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The original motivation for the Klein-Gordon equation $$(\square + m^2)\varphi = 0$$ and the Dirac equation $$(i\hbar\gamma^\mu\partial_\mu - mc)\psi = 0$$ were such that $\varphi$ and $\psi$ were to be interpreted as wavefunctions. Of course, it was quickly found that $\varphi$ cannot be interpreted as a wavefunction as it failed to represent a probability amplitude.

What I am unclear on is how one goes from viewing $\varphi$ or $\psi$ as wavefunctions to fields. For example, in quantum field theory texts they use the same equations to discuss $\varphi$ and $\psi$ as fields, but what justifies this change of view? Does anything have to be done for $\psi$ to be a Dirac field instead of a Dirac wavefunction?

This change of view happens before any quantization occurs, so I am led to believe this is simply a change of view and there is nothing more to it. Is this correct?

Note I have already read this similar post, but the answer only describes the difference between a wave function and field and not what justifies the change in point of view.

Qmechanic
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CBBAM
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3 Answers3

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There is a fundamental difference between fields (quantum or not) and corresponding wavefunctions.

A wave function has only positive energies when expanding it in modes, a field solution has both positive and negative energies.

This difference has a crucial impact on the notion of localisation. Relativistic wavefunctions contrarily to fields cannot be localised in space at fixed time. This is a source of all difficulties in defining the notion of position for a quantum relativistic particle.

(All that generalises to QFT in curved spacetime even in absence of a preferred notion of energy, but this discussion would be a bit technical and I prefer not to describe it now.)

When passing to second quantization the object to promote to an operator is a field and not a wave function in the relativistic theory.

Valter Moretti
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You are correct that there is no mathematical change in viewing them as classical fields instead of quantum wave functions. It is only our perspective on what they are useful for that changes. They are useful for creating quantum operators, rather than for describing wave functions. When we quantize, we create operators that obey the same equations as the classical field equations but with the additional property that they also obey canonical (anti)commutation relations.

Travis
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It is not just a change of view. Although there are cases where the field and the wave function represent more or less the same thing, in general they represent two different things. Apart from the differences in their formal description in terms of the formalism, there are also conceptual differences. Here, I'll focus on the conceptual aspects.

Starting with fields, one can remind ourselves that classical fields are solutions of the equations of motion. In the quantum case, one finds that the equations of motion are often the same as those for the classical fields. However, we also have the Schroedinger equation for the quantum wave functions. The difference is that, unlike in the classical case, quantum states can consist of multiple excitations (I'm avoiding using the term "particles"). Each excitation can be seen as a solution of the equations of motion and carries a full compliment of all the degrees of freedom. These solutions are the fields, as discussed above. Therefore, wave functions in general consist of multiple fields. The way Weinberg phrased it is to say that wave functions are functionals of fields.

flippiefanus
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