How is the Dirac field classical before second quantization?

Question

Maybe this is why I can't seem to understand the first thing about QFT. In the article on second quantization, they say that the name shouldn't really be "second quantization", because:

One is not quantizing "again", as the term "second" might suggest; the field that is being quantized is not a Schrödinger wave function that was produced as the result of quantizing a particle, but is a classical field (such as the electromagnetic field or Dirac spinor field), essentially an assembly of coupled oscillators, that was not previously quantized.

WHAT??? I thought the Dirac equation was just the relativistic Schrodinger equation, albeit with the added benefit of spin and relativistic corrections. It still gives discrete energy states, involves complex numbers, etc. etc. Whereas, or so I thought, a classical field is something where you can directly measure the value at any point, like you can measure the electric/magnetic field by putting a stationary/moving charge there.

So this must be why the QFT wavefunction is now a functional of Dirac+EM field configurations, right? But I'm still missing that key conceptual link: how does a given Dirac configuration correspond to a single physical reality (such that it is amenable to quantization!), and not a probability distribution like the Schrodinger wavefunction? How do you measure the Dirac field? Or if you can't, then why doesn't that matter, and how do the structures of QFT connect to experiment?

Answer 1

Just to put it in order, the fields are called classical not because they are directly measurable (electromagnetic vector potential is classical but not measurable, either) but because they are just (c-number) fields, like $$ \psi: \mathbb R^n \rightarrow \mathbb C \qquad \text{or equivalently} \qquad \psi(x) \in \mathbb C \quad \text{for} \quad x \in \mathbb R^n, $$ in opposite to quantum fields which are operator valued $$ \hat \psi(x): \mathcal F \rightarrow \mathcal F $$ for every point $x \in \mathbb R^n$ in space where $\mathcal F$ is the Fock space in which they act.

In other words, classical (Schrödinger, Dirac) wavefunction $\psi$ is an element of a Hilbert space $\mathcal H$ itself, $\psi \in \mathcal H$, whereas a quantum (Schrödinger, Dirac) field $\hat \psi(x)$ is an operator in Fock space $\mathcal F$ (which is mathematically also a Hilbert space).

Regarding first and second quantization, in Hamiltonian mechanics one postulates the Poisson brackets for the position and momentum to become commutators of the position and momentum operators $$ \{q^i, p_j\} = \delta^i_j \qquad \rightarrow \qquad [Q^i, P_j] = i\, \hbar\, \delta^i_j $$ while in field theory one postulates the Poisson brackets of the field and its canonical momentum to become the commutators of the field and its momentum operators $$ \{\phi(x), \pi(y)\} = \delta(x-y) \qquad \rightarrow \qquad [\Phi(x), \Pi(y)] = i\, \hbar\, \delta(x-y). $$ In both cases one gets an algebra of operators and looks for its representations. In the first case it is the Hilbert space $\mathcal H$, in the second the Fock space $\mathcal F$. Elements of $\mathcal H$ are (Schrödinger, Dirac) first quantized wavefunctions $\psi$ which, treated as classical fields, are second quantized to become operators in Fock space $\mathcal F$.

I personally prefer the names quantum mechanics and quantum field theory as quantized versions of classical mechanics and classical field theory, respectively.

Answer 2

Any of these wave equations - Maxwell, Schrödinger, Dirac, squared Dirac, Pauli, Klein-Gordon - are classical field equations. Their solutions are classical fields that behave like representations of the Poincaré group. It is the interpretation of these classical fields that makes the theory 'quantum'. For example, I interpret the electromagnetic potential as giving the expectation value of photon number, energy etc. This is how I deal with photon shot noise.

A quantum field is a linear combination of harmonic oscillator creation/annihilation operators, where the coefficients are taken from a wave function.

Answer 3

I guess there's no wisdom(at least physical one!) in the answers given. I don't care about the mathematical interpretation of a quantity to call it classical or quantum mechanical, but I'm pretty much sure that physics determines the difference between classical and quantum mechanical!

Answers like saying, quantum fields are operator-valued distributions and classical ones are c-numbers are merely arguments and don't mean anything physically meaningful!

Your question is quite wise due to its very honest standpoint and I have also an honest answer for it:

Classical Hamiltonian equations for quantum systems

Here you can see how one can build a classical theory out of quantum mechanical Hamiltonian.

To me, this is precisely how one reaches a classical field theory for Dirac spinors! You can choose your eigenbasis to be position or momentum or whatever you wish and then establish Dirac's lagrangian quite formally!

I guess everybody agrees that the complex root of a probability distribution is for sure a classical field since it assigns a number(complex or real or vector or etc) to every point of space. This is the very definition of a field.

This way one reaches a classical system with an infinite number of degrees of freedom("classical" oscillators attached to every point of space) that can be quantized in the following step(which means the oscillators(field values) will no longer be considered classical). It fills the gap between first and second quantization though I personally think one is always quantizing a classical system at each level, so there's no sense in saying on does a second quantization which has connotations like quantizing an already quantized system which is not the case!

No need for mathematics as far as there are simple physically meaningful interpretations :)