Method of quantum characteristics - AbsoluteAstronomy.com

In quantum mechanics, quantum characteristics are phase-space trajectories that arise in the deformation quantization

Deformation theory

In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach...

through the Weyl-Wigner transform of Heisenberg operators of canonical coordinates and momenta. These trajectories obey the Hamilton’s equations in quantum form and play the role of characteristics

Method of characteristics

In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation...

in terms of which time-dependent Weyl's symbols of quantum operators can be expressed. In classical limit, quantum characteristics turn to classical trajectories. The knowledge of quantum characteristics is equivalent to the knowledge of quantum dynamics.

Weyl-Wigner association rule

In Hamiltonian dynamics

Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

classical systems with

degrees of freedom are described by

canonical coordinates and momenta

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

Poisson bracket

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...

relations

The skew-symmetric matrix

where

is the

identity matrix, defines nondegenerate 2-form in the phase space.
The phase space acquires thereby the structure of a symplectic manifold

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...

. 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

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

leave the distance between two points invariant.
Canonical transformations in symplectic manifold leave the areas invariant.

In quantum mechanics, the canonical variables

are associated to operators of canonical coordinates and momenta

These operators act in Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

and obey commutation relations

The Weyl’s association rule extends the correspondence

to arbitrary phase-space functions and operators.

Taylor expansion

A one-sided association rule

was formulated by Weyl initially with the help of Taylor expansion

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

of functions of operators of the canonical variables

The operators

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

is called Weyl's symbol of operator

.

Under the reverse association

, the density matrix

Density matrix

In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...

turns to Wigner function

Wigner quasi-probability distribution

The Wigner quasi-probability distribution is a quasi-probability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics...

. 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 Stratonovich.

Stratonovich basis

The set of operators acting in the Hilbert space is closed under multiplication of operators by

-numbers and summation. Such a set constitutes a vector space

. 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,

and

are functions and

and

are the associated operators.

The elements of basis of

are labelled by canonical variables

. The commonly used Stratonovich basis looks like

The Weyl-Wigner two-sided association rule for function

and operator

has the form

The function

provides coordinates of the operator

in the basis

. The basis is complete and orthogonal:

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

Star-product

The set of operators

is closed under the multiplication of operators. The vector space

is endowed thereby with an associative algebra structure. Given two functions

one can construct a third function

called

-product

or Moyal product

Moyal product

In mathematics, the Moyal product, named after José Enrique Moyal, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, ∗, on the functions on ℝ2n, equipped with its Poisson bracket .This particular star product is also sometimes called...

. 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

is known under the name of Moyal bracket

Moyal bracket

In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product.The Moyal Bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a lengthy dispute with Dirac...

. This is the Weyl's symbol of commutator. In the classical limit Moyal bracket becomes Poisson bracket. Moyal bracket is quantum deformation

Deformation theory

of Poisson bracket.

Quantum characteristics

The correspondence

shows that coordinate transformations in the phase space are accompanied by transformations of operators of the canonical coordinates and momenta and vice versa. Let

be the evolution operator,

and

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

Heisenberg picture

In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...

, the operators of the canonical variables are transformed as

The phase-space coordinates

that correspond to new operators

in the old basis

are given by

with the initial conditions

The functions

define quantum phase flow. In the general case, it is canonical to first order in

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

. Transformations

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

Using the Taylor expansion, the transformation of function

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

around

is formally well defined and involves even powers of

only.
This equation shows that, given quantum characteristics are constructed, physical observables can be found without further addressing to Hamiltonian.
The functions

play the role of characteristics similarly to classical characteristics

Method of characteristics

used to solve classical Liouville equation

Liouville's theorem (Hamiltonian)

In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics...

Quantum Liouville equation

Wigner transform of the evolution equation for the density matrix in the Schrödinger representation leads quantum Liouville equation for the Wigner function. 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

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

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,

is not conserved along quantum characteristics.

Summary

Table compares properties of characteristics in classical and quantum mechanics. PDE and ODE are partial differential equations

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

and ordinary differential equations

Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

, respectively. The quantum Liouville equation is the Weyl-Wigner transform of the von Neumann evolution equation for the density matrix in Schrödinger representation

Schrödinger picture

In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time...

. The quantum Hamilton's equations are the Weyl-Wigner transforms of the evolution equations for operators of the canonical coordinates and momenta in Heisenberg representation

Heisenberg picture

.

In classical systems, characteristics

satisfy usually first-order ODE, e.g., classical Hamilton's equations, and solve first-order PDE, e.g., classical Liouville equation. Functions

are characteristics also, despite both

and

obey infinite-order PDE.

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| Liouville equation
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| Finite-order PDE
| Infinite-order PDE
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| Hamilton's equations
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| Finite-order ODE
| Infinite-order PDE
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| Initial conditions
| Initial conditions
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| Composition law
| -composition law
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| Conservation of Poisson bracket
| Conservation of Moyal bracket
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| Energy conservation
| Energy conservation
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| Solutions to Liouville equation
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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

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

and is not captured by the expansion. Quantum characteristics are distinct from trajectories of the de Broglie - Bohm theory.

Textbooks

H. Weyl, The Theory of Groups and Quantum Mechanics, (Dover Publications, New York Inc., 1931).
V. I. Arnold, Mathematical Methods of Classical Mechanics, (2-nd ed. Springer-Verlag, New York Inc., 1989).
M. V. Karasev and V. P Maslov, Nonlinear Poisson Brackets, (Nauka, Moscow, 1991).

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