Antiparticles naturally arise when studying the Dirac equation within quantum field theory. Recall that we may expand a Dirac spinor field as a plane wave, namely,
ψ=2∑s=1∫d3p(2π)31√2Ep[bspus(p)eipx+cs†pvs(p)e−ipx]
and similarly for the conjugate field. Notice the appearance of two distinct creation and annihilation operators; these give rise to the electron and positron, the antiparticle.
The Dirac spinor transforms under a representation of the double cover of SL(2,C) which is a reducible representation. Hence we may propose a decomposition or ansatz,
ψ=u(p)e−ipx
where u(p) is a four-component Dirac spinor which may be broken down into a set of two-component spinors known as Weyl spinors (and with a reality condition, Majorana spinors):
u(p)=(√p⋅σξ√p⋅σξ)
for ξ†ξ=1. The antiparticle, a positron, corresponds to a negative frequency solution, namely,
v(p)=(√p⋅ση√p⋅ση)
where ψ=v(p)e+ipx instead. Notice both solutions have positive energy, as
E=∫d3xT00=∫d3xˉψ(m−γi∂i)ψ≥0
(The above expression is obtained by applying Noether's theorem to the spacetime translation symmetry giving rise to energy-momentum tensor.)
Both the electron and positron are fermions, obey the same quantum field theory, and satisfy Fermi-Dirac statistics which - roughly - dictate we quantize the theory using anti-commutation relations rather than commutation relations, otherwise we would obtain a Hamiltonian unbounded from below.
This post imported from StackExchange Physics at 2014-05-04 11:26 (UCT), posted by SE-user JamalS