Quantum Mechanics for Mathematicians

satisfying the equation

−p^20 +p^21 +p^22 +p^23 =m^2

It is not too difficult to see that the stabilizer group of the orbit isK(0, 0 , 0 ,m)=
SO(2,1). This is isomorphic to the groupSL(2,R), and it has no finite dimen-
sional unitary representations. These orbits correspond physically to “tachyons”,
particles that move faster than the speed of light, and there is no known way
to consistently incorporate them in a conventional theory.

42.3.4 The zero orbit

The simplest case where the CasimirP^2 is zero is the trivial case of a point
p= (0, 0 , 0 ,0). This is invariant under the full Lorentz group, so the orbit
O(0, 0 , 0 ,0)is just a single point and the stabilizer groupK(0, 0 , 0 ,0)is the entire
Lorentz groupSO(3,1). For each finite dimensional representation ofSO(3,1),
one gets a corresponding finite dimensional representation of the Poincar ́e group,
with translations acting trivially. These representations are not unitary, so not
usable for our purposes. Note that these representations are not distinguished
by the value of the second CasimirW^2 , which is zero for all of them.

42.3.5 Positive energy null orbits

One hasP^2 = 0 not only for the zero-vector in momentum space, but for a
three dimensional set of energy-momentum vectors, called the null-cone. By
the term “cone” one means that if a vector is in the space, so are all products
of the vector times a positive number. Vectorsp= (p 0 ,p 1 ,p 2 ,p 3 ) are called
“light-like” or “null” when they satisfy

p^2 =−p^20 +p^21 +p^22 +p^23 = 0

One such vector isp= (|p|, 0 , 0 ,|p|) and the orbit of the vector under the action
of the Lorentz group will be the upper half of the full null-cone, the half with
energyp 0 >0, satisfying

p 0 =

√

p^21 +p^22 +p^23

It turns out that the stabilizer groupK|p|, 0 , 0 ,|p|ofp= (|p|, 0 , 0 ,|p|) isE(2),
the Euclidean group of the plane. One way to see this is to use the matrix
representation 42.1 which explicitly gives the action of the Poincar ́e Lie algebra
on Minkowski space vectors, and note that

l 3 ,l 1 +k 2 ,l 2 −k 1

each act trivially on (|p|, 0 , 0 ,|p|).l 3 is the infinitesimal spatial rotation about
the 3-axis. Defining

b 1 =

1

√

2

(l 1 +k 2 ), b 2 =

1

√

2

(l 2 −k 1 )