nLab infinite-temperature thermal field theory

Idea

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 $T$.

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 $X_3 \times S^1_\beta$ whose compact/periodic Euclidean time runs along a circle $S^1_\beta$ of circumference the inverse temperature $\beta = 1/T$, the limit of infinite temperature corresponds to the limit $\beta \to 0$ in which this circle fiber shrinks away.

In terms of the thermal Euclidean field theory on $X_3 \times S^1_\beta$ 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 $X_3$ (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).

References

The expansion of thermal field theory around the infinite-temperature-limit (i.e. around $\beta = 1/T = 0$, 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)

Review:

• Alberto Salvio: Introduction to Thermal Field Theory [arXiv:2411.02498]