Editing Symplectic vector space

{{Short description|Mathematical concept}}
In [[mathematics]], a '''symplectic vector space''' is a [[vector space]] <math>V</math> over a [[Field (mathematics)|field]] <math>F</math> (for example the real numbers <math>\mathbb{R}</math>) equipped with a symplectic [[bilinear form]].

A '''symplectic bilinear form''' is a [[map (mathematics)|mapping]] <math>\omega : V \times V \to F</math> that is
; [[bilinear form|Bilinear]]: [[linear map|Linear]] in each argument separately;
; [[alternating form|Alternating]]: <math>\omega(v, v) = 0</math> holds for all <math>v \in V</math>; and
; [[Nondegenerate form|Non-degenerate]]: <math>\omega(v, u) = 0</math> for all <math>v \in V</math> implies that <math>u = 0</math>.

If the underlying [[field (mathematics)|field]] has [[characteristic (algebra)|characteristic]] not 2, alternation is equivalent to [[skew symmetry|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 bilinear form|symmetric form]], but not vice versa. 

Working in a fixed [[basis (linear algebra)|basis]], <math>\omega</math> can be represented by a [[matrix (mathematics)|matrix]]. The conditions above are equivalent to this matrix being [[skew-symmetric matrix|skew-symmetric]], [[nonsingular matrix|nonsingular]], and [[hollow matrix#Diagonal entries all zero|hollow]] (all diagonal entries are zero). This should not be confused with a [[symplectic matrix]], which represents a symplectic transformation of the space. If <math>V</math> is [[finite-dimensional]], then its dimension must necessarily be [[even number|even]] since every skew-symmetric, hollow matrix of odd size has [[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==
{{Further|Symplectic matrix#Symplectic transformations}}

The standard symplectic space is <math>\mathbb{R}^{2n}</math> with the symplectic form given by a [[nonsingular matrix|nonsingular]], [[skew-symmetric matrix]]. Typically <math>\omega</math> is chosen to be the [[block matrix]]

:<math>\omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}</math>

where ''I''<sub>''n''</sub> is the {{nowrap|''n'' × ''n''}} [[identity matrix]]. In terms of basis vectors {{nowrap|(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'', ''y''<sub>1</sub>, ..., ''y<sub>n</sub>'')}}:

:<math>\begin{align}
  \omega(x_i, y_j) = -\omega(y_j, x_i) &= \delta_{ij}, \\
  \omega(x_i, x_j) =  \omega(y_i, y_j) &= 0.
\end{align}</math>

A modified version of the [[Gram–Schmidt process]] shows that any finite-dimensional symplectic vector space has a basis such that <math>\omega</math> takes this form, often called a '''''Darboux basis''''' or [[symplectic basis]].

'''Sketch of process:''' 

Start with an arbitrary basis <math>v_1, ..., v_n</math>, and represent the dual of each basis vector by the [[dual basis]]: <math>\omega(v_i, \cdot) = \sum_j \omega(v_i, v_j) v_j^*</math>. This gives us a <math>n\times n</math> matrix with entries <math>\omega(v_i, v_j)</math>. Solve for its null space. Now for any <math>(\lambda_1, ..., \lambda_n)</math> in the null space, we have <math>\sum_i \omega(v_i, \cdot) = 0</math>, so the null space gives us the degenerate subspace <math>V_0</math>. 

Now arbitrarily pick a complementary <math>W</math> such that <math>V = V_0 \oplus W</math>, and let <math>w_1, ..., w_m</math> be a basis of <math>W</math>. Since <math>\omega(w_1, \cdot) \neq 0</math>, and <math>\omega(w_1, w_1) = 0</math>, WLOG <math>\omega(w_1, w_2 ) \neq 0</math>. Now scale <math>w_2</math> so that <math>\omega(w_1, w_2) =1</math>. Then define <math>w' = w - \omega(w, w_2) w_1 + \omega(w, w_1) w_2</math> for each of <math>w = w_3, w_4, ..., w_m</math>. Iterate.

Notice that this method applies for symplectic vector space over any field, not just the field of real numbers.

'''Case of real or complex field:'''

When the space is over the field of real numbers, then we can modify the modified Gram-Schmidt process as follows: Start the same way. Let <math>w_1, ..., w_m</math> be an orthonormal basis (with respect to the usual inner product on <math>\R^n</math>)  of <math>W</math>. Since <math>\omega(w_1, \cdot) \neq 0</math>, and <math>\omega(w_1, w_1) = 0</math>, WLOG <math>\omega(w_1, w_2 ) \neq 0</math>. Now multiply <math>w_2</math> by a sign, so that <math>\omega(w_1, w_2) \geq 0</math>. Then define <math>w' = w - \omega(w, w_2) w_1 + \omega(w, w_1) w_2</math> for each of <math>w = w_3, w_4, ..., w_m</math>, then scale each <math>w'</math> so that it has norm one. Iterate.

Similarly, for the field of complex numbers, we may choose a unitary basis. This proves the [[Skew-symmetric matrix#Spectral theory|spectral theory of antisymmetric matrices]].

=== Lagrangian form ===
There is another way to interpret this standard symplectic form. Since the model space '''R'''<sup>2''n''</sup> 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''<sup>∗</sup> its [[dual space]]. Now consider the [[direct sum of vector spaces|direct sum]] {{nowrap|1=''W'' = ''V'' ⊕ ''V''<sup>∗</sup>}} of these spaces equipped with the following form:

:<math>\omega(x \oplus \eta, y \oplus \xi) = \xi(x) - \eta(y).</math>

Now choose any [[Basis (linear algebra)|basis]] {{nowrap|(''v''<sub>1</sub>, ..., ''v''<sub>''n''</sub>)}} of ''V'' and consider its [[dual space|dual basis]]

:<math>\left(v^*_1, \ldots, v^*_n\right).</math>

We can interpret the basis vectors as lying in ''W'' if we write {{nowrap|1=''x''<sub>''i''</sub> = (''v''<sub>''i''</sub>, 0) and ''y''<sub>''i''</sub> = (0, ''v''<sub>''i''</sub><sup>∗</sup>)}}. Taken together, these form a complete basis of ''W'',

:<math>(x_1, \ldots, x_n, y_1, \ldots, y_n).</math>

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 {{nowrap|''V'' ⊕ ''V''<sup>∗</sup>}}. 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 [[#Subspaces|as defined below]], then a choice of basis {{nowrap|(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'')}} defines a dual basis for a complement, by {{nowrap|1=''ω''(''x''<sub>''i''</sub>, ''y''<sub>''j''</sub>) = ''δ''<sub>''ij''</sub>}}.

===Analogy with complex structures===
Just as every symplectic structure is isomorphic to one of the form {{nowrap|''V'' ⊕ ''V''<sup>∗</sup>}}, every [[linear complex structure|''complex'' structure]] on a vector space is isomorphic to one of the form {{nowrap|''V'' ⊕ ''V''}}. Using these structures, the [[tangent bundle]] of an ''n''-manifold, considered as a 2''n''-manifold, has an [[almost complex structure]], and the [[cotangent bundle|''co''tangent bundle]] of an ''n''-manifold, considered as a 2''n''-manifold, has a symplectic structure: {{nowrap|1=''T''<sub>∗</sub>(''T''<sup>∗</sup>''M'')<sub>''p''</sub> = ''T''<sub>''p''</sub>(''M'') ⊕ (''T''<sub>''p''</sub>(''M''))<sup>∗</sup>}}.

The complex analog to a Lagrangian subspace is a [[real subspace|''real'' subspace]], a subspace whose [[complexification]] is the whole space: {{nowrap|1=''W'' = ''V'' ⊕ ''J'' ''V''}}. As can be seen from the standard symplectic form above, every symplectic form on '''R'''<sup>2''n''</sup> is isomorphic to the imaginary part of the standard complex (Hermitian) inner product on '''C'''<sup>''n''</sup> (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'', {{nowrap|''ω'' ∈ Λ<sup>2</sup>(''V'')}}. Then ''ω'' is non-degenerate if and only if ''n'' is even and {{nowrap|1=''ω''<sup>''n''/2</sup> = ''ω'' ∧ ... ∧ ''ω''}} is a [[volume form]].  A volume form on a ''n''-dimensional vector space ''V'' is a non-zero multiple of the ''n''-form {{nowrap|''e''<sub>1</sub><sup>∗</sup> ∧ ... ∧ ''e''<sub>''n''</sub><sup>∗</sup>}} where {{nowrap|''e''<sub>1</sub>, ''e''<sub>2</sub>, ..., ''e''<sub>''n''</sub>}} is a basis of ''V''.

For the standard basis defined in the previous section, we have

:<math>\omega^n = (-1)^\frac{n}{2} x^*_1 \wedge \dotsb \wedge x^*_n \wedge y^*_1 \wedge \dotsb \wedge y^*_n.</math>

By reordering, one can write

:<math>\omega^n = x^*_1 \wedge y^*_1 \wedge \dotsb \wedge x^*_n \wedge y^*_n.</math>

Authors variously define ''ω''<sup>''n''</sup> or (−1)<sup>''n''/2</sup>''ω''<sup>''n''</sup> 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 (mathematics)|orientation]] on the symplectic vector space {{nowrap|(''V'', ''ω'')}}.

==Symplectic map==
Suppose that {{nowrap|(''V'', ''ω'')}} and {{nowrap|(''W'', ''ρ'')}} are symplectic vector spaces. Then a [[linear map]] {{nowrap|1=''f'' : ''V'' → ''W''}} is called a '''symplectic map''' if the [[pullback (differential geometry)|pullback]] preserves the symplectic form, i.e. {{nowrap|1=''f''{{i sup|∗}}''ρ'' = ''ω''}}, where the pullback form is defined by {{nowrap|1=(''f''{{i sup|∗}}''ρ'')(''u'', ''v'') = ''ρ''(''f''(''u''), ''f''(''v''))}}. Symplectic maps are volume- and orientation-preserving.

==Symplectic group==
If {{nowrap|1=''V'' = ''W''}}, then a symplectic map is called a '''linear symplectic transformation''' of ''V''. In particular, in this case one has that {{nowrap|1=''ω''(''f''(''u''), ''f''(''v'')) = ''ω''(''u'', ''v'')}}, and so the [[linear transformation]] ''f'' preserves the symplectic form.  The set of all symplectic transformations forms a [[group (mathematics)|group]] and in particular a [[Lie group]], called the [[symplectic group]] and denoted by Sp(''V'') or sometimes {{nowrap|Sp(''V'', ''ω'')}}.  In matrix form symplectic transformations are given by [[symplectic matrix|symplectic matrices]].

==Subspaces==
Let ''W'' be a [[linear subspace]] of ''V''. Define the '''symplectic complement''' of ''W'' to be the subspace
:<math>W^\perp = \{v \in V \mid \omega(v,w) = 0 \mbox{ for all } w \in W\}.</math>

The symplectic complement satisfies:
:<math>\begin{align}
  \left(W^\perp\right)^\perp &= W \\
       \dim W + \dim W^\perp &= \dim V.
\end{align}</math>

However, unlike [[orthogonal complement]]s, ''W''<sup>⊥</sup> ∩ ''W'' need not be 0. We distinguish four cases:
* ''W'' is '''symplectic''' if {{nowrap|1=''W''<sup>⊥</sup> ∩ ''W'' = {0}}}. This is true [[if and only if]] ''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 {{nowrap|''W'' ⊆ ''W''<sup>⊥</sup>}}. This is true if and only if ''ω'' restricts to 0 on ''W''. Any one-dimensional subspace is isotropic.
* ''W'' is '''coisotropic''' if {{nowrap|''W''<sup>⊥</sup> ⊆ ''W''}}. ''W'' is coisotropic if and only if ''ω'' descends to a nondegenerate form on the [[Quotient space (linear algebra)|quotient space]] ''W''/''W''<sup>⊥</sup>. Equivalently ''W'' is coisotropic if and only if ''W''<sup>⊥</sup> is isotropic. Any [[codimension]]-one subspace is coisotropic.
* ''W'' is '''Lagrangian''' if {{nowrap|1=''W'' = ''W''<sup>⊥</sup>}}. 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 '''R'''<sup>2''n''</sup> above,
* the subspace spanned by {''x''<sub>1</sub>, ''y''<sub>1</sub>} is symplectic
* the subspace spanned by {''x''<sub>1</sub>, ''x''<sub>2</sub>} is isotropic
* the subspace spanned by {''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>, ''y''<sub>1</sub>} is coisotropic
* the subspace spanned by {''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>} is Lagrangian.

==Heisenberg group==
{{main|Heisenberg group}}
A [[Heisenberg group]] can be defined for any symplectic vector space, and this is the typical way that [[Heisenberg group]]s arise.

A vector space can be thought of as a commutative Lie group (under addition), or equivalently as a commutative [[Lie algebra]], meaning with trivial Lie bracket. The Heisenberg group is a [[central extension (mathematics)|central extension]] of such a commutative Lie group/algebra: the symplectic form defines the commutation, analogously to the [[canonical commutation relation]]s (CCR), and a Darboux basis corresponds to [[canonical coordinate]]s&nbsp;– in physics terms, to [[momentum operator]]s and [[position operator]]s.

Indeed, by the [[Stone–von Neumann theorem]], 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 ring|group algebra]] of (the dual to) a vector space is the [[symmetric algebra]], 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 quantization|deformation]].

Formally, the symmetric algebra of a vector space ''V'' over a field ''F'' is the group algebra of the dual, {{nowrap|1=Sym(''V'') := ''F''[''V''<sup>∗</sup>]}}, and the Weyl algebra is the group algebra of the (dual) Heisenberg group {{nowrap|1=''W''(''V'') = ''F''[''H''(''V''<sup>∗</sup>)]}}. Since passing to group algebras is a [[contravariant functor]], the  central extension map {{nowrap|''H''(''V'') → ''V''}} becomes an inclusion {{nowrap|Sym(''V'') → ''W''(''V'')}}.

==See also==

* A [[symplectic manifold]] is a [[smooth manifold]] with a smoothly-varying ''closed'' symplectic form on each [[tangent space]].
* [[Maslov index]]
* A [[symplectic representation]] is a [[group representation]] 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 }} [https://authors.library.caltech.edu/25029/1/FoM2.pdf 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 

[[Category:Linear algebra]]
[[Category:Symplectic geometry]]
[[Category:Bilinear forms]]