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Combinatorics and Boson normal ordering: A gentle introduction
1.P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, New York, 1982), 4th ed., Chap. 4.
2.L. E. Ballentine, Quantum Mechanics: Modern Development (World Scientific, Singapore, 1998).
4.J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).
5.J. R. Klauder and B.-S. Skagerstam, Coherent States. Application in Physics and Mathematical Physics (World Scientific, Singapore, 1985);
6.W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1990).
7.L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U.P., Cambridge, 1995).
8.W. P. Schleich, Quantum Optics in Phase Space (Wiley, Berlin, 2001).
9.J. Katriel, “Combinatorial aspects of boson algebra,” Lett. Nuovo Cimento 10, 565–567 (1974).
10.G. Dobiński, “Summierung der Reihe für ,” Grunert Archiv (Arch. für M. und Physik 61, 333–336 (1877);
10.G.-C. Rota, “The number of partitions of a set,” Amer. Math. Monthly 71, 498–504 (1964).
11.H. S. Wilf, Generatingfunctionology (Academic, New York, 1994), 2nd ed.
12.Coherent states are not orthogonal for different and the overlapping factor is . They constitute an overcomplete basis in the sense of resolution of the identity .
13.The double dot notation is almost universal in quantum optics and quantum field theory. Nevertheless some authors, for example, Ref. 6, use an alternative notation.
14.Careless use of the double dot notation may lead to inconsistencies, for example if and , we have , but . Such problems are eliminated if a rigorous definition, beyond this note is given. See A. I. Solomon, P. Blasiak, G. H. E. Duchamp, A. Horzela, and K. A. Penson, unpublished.
15.Wick’s theorem is usually formulated for the time ordered, also called chronologically ordered, products of field operators.
15.See, for example, J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw-Hill, New York, 1965), Chap. 17. In our context operators do not depend on the space-time coordinate and chronological ordering reduces to the normal ordering procedure discussed in this paper. See for example, Ref. 2.
16.R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962–982 (1967);
16.C. L. Mehta, “Ordering of the exponential of a quadratic in boson operators. I. Single mode case,” J. Math. Phys. 18, 404–407 (1977).
16.See also A. DasGupta, “Disentanglement formulas: An alternative derivation and some applications to squeezed coherent states,” Am. J. Phys. 64, 1422–1427 (1996).
17.J. Katriel, “Bell numbers and coherent states,” Phys. Lett. A 237, 159–161 (2000);
17.J. Katriel, “Coherent states and combinatorics,” J. Opt. B: Quantum Semiclassical Opt. 237, S200–S203 (2002).
18.L. Comtet, Advanced Combinatorics (Reidel, Dordrecht, 1974), Chap. 5;
18.J. Riordan, An Introduction to Combinatorial Analysis (Wiley, NY, 1984), Chap. 2.7;
18.R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics (Addison-Wesley, MA, 1994), Chap. 6.1.
19.The simplest representation acts on the space of polynomials and is defined by and . It may be naturally extended to the space of formal power series, see Refs. 11 and 36.
20.For convenience and to avoid inaccuracy, the definitions of Stirling and Bell numbers are usually extended by the following conventions: and for or .
21.P. Blasiak, A. Horzela, K. A. Penson, and A. I. Solomon, “Dobiński-type relations: Some properties and physical applications,” J. Phys. A 37, 4999–5006 (2006).
22.Because the generating functions are formal series, the question of convergence does not arise.
23.P. Blasiak, “Combinatorics of boson normal ordering and some applications,” Concepts of Physics 1, 177–278 (2004), arXiv:quant-ph/0510082.
24.We suggest derivation of these formulas as a problem to be solved by students during classes on the Fock space methods. Detailed calculations may be found in Refs. 23 and 31.
25.M. A. Méndez, P. Blasiak, and K. A. Penson, “Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem,” J. Math. Phys. 46, 083511–1 (2005).
26.A. I. Solomon, G. H. E. Duchamp, P. Blasiak, A. Horzela, and K. A. Penson, “Normal order: Combinatorial graphs,” in Proceedings of 3rd International Symposium on Quantum Theory and Symmetries, edited by P. C. Argyres, T. J. Hodges, F. Mansouri, J. J. Scanio, P. Suranyi, and L. C. R. Wijewardhana (World Scientific, Singapore, 2004), pp. 527–536, arXiv:quant-ph/0402082;
26.A. Varvak, “Rook numbers and the normal ordering problem,” J. Combin. Theory Ser. A 112, 292–307 (2005).
27.The conjugate operators and , satisfying , are subject to the Heisenberg uncertainty relation . For the coherent state the product of uncertainties exactly equals . These are the only states with this property that additionally have equal uncertainties (in general, we obtain the squeezed states (Refs. 5 and 7).
28.A detailed proof of this bijection should take into consideration the specific structure of contractions between blocks which make the order in such constructed class irrelevant.
29.D. Stoler, “Generalized coherent states,” Phys. Rev. D 2, 2309–2312 (1971);
29.A useful review is V. V. Dodonov, “‘Nonclassical’ states in quantum optics: A ‘squeezed’ review of the first 75 years,” J. Opt. B: Quantum Semiclassical Opt. 4, R1–R33 (2002).
32.G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela, and P. Blasiak, “One-parameter groups and combinatorial physics,” in Proceedings of 3rd International Workshop on Contemporary Problems in Mathematical Physics, edited by J. Govaerts, M. N. Hounkonnou, and A. Z. Msezane (World Scientific, Singapore, 2004), pp. 439–449, arXiv:quant-ph/0401126.
33.P. Blasiak, A. Horzela, K. A. Penson, G. H. E. Duchamp, and A. I. Solomon, “Boson normal ordering via substitutions and Sheffer-type polynomials,” Phys. Lett. A 37, 108–116 (2005);
33.P. Blasiak, G. Dattoli, A. Horzela, and K. A. Penson, “Representations of monomiality principle with Sheffer-type polynomials and boson normal ordering,” Phys. Lett. A 352, 7–12 (2006).
34.J. Katriel and M. Kibler, “Normal ordering for deformed boson operators and operator-valued deformed Stirling numbers,” J. Phys. A 25, 2683–2691 (1992);
34.see also P. Blasiak, A. Horzela, K. A. Penson, and A. I. Solomon, “Deformed bosons: Combinatorics of normal ordering,” Czech. J. Phys. 54, 1179–1184 (2004).
35.M. Abramovitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 24.
36.S. Roman, The Umbral Calculus (Academic Press, Orlando, 1984).
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