Classical Dirac equation

Question

In QFT, electromagnetism is represented by the quantum field $\hat{A}_\mu$, and fermions (matter) by the quantum field $\hat\psi$. The same kind of formalism is used for both phenomena, even if the methods of quantization might be somewhat different.

On the other hand - classically we use the classical version of $A_\mu$ to describe the electromagnetic interaction, but to describe classical electrons, one "dispenses with" the Dirac field $\psi$ altogether and uses classical/relativistic mechanics, in which the electron is represented by a trajectory $x^\mu(\tau)$.

If we accept the quantum theory as fundamental, and independend of the existence of any classical approximation - this distinction feels odd.

Why is it that one kind of field remains a field classically, while another type of field becomes a trajectory (or multiple ones)?

So my questions essentially:

• Is there any kind of clear reason for that?

• Is there any situation where treating the Dirac equation as a classical field equation allows one to derive useful/physically true results about classical electrons? Maybe a statistical description?

It is especially the latter in which I am interested in.

Answer 1

This is a well-formulated question. You are using the "reverse logic" in which any classical theory is a limit of the correct quantum theory, but this logic has limitations, and you ask these questions about one of them:

1. At classical level, the electron is a point particle (elementary charge) and no more than that. As of such, according to the tenets of classical electromagnetism in flat or curved spacetime, it could be described by a point of mass m and charge (-e) located at a certain point P of a Lorentzian manifold which is locally parametrized by a set of 4 functions $x^{\mu} (\tau)$. There is no proper field interpretation to give to a massive particle because classical field theory has a clear dichotomy field-particle. In classical field theory, the electron is a source term for the electromagnetic field, thus it enters the Maxwell equations by the 1-form called electric current density $j$. You can say that this $j$ is a "field" (it is mathematically a 4-vector field), for it allows the interaction electron-em field to be written as a Lagrangian density (i.e. a 4-D integration, not a 1-D one as in the case of $x^{\mu}$). This story of classical electromagnetism is neatly described by PAM Dirac in his GR booklet of about 80 pages from 1975.

2. A classical ("dequantized") Dirac field exists only in introductory QFT texts to illustrate the concept of field quantization starting from the "Dirac Lagrangian density". There is a very clear reason why a Dirac field cannot exist classically. The number 4 of fields at each point in spacetime is a consequence of the existence of spin 1/2 (thus a quantum effect) and parity invariance (this is also quantum-based). Indeed, the $j$ in point 1. is $\bar{\psi}\gamma\psi$, but the physical interpretation of each of the 3 terms requires quantum mechanics.

Answer 2

The distinction between quantum fields and classical trajectories arises from the different scales at which we observe and describe physical phenomena. In quantum field theory (QFT), both electromagnetic fields and matter are represented as quantum fields, as you correctly pointed out. However, the classical limit of these fields manifests differently due to their intrinsic properties and the way they interact with the environment.

The electromagnetic field, represented by A^μ in QFT, retains its field-like nature in the classical limit because it is a bosonic field with long-range interactions. This allows for coherent superpositions of many photons, leading to classical wave-like behavior even at macroscopic scales.

On the other hand, the fermionic field ψ, which represents matter particles like electrons, behaves differently in the classical limit. Fermions obey the Pauli exclusion principle, which prevents them from occupying the same quantum state. This property, combined with the process of decoherence, leads to the localization of fermions and the emergence of classical particle-like behavior at macroscopic scales.

The transition from quantum fields to classical trajectories for matter particles is not a fundamental distinction but rather a consequence of our observational scale and the interactions with the environment. As Einstein noted, "Guided by purely formal points of view, H. A. Lorentz was the first to introduce the hypothesis" of treating electrons as rigid bodies with classical trajectories. This approximation works well for many practical purposes but breaks down when quantum effects become significant.

Regarding your specific questions:

The reason for the different classical treatments of electromagnetic and matter fields is rooted in their quantum properties and how they interact with the environment. The coherent nature of bosonic fields allows for classical field descriptions, while the exclusion principle and decoherence lead to particle-like behavior for fermions at classical scales. Treating the Dirac equation as a classical field equation can indeed yield useful results in certain situations. For instance, in the study of plasmas or electron beams, a statistical description based on the classical Dirac field can provide insights into collective behavior and wave-like phenomena of electron ensembles. This approach bridges the gap between fully quantum and purely classical descriptions, allowing for a more nuanced understanding of electron dynamics in intermediate regimes. It's worth noting that the development of quantum electrodynamics (QED) and the standard model of particle physics has shown that all fundamental particles, including electrons, are best described as excitations of quantum fields. The classical trajectory description is an approximation that emerges under certain conditions.