What's symplectic space?

Let's start with an even dimensional Euclidean space R^2n, and a map from two vectors in that space to the reals

ω: R^2n × R^2n -> R.

It has to be linear in both arguments

ω(au,v) = ω(u,av) = aω(u,v),

be alternating or anti-symmetric

ω(u,v) = -ω(u,v),

and it must be nondegenerate in the sense that for any u we can find a v such that ω(u,v) is nonzero.

en.wikipedia.org/wiki/Symplect

1/

The popular scalar product can be defined in almost the same way, except that it must be symmetric or commutative instead of alternating:

a•b = b•a

Just as the scalar product measures angle and distance in a funny way, ω measures area in a funny way!

Making ω is bilinear is a strong requirement: it implies that, given a basis, ω can be represented as a matrix! Which then has to be skew-symmetric, and nonsingular.

And just like with vector spaces we can do a change of basis.

2/

A Darboux basis is the symplectic substitute for a standard basis. Its elements come in two kinds, let's call them x_i and y_i for now, and let's put them through ω to see what's so simple about them:

ω(x_i,y_j) = -ω(y_j,x_i) = 1
ω(χ_i,x_j) = ω(y_i,y_j) = 0

Remember? A Hamiltonian is an energy function which, if you're lucky, is a sum of potential energy, depending on position q_i, and kinetic energy, depending on momentum p_i...

Symplectic space doesn't come with an interpretetation!

/3

That shouldn't equal to 1 but to δ_ij: the Kronecker delta function, which is 1 only if i=j and 0 otherwise.

Which means ω is always zero except when the arguments correspond, having the same index, but belonging to opposite halfs:

ω(x_i,y_i) = 1
ω(y_i,x_i) = -1

wp> The standard symplectic space is R^2n with the symplectic form given by a nonsingular, skew-symmetric matrix. Typically ω is chosen to be the block matrix

ω =
| 0 -Id |
| Id 0 |

wp> where Id is the n×n identity matrix.

en.wikipedia.org/wiki/Symplect

/4

Argh. That minus sign? It should be negating the other Id block matrix! Tooting maths right in one take is difficult. Well, it shouldn't hurt us too much. I think...

Let's do some more interpretation. The external product or wedge product is ideal to understand the meaning of ω!

It is a nice way to calculate areas spanned by a pairs of vectors. If you look here for details

en.wikipedia.org/wiki/Exterior

I hope you can see that such an area is expressed as a sum of components, of shadows cast on all possible base planes (e.g. the one named e1^e2 spanned by e1 and e2).

Using the basis from earlier we find

ω(u,v) = sum_i u_i ^ v_i

5/

@RefurioAnachro This needs some \(\LaTeX\):

\[
\omega = \begin{vmatrix}
0& I_d \\
-I_d &0
\end{vmatrix}
\]

(click on the timestamp of this toot to see this rendered.)

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