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So, there are a few classes that I'm either taking or sitting in on, for which I'd like to make sure I understand the concepts. I'm going to try to write down the concepts, as I understand them, and possibly a few questions. If you spot anything patently wrong, feel free to shout at me.

QFT
The need for a theory in which the number of (each type of) particle(s) is not constant/fixed leads to second quantization. In the non-relativistic case, where time is treated strictly as a parameter, for each time t we consider the space

A_p = { |N>_p means that N particles have momentum p}; tensor over all possible momenta A_p

Or we can represent it as the space

B_N = {|p_1 p_2 ... p_N>: There are N particles, and particle i has momentum p_i}; direct sum over all numbers of particles B_N

The resultant space, A, is the same regardless of construction. We equip A with particle creation and annihilation operators a_p, b_p = a_p(conjugate), so that the creation operator b_p adds a particle of momentum p, and a_p removes a particle with momentum p, if any, and multiplies by the number of particles with momentum p. Mathematically,

b_p: A_p -> A_p with |N>_p -> |N+1>_p
a_p: A_p -> A_p with |N>_p -> N x |N-1>_p
or
b_p: B_N -> B_{N+1} with |p_1...p_N> -> |p p_1...p_N>
a_p: B_N -> B_{N-1}+{0} with |p_1...p_N> ->
sum over all i{ delta(p-p_i) |p_1...remove p_i...p_N> }

And we extend a_p and b_p to all of A by either the direct sum or tensor properties, depending on representation/construction.

First question: Is it the direct sum or direct product to use in the B_N construction? I know they may have different properties if the index is infinite (such as in this case).

Classically, the free-particle Hamiltonian would be
H = sum over all particles { p^2/2m }
For now, let us assume identical particles (bosons), so that the mass is the same for all of them. However, we would like to quantize this Hamiltonian, so that it now acts on the space A. We note that for the operators a_p, b_p, [a_p,b_p] = 1 and b_p(a_p(f)) = #{particles in the state f with momentum p} f, where f is an arbitrary state in A. Thus our quantization leads us to the non-interacting particle Hamiltonian
H = sum over all momenta { p^2/2m b_p a_p }
Since, when applied to a state, it will return the classical Hamiltonian.

Question 2: Is the space A a Hilbert space, or close to it, like we have in normal QM? How can we relate the states in A with what we classically think of as wavefunctions; i.e. for f in B_N, what would be the collection of wavefunctions psi_1,...psi_N which correspond to it?

We went over how to get from here to time-evolution, and get the second-quantization Schrödinger equation, and introduce interactions, but I'd rather ponder this stuff for now.

Haha, I came up with the same conclusion about direct sum soon after I made the post. Yeah, allowing infinite-particle states would be bad.

As far as the algebraic structure thing goes, Wikipedia claims that this graduated space is called a Fock space. I don't know what Clifford algebras are...the closest analogue I could come up with was a Grassmann algebra.

Wiki:
Technically, the Fock space is the Hilbert space made from the direct sum of tensor products of single-particle Hilbert spaces.

So, they treat my B_N as a tensor product of B_1s, throw in the 0-particle state, and take the direct sum over all of those. Going in this way, the B_1 single-particle state is itself the Hilbert space of normal QM, i.e. the Hilbert space of wavefunctions. So, I guess that answers question 2...but it seems to assume more than I wanted. Still, it's nice to have that connection.

Sometime tomorrow I think I'm going to compile what I got from the second lecture, and put it as an edit to the first post.