QuantumPhysics.dvi

will always generate rotations

δaqℓ={qℓ,La}=

∑

m

εaℓmqm (9.14)

whether or not rotations are actually a symmetry of a certain Lagrangian or not.

9.4 Symmetries in Quantum Physics

In quantum mechanics, the (only) quantities that can be measuredare the eigenvalues of

observables, and the probabilities for one state to overlap with another state. A transfor-

mation in quantum mechanics is a linear (or anti-linear for time reversal) map from the

Hilbert spaceHinto itself. A quantum mechanical symmetry must preserve the outcome

of all observations, and hence must leave all probabilities unchanged. In the case of linear

transformations, we must have the following transformation law onstates,

|φ〉→|φ′〉=g|φ〉 〈ψ′|φ′〉=〈ψ|g†g|φ〉=〈ψ|φ〉

|ψ〉→|ψ′〉=g|ψ〉 g†g=I (9.15)

Thus, to be a symmetry, a linear transformation must be unitary. On observables, the action

ofgis by conjugation,

A→A′=g†Ag (9.16)

Sincegis unitary, the set of all eigenvalues (i.e. the spectrum) ofA′exactly coincides with

the set of all eigenvalues ofA.

From the above point of view, all unitary transformations are candidates to be sym-

metries, as they represent a general change of orthonormal basis in Hilbert space. Certain

authors (see Weinberg, volume I) indeed leave the definition of a symmetry this general.

Once the dynamics of the system is further determined by a Hamiltonian, however, it be-

comes more natural to have a more dynamics-related definition of symmetry. Dynamics may

be formulated either in the Schr ̈odinger or in the Heisenberg pictures, and a symmetry then

operates as follows. In either case, we shall assume that we perform a unitary transformation

gwhich hasno explicit time dependence.

In the Schr ̈odinger picture, any state|φ(t)〉satisfying the Schr ̈odinger equation with the

HamiltonianH, is transformed to a stateg|φ(t)〉which must satisfy the Schr ̈odinger equation

for the same Hamiltonian. As a result, we must have for all states|φ(t)〉,

i ̄h

∂

∂t

|φ(t)〉 = H|φ(t)〉

i ̄h

∂

∂t

(

g|φ(t)〉

)

= H

(

g|φ(t)〉

)

(9.17)