I don’t much care for breezy optimism, in higher math nor anywhere else; but this email is from a dear friend (whose first language was not English) and fine mathematician, so I suppose I ought to post it:
“Hilbert space still allows you to hear. Hilbert space as defined by Kolmogorov and Fomin’s textbook on functional analysis contains this example:
“All the infinite sequences of complex numbers c_n, 1 <= n <= infinity, on assumption that the sum of their squares of moduli is __convergent__ !
“It means that the actual vectors of Hilbert space are ‘concentrated’ in the finite numbers ‘n’, so that the fraction (energy, probablility) related to very large ‘n’ is negligibly small.
“So, while the Hilbert space is formally infinitely-dimensional, real life in them plays out in some finite number of dimensions. The latter may grow from one problem to the other.
“Never in my professionl life have I encounered the case where essential physics required more than one million of dimensions (actually 1000 with respect to x-coordinate and 1000 with respect to y-coordinate, while evolution with respect to z-coordinate and with respect to time was governed by the wave equation itself). Even then the main ‘energy, probability’ was concentrated in around 100 x 100 = 10,000 really strong components, while the rest of the components served for the slight refinement of interference phenomena.
“Do not make formal sign of infinity lure you into despair. Life is short in time and in space, including Hilbert space!”
[Derb] Layman’s translation: If a tree falls in the forest, in a universe of infinitely many spatial dimensions (as opposed to our niggardly three), it _will_ make a noise sufficient to wake any hamsters napping in the vicinity.
I’m sure that we are all glad to know that a million dimensions suffices for all practical purposes.