Dirac spinor and antiparticles

[Lapidus]

An electron field is a superposition of two four-component Dirac spinors, one of them multiplied with a creation operator and an exponential with negative energy, the other multiplied with an annihilation operator and an exponential with positive energy.

So I assume one Dirac spinor creates a particle (electron), the other annihaltes an antiparticle (positron). The conjugated electron field does vice versa.

But then each of these two spinors consists of two Weyl spinors, i.e. each Dirac spinor represents two electrons (up and down or left-chiral and right-chiral) and two positrons (up and down or left-chiral and right-chiral).

So why do we need then two Dirac spinors (a superposition of them) to account for electrons and positrons? How and why do these 8 (2x4) components describe electrons and positrons? Does the Dirac equation restrict and reduce the number of compents somewhat? How?


thank you

 

[The_Duck]

Yes, it's the Dirac equation. In momentum space it is

$(p_\mu \gamma^\mu + m) \psi(p) = 0$.

If you write out the form of the Dirac field in terms of creation and annihilation operators, this gives a condition on the 4-component spinors that multiply the creation and annihilation operators. Namely, $u(p)$ has to satisfy the momentum-space Dirac equation, and $\bar{v}(p)$ has to satisfy its adjoint. For a given $p$, this 4x4 matrix equation only has two solutions (the 4x4 matrix on the left-hand side has two zero eigenvalues). As a result, each of these 4-component spinors only really has two independent components. This gets rid of half of the degrees of freedom. Only 4 remain, as expected.

 

[PRB147]

(1) 4×4 in Dirac equation is the requirement of parity conservation (Lorentz group representation).}
(2) secondly, in momentum space,according to mass-energy relation to which every component of spinor should obey, both u(p) and v(p) have two independent component, not four, as explained by The_Duck
(3) The choice of v(p) usually consider the charge conjugation, see book by Peskin

 

[Lapidus]

Thank you!

A follow-up question. Only the four-component Dirac spinor preserves parity and thus also improper Lorentz transformations. So chiral theories violate (improper) Lorentz invariance. Isn't that somewhat a problem, i.e. does not Nature always and anywhere demand Lorentz invariance to be kept?

 

[andrien]

So chiral theories violate (improper) Lorentz invariance. Isn't that somewhat a problem, i.e. does not Nature always and anywhere demand Lorentz invariance to be kept?

Be sure here lorentz transformation can be build up from infinitesimal ones i.e. parity is not a lorentz transformation.Weyl spinors since refer to a different chirality they don't preserve parity since they are either left handed or right handed. dirac spinor is however parity preserving.

 

[The_Duck]

So chiral theories violate (improper) Lorentz invariance. Isn't that somewhat a problem, i.e. does not Nature always and anywhere demand Lorentz invariance to be kept?

We can't deduce the symmetry group of Nature a priori: Nature can violate parity if she wants to. And apparently she does: parity is not a symmetry of the weak interaction. It's perfectly consistent for a theory to be invariant under all proper Lorentz transformations but not invariant under parity transformations.

As a side note, most people would call the weak interaction "Lorentz invariant" without a second thought, so when people say "Lorentz invariant" they are not necessarily requiring invariance under improper Lorentz transformations.