{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:20:32Z","timestamp":1760242832894,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2016,8,23]],"date-time":"2016-08-23T00:00:00Z","timestamp":1471910400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We elucidate how Quantum Thermodynamics at temperature T emerges from pure and classical     S U ( 2 )     Yang\u2013Mills theory on a four-dimensional Euclidean spacetime slice      S 1   \u00d7    R  3     . The concept of a (deconfining) thermal ground state, composed of certain solutions to the fundamental, classical Yang\u2013Mills 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\u2014Harrington\u2013Shepard (HS) (anti)calorons\u2014yield an accurate a priori estimate of the thermal ground state in terms of spatially coarse-grained centers, each containing one quantum of action \u210f localized at its inmost spacetime point, which induce an inert adjoint scalar field     \u03d5     (    | \u03d5 |     spatio-temporally constant). The field     \u03d5    , in turn, implies an effective pure-gauge configuration,     a \u03bc gs    , accurately describing HS (anti)caloron overlap. Spatial homogeneity of the thermal ground-state estimate     \u03d5 ,  a \u03bc gs      demands that (anti)caloron centers are densely packed, thus representing a collective departure from (anti)selfduality. Effectively, such a \u201cnervous\u201d microscopic situation gives rise to two static phenomena: finite ground-state energy density     \u03c1 gs     and pressure     P gs     with      \u03c1 gs  = \u2212  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     | \u03d5 |     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     \u03f5 0     and magnetic permeability     \u03bc 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     \u03f5 0     and     \u03bc 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\u2013Einstein distribution. Small correlative ramifications associate with effective radiative corrections, e.g., in terms of polarization tensors. We comment on an     S U ( 2 )     \u00d7     S U ( 2 )     based gauge-theory model, describing wave- and particle-like aspects of electromagnetic disturbances within the so far experimentally\/observationally investigated spectrum.<\/jats:p>","DOI":"10.3390\/e18090310","type":"journal-article","created":{"date-parts":[[2016,8,23]],"date-time":"2016-08-23T10:18:55Z","timestamp":1471947535000},"page":"310","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["SU(2) Yang\u2013Mills Theory: Waves, Particles, and Quantum Thermodynamics"],"prefix":"10.3390","volume":"18","author":[{"given":"Ralf","family":"Hofmann","sequence":"first","affiliation":[{"name":"Institute for Photon Science and Synchrotron Radiation, Karlsruhe Institute of Technology, Eggenstein-Leopoldshafen 76344, Germany"},{"name":"Institut f\u00fcr Theoretische Physik, Universit\u00e4t Heidelberg, Philosophenweg 16, Heidelberg 69120, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,8,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"373121","DOI":"10.5402\/2012\/373121","article-title":"Emergent Inert Adjoint Scalar Field in SU(2) Yang\u2013Mills Thermodynamics due to Coarse-Grained Topological Fluctuations","volume":"2012","author":"Herbst","year":"2012","journal-title":"ISRN High Energy Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"4123","DOI":"10.1142\/S0217751X05023931","article-title":"Nonperturbative approach to Yang\u2013Mills thermodynamics","volume":"20","author":"Hofmann","year":"2005","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Hofmann, R. (2016). The Thermodynamics of Quantum Yang\u2013Mills Theory: Theory and Application, World Scientific. [2nd ed.].","DOI":"10.1142\/9997"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2122","DOI":"10.1103\/PhysRevD.17.2122","article-title":"Periodic Euclidean solutions and the finite-temperature Yang\u2013Mills gas","volume":"17","author":"Harrington","year":"1978","journal-title":"Phys. Rev. D"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1016\/0370-2693(80)90961-2","article-title":"A simple formalism for the BPS monopole","volume":"90","author":"Nahm","year":"1980","journal-title":"Phys. Lett. B"},{"key":"ref_6","unstructured":"Nahm, W. (1981). Structural Elements in Particle Physics and Statistical Mechanics, Springer."},{"key":"ref_7","unstructured":"Nahm, W. (1982). Monopoles in Quantum Field Theory, World Scientific."},{"key":"ref_8","unstructured":"Nahm, W. (1983). Group Theoretical Methods in Physics, Springer."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1016\/S0370-2693(98)00411-0","article-title":"Exact T-duality between calorons and Taub-NUT spaces","volume":"428","author":"Kraan","year":"1998","journal-title":"Phys. Lett. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1016\/S0550-3213(98)00590-2","article-title":"Periodic instantons with non-trivial holonomy","volume":"533","author":"Kraan","year":"1998","journal-title":"Nucl. Phys. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"025011","DOI":"10.1103\/PhysRevD.58.025011","article-title":"SU(2) calorons and magnetic monopoles","volume":"58","author":"Lee","year":"1998","journal-title":"Phys. Rev. D"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"036003","DOI":"10.1103\/PhysRevD.70.036003","article-title":"Quantum weights of dyons and of instantons","volume":"70","author":"Diakonov","year":"2004","journal-title":"Phys. Rev. D"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3432","DOI":"10.1103\/PhysRevD.14.3432","article-title":"Computation of the quantum effects due to a four-dimensional pseudoparticle","volume":"14","year":"1976","journal-title":"Phys. Rev. D"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Jackiw, R., and Rebbi, C. (1976). Conformal properties of a Yang\u2013Mills pseudoparticle. Phys. Rev. D, 14.","DOI":"10.1103\/PhysRevD.14.517"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1213","DOI":"10.1142\/S0217751X07035227","article-title":"Radiative corrections and the one-loop polarization tensor of the massless mode in SU(2) Yang\u2013Mills thermodynamics","volume":"22","author":"Schwarz","year":"2007","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1002\/andp.20095210407","article-title":"Thermal photon dispersion law and modified black-body spectra","volume":"18","author":"Ludescher","year":"2009","journal-title":"Ann. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"197197","DOI":"10.1155\/2015\/197197","article-title":"Thermal ground state and nonthermal probes","volume":"2015","author":"Grandou","year":"2015","journal-title":"Adv. Math. Phys."},{"key":"ref_18","unstructured":"Hofmann, R. (2015). Electromagnetic Waves and Photons."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1166\/qm.2012.1004","article-title":"The Quantum of Action and Finiteness of Radiative Corrections: Deconfining SU(2) Yang\u2013Mills Thermodynamics","volume":"1","author":"Kaviani","year":"2012","journal-title":"Quantum Matter"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/j.aop.2014.04.024","article-title":"One-loop photon-photon scattering in a thermal, deconfining SU(2) Yang\u2013Mills plasma","volume":"347","author":"Krasowski","year":"2014","journal-title":"Ann. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"045026","DOI":"10.1103\/PhysRevD.83.045026","article-title":"The \u210f expansion in quantum field theory","volume":"83","author":"Brodsky","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Iliopoulos, J., Itzykson, C., and Martin, A. (1975). Functional methods and perturbation theory. Rev. Mod. Phys., 47.","DOI":"10.1103\/RevModPhys.47.165"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/S0370-2693(02)01246-7","article-title":"Quantum corrections to the Reissner\u2013Nordstr\u00f6m and Kerr\u2013Newman metrics","volume":"529","author":"Donoghue","year":"2002","journal-title":"Phys. Lett. B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"084005","DOI":"10.1103\/PhysRevD.68.084005","article-title":"Quantum corrections to the Schwarzschild and Kerr metrics","volume":"68","author":"Donoghue","year":"2003","journal-title":"Phys. Rev. D"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"201602","DOI":"10.1103\/PhysRevLett.93.201602","article-title":"Classical Physics and Quantum Loops","volume":"93","author":"Holstein","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Gross, D.J., Pisarski, R.D., and Yaffe, L.G. (1981). QCD and instantons at finite temperature. Rev. Mod. Phys., 53.","DOI":"10.1103\/RevModPhys.53.43"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/0550-3213(74)90486-6","article-title":"Magnetic monopoles in unified gauge theories","volume":"79","year":"1974","journal-title":"Nucl. Phys. B"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Dolan, L., and Jackiw, R. (1974). Symmetry behavior at finite temperature. Phys. Rev. D, 9.","DOI":"10.1103\/PhysRevD.9.3320"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"085022","DOI":"10.1103\/PhysRevD.76.085022","article-title":"Linear growth of the trace anomaly in Yang\u2013Mills thermodynamics","volume":"76","author":"Giacosa","year":"2007","journal-title":"Phys. Rev. D"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/0550-3213(71)90395-6","article-title":"Renormalization of massless Yang\u2013Mills fields","volume":"33","year":"1971","journal-title":"Nucl. Phys. B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/0550-3213(72)90279-9","article-title":"Regularization and renormalization of gauge fields","volume":"44","author":"Veltman","year":"1972","journal-title":"Nucl. Phys. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1016\/S0550-3213(72)80021-X","article-title":"Combinatorics of gauge fields","volume":"50","author":"Veltman","year":"1972","journal-title":"Nucl. Phys. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2343","DOI":"10.1142\/S0217732307024413","article-title":"Irreducible three-loop contributions to the pressure in Yang\u2013Mills thermodynamics","volume":"22","author":"Kaviani","year":"2007","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1007\/s13538-012-0062-5","article-title":"Loop Expansion in Yang\u2013Mills Thermodynamics","volume":"42","author":"Hofmann","year":"2012","journal-title":"Braz. J. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"904","DOI":"10.1002\/andp.201000073","article-title":"Modification of black-body radiance at low temperatures and frequencies","volume":"522","author":"Falquez","year":"2010","journal-title":"Ann. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1002\/andp.20095210905","article-title":"Low-frequency line temperatures of the CMB","volume":"18","author":"Hofmann","year":"2009","journal-title":"Ann. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1038\/nphys2793","article-title":"The fate of statistical isotropy","volume":"9","author":"Hofmann","year":"2013","journal-title":"Nat. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Szopa, M., and Hofmann, R. (2008). A model for CMB anisotropies on large angular scales. J. Cosmol. Astropart. Phys., 2008.","DOI":"10.1088\/1475-7516\/2008\/03\/001"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Fixsen, D.J., Kogut, A., Levin, S., Limon, M., Lubin, P., Mirel, P., Seiffert, M., Singal, J., Wollack, E., and Villela, T. (2011). ARCADE 2 measurement of the extra-galactic sky temperature at 3\u201390 GHz. Astrophys. J., 734.","DOI":"10.1088\/0004-637X\/734\/1\/5"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1002\/andp.201400197","article-title":"Relic photon temperature versus redshift and the cosmic neutrino background","volume":"527","author":"Hofmann","year":"2015","journal-title":"Ann. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"A16","DOI":"10.1051\/0004-6361\/201321591","article-title":"Planck 2013 results. XVI. Cosmological parameters","volume":"571","author":"Ade","year":"2014","journal-title":"Astron. Astrophys."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Becker, R.H., Fan, X., White, R.L., Strauss, M.A., Narayanan, V.K., Lupton, R.H., Gunn, J.E., Annis, J., Bahcall, N.A., and Brinkmann, J. (2001). Evidence for Reionization at z \u223c 6: Detection of a Gunn-Peterson Trough in a z = 6.28 Quasar. Astron. J., 122.","DOI":"10.1086\/324231"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1166\/qm.2012.1014","article-title":"Charge-density waves in deconfining SU(2) Yang\u2013Mills thermodynamics","volume":"1","author":"Falquez","year":"2012","journal-title":"Quantum Matter"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1103\/RevModPhys.74.775","article-title":"Origin of galactic and extragalactic magnetic fields","volume":"74","author":"Widrow","year":"2002","journal-title":"Rev. Mod. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/9\/310\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:29:02Z","timestamp":1760210942000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/9\/310"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,8,23]]},"references-count":44,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2016,9]]}},"alternative-id":["e18090310"],"URL":"https:\/\/doi.org\/10.3390\/e18090310","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2016,8,23]]}}}