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Thermal Ground State and Nonthermal Probes

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Thermal Ground State and Nonthermal Probes Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2015, Article ID 197197, 8 pages http://dx.doi.org/10.1155/2015/197197 Research Article Thermal Ground State and Nonthermal Probes Thierry Grandou1 and Ralf Hofmann2 1 Institut Non Lineaire´ de Nice, 1361 route des Lucioles, Sophia Antipolis, 06560 Valbonne, France 2Institut fur¨ Theoretische Physik, Universitat¨ Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany Correspondence should be addressed to Ralf Hofmann; [email protected] Received 31 August 2015; Accepted 27 October 2015 Academic Editor: Ivan Avramidi Copyright © 2015 T. Grandou and R. Hofmann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Euclidean formulation of SU(2) Yang-Mills thermodynamics admits periodic, (anti)self-dual solutions to the fundamental, classical equation of motion which possess one unit of topological charge: (anti)calorons. A spatial coarse graining over the central region in a pair of such localised field configurations with trivial holonomy generates an inert adjoint scalar field ,effectively describing the pure quantum part of the thermal ground state in the induced quantum field theory. Here we show for the limit of zero holonomy how (anti)calorons associate a temperature independent electric permittivity and magnetic permeability to the thermal (2) ground state of SU CMB, the Yang-Mills theory conjectured to underlie the fundamental description of thermal photon gases. 1. Introduction and Minkowskian time evolution as expressed by a time- periodic SU(2) (anti)self-dual gauge field configuration—a Quantum Mechanics is a highly efficient framework to (anti)caloron—whose action ℏ is associated with one unit describe the subatomic world [1–3], including coherence of winding about a central spacetime point gives rise to phenomena that extend to macroscopic length and time indeterminism in the process it mediates. That each unit −1 scales [4–10]. The key quantity to describe deviations from of action assigned to (anti)calorons of radius =||, classicalbehaviorisPlanck’squantumofactionℏ=ℎ/2= 16 which dominate the emergence of the thermal ground state, 6.58 × 10 eV s which determines the fundamental interac- equals ℏ follows from the value of the coupling in the tion between charged matter and the electromagnetic field induced, effective, thermal quantum field theory [13–17] of andthusalsotheshapeofblackbodyspectrabyrelating the deconfining phase in SU(2) Yang-Mills thermodynamics. frequency andwavevectork to particle-like energy =ℏ The coupling ,inturn,obeysanevolutionintemperature and momentum p =ℏk and by appeal to Bose-Einstein (flat almost everywhere) which represents the validity of statistics. In Quantum Mechanics, ℏ sets the strength of Legendre transformations in the effective ensemble where the multiplicative noncommutativity for a pair of canonically thermal ground state coexists with massive (adjoint Higgs conjugate variables such as position and momentum, imply- mechanism) and massless (intact U(1))thermalfluctuations. ing the respective uncertainty relations. The thermal ground state thus is a spatially homogeneous Despite being generally accepted as a universal constant ensemble of quantum fluctuations carried by (anti)caloron of nature and in spite of the fact that we are able to efficiently centers.Atthesametime,aswewillsee,thisstateprovides compute quantum mechanical amplitudes and quantum electric and magnetic dipole densities supporting the prop- statistical averages for a vast variety of processes in parti- agation of certain electromagnetic waves in an SU(2) Yang- Λ∼10−4 (2) cle collisions, atoms, and molecules, extended condensed- Mills of scale eV, SU CMB [18]. matter systems, and astrophysical objects to match exper- In the present work, we establish this link between iment and observation very well, one should remain curi- quantised action, represented by ,andclassicalwaveprop- ous concerning the principle mechanism that causes the agationenabledbythevacuumparameters0 and 0 in emergence of a universal quantum of action. In [11, 12] it terms of the central and peripheral structure of a trivial- was argued that the irreconcilability of classical Euclidean holonomy (anti)caloron, respectively. That is, by allowing 2 Advances in Mathematical Physics afictitioustemperature to represent the energy density of (unitary-Coulomb) characterised by a massless mode (, an electromagnetic wave (nonthermal, external probe) via unbroken U(1) subgroup of SU(2)) and two thermal quasi- ± the thermal ground state through which it propagates we ask particle modes of equal mass = 2|| ( ,massinduced what this implies for 0 and 0.Asaresult,both0 and 0 by adjoint Higgs mechanism) which propagate thermally, neither depend on nor, as we will argue, on a singled-out that is, on-shell only. Interactions within this propagating inertial frame provided that a certain condition on intensity sector are mediated by isolated (anti)calorons whose action is and frequency of the wave is satisfied. This is a first step in argued to be ℏ [11, 12]. Judged in terms of inclusive quantities reviving the concept of the luminiferous aether, albeit now such as radiative corrections to the one-loop pressure or with the goal of constructing a Poincareinvariantobject.´ the energy density of blackbody radiation, these interactions This paper is organised as follows. In the next section are feeble, and their expansion into 1-PI irreducible bubble we shorty discuss key features of the effective theory for diagrams is conjectured to terminate at a finite number of ± the deconfining phase of SU(2) Yang-Mills thermodynamics. loops [18]. However, spectrally seen, the effects of inter- Section 3 contains a reminder to principles in interpreting acting with lead to severe consequences at low frequencies a Euclidean field configuration in terms of Minkowskian and temperatures comparable to the critical temperature observables. In a next step, general facts are reviewed on where screened (anti)monopoles, released by (anti)caloron Euclidean, periodic, (anti)self-dual field configurations of dissociation upon large-holonomy deformations [20], rapidly charge modulus unity concerning the central locus of action, become massless and thus start to condense. their holonomy, and their behaviour under semiclassical deformation. Finally, we review the anatomy of a zero- 3. Caloron Structure holonomy Harrington-Shepard (HS) caloron in detail, point- ing out its staticity for spatial distances from the center 3.1. Euclidean Field Theory and Interpretable Quantities. Non- that exceed the inverse of temperature, and discuss which trivial solutions to an elliptic differential equation, such as static charge configuration it resembles depending on two the Euclidean Yang-Mills equation ] =0,nolonger distinct spatial distance regimes. In Section 4 we briefly are solutions of the corresponding hyperbolic equation upon review the postulate that an SU(2) Yang-Mills theory of analytic continuation 4 ≡ → 0 (Wick rotation). To Λ∼10−4 (2) scale eV, SU CMB, describes thermal photon endow meaning to quantities computed on classical field con- gases [18]. Subsequently, the large-distance regime in an HS figurations on a 4D Euclidean spacetime in SU(2) Yang-Mills (anti)caloron is considered in order to deduce an expression thermodynamics in terms of observables in a Minkowskian for 0 based on knowledge about the electric dipole moment spacetimewethusmustinsistthatthesequantitiesarenot −1 provided by a (anti)caloron of radius radius =|| ,the affected by the Wick rotation. That is, to assign a real-world size of the spatial coarse-graining volume cg, and the fact interpretation to a Euclidean quantity it needs to be (i) that the energy density of the probe must match that of either stationary (not depend on ) or (ii) associated with the thermal ground state. As a result, 0 and 0 turn out an instant in Euclidean spacetime because, by exploiting time to be -independent, the former representing an electric translational invariance of the Yang-Mills action, this instant charge, large on the scale of the electron charge, of the canbepickedas( = 0, x) in Euclidean spacetime. fictitious constituent monopoles giving rise to the associated dipole density. Zooming in to smaller spatial distances to the 3.2. Review of General Facts. If not stated otherwise we work center, the HS (anti)caloron exhibits isolated (anti)self-dual in supernatural units, ℏ== =1,whereℏ is the monopoles. For them to turn into dipoles shaking by the reduced quantum of action, is the speed of light in vacuum, probe fields is required. We then show that the definitions and is Boltzmann’s constant. A trivial-holonomy caloron 3 of 0 and 0, which were successfully applied to the large- of topological charge unity on the cylinder 1 × R ,where distance regime, become meaningless. Finally, our results 1 isthecircleofcircumference≡1/( temperature) are discussed. Section 5 summarises the paper and discusses describing the compactified Euclidean time dimension (0≤ how a universality of 0 and 0 for the entire, experimentally ≤), is constructed by an appropriate superposition investigated electromagnetic spectrum could emerge. of charge-one singular-gauge instanton prepotentials [21, 22] with the temporal coordinate of their instanton centers 2. Sketch of Deconfining SU(2) equidistantly
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