Wightman's axioms

The Wightman axioms constitute one of the existing approaches to construct a rigorous quantum field theory, which combines relativistic requirements and quantum principles.

W0 (assumptions of relativistic quantum mechanics)

Quantum mechanics was formalized by John von Neumann, while the way in which symmetries are reflected in it is described by a famous theorem of Eugene Paul Wigner. This is to build on Eugene Paul Wigner's successful description of relativistic particles in his famous 1939 paper. Thus, the pure states are given by the rays of some complex separable Hilbert space. (For which the dot product will be denoted <Ψ| Φ>).

In elementary wave mechanics, the total phase of a wave function Ψ is not observable. In general quantum mechanics, this idea leads to the postulate that given a vector Ψ in Hilbert space, all vectors that differ from Ψ by a non-zero complex multiple (ray containing Ψ) must represent the same pure state of the system. Geometrically, we say that the relevant space is the set of rays, known as the Hilbert projective space. The interpretation of the dot product in terms of probability (width) means that, by convention, we need only consider vectors of unit length. Notice that the rays themselves do not form a linear space (but a projective variety). A unit vector Ψ on a given ray Ψ can be used to represent the physical state more conveniently than Ψ itself, although it is ambiguous in phase (complex multiple of unit modulus). The transition probability between two rays Ψ and Φ can be defined in terms of the vector representatives Ψ and Φ by:

P( ,≈ ≈ )=日本語 ,≈ ≈ 日本語2{displaystyle P(PsiPhi)=ёlangle PsiPhi rangle Δ^{2}}

y is independent of which vector representatives of Ψ and Φ are chosen. Wigner postulated that the probability of the transition between the states must be equal for all observers related by a transformation of special relativity. More generally, he considered the statement that a theory is invariant under a group G to be expressed in terms of the invariance of the transition probability between any two rays. The statement postulates that the group acts on the set of rays, that is, on projective space. Let (a, L) be an element of the Poincaré group (the Lorentz inhomogeneous group), thus, a is a tetra-vector Lorentz real representing the change of the origin of space-time

x x− − a{displaystyle xmapsto x-a}

(x in Lorentz space = R⁴)

y L is a Lorentz transformation, which can be defined as a linear transformation of four-dimensional space-time that preserves the distance c²t²- x .Lorentz x of each vector (c t, x). Then the theory is invariant under the Poincaré group if for every ray Ψ of the Hilbert space and every element (a, L) of the group a transformed ray Ψ(a, L) is given, and the transition probability (amplitude) is left unchanged by the transformation:

<Ψ(a, L)| Φ(a, L)> = <Ψ| Φ>

Wigner's first theorem is that, under these conditions, one can more conveniently express invariance in terms of linear or anti-linear operators (in fact, unitary or anti-unitary); the symmetry operator on the ray projective space can be raised to the underlying Hilbert space. Doing this for each element of the group (a, L), we get a family of unitary or anti-unitary operators U(a, L) on our Hilbert space, such that the ray Ψ transformed by (a, L) is equal to the ray containing U(a, L)Ψ. If we restrict attention to the elements of the group connected with the identity, then the anti-unitary case does not occur. Let (a,L) and (b,M) be two Poincaré transformations, and denote their group product as (a, L).(b, M) from the physical interpretation we see that the ray containing U(a, L) [U(b, M)Ψ] (for any Ψ) must be the ray containing U((a, L).(b, M))Ψ. Therefore these two vectors differ by a phase, which can depend on the two elements of the group (a,L) and (b,M). These two vectors need not be equal, however. In fact, for particles with spin 1/2, they cannot be the same for all the elements of the group. By further use of arbitrary phase changes, Wigner proved that the product of the unitary representation operators obeys:

U(a,L)U(b,M)=(+/− − )U((a,L).(b,M)){displaystyle U(a,L)U(b,M)=(+/-)U(a,L)}

instead of the group law. For integer spin particles (pions, photons, gravitons...) one can remove +/- by further phase changes, but for half integer spin representations, we can't, and it changes discontinuously as we rotate around any axis by an angle of 2π. We can, however, construct a covering group representation of the Poincaré group, called SL(2, C) this has elements (a,A) as before, a is a tetra-vector, but now A is a complex 2 by 2 matrix with unit determinant. We denote the unitary operators we obtain by U(a, A), and these give us a continuous, unitary, true representation in which the collection of U(a, A) obeys the group law SL(2, C).

Due to the sign change under 2π rotations, the Hermitian operators that transform like spin 1/2, 3/2 etc. cannot be observable. This is shown as the univalence superselection rules: the phases between spin states 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 total phase of a state vector. Regarding the observables, and the states |v>, we obtain a representation U(a, L) of the Poincaré group, in integer spin subspaces, and U(a, A) of the SL(2, C) on subspaces of the semiinteger, which works according to the following interpretation:

An ensemble corresponding to U(a, L)|v> must be interpreted with respect to the coordinates x' = L^-1(x-a) in exactly the same way that a set corresponding to |v> is interpreted with respect to the x coordinates and similarly for odd subspaces.

The space-time translation group is commutative, so the operators can be simultaneously diagonalized. The generators of these groups give us four self-adjoint operators, P₀, P_j, j=1,2,3, which are transformed under the homogeneous group as a tetra-vector, called the energy-momentum tetra-vector.

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

P0≥ ≥ 0{displaystyle P_{0}geq 0}...... P02− − PjPj≥ ≥ 0{displaystyle P_{0}{2}-P_{j}P_{j}geq 0}.

The third part of the axiom is that there is a unique state, represented by a ray in Hilbert space, which is invariant under the action of the Poincaré group. It's called a void.

W1 (assumptions about the domain and continuity of the field)

for each test function f, there exists a set of operators A₁(f),..., A _n(f) which, together with their adjoints, are defined on a dense subset of the Hilbert space of states, containing the void. The A fields are operator-valued tempered distributions. The Hilbert space of states is generated by field polynomials acting in a vacuum (cyclicity condition).

W2 (field transformation law)

Fields are operators that under the action of the Poincaré group (this would correspond, for example, to a change in the reference system), the operator associated with the field must be transformed according to an S representation of the group of Lorentz (or its universal spanner SL(2, C) if the spin is not an integer):

U(a,.... )† † A(x)U(a,.... )=S(.... )A(.... − − 1(x− − a)){displaystyle U(a,Lambda)^{dagger }A(x)U(a,Lambda)=S(Lambda)A(Lambda ^{-1}(x-a)})}

where:

The pair (a,.... )한 한 P{displaystyle (a,Lambda)in {mathcal {P}}} is an element of the Poincaré group, being a a time-space translation and consuming an element of the Lorentz group.

W3 (local commutativity or microscopic causation)

If the brackets of two fields are separated by a "space type" interval, that is:

fi(x)gj(and)=0  MIL MIL α α β β (xα α − − andα α )(xβ β − − andβ β )≥ ≥ 0i,j한 한 {1,...... ,r!,x,and한 한 R4{displaystyle {begin{matrix}f_{i}(x)g_{j}(y)=0Leftarrow eta _{alpha beta }(x^{alpha}{alpha}{alpha}{alphah}{beta bb}{beta }{geq 0x,j

Then the associated fields commute (if we are dealing with bosonic fields) or anticommutate (if we are dealing with fermionic fields):

[chuckles]φ φ (f),φ φ (g)]± ± =[chuckles]φ φ (f),φ φ † † (g)]± ± =0{displaystyle [phi (f),phi (g)]_{pm }=[phi (f),phi ^{dagger }(g)]_{pm }=0}

Mass gap

The cyclicity of a vacuum, and the uniqueness of a vacuum are sometimes considered separately. Also, there is a property of asymptotic completeness - that the hilbert space of states is generated by the asymptotic spaces Hⁱⁿ and H^out, appearing in the scatter matrix S. The other important property of field theory that is not required by the axioms is mass jump - that the energy-momentum spectrum has a jump between zero and some positive number.

From these axioms, certain general theorems follow:

• The connection between the spine and the statistic - the fields that are transformed according to the semi-entero thorn anticommutes, while the entire thorn commutes (Axiom W3).
• The theorem CPT - there is a general symmetry under change of parity, of reversal of the anti-particles and of the investment of time (no such symmetries exist separately in nature, as experimentally has been verified)

Arthur Wightman showed that the expectation value distributions of the vacuum, satisfying a certain set of properties that follow from the axioms, are sufficient to reconstruct field theory - Wightman's reconstruction theorem, including the existence of a state of emptiness; he did not find the condition of the expectation values of the void that guaranteed the uniqueness of the void; this condition, the cluster property, was found later by Jost, Hepp, Ruelle, and Steinmann.

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

Haag's theorem says that there can be no image of interaction, that we cannot use the Fock space of non-interacting particles as a Hilbert space - in the sense that we would identify Hilbert spaces via field polynomials acting on a empty at some point.

Currently, there is no proof that these axioms can be satisfied for gauge theories in four dimensions, so the standard model is not definitively supported. There is a million dollar prize for a proof that these axioms can be satisfied for gauge theories, with the additional requirement of jump mass.