infinite-temperature thermal field theory



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Broadly speaking, high- or infinite-temperature thermal field theory is thermal field theory in or around the limiting case of very high or (practically) infinite temperature TT.

This is relevant for example in the study of extreme phases of matter such as the quark-gluon plasma of QCD (Blaizot-Iancu-Rebhan 03, section 2.2.4, Blaizot 04, around p. 17).

More concretely, with the formulation of thermal field theory via Wick rotation as a Euclidean field theory on a Riemannian manifold X 3×S β 1X_3 \times S^1_\beta whose compact/periodic Euclidean time runs along a circle S β 1S^1_\beta of circumference the inverse temperature β=1/T\beta = 1/T, the limit of infinite temperature corresponds to the limit β0\beta \to 0 in which this circle fiber shrinks away.

In terms of the thermal Euclidean field theory on X 3×S β 1X_3 \times S^1_\beta therefore the infinite-temperature limit is given by Kaluaza-Klein-type dimensional reduction along this circle fiber to a 3-dimensional Euclidean field theory on X 3X_3 (Ginsparg 80, Appelquist-Pisarski 81, Nadkarni 83, Jourjine 84, Nadkarni 88).

As usual with KK-reduction, some care must be exercised to ensure that the compactified theory is itself still a local field theory. Depending on how exactly one proceeds this may be subtle (Landsman 89), but there exist robust approaches (Reisz 92, Kajantie-Laine-Rummukainen-Shaposhnikov 96).


The expansion of thermal field theory around the infinite-temperature-limit (i.e. around β=1/T=0\beta = 1/T = 0, i.e. KK-reduction in compact/periodic Euclidean time) is discussed in

  • Paul Ginsparg, First and second order phase transitions in gauge theories at finite temperature, Nuclear Physics B Volume 170, Issue 3, 15 December 1980, Pages 388-408 (doi:10.1016/0550-3213(80)90418-6)

  • Thomas Appelquist, Robert Pisarski, High-temperature Yang-Mills theories and three-dimensional quantum chromodynamics, Phys. Rev. D 23, 2305 (1981) (doi:10.1103/PhysRevD.23.2305)

  • Sudhir Nadkarni, Dimensional reduction in finite-temperature quantum chromodynamics, Phys. Rev. D 27, 917 (1983) (doi:10.1103/PhysRevD.27.917)

  • Sudhir Nadkarni, Dimensional reduction in finite-temperature quantum chromodynamics. II, Phys. Rev. D 38, 3287 (1988) (doi:10.1103/PhysRevD.38.3287)

  • Alexander N Jourjine, Quantum field theory in the infinite temperature limit, Annals of Physics Volume 155, Issue 2, July 1984, Pages 305-332 (doi:10.1016/0003-4916(84)90003-4)

  • Klaas Landsman, Limitations to dimensional reduction at high temperature, Nuclear Physics B Volume 322, Issue 2, 14 August 1989, Pages 498-530 (doi:10.1016/0550-3213(89)90424-0)

  • T. Reisz, Realization of dimensional reduction at high temperature, Z. Phys. C - Particles and Fields (1992) 53: 169 (doi:10.1007/BF01483886)

  • Eric Braaten, Solution to the Perturbative Infrared Catastrophe of Hot Gauge Theories, Phys. Rev. Lett. 74, 2164 (1995) (doi:10.1103/PhysRevLett.74.2164)

  • K. Kajantie, M. Laine, K. Rummukainen, M. Shaposhnikov, Generic rules for high temperature dimensional reduction and their application to the standard model, Nuclear Physics B Volume 458, Issues 1–2, 1 January 1996, Pages 90-136 (doi:10.1016/0550-3213(95)00549-8)

and specifically with an eye to discussion of the quark-gluon plasma in

  • Jean-Paul Blaizot, Edmond Iancu, Anton Rebhan, section 2.2.4 of Thermodynamics of the high temperature quark gluon plasma, Quark–Gluon Plasma 3, pp. 60-122 (2004) (arXiv:hep-ph/0303185, spire:615570)

  • Jean-Paul Blaizot, around p. 17 of Thermodynamics of the high temperature Quark-Gluon Plasma, AIP Conf. Proc. 739, 63-96 (2004) (doi:10.1063/1.1843592)

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