## Elliptic thermal correlation functions and modular forms in a globally conformal invariant QFT.(English)Zbl 1111.81112

Summary: Global conformal invariance (GCI) of quantum field theory (QFT) in two and higher space-time dimensions implies the Huygens’ principle, and hence, rationality of correlation functions of observable fields. The conformal Hamiltonian H has discrete spectrum assumed here to be finitely degenerate. We then prove that thermal expectation values of field products on compactified Minkowski space can be represented as finite linear combinations of basic (doubly periodic) elliptic functions in the conformal time variables (of periods 1 and $$\tau$$) whose coefficients are, in general, formal power series in $$q^{\frac12} = e^{i\pi\tau}$$ involving spherical functions of the ”space-like” fields’ arguments. As a corollary, if the resulting expansions converge to meromorphic functions, then the finite temperature correlation functions are elliptic. Thermal 2-point functions of free fields are computed and shown to display these features. We also study modular transformation properties of Gibbs energy mean values with respect to the (complex) inverse temperature $$\tau(\text{lm}\tau = \frac{\beta}{2\pi} > 0)$$. The results are used to obtain the thermodynamic limit of thermal energy densities and correlation functions.

### MSC:

 81T05 Axiomatic quantum field theory; operator algebras 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 11Z05 Miscellaneous applications of number theory
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### References:

 [1] Atiyah M. F., Introduction to Commutative Algebra (1969) · Zbl 0175.03601 [2] DOI: 10.1073/pnas.83.10.3068 · Zbl 0613.17012 [3] DOI: 10.1007/BF02749678 · Zbl 0152.46304 [4] Bros J., Nuclear Phys. 423 pp 291– [5] DOI: 10.1007/BF01454978 · Zbl 0626.46064 [6] DOI: 10.1007/BF02096782 · Zbl 0773.47007 [7] DOI: 10.1007/s00220-003-0839-z · Zbl 1037.81063 [8] DOI: 10.2307/1968455 · Zbl 0014.08004 [9] Dowker J. S., Nuclear Phys. 638 pp 405– · Zbl 0997.81114 [10] DOI: 10.1090/surv/088 [11] Waldschmidt M., From Number Theory to Physics (1995) [12] Gursey F., Phys. Rev. 7 pp 2414– [13] DOI: 10.1007/978-3-642-61458-3 [14] DOI: 10.1007/BF01646342 · Zbl 0171.47102 [15] DOI: 10.1023/A:1023379022112 · Zbl 1020.83026 [16] Hortacsu M., Phys. Rev. 5 pp 2519– [17] Jost R., The General Theory of Quantized Fields (1965) · Zbl 0127.19105 [18] DOI: 10.1090/ulect/010 [19] DOI: 10.1007/s00220-003-0979-1 · Zbl 1043.81063 [20] DOI: 10.1103/PhysRev.60.252 · Zbl 0027.28505 [21] DOI: 10.1007/978-1-4612-4752-4 [22] DOI: 10.1007/BF01608988 [23] DOI: 10.1007/BF01613145 · Zbl 0352.22012 [24] DOI: 10.1007/978-0-8176-4578-6 · Zbl 0549.14014 [25] DOI: 10.1007/s00220-004-1133-4 · Zbl 1125.17010 [26] DOI: 10.1088/0305-4470/35/12/319 · Zbl 1041.81097 [27] Nikolov N. M., Nucl. Phys. 670 pp 373– · Zbl 1058.81054 [28] DOI: 10.1007/s002200100414 · Zbl 0985.81055 [29] Schroer B., Phys. Lett. 506 pp 337– · Zbl 0977.81131 [30] DOI: 10.1090/S0002-9904-1971-12815-X · Zbl 0242.53009 [31] DOI: 10.1007/BF01856877 [32] Streater R. F., PCT, Spin and Statistics and All That (2000) · Zbl 1026.81027 [33] DOI: 10.1007/3540171630_96 [34] Uhlmann A., Acta Phys. Pol. 24 pp 293– [35] DOI: 10.1007/978-3-322-90166-8_2 [36] DOI: 10.1090/S0894-0347-96-00182-8 · Zbl 0854.17034
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