QuantumPhysics.dvi

Since, by assumption,ghas no time dependence, this requires [g,H]|φ(t)〉= 0 for all states

|φ(t)〉, which can only happen is

[g,H] = 0 ⇒

dg

dt

= 0 (9.18)

in other words,gmust be conserved in time.

The derivation in the Heisenberg picture is analogous. Any observable A(t) satisfying

the Heisenberg equation is transformed into an observableg†A(t)gwhich must also satisfy

the Heisenberg equation, so that

i ̄h

∂

∂t

A(t) = [A(t),H]

i ̄h

∂

∂t

(

g†A(t)g

)

= [gA(t)g†,H] (9.19)

Sinceghas no explicit time dependence, this requires

[

A,g[g†,H]

]

= 0 (9.20)

which in turn again requires that [g,H] = 0. It is possible to extend this result to the case

wheregdoes have explicit time dependence as well; the correct equation is then that ̇g= 0.

Continuous symmetries allow us to consider infinitesimal symmetry transformations.

Sincegis unitary and parametrized by a continuous parameter, we may expand the trans-

formation around the identity,

g=I−iεG+O(ε^2 ) (9.21)

whereGmust be a self-adjoint operator, and thus an observable. The transformation rule

on observables is deduced directly from the finite transformation lawA→A′=g†Ag, and

is given by

g†Ag−A=iε[G,A] +O(ε^2 ) (9.22)

One defines the infinitesimal transformation by

δA=i[G,A] (9.23)

The composition of two infinitesimal symmetry transformationsδ 1 andδ 2 is given by their

associated conserved chargesG 1 andG 2 by,

(δ 1 δ 2 −δ 2 δ 1 )A = −[G 1 ,[G 2 ,A]] + [G 2 ,[G 1 ,A]]

= −i[G,A] (9.24)