Finite-temperature effects on the Gaussian effective potential

G. A. Hajj; P. M. Stevenson

doi:10.1103/PhysRevD.37.413

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Finite-temperature effects on the Gaussian effective potential

G. A. Hajj and P. M. Stevenson

T. W. Bonner Laboratories, Physics Department, Rice University, Houston, Texas 77251-1892

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Phys. Rev. D 37, 413 – Published 15 January, 1988

DOI: https://doi.org/10.1103/PhysRevD.37.413

Abstract

We derive the finite-temperature Gaussian effective potential (FTGEP) from first principles, resolving some confusions arising in earlier treatments. The advantages of the FTGEP approach over the conventional loop expansion, even for high temperatures, are illustrated in a quantum-mechanical example. Results are presented for $φ^{6}$ theory in two and three dimensions and for the ‘‘autonomous’’ version of λ $φ^{4}$ theory in four dimensions.

References (27)

D. A. Kirzhnits and A. D. Linde, Phys. Lett. 42B, 471 (1972).
L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974); C. W. Bernard, ibid. 9, 3312 (1974); S. Weinberg, ibid. 9, 3357 (1974).
T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); S. J. Chang, ibid. 12, 1071 (1975); G. Rosen, Phys. Rev. 173, 1632 (1968); L. I. Schiff, ibid. 130, 458 (1963). A more complete list of references can be found in Ref. 6.
W. A. Bardeen and M. Moshe, Phys. Rev. D 28, 1372 (1983).
P. M. Stevenson, Phys. Rev. D 30, 1712 (1984).
P. M. Stevenson, Phys. Rev. D 32, 1389 (1985); Z. Phys. C 24, 87 (1984).
P. M. Stevenson and I. Roditi, Phys. Rev. D 33, 2305 (1986).
P. M. Stevenson, B. Alle`s and R. Tarrach, Phys. Rev. D 35, 2407 (1987).
I. Roditi, Phys. Lett. 169B, 264 (1986).
B. Alle`s and R. Tarrach, Phys. Rev. D 33, 1718 (1986); ibid. 34, 664(E) (1986).
W. A. Bardeen and M. Moshe, Phys. Rev. D 34, 1229 (1986).
I. Roditi, Phys. Lett. B 177, 85 (1986).
A. Okopinska, Phys. Rev. D 36, 2415 (1987).
M. Consoli and A. Ciancitto, Nucl. Phys. B254, 653 (1985); Y. Brihaye and M. Consoli, Nuovo Cimento 94A, 1 (1986).
P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986).
P. M. Stevenson, Z. Phys. C 35, 467 (1987).
K. G. Wilson and J. Kogut, Phys. Rep. 12C, 75 (1974); G. A. Baker and J. M. Kincaid, Phys. Rev. Lett. 42, 1431 (1979); B. Freedman, P. Smolensky and D. Weingarten, Phys. Lett. 113B, 481 (1982); D. J. E. Callaway and R. Petronzio, Nucl. Phys. B240, 577 (1984); I. A. Fox and I. G. Halliday, Phys. Lett. 159B, 148 (1985); C. B. Lang, Nucl. Phys. B265, 630 (1986).
M. Aizenman, Phys. Rev. Lett. 47, 1 (1981); Commun. Math. Phys. 86, 1 (1982) J. Fröhlich, Nucl. Phys. B200, 281 (1982), and references therein. A. Sokal, Ann. Inst. Henri Poincaré 37, 317 (182).
For some interesting remarks expressing reservations about the ``triviality'' claims, see N. N. Khuri, Phys. Rev. D 28, 3041 (1983); G. Gallavotti and V. Rivasseau, Ann. Inst. Henri Poincaré 40, 185 (1984).
A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1965); P. M. Morse, Thermal Physics (Benjamin, New York, 1969).
See, for example, L. I. Schiff, Quantum Mechanics (McGraw-Hill, Kogakusha, Tokyo, 1968).
Details will be presented in G. A. Hajj, Ph.D. thesis, Rice University (unpublished).
It is curious that their version of $I_{1}^{FT}$ looks like a natural generalization of the canonical form of $I_{1}$ , (2.7), while the correct result is a natural generalization of the covariant form (3.10).
G. M. Shore (private communication).
F. T. Hioe and E. W. Montroll, J. Math. Phys. 16, 1945 (1975).
The names have been changed from alpha, beta to $α_{4}$ , $α_{6}$ to avoid confusion with β == 1 / kT.
There is some indication, from a dimensional-continuation viewpoint, that negative values of $m_{0}^{2}$ are not allowed. See Ref. 16.

[c1] D. A. Kirzhnits and A. D. Linde, Phys. Lett. 42B, 471 (1972).

[c2] L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974); C. W. Bernard, ibid. 9, 3312 (1974); S. Weinberg, ibid. 9, 3357 (1974).

[c3] T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); S. J. Chang, ibid. 12, 1071 (1975); G. Rosen, Phys. Rev. 173, 1632 (1968); L. I. Schiff, ibid. 130, 458 (1963). A more complete list of references can be found in Ref. 6.

[c4] W. A. Bardeen and M. Moshe, Phys. Rev. D 28, 1372 (1983).

[c5] P. M. Stevenson, Phys. Rev. D 30, 1712 (1984).

[c6] P. M. Stevenson, Phys. Rev. D 32, 1389 (1985); Z. Phys. C 24, 87 (1984).

[c7] P. M. Stevenson and I. Roditi, Phys. Rev. D 33, 2305 (1986).

[c8] P. M. Stevenson, B. Alle`s and R. Tarrach, Phys. Rev. D 35, 2407 (1987).

[c9] I. Roditi, Phys. Lett. 169B, 264 (1986).

[c10] B. Alle`s and R. Tarrach, Phys. Rev. D 33, 1718 (1986); ibid. 34, 664(E) (1986).

[c11] W. A. Bardeen and M. Moshe, Phys. Rev. D 34, 1229 (1986).

[c12] I. Roditi, Phys. Lett. B 177, 85 (1986).

[c13] A. Okopinska, Phys. Rev. D 36, 2415 (1987).

[c14] M. Consoli and A. Ciancitto, Nucl. Phys. B254, 653 (1985); Y. Brihaye and M. Consoli, Nuovo Cimento 94A, 1 (1986).

[c15] P. M. Stevenson and R. Tarrach, Phys. Lett. B 176, 436 (1986).

[c16] P. M. Stevenson, Z. Phys. C 35, 467 (1987).

[c17] K. G. Wilson and J. Kogut, Phys. Rep. 12C, 75 (1974); G. A. Baker and J. M. Kincaid, Phys. Rev. Lett. 42, 1431 (1979); B. Freedman, P. Smolensky and D. Weingarten, Phys. Lett. 113B, 481 (1982); D. J. E. Callaway and R. Petronzio, Nucl. Phys. B240, 577 (1984); I. A. Fox and I. G. Halliday, Phys. Lett. 159B, 148 (1985); C. B. Lang, Nucl. Phys. B265, 630 (1986).

[c18] M. Aizenman, Phys. Rev. Lett. 47, 1 (1981); Commun. Math. Phys. 86, 1 (1982) J. Fröhlich, Nucl. Phys. B200, 281 (1982), and references therein. A. Sokal, Ann. Inst. Henri Poincaré 37, 317 (182).

[c19] For some interesting remarks expressing reservations about the ``triviality'' claims, see N. N. Khuri, Phys. Rev. D 28, 3041 (1983); G. Gallavotti and V. Rivasseau, Ann. Inst. Henri Poincaré 40, 185 (1984).

[c20] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1965); P. M. Morse, Thermal Physics (Benjamin, New York, 1969).

[c21] See, for example, L. I. Schiff, Quantum Mechanics (McGraw-Hill, Kogakusha, Tokyo, 1968).

[c22] Details will be presented in G. A. Hajj, Ph.D. thesis, Rice University (unpublished).

[c23] It is curious that their version of $I_{1}^{FT}$ looks like a natural generalization of the canonical form of $I_{1}$ , (2.7), while the correct result is a natural generalization of the covariant form (3.10).

[c24] G. M. Shore (private communication).

[c25] F. T. Hioe and E. W. Montroll, J. Math. Phys. 16, 1945 (1975).

[c26] The names have been changed from alpha, beta to $α_{4}$ , $α_{6}$ to avoid confusion with β == 1 / kT.

[c27] There is some indication, from a dimensional-continuation viewpoint, that negative values of $m_{0}^{2}$ are not allowed. See Ref. 16.

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