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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a symplectic vector space is a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
''V'' over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
''F'' (for example the real numbers R) equipped with a symplectic
bilinear form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. A symplectic bilinear form is a
mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ...
that is ; Bilinear:
Linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...

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 definition when ...
: for all implies that . If the underlying
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
has
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
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 Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
, ''ω'' can be represented by a
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
. The conditions above are equivalent to this matrix being skew-symmetric,
nonsingularIn linear algebra, an ''n''-by-''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
, and hollow. This should not be confused with a
symplectic matrixIn mathematics, a symplectic matrix is a 2n\times 2n matrix M with real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. ...
, 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 (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...
, then its dimension must necessarily be
even Even may refer to: General * Even (given name)''Even'' is a Norwegian given name coming from Old Norse ''Eivindr'' (existing as ''Eivindur'' in Iceland). Another common name derived from Old Norse ''Eivindr'' is the Norwegianized ''Eivind''. ''Ei ...
since every skew-symmetric, hollow matrix of odd size has
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

determinant
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
nonsingularIn linear algebra, an ''n''-by-''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
,
skew-symmetric matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. Typically ''ω'' is chosen to be the
block matrix In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
:\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'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

identity matrix
. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

Gram–Schmidt process
shows that any finite-dimensional symplectic vector space has a basis such that ''ω'' takes this form, often called a ''Darboux basis'', or
symplectic basisIn linear algebra, a standard symplectic basis is a basis _i, _i of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form \omega, such that \omega(_i, _j) = 0 = \omega(_i, _j), \omega(_i, _j) = \delta_. A s ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. Now consider the
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of these spaces equipped with the following form: :\omega(x \oplus \eta, y \oplus \xi) = \xi(x) - \eta(y). Now choose any
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
of ''V'' and consider its
dual basis In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
:\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 Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

tangent bundle
of an ''n''-manifold, considered as a 2''n''-manifold, has an
almost complex structureIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. 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 Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
on the symplectic vector space .


Symplectic map

Suppose that and are symplectic vector spaces. Then a
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

linear map
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 (disambiguation), pushforward. Precomposition Precomposition with a Function (mathematics), function probabl ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
''f'' preserves the symplectic form. The set of all symplectic transformations forms a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
and in particular a
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, called the
symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical Group (mathematics), groups, denoted and for positive integer ''n'' and field (mathematics), field F (usually C or R). The lat ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s, ''W'' ∩ ''W'' need not be 0. We distinguish four cases: * ''W'' is symplectic if . This is true
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
''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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
-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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
can be defined for any symplectic vector space, and this is the typical way that
Heisenberg group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, 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 Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking i ...
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 mechanics, Hamiltoni ...
s – in physics terms, to
momentum operator In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum ph ...
s and
position operator In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum ph ...
s. Indeed, by the
Stone–von Neumann theoremIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and the group algebra of the Heisenberg group (of the dual) is the
Weyl algebra In abstract algebra, the Weyl algebra is the ring of differential operator 300px, A harmonic function defined on an annulus. Harmonic functions are exactly those functions which lie in the kernel of the Laplace operator, an important differe ...
: one can think of the central extension as corresponding to quantization or
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (mechanics), such changes considered and analyzed as displacements of continuum bodies. * Defo ...
. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, the central extension map becomes an inclusion .


See also

* A
symplectic manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral ...
is a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
with a smoothly-varying ''closed'' symplectic form on each
tangent space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. * Maslov index * A symplectic representation is a
group representation In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
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