Wightman axioms
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In
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, the Wightman axioms (also called Gårding–Wightman axioms), named after
Arthur Wightman Arthur Strong Wightman (March 30, 1922 – January 13, 2013) was an American mathematical physicist. He was one of the founders of the axiomatic approach to quantum field theory, and originated the set of Wightman axioms. With his rigorous treatm ...
, are an attempt at a mathematically rigorous formulation of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
. 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. The axioms exist in the context of constructive quantum field theory and are meant to provide a basis for rigorous treatment of quantum fields and strict foundation for the perturbative methods used. One of the Millennium Problems is to realize the Wightman axioms in the case of Yang–Mills fields.


Rationale

One basic idea of the Wightman axioms is that there is a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, upon which the Poincaré group acts unitarily. In this way, the concepts of energy, momentum, angular momentum and center of mass (corresponding to boosts) are implemented. There is also a stability assumption, which restricts the spectrum of the
four-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
to the positive light cone (and its boundary). However, this isn't enough to implement
locality Locality may refer to: * Locality (association), an association of community regeneration organizations in England * Locality (linguistics) * Locality (settlement) * Suburbs and localities (Australia), in which a locality is a geographic subdivis ...
. For that, the Wightman axioms have position-dependent operators called quantum fields, which form covariant
representations of the Poincaré group ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
. Since quantum field theory suffers from ultraviolet problems, the value of a field at a point is not well-defined. To get around this, the Wightman axioms introduce the idea of smearing over a test function to tame the UV divergences, which arise even in a
free field theory In physics a free field is a field without interactions, which is described by the terms of motion and mass. Description In classical physics, a free field is a field whose equations of motion are given by linear partial differential equati ...
. Because the axioms are dealing with unbounded operators, the domains of the operators have to be specified. The Wightman axioms restrict the causal structure of the theory by imposing either commutativity or anticommutativity between spacelike separated fields. They also postulate the existence of a Poincaré-invariant state called the
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
and demand it to be unique. Moreover, the axioms assume that the vacuum is "cyclic", i.e., that the set of all vectors obtainable by evaluating at the vacuum-state elements of the polynomial algebra generated by the smeared field operators is a dense subset of the whole Hilbert space. 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 in smeared fields over test functions with support in an open set in Minkowski space whose causal closure is the whole Minkowski space.


Axioms


W0 (assumptions of relativistic quantum mechanics)

Quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
is described according to von Neumann; in particular, the pure states are given by the rays, i.e. the one-dimensional subspaces, of some separable complex
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. In the following, the scalar product of Hilbert space vectors Ψ and Φ is denoted by \langle\Psi, \Phi\rangle, and the norm of Ψ is denoted by \lVert\Psi\rVert. The transition probability between two pure states ¨and ¦can be defined in terms of non-zero vector representatives Ψ and Φ to be : P\big(
Psi Psi, PSI or Ψ may refer to: Alphabetic letters * Psi (Greek) (Ψ, ψ), the 23rd letter of the Greek alphabet * Psi (Cyrillic) (Ѱ, ѱ), letter of the early Cyrillic alphabet, adopted from Greek Arts and entertainment * "Psi" as an abbreviation ...
Phi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
big) = \frac and is independent of which representative vectors Ψ and Φ are chosen. 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. Wigner postulated the transition probability between states to be the same to all observers related by a transformation of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
. 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
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
origin ''x'' ↦ ''x'' − ''a'', where ''x'' is in the Minkowski space ''M''4, and ''L'' is a Lorentz transformation, which can be defined as a linear transformation of four-dimensional spacetime preserving the Lorentz distance ''c''2''t''2 − ''x''â‹…''x'' of every vector (''ct'', ''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: : \langle \Psi(a, L), \Phi(a, L) \rangle = \langle\Psi, \Phi\rangle. Wigner's theorem says that under these conditions, the transformation on the Hilbert space are either linear or anti-linear operators (if moreover they preserve the norm, then they are
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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 ; from the physical interpretation we see that the ray containing ''U''(''a'', ''L'') 'U''(''b'', ''M'')ψmust (for any ψ) be the ray containing ''U''((''a'', ''L'')â‹…(''b'', ''M''))ψ (associativity of the group operation). Going back from the rays to the Hilbert space, these two vectors may differ by a phase (and not in norm, because we choose unitary operators), which can depend on the two group elements (''a'', ''L'') and (''b'', ''M''), i.e. we don't have a representation of a group but rather a
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where GL(' ...
. These phases can't always be cancelled by redefining each ''U''(''a''), example for particles of spin 1/2. Wigner showed that the best one can get for Poincare group is : U(a, L) U(b, M) = \pm U\big((a, L) \cdot (b, M)\big), i.e. the phase is a multiple of \pi. 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 the sign changes discontinuously as we go round any axis by an angle of 2Ï€. 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 Ã— 2 matrix with unit determinant. We denote the
unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the con ...
s 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 2Ï€,
Hermitian operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
s 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
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corresponding to ''U''(''a'', ''L''), ''v''⟩ is to be interpreted with respect to the coordinates x' = L^(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 spacetime 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, \quad 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 the Hilbert space, which is invariant under the action of the Poincaré group. It is called a vacuum.


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 Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose de ...
. The Hilbert state space is spanned by the field polynomials acting on the vacuum (cyclicity condition).


W2 (transformation law of the field)

The fields are covariant under the action of Poincaré group and 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\big(L^(x - a)\big).


W3 (local commutativity or microscopic causality)

If the supports of two fields are
space-like In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
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^\text and H^\text, 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.


Consequences of the axioms

From these axioms, certain general theorems follow: *
CPT theorem Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and T ...
— 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 Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
and statistic — fields that 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 transformation In quantum field theory, the Klein transformation is a redefinition of the fields to amend the spin-statistics theorem. Bose–Einstein Suppose φ and χ are fields such that, if ''x'' and ''y'' are spacelike-separated points and ''i'' and ''j' ...
s. See
parastatistics In quantum mechanics and statistical mechanics, parastatistics is one of several alternatives to the better known particle statistics models (Bose–Einstein statistics, Fermi–Dirac statistics and Maxwell–Boltzmann statistics). Other alte ...
and also the ghosts in BRST. * The impossibility of superluminal communication – if two observers are spacelike separated, then the actions of one observer (including both measurements and changes to the Hamiltonian) do not affect the measurement statistics of the other observer.
Arthur Wightman Arthur Strong Wightman (March 30, 1922 – January 13, 2013) was an American mathematical physicist. He was one of the founders of the axiomatic approach to quantum field theory, and originated the set of Wightman axioms. With his rigorous treatm ...
showed that the
vacuum expectation value In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. ...
distributions, satisfying certain set of properties, which follow from the axioms, are sufficient to reconstruct the field theory —
Wightman reconstruction theorem Wightman may refer to: *Andy Wightman, Scottish Green MSP and writer *Arthur Wightman (1922–2013), American theoretical physicist *Brian Wightman (born 1976), Australian politician *Bruce Wightman (1925–2009), actor who co-founded the Dracula ...
, 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 Klaus Hepp (born 11 December 1936) is a German-born Swiss theoretical physicist working mainly in quantum field theory. Hepp studied mathematics and physics at Westfälischen Wilhelms-Universität in Münster and at the Eidgenössischen Technis ...
, David Ruelle and
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. 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 While working on the mathematical physics of an interacting, relativistic, quantum field theory, Rudolf Haag developed an argument against the existence of the interaction picture, a result now commonly known as Haag’s theorem. Haag’s origina ...
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.


Relation to other frameworks and concepts in quantum field theory

The Wightman framework does not cover infinite-energy states like finite-temperature states. Unlike
local quantum field theory The Haag–Kastler axiomatic framework for quantum field theory, introduced by , is an application to local quantum physics of C*-algebra theory. Because of this it is also known as algebraic quantum field theory (AQFT). The axioms are stated in t ...
, the Wightman axioms restrict the causal structure of the theory explicitly by imposing either commutativity or anticommutativity between spacelike separated fields, instead of deriving the causal structure as a theorem. If one considers a generalization of the Wightman axioms to dimensions other than 4, this (anti)commutativity postulate rules out
anyon In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchangi ...
s and
braid statistics In mathematics and theoretical physics, braid statistics is a generalization of the spin statistics of bosons and fermions based on the concept of braid group. While for fermions (Bosons) the corresponding statistics is associated to a phase ...
in lower dimensions. The Wightman postulate of a unique vacuum state 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 In quantum mechanics, superselection extends the concept of selection rules. Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables. It was originall ...
. The cyclicity of the vacuum demanded by the Wightman axioms means that they describe only the superselection sector of the vacuum; again, this is not a great loss of generality. 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. The Wightman framework does not cover
effective field theories In physics, an effective field theory is a type of approximation, or effective theory, for an underlying physical theory, such as a quantum field theory or a statistical mechanics model. An effective field theory includes the appropriate degrees ...
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 off with a "Hilbert space" with an indefinite norm (hence not truly a Hilbert space, which requires a positive-definite norm, but physicists call it a Hilbert space nonetheless), and the physical states and physical operators belong to a
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
. 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.) The Wightman axioms can be rephrased in terms of a state called a
Wightman functional In mathematics, a Borchers algebra or Borchers–Uhlmann algebra or BU-algebra is the tensor algebra of a vector space, often a space of smooth test functions. They were studied by , who showed that the Wightman distributions of a quantum fiel ...
on a
Borchers algebra In mathematics, a Borchers algebra or Borchers–Uhlmann algebra or BU-algebra is the tensor algebra of a vector space, often a space of smooth test functions. They were studied by , who showed that the Wightman distributions of a quantum fiel ...
equal to the tensor algebra of a space of test functions.


Existence of theories that satisfy the axioms

One can generalize the Wightman axioms to dimensions other than 4. In dimension 2 and 3, interacting (i.e. non-free) theories that satisfy the axioms have been constructed. Currently, there is no proof that the Wightman axioms can be satisfied for interacting theories in dimension 4. In particular, the
Standard model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
of particle physics has no mathematically rigorous foundations. There is a million-dollar prize for a proof that the Wightman axioms can be satisfied for gauge theories, with the additional requirement of a mass gap.


Osterwalder–Schrader reconstruction theorem

Under certain technical assumptions, it has been shown that a Euclidean QFT can be Wick-rotated into a Wightman QFT, see Osterwalder–Schrader theorem. This theorem is the key tool for the constructions of interacting theories in dimension 2 and 3 that satisfy the Wightman axioms.


See also

*
Haag–Kastler axioms Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by . The axioms are stated in te ...
*
Hilbert's sixth problem Hilbert's sixth problem is to axiomatize those branches of physics in which mathematics is prevalent. It occurs on the widely cited list of Hilbert's problems in mathematics that he presented in the year 1900. In its common English translation, ...
*
Axiomatic quantum field theory Axiomatic quantum field theory is a mathematical discipline which aims to describe quantum field theory in terms of rigorous axioms. It is strongly associated with functional analysis and operator algebras, but has also been studied in recent years ...
*
Local quantum field theory The Haag–Kastler axiomatic framework for quantum field theory, introduced by , is an application to local quantum physics of C*-algebra theory. Because of this it is also known as algebraic quantum field theory (AQFT). The axioms are stated in t ...


References

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Further reading

*
Arthur Wightman Arthur Strong Wightman (March 30, 1922 – January 13, 2013) was an American mathematical physicist. He was one of the founders of the axiomatic approach to quantum field theory, and originated the set of Wightman axioms. With his rigorous treatm ...
, "Hilbert's sixth problem: Mathematical treatment of the axioms of physics", in F. E. Browder (ed.): Vol. 28 (part 1) of ''Proc. Symp. Pure Math.'', Amer. Math. Soc., 1976, pp. 241–268. * Res Jost, ''The general theory of quantized fields'', Amer. Math. Soc., 1965. Axiomatic quantum field theory