- using equations 36.15 formally, witha(α) anda†(α) the objects that are
well-defined, for some specified class of functionsα, generallyS(R). In this
case thea(p),a†(p) are often described as “operator-valued distributions”.
The non-zero commutators ofa(p),a†(p) can be written as
[a(p),a†(p′)] =δ(p−p′)a formula that should be interpreted as meaning either a continuum limit of
[a(pj),a†(pk)] =δjkor
[a(α 1 ),a†(α 2 )] =[∫
α 1 (p)a(p)dp,∫
α 2 (p′)a(p′)dp′]
=
∫ ∫
α 1 (p)α 2 (p′)δ(p−p′)dpdp′=
∫
α 1 (p)α 2 (p)dp=〈α 1 ,α 2 〉for some class (e.g.S(R) orL^2 (R)) of functions for which the inner product of
α 1 andα 2 makes sense.
While we have defined here first the quantum theory in terms of a state space
and operators, one could instead start by writing down a classical theory, with
dual phase spaceH 1. This is already a complex vector space with Hermitian
inner product, so we are in the situation described for the finite dimensional
case in section 26.4. We need to apply Bargmann-Fock quantization in the
manner described there, introducing a complex conjugate spaceH 1 , as well
as a symplectic structure and indefinite Hermitian inner product onH 1 ⊕H 1.
Restricted toH 1 the Hermitian inner product will be the given one, and the
symplectic structure will be its imaginary part.
If we denote byA(α)∈ H 1 the solution of the Schr ̈odinger equation with
Fourier transform of initial data given byα, and byA(α)∈H 1 the conjugate
solution of the conjugate Schr ̈odinger equation, the Poisson bracket relations
are then
{A(α 1 ),A(α 2 )}={A(α 1 ),A(α 2 )}= 0, {A(α 1 ),A(α 2 )}=i〈α 1 ,α 2 〉 (36.16)Quantization then takes
A(α)→−ia†(α), A(α)→−ia(α)wherea†(α) anda(α) are given by equations 36.12 and 36.13. This gives a repre-
sentation of the Lie algebra relations 36.16 for an infinite dimensional Heisenberg
Lie algebra.
As with annihilation and creation operators, adopting a notation that for-
mally extends the state space toH 1 =S′(R), we define
A(p) =A(δ(p′−p)), A(p) =A(δ(p′−p))