Quantum mechanics is described according to von Neumann, and the theory of symmetry is described according to Wigner. This is to take advantage of the successful description of relativistic particles by Eugene_Paul_Wigner in his famous paper of 1939. Thus, the pure states are given by the unit rays of some separable complex Hilbert space, in which the scalar product will be denoted by

.

Recall that in elementary wave mechanics, the overall phase of a wave-function is not observable. In general quantum mechanics, this idea leads to the postulate that given a vector psi in Hilbert space, all vectors differing from by a complex non-zero multiple (=the ray containing should represent the same state of the system. Geometrically, we say that the relevant space is the set of rays, known as the projective Hilbert space. The interpretation of the scalar product in terms of probability means that, by convention, we need consider only rays of unit length, so Wigner starts with the set of unit rays. Note that the rays do not themselves form a linear space. A vector in a given unit ray might be used to represent the physical state more conveniently than itself, but is ambiguous up to a phase (complex multiple of unit modulus). The transition probability between two rays and can be defined in terms of vector representatives and to be

- (x in Lorentz space = )

- Psi(a,L).Phi(a,L) = Psi.Phi

Because of the sign-change under rotations by 2 pi, Hermitian operators transforming as spin 1/2, 3/2 etc cannot be observables. This shows up as the *univalence superselection rule*: phases between states of spin 0,1,2 etc and those of spin 1/2,3/2 etc., are not observable. This rule is in addition to the non-observability of the overall phase of a state vector.
Concerning the observables, and states |v), we get a
representation U(a,L) of Poincaré group, on integer spin subspaces, and U(a,A) of the inhomogeneous SL(2,C) on half-odd-integer subspaces, which acts according to the following interpretation:

An ensemble corresponding to U(a,L)|v) is to be interpreted with respect to the coordinates in exactly the same way as an ensemble corresponding to |v) is interpreted with respect to the coordinates x; and similarly for the odd subspaces.

The group of space-time translations is commutative, and so the operators can be simultaneously diagonalised. The generators of these groups give us four self-adjoint operators, , j=1,2,3, which transform under the homogeneous group as a four-vector, called the energy-momentum four-vector.

The second part of the zeroth axiom of Wightman is that the representation U(a,A) fulfills the spectral condition - that the simultaneous spectrum of energy-momentum is contained in the forward cone:

- ............... .
- The third part of the axiom is that there is a unique state, represented by a ray in the Hilbert space, which is invariant under the action of the Poincare group. It is called a vacuum.

For each test function f, there exists a set of operators which, together with their adjoints, are defined on a dense subset of the Hilbert state space, containing the vacuum. The fields A are operator valued tempered distributions. The Hilbert state space is spanned by the field polynomials acting on the vacuum (cyclicity condition).

The fields are covariant under the action of Poincaré group, and they transform according to some representation S of the Lorentz group, or SL(2,C) if the spin is not integer:

If the supports of two fields are space-like separated, then the fields either commute or anticommute

Cyclicity of a vacuum, and uniqueness of a vacuum are sometimes considered separately. Also, there is property of asymptotic completeness - that hilbert state space is spanned by the asymptotic spaces and , appearing in the collision S matrix. The other important property of field theory is mass gap which is not required by the axioms - that energy-momentum spectrum has a gap between zero and some positive number.

From these axioms, certain general theorems follow:

- Connection between spin and statistic - fields which transform according to half integer spin anticommute, while those with integer spin commute (axiom W3)
- PCT theorem - there is general symmetry under change of parity, particle-antiparticle reversal and time inversion (none of these symmetries alone exists in nature, as it turns out)

If the theory has a mass gap, i.e. there are no masses between 0 and some constant greater than zero, then vacuum expectation distributions are asymptotically independent in distant regions.

Haag's theorem says that there can be no interaction picture - that we cannot use the Fock space of noninteracting particles as a Hilbert space - in the sense that we would identify Hilbert spaces via field polynomials acting on a vacuum at a certain time.

Currently, there is no proof that these axioms can be satisfied for gauge theories in dimension 4 - Standard model thus has no strict foundations. There is a million dollar prize for a proof that these axioms can be satisfied for gauge theories, with the additional requirement of a mass gap.

Princeton University Press, Landmarks in Mathematics and Physics, 2000.