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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 ...
, a symplectic vector space is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
. A symplectic bilinear form is a mapping that is ; Bilinear:
Linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
in each argument separately; ; Alternating: holds for all ; and ;
Non-degenerate In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent definit ...
: for all implies that . If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, ''ω'' can be represented by a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
. The conditions above are equivalent to this matrix being skew-symmetric,
nonsingular In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplic ...
, and hollow (all diagonal entries are zero). This should not be confused with a
symplectic matrix In mathematics, a symplectic matrix is a 2n\times 2n matrix M with real entries that satisfies the condition where M^\text denotes the transpose of M and \Omega is a fixed 2n\times 2n nonsingular, skew-symmetric matrix. This definition can be e ...
, which represents a symplectic transformation of the space. If ''V'' is
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
, then its dimension must necessarily be
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...
since every skew-symmetric, hollow matrix of odd size has
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
zero. Notice that the condition that the matrix be hollow is not redundant if the characteristic of the field is 2. A symplectic form behaves quite differently from a symmetric form, for example, the scalar product on Euclidean vector spaces.


Standard symplectic space

The standard symplectic space is R2''n'' with the symplectic form given by a
nonsingular In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplic ...
,
skew-symmetric matrix In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, i ...
. Typically ''ω'' is chosen to be the
block matrix In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original mat ...
:\omega = \begin 0 & I_n \\ -I_n & 0 \end where ''I''''n'' is the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
. In terms of basis vectors : :\begin \omega(x_i, y_j) = -\omega(y_j, x_i) &= \delta_, \\ \omega(x_i, x_j) = \omega(y_i, y_j) &= 0. \end A modified version of the
Gram–Schmidt process In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space equipped with the standard inner produ ...
shows that any finite-dimensional symplectic vector space has a basis such that ''ω'' takes this form, often called a ''Darboux basis'' or symplectic basis. There is another way to interpret this standard symplectic form. Since the model space R2''n'' used above carries much canonical structure which might easily lead to misinterpretation, we will use "anonymous" vector spaces instead. Let ''V'' be a real vector space of dimension ''n'' and ''V'' its
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
. Now consider the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of these spaces equipped with the following form: :\omega(x \oplus \eta, y \oplus \xi) = \xi(x) - \eta(y). Now choose any basis of ''V'' and consider its
dual basis In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimension of ''V''), the dual set of ''B'' is a set ''B''∗ of vectors in the dual space ''V''∗ with the ...
:\left(v^*_1, \ldots, v^*_n\right). We can interpret the basis vectors as lying in ''W'' if we write . Taken together, these form a complete basis of ''W'', :(x_1, \ldots, x_n, y_1, \ldots, y_n). The form ''ω'' defined here can be shown to have the same properties as in the beginning of this section. On the other hand, every symplectic structure is isomorphic to one of the form . The subspace ''V'' is not unique, and a choice of subspace ''V'' is called a polarization. The subspaces that give such an isomorphism are called Lagrangian subspaces or simply Lagrangians. Explicitly, given a Lagrangian subspace as defined below, then a choice of basis defines a dual basis for a complement, by .


Analogy with complex structures

Just as every symplectic structure is isomorphic to one of the form , every ''complex'' structure on a vector space is isomorphic to one of the form . Using these structures, the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of an ''n''-manifold, considered as a 2''n''-manifold, has an
almost complex structure In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex ...
, and the ''co''tangent bundle of an ''n''-manifold, considered as a 2''n''-manifold, has a symplectic structure: . The complex analog to a Lagrangian subspace is a ''real'' subspace, a subspace whose
complexification In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...
is the whole space: . As can be seen from the standard symplectic form above, every symplectic form on R2''n'' is isomorphic to the imaginary part of the standard complex (Hermitian) inner product on C''n'' (with the convention of the first argument being anti-linear).


Volume form

Let ''ω'' be an
alternating bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear ...
on an ''n''-dimensional real vector space ''V'', . Then ''ω'' is non-degenerate if and only if ''n'' is even and is a
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
. A volume form on a ''n''-dimensional vector space ''V'' is a non-zero multiple of the ''n''-form where is a basis of ''V''. For the standard basis defined in the previous section, we have :\omega^n = (-1)^\frac x^*_1 \wedge \dotsb \wedge x^*_n \wedge y^*_1 \wedge \dotsb \wedge y^*_n. By reordering, one can write :\omega^n = x^*_1 \wedge y^*_1 \wedge \dotsb \wedge x^*_n \wedge y^*_n. Authors variously define ''ω''''n'' or (−1)''n''/2''ω''''n'' as the standard volume form. An occasional factor of ''n''! may also appear, depending on whether the definition of the alternating product contains a factor of ''n''! or not. The volume form defines an orientation on the symplectic vector space .


Symplectic map

Suppose that and are symplectic vector spaces. Then a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
is called a symplectic map if the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...
preserves the symplectic form, i.e. , where the pullback form is defined by . Symplectic maps are volume- and orientation-preserving.


Symplectic group

If , then a symplectic map is called a linear symplectic transformation of ''V''. In particular, in this case one has that , and so the
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
''f'' preserves the symplectic form. The set of all symplectic transformations forms a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
and in particular a
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 addi ...
, called the
symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gro ...
and denoted by Sp(''V'') or sometimes . In matrix form symplectic transformations are given by symplectic matrices.


Subspaces

Let ''W'' be a
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
of ''V''. Define the symplectic complement of ''W'' to be the subspace :W^\perp = \. The symplectic complement satisfies: :\begin \left(W^\perp\right)^\perp &= W \\ \dim W + \dim W^\perp &= \dim V. \end However, unlike
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
s, ''W'' ∩ ''W'' need not be 0. We distinguish four cases: * ''W'' is symplectic if . This is true
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
''ω'' restricts to a nondegenerate form on ''W''. A symplectic subspace with the restricted form is a symplectic vector space in its own right. * ''W'' is isotropic if . This is true if and only if ''ω'' restricts to 0 on ''W''. Any one-dimensional subspace is isotropic. * ''W'' is coisotropic if . ''W'' is coisotropic if and only if ''ω'' descends to a nondegenerate form on the quotient space ''W''/''W''. Equivalently ''W'' is coisotropic if and only if ''W'' is isotropic. Any
codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals ...
-one subspace is coisotropic. * ''W'' is Lagrangian if . A subspace is Lagrangian if and only if it is both isotropic and coisotropic. In a finite-dimensional vector space, a Lagrangian subspace is an isotropic one whose dimension is half that of ''V''. Every isotropic subspace can be extended to a Lagrangian one. Referring to the canonical vector space R2''n'' above, * the subspace spanned by is symplectic * the subspace spanned by is isotropic * the subspace spanned by is coisotropic * the subspace spanned by is Lagrangian.


Heisenberg group

A
Heisenberg group 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 ...
can be defined for any symplectic vector space, and this is the typical way that
Heisenberg group 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 ...
s arise. A vector space can be thought of as a commutative Lie group (under addition), or equivalently as a commutative
Lie algebra 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 ...
, meaning with trivial Lie bracket. The Heisenberg group is a central extension of such a commutative Lie group/algebra: the symplectic form defines the commutation, analogously to the
canonical commutation relation In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p_ ...
s (CCR), and a Darboux basis corresponds to
canonical coordinate 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 cl ...
s – in physics terms, to
momentum operator In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimensi ...
s and
position operator In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. When the position operator is considered with a wide enough domain (e.g. the space of tempered distributions), its eigenvalues ...
s. Indeed, by the
Stone–von Neumann theorem In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named afte ...
, every representation satisfying the CCR (every representation of the Heisenberg group) is of this form, or more properly unitarily conjugate to the standard one. Further, the group algebra of (the dual to) a vector space is the
symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
, and the group algebra of the Heisenberg group (of the dual) is the Weyl algebra: one can think of the central extension as corresponding to quantization or deformation. Formally, the symmetric algebra of a vector space ''V'' over a field ''F'' is the group algebra of the dual, , and the Weyl algebra is the group algebra of the (dual) Heisenberg group . Since passing to group algebras is a
contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
, the central extension map becomes an inclusion .


See also

* A
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sym ...
is a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
with a smoothly-varying ''closed'' symplectic form on each
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
. * Maslov index * A
symplectic representation In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (''V'', ''ω'') which preserves the symplectic form ''ω''. Here ''ω'' is a nondegenerate ...
is a
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
where each group element acts as a symplectic transformation.


References

* Claude Godbillon (1969) "Géométrie différentielle et mécanique analytique", Hermann *{{cite book , authorlink=Ralph Abraham (mathematician) , first1=Ralph , last1=Abraham , first2=Jerrold E. , last2=Marsden , authorlink2=Jerrold E. Marsden , title=Foundations of Mechanics , year=1978 , publisher=Benjamin-Cummings , location=London , isbn=0-8053-0102-X , chapter=Hamiltonian and Lagrangian Systems , pages=161–252 , edition=2nd }
PDF
* Paulette Libermann and Charles-Michel Marle (1987) "Symplectic Geometry and Analytical Mechanics", D. Reidel * Jean-Marie Souriau (1997) "Structure of Dynamical Systems, A Symplectic View of Physics", Springer Linear algebra Symplectic geometry Bilinear forms