I “grew up” in mathematical sense learning about Fock spaces and some nice things one can do with them. There always was a mysterious physical term “second quantization” attached to the subject (wikipedia). Recently I was reading Felix Berezin’s lecture notes from 1966-67; they are called “Lectures on Statistical Physics”, in English: translated from the Russian and edited by Dimitry Leites), in Russian: MCCME 2008. The full English text can be readily found.
Chapter 25 of these lecture notes contains a clear and historic description of second quantization, in a unified way for Bosons and Fermions. Let me briefly record how this is done.
Personal remark.
Mainly I learned material about Fock spaces directly from my advisor, together with reading Okounkov’s papers (1, 2, 3, 4; the last one is together with Reshetikhin), attending Neretin’s graduate-level lectures at MSU, and writing something on my own (1, 2). With these modern references in mind, I was surprised to find that the understanding of the second quantization dates back (in almost the same words) to at least mid-1960s.
The first quantization refers to the usual quantization of classical mechanics, and the result of this process is the quantum mechanics. This wikipedia article contains lots of details on history and postulates of quantum mechanics.
I think of the first quantization as of a process of replacing a classical particle (having coordinate and momentum) by an element \(f\) in a complex Hilbert space \(\mathcal{L}\). Usually, as \(\mathcal{L}\) one takes \(L^2(\mathbb{R}^{d})\), and then the function \(|f|^{2}\) is the probability density of the position/momentum of the particle.
The coordinate and momentum are replaced by operators in this Hilbert space \(\mathcal{L}\) (these operators do not commute).
Thus, the quantization is the process of replacing classical physical quantities by noncommutative operators.
We will also need the following remark:
Remark.
Operators in \(\mathcal{L}=L^2(\mathbb{R}^{d})\) can be written in an integral form with a kernel \(K(\xi\mid\eta)\), \(\xi,\eta\in\mathbb{R}^{d}\):
\[\displaystyle (Kf)(\xi)=\int_{\mathbb{R}^{d}} K(\xi\mid\eta) f(\eta)d\eta.\]
The second quantization procedure takes the single-particle space \(\mathcal{L}\), and lifts it to a Fock space, fermionic or bosonic.
The Fock space \(\mathcal{H}\) is a Hilbert space consisting of functions depending on zero-particle, single-particle, two-particle, three-particle, etc., configurations. That is, an element of \(\mathcal{H}=\overline{\bigoplus_{n=0}^{\infty} S_n^{\pm} \mathcal{L}}\) represents a multi-particle configuration (state) in the same sense as an element of \(\mathcal{L}\) represents a single-particle configuration. Here \(S_n=S_n^+\) is the symmetrization (for Bosons) and \(S_n=S_n^-\) is the anti-symmetrization (for Fermions), and the bar in the definition of \(\mathcal{H}\) means Hilbert completion.
In the Fock space there are creation and annihilation operators \(a^{*}(f)\) and \(a(f)\), respectively, which are defined for any \(f\in\mathcal{L}\). Up to functional-analytic technicalities, these operators must satisfy the following two conditions:
The operator \(a^*(f^*)\) is conjugate to \(a(f)\) (here \(f^*\) is the complex conjugate function).
The creation and annihilation operators satisfy
in the fermionic case:
\(a(f)a(g)+a(g)a(f)=a^*(f)a^*(g)+a^*(g)a^*(f)=0\),
\(a(f)a^*(g)+a^*(g)a(f)=(f,g^*),\) where \((f,g^*)\) is the inner product in \(\mathcal{L}\).
in the bosonic case:
\(a(f)a(g)-a(g)a(f)=a^*(f)a^*(g)-a^*(g)a^*(f)=0\),
\(a(f)a^*(g)-a^*(g)a(f)=(f,g^*),\) where \((f,g^*)\) is the inner product in \(\mathcal{L}\).
When \(\mathcal{L}=L^2(\mathbb{R}^{d})\), it is natural to write the creation and annihilation operators as integral operators
\[a(f)=\int_{\mathbb{R}^{d}} f(\xi)a(\xi)d\xi,\qquad a^*(f)=\int_{\mathbb{R}^{d}} f(\xi)a^*(\xi)d\xi.\]Here \(a(\xi)\), \(a^*(\xi)\) are the “delta-versions” (i.e., generalized function versions) of the creation and annihilation operators. They satisfy (e.g., for the fermionic case):
\[a(\xi)a(\eta)+a(\eta)a(\xi) =a^*(\xi)a^*(\eta)+a^*(\eta)a^*(\xi)=0, \qquad a(\xi)a^*(\eta)+a^*(\eta)a(\xi)=\delta(\xi-\eta).\]Having an integral operator \(K\) with kernel \(K(\xi\mid \eta)\) in the single-particle space \(\mathcal{L}=L^2(\mathbb{R}^{d})\) (see Remark above), one can define the corresponding operator \(\tilde K\) in the Fock space \(\mathcal{H}\) as follows:
\[\displaystyle \tilde K:=\int_{\mathbb{R}^{d}} \int_{\mathbb{R}^{d}} K(\xi\mid \eta)a^*(\xi)a(\eta)d\xi d\eta.\]Since this formula resembles the following formula in \(\mathcal{L}=L^2(\mathbb{R}^{d})\)
\[\displaystyle (K f,f)=\int_{\mathbb{R}^{d}} \int_{\mathbb{R}^{d}} f(\xi)f^*(\eta)d\xi d\eta,\]the term second quantization was invented to call the operator \(\tilde K\) in the Fock space. This quantization is “second”, because an operator \(K\) in \(\mathcal{L}=L^2(\mathbb{R}^{d})\) itself can come from a “first” quantization of a classical physical quantity.
Example.
If the operator \(K\) in \(\mathcal{L}=L^2(\mathbb{R}^{d})\) is the identity operator, that is, its kernel is \(K(\xi\mid\eta)=\delta(\xi-\eta)\), then its second quantization \(\tilde K\) is the particle number operator
\[\displaystyle N=\int_{\mathbb{R}^{d}}a^*(\xi)a(\xi)d\xi\]Informally: for each point \(\xi\) in the space \(\mathbb{R}^{d}\), annihilate a particle at this point (if there is no particle, this gives zero). Then create a particle at the same point. After that, sum over all \(\xi\in\mathbb{R}^{d}\).