Abstract
We elucidate how Quantum Thermodynamics at temperature T emerges from pure and classical S U ( 2 ) Yang–Mills theory on a four-dimensional Euclidean spacetime slice S 1 × R 3 . The concept of a (deconfining) thermal ground state, composed of certain solutions to the fundamental, classical Yang–Mills equation, allows for a unified addressation of both (classical) wave- and (quantum) particle-like excitations thereof. More definitely, the thermal ground state represents the interplay between nonpropagating, periodic configurations which are electric-magnetically (anti)selfdual in a non-trivial way and possess topological charge modulus unity. Their trivial-holonomy versions—Harrington–Shepard (HS) (anti)calorons—yield an accurate a priori estimate of the thermal ground state in terms of spatially coarse-grained centers, each containing one quantum of action ℏ localized at its inmost spacetime point, which induce an inert adjoint scalar field ϕ ( | ϕ | spatio-temporally constant). The field ϕ , in turn, implies an effective pure-gauge configuration, a μ gs , accurately describing HS (anti)caloron overlap. Spatial homogeneity of the thermal ground-state estimate ϕ , a μ gs demands that (anti)caloron centers are densely packed, thus representing a collective departure from (anti)selfduality. Effectively, such a “nervous” microscopic situation gives rise to two static phenomena: finite ground-state energy density ρ gs and pressure P gs with ρ gs = − P gs as well as the (adjoint) Higgs mechanism. The peripheries of HS (anti)calorons are static and resemble (anti)selfdual dipole fields whose apparent dipole moments are determined by | ϕ | and T, protecting them against deformation potentially caused by overlap. Such a protection extends to the spatial density of HS (anti)caloron centers. Thus the vacuum electric permittivity ϵ 0 and magnetic permeability μ 0 , supporting the propagation of wave-like disturbances in the U ( 1 ) Cartan subalgebra of S U ( 2 ) , can be reliably calculated for disturbances which do not probe HS (anti)caloron centers. Both ϵ 0 and μ 0 turn out to be temperature independent in thermal equilibrium but also for an isolated, monochromatic U ( 1 ) wave. HS (anti)caloron centers, on the other hand, react onto wave-like disturbances, which would resolve their spatio-temporal structure, by indeterministic emissions of quanta of energy and momentum. Thermodynamically seen, such events are Boltzmann weighted and occur independently at distinct locations in space and instants in (Minkowskian) time, entailing the Bose–Einstein distribution. Small correlative ramifications associate with effective radiative corrections, e.g., in terms of polarization tensors. We comment on an S U ( 2 ) × S U ( 2 ) based gauge-theory model, describing wave- and particle-like aspects of electromagnetic disturbances within the so far experimentally/observationally investigated spectrum.
Highlights
Boltzmann’s statistical approach to kinetic gas theory can be considered an anticipation of Quantum Physics
The arrow of time, expressing the asymptotic attainment of an ergodic equilibrium state of maximum entropy as a consequence of f ’s evolution via Boltzmann’s Equation (1), is a direct consequence of the indeterminism inherent to the collision integral, and our modern understanding of molecular chaos is that this integral be expanded into positive powers of h—Planck’s quantum of action (In the formal limit h → 0 the quantity ∂t f |coll is given by Classical Mechanics or vanishes)
Where n B ( x ) denotes the Bose–Einstein distribution function. In this contribution we have given a sketchy overview on how the thermal ground state emerges in SU (2) Yang–Mills theory in terms of a spatial coarse-graining over the field-strength correlation within the center of an electric-magneticallyselfdualcaloron gauge-field configuration of topological charge modulus unity and trivial holonomy [4], giving rise to an effective inert scalar field φ, gs and a pure-gauge solution aμ of the effective Yang–Mills field equations, sourced by φ
Summary
Boltzmann’s statistical approach to kinetic gas theory can be considered an anticipation of Quantum Physics. Equation (1) would describe a time-reversal invariant evolution of f like in Classical Mechanics (that is, if f (~x, ~x , t) satisfies. The collision integral’s intrinsic indeterminism, expressed through probabilistic changes from initial to final scattering states (molecular chaos), selfconsistently underlies the concept of the probability distribution f and its time-reversal non-invariant evolution. The arrow of time, expressing the asymptotic attainment of an ergodic (thermal) equilibrium state of maximum entropy as a consequence of f ’s evolution via Boltzmann’s Equation (1), is a direct consequence of the indeterminism inherent to the collision integral, and our modern understanding of molecular chaos is that this integral be expanded into positive powers of h—Planck’s (reduced) quantum of action (In the formal limit h → 0 the quantity ∂t f |coll is given by Classical Mechanics or vanishes)
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