Quantum Mechanics for Mathematicians

Requiring that these coordinates behave just like position and momentum co-
ordinates in the finite dimensional case, we can specify the Poisson bracket and
thus the symplectic form by

{Aj(x),Ak(x′)}={Ej(x),Ek(x′)}= 0

{Aj(x),Ek(x′)}=−δjkδ^3 (x−x′) (46.7)

If we then take as Hamiltonian function

h=

1

2

∫

R^3

(|E|^2 +|B|^2 )d^3 x (46.8)

Hamilton’s equations become

∂A

∂t

(t,x) ={A(t,x),h}=−E(x) (46.9)

and
∂E
∂t

(t,x) ={E(t,x),h}

which one can show is just the Maxwell equation 46.3

∂E

∂t

=∇×B

The final Maxwell’s equation, Gauss’s law (46.4), does not appear in the Hamil-
tonian formalism as an equation of motion. In later sections we will see several
different ways of dealing with this problem.
For the Yang-Mills case, in temporal gauge we can again take as initial data
(A(x),−E(x)), where these are now matrix-valued. For the Hamiltonian, we
can use the trace function on matrices and take

h=

1

2

∫

R^3

tr(|E|^2 +|B|^2 )d^3 x (46.10)

since
〈X 1 ,X 2 〉=tr(X 1 X 2 )

is a non-degenerate, positive,SU(2) invariant inner product onsu(2). One of
Hamilton’s equations is then equation 46.9, which is also just the definition of
the Yang-Mills electric field whenA 0 = 0 (see equation 45.3).
The other Hamilton’s equation can be shown to be
∂Ej
∂t

(t,x) ={Ej(t,x),h}

=(∇×B)j−iejkl[Ak,Bl] (46.11)

whereBis the Yang-Mills magnetic field (45.4). If a covariant derivative acting
on fields valued insu(2) is defined by

∇A(·) =∇(·)−ie[A,·]