Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T11:20:56.021Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence theorems for Segal quantization via spectral theory in Krein space

Published online by Cambridge University Press:  17 February 2009

P. Broadbridge
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Segal's unitarizing complex structure J is shown, in the Fermi-Dirac case, to be the orthogonal component in the polar decomposition of the real skew adjoint generator of classical dynamics. It is proven that in the Bose-Einstein case, the classical symplectic dynamics cannot be unitarized unless the generator is similar to a real skew adjoint operator.

With the classical Hamiltonian strictly positive, J is the pseudo-orthogonal component in the polar decomposition of the generator, using spectral theory in Krein space with indefinite metric. Thus, J can be expressed simply in terms of the projection E(0) onto the subspace of classical solutions with negative frequency. This complements the physicists' experience that conceptual difficulties arise when dynamically invariant separation of positive and negative frequency solutions is impossible.

Type
Research Article
Information
The ANZIAM Journal , Volume 24 , Issue 4 , April 1983 , pp. 439 - 460
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Ashtekar, A. and Magon, A., “Quantum fields in curved space-time”, Proc. Roy. Soc. London Ser. A 346 (1975), 375–94.Google Scholar
[2]Bognar, J., Indefinite inner product spaces (Springer, Berlin, 1974).CrossRefGoogle Scholar
[3]Broadbridge, P., “Problems in the quantization of quadratic Hamiltonians”, Hadronic J. 4 (1981), 899–948.Google Scholar
[4]Broadbridge, P. and Hurst, C. A., “Existence of a complex stucture for quadratic Hamiltonians?” Ann. Physics 131 (1981), 104–117.CrossRefGoogle Scholar
[5]Broadbridge, P. and Hurst, C. A., “Fermi-Dirac quantization of linear systems”, Ann. Physics 137 (1981), 86–103.CrossRefGoogle Scholar
[6]Broadbridge, P. and Hurst, C. A., “Canonical forms for quadratic Hamiltonians”, Phys. A 108 (1981), 39–62.CrossRefGoogle Scholar
[7]Cook, J. M., “The mathematics of second quantization”, Trans. Amer. Math. Soc. 74 (1953), 222–245.CrossRefGoogle Scholar
[8]Dirac, P.A.M., The principles of quantum mechanics (Oxford University Press, London, 4th edition, 1958).CrossRefGoogle Scholar
[9]Fulling, S. A., “Varieties of instability of a boson field in an external potential”, Phys. Rev. D14 (1976), 1939–1944.Google Scholar
[10]Gallone, F. and Sparzani, A., “Segal quantization”, J. Math. Phys. 20 (1979), 1375–1384.CrossRefGoogle Scholar
[11]Gallone, F. and Sparzani, A., “On the uniqueness of the Segal quantization of linear Bose systems”, J. Phys. A 14 (1981), 1341–1350.Google Scholar
[12]Garding, L. and Wightman, A. S., “Representations of the anticommutation relations”, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 617–621.CrossRefGoogle ScholarPubMed
[13]Harvey, B. N., “A spectral theorem for J. nonnegative operators”, Trans. Amer. Math. Soc. 257 (1980), 387–396.Google Scholar
[14]Kay, B. S., “A uniqueness result in the Segal-Weinless approach to linear Bose fields”, J. Math. Phys. 20 (1979), 1712–1714.CrossRefGoogle Scholar
[15]Krein, M. G. and Smul'jan, Ju. L., “J-polar representation of plus-operators”, Mat. Issled. 1 (1966), 172–210;Google Scholar
English transl., Amer. Math. Soc. Transl. (2)85 (1969), 115–143.Google Scholar
[16]Maurin, K., General eigenfunction expansions and unitary representations of topological groups (Polish Scientific Publishers, Warsaw, 1968).Google Scholar
[17]Naimark, M. A., “An analogue of Stone's theorem in a space with an indefinite metric”, Soviet Math. Dokl. 7(1966), 1366–1368.Google Scholar
[18]Paneitz, S. M., “Unitarization of symplectics and stability for causal differential equations in Hilbert space”, J. Funct. Anal. 41 (1981), 315–326.CrossRefGoogle Scholar
[19]Reed, M. and Simon, B., Methods of modern mathematical physics II. (Academic Press, New York, 1975).Google Scholar
[20]Segal, I. E., ‘Foundations of the theory of dynamical systems of infinitely many degrees of freedom I”, Mat.-Fys. Medd. Danske Vid. Selsk. 31 (1959), No. 12.Google Scholar
[21]Segal, I. E., “Mathematical characterization of the physical vacuum for a linear Bose-Einstein field”, Illinois J. Math. 6 (1962), 500–523.CrossRefGoogle Scholar
[22]Segal, I. E., Mathematical problems of relativistic physics (Amer. Math. Soc., Providence, R. I., 1963).Google Scholar
[23]Segal, I. E., “Stability theory and quantization”, in Differential geometric methods in mathematical physics, Lecture Notes in Physics 836 (Springer, Berlin, 1981).Google Scholar
[24]Shale, D. and Stinespring, W. F., “States of the Clifford algebra”, Ann. of Math. (2) 80 (1964), 365–381.CrossRefGoogle Scholar
[25]Weinless, M., “Existence and uniqueness of the vacuum for linear quantized fields”, J. Funct. Anal. 4 (1969), 350–379.CrossRefGoogle Scholar
[26]Whittaker, E. T., Analytical dynamics of particles and rigid bodies, Section 192 (Cambridge University Press, London, 4th edition, 1959).Google Scholar