Yang–Mills Existence And Mass Gap
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The Yang–Mills existence and mass gap problem is an unsolved problem in
mathematical physics Mathematical physics is 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 the de ...
and
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and one of the seven
Millennium Prize Problems The Millennium Prize Problems are seven well-known complex mathematics, mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem ...
defined by the Clay Mathematics Institute, which has offered a prize of $1,000,000 USD for its solution. The problem is phrased as follows: :''Yang–Mills Existence and Mass Gap.'' Prove that for any compact simple
gauge group A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle P\to X with a structure Lie group G, a gauge group is defined to be a group of its vertical ...
G, a
non-trivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or a particularly simple object possessing a given structure (e.g., group, topological space). The noun triviality usual ...
quantum
Yang–Mills theory Yang–Mills theory is a quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is a gauge theory based on a special un ...
exists on \mathbb^4 and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in , and . In this statement, a quantum
Yang–Mills theory Yang–Mills theory is a quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is a gauge theory based on a special un ...
is a non-abelian
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
similar to that underlying the
Standard Model The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
of
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
; \mathbb^4 is Euclidean 4-space; the mass gap Δ is the mass of the least massive particle predicted by the theory. Therefore, the winner must prove that: * Yang–Mills theory exists and satisfies the standard of rigor that characterizes contemporary
mathematical physics Mathematical physics is 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 the de ...
, in particular constructive quantum field theory, and * The mass of all particles of the force field predicted by the theory are strictly positive. For example, in the case of G=SU(3)—the strong nuclear interaction—the winner must prove that glueballs have a lower mass bound, and thus cannot be arbitrarily light. The general problem of determining the presence of a mass gap (a special case of a
spectral gap In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to oth ...
) in a system is known to be undecidable, meaning no computer algorithm exists that can find the answer programmatically.


Background

The problem requires the construction of a QFT satisfying the Wightman axioms and showing the existence of a mass gap. Both of these topics are described in sections below.


Wightman axioms

The Millennium problem requires the proposed Yang–Mills theory to satisfy the Wightman axioms or similarly stringent axioms. There are four axioms: ;W0 (assumptions of relativistic quantum mechanics)
Quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
is described according to
John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
; in particular, the
pure state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
s are given by the rays, i.e. the one-dimensional subspaces, of some separable complex
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
. The Wightman axioms require that the Poincaré group acts unitarily on the Hilbert space. In other words, a change of reference frame (position, velocity, rotation) must be a
unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Non-trivial examples include rotations, reflections, and the Fourier operator. Unitary operators generalize unitar ...
, or a surjective operator which preserves the inner product, which can be viewed as an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
on a Hilbert space. Among the implications of this is the fact that the probability of an event must not change with a change of reference frame, as the probability of an event occurring is its inner product with itself. This requirement can also be stated in other words to mean that the Wightman axioms have position dependent operators called ''quantum fields'' which form covariant representations of the Poincaré group. The group of space-time 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. Perhaps most familiar as a pr ...
, 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 a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for al ...
s, P_j,j=0,1,2,3, which transform under the homogeneous group as a
four-vector In special relativity, a four-vector (or 4-vector, sometimes Lorentz 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 vect ...
, 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. ;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). ;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 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 physi ...
, or SL(2,C) if the spin is not integer: :U(a,L)^A(x)U(a,L)=S(L)A(L^(x-a)). ;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 the property of asymptotic completeness—that the Hilbert state space is spanned by the asymptotic spaces H^ and H^, appearing in the collision S matrix. The other important property of field theory is the mass gap which is not required by the axioms—that the energy-momentum spectrum has a gap between zero and some positive number.


Mass gap

In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, the mass gap is the difference in energy between the vacuum and the next lowest
energy state A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The ...
. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle. For a given real field \phi(x), we can say that the theory has a mass gap if the two-point function has the property :\langle\phi(0,t)\phi(0,0)\rangle\sim \sum_nA_n\exp\left(-\Delta_nt\right) with \Delta_0>0 being the lowest energy value in the spectrum of the Hamiltonian and thus the mass gap. This quantity, easy to generalize to other fields, is what is generally measured in lattice computations. It was proved in this way that
Yang–Mills theory Yang–Mills theory is a quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is a gauge theory based on a special un ...
develops a mass gap on a lattice...


Importance of Yang–Mills theory

Most known and nontrivial (i.e. interacting) quantum field theories in 4 dimensions are effective field theories with a cutoff scale. Since the
beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^ ...
is positive for most models, it appears that most such models have a
Landau pole In physics, the Landau pole (or the Moscow zero, or the Landau ghost) is the momentum (or energy) scale at which the coupling constant (interaction strength) of a quantum field theory becomes infinite. Such a possibility was pointed out by the ph ...
as it is not at all clear whether or not they have nontrivial UV fixed points. This means that if such a QFT is well-defined at all scales, as it has to be to satisfy the axioms of axiomatic quantum field theory, it would have to be trivial (i.e. a free field theory). Quantum Yang–Mills theory with a non-abelian
gauge group A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle P\to X with a structure Lie group G, a gauge group is defined to be a group of its vertical ...
and no quarks is an exception, because asymptotic freedom characterizes this theory, meaning that it has a trivial UV fixed point. Hence it is the simplest nontrivial constructive QFT in 4 dimensions. (
QCD In theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in ...
is a more complicated theory because it involves
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nucleus, atomic nuclei ...
s.)


Quark confinement

At the level of rigor of
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, it has been well established that the quantum Yang–Mills theory for a non-abelian
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
exhibits a property known as confinement; though proper
mathematical physics Mathematical physics is 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 the de ...
has more demanding requirements on a proof. A consequence of this property is that above the confinement scale, the color charges are connected by chromodynamic flux tubes leading to a linear potential between the charges. Hence isolated color charge and isolated
gluon A gluon ( ) is a type of Massless particle, massless elementary particle that mediates the strong interaction between quarks, acting as the exchange particle for the interaction. Gluons are massless vector bosons, thereby having a Spin (physi ...
s cannot exist. In the absence of confinement, we would expect to see massless gluons, but since they are confined, all we would see are color-neutral bound states of gluons, called glueballs. If glueballs exist, they are massive, which is why a mass gap is expected.


References


Further reading

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External links


The Millennium Prize Problems: Yang–Mills and Mass Gap
{{DEFAULTSORT:Yang-Mills Existence And Mass Gap Millennium Prize Problems Gauge theories Quantum chromodynamics Unsolved problems in mathematics Unsolved problems in physics