Geometric Interpretation of Bargmann-Wigner Equation

[Markus Hanke]

Hi all,

Consider the Bargmann-Wigner equation, which is the relativistic wave equation for particles of arbitrary spin (i.e. valid for both bosons and fermions). This is a set of coupled differential equations for the components of the wave function, which is an object the type of which is in turn a representation of the Lorentz group. So for example, this would be a bispinor for the Dirac field, or a Rarita-Schwinger spinor for spin-3/2 particles, and so on. This equation can be generalised to curved spacetime backgrounds.

I’ve struggled for a long time with the attempt to come up with some kind of geometric interpretation for this equation (or even just for the Dirac equation), in the same way as one can find geometric interpretations for the GR equations. Thus far unsuccessfully. Does anyone here know if such a geometric interpretation exists? Of course I know that it does not necessarily need to exist, but it would be really helpful if it did.

With wave equations being in some sense representations of the Lorentz group, it ought to be possible to somehow bring this back to rotations in spacetime, though I struggle to find an intuitive, visualisable interpretation.

[joigus]

That is a very interesting question. Not that I can be of any great help, but if I wanted to go in that direction I would study very deeply 2-spinor formulation of GR and from there try to relate both languages through torsion perhaps. I assume you're talking about the Dirac equation and its higher-order spinor counterparts without 2nd-quantisation.

[Markus Hanke]

1 hour ago, joigus said:

That is a very interesting question. Not that I can be of any great help, but if I wanted to go in that direction I would study very deeply 2-spinor formulation of GR and from there try to relate both languages through torsion perhaps. I assume you're talking about the Dirac equation and its higher-order spinor counterparts without 2nd-quantisation.

Having a geometric interpretation of the Dirac equation would certainly be a good start, since it is a special case of Bargmann-Wigner for spin-½. There has to be a way to do that, somehow.

[joigus]

38 minutes ago, Markus Hanke said:

Having a geometric interpretation of the Dirac equation would certainly be a good start, since it is a special case of Bargmann-Wigner for spin-½. There has to be a way to do that, somehow.

There's also work by geometers Michael Atiyah, Isidore Singer, and others which could provide some insights. More topology than local geometry, though. They studied the Dirac equation, harmonic spinors, and such.

Atiyah's words are not very encouraging. He said 'I don't know what a spinor is.' And he was a geometer extraordinaire.

What's most confusing to me is that the connection in GR has to do with infinitesimally shifting points. But the connection for spinors shifts field identities, not points.

I'm assuming that for r-rank spinors it's a tensor product version of the same thing.

I don't know what your goal is, but I would keep it as simple as possible. 1-rank spinors, take everything to a Weyl spinor language, and see if I can make sense of connections, metrics, etc.

I don't even know if I'm being remotely helpful.

As a final reflection, I am of the idea that physicists must change the point of view from the geometer's (God's view: you're given the manifold; you can see all) to the cartographer (you're mapping the territory locally).