Abstract
The structure of the thermal equilibrium state of a weakly interacting Bose gas is of current interest. We calculate the density matrix of that state in two ways. The most effective method, in terms of yielding a simple, explicit answer, is to construct a generating function within the traditional framework of quantum statistical mechanics. The alternative method, arguably more interesting, is to construct the thermal state as a vector state in an artificial system with twice as many degrees of freedom. It is well known that this construction has an actual physical realization in the quantum thermodynamics of black holes, where the added degrees of freedom correspond to the second sheet of the Kruskal manifold and the thermal vector state is a state of the Unruh or the Hartle–Hawking type. What is unusual about the present work is that the Bogolubov transformation used to construct the thermal state combines in a rather symmetrical way with Bogolubov's original transformation of the same form, used to implement the interaction of the nonideal gas in linear approximation. In addition to providing a density matrix, the method makes it possible to calculate efficiently certain expectation values directly in terms of the thermal vector state of the doubled system.
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REFERENCES
J. D. Bekenstein, Phys. Rev. D 7, 2333(1973); 12, 3077(1975).
J. D. Bekenstein and A. Meisels, Phys. Rev. D 15, 2775(1977).
S. W. Hawking, Comm. Math. Phys. 43, 199(1975); Phys. Rev. D 13, 191(1976); Phys. Rev. D 14, 2460(1976).
B. S. DeWitt, Phys. Rep. 19, 295(1975). R. M. Wald, Comm. Math. Phys. 45, 9(1975). R. M. WaldPhys. Rev. D 13, 3176(1976). L. Parker, Phys. Rev. D 12, 1519(1975).
W. G. Unruh, Phys. Rev. D 14, 870(1976).
J. B. Hartle and S. W. Hawking, Phys. Rev. D 13, 2188(1976).
S. A. Fulling, J. Phys. A 10, 917(1977).
G. W. Gibbons and M. J. Perry, Phys. Rev. Lett. 36, 985(1976); Proc. Roy. Soc. A 358, 467(1978).
G. L. Sewell, Ann. Phys. (N.Y.) 141, 201(1982).
S. A. Fulling and S. N. M. Ruijsenaars, Phys. Rep. 152, 135(1987). B. S. Kay and R. M. Wald, Phys. Rep. 207, 49(1991).
R. Kubo, J. Phys. Soc. Japan 12, 570(1957). P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342(1959).
R. Haag, N. M. Hugenholtz, and M. Winnink, Comm. Math. Phys. 5, 215(1967).
L. Parker, Nature 261, 20(1976).
W. Israel, Phys. Lett. A 57, 107(1976).
H. Araki and E. J. Woods, J. Math. Phys. 4, 637(1963).
Y. Takahashi and H. Umezawa, Collective Phenom. 2, 55(1975). H. Umezawa, H. Matsumoto, and M. Tachiki, Thermo Field Dynamics and Condensed States (North-Holland, Amsterdam, 1982).
J. J. Bisognano and E. H. Wichmann, J. Math. Phys. 16, 985(1975); 17, 303(1976).
G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738(1977).
N. N. Bogolubov, J. Phys. (USSR) 11, 23(1947).
N. N. Bogoliubov, Lectures on Quantum Statistics, Vol. 1 (Gordon & Breach, New York, 1967) (translation of Ukrainian original, 1949).
V. A. Zagrebnov and J.-B. Bru, Phys. Rep. 350, 291(2001).
V. V. Kocharovsky, Vl. V. Kocharovsky, and M. O. Scully, Phys. Rev. A 61, 053606(2000).
E. A. Cornell and C. E. Wieman, Rev. Mod. Phys. 74, 875(2002). W. Ketterle, Rev. Mod. Phys. 74, 1131(2002).
A. Brunello, F. Dalfovo, L. Pitaevski, and S. Stringari, Phys. Rev. Lett. 85, 4422(2000). J. M. Vogels, K. Xu, C. Raman, J. R. Abo-Shaeer, and W. Ketterle, Phys. Rev. Lett. 88, 060402(2002).
B.-G. Englert, S. A. Fulling, and M. D. Pilloff, Optics Comm. 208, 139(2002).
D. E. Knuth, The Art of Computer Programming, Vol. 1, 2nd edn. (Addison–Wesley, Reading, 1973), Sec. 1.2.6.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Sec. 8.96.
M. Petkovšek, H. S. Wilf, and D. Zeilberger, A=B (Peters, Wellesley, 1996).
M. A. Rashid and A. Mahmood, J. Phys. A 34, 8185(2001).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
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Fulling, S.A., Englert, BG. & Pilloff, M.D. Interacting Bosons at Finite Temperature: How Bogolubov Visited a Black Hole and Came Home Again. Foundations of Physics 33, 87–110 (2003). https://doi.org/10.1023/A:1022819825765
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DOI: https://doi.org/10.1023/A:1022819825765