# Symplectic vector space

In

mathematics , a**symplectic vector space**is avector space "V" equipped with anondegenerate ,skew-symmetric ,bilinear form ω called the**symplectic form**.Explicitly, a symplectic form is a bilinear form ω : "V" × "V" →

**R**which is

* "Skew-symmetric": ω("u", "v") = −ω("v", "u") for all "u", "v" ∈ "V",

* "Nondegenerate": if ω("u", "v") = 0 for all "v" ∈ "V" then "u" = 0.Working in a fixed basis, ω can be represented by a matrix. The two conditions above say that this matrix must be skew-symmetric and nonsingular. This is "not" the same thing as asymplectic matrix , which represent a symplectic transformation of the space.If "V" is

finite-dimensional then its dimension must necessarily be even since every skew-symmetric matrix of odd size hasdeterminant zero.A nondegenerate skew-symmetric bilinear form behaves quite differently from a nondegenerate "symmetric" bilinear form, such as the dot product on Euclidean vector spaces. With a Euclidean inner product "g", we have "g"("v","v") > 0 for all nonzero vectors "v", whereas a symplectic form ω satisfies ω("v","v") = 0.

**tandard symplectic space**The standard symplectic space is

**R**^{2"n"}with the symplectic form given by a nonsingular,skew-symmetric matrix . Typically ω is chosen to be theblock matrix :$omega\; =\; egin\{bmatrix\}\; 0\; I\_n\; \backslash \; -I\_n\; 0\; end\{bmatrix\}$where "I"

_{"n"}is the "n" × "n"identity matrix . In terms of basis vectors:$(x\_1,\; ldots,\; x\_n,\; y\_1,\; ldots,\; y\_n)$::$omega(x\_i,\; y\_j)\; =\; -omega(y\_j,\; x\_i)\; =\; delta\_\{ij\},$:$omega(x\_i,\; x\_j)\; =\; omega(y\_i,\; y\_j)\; =\; 0.,$

A modified version of the

Gram-Schmidt process shows that any finite-dimensional symplectic vector space has such a basis, often called a**"Darboux basis**".There is another way to interpret this standard symplectic form. Since the model space

**R**^{"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"^{∗}itsdual space . Now consider thedirect sum "W" := "V" ⊕ "V"^{∗}of these spaces equipped with the following form::$omega(x\; oplus\; eta,\; y\; oplus\; xi)\; =\; xi(x)\; -\; eta(y).$

Now choose any basis ("v"

_{1}, …, "v"_{"n"}) of "V" and consider its dual basis:$(v^*\_1,\; ldots,\; v^*\_n).$

We can interpret the basis vectors as lying in "W" if we write"x"

_{"i"}= ("v"_{"i"}, 0) and "y"_{"i"}= (0, v_{"i"}^{∗}). Taken together, these form a complete basis of "W",:$(x\_1,\; ldots,\; x\_n,\; y\_1,\; ldots,\; y\_n).$

The form $omega$ defined here can be shown to have the same properties as in the beginning of this section.

**Volume form**Let ω be a

form on a "n"-dimensional real vector space "V", ω ∈ Λ^{2}("V"). Then ω is non-degenerate if and only if "n" is even, and ω^{"n"/2}= ω ∧ … ∧ ω is avolume form . A volume form on a "n"-dimensional vector space "V" is a non-zero multiple of the (unique) "n"-form "e"_{1}^{∗}∧ … ∧ "e"_{"n"}^{∗}where the "e"_{"i"}are standard basis vectors on "V".For the standard basis defined in the previous section, we have

:$omega^n=(-1)^\{n/2\}\; x^*\_1wedgeldots\; wedge\; x^*\_nwedge\; y^*\_1wedge\; ldots\; wedge\; y^*\_n.$

By reordering, one can write

:$omega^n=\; x^*\_1wedge\; y^*\_1wedge\; ldots\; wedge\; x^*\_nwedge\; y^*\_n.$

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 thealternating product contains a factor of "n"! or not. The volume form defines an orientation on the symplectic vector space ("V", ω).**ymplectic map**Suppose that $(V,omega)$ and $(W,\; ho)$ are symplectic vector spaces. Then a

linear map $f:V\; ightarrow\; W$ is called a**symplectic map**if and only if the pullback $f^*$ preserves the symplectic form, that is, if $f^*\; ho=omega$. The pullback form is defined by:$f^*\; ho(u,v)=\; ho(f(u),f(v))$

and thus "f" is a symplectic map

if and only if :$ho(f(u),f(v))=omega(u,v)$

for all "u" and "v" in "V". In particular, symplectic maps are volume-preserving, orientation-preserving, and are

isomorphism s.**ymplectic group**If "V" = "W", then a symplectic map is called a

**linear symplectic transformation**of "V". In particular, in this case one has that:$omega(f(u),f(v))\; =\; omega(u,v)$,

and so the

linear transformation "f" preserves the symplectic form. The set of all symplectic transformations forms a group and in particular aLie group , called thesymplectic group and denoted by Sp("V") or sometimes Sp("V",ω). In matrix form symplectic transformations are given by symplectic matrices.**ubspaces**Let "W" be a

linear subspace of "V". Define the**symplectic complement**of "W" to be the subspace:$W^\{perp\}\; =\; \{vin\; V\; mid\; omega(v,w)\; =\; 0\; mbox\{\; for\; all\; \}\; win\; W\}.$The symplectic complement satisfies:$(W^\{perp\})^\{perp\}\; =\; W$and:$dim\; W\; +\; dim\; W^perp\; =\; dim\; V.$However, unlikeorthogonal complement s, "W"^{⊥}∩ "W" need not be 0. We distinguish four cases:*"W" is

**symplectic**if "W"^{⊥}∩ "W" = {0}. This is trueif 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 "W" ⊆ "W"^{⊥}. This is true if and only if ω restricts to 0 on "W". Any one-dimensional subspace is isotropic.

*"W" is**coisotropic**if "W"^{⊥}⊆ "W". "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. Anycodimension -one subspace is coisotropic.

*"W" is**Lagrangian**if "W" = "W"^{⊥}. 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**^{2"n"}above,

* the subspace spanned by {"x"_{1}, "y"_{1}} is symplectic

* the subspace spanned by {"x"_{1}, "x"_{2}} is isotropic

* the subspace spanned by {"x"_{1}, "x"_{2}, …, "x"_{"n"}, "y"_{1}} is coisotropic

* the subspace spanned by {"x"_{1}, "x"_{2}, …, "x"_{"n"}} is Lagrangian.**Properties**Note that the symplectic form resembles the

canonical commutation relation s. As a result, the additive group of a symplectic vector space has a central extension, this central extension is theHeisenberg group .**ee also*** A

symplectic manifold is asmooth manifold with a smoothly-varying "closed" symplectic form on eachtangent space

*Maslov index

* Asymplectic representation is agroup representation where each group element acts as a symplectic transformation.**References***

Ralph Abraham and Jarrold E. Marsden, "Foundations of Mechanics", (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X "See chapter 3".

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