# Dirac equation

OK, that's true, but you still don't need to know the entire history of the particles. You just need to know about at the time(s) at which their worldlines intersect the light cones of the point.
You didn't read the remainder of my post.

Or are there many other problems/difficulties/unknowns introduced by the Dirac solution ?
Because of the very high importance of the Dirac equation, I think you're certainly justified in asking to be presented with a higher level of rigor and views from various perspectives. Here are some of the less subtle difficulties presented by several authors:

1. "The Lorentz group, being 'non-compact', has no faithful, finite-dimensional representations that are unitary" Peskin and Schroeder p. 41. The Dirac equation is Lorentz invariant under boosts (expect for the mass term) and so lacks a unitary representation. Thaller (below) adds further detail on p. 386 that that applies to 4 dimensional complex number space $\mathbb C^4$ but not Hilbert space $\mathbb L^2(\mathbb R^3)^4$.

2. "In the presence of external fields, the Dirac equation cannot be invariant under Lorentz transformations" B. Thaller "Advanced Visual Quantum Mechanics" p. 377.

3. "The Klein paradox occurs for a high one-dimensional electrostatic potential step" Thaller p. 395.

4. The rotation symmetry of the Dirac equation is not complete or correct. Calculations for the magnetic field for directions not orthogonal to the plane of the trajectory fail to match experimental results. L. de Broglie "L 'electron magnetique" 1934 p. 138.

5. "The most important failure of the model seems to be that the magnitude of the resultant orbital angular momentum of an electron moving in an orbit in a central field of force is not a constant, as the model leads one to expect." P. A. M. Dirac "The Quantum Theory of the Electron" 1928 p. 610.

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dextercioby
Homework Helper
I'm sure I've adressed this issue before. The so-called Dirac Lagrangian which produces the Dirac eqns. for the spinor and its adjoint is an artefact, because the "objects" (mathematical items) Psi and Psibar are purely quantum theoretical without any classical counterparts, while for example the g_{ab} from the Hilbert-Einstein action is a purely classical object. This artefact is known to be useful if one is using the path integral approach to the quantum Dirac field, because this method requires the existence of a <classical> action/Lagrangian. So we simply 'manufacture' this Lagrangian and make the "objects" in it anticommutative, instead of commutative, anticommutativity being a feature without any connection to classical physics.

Lagrangian and make the "objects" in it anticommutative, instead of commutative, anticommutativity being a feature without any connection to classical physics.
I'm not sure I quite agree with that. I recently found that the 3 x 3 matrix representation of the curl operation (harkening all the way back to J. C. Maxwell) anti-commutes with the vector or diagonal vector-matrix it's applied to.

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I'm sure I've adressed this issue before. The so-called Dirac Lagrangian which produces the Dirac eqns. for the spinor and its adjoint is an artefact, because the "objects" (mathematical items) Psi and Psibar are purely quantum theoretical without any classical counterparts, while for example the g_{ab} from the Hilbert-Einstein action is a purely classical object. This artefact is known to be useful if one is using the path integral approach to the quantum Dirac field, because this method requires the existence of a <classical> action/Lagrangian. So we simply 'manufacture' this Lagrangian and make the "objects" in it anticommutative, instead of commutative, anticommutativity being a feature without any connection to classical physics.
With all due respect, I did not see any solid arguments supporting your point of view in your previous posts, and I don't see such arguments in your latest post. You just say without any proof (or maybe a reference to such proof) that "Psi and Psibar are purely quantum theoretical without any classical counterparts", although, as I said, one can treat the Dirac equation as an equation describing a classical field. The (non-second-quantized) Dirac equation is not an ideal theory, but it is a damn good theory. Your opinion is just your opinion, not a fact.

Jano L.
Gold Member
...one can treat the Dirac equation as an equation describing a classical field.
But what do you mean by " classical field " ? Psi is a complex-valued four-component function. There is no such field in classical physics. Why call it classical?

Perhaps it would be better to agree on some less ear-earing expression, say "Psi is a c - number field quantity."

But what do you mean by " classical field " ?
The same as the author of the book I quoted in post 28 in this thread (and I can assure you , this is a pretty standard parlance): "This chapter is devoted to a brief summary of classical field theory. The reader should not be surprised by such formulations as, for example, the ‘classical’ Dirac field which describes the spin ½ particle. Our ultimate goal is quantum field theory and our classical fields in this chapter are not necessarily only the fields which describe the classical forces observed in Nature."

Jano L. said:
Psi is a complex-valued four-component function. There is no such field in classical physics. Why call it classical?
So why the real four-component potential of electromagnetic field can be regarded as classical, but the complex-valued four-component function of the Dirac equation cannot?

Furthermore, in a general case (if a certain linear function of electromagnetic field does not vanish identically), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component. Moreover, this remaining component can be made real by a gauge transform. (Journal of Mathematical Physics, 52, 082303 (2011) (http://jmp.aip.org/resource/1/jmapaq/v52/i8/p082303_s1 [Broken] or http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf )). So you may replace Psi by just one real function. As for "no such field in classical physics"... My feeling is this argument has only historical significance nowadays.

Jano L. said:
Perhaps it would be better to agree on some less ear-earing expression, say "Psi is a c - number field quantity."
Your expression is certainly OK, but that does not mean any other expression is wrong. However, the phrase that I questioned - "one can build classical theories of electrons, but not with spinors", does not seem correct to me.

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Jano L.
Gold Member
So why the real four-component potential of electromagnetic field can be regarded as classical, but the complex-valued four-component function of the Dirac equation cannot?
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.

I recently found that the 3 x 3 matrix representation of the curl operation (harkening all the way back to J. C. Maxwell) anti-commutes with the vector or diagonal vector-matrix it's applied to.
I have to correct myself on this. The relationship is actually $\mathsf A \mathsf B + \mathsf B^t \mathsf A = 0$

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
Homework Helper
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).

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).

dextercioby said:
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?