In
mathematical physics
Mathematical physics refers to the development of 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 the developm ...
, 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
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
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
In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
(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.
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 equ ...
. Because the axioms are dealing with
unbounded operator In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.
The ter ...
s, 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 ...
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
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
) by polynomials in smeared fields over test functions with support in an open set in
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
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 Von Neumann may refer to:
* John von Neumann (1903–1957), a Hungarian American mathematician
* Von Neumann family
* Von Neumann (surname), a German surname
* Von Neumann (crater), a lunar impact crater
See also
* Von Neumann algebra
* Von Ne ...
; in particular, the
pure state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
s 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
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
of Hilbert space vectors Ψ and Φ is denoted by
, and the norm of Ψ is denoted by
. The transition probability between two pure states
�and
�can be defined in terms of non-zero vector representatives Ψ and Φ to be
:
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
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...
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 law ...
. 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 Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
(the inhomogeneous Lorentz group). Thus, ''a'' is a real Lorentz
four-vector
In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
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 differ ...
origin ''x'' ↦ ''x'' − ''a'', where ''x'' is in the Minkowski space ''M''
4, and ''L'' is a
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
, which can be defined as a linear transformation of four-dimensional spacetime preserving the Lorentz distance ''c''
2''t''
2 − ''x''⋅''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:
:
Wigner's theorem
Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT are represented on the Hilbert sp ...
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
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation In mathematics, a unitary representation of a grou ...
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'')⋅(''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 ...
. 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
:
i.e. the phase is a multiple of
. 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 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 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 co ...
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 i ...
s transforming as spin 1/2, 3/2 etc., cannot be
observable
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
s. This shows up as the ''univalence
superselection
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 origina ...
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
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
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 spacetime translations is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, and so the operators can be simultaneously diagonalised. The generators of these groups give us four
self-adjoint 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 its ...
s
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 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
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 differentiate functions whose derivatives d ...
. 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
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
and transform according to some representation ''S'' of the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
, or SL(2, C) if the spin is not integer:
:
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 dif ...
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.
Consequences of the axioms
From these axioms, certain general theorems follow:
*
CPT 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
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 ...
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 and also the ghosts in
BRST.
* The impossibility of
superluminal communication
Superluminal communication is a hypothetical process in which information is sent at faster-than-light (FTL) speeds. The current scientific consensus is that faster-than-light communication is not possible, and to date it has not been achieved in ...
– 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, including the existence of a
vacuum state
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
; 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 Techni ...
,
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
The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intr ...
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 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 exchan ...
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
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
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 origina ...
.
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 view ...
. 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 on a
Borchers algebra 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 ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...
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
*
Hilbert's sixth problem
*
Axiomatic quantum field theory
*
Local quantum field theory
References
{{reflist
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