Symplectic vector space

In mathematics, a symplectic vector space is a vector space "V" equipped with a nondegenerate, 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 a symplectic 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 has determinant 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 the block matrix

: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"^∗ its dual space. Now consider the direct 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 ∈ Λ²("V"). Then ω is non-degenerate if and only if "n" is even, and ω^"n"/2 = ω ∧ … ∧ ω is a volume 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"/2ω^"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 ("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 isomorphisms.

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 a Lie group, called the symplectic 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, unlike orthogonal complements, "W"^⊥ ∩ "W" need not be 0. We distinguish four cases:

*"W" is symplectic if "W"^⊥ ∩ "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 "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. Any codimension-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 relations. As a result, the additive group of a symplectic vector space has a central extension, this central extension is the Heisenberg group.

ee 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

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