Method of quantum characteristics

Weyl-Wigner association rule

In Hamiltonian dynamics, classical systems with n degrees of freedom are described by 2 n canonical coordinates and momenta

that form a coordinate system in the phase space. These variables satisfy the Poisson bracket relations

The skew-symmetric matrix I k l ,

where E n is the n × n identity matrix, defines nondegenerate 2-form in the phase space. The phase space acquires thereby the structure of a symplectic manifold. The phase space is not metric space, so distance between two points is not defined. The Poisson bracket of two functions can be interpreted as the oriented area of a parallelogram whose adjacent sides are gradients of these functions. Rotations in Euclidean space leave the distance between two points invariant. Canonical transformations in symplectic manifold leave the areas invariant.

In quantum mechanics, the canonical variables ξ are associated to operators of canonical coordinates and momenta

These operators act in Hilbert space and obey commutation relations

Weyl’s association rule extends the correspondence ξ i → ξ ^ i to arbitrary phase-space functions and operators.

Taylor expansion

A one-sided association rule f ( ξ ) → f ^ was formulated by Weyl initially with the help of Taylor expansion of functions of operators of the canonical variables

The operators ξ ^ do not commute, so the Taylor expansion is not defined uniquely. The above prescription uses the symmetrized products of the operators. The real functions correspond to the Hermitian operators. The function f ( ξ ) is called Weyl's symbol of operator f ^ .

Under the reverse association f ( ξ ) ← f ^ , the density matrix turns to the Wigner function. Wigner functions have numerous applications in quantum many-body physics, kinetic theory, collision theory, quantum chemistry.

A refined version of the Weyl-Wigner association rule is proposed by Groenewold and Stratonovich.

Groenewold-Stratonovich basis

The set of operators acting in the Hilbert space is closed under multiplication of operators by c -numbers and summation. Such a set constitutes a vector space V . The association rule formulated with the use of the Taylor expansion preserves operations on the operators. The correspondence can be illustrated with the following diagram:

Here, f ( ξ ) and g ( ξ ) are functions and f ^ and g ^ are the associated operators.

The elements of basis of V are labelled by canonical variables ξ i . The commonly used Stratonovich basis looks like

The Weyl-Wigner two-sided association rule for function f ( ξ ) and operator f ^ has the form

The function f ( ξ ) provides coordinates of the operator f ^ in the basis B ^ ( ξ ) . The basis is complete and orthogonal:

Alternative operator bases are discussed also. The freedom in choice of the operator basis is better known as the operator ordering problem.

Star-product

The set of operators O p ( L 2 ( R n ) ) is closed under the multiplication of operators. The vector space V is endowed thereby with an associative algebra structure. Given two functions

one can construct a third function

called ⋆ -product or Moyal product. It is given explicitly by

where

is the Poisson operator. The ⋆ -product splits into symmetric and skew-symmetric parts

The ∘ -product is not associative. In the classical limit ∘ -product becomes the dot-product. The skew-symmetric part f ∧ g is known under the name of Moyal bracket. This is the Weyl's symbol of commutator. In the classical limit Moyal bracket becomes Poisson bracket. Moyal bracket is quantum deformation of Poisson bracket.

Quantum characteristics

The correspondence ξ ↔ ξ ^ shows that coordinate transformations in the phase space are accompanied by transformations of operators of the canonical coordinates and momenta and vice versa. Let U ^ be the evolution operator,

and H ^ is Hamiltonian. Consider the following scheme:

Quantum evolution transforms vectors in the Hilbert space and, upon the Wigner association rule, coordinates in the phase space. In Heisenberg representation, the operators of the canonical variables are transformed as

The phase-space coordinates ξ ´ i that correspond to new operators ξ ^ i ´ in the old basis B ^ ( ξ ) are given by

with the initial conditions

The functions q i ( ξ , τ ) define quantum phase flow. In the general case, it is canonical to first order in τ .

Star-function

The set of operators of canonical variables is complete in the sense that any operator can be represented as a function of operators ξ ^ . Transformations

induce under the Wigner association rule transformations of phase-space functions:

Using the Taylor expansion, the transformation of function f ( ξ ) under the evolution can be found to be

Composite function defined in such a way is called ⋆ -function. The composition law differs from the classical one. However, semiclassical expansion of f ( ⋆ q ( ξ , τ ) ) around f ( q ( ξ , τ ) ) is formally well defined and involves even powers of ℏ only. This equation shows that, given quantum characteristics are constructed, physical observables can be found without further addressing to Hamiltonian. The functions q i ( ξ , τ ) play the role of characteristics similarly to classical characteristics used to solve classical Liouville equation.

Quantum Liouville equation

The Wigner transform of the evolution equation for the density matrix in the Schrödinger representation leads to a quantum Liouville equation for the Wigner function. The Wigner transform of the evolution equation for operators in the Heisenberg representation,

leads to the same equation with the opposite (plus) sign in the right-hand side:

⋆ -function solves this equation in terms of quantum characteristics:

Similarly, the evolution of the Wigner function in the Schrödinger representation is given by

The Liouville theorem of classical mechanics fails, to the extent that, locally, the "probability" density in phase space is not preserved in time.

Quantum Hamilton's equations

Quantum Hamilton's equations can be obtained applying the Wigner transform to the evolution equations for Heisenberg operators of canonical coordinates and momenta

The right-hand side is calculated like in the classical mechanics. The composite function is, however, ⋆ -function. The ⋆ -product violates canonicity of the phase flow beyond the first order in τ .

Conservation of Moyal bracket

The antisymmetrized products of even number of operators of canonical variables are c-numbers as a consequence of the commutation relations. These products are left invariant by unitary transformations and, in particular,

Phase-space transformations induced by the evolution operator preserve the Moyal bracket and do not preserve the Poisson bracket, so the evolution map

is not canonical. Transformation properties of canonical variables and phase-space functions under unitary transformations in the Hilbert space have important distinctions from the case of canonical transformations in the phase space:

Composition law

Quantum characteristics can hardly be treated visually as trajectories along which physical particles move. The reason lies in the star-composition law

which is non-local and is distinct from the dot-composition law of classical mechanics.

Energy conservation

The energy conservation implies

where

is Hamilton's function. In the usual geometric sense, H ( ξ ) is not conserved along quantum characteristics.

Summary

The origin of the method of characteristics can be traced back to Heisenberg’s matrix mechanics. Suppose that we have solved in the matrix mechanics the evolution equations for the operators of the canonical coordinates and momenta in the Heisenberg representation. These operators evolve according to

It is known that for any operator f ^ one can find a function f(ξ) through which f ^ is represented in the form f ( ξ ^ ) . The same operator f ^ at time τ is equal to

This equation shows that ξ ^ ( τ ) are characteristics that determine the evolution for all of the operators in Op(L²(Rⁿ)). This property is fully transferred to the phase space upon deformation quantization and, in the limit of ħ → 0, to the classical mechanics.

Table compares properties of characteristics in classical and quantum mechanics. PDE and ODE are partial differential equations and ordinary differential equations, respectively. The quantum Liouville equation is the Weyl-Wigner transform of the von Neumann evolution equation for the density matrix in Schrödinger representation. The quantum Hamilton equations are the Weyl-Wigner transforms of the evolution equations for operators of the canonical coordinates and momenta in Heisenberg representation.

In classical systems, characteristics c i ( ξ , τ ) satisfy usually first-order ODE, e.g., classical Hamilton's equations, and solve first-order PDE, e.g., classical Liouville equation. Functions q i ( ξ , τ ) are characteristics also, despite both q i ( ξ , τ ) and f ( ξ , τ ) obey infinite-order PDE.

The quantum phase flow contains entire information on the quantum evolution. Semiclassical expansion of quantum characteristics and ⋆ -functions of quantum characteristics in power series in ℏ allows calculation of the average values of time-dependent physical observables by solving a finite-order coupled system of ODE for phase space trajectories and Jacobi fields. The order of the system of ODE depends on truncation of the power series. The tunneling effect is nonperturbative in ℏ and is not captured by the expansion. The density of the quantum probability fluid is not preserved in phase-space, and there is no unambiguously defined notion of trajectories for quantum systems, as the quantum fluid "diffuses". Quantum characteristics must therefore be distinguished from both the trajectories of the de Broglie - Bohm theory and the trajectories of the path-integral method in phase space for the amplitudes and the Wigner function. So far, only a few quantum systems have been explicitly solved using the method of quantum characteristics.

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