Historical research guide · 1109 · 2005
Nonperturbative approach to Yang–Mills thermodynamics
Ralf Hofmann
In one minute
Ralf Hofmann sets out to do something the textbooks say is hard: work out what a strongly interacting gauge theory does at a given temperature, analytically, without the usual trick of expanding in a small coupling. He treats the SU(2) and SU(3) theories — the mathematics behind the weak force and the strong force — and finds that each has three phases, and that in each phase the vacuum itself is a real, macroscopic, non-fluctuating object built out of topological lumps: calorons in the hot phase, condensed magnetic monopoles in the middle phase, condensed vortex loops in the cold one. The vacuum is not a blank background in this picture; it has an energy density, a pressure, and a temperature evolution you can compute. In the hot phase its pressure comes out exactly equal and opposite to its energy density, which is the equation of state cosmologists write for dark energy. Hofmann then proposes, as an outlook, that the photon is the massless mode of one such theory whose scale is set by the temperature of the microwave background.
Connection
Chapter 2 argues the vacuum is a structured medium rather than an empty stage, and this is one of the few places where that structure is derived rather than assumed — a ground state with a computable energy density and a negative pressure, falling straight out of Yang–Mills theory. Chapter 13 is where the site keeps frameworks that try to put the vacuum, the particles and the dark sector on one footing, and Hofmann’s is among the most complete of them.
From the source
SU(2) and SU(3) Yang–Mills theory each come in three phases — a deconfining or electric one, a preconfining or magnetic one, and a confining or center one — and in every phase a macroscopic, inert scalar field emerges that fixes the physics of the ground state and the properties of its excitations.Abstract; Sections 2, 3 and 4; conclusions in Section 8
The ground state of the deconfining phase, assembled from caloron and anticaloron systems together with a pure-gauge configuration, carries a pressure equal and opposite to its energy density: the pressure is minus four pi times the cube of the Yang–Mills scale times the temperature. That is the equation of state written for a cosmological constant, and it is what makes the otherwise hidden Yang–Mills scale gravitationally measurable.Section 2.2.1, the paragraph following Equation 74
The preconfining ground state is a Bose condensate of thermalised magnetic monopole and antimonopole systems, and evaluated on it the Polyakov loop is trivial, so the electric two-fold and three-fold degeneracies of the hot phase are gone and this phase confines fundamental test charges.Sections 3.1 and 3.2, opening paragraph and Equation 138; Section 3.4
Across the last transition the statistics of the excitations changes from bosonic to fermionic: the confining phase carries massless or massive spin-one-half fermions, identified with single and self-intersecting center-vortex loops, and the number of available fermion states grows fast enough with mass to make that transition a Hagedorn one.Abstract; Sections 4.2 to 4.4; conclusions in Section 8
Hofmann proposes as an outlook that the photon is the one massless excitation of an SU(2) theory sitting exactly on its electric–magnetic boundary with a scale set by the microwave-background temperature of 2.728 kelvin, giving a Yang–Mills scale of 1.065 times ten to the minus four electronvolts and a ground-state energy density of the fourth power of 2.444 times ten to the minus four electronvolts — about 0.36 per cent of the accepted dark-energy density, the rest coming from a slowly rolling Planck-scale axion.Section 7.2, subsection SU(2) CMB; Section 7.1, Equations 201 to 204
Where this approach and lattice simulations disagree is near the electric–magnetic transition, and Hofmann argues the lattice is the party that cannot be trusted there, because a finite spatial lattice cuts off correlations longer than its own box while the physical correlation length is growing without bound — so a lattice run large enough to resolve that region is the measurement that would decide between them.Section 6.2.1, SU(2) case; Sections 6.1 and 5
The way in
https://doi.org/10.1142/s0217751x05023931How to cite it
Ralf Hofmann (2005) Nonperturbative approach to Yang–Mills thermodynamics doi:10.1142/s0217751x05023931