Abstract
For relativistic particles with spin 1/2, which are described by the Dirac equation, a semiclassical trace formula is introduced that incorporates expectation values of observables in eigenstates of the Dirac-Hamiltonian. Furthermore, the semiclassical limit of an average of expectation values is expressed in terms of a classical average of the corresponding classical observable.
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REFERENCES
M. C. Gutzwiller, ``Periodic orbits and classical quantization conditions,'' J. Math. Phys. 12, 343–358 (1971).
M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).
M. C. Gutzwiller, ``The anisotropic Kepler problem in two dimensions,'' J. Math. Phys. 14, 139–152 (1973).
Focus Issue on Periodic Orbit Theory, Chaos 2(1), 1–158 (1992).
R. Aurich, C. Matthies, M. Sieber, and F. Steiner, ``Novel rule for quantizing chaos,'' Phys. Rev. Lett. 68, 1629–1632 (1992).
J. H. Hannay and A. M. Ozorio de Almeida, ``Periodic orbits and a correlation function for the semiclassical density of states,'' J. Phys. A: Math. Gen. 17, 3429–3440 (1984).
M. V. Berry, ``Semiclassical theory of spectral rigidity,'' Proc. R. Soc. London Ser. A 400, 229–251 (1985).
O. Bohigas, M. J. Giannoni, and C. Schmit, ``Characterization of chaotic quantum spectra and universality of level fluctuation laws,'' Phys. Rev. Lett. 52, 1–4 (1984).
Y. Colin de Verdiè re, ``Spectre du Laplacian et longueurs des gè odèsiques pèriodiques I,'' Composito Mathematica 27, 83–10 (1973).
J. J. Duistermaat and V. W. Guillemin, ``The spectrum of positive elliptic operators and periodic bicharacteristics,'' Inv. Math. 29, 39–79 (1975).
E. Meinrenken, ``Semiclassical principal symbols and Gutzwiller's trace formula,'' Rep. Math. Phys. 31, 279–295 (1992).
T. Paul and A. Uribe, ``The semi-classical trace formula and propagation of wave packets,'' J. Funct. Anal. 132, 192–249 (1995).
M. Combescure, J. Ralston, and D. Robert, ``A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition,'' Commun. Math. Phys. 202, 463–480 (1999).
M. Wilkinson, ``Random matrix theory in semiclassical quantum mechanics of chaotic systems,'' J. Phys. A: Math. Gen. 21, 1172–1190 (1988).
J. Bolte and S. Keppeler, ``Semiclassical time evolution and trace formula for relativistic spin-1/2 particles,'' Phys. Rev. Lett. 81, 1987–1991 (1998).
J. Bolte and S. Keppeler, ``A semiclassical approach to the Dirac equation,'' Ann. Phys. (N.Y.) 271, 125–162 (1999).
B. Thaller, The Dirac Equation (Springer, Berlin, Heidelberg, 1992).
M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, London Mathematical Society Lecture Notes Series, Vol. 268 (Cambridge University Press, Cambridge, 1999).
J. Bolte and R. Glaser, ``Quantum ergodicity for Pauli Hamiltonians with spin 1/2,'' Non-linearity 13, 1987–2003 (2000).
C. Emmrich and A. Weinstein, ``Geometry of the transport equation in multicomponent WKB approximations,'' Commun. Math. Phys. 176, 701–711 (1996).
R. Brummelhuis and J. Nourrigat, ``Scattering amplitude for Dirac operators,'' Commun. Part. Diff. Equations 24, 377–394 (1999).
D. Robert, Autour de l'Approximation Semi-Classique (Birkhäuser, Boston, 1987).
M. C. Gutzwiller, ``Phase-integral approximation in momentum space and the bound states of an atom,'' J. Math. Phys. 8, 1979–2000 (1967).
S. I. Rubinow and J. B. Keller, ``Asymptotic solution of the Dirac equation,'' Phys. Rev. 131, 2789–2796 (1963).
H. Spohn, ``Semiclassical limit of the Dirac equation and spin precession,'' Ann. Phys. (N.Y.) 282, 420–431 (2000).
J. J. Duistermaat, Fourier Integral Operators (Birkhaäuser, Boston, 1996).
B. Helffer, A. Martinez, and D. Robert, ``Ergodicité et limite semi-classique,'' Commun. Math. Phys. 109, 313–326 (1987).
R. Brummelhuis, T. Paul, and A. Uribe, ``Spectral estimates around a critical level,'' Duke Math. J. 78, 477–530 (1995).
V. Guillemin, ``Some classical theorems in spectral theory revisited,'' in Seminar on Singularties of Solutions of Linear Partial Differential Equations, L. Hörmander, ed., Ann. Math. Stud., Vol. 91 (Princeton University Press, Princeton, 1979), pp. 219–259.
J. Bolte and S. Keppeler, ``Semiclassical form factor for chaotic systems with spin 1/2,'' J. Phys. A: Math. Gen. 32, 8863–8880 (1999).
B. Eckhardt, S. Fishman, J. P. Keating, O. Agam, J. Main, and K. Müller, ``Approach to ergodicity in quantum wave functions,'' Phys. Rev. E 52, 5893–5903 (1995).
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Bolte, J. Semiclassical Expectation Values for Relativistic Particles with Spin 1/2. Foundations of Physics 31, 423–444 (2001). https://doi.org/10.1023/A:1017502906292
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DOI: https://doi.org/10.1023/A:1017502906292