Second quantization is quantizing the wave function?

Question

How right or wrong to view second quantization as quantizing the wave function? Here, quantizing the wave function means taking a wave function $\psi(x,t)$ resulting from a first quantized theory and promote it to a operator-valued field $\hat{\psi}(x, t)$.

Many modern texts point out that second quantization is actually bad terminology that is kept only for historical reason. For example, both Altland & Simons' Condensed Matter Field Theory and David Tong's Lecture Notes on QFT explained that we are never quantizing anything twice, and second quantization is quantizing a classical field once and only once.

On the other hand, Coleman's Introduction to Many-body Physics gave a hestusic derivation of second quantization by considering the wave function of a Bose-Einstein condesnsate, dervied from it a Hamilton-like equation hence the second quantizated Hamiltonian, and "second quantized" the condensate wave function to a field operator.

A similar argument could be found in Mahan's Many-Particle Physics. And then, Nastase's Introduction to Quantum Field Theory (which is quite a modern book) claims directly second quantization is quantizing the wave function.

How to reconcile all these conflicting perspective, and what's the modern viewpoint on second quantization?

Answer 1

IMHO, the two views are not really conflicting with each other.

As the other answer has already pointed out, "second quantization" is quantization of fields. This field could be either electromagnetic field, or wave function, the displacement of atoms in a solid, wave in plasma, etc. Mathematically second quantization can be applied to any field, described by an appropriate field equation or equivalent formalism... although in some cases such procedure may have no physical meaning - like applying it to waves in air, which are a continuum approximation for inherently macroscopic variables like pressure and density (i.e., averaged over many particles.)

The term second quantization arises from thinking of fermions - mostly electrons, for which the "first quantization" meant describing them by a wave equation (rather than Newton-like equations), i.e., treating them as fields rather than particles. On the other hand, since electromagnetic field is already a field in classical physics, applying "second quantization" actually means simply quantizing this field (aka "first quantization".)

In this context it is worth mentioning another misleading concept - rarely mentioned explicitly, but often implicit in discussions - that of wave-particle duality. It is wrong to think that electrons, photons, and other particles "exhibit particle or wave properties according to the experimental circumstances" - as one often hears in basic QM courses or popular texts. Indeed, quantizing EM field does not transform photons into particles obeying some kind of Newton-like laws, moving along defined trajectories, or occupying a specific region of space, whatever are "the experimental circumstances". However, it does assign to EM field some properties that a classical field does not possess, while classical particles do - most importantly, photons can be counted (which, e.g., solves the problem of black body radiation.)

See also:
How does quantization arise in quantum mechanics? - a more detailed answer, but written from a different perspective.
How can blackbody radiation be explained by quantization?

Answer 2

How right or wrong to view second quantization as quantizing the wave function?

It is absolutely wrong. "Second quantization" involves the quantization of fields, AKA systems with infinitely many degrees of freedom.