One important distinction of the electromagnetic fields versus the Dirac fields is that the equations are second order.
The extension to the many particle case leads to a proliferation of functions akin to the rapid number of increasing spin states for multiple Dirac fields.
The previous notation we used to distinguish coordinate order for the Dirac fields is not available here because of the more complicated index structure and we replace A and C as field labels with [[PSI].
The one additional conserved quantity that Dirac fields have is "norm" associated with the complex global phase freedom.
It should now not be surprising that a similar situation arises for the Dirac fields.
with a + b = 1 and a - b = 0 so a = b = 1/4 but this will turn out not to be the interesting solution for coupling of KG to a positive energy Dirac field.
They include thorough analyses of the data and the emergent theories as they cover quantized Dirac fields
, the standard model, three-generation mixing, neutrino interactions, massive neutrinos, neutrino oscillations in vacuum and the associated theory, neutrino oscillations in matter, solar neutrinos, atmospheric neutrinos, terrestrial neutrino oscillation experiments, phenomenology of three-neutrino mixing, direct measurements of neutrino mass, supernova neutrinos, cosmology, and relic neutrinos.
Initially we need to reconsider some aspects of the particular fields in our study: the metric, electromagnetic and Dirac fields.
The only object directly coupling to the free Dirac fields is [gamma].
In the case of the Dirac field [psi] and the electromagnetic field A each has a set of gauge transformations as free fields.
One way the Dirac field is incorporated into curved spacetime is to fix [[gamma].
In particular, it can be applied to spin-1/2 Dirac fields
, with similar conclusions, though, technically, it is more cumbersome to write down the action for spinors in curved spaces.