Quantum Mechanics for Mathematicians

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