Quantum Mechanics for Mathematicians

• Φ(φ(x),π(x)): the solution with initial dataφ(x),π(x) att= 0.


• A(α(p)): the solution with initial data att= 0 specified by the com-
  plex functionα(p) on momentum space, related toφ(x),π(x) by equation
  43.11.

Quantization will take these to operatorsΦ(̂φ(x),π(x)),Â(α(p)).
We can also define versions of the above that are distributional objects cor-
responding to taking the functionsφ(x),π(x),α(p) to be delta-functions:

• Φ(x): the distributional solution with initial dataφ(x′) =δ(x′−x),π(x) =
  0. One then writes

Φ(φ(x),0) =

∫

R^3

Φ(x)φ(x′)d^3 x′

• Π(x): the distributional solution with initial dataφ(x′) = 0,π(x′) =
  δ(x′−x). One then writes

Φ(0,π(x)) =

∫

R^3

Π(x)π(x′)d^3 x′

• A(p): the distributional solution with initial dataα(p′) =δ(p′−p). One
  then writes
  A(α(p)) =

∫

R^3

A(p)α(p′)d^3 p′

43.2 The symplectic and complex structures onM.

Taking as dual phase spaceMthe space of solutions of the Klein-Gordon equa-
tion, one way to write elements of this space is as pairs of functions (φ,π) on
R^3. The symplectic structure is then given by

Ω((φ 1 ,π 1 ),(φ 2 ,π 2 )) =

∫

R^3

(φ 1 (x)π 2 (x)−π 1 (x)φ 2 (x))d^3 x (43.12)

and the (φ(x),π(x)) can be thought of as pairs of conjugate coordinates anal-
ogous to the pairsqj,pjbut with a continuous indexxinstead of the discrete
indexj. Also by analogy with the finite dimensional case, the symplectic struc-
ture can be written in terms of the distributional fields Φ(x),Π(x) as

{Φ(x),Π(x′)}=δ^3 (x−x′), {Φ(x),Φ(x′)}={Π(x),Π(x′)}= 0 (43.13)

We now have a dual phase spaceMand a symplectic structure on it, so in
principle could quantize using an infinite dimensional version of the Schr ̈odinger
representation, treating the valuesφ(x) as an infinite number of position-like
coordinates, and taking states to be functionals of these coordinates. It is