Another formulation of the Wick’s theorem. Farewell, pairing?

Igor V. Beloussov 1
  • 1 Institute of Applied Physics, Academy of Sciences of Moldova, 5 Academy Str., Kishinev, 2028, Republic of Moldova


The algebraic formulation of Wick’s theorem that allows one to present the vacuum or thermal averages of the chronological product of an arbitrary number of field operators as a determinant (permanent) of the matrix is proposed. Each element of the matrix is the average of the chronological product of only two operators. This formulation is extremely convenient for practical calculations in quantum field theory, statistical physics, and quantum chemistry by the standard packages of the well known computer algebra systems.

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  • [1] S. Weinberg, The Quantum Theory of Fields: V. 1, Foundations, Cambridge University Press, Cambridge, 1995.

  • [2] S. S. Schweber, An Introduction to Relativistic Quantum Field Theory, John Weatherhill, Tokyo, 1961.

  • [3] M. E. Peskin, D. V. Schroeder, Introduction to Quantum Field Theory, Addison-Wesley, Redwood City, 1995.

  • [4] N. N. Bogoliubov, D. V. Shirkov, Introduction to the Theory of Quantized Fields, John Willey & Sons, New York, 1980.

  • [5] J. W. Negele, H. Orland, Quantum Many-Particle Systems, Westview Press, 1998.

  • [6] A. A. Abrikosov, L. P. Gor‘kov, I. E. Dzyaloshinski, Methods of QuantumField Theory in Statistical Physics, Dover Publications, Inc., New York, 1963.

  • [7] A. L. Fetter, J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, 1971.

  • [8] A. Szabo, N. S. Ostlund, Modern Quantum Chemistry, Macmillan, New York, 1982.

  • [9] G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Co., New York, 1961.

  • [10] M. Veltman, D. N. Williams, “Schoonschip ’91”, hep-ph/9306228.

  • [11] H. Strubbe,Manual for SCHOONSCHIP a CDC 6000/7000 programfor symbolic evaluation of algebraic expressions, Comput. Phys. Commun. 8 (1974) 1–30.

  • [12] The feyncalc home page, entry point to information about feyncalc and related programs for high energy physics. URL

  • [13] A. Hsieh, HIP: Symbolic High-Energy Physics Calculations, Computers in Physics, 6 (1992) 253–261.

  • [14] A. C. Hearn, REDUCE User’s Manual, Version 3.8, Santa Monica, CA, USA, 2004.

  • [15] J. A. M. Vermaseren, Symbolic Manipulation with FORM, Computer Algebra Nehterlands, Amsterdam, 1991.

  • [16] L. Brücher, J. Franzkowski, D. Kreimer, xloops — automated Feynman diagram calculation, Comput. Phys. Commun. 115 (1998) 140–160.

  • [17] R. ˘Zitko, SNEG – Mathematica package for symbolic calculations with second-quantization-operator expressions, Comput. Phys. Commun. 182 (2011) 2259–2264.

  • [18] C. L. Janssen, H. F. Schaefer. III, The automated solution of second quantization equations with applications to the coupled cluster approach, Theor. Chim. Acta. 79 (1991) 1–42.

  • [19] S. Hirata, Tensor contraction engine: abstraction and automated parallel implementation of configuration-interaction, coupled-cluster, and many-body perturbation theories, J. Phys. Chem. A 107 (2003) 9887–9897.

  • [20] I. Lindgren, J. Morrison, Atomic Many–Body Theory, 2nd ed., Springer–Verlag, Berlin, 1986.

  • [21] A. Derevianko, Post-Wick theorems for symbolic manipulation of second-quantized expressions in atomic many-body perturbation theory, J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 074001.

  • [22] A. Derevianko, Post-wick theorems for symbolic manipulation of second-quantized expressions in atomic many-body perturbation theory, arXiv 0910.3613v1.

  • [23] S. Wolfram, The Mathematica Book, 5th ed., Wolfram Media, Champaign, USA, 2003.

  • [24] Maple 15 Programming Guide, Maplesoft, 2011. URL

  • [25] S. Wieder, Introduction to Mathcad for Scientists and Engineers, Mcgraw-Hill College, 1992.


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