as well as the same product in the opposite order, and then comparing the
results.
Note that, for the Schr ̈odinger representation, we have
ΓS
(((
0
0
)
,z+ 2π))
= ΓS
(((
0
0
)
,z))
so the representation operators are periodic with period 2πin thez-coordinate.
Some authors choose to define the Heisenberg groupH 3 as notR^2 ⊕R, but
R^2 ×S^1 , building this periodicity automatically into the definition of the group,
rather than the representation.
We have seen that the Fourier transformFtakes the Schr ̈odinger represen-
tation to a unitarily equivalent representation ofH 3 , in terms of functions ofp
(the momentum space representation). The equivalence is given by a change
ΓS(g)→Γ ̃S(g) =FΓS(g)F ̃in the representation operators, with the Plancherel theorem (equation 11.5
ensuring thatFandF ̃=F−^1 are unitary operators.
In typical physics quantum mechanics textbooks, one often sees calculations
made just using the Heisenberg commutation relations, without picking a spe-
cific representation of the operators that satisfy these relations. This turns out
to be justified by the remarkable fact that, for the Heisenberg group, once one
picks the constant with whichZacts, all irreducible representations are unitar-
ily equivalent. By unitarity this constant is−ic,c∈R. We have chosenc= 1,
but other values of c would correspond to different choices of units.
In a sense, the representation theory of the Heisenberg group is very sim-
ple: there’s only one irreducible representation. This is very different from the
theory for even the simplest compact Lie groups (U(1) andSU(2)) which have
an infinity of inequivalent irreducibles labeled by weight or by spin. Represen-
tations of a Heisenberg group will appear in different guises (we’ve seen two,
will see another in the discussion of the harmonic oscillator, and there are yet
others that appear in the theory of theta-functions), but they are all unitarily
equivalent, a statement known as the Stone-von Neumann theorem. Some good
references for this material are [90], and [41]. In depth discussions devoted to
the mathematics of the Heisenberg group and its representations can be found
in [51], [26] and [94].
In these references can be found a proof of the (not difficult)
Theorem.The Schr ̈odinger representationΓSdescribed above is irreducible.
and the much more difficult
Theorem(Stone-von Neumann).Any irreducible representationπof the group
H 3 on a Hilbert space, satisfying
π′(Z) =−i 1is unitarily equivalent to the Schr ̈odinger representation(ΓS,L^2 (R)).