Phase-space formulation

The phase-space distribution f(x, p) of a quantum state is a quasiprobability distribution. In the phase-space formulation, the phase-space distribution may be treated as the fundamental, primitive description of the quantum system, without any reference to wave functions or density matrices.[7]

There are several different ways to represent the distribution, all interrelated.[8][9] The most noteworthy is the Wigner representation, W(x, p), discovered first.[4] Other representations (in approximately descending order of prevalence in the literature) include the Glauber–Sudarshan P,[10][11] Husimi Q,[12] Kirkwood–Rihaczek, Mehta, Rivier, and Born–Jordan representations.[13][14] These alternatives are most useful when the Hamiltonian takes a particular form, such as normal order for the Glauber–Sudarshan P-representation. Since the Wigner representation is the most common, this article will usually stick to it, unless otherwise specified.

The phase-space distribution possesses properties akin to the probability density in a 2n-dimensional phase space. For example, it is real-valued, unlike the generally complex-valued wave function. We can understand the probability of lying within a position interval, for example, by integrating the Wigner function over all momenta and over the position interval:

${\displaystyle \operatorname {P} [a\leq X\leq b]=\int _{a}^{b}\int _{-\infty }^{\infty }W(x,p)\,dp\,dx.}$

If Â(x, p) is an operator representing an observable, it may be mapped to phase space as A(x, p) through the Wigner transform. Conversely, this operator may be recovered by the Weyl transform.

The expectation value of the observable with respect to the phase-space distribution is[2][15]

${\displaystyle \langle {\hat {A}}\rangle =\int A(x,p)W(x,p)\,dp\,dx.}$

A point of caution, however: despite the similarity in appearance, W(x, p) is not a genuine joint probability distribution, because regions under it do not represent mutually exclusive states, as required in the third axiom of probability theory. Moreover, it can, in general, take negative values even for pure states, with the unique exception of (optionally squeezed) coherent states, in violation of the first axiom.

Regions of such negative value are provable to be "small": they cannot extend to compact regions larger than a few ħ, and hence disappear in the classical limit. They are shielded by the uncertainty principle, which does not allow precise localization within phase-space regions smaller than ħ, and thus renders such "negative probabilities" less paradoxical. If the left side of the equation is to be interpreted as an expectation value in the Hilbert space with respect to an operator, then in the context of quantum optics this equation is known as the optical equivalence theorem. (For details on the properties and interpretation of the Wigner function, see its main article.)

An alternative phase-space approach to quantum mechanics seeks to define a wave function (not just a quasiprobability density) on phase space, typically by means of the Segal–Bargmann transform. To be compatible with the uncertainty principle, the phase-space wave function cannot be an arbitrary function, or else it could be localized into an arbitrarily small region of phase space. Rather, the Segal–Bargmann transform is a holomorphic function of ${\displaystyle x+ip}$. There is a quasiprobability density associated to the phase-space wave function; it is the Husimi Q representation of the position wave function.

The fundamental noncommutative binary operator in the phase-space formulation that replaces the standard operator multiplication is the star product, represented by the symbol ★.[1] Each representation of the phase-space distribution has a different characteristic star product. For concreteness, we restrict this discussion to the star product relevant to the Wigner–Weyl representation.

For notational convenience, we introduce the notion of left and right derivatives. For a pair of functions f and g, the left and right derivatives are defined as

${\displaystyle {\begin{aligned}f{\overset {_{\gets }}{\partial }}_{x}g&={\frac {\partial f}{\partial x}}\cdot g,\\f{\vec {\partial }}_{x}g&=f\cdot {\frac {\partial g}{\partial x}}.\end{aligned}}}$

The differential definition of the star product is

${\displaystyle f\star g=f\,\exp {\left({\frac {i\hbar }{2}}\left({\overset {_{\gets }}{\partial }}_{x}{\vec {\partial }}_{p}-{\overset {_{\gets }}{\partial }}_{p}{\vec {\partial }}_{x}\right)\right)}g,}$

where the argument of the exponential function can be interpreted as a power series. Additional differential relations allow this to be written in terms of a change in the arguments of f and g:

${\displaystyle {\begin{aligned}(f\star g)(x,p)&=f\left(x+{\tfrac {i\hbar }{2}}{\vec {\partial }}_{p},p-{\tfrac {i\hbar }{2}}{\vec {\partial }}_{x}\right)\cdot g(x,p)\\&=f(x,p)\cdot g\left(x-{\tfrac {i\hbar }{2}}{\overset {_{\gets }}{\partial }}_{p},p+{\tfrac {i\hbar }{2}}{\overset {_{\gets }}{\partial }}_{x}\right)\\&=f\left(x+{\tfrac {i\hbar }{2}}{\vec {\partial }}_{p},p\right)\cdot g\left(x-{\tfrac {i\hbar }{2}}{\overset {_{\gets }}{\partial }}_{p},p\right)\\&=f\left(x,p-{\tfrac {i\hbar }{2}}{\vec {\partial }}_{x}\right)\cdot g\left(x,p+{\tfrac {i\hbar }{2}}{\overset {_{\gets }}{\partial }}_{x}\right).\end{aligned}}}$

It is also possible to define the ★-product in a convolution integral form,[16] essentially through the Fourier transform:

${\displaystyle (f\star g)(x,p)={\frac {1}{\pi ^{2}\hbar ^{2}}}\,\int f(x+x',p+p')\,g(x+x'',p+p'')\,\exp {\left({\tfrac {2i}{\hbar }}(x'p''-x''p')\right)}\,dx'dp'dx''dp''.}$

(Thus, e.g.,[7] Gaussians compose hyperbolically:

${\displaystyle \exp {\big (}{-a}(x^{2}+p^{2}){\big )}\star \exp {\big (}{-b}(x^{2}+p^{2}){\big )}={\frac {1}{1+\hbar ^{2}ab}}\exp \left(-{\frac {a+b}{1+\hbar ^{2}ab}}(x^{2}+p^{2})\right),}$

or

${\displaystyle \delta (x)\star \delta (p)={\frac {2}{h}}\exp \left(2i{\frac {xp}{\hbar }}\right),}$

etc.)

The energy eigenstate distributions are known as stargenstates, ★-genstates, stargenfunctions, or ★-genfunctions, and the associated energies are known as stargenvalues or ★-genvalues. These are solved, analogously to the time-independent Schrödinger equation, by the ★-genvalue equation,[17][18]

${\displaystyle H\star W=E\cdot W,}$

where H is the Hamiltonian, a plain phase-space function, most often identical to the classical Hamiltonian.

The time evolution of the phase space distribution is given by a quantum modification of Liouville flow.[2][9][19] This formula results from applying the Wigner transformation to the density matrix version of the quantum Liouville equation, the von Neumann equation.

In any representation of the phase space distribution with its associated star product, this is

${\displaystyle {\frac {\partial f}{\partial t}}=-{\frac {1}{i\hbar }}\left(f\star H-H\star f\right),}$

or, for the Wigner function in particular,

${\displaystyle {\frac {\partial W}{\partial t}}=-\{\{W,H\}\}=-{\frac {2}{\hbar }}W\sin \left({\frac {\hbar }{2}}({\overset {_{\gets }}{\partial }}_{x}{\vec {\partial }}_{p}-{\overset {_{\gets }}{\partial }}_{p}{\vec {\partial }}_{x})\right)H=-\{W,H\}+O(\hbar ^{2}),}$

where {{ , }} is the Moyal bracket, the Wigner transform of the quantum commutator, while { , } is the classical Poisson bracket.[2]

This yields a concise illustration of the correspondence principle: this equation manifestly reduces to the classical Liouville equation in the limit ħ → 0. In the quantum extension of the flow, however, the density of points in phase space is not conserved; the probability fluid appears "diffusive" and compressible.[2] The concept of quantum trajectory is therefore a delicate issue here.[20] See the movie for the Morse potential, below, to appreciate the nonlocality of quantum phase flow.

N.B. Given the restrictions placed by the uncertainty principle on localization, Niels Bohr vigorously denied the physical existence of such trajectories on the microscopic scale. By means of formal phase-space trajectories, the time evolution problem of the Wigner function can be rigorously solved using the path-integral method[21] and the method of quantum characteristics,[22] although there are severe practical obstacles in both cases.