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May1-08, 07:03 AM
P: 78
Quote Quote by jdstokes View Post
I believe the Dirac equation is an example of an equation in physics which is essentially an inhomogeneous sum of differential forms: [itex]\gamma^\mu \partial_\mu \psi[/itex] has different degree to [itex]\bar{m}\psi[/itex], yet they appear as a sum.
No, neither of those terms can be represented by a differential form or any sort of sum of differential forms. This is evident from the fact that a differential form of any order is invariant under a rotation about any axis by 2\pi. Both [itex]\gamma^\mu \partial_\mu \psi[/itex] and [itex]\bar{m}\psi[/itex] change sign under such a rotation. The two quantities transform the same way under infinitesimal lorentz transformation, and that is the key consistency test for adding them together.

Spinors take some getting used to, particularly after the dogmatically geometric construction of general relativity. I've gradually come to think of spinors as a generalisation of the complex scalar field. A complex scalar field (such as a Klein-Gordon field) [itex] \phi = |\phi| e^{i\alpha} [/itex] also has quite some ambiguity in its definition: you can redefine it by a complex phase which varies arbitrarily in spacetime. (...equivalent to introducing an electromagnetic potential. This is the electromagnetic "U(1) gauge covariance"). So it's really impossible to give physical meaning to the phase of a complex field, since it can be freely redefined without changing the physical consequences of the theory. In the same way, the complex numbers which comprise a spinor don't necessarily have a direct geometrical meaning, since they can also be redefined locally, according to their symmetry group, SL(2). (...equivalent to introducing a 'spin connection').

I'd be happy to hear if anyone has an opinion on this interpretation...... :-)