Weyl and Dirac spinors

Question

I know that the Dirac spinor is composed of two Weyl spinors and each of the Weyl spinors also has two components. Can I see its two components as two different wave functions? Can I see four different components of the Dirac spinor as four different wave functions? Let´s say that the second component of Weyl spinor is zero. Does this spinor correspond to the wave function which has complex numbers as its values?

Answer 1

Dirac above all proposed the Dirac-equation because it allows to construct a 4-current $j^\mu = \overline{\psi}\gamma^\mu \psi$ whose zero component $j^0 = \overline{\psi}\gamma^\mu \psi =\psi^\dagger \psi $ fulfills the role of a probability density since it is positive-definite. Furthermore $j^\mu$ fulfills the continuity equation $\partial_\mu j^\mu=0$ as it should be for a probability current.

If the zero component of the Dirac probability current $j^0$ is expanded we get :

$\psi^\dagger \psi = (\psi^\ast_1, \psi^\ast_2, \psi^\ast_3, \psi^{\ast}_4) \left(\begin{array}{c} \psi_1 \\ \psi_2 \\ \psi_3 \\ \psi_4 \end{array}\right) $

In the probability density all 4 components of $\psi$ participate, therefore the normalisation of the Dirac- wave function looks like this :

$\int_{-\infty}^\infty d^3 x\, \psi^\dagger \psi = \int_{-\infty}^\infty d^3 x (\psi^\ast_1\psi_1 + \psi^\ast_2\psi_2 + \psi^\ast_3 \psi_3+ \psi^\ast_4 \psi_4 )= 1$

In this respect considering only a subset of components, the first two or the second two components would no longer fulfill the normalisation condition of a wave-function, i.e. it would no longer be a genuine wave-function. This also means that positive frequency (energy) and negative frequency (energy) solutions can mix. For instance wave packet of originally only positive energy solutions can evolve partially into negative energy solutions, in particular if it is localised in an area smaller or equal of the size of the Compton wavelength. This is actually difficult to interpret. Well, one could assume that the forced localisation of the wave packet requires an external field which can apparently lead to the creations of positrons. However, taking into account this observation we no longer deal with an one-particle theory, whereas the concept of a wave-function is actually based on this (more details in Bjorken & Drell,Vol.1, Relativistic Quantum mechanics).

This is by the way one of the reasons why the concept of a wave-function has only a very limited sense in relativistic quantum mechanics, QFT-experts prefer no longer the use of wave-functions in QFT.

So the "Dirac spinor" has indeed properties of a wave-function, but only if the localisation of its wave packets is rather relaxed and the potential energies (bound states) where it is used are small with respect to the energy equivalent of the electron mass.

Another general property of a wave-function is that it contains the complete information of the system. However, using only 2 Weyl-components of it would no longer ensure that this is possible.

EDIT By the way, a particle described by a Weyl-spinor fulfilling the Weyl-equation is massless and therefore not localizable (like the photon). A proper wave function for massless particles actually does not exist.