Abstract
The representation of the observables of a physical system as self-adjoint operators on Hilbert space is traditionally motivated by making reference to the correspondence principle or by invoking some quantisation scheme, such as the transcription of the Lie algebra structure from phase space to Hilbert space. There exists an alternative approach which refers directly to the operationally relevant features of these observables. This relativistic approach has the advantage of being open to the consideration of unsharp observables, which turns out necessary in some cases where sharp ‘observables’ simply do not exist or are not amenable to measurements.
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Chapter III.
H. Weyl. The Theory of Groups and Quantum Mechanics. Dover Publications, Inc., New York, 1950. German original 1931.
E.P. Wigner. Unitary representations of the inhomogeneous Lorentz group. Annals of Mathematics 40, 149–204, 1939.
G.W. Mackey. The Theory of Unitary Group Representations in Physics, Probability, and Number Theory. Benjamin/Cummings, Reading, Massachusetts, 1978.
A.S. Wightman. On the localizability of quantum mechanical systems. Reviews of Modern Physics 34, 845–872, 1962.
J.-M. Lévy-Leblond. Galilei group and Galilean invariance. In: Group Theory and Its Applications. Ed. E.M. Loebl, Academic Press, New York, pp. 221–299, 1971.
B. Simon. Quantum dynamics: from automorphism to Hamiltonian. In: Studies in Mathematical Physics. Essays in Honor of Valentine Bargman. Eds. E.H. Lieb, B. Simon and A.S. Wightman, Princeton Series in Physics, Princeton University Press, pp. 327–349, 1976.
S.T. Ali. A geometrical property of POV-measures and systems of covariance. In: Differential Geometric Methods in Mathematical Physics. Eds. H.-D. Doebner, S.I. Anderson and H.R. Petri, Lecture Notes in Mathematics, Vol. 905, Springer-Verlag, Berlin, pp. 207–228, 1982.
S.T. Ali, E. Prugovečki. Systems of imprimitivity and representations of quantum mechanics in fuzzy phase spaces. Journal of Mathematical Physics 18, 219–228, 1977.
E.B. Davies. On repeated measurements of continuous observables. Journal of Functional Analysis 6, 318–346, 1970.
R. Werner. Screen observables in relativistic and nonrelativistic quantum mechanics. Journal of Mathematical Physics 27, 793–803, 1986.
Y. Aharonov, D. Bohm. Time in the quantum theory and the uncertainty relation for time and energy. Physical Review 122, 1649–1658, 1961.
A. Peres. Measurement of time by quantum clocks. American Journal of Physics 48, 552–557, 1980.
E.P. Wigner. On the time-energy uncertainty relation. In: Aspects of Quantum Theory. Eds. A. Salam and E.P. Wigner, Cambridge University Press, Cambridge, pp. 237–247, 1972.
A. Barchielli. Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics. Quantum Optics 2, 423–441, 1990. V.P. Belavkin. A continuous counting observation and posterior quantum dynamics. Journal of Physics A 22, 1109–1114, 1989.
R.L. Hudson, K.R. Parthasarathy. Quantum Ito’s formula and stochastic evolutions. Communications in Mathematical Physics 93, 301–323, 1984.
K.R. Parthasarathy. An Introduction to Quantum Stochastic Calculus. Monographs in Mathematics, Vol. 85, Birkhäuser Verlag, Basel, 1992.
W. Schleich, S.M. Barnett (Eds.). Quantum Phase and Phase Dependent Measurements. Special Issue, Physica Scripta T48, 1993.
R.G. Newton. Quantum action-variables for harmonic oscillators. Annals of Physics 124, 327–346, 1980.
A. Galindo. Phase and number. Letters in Mathematical Physics 8, 495–500, 1984. J.C. Garrison and J. Wong. Canonically conjugate pairs, uncertainty relations, and phase operators. Journal of Mathematical Physics 11, 2242–2249, 1970.
P. Carruthers and M.M. Nieto. Phase and angle variables in quantum mechanics. Reviews of Modem Physics 40, 411–440, 1968.
D.T. Pegg, S.M. Barnett. Phase properties of the quantized single-mode electromagnetic field. Physical Review A 39, 1665–1675, 1989.
S.M. Barnett and D.T. Pegg. Quantum theory of rotation angles. Physical Review A 41, 3427–3435, 1990.
V.N. Popov and V.S. Yarunin. Photon phase operator. Theoretical and Mathematical Physics 89, 1292–1297, 1992.
R. Glauber. The quantum theory of optical coherence. Physical Review 130, 2529–2539, 1963. Coherent and incoherent states of the radiation field. Physical Review 131, 2766–2788, 1963.
M. Grabowski. New observables in quantum optics and entropy. In: Symposium on the Foundations of Modern Physics 1993. Eds P. Busch, P. Lahti, and P. Mittelstaedt, World Scientific, pp 182–191.
P. Busch, M. Grabowski, P. Lahti. Who is afraid of POV measures? Unified approach to quantum phase observables. Annals of Physics 237, 1–11, 1995.
T.D. Newton, E.P. Wigner. Localized states for elementary systems. Reviews in Modern Physics 21, 400–406, 1949.
K. Kraus. Position observable of the photon. In: The Uncertainty Principle and Foundations of Quantum Mechanics. Eds. W.C. Price and S.S. Chissick, John Wiley & Sons, New York, pp. 293–320, 1976. S.T. Ali, G.G. Emch. Fuzzy observables in quantum mechanics. Journal of Mathematical Physics 15, 176–182, 1974.
J.A. Brooke, F.E. Schroeck. Localization of the photon on phase space. Preprint, 1994.
Further Reading
S.T. All Survey of quantization methods. In: Classical and Quantum Systems — Foundations and Symmetries. Eds. H.D. Doebner, W. Scherer, and F.E. Schroeck, Jr., World Scientific, Singapore, 1993.
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(1995). Observables. In: Operational Quantum Physics. Lecture Notes in Physics Monographs, vol 31. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-49239-9_3
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