Yang–Mills existence and mass gap

The Millennium problem requires the proposed Yang–Mills theory to satisfy the Wightman axioms or similarly stringent axioms.[1] There are four axioms:

W0 (assumptions of relativistic quantum mechanics)

Quantum mechanics 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.

The Wightman axioms require that the Poincaré group acts unitarily on the Hilbert space. In other words, they have position dependent operators called quantum fields which form covariant representations of the Poincaré group.

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, ${\displaystyle P_{j},j=0,1,2,3}$, 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:

${\displaystyle P_{0}\geq 0,\;\;\;\;P_{0}^{2}-P_{j}P_{j}\geq 0.}$

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 ${\displaystyle A_{1}(f),\ldots ,A_{n}(f)}$ 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, or SL(2,C) if the spin is not integer:

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

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 ${\displaystyle H^{in}}$ and ${\displaystyle H^{out}}$, 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.

In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. 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 ${\displaystyle \phi (x)}$, we can say that the theory has a mass gap if the two-point function has the property

${\displaystyle \langle \phi (0,t)\phi (0,0)\rangle \sim \sum _{n}A_{n}\exp \left(-\Delta _{n}t\right)}$

with ${\displaystyle \Delta _{0}>0}$ 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 develops a mass gap on a lattice.[6][7]

At the level of rigor of theoretical physics, it has been well established that the quantum Yang–Mills theory for a non-abelian Lie group exhibits a property known as confinement; though proper mathematical physics 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 free color charge and free gluons 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.