78 From Classical Mechanics to Quantum Field Theory. A Tutorial
product of the two spaces of phases. In that case, the dimension would be thesum,
rather than the product, of the dimensions of the component spaces.
(b)H 1 ⊗H 2 contains the so-calledentangled states. They are states represented
by vectorsnot factorized asψ 1 ⊗ψ 2 ,buttheyarelinear combinationsof such
vectors. Suppose the whole state is represented by the entangled state
Ψ=1
√
2
(ψa⊗φ+ψa′⊗φ′),where A 1 ψa =aψaandA 1 ψa′ =a′ψa′ for a certain observableA 1 of the part
S 1 of the total system. Performing a measurement ofA 1 onS 1 , due to the col-
lapse of state phenomenon, we automatically act one the whole state and on the
part describingS 2. As a matter of fact, up to normalization, the state of the full
system after the measurement ofA 1 will beψa⊗φif the outcome ofA 1 isa,or
it will beψa′⊗φ′if the outcome ofA 1 isa′. It could happen that the two mea-
surement apparatuses, respectively measuringS 1 andS 2 , are localized very far in
the physical space. Therefore acting onS 1 by measuringA 1 , we “instantaneously”
produce a change ofS 2 which can be seen performing mesurements on it, even if
the measurement apparatus ofS 2 is very far from the one ofS 1 .Thisseemsto
contradict the fundamental relativistic postulate, thelocalitypostulate, that there
is a maximal speed, the one of light, for propagating physical information. After
the famous analysis of Bell, improving the original one by Einstein, Podolsky and
Rosen, the phenomenon has been experimentally observed. Locality is truly vio-
lated, but in a such subtle way which does not allows superluminal propagation of
physical information. Non-locality of QM is nowadays widely accepted as a real
and fundamental feature of Nature[1; 2; 6].
Example 2.1.10.An electron also possesses anelectric charge. That is another
internalquantum observable,Q, with two values±e,wheree<0 is the elementary
electrical charge of an electron. Sothere are two types of electrons.Proper elec-
trons, whose internal state of charge is an eigenvector ofQwith eigenvalue−eand
positrons, whose internal state of charge is a eigenvector ofQwith eigenvaluee.
The simplest version of the internal Hilbert space of the electrical charge is there-
foreHcwhich^1 , again, is isomorphic toC^2. With this representationQ=eσ 3.
The full Hilbert space of an electron must contain a factorHs⊗Hc. Obviously
this is by no means sufficient to describe an electron, since we must introduce at
least the observables describing the position of the particle in the physical space
at rest with a reference (inertial) frame.
(^1) As we shall say later, in view of asuperselection rulenot all normalized vectors ofHcrepresent
(pure) states.