Abstract
Employing positive-definiteness arguments we analyse Boson field states, which combine classical and quantum mechanical features (signal and noise), in a constructive manner. Mathematically, they constitute Bauer simplexes within the convex, weak-*-compact state space of the C*-Weyl algebra, defined by a presymplectic test function space (smooth one-Boson wave functions) and are affinely homeomorphic to a state space of a classical field. The regular elements are expressed in terms of weak distributions (probability premeasures) on the dual test function space. The Bauer simplex arising from the bare vacuum is shown to generalize the quantum optical photon field states with positive P-functions.
Similar content being viewed by others
References
Alfsen, E. M.: Compact Convex Sets and Boundary Integrals, Springer-Verlag, Berlin, 1971.
Binz, E., Honegger, R. and Rieckers, A.: Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic form, Preprint Mannheim, Tübingen, 2003.
Binz, E., Honegger, R. and Rieckers, A.: Field-theoretic Weyl quantization of large Poisson algebras, Preprint Mannheim, Tübingen, 2003.
Binz, E., Honegger, R. and Rieckers, A.: Field-theoretic Weyl quantization as a strict and continuous deformation quantization, Preprint Mannheim, Tübingen, 2003.
Bratteli, O. and Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics I, Springer-Verlag, New York, 1987.
Bratteli, O. and Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics II, Springer-Verlag, New York, 1981.
Evans, D. E. and Lewis, J. T.: Dilations of Irreversible Evolutions in Algebraic Quantum Theory. Comm. Dublin Inst. Adv. Stud. Ser. A (Theoret. Phys.) 24, Dublin Institute for Advanced Studies, Dublin, 1977.
Gelfand, I. M. and Vilenkin, N. Ya.: Generalised Functions IV, Academic Press, New York, 1964.
Glauber, R. J.: The quantum theory of optical coherence, Phys. Rev. 130 (1963), 2529–2539.
Glauber, R. J.: Coherent and incoherent states of the radiation field, Phys. Rev. 131 (1963), 2766–2788.
Glauber, R. J.: Optical coherence and photon statistics, In: C. de Witt, A. Blandin and C. Cohen-Tannoudji (eds), Quantum Optics and Electronics, Gordon and Breach, New York, 1995, 1964.
Hewitt, E. and Ross, K. A.: Abstract Harmonic Analysis I, II, Springer-Verlag, New York, 1963, 1970.
Hida, T.: Brownian Motion, Springer-Verlag, New York, 1980.
Hillery, M.: Classical pure states are coherent states, Phys. Lett. A 111 (1985), 409–411.
Honegger, R. and Rieckers, A.: The general form of non-Fock coherent boson states, Publ. RIMS Kyoto Univ. 26 (1990), 397–417.
Honegger, R. and Rieckers, A.: Squeezed variances of smeared Boson fields, Helv. Phys. Acta 70 (1997), 507–541.
Honegger, R. and Rieckers, A.: Squeezing of optical states on the CCR-algebra, Publ. RIMS Kyoto Univ. 33 (1997), 869–892.
Honegger, R. and Rieckers, A.: Construction of classical and non-classical coherent photon states, Ann. Phys. 289 (2001), 213–231.
Honegger, R. and Rieckers, A.: Some continuous field quantizations, equivalent to the C*-Weyl quantization, Preprint Mannheim, Tübingen, 2003.
Honegger, R. and Rieckers, A.: State quantizations associated with the field-theoretic C*-Weyl algebra, Preprint Mannheim, Tübingen, 2003.
Kadison, R. V. and Ringrose, J. R.: Fundamentals of the Theory of Operator Algebras I,II, Academic Press, New York, 1983, 1986.
Kastler, D.: The C-algebras of a free Boson field, Comm. Math. Phys. 1 (1965), 14–48.
Klauder, J. R. and Sudarshan, E. C. G.: Fundamentals of Quantum Optics, Benjamin, New York, 1968.
Landsman, N. P.: Mathematical Topics Between Classical and Quantum Mechanics, Springer-Verlag, New York, 1998.
Loudon, R.: The Quantum Theory of Light, Clarendon Press, Oxford, 1979.
Loudon. R. and Knight, P. L.: Squeezed light, J. Modern Option 34 (1987), 709–759.
Ma, X. and Rhodes, W.: Multimode squeeze operators and squeezed states, Phys. Rev. A 41 (1990), 4625–4631.
Manuceau, J., Sirugue, M., Testard, D. and Verbeure, A.: The smallest C*-algebra for canonical commutation relations. Comm. Math. Phys. 32 (1973), 231–243.
Meystre, P. and Sargent, M. III: Elements of Quantum Optics, Springer-Verlag, New York, 1990.
Nussenzveig, H. M.: Introduction to Quantum Optics, Gordon and Breach, London, 1973.
Petz, D.: An Invitation to the Algebra of Canonical Commutation Relations, Leuven Notes in Math. Theoret. Phys. Vol. 2, Leuven Univ. Press, Leuven, Belgium, 1990.
Richter, H.: Wahrscheinlichkeitstheorie, 2nd edn, Springer-Verlag, Berlin, 1966.
Rieffel, M. A.: Quantization and C*-algebras, In: R. S. Doran (eds), C*-Algebras: 1943–1993, Contemp. Math. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 67–97.
Robinson, P. L.: Symplectic pathology, Quart. J. Math. Oxford 44 (1993), 101–107.
Skorokhod, A. V.: Integration in Hilbert Space, Springer-Verlag, New York, 1974.
Slawny, J.: On factor representations and the C*-algebra of the canonical commutation relations, Comm. Math. Phys. 24 (1971), 151–170.
Takesaki, M.: Theory of Operator Algebras I, Springer-Verlag, New York, 1979.
Titulaer, U. M. and Glauber, R. J.: Correlation functions and coherent fields, Phys. Rev. B 140 (1965), 676–682.
Umemura, Y.: Measures on infinite dimensional vector spaces, Publ. RIMS Kyoto Univ. A 1 (1965), 1–47.
Vogel, W., Welsch, D.-G. and Wallentowitz, S.: Quantum Optics, an Introduction, Wiley-VCH, Berlin, 2001.
Walls, D. F. and Milburn, G. J.: Quantum Optics, Springer-Verlag, New York, 1994.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Honegger, R., Rieckers, A. Partially Classical States of a Boson Field. Letters in Mathematical Physics 64, 31–44 (2003). https://doi.org/10.1023/A:1024972815695
Issue Date:
DOI: https://doi.org/10.1023/A:1024972815695