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J 1 J 1 J 2 J 1 J 1 J 2 J 3 2 3
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k 1 terms 1 k terms
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The different terms in A give the same symmetrized operator Thus,
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S(A) = 1 S J 1 1 1 J 2 +1 J 3 3 2
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Coherent States in Quantum Physics Jean-Pierre Gazeau Copyright 2009 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 978-3-527-40709-5
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Appendix E Symmetrization of the Commutator
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= 1
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( 1 1)!( 2 + 1)! 3 ! l!
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J u1 J ul
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Similarly, for B, S(B) = 2 Now we calculate I = J 3 , S( J 1 J 2 J 3 ) 2 3 1 = 1 ! 2 ! 3 ! l!
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( 1 + 1)!( 2 1)! 3 ! l!
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The sum splits into two parts, according to the value of u k = 1 or 2: I =A +B , with A = 1 ! 2 ! 3 ! l! J u 1 J u k 1 J 2 J u k+1 J u l
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u U 1 , 2 , 3 k|u k =1
and B = 1 ! 2 ! 3 ! l! J u 1 J u k 1 J 1 J u k+1 J u l
u U 1 , 2 , 3 k|u k =2
Let us examine the constituents of A There are of the form J u 1 J u l , with u U 1 1, 2 +1, 3 Their number is l!/( 1 ! 2 ! 3 !) ~ 1 , but they are not all different Each monomial emerges from a term where a J 1 has been transformed into a J 2 Since there are 2 +1 occurrences of J 2 in each term, each monomial appears 2 +1 times We now group these identical terms: A = 1 ! 2 ! 3 ! ( 2 + 1) l! J u1 J ul
It remains to determine the de nition set of the summation Let us rst estimate the number of its terms, namely, N = 1 l! l! = 1 ! 2 ! 3 ! 2 + 1 ( 1 1)!( 2 + 1)! 3 !
This is the number of elements in U 1 1, 2 +1, 3 On the other hand, all the elements of U 1 1, 2 +1, 3 appear In the contrary case, the retransformation of a J 2 into a J 1 would provide some elements not appearing in I, which cannot be true It re-
Appendix E Symmetrization of the Commutator
sults that the sum comprises exactly all symmetrized expressions of J 1 1 1 J 2 +1 J 3 3 2 Thus,
1 ! 2 ! 3 ! ( 2 + 1) l!
J u1 J ul
u U 1 1, 2 +1, 3
= 1
( 1 1)!( 2 + 1)! 3 ! l!
J u1 J ul
u U 1 1, 2 +1, 3
= S(A) The application of the same method to B leads to the proof
References
1 Schr dinger, E (1926) Der stetige bergang von der Mikro- zur Makromechanik, Naturwiss, 14, 664 2 Klauder, JR (1960) The action option and the Feynman quantization of spinor elds in terms of ordinary c-numbers, Ann Phys, 11, 123 3 Klauder, JR (1963) Continuous Representation theory I Postulates of continuous-representation theory, J Math Phys, 4, 1055 4 Klauder, JR (1963) Continuous Representation theory II Generalized relation between quantum and classical dynamics, J Math Phys, 4, 1058 5 Glauber, RJ (1963) Photons correlations, Phys Rev Lett, 10, 84 6 Glauber, RJ (1963) The quantum theory of optical coherence, Phys Rev, 130, 2529 7 Glauber, RJ (1963) Coherent and incoherent states of radiation eld, Phys Rev, 131, 2766 8 Sudarshan, ECG (1963) Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams, Phys Rev Lett, 10, 277 9 Zhang, W-M, Feng, DH, and Gilmore, R (1990) Coherent states: Theory and some applications, Rev Mod Phys, 26, 867 10 Perelomov, AM (1986) Generalized Coherent States and their Applications, Springer-Verlag, Berlin 11 Syad Twareque Ali, Antoine, JP, and Gazeau, JP (2000) Coherent states,
wavelets and their generalizations Graduate Texts in Contemporary Physics, SpringerVerlag, New York Dodonov, VV (2002) Nonclassical states in quantum optics: a squeezed review of the rst 75 years, J Opt B: Quantum Semiclass Opt, 4, R1 Dodonov, VV and Man ko, VI (eds) (2003) Theory of Nonclassical States of Light, Taylor & Francis, London, New York Vourdas, A (2006) Analytic representations in quantum mechanics, J Phys A, 39, R65 Klauder, JR and Skagerstam, BS (eds) (1985) Coherent states Applications in physics and mathematical physics, World Scienti c Publishing Co, Singapore Feng, DH, Klauder, JR and Strayer, M (eds) (1994) Coherent States: Past, Present and Future, Proceedings of the 1993 Oak Ridge Conference World Scienti c, Singapore Landau, LD and Lifshitz, EM (1958) Quantum Mechanics Non-relativistic Theory, Pergamon Press, Oxford Magnus, W, Oberhettinger, F and Soni, RP (1966) Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, Berlin, Heidelberg and New York Lieb, EH (1973) The classical limit of quantum spin systems, Commun Math Phys, 31, 327
Coherent States in Quantum Physics Jean-Pierre Gazeau Copyright 2009 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 978-3-527-40709-5
References 20 Berezin, FA (1975) General concept of quantization, Commun Math Phys, 40, 153 21 von Neumann, J (1955) Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, NJ, English translation by RT Byer 22 Torr sani, B (1995) Analyse continue par ondelettes, EDP Sciences, Paris 23 Cohen, L (1994) Time-frequency analysis: theory and applications, Prentice Hall Signal Processing Series, NJ 24 Mallat, SG (1998) A Wavelet Tour of Signal Processing, Academic Press, New York, NY 25 Scully, MO and Suhail Zubairy, M (1997) Quantum Optics, Cambridge Univ Press, Cambridge 26 Schleich, WP (2001) Quantum optics in Phase space Wiley-VCH Verlag GmbH 27 Feynman, RP (1948) Space-time approach to nonrelativistic quantum mechanics, Rev Mod Phys, 20, 367 28 Klauder, JR (2003) The Feynman Path Integral: An Historical Slice, in A Garden of Quanta (eds Arafune, J et al), World Scienti c, Singapore, pp 55 76 29 Feynman, RP (1951) An operator calculus having applications in quantum electrodynamics, Phys Rev, 84, 108 30 Daubechies, I, and Klauder, JR (1985) Quantum mechanical path integrals with Wiener measures for all polynomial Hamiltonians II, J Math Phys, 26, 2239 31 dos Santos, LC and de Aguiar, MAM (2006) Coherent state path integrals in the Weyl representation, J Phys A: Math Gen, 39, 13465 32 Preskill, J (2008) Quantum Computation and Information, Caltech PH-229 Lecture Notes http://wwwtheorycaltechedu/ ~preskill/ph219/ph219_2008-09 33 Nielsen, MA and Chuang, IL (2000) Quantum Computation and Quantum Information, Cambridge University Press, Cambridge 34 Audenaert, KMR (2007) Mathematical Aspects of Quantum Information Theory, in Physics and Theoretical Computer Science, (eds Gazeau, J-P et al), IOS Press, pp 3 24 35 Holevo, AS (2001) Statistical Structure of Quantum Theory, Springer-Verlag, Berlin 36 Fuchs, CA (1996) Distinguishability and Accessible Information in Quantum Theory, PhD thesis, University of New Mexico 37 Peres, A (1995) Quantum Theory: Concepts and Methods, Kluwer Academic Publishers, Dordrecht 38 Helstrom, CW (1976) Quantum Detection and Estimation Theory, Academic Press, New York 39 Bennett, CH and Brassard, G (1984) in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, IEEE, New York, p 175 40 Cook, RL, Martin, PJ and Geremia, JM (2007) Optical coherent state discrimination using a closed-loop quantum measurement, Nature, 446, 774 41 Dolinar, S (1973) Quaterly Progress Report Tech Rep 111, Research Laboratory of Electronics, MIT, p 115, unpublished 42 Geremia, JM (2004) Distinguishing between optical coherent states with imperfect detection, Phys Rev A, 70, 062303-1 43 Loudon, R (1973) The Quantum Theory of Light, Oxford Univ Press, Oxford 44 Kennedy, RS (1972) Quaterly Progress Report Tech Rep 110, Research Laboratory of Electronics, MIT, p 219, unpublished 45 Sasaki, M and Hirota, O (1996) Optimal decision scheme with a unitary control process for binary quantum-state signals, Phys Rev A, 54, 2728 46 Dolinar, S (1976) PhD thesis, Massachussets Institute of Technology 47 Bertsekas, DP (2000) Dynamic Programming and Optimal Control, vol 1, Athena Scienti c, Belmont, MA 48 Wittman, C, Takeoka, M, Cassemiro, KN, Sasaki, M, Leuchs, G and Andersen, UL (2008) Demonstration of near-optimal discrimination of optical coherent states, arXiv:08094953v1 [quantph] 49 Takeoka, M, and Sasaki, M (2008) Discrimination of the binary coherent signal: Gaussian-operation limit and simple non-Gaussian near optimal receivers, Phys Rev A, 78, 022320-1 50 Kotz, S, Balakrishnan, N, Read, CB and Vidakovic, B (2006) editors-in-chief: under the entries Bayesian Inference and
References Conjugate Families of Distributions Encyclopaedia of Statistical Sciences, 2nd edn, vol 1, John Wiley & Sons, Inc, Hoboken, NJ Box, GEP and Tiao, GC (1973) Bayesian Inference in Statistical Analysis, Wiley Classics Library, Wiley-Interscience Heller, B and Wang, M (2004) Posterior distributions on certain parameter spaces by using group theoretic methods adopted from quantum mechanics, Univ of Chicago, Dept of Statistics Technical Report Series, Tech Rep No 546, p 26 Heller, B and Wang, M (2007) Group invariant inferred distributions via noncommutative probability, in Recent Developments in Nonparametric Inference and Probability, IMS Lecture NotesMonograph Series, 50, p 1 Heller, B and Wang, M (2007) Posterior distribution for negative binomial parameter p using a group invariant prior, Stat Probab Lett, 77, 1542 Ali, ST, Gazeau, J-P and Heller, B (2008) Coherent states and Bayesian duality, J Phys A: Math Theor, 41, 365302 Daubechies, I (1992) Ten Lectures on Wavelets, SIAM, Philadelphia Klauder, JR (1996) Coherent states for the hydrogen atom, J Phys A: Math Gen, 29, L293 Gazeau, J-P and Klauder, JR (1999) Coherent states for systems with discrete and continuous spectrum, J Phys A: Math Gen, 32, 123 Gazeau, J-P and Monceau, P (2000) Generalized coherent states for arbitrary quantum systems in Conf rence Mosh Flato 1999 Quantization, Deformations, and Symmetries, vol II (eds Dito, G and Sternheimer, D), Kluwer, Dordrecht, pp 131 144 Busch, P, Lahti, P and Mittelstaedt, P (1991) The Quantum Theory of Measurement, LNP vol m2, Springer-Verlag, Berlin, second revised edition 1996 Ali, ST, Antoine, J-P, and Gazeau, J-P (1993) Continuous frames in Hilbert spaces, Ann Phys, 222, 1 De Bi vre, S and Gonz lez, JA (1993) Semiclassical behaviour of coherent states on the circle, in Quantization and Coherent States Methods in Physics (eds Odzijewicz, A et al), World Scienti c, Singapore Kowalski, K, Rembieli ski, J and Pan paloucas, LC (1996) Coherent states for a quantum particle on a circle, J Phys A: Math Gen, 29, 4149 Kowalski, K and Rembielinski, J (2002) Exotic behaviour of a quantum particle on a circle, Phys Lett A, 293, 109 Kowalski, K and Rembielinski, J (2002) On the uncertainty relations and squeezed states for the quantum mechanics on a circle, J Phys A: Math Gen, 35, 1405 Kowalski, K and Rembielinski, J (2003) Reply to the Comment on On the uncertainty relations and squeezed states for the quantum mechanics on a circle , J Phys A: Math Gen, 36, 5695 Gonz lez, JA, and del Olmo, MA (1998) Coherent states on the circle, J Phys A: Math Gen, 31, 8841 Hall, BC, and Mitchell, JJ (2002) Coherent states on spheres, J Math Phys, 43, 1211 Radcliffe, JM (1971) Some properties of spin coherent states, J Phys A, 4, 313 Gilmore, R (1972) Geometry of symmetrized states, Ann Phys (NY), 74, 391 Gilmore, R (1974) On properties of coherent states, Rev Mex Fis, 23, 143 Perelomov, AM (1972) Coherent states for arbitrary Lie group, Commun Math Phys, 26, 222 Hamilton, WR (1844) On a new species of imaginary quantities connected with a theory of quaternions, Proc R Ir Acad, 2, 424 Talman, JD (1968) Special Functions, A Group Theoretical Approach, WA Benjamin, New York, Amsterdam Edmonds, AR (1968) Angular Momentum in Quantum Mechanics, 2nd ed, Princeton University Press, Princeton, NJ, rev printing Campbell, WB (1971) Tensor and spinor spherical harmonics and the spins harmonics s Y lm ( , ), J Math Phys, 12, 1763 Newman, ET and Penrose, R (1966) Note on the Bondi-Metzner-Sachs Group, J Math Phys, 7, 863
69 70
56 57
71 72
References 78 Hu, W and White, M (1997) CMB anisotropies: total angular momentum method, Phys Rev D, 56, 596 79 Carruthers, P and Nieto, MM (1965) Coherent states and the forced quantum oscillator, Am J Phys, 33, 537 80 Wang, YK and Hioe, FT (1973) Phase transition in the Dicke maser model, Phys Rev A, 7, 831 81 Hepp, K and Lieb, EH (1973) Equilibrium statistical mechanics of matter interacting with the quantized radiation eld, Phys Rev A, 8, 2517 82 Erd lyi, A (1987) Asymptotic Expansions, Dover, New York 83 Fuller, W and Lenard, A (1979) Generalized quantum spins, coherent states, and Lieb inequalities, Commun Math Phys, 67, 69 84 Biskup, M, Chayes, L and Starr, S (2007) Quantum spin systems at positive temperature, Commun Math Phys, 269, 611 85 Encyclopedia of Laser Physics and Technology, http://wwwrp-photonicscom/ encyclopediahtml 86 Dicke, RH (1954) Coherence in Spontaneous Radiation Processes, Phys Rev, 93, 99 87 Rehler, NE and Eberly, JH (1971) Superradiance, Phys Rev A, 3, 1735 88 Bonifacio, R, Schwendimann, P and Haake, F (1971) Quantum statistical theory of superradiance I, Phys Rev A, 4, 302 89 Bonifacio, R, Schwendimann, P and Haake, F (1971) Quantum statistical theory of superradiance II, Phys Rev A, 4, 854 90 Berezin, FA (1971) Wick and anti-Wick symbols of operators, Mat Sb, 86, 578, (english transl in (1971) Math USSR-Sb, 15, 577) 91 Berezin, FA (1972) Covariant and contravariant symbols of operators, SSSR Ser Mat, 36, 1134, (english transl in (1973) Math USSR-Izv, 6, 1117) 92 Thirring, WE and Harrell, EM (2004) Quantum Mathematical Physics, Springer, Berlin 93 Gazeau, J-P and Hussin, V (1992) Poincar contraction of SU (1, 1) Fock Bargmann structure, J Phys A: Math Gen, 25, 1549 Gazeau, J-P and Renaud, J (1993) Lie algorithm for an interacting SU (1, 1) elementary system and its contraction, Ann Phys (NY), 222, 89 Gazeau, J-P and Renaud, J (1993) Relativistic harmonic oscillator and space curvature, Phys Lett A, 179, 67 Doubrovine, B, Novikov, S and Famenko, A (1982) G om trie Contemporaine, M thodes et Applications, (1 re Partie) Mir, Moscow Bargmann, V (1947) Irreducible unitary representations of the Lorentz group, Ann Math, 48, 568 Gelfand, I and Neumark, M (1947) Unitary representations of the Lorentz group, Acad Sci USSR J Phys, 10, 93 Vilenkin, NJ and Klimyk, AU (1991) Representations of Lie Groups and Special Functions, Kluwer Academic, Boston Puk nszky, L (1964) The Plancherel formula for the universal covering group of SL(R, 2), Math Ann, 156, 96 Basu, D (2007) The Plancherel formula for the universal covering group of SL(2, R) revisited, arXiv:07102224v3 [hep-th] Miller Jr, W (1968) Lie Theory and Special Functions, Academic, New York, Chap I and V Gazeau, J-P and Maquet, A (1979) Bound states in a Yukawa potential: A Sturmian group-theoretical approach, Phys Rev A, 20, 727 Casta eda, J A, Hern ndez, M A, and J uregui, R (2008) Continuum effects on the temporal evolution of anharmonic coherent molecular vibrations, Phys Rev A, 78, 78 Benedict, MG, and Molnar, B (1999) Algebraic construction of the coherent states of the Morse potential based on supersymmetric quantum mechanics, Phys Rev A, 60, R1737 Molnar, B, Benedict, MG, and Bertrand, J (2001) Coherent states and the role of the af ne group in the quantum mechanics of the Morse potential, J Phys A: Math Gen, 34, 3139 Ferapontov, EV and Veselov, AP (2001) Integrable Schr dinger operators with
References magnetic elds: Factorization method on curved surfaces, J Math Phys, 42, 590 Mouayn, Z (2003) Characterization of hyperbolic Landau states by coherent state transforms, J Phys A: Math Gen, 36, 8071 Aslaksen, EW and Klauder, JR (1968) Unitary representations of the af ne group, J Math Phys, 9, 206 Aslaksen, EW and Klauder, JR (1969) Continuous representation theory using the af ne group, J Math Phys, 10, 2267 P schl, G and Teller, E (1933) Bemerkungen zur Quantenmechanik des anharmonischen Oszillators, Z Phys, 83, 143 Antoine, J-P, Gazeau, J-P, Monceau, P, Klauder, JR, Penson, KA (2001) Temporally stable coherent states for in nite well, J Math Phys, 42, 2349 Barut, AO and Girardello, L (1971) New coherent states associated with non compact groups, Commun Math Phys, 21, 41 Fl gge, S (1971) Practical Quantum Mechanics I, Springer-Verlag, Berlin, Heidelberg and New York Akhiezer, NI (1965) The classical moment problem Oliver & Boyd LTD, Edinburgh and London Simon, B (1998) The classical moment problem as a self-adjoint nite difference operator, Adv Math, 137, 82 Besicovitch, AS (1932) Almost Periodic Functions, Cambridge University Press, Cambridge Nieto, MM and Simmons, Jr, LM (1987) Coherent states for general potentials, Phys Rev Lett, 41, 207 Aronstein, DL and Stroud, CR (1997) Fractional wave-function revivals in the in nite square well, Phys Rev A, 55, 4526 Averbuch, IS and Perelman, NF (1989) Fractional revivals: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics, Phys Lett A, 139, 449 Kinzel, W (1995) Bilder elementarer Quantenmechanik, Phys Bl, 51, 1190 Matos Filho, RL and Vogel, W (1996) Nonlinear coherent states, Phys Rev A, 54, 4560 123 Gro mann, F, Rost, J-M and Schleich, WP (1997) Spacetime structures in simple quantum systems, J Phys A: Math Gen, 30, L277 124 Stifter, P, Lamb, Jr, WE and Schleich, WP (1997) The particle in the box revisited, in Proceedings of the Conference on Quantum Optics and Laser Physics (eds Jin, L and Zhu, YS) World Scienti c, Singapore 125 Marzoli, I, Friesch, OM and Schleich, WP (1998) Quantum carpets and Wigner functions, in Proceedings of the 5th Wigner Symposium (Vienna,1997) (eds Kasperkovitz, P and Grau, D) World Scienti c, Singapore, pp 323 329 126 Bluhm, R, Kostelecky, VA and Porter, JA (1996) The evolution and revival structure of localized quantum wave packets, Am J Phys, 64, 944 127 Robinett, RW (2000) Visualizing the collapse and revival of wave packets in the in nite square well using expectation values, Am J Phys, 68, 410 128 Robinett, RW (2004) Quantum wave packet revivals, Phys Rep, 392, 1 129 Mandel, L (1979) Sub-Poissonian photon statistics in resonance uorescence, Opt Lett, 4, 205 130 Pe ina, J (1984) Quantum Statistics of r Linear and Nonlinear Optical Phenomena, Reidel, Dordrecht 131 Solomon, AI (1994) A characteristic functional for deformed photon phenomenology, Phys Lett A, 196, 29 132 Katriel, J and Solomon, AI (1994) Nonideal lasers, nonclassical light, and deformed photon states, Phys Rev A, 49, 5149 133 Pe ina, J (ed) (2001) Coherence and r Statistics of Photons and Atoms, Wiley Interscience 134 Bonneau, G, Faraut, J, and Valent, G (2001) Self-adjoint extensions of operators and the teaching of quantum mechanics, Am J Phys, 69, 322 135 Voronov, BL, Gitman, DM and Tyutin, IV (2007) Constructing quantum observables and self-adjoint extensions of symmetric operators I, Russ Phys J, 50(1), 1 136 Voronov, BL, Gitman, DM and Tyutin, IV (2007) Constructing quantum
121 122
References observables and self-adjoint extensions of symmetric operators II, Russ Phys J, 50(9), 3 Voronov, BL, Gitman, DM and Tyutin, IV (2001) Constructing quantum observables and self-adjoint extensions of symmetric operators III, Russ Phys J, 51(2), 645 Fox, R and Choi, MH (2000) Generalized coherent states and quantumclassical correspondence, Phys Rev A, 61, 032107-1 Hollenhorst, JN (1979) Quantum limits on resonant-mass gravitational-radiation detectors, Phys Rev D, 19, 1669 Yuen, HP (1976) Two-photon coherent states of the radiation eld, Phys Rev A, 13, 2226 Stoler, D (1970) Equivalence classes of minimum uncertainty packets, Phys Rev D, 1, 3217 Stoler, D (1971) Equivalence classes of minimum uncertainty packets II, Phys Rev D, 4, 1925 Lu, EYC (1971) New coherent states of the electromagnetic eld, Nuovo Cim Lett, 2, 1241 Slusher, RE, Hollberg, LW, Yurke, B, Mertz, JC and Valley, JF (1985) Observation of squeezed states generated by four-wave mixing in an optical cavity, Phys Rev Lett, 55, 2409 Caves, CV (1981) Quantum-mechanical noise in an interferometer, Phys Rev D, 23, 1693 Walls, DF (1983) Squeezed states of light, Nature, 306, 141 Nieto, MM (1997) The Discovery of Squeezed States In 1927, in Proceedings of the 5th International Conference on Squeezed States and Uncertainty Relations, Balatonfured, 1997 (eds Han, D et al) NASA/CP-1998-206855 Dell-Annoa, F, De Siena, S, and Illuminatia, F (2006) Multiphoton quantum optics and quantum state engineering, Phys Rep, 428, 53 Gilmore, R and Yuan, JM (1987) Group theoretical approach to semiclassical dynamics: Single mode case, J Chem Phys, 86, 130 Gilmore, R and Yuan, JM (1989) Group theoretical approach to semiclassical dynamics: Multimode case, J Chem Phys, 91, 917 Gazdy, B and Micha, DA (1985) The linearly driven parametric oscillator: Application to collisional energy transfer, J Chem Phys, 82, 4926 Gazdy, B and Micha, DA (1985) The linearly driven parametric oscillator: Its collisional time-correlation function, J Chem Phys, 82, 4937 Helgason, S (1962) Differential Geometry and Symmetric Spaces, Academic Press, New York Klauder, JR (2000) Beyond Conventional Quantization, Cambridge University Press, Cambridge Ali, ST, Engli , M (2005) Quantization methods: a guide for physicists and analysts, Rev Math Phys, 17, 391 Hall, BC (2005) Holomorphic methods in analysis and mathematical physics, First Summer School in Analysis and Mathematical Physics (Cuernavaca Morelos, 1998), Contemp Math260 1, (Am Math Soc, Providence, RI, 2000) Cahill, KE (1965) Coherent-State Representations for the Photon Density, Phys Rev, 138, 1566 Miller, MM (1968) Convergence of the Sudarshan expansion for the diagonal coherent-state, J Math Phys, 9, 1270 Klauder, JR and Sudarshan, ECG (1968) Fundamentals of Quantum Optics, Dover Publications Chakraborty, B, Gazeau, J-P and Youssef, A (2008) Coherent state quantization of angle, time, and more irregular functions and distributions, submitted, arXiv:08051847v2 [quant-ph] Van Hove, L (1961) Sur le probl me des relations entre les transformations unitaires de la M canique quantique et les transformations canoniques de la M canique classique, Bull Acad R Belg, Cl Sci, 37, 610 Kastrup, HA (2007) A new look at the quantum mechanics of the harmonic oscillator, Ann Phys (Leipzig), 7 8, 439 Dirac, PAM (1927) The Quantum Theory of Emission and Absorption of Radiation, Proc R Soc Lond Ser A, 114, 243
146 147
References 164 Schwartz, L (2008) Mathematics for the Physical Sciences, Dover Publications 165 Bu ek, V, Wilson-Gordon, AD, Knight, PL and Lai, WK (1992) Coherent states in a nite-dimensional basis: Their phase properties and relationship to coherent states of light, Phys Rev A, 45, 8079 166 Kuang, LM, Wang, FB and Zhou, YG (1993) Dynamics of a harmonic oscillator in a nite-dimensional Hilbert space, Phys Lett A, 183, 1 167 Kuang, LM, Wang, FB and Zhou, YG (1994) Coherent states of a harmonic oscillator in a nite-dimensional Hilbert space and their squeezing properties, J Mod Opt, 41, 1307 168 Kehagias, AA and Zoupanos, G (1994) Finiteness due to cellular structure of RN I Quantum Mechanics, Z Phys C, 62, 121 169 Polychronakos, AP (2001) Quantum Hall states as matrix Chern-Simons theory, JHEP, 04, 011 170 Miranowicz, A, Ski, W and Imoto, N: Quantum-Optical States in Finite-Dimensional Hilbert Spaces 1 General Formalism, in Modern Nonlinear Optics (ed Evans, MW) Adv Chem Phys, 119 (I) (2001), 155, John Wiley & Sons, New York; quant-ph/0108080; ibidem p 195; quant-ph/0110146 171 Gazeau, J-P, Josse-Michaux, FX and Monceau, P (2006) Finite dimensional quantizations of the (q, p) plane: new space and momentum inequalities, Int J Mod Phys B, 20, 1778 172 Lubinsky, DS (1987) A survey of general orthogonal polynomials for weights on nite and in nite intervals, Act Appl Math, 10, 237 173 Lubinsky, DS (1993) An update on orthogonal polynomials and weighted approximation on the real line, Act Appl Math, 33, 121 174 Landsman, NP (2006) Between classical and quantum, in Handbook of the Philosophy of Science, vol 2: Philosophy of Physics (eds Earman, J and Butter eld, J), Elsevier, Amsterdam 175 Woodhouse, NJM (1992) Geometric Quantization 2nd edn, Clarendon Press, Oxford 176 Dito, G and Sternheimer, D (2002) Deformation quantization: genesis, developments and metamorphoses, in Deformation quantization (ed Halbout, G), IRMA Lectures in Math Theor Phys, Walter de Gruyter, p 9 177 Karaali, G: Deformation quantization a brief survey, http://pages pomonaedu/~gk014747/research/ deformationquantizationpdf 178 Madore, J (1995) An Introduction to Noncommutative Differential Geometry and its Physical Applications, Cambridge University Press, Cambridge 179 Connes, A and Marcolli, M (2006) A walk in the noncommutative garden, arXiv:math/0601054v1 Connes, A and Marcolli, M (2006) A walk in the noncommutative garden, arXiv:math/0601054v1 180 Taylor, W (2001) M(atrix) Theory, Rev Mod Phys, 73, 419 181 Deltheil, R (1926) Probabilit s g om triques, Trait de Calcul des Probabilit s et de ses Applications par mile Borel, Tome II Gauthiers-Villars, Paris 182 Filippov, AT, Isaev, AP and Kurdikov, AB (1996) Paragrassmann Algebras, Discrete Systems and Quantum Groups, Problems in Modern Theoretical Physics, dedicated to the 60th anniversary of the birthday of AT Filippov, Dubna 96 212, p 83 183 Isaev, AP (1996) Paragrassmann integral, discrete systems and quantum groups (arXiv:q-alg/9609030) 184 Daoud, M and Kibler M (2002) A fractional supersymmetric oscillator and its coherent states, in Proceedings of the International Wigner Symposium, Istanbul, Aout 1999 (eds Arik, M et al), Bogazici University Press, Istanbul 185 Majid, S and Rodriguez-Plaza, M (1994) Random walk and the heat equation on superspace and anyspace, J Math Phys, 35, 3753 186 Cotfas, N and Gazeau, J-P (2008) Probabilistic aspects of nite frame quantization, arXiv:08030077v2 [math-ph] 187 Garc a de L on, P and Gazeau, J-P (2007) Coherent state quantization and phase operator, Phys Lett A, 361, 301
References 188 Carruthers, P and Nieto, MM (1968) Phase and Angle Variables in Quantum Mechanics, Rev Mod Phys, 40, 411 189 Louisell, WH (1963) Amplitude and phase uncertainty relations, Phys Lett, 7, 60 190 Susskind, L and Glogower, J (1964) Quantum mechanical phase and time operator, Physics, 1, 49 191 Popov, VN and Yarunin, VS (1973) Quantum and quasi-classical states of the photon phase operator, Vestnik Leningrad University, 22, 7 192 Popov, VN and Yarunin, VS (1992) Quantum and Quasi-classical States of the Photon Phase Operator, J Mod Opt, 39, 1525 193 Barnett, SM and Pegg, DT (1989) Phase in quantum optics, J Mod Opt, 36, 7 194 Busch, P, Grabowski, M and Lahti, PJ (1995) Who is afraid of POV measures Uni ed approach to quantum phase observables, Ann Phys (NY), 237, 1 195 Busch, P, Lahti, P, Pellonpa, J-P and Ylinen, K (2001) Are number and phase complementary observables J Phys A: Math Gen, 34, 5923 196 Dubin, DA, Hennings, MA and Smith, TB (1994) Quantization in polar coordinates and the phase operator, Publ Res Inst Math Sci, 30, 479 197 Busch, P, Hennings, MA and Smith, TB (2000) Mathematical Aspects of Weyl Quantization and Phase, World Scienti c, Singapore 198 Gazeau, J-P, Piechocki, W (2004) Coherent state quantization of a particle in de Sitter space, J Phys A: Math Gen, 37, 6977 199 Garcia de Leon, P, Gazeau, J-P and Qu va, J (2008) The in nite well revisited: coherent states and quantization, Phys Lett A, 372, 3597 200 Garcia de Leon, P, Gazeau, J-P, Gitman, D, and Qu va, J (2009) In nite quantum well: on the quantization problem, Quantum Wells: Theory, Fabrication and Applications Nova Science Publishers, Inc 201 Ashtekar, A, Fairhurst, S and Willis, JL (2003) Quantum gravity, shadow states, and quantum mechanics, Class Quantum Grav, 20, 1031 202 L vy-Leblond, JM (2003) Who is afraid of Nonhermitian Operators A quantum description of angle and phase, Ann Phys, 101, 319 203 Rabeie, A, Huguet, E and Renaud, J (2007) Wick ordering for coherent state quantization in 1 + 1 de Sitter space, Phys Lett A, 370, 123 204 Znojil, M (2001) PT -symmetric square well, Phys Lett A, 285, 7 205 Ali, ST, Engli , M and Gazeau, J-P (2004) Vector coherent states from Plancherel s theorem, Clifford algebras and matrix domains, J Phys A: Math Gen, 37, 6067 206 Reed, M and Simon, B (1980) Methods of Modern Mathematical Physics, Functional Analysis vol I, Academic Press, New York 207 Lagarias, J (2000) Mathematical Quasicrystals and the Problem of Diffraction, in Directions in Mathematical Quasicrystals (eds Baake, M and Moody, RV), CRM monograph Series, Am Math Soc, Providence, RI 208 Bouzouina, A, and De Bi vre, S (1996) Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Commun Math Phys, 178, 83 209 Abramowitz, M and Stegun, IA (1972) Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series 55 210 Calderon, AP and Vaillancourt, R (1972) A class of bounded pseudodifferential operators, Proc Natl Acad Sci USA, 69, 1185 211 Bouzouina, A and De Bi vre, S (1998) Equidistribution des valeurs propres et ergodicit semi-classique de symplectomorphismes du tore quanti s, C R Acad Sci, S rie I, 326, 1021 212 Gazeau, J-P, Huguet, E, Lachi ze Rey, M and Renaud, J (2007) Fuzzy spheres from inequivalent coherent states quantizations, J Phys A: Math Theor, 40, 10225 213 Grosse, H and Pre najder, P (1993) The construction of non-commutative manifolds using coherent states, Lett Math Phys, 28, 239 214 Kowalski, K and Rembielinski, J (2000) Quantum mechanics on a sphere and coherent states, J Phys A: Math Gen, 33, 6035
References 215 Kowalski, K and Rembielinski, J (2001) The Bargmann representation for the quantum mechanics on a sphere, J Math Phys, 42, 4138 216 Hall, B and Mitchell, JJ (2002) Coherent states on spheres, J Math Phys, 43, 1211 217 Freidel, L and Krasnov, K (2002) The fuzzy sphere -product and spin networks, J Math Phys, 43, 1737 218 Gazeau, J-P, Mourad, J and Qu va, J (2009) Fuzzy de Sitter space-times via coherent states quantization, in Proceedings of the XXVIth Colloquium on Group Theoretical Methods in Physics, New York, USA, 2006 (eds Birman, J and Catto, S) to appear 219 Gazeau, J-P (2007) An Introduction to Quantum Field Theory in de Sitter space-time, in Cosmology and Gravitation: XIIth Brazilian School of Cosmology and Gravitation (eds Novello, M and Perez-Bergliaffa, SE), AIP Conference Proceedings 910, 218 220 Sinai, YG (1992) Probability Theory An introductory Course, Springer Textbook, Springer-Verlag, Berlin 221 Johnson, NL and Kotz, S (1969) Discrete Distributions, John Wiley & Sons, Inc, New York 222 Ross, SM (1985) Introduction to Probability Models Academic Press, New York 223 Douglas, JB (1980) Analysis with Standard Contagious Distributions, International Co-operative Publishing House, Fairland, Maryland 224 Vilenkin, NY (1968) Special Functions and the Theory of Group Representations, Am Math Soc, Providence, RI 225 Barut, AO and Raczka, R (1977) Theory of Group Representations and Applications, PWN, Warszawa 226 Kirillov, AA (1976) Elements of the Theory of Representations, Springer-Verlag, Berlin 227 Arfken, G (1985) Spherical Harmonics and Integrals of the Products of Three Spherical Harmonics 126 and 129 in Mathematical Methods for Physicists, p 680 and p 698, 3rd ed, Academic Press, Orlando, FL 228 Weisstein, EW Spherical Harmonic, MathWorld A Wolfram Web Resource http://mathworldwolframcom/ SphericalHarmonichtml