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Formal motivation of using the Dirac equation

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As is well-known, in 1928 Dirac has derived the same name equation by using the requirement of constructing a relativistic covariant equation describing the function $\psi$ with corresponding positive valued conserved density, which is interpreted as the probability density to find the one particle in the given point of space. But, as we clearly understand now, the localization of the particle with the wave function obeying any relativistic wave equation leads to creation of particle-antiparticle pairs, and therefore the one-particle wave function interpretation of $\psi$ fails.

Another reason to assume the Dirac equation is that, allegedly, only it describes the spin one half particle. But this is not correct, since actually the Dirac spinor $\psi$ is the direct sum of the two truly irreducible representations of the (double covering of the) Lorentz (Poincare) group - the two-dimensional spinors obeying the Klein-Gordon equation, each of which corresponds to the spin one half. Therefore, this formal argument also doesn't work.

As for me, I can only find the following formal argument to be true. The two-dimensional representations mentioned above are not invariant under $P, T, C$ transformations: these transformations convert one representation into another one. In order to work with P-, T-, C-invariant theory of free spin one half particles, one need to take the direct sum of these representations.

Does anyone know any other formal argument why do we need to use the Dirac equation? I don't ask about experimental motivation (the Dirac equation gives correct hydrogen atom spectrum corrections, while the KG one doesn't, and so on).

asked Sep 9 in Theoretical Physics by NAME_XXX (1,010 points) [ revision history ]
recategorized Sep 12 by Dilaton

Without experimental motivation, your CPT argument does not make sense either since CPT are "experimental" notions.

2 Answers

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There is a very interesting story here which recently has become very relevant in the theory of topological insulators and Weyl semimetals. It is a version of the chiral anomaly: a single chiral fermion in 3+1D cannot be consistently coupled to a U(1) gauge field. One statement of this theorem is due to Nielsen and Ninomiya and you can read about it here.

You may ask why we care about coupling to a U(1) gauge field. The corresponding current is the particle number. If this current has a 't Hooft anomaly, then it's impossible to give a UV completion where the fermion is created by a local operator. In practice, this means that the other chirality is floating around somewhere, possibly with a very large mass, as in Pauli-Villars regularization, or at a very large momentum, as in Weyl semimetals. So in any case, if you take UV locality as a requirement, the Weyl physics is at most a low energy phenomenon emergent from a Dirac system.

Also I would disagree with Vladimir Kalitvianski assuming that we already agree to be talking about quantum mechanics and say that CPT is a result of unitarity.

answered Sep 11 by Ryan Thorngren (1,605 points) [ no revision ]

@RyanThorngren: If we agree that we deal with physically induced notions (space, time, charge, QM, etc.), then many things are physically induced. In particular, if the Dirac equation gave a wrong dispersion law for a free particle, it would be abandoned.

+ 1 like - 0 dislike

Dirac's equation is a consequence of the representation theory of the Poincare group for spin 1/2, together with the PCT-theorem, which states that a relativistic QFT must be invariant under PCT. Alternatively, one can work out the requirements for a free field and findes the need for the doubling. See Weinberg's QFT book.

answered Sep 12 by Arnold Neumaier (12,355 points) [ no revision ]

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