From Classical Mechanics to Quantum Field Theory

158 From Classical Mechanics to Quantum Field Theory. A Tutorial

the standard procedure (which is nothing but the trace procedure with respect to
Tψ:=〈ψ, 〉ψ!)

ρψ(P)=〈ψ,Pψ〉 P∈LR(H).

In this case however, sincePPk=PkPandψk=Pkψkwe have

ρψ(P)=

∑

k

∑

h

ckch〈ψk,PkPPhψk〉=

∑

k

∑

h

ckch〈ψk,PPkPhψk〉

=

∑

k

∑

h

ckch〈ψ,PPkψ〉δkh=

∑

k

|ck|^2 〈ψk,Pψk〉=tr(Tψ′P)

where

Tψ′=

∑

k∈K

|ck|^2 〈ψk,〉ψk.

We conclude that the apparent pure stateψ and the mixed stateTψ′ cannot be
distinguished, just because the algebraRis too small to make a difference. Actually
they define the same state at all and this is an elementary case of (c) in the above
theorem withT 1 =〈ψ, 〉ψandT 2 =Tψ′.
This discussion, in the language of physicist is often stated as follows:
No coherent superpositionsψ=

∑

k∈Kckψkof pure statesψk∈Hkof different
coherent sectors are possible, only incoherent superpositions

∑

k∈K|ck|

(^2) 〈ψk, 〉ψk
are allowed.
(b)It should be clear that the one-to-one correspondence between pure states
and atomic elementary observables (one dimensional projectors) here does not
work. Consequently, notions likeprobability amplitudemust be handled with great
care. In general, however, everything goes right if staying in a fixed superselection
sectorHkwhere the said correspondence exists.
2.3.6 Quantum symmetries: unitary projective representations
The notion of symmetry in QM is quite abstract. Actually there are three dis-
tinct ideas, respectively by Wigner, Kadison and Segal[ 23 ].Herewefocusonthe
first pair only. Physically speaking, asymmetryis an active transformation on
the quantum system changing its state. It is supposed that this transformation
preserves some properties of the physical system and here we have to distinguish
between the two aforementioned cases. However in both cases the transformation
is required to be reversible (injective) and to cover (surjective) the space of the
states. Symmetries are supposed to mathematically describe some concrete trans-
formation acting on the physical system. Sometimes their action, in practice, can
be cancelled by simply changing the reference frame. This is not the general case
however, even if this class of symmetries play a relevant role in physics.