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I have a question for people who also have studied Quantum Field Theory. To quantize the Dirac field is mathematical quite straight forward, at least with canonical quantization. “Just” make the field an operator and apply the correct commutation relationship.

I also understand the quantization of the usual fields of course, like the electromagnetic field (vector fields) and scalar fields. But outside of the mathematical procedure, I do have a problem (with fermions): understanding why we “originally” could classify the Dirac spinor wavefunction into a field. Because originally, coming from OG quantum mechanics, the wavefunction is just the projection of the state vector onto the position eigenstate vector. So, it’s really just the state of the system. And then going to relativistic quantum mechanics, we obviously treat space and time equally, so the spacial coordinates are no longer observables and have no operator, and that is somehow fine (even though it is confusing to why the classical QM even worked, by treating position as observable).

But what I don’t understand at all is that in QFT, we reclassify the wavefunction as a field, and then we (again?) introduce Fock states, which are Hilbert space quantum states as usual. So now we have two different things, the “wavefunction” field, and the Fock states? How does it make sense to do this? How can the wavefunction become a field, and then we get a new type of quantum state vector thing?

It is like the wavefunction and the quantum state became separate things, and maybe it’s due to treating position equally to time, and consequently removing the position eigenstates, leading to the “disconnection” between state vector and the wavefunction, but to be honest I don’t know at all.

And please don’t answer with just “the Dirac field is not the wavefunction, they are different things”. Obviously they are different, but the Dirac field came from the concept of wavefunction, and it follows the Dirac equation, which was the relativistic generalization of normal QM, and originally was classified as a wavefunction. I don’t understand what the logic is behind redefining it to be a field, and still having quantum states, and them being seperate things.

This really confuses me and after year of trying to figure it out, I haven’t really gotten any further, so if anyone here knows, please help.

Elements In Space
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Zethrun
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2 Answers2

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First of all, no doubt you know what a classical field theory is like: just think E&M. In contrast to classical mechanics of discrete objects, where the dynamical variables are something like (position of particle 1), (position of particle 2)..., a field theory have something like (E field at this point in space), (E field at another point in space)... And you get a QFT when you try to quantize the classical field theory.

(A reason why you want a QFT over single particle QM is that particle creation/annihilation comes very natural in QFT.)

Historically you start with the Dirac equation, and it seems to do a good job describing a free electron. Now you want to write down a QFT that describes interacting electrons. A sensible requirement is that the free part of the theory gives particle solutions that behave just like the first-quantized Dirac fermions.

So, conceptually what you do is you invent a field theory (out of thin air, so to speak), and demand that its quantized particle solution takes a certain form. What you end up with is the Dirac QFT. You don't quantize the Dirac wavefunction again (even though it positively looks like you are just doing it).

With the hindsight of renormalization group and all that, a modern view point is that the low-energy effective theory for a spinor object can only take a few different forms, and the Dirac theory is one of the few possibilities. Even if you don't want to take any input from the Dirac equation, we now know that Dirac QFT is pretty much forced on us anyway.

T.P. Ho
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Firstly, we do not need second quantization to describe multiparticle systems - one could do with Slater determinants and permanents, as in the non-relativistic quantum mechanics textbooks... but it is very cumbersome.

The advantage of the second quantization when applied to Dirac and other fields that are classically viewed as "particles", is that it allows to treat them on equal footing with the electromagnetic field, which is classically a wave, and for which second quantization is really first quantization.

Physically second quantization amounts to giving a field particle properties (those which still hold in quantum mechanics) - notably counting particles.

For more discussion see How does quantization arise in quantum mechanics?

Roger V.
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