

Remark 3.

Using the expressions for the Wigner function, it is simple to see that fn​(x,p)=Fn​(ζ)f_{n}(x,p)=F_{n}(\zeta) and the other form of the Wigner state is Fn/2​(2​i​z)F_{n/2}(2iz). We have not considered the statement of total probability here.

5 Matrix Wigner Function

We now examine finite matrix groups that have similar properties to the functions we generated in the previous calculations. Some results are available, but in general research is only thinly available on the topic. Recent advances may be found in Seyfarth et. al seyfarth2020wigner . One way in which to generate matrix groups with an equivalent structure to the previous sets of special functions is to use the creation and annihilation representation of the coherent state. The matrix calculus is then determined by the displacement and squeezing of the distribution. We shall give a brief summary of known results. The aim is to determine the basic determining equations which can be used to characterise a Wigner function via the continuous differential operators, the final part of this paper shall then show that there exist similar types of relationships for particular special matrices. Briefly, we hope to address the following:

Proposition 5.1.

Is there a matrix equivalent for the Wigner function on SU(1,1)?

Proposition 5.2.

What is the correct way in which to model the dynamic evolution of the state space?

5.1 Displacement Operator

Proposition 5.3.

The displacement operator defines the phase space version of quantum mechanics.

Corollary 5.4.

The Stone-von Neumann theorem emerges as a result of braiding relationships.

Proposition 5.5.

The group representation theory of phase space quantum mechanics can be generated through the application of the displacement operator.

Proof.

In the hyperbolic plane, one important operator is that of displacement. This operator is a Weyl transform in the creation and annihilation operators as shown in potovcek2015exponential :

Definition 5.6.
D^​(α)=exp⁡(α​a^†−α∗​a^)\hat{D}(\alpha)=\exp\left(\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}\right) (102)

The composition formula is:

Lemma 5.7.
D^​(α)​D^​(α′)=exp⁡(12​(α​α′⁣∗−α∗​α′))​D^​(α+α′)\hat{D}(\alpha)\hat{D}(\alpha^{\prime})=\exp\left(\dfrac{1}{2}(\alpha\alpha^{\prime*}-\alpha^{*}\alpha^{\prime})\right)\hat{D}(\alpha+\alpha^{\prime}) (103)

In terms of the position and momentum coordinates it is simple to show such laws as:

Lemma 5.8.
D^​(x,p)​D^​(x′,p′)=exp⁡(i​(p​x′−x​p′)/2)​D^​(x+x′,p+p′)\hat{D}(x,p)\hat{D}(x^{\prime},p^{\prime})=\exp\left(i(px^{\prime}-xp^{\prime})/2\right)\hat{D}(x+x^{\prime},p+p^{\prime}) (104)

Other identities include the braiding relationship:

Theorem 5.9.
D^​(α)​D^​(β)=e(α​β∗−β​α∗)/2​D^​(β)​D^​(α)\hat{D}(\alpha)\hat{D}(\beta)=e^{(\alpha\beta^{*}-\beta\alpha^{*})/2}\hat{D}(\beta)\hat{D}(\alpha) (105)

which are easily derived using some elementary applications of the commutation rules for creation and annihilation operators. We assume as always the standard boson algebra:

Definition 5.10.
a^=12​(x^+i​p^),a^†=12​(x^−i​p^)\hat{a}=\dfrac{1}{\sqrt{2}}\left(\hat{x}+i\hat{p}\right),\hat{a}^{\dagger}=\dfrac{1}{\sqrt{2}}\left(\hat{x}-i\hat{p}\right) (106)
[a^,a^†]=1,[x^,p^]=i​1\left[\hat{a},\hat{a}^{\dagger}\right]=1,\left[\hat{x},\hat{p}\right]=i1 (107)

Using the BCH formula it is possible to develop a number of identities including:

Theorem 5.11.
D^​(α)=e−|α|2/2​eα​a^†​e−α∗​a^\hat{D}(\alpha)=e^{-|\alpha|^{2}/2}e^{\alpha\hat{a}^{\dagger}}e^{-\alpha^{*}\hat{a}} (108)

We know from Weyl’s transformation law that there is a representation of the Wigner function that corresponds to this algebra. The authors in potovcek2015exponential showed that one way to realise it is to use the expression:

Δ^​(x,p)=1(2​π)2​∫−∞+∞∫−∞+∞ei​(p​x′−x​p′)​D^​(x,p)​𝑑x′​𝑑p′\hat{\Delta}(x,p)=\dfrac{1}{(2\pi)^{2}}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}e^{i(px^{\prime}-xp^{\prime})}\hat{D}(x,p)dx^{\prime}dp^{\prime} (109)

They also proved the similarity type relation:

D^​(x,p)​Π^​D^†​(x,p)=D^​(x,p)​D^​(x,p)​Π^\hat{D}(x,p)\hat{\Pi}\hat{D}^{\dagger}(x,p)=\hat{D}(x,p)\hat{D}(x,p)\hat{\Pi} (110)
=D^​(2​x,2​p)​Π^=Π^​D^†​(2​x,2​p)=\hat{D}(2x,2p)\hat{\Pi}=\hat{\Pi}\hat{D}^{\dagger}(2x,2p) (111)

where the operator Π^\hat{\Pi} is the parity operator which acts on the state via

π​Δ^​(x,p)=D^​(x,p)​Π^​D^†​(x,p)\pi\hat{\Delta}(x,p)=\hat{D}(x,p)\hat{\Pi}\hat{D}^{\dagger}(x,p) (112)

Some relevant formula are the completeness relationship:

∫−∞+∞∫−∞+∞D^​(x,p)​𝑑x′​𝑑p′=𝟏\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\hat{D}(x,p)dx^{\prime}dp^{\prime}=\mathbf{1} (113)

Initial conditions are specified through π​Δ^​(0,0)=Π^\pi\hat{\Delta}(0,0)=\hat{\Pi}. The action on the wave function is:

Theorem 5.12.
[Δ^​(x,p)​ψ]​(χ)=1π​e2​i​p​(χ−x)​ψ​(2​x−χ)[\hat{\Delta}(x,p)\psi](\chi)=\dfrac{1}{\pi}e^{2ip(\chi-x)}\psi(2x-\chi) (114)

Q.E.D. ∎

These types of formulae are sufficient to determine the Wigner function on the hyperboloid. We seek a finite matrix version of this above structure, especially the relationship between the parity and displacement operator.

5.2 Squeeze Transform

The other related transform on this space is given by a squeeze operator, which can be seen as the Wigner-Weyl transform of a parametric down conversion.

Proposition 5.13.

The squeeze operator performs a similar role to the displacement operator. Braiding relationships and isomorphisms show that this is intimately related to the parity operator in the phase space. This operator plays a key role in determining the structure of the algebra associated to this representation of phase space quantum mechanics.

Proposition 5.14.

It is possible to generate the SU(1,1) algebra by using relationships between creation and annihilation operators.

Corollary 5.15.

It is possible to define the basic relationships we desire for a matrix Wigner function through the laws of the parity operator.

Proof.

The squeeze operator is a quadratic in the creation and annihilation operators.

Definition 5.16.
S^​(z)=exp⁡(12​(z∗​a^2−z​a^†2))\hat{S}(z)=\exp\left(\dfrac{1}{2}\left(z^{*}\hat{a}^{2}-z\hat{a}^{\dagger 2}\right)\right) (115)
D^​(α)​S^​(z)=S^​(z)​(S^†​(z)​D^​(α)​S^​(z))=S^​(z)​D^​(γ)\hat{D}(\alpha)\hat{S}(z)=\hat{S}(z)\left(\hat{S}^{\dagger}(z)\hat{D}(\alpha)\hat{S}(z)\right)=\hat{S}(z)\hat{D}(\gamma) (116)

There is another braiding relationship between the squeeze and displacement operators:

Lemma 5.17.
γ=α​cosh⁡r+α∗​ei​θ​sinh⁡r\gamma=\alpha\cosh r+\alpha^{*}e^{i\theta}\sinh r (117)

It is not too hard to show that the creation and annhilation operators act as rotations in the hyperbolic space:

Theorem 5.18.
S^†​(z)​a^​S^​(z)=a^​cosh⁡r−a^†​ei​θ​sinh⁡r,z=r​ei​θ\hat{S}^{\dagger}(z)\hat{a}\hat{S}(z)=\hat{a}\cosh r-\hat{a}^{\dagger}e^{i\theta}\sinh r,z=re^{i\theta} (118)

Writing now the displacement operator in phase space coordinates, we find the Glauber-Sudarshan glauber1955time representation:

D^​(α)=exp⁡(i​(p​x^−x​p^))\hat{D}(\alpha)=\exp\left(i(p\hat{x}-x\hat{p})\right) (119)

Using the same type of substitution, the squeeze operator takes the form:

S^​(α)=exp⁡(i2​(p2​(x^2−p^2)−x2​{x^,p^}))\hat{S}(\alpha)=\exp\left(\dfrac{i}{2}\left(\dfrac{p}{\sqrt{2}}(\hat{x}^{2}-\hat{p}^{2})-\dfrac{x}{\sqrt{2}}\left\{\hat{x},\hat{p}\right\}\right)\right) (120)

which is of the form of a Wigner-Weyl transform:

S^​(α)=exp⁡(i2​(p​𝒫^−x​𝒳^))\hat{S}(\alpha)=\exp\left(\dfrac{i}{2}\left(p\hat{\mathcal{P}}-x\hat{\mathcal{X}}\right)\right) (121)

We can use this form of the transform to derive the symmetry properties of the system:

[a^​(α)a^†​(α)]=[cosh⁡r−ei​θ​sinh⁡r−e−i​θ​sinh⁡rcosh⁡r]​[a^a^†]\left[\begin{array}[]{c}\hat{a}(\alpha)\\ \hat{a}^{\dagger}(\alpha)\end{array}\right]=\left[\begin{array}[]{cc}\cosh r&-e^{i\theta}\sinh r\\ -e^{-i\theta}\sinh r&\cosh r\end{array}\right]\left[\begin{array}[]{c}\hat{a}\\ \hat{a}^{\dagger}\end{array}\right] (122)
U^​(α)​𝐛=𝐛′\hat{U}(\alpha)\mathbf{b}=\mathbf{b}^{\prime} (123)
Theorem 5.19.

In terms of the phase space coordinates, the transformation takes the form:

U^​(α)=[cosh⁡r−cos⁡θ​sinh⁡rsin⁡θ​sinh⁡rsin⁡θ​sinh⁡rcosh⁡r+cos⁡θ​sinh⁡r]\hat{U}(\alpha)=\left[\begin{array}[]{cc}\cosh r-\cos\theta\sinh r&\sin\theta\sinh r\\ \sin\theta\sinh r&\cosh r+\cos\theta\sinh r\end{array}\right] (124)
detU^​(α)=1,U^−1​(r,θ)=U^​(−r,θ)\det\hat{U}(\alpha)=1,\hat{U}^{-1}(r,\theta)=\hat{U}(-r,\theta) (125)

For the angle chosen to be θ=0\theta=0 we have action upon the phase space coordinates in the form:

U^​(r)=[cosh⁡r−sinh⁡r00cosh⁡r+sinh⁡r]=[e−r00er]\hat{U}(r)=\left[\begin{array}[]{cc}\cosh r-\sinh r&0\\ 0&\cosh r+\sinh r\end{array}\right]=\left[\begin{array}[]{cc}e^{-r}&0\\ 0&e^{r}\end{array}\right]
[x^′p^′]=[e−r​x^er​p^]\left[\begin{array}[]{c}\hat{x}^{\prime}\\ \hat{p}^{\prime}\end{array}\right]=\left[\begin{array}[]{c}e^{-r}\hat{x}\\ e^{r}\hat{p}\end{array}\right] (126)

From this perspective, we can see how a squeeze in one parameter results in the broadening of the other, which is an aspect of the Heisenberg uncertainty principle. We seek a more direct way to access the implied symmetry of the Wigner function. Using the following results from quantum optics potovcek2015exponential , the Wigner operator is given by:

Definition 5.20.
w^​(α)=1π2​∫ℂexp⁡(α​β∗−β​α∗)​D^​(β)​𝑑β\hat{w}(\alpha)=\dfrac{1}{\pi^{2}}\int_{\mathbb{C}}\exp\left(\alpha\beta^{*}-\beta\alpha^{*}\right)\hat{D}(\beta)d\beta (127)

The Wigner operator follows the displacement law of quantum optics via isometric transformation, viz.

Definition 5.21.
w^​(α)=D^​(α)​w^​(0)​D^†​(α)\hat{w}(\alpha)=\hat{D}(\alpha)\hat{w}(0)\hat{D}^{\dagger}(\alpha) (128)

Similarly to the original postulates, we have the Wigner operator is Hermitian, via w^​(α)=w^†​(α)\hat{w}(\alpha)=\hat{w}^{\dagger}(\alpha). Identically to the calculation from quantum optics potovcek2015exponential , the initial condition of the Wigner operator is the parity:

w^​(0)=∫ℂD^​(β)​𝑑β=2​P^\hat{w}(0)=\int_{\mathbb{C}}\hat{D}(\beta)d\beta=2\hat{P} (129)

where the parity is P^=(−1)a^†​a^=(−1)N^\hat{P}=\left(-1\right)^{\hat{a}^{\dagger}\hat{a}}=(-1)^{\hat{N}}. The connection between the Wigner distribution and this operator is given by the tracial relation:

WA^​(α)=𝐓𝐫​[A^​w^​(α)]W_{\hat{A}}(\alpha)=\mathbf{Tr}\left[\hat{A}\hat{w}(\alpha)\right] (130)

The inverse to this map is:

A^=12​π2​∫ℂw^​(α)​WA^​(α)​𝑑α\hat{A}=\dfrac{1}{2\pi^{2}}\int_{\mathbb{C}}\hat{w}(\alpha)W_{\hat{A}}(\alpha)d\alpha (131)

For the density matrix, we can associate the Wigner function via:

Theorem 5.22.
Wρ^​(α)=𝐓𝐫​[ρ^​w^​(α)]W_{\hat{\rho}}(\alpha)=\mathbf{Tr}\left[\hat{\rho}\hat{w}(\alpha)\right] (132)

We have that ρ^†=ρ^\hat{\rho}^{\dagger}=\hat{\rho} so Wρ^∗​(α)=Wρ^​(α)W_{\hat{\rho}}^{*}(\alpha)=W_{\hat{\rho}}(\alpha), i.e. is a real function. The displaced density matrix can be written:

ρ^′=D^​(α′)​ρ^​D^†​(α′)\hat{\rho}^{\prime}=\hat{D}(\alpha^{\prime})\hat{\rho}\hat{D}^{\dagger}(\alpha^{\prime}) (133)

If we substitute this into the formula for the Wigner function and rearrange, we find:

Theorem 5.23.
Wρ^′​(α)=Wρ^​(α−α′)W_{\hat{\rho}^{\prime}}(\alpha)=W_{\hat{\rho}}(\alpha-\alpha^{\prime}) (134)

∎

This is all the basic structure we need to identify a Wigner function. We can see how the different parts of this operator work together. In particular the relationship between the trace and the phase space function is of interest. The parity operator obviously enters in a fundamental way and defines the symmetries of this operator. We shall now show how finite matrices enter as a complementary representation to the continuous systems we have considered so far.

6 Pseudounitary Matrices and SU(1,1)

We shall now demonstrate one simple way in which a finite version of the Wigner distribution can be found. This is achieved by using the pseudounitary counterparts of SU(2), i.e. SU(1,1). The most basic representation of this is given by 2x2 matrices; the idea is that if we can find ways in which to represent relations using small matrices, we can generalise to any other representation.

Definition 6.1.

By examining the structure of the parametric algebra, and defining the group generators as K^+=12​a^†2,K^−=12​a^2\hat{K}_{+}=\dfrac{1}{2}\hat{a}^{\dagger 2},\hat{K}_{-}=\dfrac{1}{2}\hat{a}^{2}, it is not too hard to show that the boson algebra goes over into SU(1,1):

[K^−,K^+]=14​[a^2,a^†2]=12​(1+2​a^†​a^)=K^0\left[\hat{K}_{-},\hat{K}_{+}\right]=\dfrac{1}{4}\left[\hat{a}^{2},\hat{a}^{\dagger 2}\right]=\dfrac{1}{2}\left(1+2\hat{a}^{\dagger}\hat{a}\right)=\hat{K}_{0} (135)
[K^0,K^±]=±2​K^±\left[\hat{K}_{0},\hat{K}_{\pm}\right]=\pm 2\hat{K}_{\pm} (136)

Recognising that this is the SU(1,1) algebra, we can use the Pauli matrix representation:

Definition 6.2.
k^0=σ^z=[100−1],k^+=i​σ^+=[0i00]\hat{k}_{0}=\hat{\sigma}_{z}=\left[\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right],\hat{k}_{+}=i\hat{\sigma}_{+}=\left[\begin{array}[]{cc}0&i\\ 0&0\end{array}\right] (137)
k^−=i​σ^−=[00i0]\hat{k}_{-}=i\hat{\sigma}_{-}=\left[\begin{array}[]{cc}0&0\\ i&0\end{array}\right] (138)
Proposition 6.3.

Using the finite representation of the hyperbolic algebra defined through the parabolic operators, the Wigner function can be written as an isomorphic, unitary transformation of the parity operator.

Proposition 6.4.

The parity laws can be solved for a finite matrix representation of the displacement operator.

Proof.

Applying decomposition formulae to this representation of the fundamental algebra, it is possible to show the following:

Lemma 6.5.
S^​(ξ)=exp⁡(r​Q^​(θ)​J^​Q^†​(θ))=Q^​(θ)​exp⁡(r​J^)​Q^†​(θ)\hat{S}(\xi)=\exp\left(r\hat{Q}(\theta)\hat{J}\hat{Q}^{\dagger}(\theta)\right)=\hat{Q}(\theta)\exp\left(r\hat{J}\right)\hat{Q}^{\dagger}(\theta) (139)
=[cosh⁡ri​ei​θ​sinh⁡r−i​e−i​θ​sinh⁡rcosh⁡r]=\left[\begin{array}[]{cc}\cosh r&ie^{i\theta}\sinh r\\ -ie^{-i\theta}\sinh r&\cosh r\end{array}\right] (140)
=exp⁡(−e−i​θ​tanh⁡r​k^−)​exp⁡(ln⁡cosh⁡r​k^0)​exp⁡(ei​θ​tanh⁡r​k^+)=\exp\left(-e^{-i\theta}\tanh r\hat{k}_{-}\right)\exp\left(\ln\cosh r\hat{k}_{0}\right)\exp\left(e^{i\theta}\tanh r\hat{k}_{+}\right) (141)

This expression is equivalent to the Gauss LDU decomposition and gives the fundamental relationship for any representation of this hyperbolic algebra. Seyfarth et. al seyfarth2020wigner have established the following relationships using the method of coherent states.

Theorem 6.6.

The parity inversion theorem:

D^​(α)​P^​(α)​D^†​(α)=P^​(α)​D^​(−2​α)=D^​(2​α)​P^​(α)=w^​(α)\hat{D}(\alpha)\hat{P}(\alpha)\hat{D}^{\dagger}(\alpha)=\hat{P}(\alpha)\hat{D}(-2\alpha)=\hat{D}(2\alpha)\hat{P}(\alpha)=\hat{w}(\alpha) (142)

which we recognise as the property derived in the previous sections.

Theorem 6.7.

The formula for the Wigner operator in terms of displacement:

w^​(α,β)=22​(−1)a^†​a^+b^†​b^​D^​(−2​α)​D^†​(−2​β)\hat{w}(\alpha,\beta)=2^{2}(-1)^{\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}}\hat{D}(-2\alpha)\hat{D}^{\dagger}(-2\beta) (143)
Theorem 6.8.

The composition formula:

w^​(α,β)=w^​(α)​w^​(β)\hat{w}(\alpha,\beta)=\hat{w}(\alpha)\hat{w}(\beta) (144)
Theorem 6.9.

The Wigner operator in terms of displacement of the parity:

w^​(α)=D^​(α)​[2​(−1)a^†​a^]​D^†​(α)\hat{w}(\alpha)=\hat{D}(\alpha)\left[2(-1)^{\hat{a}^{\dagger}\hat{a}}\right]\hat{D}^{\dagger}(\alpha) (145)

We wish to establish a matrix theory of the Wigner operator independent of the mechanics of coherent states. There are advantages to doing this, as a concrete matrix representation avoids many of the pitfalls involved in creation-annihilation operator calculus. If we can specify the matrix form of the parity operator, we can work the rest out.

Theorem 6.10.
P^​(Φ)=exp⁡(i​Φ2​k^0)=[ei​Φ/200e−i​Φ/2]\hat{P}(\Phi)=\exp\left(i\dfrac{\Phi}{2}\hat{k}_{0}\right)=\left[\begin{array}[]{cc}e^{i\Phi/2}&0\\ 0&e^{-i\Phi/2}\end{array}\right] (146)

If we use this form of the parity operator, any changes due to displacement will be absorbed into the constant in the argument. This is, intuitively, the formula for a displaced version of parity; we can show this by writing out the displacement via:

Theorem 6.11.
D^​(ξ)=[cos⁡re−i​θ​sin⁡r−ei​θ​sin⁡rcos⁡r]=[αβ−β∗α]\hat{D}(\xi)=\left[\begin{array}[]{cc}\cos r&e^{-i\theta}\sin r\\ -e^{i\theta}\sin r&\cos r\end{array}\right]=\left[\begin{array}[]{cc}\alpha&\beta\\ -\beta^{*}&\alpha\end{array}\right] (147)

Using basic matrix multiplication, it is not too difficult to show the identity:

D^​(ξ)​P^​(Φ)​D^†​(ξ)\hat{D}(\xi)\hat{P}(\Phi)\hat{D}^{\dagger}(\xi)
=[ei​Φ/2​cos2⁡r+e−i​Φ/2​sin2⁡r−i​sin⁡Φ2​sin⁡2​r​e−i​θ−i​sin⁡Φ2​sin⁡2​r​ei​θe−i​Φ/2​cos2⁡r+ei​Φ/2​sin2⁡r]=\left[\begin{array}[]{cc}e^{i\Phi/2}\cos^{2}r+e^{-i\Phi/2}\sin^{2}r&-i\sin\dfrac{\Phi}{2}\sin 2re^{-i\theta}\\ -i\sin\dfrac{\Phi}{2}\sin 2re^{i\theta}&e^{-i\Phi/2}\cos^{2}r+e^{i\Phi/2}\sin^{2}r\end{array}\right] (148)

and also the parity inversion theorem:

Theorem 6.12.
P^​(Φ)​D^​(−2​ξ)=P^​(Φ)​D^†2​(ξ)\hat{P}(\Phi)\hat{D}(-2\xi)=\hat{P}(\Phi)\hat{D}^{\dagger 2}(\xi) (149)

By calculating the other side of the theorem, we can fix the value of the constant in the parity operator. In this representation, the matrices are unitary. The other part of the parity inversion theorem reads as:

D^2​(ξ)​P^​(Φ)=[ei​Φ/2​cos⁡2​rsin⁡2​r​e−i​(Φ/2+θ)−sin⁡2​r​ei​(Φ/2+θ)e−i​Φ/2​cos⁡2​r]\hat{D}^{2}(\xi)\hat{P}(\Phi)=\left[\begin{array}[]{cc}e^{i\Phi/2}\cos 2r&\sin 2re^{-i(\Phi/2+\theta)}\\ -\sin 2re^{i(\Phi/2+\theta)}&e^{-i\Phi/2}\cos 2r\end{array}\right] (150)

So if we choose the constant parameter to be Φ=π\Phi=\pi we can solve the expression:

D^​(ξ)​P^​(π)​D^†​(ξ)=D^2​(ξ)​P^​(π)=P^​(π)​D^†2​(−ξ)\hat{D}(\xi)\hat{P}(\pi)\hat{D}^{\dagger}(\xi)=\hat{D}^{2}(\xi)\hat{P}(\pi)=\hat{P}(\pi)\hat{D}^{\dagger 2}(-\xi) (151)

The parity operator is then given by:

Lemma 6.13.
w^​(0)=P^​(π)=[i00−i]\hat{w}(0)=\hat{P}(\pi)=\left[\begin{array}[]{cc}i&0\\ 0&-i\end{array}\right] (152)

which can also be realised by using the resolution of identity:

w^​(0)=∑k(−1)k​|ki⟩​⟨ki|\hat{w}(0)=\sum_{k}(-1)^{k}\left|k_{i}\right\rangle\left\langle k_{i}\right| (153)

If we look at the group homomorphism of the Wigner operator, we can write:

Theorem 6.14.
Wρ^​(ζ′)=𝐓𝐫​(ρ^​T^​w^​(ζ)​T^†)W_{\hat{\rho}}(\zeta^{\prime})=\mathbf{Tr}\left(\hat{\rho}\hat{T}\hat{w}(\zeta)\hat{T}^{\dagger}\right) (154)
=𝐓𝐫​(ρ^​w^​(g−1​ζ))=Wρ^​(g−1​ζ)=\mathbf{Tr}\left(\hat{\rho}\hat{w}(g^{-1}\zeta)\right)=W_{\hat{\rho}}(g^{-1}\zeta) (155)

Q.E.D. ∎

Proposition 6.15.

The squeeze operator can be solved in finite matrix form using the insights of the previous proof.

Proposition 6.16.

This is equivalent, on SU(1,1), to the Fourier transform of the parity operator.

Corollary 6.17.

This can be written in a form which resembles the characteristic function of a random variable, with the random variable replaced by a unitary matrix.

Proof.

The most general operator in SU(1,1) is given by:

Definition 6.18.
T^=S^​(ξ)​ei​Φ​K^0​S^−1​(ξ)\hat{T}=\hat{S}(\xi)e^{i\Phi\hat{K}_{0}}\hat{S}^{-1}(\xi) (156)

Evaluating, we find the matrix representation for the squeeze operator in the pseudounitary case:

Lemma 6.19.
S^​(ξ)=[cosh⁡τ2−i​ei​χ​sinh⁡τ2i​e−i​χ​sinh⁡τ2cosh⁡τ2]\hat{S}(\xi)=\left[\begin{array}[]{cc}\cosh\dfrac{\tau}{2}&-ie^{i\chi}\sinh\dfrac{\tau}{2}\\ ie^{-i\chi}\sinh\dfrac{\tau}{2}&\cosh\dfrac{\tau}{2}\end{array}\right] (157)

The rest of the parts of the group isomorphism are easily calculated:

ei​Φ​K^0=[ei​Φ00e−i​Φ]e^{i\Phi\hat{K}_{0}}=\left[\begin{array}[]{cc}e^{i\Phi}&0\\ 0&e^{-i\Phi}\end{array}\right] (158)
T^​(g)=[cos⁡Φ+i​sin⁡Φ​cosh⁡τ−ei​χ​sin⁡Φ​sinh⁡τ−e−i​χ​sin⁡Φ​sinh⁡τcos⁡Φ−i​sin⁡Φ​cosh⁡τ]\hat{T}(g)=\left[\begin{array}[]{cc}\cos\Phi+i\sin\Phi\cosh\tau&-e^{i\chi}\sin\Phi\sinh\tau\\ -e^{-i\chi}\sin\Phi\sinh\tau&\cos\Phi-i\sin\Phi\cosh\tau\end{array}\right] (159)

where we note that detT^=1\det\hat{T}=1, so this is a hyperbolic rotation. The inverse matrix is given by the element:

T^​(g−1)=[cos⁡Φ−i​sin⁡Φ​cosh⁡τei​χ​sin⁡Φ​sinh⁡τe−i​χ​sin⁡Φ​sinh⁡τcos⁡Φ+i​sin⁡Φ​cosh⁡τ]\hat{T}(g^{-1})=\left[\begin{array}[]{cc}\cos\Phi-i\sin\Phi\cosh\tau&e^{i\chi}\sin\Phi\sinh\tau\\ e^{-i\chi}\sin\Phi\sinh\tau&\cos\Phi+i\sin\Phi\cosh\tau\end{array}\right] (160)

We shall now show a simple alternative way to access these formulae. If we examine the trace formula, we note that following Seyfarth we may write the expectation value seyfarth2020wigner :

Theorem 6.20.
Wρ^​(ξ)=𝐓𝐫​[ρ^​w^​(ξ)]=⟨(−1)μ​e2​i​μ⟩W_{\hat{\rho}}(\xi)=\mathbf{Tr}[\hat{\rho}\hat{w}(\xi)]=\left\langle(-1)^{\mu}e^{2i\mu}\right\rangle (161)

If we examine the physical meaning of this expression, we notice that we can understand the Wigner distribution as the Fourier transform of the parity operator. The matrix form of this equation will then be given by:

Theorem 6.21.
w^​(ξ)=exp⁡(i​Φ​U^​(ξ))\hat{w}(\xi)=\exp\left(i\Phi\hat{U}(\xi)\right) (162)

where U^\hat{U} is the time evolution operator of the system. We can evaluate the time evolution operator using the expansion over the generators of SU(1,1):

U^​(ξ)=K^0​cosh⁡τ+ei​χ​sinh⁡τ​K^++e−i​χ​sinh⁡τ​K^−\hat{U}(\xi)=\hat{K}_{0}\cosh\tau+e^{i\chi}\sinh\tau\hat{K}_{+}+e^{-i\chi}\sinh\tau\hat{K}_{-} (163)

Using the small-k^\hat{k} representation over the 2x2 matrices this is evaluated as the matrix:

U^​(ξ)=[cosh⁡τ−i​ei​χ​sinh⁡τ−i​e−i​χ​sinh⁡τ−cosh⁡τ]\hat{U}(\xi)=\left[\begin{array}[]{cc}\cosh\tau&-ie^{i\chi}\sinh\tau\\ -ie^{-i\chi}\sinh\tau&-\cosh\tau\end{array}\right] (164)
Theorem 6.22.

The Wigner operator in this representation can be understood as a matrix form of the characteristic function:

w^​(ξ)=exp⁡(i​Φ​U^​(ξ))\hat{w}(\xi)=\exp\left(i\Phi\hat{U}(\xi)\right) (165)
=[cos⁡Φ+i​sin⁡Φ​cosh⁡τei​χ​sin⁡Φ​sinh⁡τe−i​χ​sin⁡Φ​sinh⁡τcos⁡Φ−i​sin⁡Φ​cosh⁡τ]=\left[\begin{array}[]{cc}\cos\Phi+i\sin\Phi\cosh\tau&e^{i\chi}\sin\Phi\sinh\tau\\ e^{-i\chi}\sin\Phi\sinh\tau&\cos\Phi-i\sin\Phi\cosh\tau\end{array}\right] (166)

which matches the formula obtained by using the group decomposition law. Q.E.D. ∎

Remark 4.

As far as the author is aware, this is a new result and has not been reported in the literature. It is interesting to consider the natural generalisation of this idiom to other unitary operators. It is not known which groups will give “nice” matrix representations, but by using the obvious relationships between SU(2)~SO(3) and SU(1,1)~SO(2,1), it is not too hard to see that at least for these groups there is a possibility of extension. The general theory of these groups and matrix decompositions is beyond the scope of this paper, but by using the example of SU(3), it is possible to illustrate the complications in finding explicit Wigner matrices on higher dimensions. We can see immediately how to exploit this for other systems. We have immediately, the following:

Corollary 6.23.

The Wigner function associated to the quantum brachistochrone and time optimal control problem may be evaluated through the use of the matrix characteristic function.

Corollary 6.24.

The squeeze operator defines a Hermitian, unitary transformation and in this state space the Wigner function evolves unitarily according to the von Neumann equation.

Proof.

We begin with the basic expression for the matrix characteristic function:

Definition 6.25.
w^​(ξ)=exp⁡(i​Φ​U^​(ξ))\hat{w}(\xi)=\exp\left(i\Phi\hat{U}(\xi)\right) (167)

If we have access to a formula for the time evolution operator, this gives us a way to define the matrix Wigner operator consistently. From the theory of the quantum brachistochrone morrison2012time , see e.g. also the theory of group representations in vilenkin1978special , we have many formulae for the time evolution operator, e.g.

Lemma 6.26.
U^3=ei​ϕ​[cos⁡ϕi​sin⁡ϕi​sin⁡ϕcos⁡ϕ]\hat{U}_{3}=e^{i\phi}\left[\begin{array}[]{cc}\cos\phi&i\sin\phi\\ i\sin\phi&\cos\phi\end{array}\right] (168)

The unitary matrix we get from the quantum brachistochrone is not correct for the Wigner transform. We have detU^=1\det\hat{U}=1. Here we are required to use a unitary with the relation to the parity operator, where detU^=−1\det\hat{U}=-1. One such operator is given by the matrix:

Theorem 6.27.
U^​(ξ)=[cos⁡τ−ei​χ​sin⁡τ−e−i​χ​sin⁡τ−cos⁡τ]\hat{U}(\xi)=\left[\begin{array}[]{cc}\cos\tau&-e^{i\chi}\sin\tau\\ -e^{-i\chi}\sin\tau&-\cos\tau\end{array}\right] (169)

which is the real, unitary form of the transformed parity operator. We intend to apply the same logic as before to write down the Wigner operator for this state, and derive the Wigner distribution. Evaluating the matrix exponential as before, we find:

Theorem 6.28.
w^​(ξ)=[cos⁡Φ+i​sin⁡Φ​cos⁡τ−i​sin⁡τ​sin⁡Φ​ei​χ−i​sin⁡τ​sin⁡Φ​e−i​χcos⁡Φ−i​sin⁡Φ​cos⁡τ]\hat{w}(\xi)=\left[\begin{array}[]{cc}\cos\Phi+i\sin\Phi\cos\tau&-i\sin\tau\sin\Phi e^{i\chi}\\ -i\sin\tau\sin\Phi e^{-i\chi}&\cos\Phi-i\sin\Phi\cos\tau\end{array}\right] (170)

We wish to extend the calculation further, by using the density matrix we can extract the Wigner function:

Wρ^​(α)=𝐓𝐫​[ρ^​w^​(α)]W_{\hat{\rho}}(\alpha)=\mathbf{Tr}\left[\hat{\rho}\hat{w}(\alpha)\right] (171)

To evaluate the density matrix for this Wigner operator, we may utilise the decomposition over the diagonal eigenstates of the squeeze operator, finding:

S^​(ξ)=exp⁡(−i​r​Q^​(θ)​J^​Q^†​(θ))=Q^​(θ)​exp⁡(−i​r​J^)​Q^†​(θ)\hat{S}(\xi)=\exp\left(-ir\hat{Q}(\theta)\hat{J}\hat{Q}^{\dagger}(\theta)\right)=\hat{Q}(\theta)\exp\left(-ir\hat{J}\right)\hat{Q}^{\dagger}(\theta) (172)

Writing out the decomposition of the squeeze operator:

S^​(ξ)=12​[i​ei​θ−i​ei​θ11]​[ei​r00e−i​r]​[−i​e−i​θ1i​e−i​θ1]\hat{S}(\xi)=\dfrac{1}{2}\left[\begin{array}[]{cc}ie^{i\theta}&-ie^{i\theta}\\ 1&1\end{array}\right]\left[\begin{array}[]{cc}e^{ir}&0\\ 0&e^{-ir}\end{array}\right]\left[\begin{array}[]{cc}-ie^{-i\theta}&1\\ ie^{-i\theta}&1\end{array}\right] (173)

From the matrix of eigenstates, we can reconstruct the density matrix:

Theorem 6.29.
Q^​(θ)=12​[i​ei​θ−i​ei​θ11]=[|+⟩,|−⟩]\hat{Q}(\theta)=\dfrac{1}{\sqrt{2}}\left[\begin{array}[]{cc}ie^{i\theta}&-ie^{i\theta}\\ 1&1\end{array}\right]=\left[\left|+\right\rangle,\left|-\right\rangle\right] (174)
H~=r​[0e+i​θ−e−i​θ0],H~​|±⟩=±r​|±⟩\tilde{H}=r\left[\begin{array}[]{cc}0&e^{+i\theta}\\ -e^{-i\theta}&0\end{array}\right],\tilde{H}\left|\pm\right\rangle=\pm r\left|\pm\right\rangle (175)
i​∂ρ^∂t=[H~,ρ^]i\dfrac{\partial\hat{\rho}}{\partial t}=\left[\tilde{H},\hat{\rho}\right] (176)

From the formula for the density matrix we can derive the result we need:

ρ^​(t)=U^​(t,0)​ρ^​(0)​U^†​(t,0)\hat{\rho}(t)=\hat{U}(t,0)\hat{\rho}(0)\hat{U}^{\dagger}(t,0) (177)
U^​(t,0)=W^​(t)​W^†​(0),U^†​(t,0)=W^​(0)​W^†​(t)\hat{U}(t,0)=\hat{W}(t)\hat{W}^{\dagger}(0),\hat{U}^{\dagger}(t,0)=\hat{W}(0)\hat{W}^{\dagger}(t) (178)

Using the expression for the transformed density, we find the isomorphism in terms of the solution matrix:

Theorem 6.30.
ρ^​(t)=W^​(t)​W^†​(0)​ρ^​(0)​W^​(0)​W^†​(t)=W^​(t)​ρ^w​(0)​W^†​(t)\hat{\rho}(t)=\hat{W}(t)\hat{W}^{\dagger}(0)\hat{\rho}(0)\hat{W}(0)\hat{W}^{\dagger}(t)=\hat{W}(t)\hat{\rho}_{w}(0)\hat{W}^{\dagger}(t) (179)

Since we know that the solution matrix follows standard dynamics, it is simple to show that this expression solves the von Neumann equation Evaluating the derivative:

Corollary 6.31.
i​∂ρ^∂t=i​∂∂t​(W^​(t)​ρ^w​(0)​W^†​(t))i\dfrac{\partial\hat{\rho}}{\partial t}=i\dfrac{\partial}{\partial t}\left(\hat{W}(t)\hat{\rho}_{w}(0)\hat{W}^{\dagger}(t)\right) (180)
=[H~,ρ^]=\left[\tilde{H},\hat{\rho}\right] (181)

Q.E.D. ∎

Remark 5.

We can therefore use this expression for the density matrix when calculating the Wigner function. Choosing an initial state under a given Hamiltonian is enough to find the density matrix in the simplified situation of a pure state. As shall be shown in the next section, it is important to make a distinction between pure and mixed states, as even in this simplified example of a two state non-commutative system their behaviour is markedly different.

7 Theory of the Projection Matrix

The remaining topic of interest relates to the integration theory of groups. We state the following:

Proposition 7.1.

The projective theory is important to the formulation of the representation theory of the matrix Wigner theory.

Proposition 7.2.

The evolution of the matrix Wigner function is more delicate than the Bloch picture of the density matrix. In particular, pure and mixed states behave very differently.

Corollary 7.3.

The projective theory of the matrix Wigner function results in a type of Hadamard transformation which may be of interest in quantum mechanics.

Proof.

Note the projection formula:

Definition 7.4.
ρ^=1A​∫Ω𝑑μ​(Ω).w^​(Ω)​Tr​[ρ^​w^​(Ω)]\hat{\rho}=\dfrac{1}{A}\int_{\Omega}d\mu(\Omega).\hat{w}(\Omega)\mathrm{Tr}\left[\hat{\rho}\hat{w}(\Omega)\right] (182)

We can also analyse the Wigner function using properties of the projection operator. From basic quantum mechanics, and assuming the system is unitary SU(2), we can write:

Example 7.5.
ρ^​(t)=12​(𝟏^+n→​(t)⋅σ)=[r100r2]\hat{\rho}(t)=\dfrac{1}{2}\left(\hat{\mathbf{1}}+\vec{n}(t)\cdot\mathbf{\sigma}\right)=\left[\begin{array}[]{cc}r_{1}&0\\ 0&r_{2}\end{array}\right] (183)

in the simplest instance. The Wigner operator is then evaluated using the Euler decomposition:

w^​(θ)=12​exp⁡(i​θ2​σ^y)​[1+a001−a]​exp⁡(−i​θ2​σ^y)\hat{w}(\theta)=\dfrac{1}{2}\exp\left(i\dfrac{\theta}{2}\hat{\sigma}_{y}\right)\left[\begin{array}[]{cc}1+\sqrt{a}&0\\ 0&1-\sqrt{a}\end{array}\right]\exp\left(-i\dfrac{\theta}{2}\hat{\sigma}_{y}\right) (184)

The result for the Wigner operator is then:

Lemma 7.6.
w^​(θ)=12​[a​cos⁡θ+1−a​sin⁡θ−a​sin⁡θ−a​cos⁡θ+1]\hat{w}(\theta)=\dfrac{1}{2}\left[\begin{array}[]{cc}\sqrt{a}\cos\theta+1&-\sqrt{a}\sin\theta\\ -\sqrt{a}\sin\theta&-\sqrt{a}\cos\theta+1\end{array}\right] (185)

Note that we satisfy the constraint equations:

Definition 7.7.
Tr​[w^​(θ)]=1,Tr​[w^2​(θ)]=12​(a+1)\mathrm{Tr}\left[\hat{w}(\theta)\right]=1,\mathrm{Tr}\left[\hat{w}^{2}(\theta)\right]=\dfrac{1}{2}(a+1) (186)

The authors in hillery1984distribution conduct analysis of the Stratonovich-Weyl operator. They give the relationship between the dimension of the space and the Wigner operator as:

Tr​[w^2​(θ)]=N=2\mathrm{Tr}\left[\hat{w}^{2}(\theta)\right]=N=2 (187)

from which we are able to determine a=3a=3. Calculating the Wigner distribution via the trace relationship gives the expression:

Wρ^​(θ)=𝐓𝐫​[ρ^​w^​(θ)]W_{\hat{\rho}}(\theta)=\mathbf{Tr}[\hat{\rho}\hat{w}(\theta)] (188)

we find the Wigner distribution:

Theorem 7.8.
Wρ^​(θ)=Tr​(12​[r100r2]​[3​cos⁡θ+1−3​sin⁡θ−3​sin⁡θ−3​cos⁡θ+1])W_{\hat{\rho}}(\theta)=\mathrm{Tr}\left(\dfrac{1}{2}\left[\begin{array}[]{cc}r_{1}&0\\ 0&r_{2}\end{array}\right]\left[\begin{array}[]{cc}\sqrt{3}\cos\theta+1&-\sqrt{3}\sin\theta\\ -\sqrt{3}\sin\theta&-\sqrt{3}\cos\theta+1\end{array}\right]\right) (189)
=32​(r1−r2)​cos⁡θ+r1+r22=\dfrac{\sqrt{3}}{2}(r_{1}-r_{2})\cos\theta+\dfrac{r_{1}+r_{2}}{2} (190)

To close the proof we must show that we can recover the operator via the integral identity:

ρ^=1A​∫Ω𝑑μ​(Ω).w^​(Ω)​Tr​[ρ^​w^​(Ω)]\hat{\rho}=\dfrac{1}{A}\int_{\Omega}d\mu(\Omega).\hat{w}(\Omega)\mathrm{Tr}\left[\hat{\rho}\hat{w}(\Omega)\right] (191)

From our initial density matrix, we know from the projection theorem that we must have:

ρ^=12​(𝟏^+r​σ^z)=12​[1+r001−r]=[r100r2]\hat{\rho}=\dfrac{1}{2}(\hat{\mathbf{1}}+r\hat{\sigma}_{z})=\dfrac{1}{2}\left[\begin{array}[]{cc}1+r&0\\ 0&1-r\end{array}\right]=\left[\begin{array}[]{cc}r_{1}&0\\ 0&r_{2}\end{array}\right] (192)

and we finally obtain:

Theorem 7.9.
∫02​π[32​r​cos⁡θ+12]​w^​(θ)​𝑑θ=π2​[1+32​r001−32​r]\int_{0}^{2\pi}\left[\dfrac{\sqrt{3}}{2}r\cos\theta+\dfrac{1}{2}\right]\hat{w}(\theta)d\theta=\dfrac{\pi}{2}\left[\begin{array}[]{cc}1+\dfrac{3}{2}r&0\\ 0&1-\dfrac{3}{2}r\end{array}\right] (193)

Note that this is a mixed state. We have ρ^2≤ρ^\hat{\rho}^{2}\leq\hat{\rho}. The theory partially fails because of this. We readjust and use the density of a pure state. A simple formula for a pure state is given by:

Definition 7.10.
ρ^=12​(𝟏^+rx​σ^x+rz​σ^z)\hat{\rho}=\dfrac{1}{2}(\hat{\mathbf{1}}+r_{x}\hat{\sigma}_{x}+r_{z}\hat{\sigma}_{z}) (194)
ρ^=12​[1+1212121−12],Tr​(ρ^)=1,ρ^2=ρ^\hat{\rho}=\dfrac{1}{2}\left[\begin{array}[]{cc}1+\dfrac{1}{\sqrt{2}}&\dfrac{1}{\sqrt{2}}\\ \dfrac{1}{\sqrt{2}}&1-\dfrac{1}{\sqrt{2}}\end{array}\right],\mathrm{Tr}(\hat{\rho})=1,\hat{\rho}^{2}=\hat{\rho}

Using the same expression for the rotation as before:

w^​(θ)=ei​θ​σ^y/2​ρ^​(0)​e−i​θ​σ^y/2\hat{w}(\theta)=e^{i\theta\hat{\sigma}_{y}/2}\hat{\rho}(0)e^{-i\theta\hat{\sigma}_{y}/2} (195)
=12​[1+22​(sin⁡θ+cos⁡θ)−22​(sin⁡θ−cos⁡θ)−22​(sin⁡θ−cos⁡θ)1−22​(sin⁡θ+cos⁡θ)]=\dfrac{1}{2}\left[\begin{array}[]{cc}1+\dfrac{\sqrt{2}}{2}\left(\sin\theta+\cos\theta\right)&-\dfrac{\sqrt{2}}{2}\left(\sin\theta-\cos\theta\right)\\ -\dfrac{\sqrt{2}}{2}\left(\sin\theta-\cos\theta\right)&1-\dfrac{\sqrt{2}}{2}\left(\sin\theta+\cos\theta\right)\end{array}\right] (196)

Evaluating the trace to extract the Wigner distribution, we find:

Tr​[w^​(θ)]=Tr​[w^​(θ)2]=1\mathrm{Tr}[\hat{w}(\theta)]=\mathrm{Tr}[\hat{w}(\theta)^{2}]=1 (197)
Wρ​(θ)=Tr​[w^​(θ)​ρ^​(0)]=12​(cos⁡θ+1)W_{\rho}(\theta)=\mathrm{Tr}[\hat{w}(\theta)\hat{\rho}(0)]=\dfrac{1}{2}(\cos\theta+1) (198)

If we then go on to calculate the integral, it is simple to show that we receive:

Theorem 7.11.
∫02​πWρ​(θ)​w^​(θ)​𝑑θ=π2​[1+1212121−12]\int_{0}^{2\pi}W_{\rho}(\theta)\hat{w}(\theta)d\theta=\dfrac{\pi}{2}\left[\begin{array}[]{cc}1+\dfrac{1}{\sqrt{2}}&\dfrac{1}{\sqrt{2}}\\ \dfrac{1}{\sqrt{2}}&1-\dfrac{1}{\sqrt{2}}\end{array}\right] (199)
=π​ρ^​(0)=\pi\hat{\rho}(0) (200)

We can see that in this instance, we recover both the normalisation and the original pure state density matrix. Q.E.D. ∎

It has been shown how to link sets of special functions and the Wigner distribution in the first part of this work. The second half has focused on matrix models which can be compared to these continuous results. We can see that in both cases, there are certain underlying themes of projection. We can see this reflected in the original expression for the Wigner distribution:

fW​(p,q)=12​π​ℏ​∫−∞+∞ei​p​s/ℏ​⟨q−s2|Ψ⟩​⟨Ψ|q+s2⟩​𝑑sf_{W}(p,q)=\dfrac{1}{2\pi\hbar}\int_{-\infty}^{+\infty}e^{ips/\hbar}\left\langle q-\dfrac{s}{2}\right|\left.\Psi\right\rangle\left\langle\Psi\right.\left|q+\dfrac{s}{2}\right\rangle ds (201)

where we can see the intersection between projective theorems of the density matrix, autocorrelation, and the Fourier transform. This concludes the calculation in this paper. We shall now discuss some topics in analysis that relate to the future developments that can be expected in this area.

8 Discussion and Conclusions

We have shown in this paper some interesting relationships between the construction of the Wigner function via the star-eigenvalue equations, and the group representations of matrices and parity operators. These results have consequences for the theory of projective operators. It is relevant to consider the direction of using these types of projections to map out spaces by projecting down to cover the different functions and matrices. The question of the correct types of projection to be used in higher dimensional spaces has not been analysed in this paper, and can be expected to be different to that of these simple contrived examples. This calculation has demonstrated that a close inspection of the spectral theory can result in much understanding of the structure of different groups. Here we have presented the basic examples of spherical and pseudospherical systems. These methods, with appropriate modifications can be shown to be applicable to such varied systems as the Morse oscillator, modified Bessel function and associated Legendre functions. This will be covered in future works to appear in the literature.

It is of interest that the particular sets of special functions uncovered in this calculation indicate a relationship between the confluent hypergeometric function, the Laguerre functions and the Hermite polynomials. Indeed, we have found a number of multiplication theorems that essentially amount to different types of convolutions in these systems. These convolutions are related to the kernel; as we have seen, the different forms of hyperbolic systems have differing eigenfunctions that make up the spectral solutions.

It is hoped that by utilising the intersection of matrix representation theory and special functions that a picture of the different families can be built up by understanding the limits between different sets of PDE systems. One way to understand this, as we have demonstrated in this paper, is through analysis of the projection operator and transform theory using Wigner functions. This is a powerful technique, and it is possible that a more general understanding of the theory of diffusion problems and stochastic processes can be found by using this perspective.

The matrix perspective in this sense requires more development, as it is not as clear as it would be desirable. The use of group representation techniques should enable these simple results to be generalised to any other system with SU(1,1) symmetry. The groups SL(2,R) and SL(2,C) form obvious extensions of the method and have relationships to SU(1,1) transforms lenz2017mehler . The important part in which more understanding is required is the difference between trace class of Wigner matrices and their determinants. As is easily shown, the mixed state we have calculated has trace class 2, the pure state trace class 1. Whether other well defined trace classes of Wigner matrices exist, and the exact analysis of the “defect” between pure and mixed states with respect to the Wigner function is of crucial importance in understanding the matrix form of these equations.

9 Future Directions

Several questions are opened up through this investigation. The details of the exact link between the Wigner function and the kernel have not been explored in this paper. If we take the kernel to be the projection state:

K​(x,x′;t)=∑ne−En​t​ψn∗​(x′)​ψn​(x)K(x,x^{\prime};t)=\sum_{n}e^{-E_{n}t}\psi_{n}^{*}(x^{\prime})\psi_{n}(x) (202)
ψn​(x)=⟨n|x⟩\psi_{n}(x)=\left\langle n|x\right\rangle (203)
ψn∗​(x′)=(⟨n|x′⟩)∗=⟨x′|n⟩\psi_{n}^{*}(x^{\prime})=(\left\langle n|x^{\prime}\right\rangle)^{*}=\left\langle x^{\prime}|n\right\rangle (204)
K(x,x′;t)=∑ne−En​t⟨x′|n⟩⟨n|x⟩=⟨x′|∑ne−En​t|n⟩⟨n||x⟩K(x,x^{\prime};t)=\sum_{n}e^{-E_{n}t}\left\langle x^{\prime}|n\right\rangle\left\langle n|x\right\rangle=\left\langle x^{\prime}\left|\sum_{n}e^{-E_{n}t}\left|n\left\rangle\right\langle n\right|\right|x\right\rangle (205)

It is tempting, in light of the formula for the Wigner function, to hope for a simple relationship such as:

fW​(p,q;t)=12​π​ℏ​∫−∞+∞ei​p​s/ℏ​K​(q−s/2,q+s/2;t)​𝑑sf_{W}(p,q;t)=\dfrac{1}{2\pi\hbar}\int_{-\infty}^{+\infty}e^{ips/\hbar}K(q-s/2,q+s/2;t)ds (206)

It is unknown if such a formula exists. This can be easily recovered by using

|Ψ⟩⟨Ψ|=∑ne−En​t|n⟩⟨n|\left|\Psi\left\rangle\right\langle\Psi\right|=\sum_{n}e^{-E_{n}t}\left|n\left\rangle\right\langle n\right| (207)

If such relationships could be developed, it would save a great deal of calculation, as we could exploit already known formulae for the kernel to find solutions to the simultaneous differential equations given through the star equations. Other interesting formulae might be found by using the expression for the resolvent/Green’s function as given by a similar eigenfunction decomposition.

Acknowledgment

This project was supported under ARC Research Excellence Scholarships at the University of Technology, Sydney. The author acknowledges useful discussions and support from Dr. Mark Craddock and Prof. Anthony Dooley.

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