There is a lot of people that do not like the name second quantization, mostly because the second quantization is often introduced in a not so clear way, at least in my opinion.

Second quantization has nothing wrong, if you see it as a precisely defined mathematical object: *it is a functor between Hilbert spaces*, that associates to the original one-particle space the suitable (anti)symmetric Fock space, to self adjoint operators $H$ their second quantization $d\Gamma(H)$ and to unitary operators $e^{−itH}$ the second quantization $\Gamma(e^{−itH})=e^{−itd\Gamma(H)}$.

The physical interpretation is quite simple; to a single particle Hilbert space $\mathscr{H}$, it is associated an Hilbert space that contains information about all the spaces with an arbitrary number of particles (the Fock space); to an operator $H$ that acts on a single particle it can be associated an operator [$d\Gamma(H)$] that acts on any space with $n$ particles as a sum of the action of H on each particle, or an operator [$\Gamma(H)$] that acts as the $n$-product of $H$, each one acting on a different particle.

With such second quantization functor however, you do not give meaning to important operators of the theory, e.g. the creation/annihilation operators; for they are not the second quantization of anything (the second quantization functor gives, roughly speaking, only operators on the Fock space that preserve the number of particles).

To introduce annihilation/creation operators one has to use "standard" **quantization**. The meaning is analogous to the quantum mechanical one, simply starting from a classical phase space that is infinite dimensional. So to functionals on that classical phase space are associated operators on a bigger space (the Fock space) almost exactly as in the usual quantization of finite dimensional phase spaces (with problems of ordering, domains of definition and so on).

But if second quantization can be defined mathematically as a functor (without entering too much into details), there is no satisfactory functorialization of the first quantization procedure, even with finite dimensional phase spaces. This is because it is not possible, roughly speaking, to define a quantization procedure that is coherent, for each phase space function, both with the symplectic structure of the classical space (Poisson brackets) and the irreducibility requirements of the Weyl commutation relations. This explains the quote of Edward Nelson that you find in the comments (this does not mean, however, that the quantization procedure is not well-defined mathematically, simply it is not an elegant categorical notion such as a functor).

This post imported from StackExchange Physics at 2015-05-13 18:54 (UTC), posted by SE-user yuggib