Solutions to this will be distributions that are non-zero only on the hyperboloid
p^20 −p^21 −p^22 −p^23 −m^2 = 0
in energy-momentum spaceR^4. This hyperboloid has two components, with
positive and negative energy
p 0 =±ωp
Ignoring one dimension, these look like
p 0
p 1 p 2
p 0 = 0
(plane)
p 0 = +
√
|p|^2 +m^2
(upper sheet)
p 0 =−
√
|p|^2 +m^2
(lower sheet)
(−m, 0 , 0 ,0)
(m, 0 , 0 ,0)
Figure 43.1: Orbits of energy-momentum vectors (m, 0 , 0 ,0) and (−m, 0 , 0 ,0)
under the Poincar ́e group action.
and are the orbits of the energy-momentum vectors (m, 0 , 0 ,0) and (−m, 0 , 0 ,0)
under the Poincar ́e group action discussed in sections 42.3.1 and 42.3.2.
In the non-relativistic case, a continuous basis of solutions of the free particle
Schr ̈odinger equation labeled byp∈R^3 was given by the functions
e−i
|p|^2
2 mteip·x
with a general solution a superposition of these given by
ψ(x,t) =
1
(2π)^3 /^2
∫
R^3
ψ ̃(p,0)e−i|p|
2
2 mteip·xd^3 p