Van der Waerden's Derivation of the Dirac Equation

[arivero]

Sakurai credits B. L. van der Waerden 1932 a pretty derivation of Dirac equation from two-component wave functions. First decompose E^2-p^2=m^2 as

[tex]
(i \hbar {\partial \over \partial x_0} + {\bf \sigma} . i \hbar \nabla)
(i \hbar {\partial \over \partial x_0} - {\bf \sigma} . i \hbar \nabla)
\phi= (mc)^2 \phi
[/tex]

This phi has two components, but it is a second order equation, so another two components are needed (say, the first derivative of phi) to fully specify a solution. Instead, we define

[tex]
\phi^R\equiv{1 \over mc}
(i \hbar {\partial \over \partial x_0} - {\bf \sigma} . i \hbar \nabla) \phi
[/tex]

and [tex]\phi^L\equiv\phi[/tex]. Then we have

[tex]
i \hbar ({\bf \sigma} . \nabla - {\partial \over \partial x_0} ) \phi^L= - m c \phi^R
[/tex]

[tex]
i \hbar (-{\bf \sigma} . \nabla - {\partial \over \partial x_0} ) \phi^R= - m c \phi^L
[/tex]

Now you see the trick. These are the usual left and right handed two-component spinors; if you define

[tex]
\psi=
\begin{pmatrix}{\phi^R + \phi^L \cr \phi^R - \phi^L}
\end{pmatrix}
[/tex]

then the equation for the four component spinor [tex]\psi[/tex] is just Dirac equation!

 

[arivero]

Why have I started this thread here, instead of at the Quantum Mechanics subforum? Because it shows what is going on in the "non commutative geometry", or "discrete 5th dimension", models of elementary particles. Weyl spinors live in two parallell sheets of space-time, and the Dirac operator connects both sheets.

EDITED: you can see how geometric is Dirac equation if you put [tex]L_0 \equiv \hbar / m c[/tex]. In this way our pair of equations become
[tex]
i L_0 ( \vec \sigma \cdot \vec \nabla - {\partial \over \partial x_0} ) \phi^L= - \phi^R
[/tex]
[tex]
i L_0 (- \vec \sigma \cdot \vec \nabla - {\partial \over \partial x_0} ) \phi^R= - \phi^L
[/tex]

We have thus a purely geometrical game, jumping across an Euclidean operator and a Minkowskian one! Planck's constant lives still there because the spinors have [tex]\hbar /2[/tex] angular momentum... but you need to reintroduce mass if you want to define such momentum, do you?

 

[arivero]

Now, I am intrigued by the following feature: the mass constant, m, does not to need to be the same when going from R to L that when going from L to R. Thus v. d. Waerden's equation seems to be a bit more general than Dirac's (EDITED: how general it is, has has been studied by Dvoeglazov). Moreover, we could want to treat us with two or three generations in the same multiplet, by generalising m to be a mass matrix M. Again, Waerden's seems to be more general than Dirac's.

 

[dextercioby]

Can u post a reference to van Waerden's original article,please?

Daniel.

 

[Hans de Vries]

Can u post a reference to van Waerden's original article,please?

Daniel.

It's from his book: B.L. van der Waerden, Gruppentheoretische Metode in der
Quanten Mechanik (Springer, Berlin, 1932), Ch. 13.

Bartel Leendert van der Waerden (1903-1996) is credited with developing
spinor analysis after Paul Ehrenfest (who came up with the name spinor)
suggested this to him.

Regards, Hans

 

[arivero]

It's from his book: B.L. van der Waerden, Gruppentheoretische Metode in der
Quanten Mechanik (Springer, Berlin, 1932), Ch. 13.

Thanks Hans! I only had the Sakurai reference (equations 3.24 to 3.29 of Advanced Quantum Mechanics). I should have expected that you have read every occurence of [tex]\hbar/mc[/tex] in the literature !

It could exist a more recent revised English translation:
Group Theory and Quantum Mechanics (Springer, Berlin, 1974)

 

[arivero]

Bartel Leendert van der Waerden (1903-1996) is credited with developing
spinor analysis after Paul Ehrenfest (who came up with the name spinor)
suggested this to him.

I am not familiar with the dotted/undotted notation for spinors, but it seems that Landau Relativistic Quantum Theory (er, the one which is not from Landau, but BLP) also follows van der Waerden derivation.

The book (BLP) explicitly says that there is not point on defining different masses m1, m2, because they can be absorbed in redefinitions of the spinor fields. But they do not consider the case of two noncommuting mass matrices M1, M2, which could be used to give different mass to L and R spinors (one can always put [M1,M2]=O(c) so that in the nonrelativistic limit both masses become equal).