Comparing Dirac and Schrodinger Equations for the Hydrogen Atom

[akhmeteli]

The em potential is a part of classical theory, and can be defined in terms of classical physics, say physical quantities rho and j, or E and B. Then it makes no harm to call it classical.

However, as far as I know, Dirac's Psi does not have a definition in terms of classical physics. Can you give one? If so, we could say Psi is a classical field. If not, the use of the word classical in my opinion is seriously hampering correct understanding of both classical and quantum theory.

Dear Jano L.,

If I am "seriously hampering correct understanding of both classical and quantum theory", I can only console myself by the thought that I am in good company:-) For example, if we look at the well-known textbook "Introduction to the theory of quantized fields" by Bogoliubov and Shirkov (I used it to study QFT many years ago), there is a chapter "Classical theory of free fields" there, containing a section "Dirac equation". At the beginning of the chapter, the authors write:" Laying out the theory of classical fields, for the sake of illustration, we will sometimes use notions related to characteristics of the relevant particles (mass, spin, etc.) It should be noted that these notion acquire their full meaning only after quantization.” (my emphasis and translation from the Russian original). So Bogoliubov and Shirkov have no problems calling the Dirac field "classical". Why should we have such problems?

Another thing. As I said, this is pretty standard parlance, and that is not coincidental. It reflects a certain (modern) view, which is quite different from the way the theory developed historically. The following article may be of interest in this respect: http://philsci-archive.pitt.edu/4097/1/Dirac_equation,_quanta_and_interactions.pdf . In particular, the following phrase from the abstract is interesting: "In this article the Dirac equation is used as a guideline to see the historical emergence of the concept of quanta, associated with the quantum field. In P. Jordan’s approach, electrons as quanta result from the quantization of a classical field described by the Dirac equation." And later:"The meaning of the simple looking Dirac equation is not as simple as we might think. Since its first formulation, its meaning has changed from a relativistic wave equation for an electron to a classical field equation from which an electron-positron quantum field is derived"

I could also add, for what it's worth, that in Dirac-Maxwell electrodynamics, Psi can be expressed as a function of a complex electromagnetic 4-potential, which yields the same electromagnetic field as the standard real electromagnetic 4-potential (http://dx.doi.org/10.1088/1742-6596/361/1/012037 and references there), so maybe spinor and electromagnetic fields are closer related or more similar than we tend to think.

 

[dextercioby]

You may adopt your own views and conventions, but there's a clear distinction between classical field theory and quantum field theory. The Dirac equation is an Euler-Lagrange equation for a fictional Lagrangian written with mathematical objects whose physical relevance one finds only through the axioms of quantum field theory and not the axioms of electromagnetism or general relativity (the only 2 classical field theories).

 

[akhmeteli]

You may adopt your own views and conventions, but there's a clear distinction between classical field theory and quantum field theory.

I referred to three sources confirming that these are not just MY "views and conventions" and maybe could mention many more solid sources confirming that (but I am not going to do that).

The Dirac equation is an Euler-Lagrange equation for a fictional Lagrangian written with mathematical objects whose physical relevance one finds only through the axioms of quantum field theory and not the axioms of electromagnetism or general relativity (the only 2 classical field theories).

You, however, just repeat some mantra, without any supporting evidence, and I don't quite understand why one should accept your mantra. For example, I don't even understand why a theory being classical or not depends on whether it has a Lagrangian or not. Why aren't equations of motion enough? As for electromagnetism and general relativity being the only 2 classical field theories, this statement seems not only baseless, but also downright incomprehensible: so, for example, continuum mechanics (say, theory of elasticity or fluid dynamics) is not a classical field theory any more?