FOUNDATIONS OF QUANTUM MECHANICS

III. 1. VON NEUMANN’S POSTULATES 43
In absence of superselection rules the entire Hilbert space is one coherent sector. Then the superposition
principle is valid in general, which says that for every two states |ψ⟩ and |ϕ⟩ the linear
combination a|ψ⟩ + b|ϕ⟩, with |a| 2 + |b| 2 = 1, is a state too. Because nature apparently imposes
superselection rules, which can sometimes be derived from symmetries as was first shown by Wick,
Wightman and Wigner (1952), the superposition principle only applies for coherent sectors. Since
superpositions of vectors from different coherent sectors do not correspond to physical states, the
state postulate has to be accordingly reformulated.
As far as composite physical systems are concerned, we say that the system is in an entangled
state iff the state vector is not factorizable, see section II. 5. In the thought experiment of EPR such an
entangled state plays the principal part. Schrödinger (1935b) was the first to show that the occurrence
of entanglement is widespread in quantum mechanics and he considered this to be the cardinal distinction
between classical mechanics and quantum mechanics. In section III. 2 we will further extend
the notion of state.
Ad 2. The question if every self - adjoint operator represents a physical quantity, has, according
to some authors, a negative answer. Wigner, for instance, asked how to measure the quantity corresponding
to the self - adjoint operator P + Q. Another example is a projector which projects on
superpositions of vectors from different coherent sectors, as we saw in Ad 1.
Also the reverse question, whether every physically meaningful quantity is represented by a self -
adjoint operator, is controversial. For some physical quantities which correspond to experimentally
clear measuring procedures, such as ‘time of decay’ in case of a radioactive atom, or the ‘phase’ of
a harmonic oscillator, no associated self - adjoint operator can be found. In later generalizations of
the formalism of quantum mechanics this problem is somewhat relieved by considering more general
mathematical constructions, the so - called positive operator valued measures, which are also capable
of representing physical quantities; see for example A.S. Holevo (1982) or Busch, Grabowski and
Lahti (1995).
Another question is which operator exactly corresponds to which quantity. Again, no commonly
accepted recipe is available here. Generally, one starts with demanding that certain classical quantities
are represented by special operators. It is standard procedure to choose position and momentum to
be these quantities and to require that the corresponding operators satisfy the canonical commutation
relation of Born and Jordan (1925), and Dirac (1925),
[P, Q] := P Q − Q P = − i 11. (III. 5)
Next, a certain ‘quantization prescription’ is chosen which can be used to construct an operator
corresponding to more general physical quantities. Dirac’s mathematical prescription of replacing
Poisson brackets by commutators is famous. Unfortunately, this prescription is inconsistent. The
alternative prescriptions for quantization which have been presented for this purpose, do not mutually
agree. We will not discuss this problem further.
Ad 4. With P ψ = |ψ⟩ ⟨ψ|, as defined in (II. 37), P ai as in (II. 54), and using the relation (II. 56),
the probability of finding a value a i ∈ Spec A, in a measurement of the physical quantity A with