Quantum Mechanics for Mathematicians

It will also give a representation ofU(1) on states, with the state space de-
composing into sectors each labeled by the integer eigenvalue of the operator
Q̂.
Instead of starting in momentum space with solutions given byA(p),B(p),
we could instead have considered position space initial data and distributional
fields

Φ(x) =

1

√

2

(Φ 1 (x) +iΦ 2 (x)), Π(x) =

1

√

2

(Π 1 (x)−iΠ 2 (x)) (44.3)

and their complex conjugatesΦ(x),Π(x). The Poisson bracket relations on such
complex fields will be

{Φ(x),Φ(x′)}={Π(x),Π(x′)}={Φ(x),Π(x′)}={Φ(x),Π(x′)}= 0

{Φ(x),Π(x′)}={Φ(x),Π(x′)}=δ(x−x′)

and the classical Hamiltonian is

h=

∫

R^3

(|Π|^2 +|∇Φ|^2 +m^2 |Φ|^2 )d^3 x

The charge functionQwould be given by

Q=−i

∫

R^3

(Π(x)Φ(x)−Π(x)Φ(x))d^3 x (44.4)

satisfying
{Q,Φ(x)}=iΦ(x), {Q,Φ(x)}=−iΦ(x)

44.2 Poincar ́e symmetry and scalar fields

Returning to the case of a single real relativistic field, the dual phase space
Mcarries an action of the Poincar ́e groupP, and the quantum field theory
will come with a unitary representation of this group, in much the same way
that the non-relativistic case came with a representation of the Euclidean group
E(3) (see section 38.3). The Poincar ́e group acts on the space of solutions to
the Klein-Gordon equation since its action on functions on space-time commutes
with the Casimir operator

P^2 =

∂^2

∂t^2

−

∂^2

∂x^21

−

∂^2

∂x^22

−

∂^2

∂x^23

This Poincar ́e group action on Klein-Gordon solutions is by the usual action
on functions
φ→u(a,Λ)φ=φ(Λ−^1 (x−a)) (44.5)

induced from the group action on Minkowski space. On fields Φ(x) the action
is
Φ→u(a,Λ)Φ = Φ(Λx+a)