>>John Baez <email@example.com> wrote at the end:
>>>Quiz question: what mathematical object is utterly fundamental to >>>quantum mechanics and has BOTH an inner product and a symplectic >>>structure? The answer is shockingly simple.
>>Well, the complex field, being a 2 dimensional vector space over the reals, >>certainly has both inner products and symplectic products over the >>reals.
>You got it. The complex numbers C - and thus every space C^n, or more >generally every Hilbert space - has both a real inner product and a >symplectic structure. It's nice to call the real inner product an >"orthogonal structure", so let me do that. The orthogonal structure on >C^n is just the real part of the complex inner product >while the symplectic structure is just the imaginary part of the complex >inner product, and they satisfy Re<v,w> = Re<w,v>, Im<v,w> = -Im<w,v>.
Neat. Reminds me of stuff I've seen in QFT texts, too, as it probably should.
Another way I thought of to look at this:
Any bilinear form <,> gives rise to symmetric and alternating forms: v.w := <v,w> + <w,v> and v^w := <v,w> - <w,v>. Even though the Hilbert space inner product is complex valued, as a consequence of its sesquilinear nature, the symmetric and alternating forms associated with it are real valued, which is what gives us our orthogonal and symplectic structures. (Actually, the alternating form is pure imaginary, but that comes to the same thing.)