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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, the Poisson bracket is an important
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
in
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
. The Poisson bracket also distinguishes a certain class of coordinate transformations, called ''
canonical transformations In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canoni ...
'', which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by q_i and p_i, respectively) that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself H =H(q, p, t) as one of the new canonical momentum coordinates. In a more general sense, the Poisson bracket is used to define a
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central ...
, of which the algebra of functions on a
Poisson manifold In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule : \ = \h + g \ . Equivalent ...
is a special case. There are other general examples, as well: it occurs in the theory of
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s, where the
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the
universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the represent ...
article. Quantum deformations of the universal enveloping algebra lead to the notion of
quantum group In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras) ...
s. All of these objects are named in honor of
Siméon Denis Poisson Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
.


Properties

Given two functions and that depend on phase space and time, their Poisson bracket \ is another function that depends on phase space and time. The following rules hold for any three functions f,\, g,\, h of phase space and time: ;
Anticommutativity In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
: \ = -\ ;
Bilinearity In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W ...
: \ = a\ + b\, \quad \ = a\ + b\, \quad a, b \in \mathbb R ; Leibniz's rule: \ = \g + f\ ;
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
: \ + \ + \ = 0 Also, if a function k is constant over phase space (but may depend on time), then \ = 0 for any f.


Definition in canonical coordinates

In canonical coordinates (also known as
Darboux coordinates Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief amon ...
) (q_i,\, p_i) on the phase space, given two functions f(p_i,\, q_i, t) and g(p_i,\, q_i, t), f(p_i,\, q_i,\, t) means f is a function of the 2N + 1 independent variables: momentum, p_; position, q_; and time, t the Poisson bracket takes the form \ = \sum_^ \left( \frac \frac - \frac \frac\right). The Poisson brackets of the canonical coordinates are \begin \ &= 0 \\ \ &= 0 \\ \ &= \delta_ \end where \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
.


Hamilton's equations of motion

Hamilton's equations of motion have an equivalent expression in terms of the Poisson bracket. This may be most directly demonstrated in an explicit coordinate frame. Suppose that f(p, q, t) is a function on the solution's trajectory-manifold. Then from the multivariable
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, \frac f(p, q, t) = \frac \frac + \frac \frac + \frac. Further, one may take p = p(t) and q = q(t) to be solutions to
Hamilton's equations Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
; that is, \begin \dot = \frac = \; \\ \dot = -\frac = \. \end Then \begin \frac f(p, q, t) &= \frac \frac - \frac \frac + \frac \\ &= \ + \frac ~. \end Thus, the time evolution of a function f on a symplectic manifold can be given as a one-parameter family of
symplectomorphism In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ...
s (i.e.,
canonical transformations In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canoni ...
, area-preserving diffeomorphisms), with the time t being the parameter: Hamiltonian motion is a canonical transformation generated by the Hamiltonian. That is, Poisson brackets are preserved in it, so that ''any time t'' in the solution to Hamilton's equations, q(t) = \exp (-t \ ) q(0), \quad p(t) = \exp (-t \) p(0), can serve as the bracket coordinates. ''Poisson brackets are canonical invariants''. Dropping the coordinates, \frac f = \left(\frac - \\right)f. The operator in the convective part of the derivative, i\hat = -\, is sometimes referred to as the Liouvillian (see
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. It asserts that ''the phase-space distribution function is constant along the trajectorie ...
).


Constants of motion

An integrable dynamical system will have
constants of motion In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a ''mathematical'' constraint, the natural consequence of the equations of motion, rather than ...
in addition to the energy. Such constants of motion will commute with the Hamiltonian under the Poisson bracket. Suppose some function f(p, q) is a constant of motion. This implies that if p(t), q(t) is a
trajectory A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traj ...
or solution to Hamilton's equations of motion, then 0 = \frac along that trajectory. Then 0 = \frac f(p,q) = \ where, as above, the intermediate step follows by applying the equations of motion and we assume that f does not explicitly depend on time. This equation is known as the
Liouville equation :''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).'' : ''For Liouville's equation in quantum mechanics, see Von Neumann equation.'' : ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gel ...
. The content of Liouville's theorem is that the time evolution of a
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
given by a distribution function f is given by the above equation. If the Poisson bracket of f and g vanishes (\ = 0), then f and g are said to be in involution. In order for a Hamiltonian system to be
completely integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
, n independent constants of motion must be in mutual involution, where n is the number of degrees of freedom. Furthermore, according to Poisson's Theorem, if two quantities A and B are explicitly time independent (A(p, q), B(p, q)) constants of motion, so is their Poisson bracket \. This does not always supply a useful result, however, since the number of possible constants of motion is limited (2n - 1 for a system with n degrees of freedom), and so the result may be trivial (a constant, or a function of A and B.)


The Poisson bracket in coordinate-free language

Let M be a symplectic manifold, that is, a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
equipped with a
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument ...
: a 2-form \omega which is both closed (i.e., its
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
d \omega vanishes) and non-degenerate. For example, in the treatment above, take M to be \mathbb^ and take \omega = \sum_^ d p_i \wedge d q_i. If \iota_v \omega is the
interior product In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of ...
or
contraction Contraction may refer to: Linguistics * Contraction (grammar), a shortened word * Poetic contraction, omission of letters for poetic reasons * Elision, omission of sounds ** Syncope (phonology), omission of sounds in a word * Synalepha, merged ...
operation defined by (\iota_v \omega)(w) = \omega(v,\, w), then non-degeneracy is equivalent to saying that for every one-form \alpha there is a unique vector field \Omega_\alpha such that \iota_ \omega = \alpha. Alternatively, \Omega_ = \omega^(d H). Then if H is a smooth function on M, the
Hamiltonian vector field In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is ...
X_H can be defined to be \Omega_. It is easy to see that \begin X_ &= \frac \\ X_ &= -\frac. \end The Poisson bracket \ \ on is a
bilinear operation In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W ...
on
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
s, defined by \ \;=\; \omega(X_f,\, X_g) ; the Poisson bracket of two functions on is itself a function on . The Poisson bracket is antisymmetric because: \ = \omega(X_f, X_g) = -\omega(X_g, X_f) = -\ . Furthermore, Here denotes the vector field applied to the function as a directional derivative, and \mathcal_ f denotes the (entirely equivalent)
Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
of the function . If is an arbitrary one-form on , the vector field generates (at least locally) a flow \phi_x(t) satisfying the boundary condition \phi_x(0) = x and the first-order differential equation \frac = \left. \Omega_\alpha \_. The \phi_x(t) will be
symplectomorphism In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ...
s (
canonical transformation In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canon ...
s) for every as a function of if and only if \mathcal_\omega \;=\; 0; when this is true, is called a
symplectic vector field In physics and mathematics, a symplectic vector field is one whose flow preserves a symplectic form. That is, if (M,\omega) is a symplectic manifold with smooth manifold M and symplectic form \omega, then a vector field X\in\mathfrak(M) in the ...
. Recalling Cartan's identity \mathcal_X\omega \;=\; d (\iota_X \omega) \,+\, \iota_X d\omega and , it follows that \mathcal_\omega \;=\; d\left(\iota_ \omega\right) \;=\; d\alpha. Therefore, is a symplectic vector field if and only if α is a closed form. Since d(df) \;=\; d^2f \;=\; 0, it follows that every Hamiltonian vector field is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations. From above, under the Hamiltonian flow , \fracf(\phi_x(t)) = X_Hf = \. This is a fundamental result in Hamiltonian mechanics, governing the time evolution of functions defined on phase space. As noted above, when , is a constant of motion of the system. In addition, in canonical coordinates (with \ \;=\; \ \;=\; 0 and \ \;=\; \delta_), Hamilton's equations for the time evolution of the system follow immediately from this formula. It also follows from that the Poisson bracket is a
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
; that is, it satisfies a non-commutative version of Leibniz's
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
: The Poisson bracket is intimately connected to the
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of the Hamiltonian vector fields. Because the Lie derivative is a derivation, \mathcal L_v\iota_w\omega = \iota_\omega + \iota_w\mathcal L_v\omega = \iota_\omega + \iota_w\mathcal L_v\omega. Thus if and are symplectic, using \mathcal_v\omega \;=\; 0, Cartan's identity, and the fact that \iota_w\omega is a closed form, \iota_\omega = \mathcal L_v\iota_w\omega = d(\iota_v\iota_w\omega) + \iota_vd(\iota_w\omega) = d(\iota_v\iota_w\omega) = d(\omega(w,v)). It follows that ,w= X_, so that Thus, the Poisson bracket on functions corresponds to the Lie bracket of the associated Hamiltonian vector fields. We have also shown that the Lie bracket of two symplectic vector fields is a Hamiltonian vector field and hence is also symplectic. In the language of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, the symplectic vector fields form a
subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operat ...
of the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of smooth vector fields on , and the Hamiltonian vector fields form an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
of this subalgebra. The symplectic vector fields are the Lie algebra of the (infinite-dimensional)
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
of
symplectomorphism In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ...
s of . It is widely asserted that the
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
for the Poisson bracket, \ + \ + \ = 0 follows from the corresponding identity for the Lie bracket of vector fields, but this is true only up to a locally constant function. However, to prove the Jacobi identity for the Poisson bracket, it is
sufficient In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
to show that: \operatorname_=\operatorname_= operatorname_f,\operatorname_g/math> where the operator \operatorname_g on smooth functions on is defined by \operatorname_g(\cdot) \;=\; \ and the bracket on the right-hand side is the commutator of operators, operatorname A,\, \operatorname B\;=\; \operatorname A\operatorname B - \operatorname B\operatorname A. By , the operator \operatorname_g is equal to the operator . The proof of the Jacobi identity follows from because, up to the factor of -1, the Lie bracket of vector fields is just their commutator as differential operators. The
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
of smooth functions on M, together with the Poisson bracket forms a
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central ...
, because it is a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
under the Poisson bracket, which additionally satisfies Leibniz's rule . We have shown that every symplectic manifold is a
Poisson manifold In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule : \ = \h + g \ . Equivalent ...
, that is a manifold with a "curly-bracket" operator on smooth functions such that the smooth functions form a Poisson algebra. However, not every Poisson manifold arises in this way, because Poisson manifolds allow for degeneracy which cannot arise in the symplectic case.


A result on conjugate momenta

Given a smooth vector field X on the configuration space, let P_X be its
conjugate momentum In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of ...
. The conjugate momentum mapping is a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
anti-homomorphism from the
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
to the Poisson bracket: \ = -P_. This important result is worth a short proof. Write a vector field X at point q in the configuration space as X_q = \sum_i X^i(q) \frac where \frac is the local coordinate frame. The conjugate momentum to X has the expression P_X(q, p) = \sum_i X^i(q) \;p_i where the p_i are the momentum functions conjugate to the coordinates. One then has, for a point (q,p) in the phase space, \begin \(q,p) &= \sum_i \sum_j \left\ \\ &= \sum_ p_i Y^j(q) \frac - p_j X^i(q) \frac \\ &= -\sum_i p_i \;
, Y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
i(q) \\ &= - P_(q, p). \end The above holds for all (q, p), giving the desired result.


Quantization

Poisson brackets
deform Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * De ...
to
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 le ...
s upon quantization, that is, they generalize to a different Lie algebra, the Moyal algebra, or, equivalently in
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, quantum
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
s. The Wigner-İnönü
group contraction In theoretical physics, Eugene Wigner and Erdal İnönü have discussed the possibility to obtain from a given Lie group a different (non-isomorphic) Lie group by a group contraction with respect to a continuous subgroup of it. That amounts to a ...
of these (the classical limit, ) yields the above Lie algebra. To state this more explicitly and precisely, the
universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the represent ...
of the
Heisenberg algebra In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ...
is the
Weyl algebra In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More prec ...
(modulo the relation that the center be the unit). The Moyal product is then a special case of the star product on the algebra of symbols. An explicit definition of the algebra of symbols, and the star product is given in the article on the
universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the represent ...
.


See also

*
Commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
* Dirac bracket *
Lagrange bracket Lagrange brackets are certain expressions closely related to Poisson brackets that were introduced by Joseph Louis Lagrange in 1808–1810 for the purposes of mathematical formulation of classical mechanics, but unlike the Poisson brackets, have fal ...
*
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 le ...
*
Peierls bracket In theoretical physics, the Peierls bracket is an equivalent description of the Poisson bracket. It can be defined directly from the action and does not require the canonical coordinates and their canonical momenta to be defined in advance. The ...
* Phase space *
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central ...
* Poisson ring *
Poisson superalgebra In mathematics, a Poisson superalgebra is a Z2- graded generalization of a Poisson algebra. Specifically, a Poisson superalgebra is an (associative) superalgebra ''A'' with a Lie superbracket : cdot,\cdot: A\otimes A\to A such that (''A'', ·,· i ...
* Poisson superbracket


Remarks


References

* * *


External links

* * {{mathworld , urlname=PoissonBracket , title=Poisson bracket, author=
Eric W. Weisstein Eric Wolfgang Weisstein (born March 18, 1969) is an American mathematician and encyclopedist who created and maintains the encyclopedias ''MathWorld'' and ''ScienceWorld''. In addition, he is the author of the '' CRC Concise Encyclopedia of M ...
Symplectic geometry Hamiltonian mechanics Bilinear maps Concepts in physics