05.09.2004 20:33
The Dirac-Hamiltonian
HD = c * \alphai pi -m c^2 \beta
has two eigenvalues +E and -E. +E for the particle solution and -E for the
antiparticle solution. Both of them are twofold degenerate to allow for the
two spin states. The helicity operator
h = \sigma p
commutes with HD and
removes the spin degeneracy. So HD and h are
together a complete set of observables that define a "pure" quantum state.
Dirac theory is a relativistic covariant formulation of quantum mechanics,
the energy |E| itself is a scalar and therefore relativistically invariant,
but helicity isn't a good quantum number for particles with mass. Wether
the spin of a particle is parallel or antiparallel to its momentum depends on
the reference frame. (If you pass by the particle by running faster than it, it
changes its helicity. Unless the particle has zero mass it is slower than light and can be bypassed.)
Isn't that somehow odd? (I mean that in a covariant theory a non-covariant
operator takes such a prominent place.) Or is this nonsense?
