Question

Here's a conceptual question I'd like to clear up. In classical electrodynamics, "waves" lie in Hilbert space, not in real space, right?

Answer 1

Let's see. Aren't they in C^3 x R (which is not a Hilbert space)

Answer 2

I'm not sure I understand. Could you clarify?

Answer 3

I mean that we write the space components x, y, z are complex numbers and time t as a real number. Hence the 4-vector
(x,y,z,t)
is a member of
\[ \mathbb{C}^3 \times \mathbb{R} \]

Answer 4

Well, I got this part. I'm not sure I understand how this isn't a Hilbert space, though.

Answer 5

By definition, a Hilbert space is a complete, COMPLEX inner product space. As the time component is not complex, this vector space isn't a Hilbert space.

But it does have many of the properties of a Hilbert space. However, I'm not sure thinking of it in terms of Hilbert formalism adds much to the situation.

Answer 6

Oh, right. Complex. XD Thanks.