Wightman axioms

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In physics the Wightman axioms are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s but they were first published only in 1964, after Haag-Ruelle scattering theory affirmed their significance.

1 Rationale
2 Axioms
3 Consequences of the axioms
- 3.1 Osterwalder-Schrader reconstruction theorem
4 See also
5 Literature

[edit] Rationale

Basically, the idea of the Wightman axioms is there is a Hilbert space upon which the Poincaré group acts unitarily. In other words, it's the space of finite energy, momentum, angular momentum and center of mass (corresponding to boosts). The Wightman framework does not cover infinite energy states like finite temperature states.

There is also a stability assumption which restricts the spectrum of the four-momentum to the positive light cone (and its boundary). However, this isn't enough to implement locality. For that, we have position dependent operators called quantum fields which form covariant representations of the Poincaré group.

Unfortunately, quantum field theory suffers from ultraviolet problems: the field at a point is not well-defined. To get around this, we introduce the idea of smearing over a test function to tame the UV divergences which arise even in a free field theory.

There is problem with the domain, because we are dealing with unbounded operators here. Unlike local quantum field theory, we will restrict the causal structure of the theory by hand by imposing either commutativity or anticommutativity between spacelike separated fields, instead of deriving it as a theorem. This misses anyons and braid statistics in lower dimensions.

We also postulate the existence of a Poincaré-invariant state called the vacuum and insist it is unique. This doesn't necessarily make the Wightman axioms inappropriate for the case of spontaneous symmetry breaking because we can always restrict ourselves to a superselection sector. We also assume the vacuum is cyclic in the polynomial algebra generated by the smeared fields. This again isn't really that much of a loss of generality because we can think of the Wightman axioms as describing the superselection sector of the vacuum only. However, this assumption does leave out finite energy states like solitons which can't be generated by a polynomial of fields smeared by test functions because a soliton, at least from a field theoretic perspective is a global structure involving topological boundary conditions at infinity. Lastly, there is the primitive causality restriction which states that any polynomial in the smeared fields can be arbitrarily accurately approximated (i.e. is the limit of operators in the weak topology) by polynomials over fields smeared over test functions with support in any open subspace of Minkowski space whose causal closure is the whole Minkowski space itself.

The Wightman framework does not cover effective field theories because there is no limit as to how small the support of a test function can be. i.e. there is no cutoff scale.

The Wightman framework also does not cover gauge theories. Even in Abelian gauge theories conventional approaches start of with a "Hilbert space" (it's not a Hilbert space, but physicists call it a Hilbert space) with an indefinite norm and the physical states and physical operators belong to a cohomology. This obviously is not covered anywhere in the Wightman framework.

[However as shown by Schwinger, Christ and Lee, Gribov, Zwanziger, Van Baal, etc., canonical quantization of gauge theories in Coulomb gauge is possible with an ordinary Hilbert space, and this might be the way to make them fall under the applicability of the axiom systematics]

[edit] Axioms

[edit] W0 (assumptions of relativistic quantum mechanics)

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. See Wigner's classification. 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
$(Ψ,Φ)$ .

In elementary wave mechanics, the overall 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 differing from Ψ by a complex non-zero multiple (the ray containing Ψ) 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 Ψ in a given unit ray Ψ might be used to represent the physical state more conveniently than Ψ itself, but is ambiguous up to a phase (complex multiple of unit modulus). The transition probability between two rays Ψ and Φ can be defined in terms of vector representatives Ψ and Φ to be

$P(\Psi , \Phi) = |\langle \Psi,\Phi\rangle|^2$

and is independent of which representative vectors, Ψ and Φ, are chosen. Wigner postulated that for the transition probability between states to be the same to all observers related by a transformation of special relativity. More generally, he considered the statement that a theory be 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 inhomogeneous Lorentz group). Thus, a is a real Lorentz four-vector representing the change of space-time origin

$x \mapsto x - a$

where x is in the Minkowski space M⁴ and L is a Lorentz transformation, which can be defined as a linear transformation of four-dimensional space-time which preserves the Lorentz distance $c 2 t 2 - x . x$ of every vector $(c t, x)$ . Then the theory is invariant under the Poincaré group if for every ray Ψ of the Hilbert space and every group element (a,L) is given a transformed ray Ψ(a,L) and the transition probability is unchanged by the transformation:

Ψ(a, L).Φ(a, L) = Ψ.Φ

The first theorem of Wigner is that under these conditions, we can express invariance more conveniently in terms of linear or anti-linear operators (indeed, unitary or antiunitary operators); the symmetry operator on the projective space of rays can be lifted to the underlying Hilbert space. This being done for each group element (a, L), we get a family of unitary or antiunitary operators U(a, L) on our Hilbert space, such that the ray Ψ transformed by (a, L) is the same as the ray containing U(a, L) ψ. If we restrict attention to elements of the group connected to the identity, then the anti-unitary case does not occur. Let (a, L) and (b, M) be two Poincaré transformations, and let us denote their group product by (a, L).(b,M); from the physical interpretation we see that the ray containing U(a, L)[U(b, M)]ψ must (for any psi) be the ray containing U((a, L). (b, M))ψ. Therefore these two vectors differ by a phase, which can depend on the two group elements (a, L) and (b, M). These two vectors do not need to be equal, however. Indeed, for particles of spin 1/2, they cannot be equal for all group elements. By further use of arbitrary phase-changes, Wigner showed that the product of the representing unitary operators obeys

$U(a,L)U(b,M)= \pm U( (a,L).(b,M) )$

instead of the group law. For particles of integer spin (pions, photons, gravitons...) one can remove the +/− sign by further phase changes, but for representations of half-odd-spin, we cannot, and that the sign changes discontinuously as we go round any axis by an angle of 2π. We can, however, construct a representation of the covering group of the Poincare group, called the inhomogeneous SL(2,C); this has elements (a, A) where as before, a is a four-vector, but now A is a complex 2 × 2 matrix with unit determinant. We denote the unitary operators we get by U(a, A), and these give us a continuous, unitary and true representation in that the collection of U(a,A) obey the group law of the inhomogeneous SL(2,C).

Because of the sign-change under rotations by 2π, 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 $x^\prime=L^{-1}(x-a)$ 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, $P 0, P j$ , 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:

$P_0\geq 0$ ............... $P_0^2 - P_jP_j\geq 0.$

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 Poincaré group. It is called a vacuum.

[edit] W1 (assumptions on the domain and continuity of the field)

For each test function f, there exists a set of operators $A_1(f),\ldots A_n(f)$ 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).

[edit] W2 (transformation law of the field)

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:

$U(a,L)^{\dagger}A(x)U(a,L)=S(L)A(L^{-1}(x-a)).$

[edit] W3 (local commutativity or microscopic causality)

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 $H i n$ and $H o u t$ , 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.

[edit] Consequences of the axioms

From these axioms, certain general theorems follow:

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)
Connection between spin and statistic — fields which transform according to half integer spin anticommute, while those with integer spin commute (axiom W3) There are actually technical fine details to this theorem. This can be patched up using Klein transformations. See parastatistics. See also the ghosts in BRST.

Arthur Wightman showed that the vacuum expectation value distributions, satisfying certain set of properties which follow from the axioms, are sufficient to reconstruct the field theory — Wightman reconstruction theorem, including the existence of a vacuum state; he did not find the condition on the vacuum expectation values guaranteeing the uniqueness of the vacuum; this condition, the cluster property, was found later by Res Jost, Klaus Hepp, David Ruelle and Othmar Steinmann.

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.