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Semiclassical Roles for Classical Orbits

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Quantum Signatures of Chaos

Part of the book series: Springer Series in Synergetics ((SSSYN))

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Abstract

The precursor of quantum mechanics due to Bohr and Sommerfeld was already seen by Einstein as applicable only to classically integrable dynamics.

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Notes

  1. 1.

    Throughout this chapter, \(\sqrt {\mathrm {i}}=\mathrm {e}^{\mathrm {i}\pi /4}\) will be understood.

  2. 2.

    Formally, the contribution of the αth classical path can be seen as arising from a Gaussian integral of the type (5.13.22) after analytic continuation to purely imaginary exponents; that continuation, achieved by the Fresnel integral (10.2.12), brings about the modulus operation and the ν α phase factors e−iπ∕2 in (10.2.13).

  3. 3.

    This is equally valid in continuous time; if we let n →∞, τ → 0 with nτ = t fixed, S (n, α)(q, q′) → S α(q, q′, t).

  4. 4.

    When the eigenvalues of are ordered as \(g^{(i+1)}_1\leq g^{(i+1)}_2\leq \ldots \leq g^{(i+1)}_i\) and those of as \(g^{(i)}_1\leq g^{(i)}_2\leq \ldots \leq g^{(i)}_{i-1}\) the corollary to the minimax theorem in question yields \(g^{(i+1)}_1\leq g^{(i)}_1\leq g^{(i+1)}_2\leq g^{(i)}_2\leq \ldots g^{(i)}_{i-1}\leq g^{(i+1)}_i\).

  5. 5.

    Somewhat arbitrarily but consistently, we shall speak of Morse indices in propagators and Maslov indices in trace formulae.

  6. 6.

    Should such nongeneric a disaster happen, there is a way out shown by Maslov: one changes to, say, the momentum representation; see next section.

  7. 7.

    Remarkably, the semiclassically approximate equality in (10.3.7) becomes a rigorous equality for the cat map , as was shown by Keating in Ref. [23].

  8. 8.

    Subtract and then divide by the product (1∕2π)2 of two mean densities, take out the delta function provided by the self-correlation terms , and change the overall sign.

  9. 9.

    The structures of (10.2.25) and (10.3.15) are similar; one might see the difference as the result of an extension of the phase space by inclusion of the pair E, t.

  10. 10.

    While this generalization is suggested by the very appearance of the prefactor in (10.3.15) and the previous footnote, and thus easy to guess, serious work is needed to actually verify the result.

  11. 11.

    Note that in contrast to the preceding chapter we here use the same symbol M for the stability matrix of maps and flows.

  12. 12.

    Departing from the convention mostly adhered to in this book, here we do not restrict ourselves to Hilbert spaces with the finite dimension N and thus do not write a factor 1∕N into the density of levels; the reader is always well advised to check from the context on the normalization of ϱ(E).

  13. 13.

    The stable and unstable manifolds of a periodic orbit are defined as the sets of points which the dynamics asymptotically carries toward the orbit as, respectively, t → +∞ and t →−∞; the reader is kindly asked to think for a moment about why L s , as well as the unstable manifold L u, are f-dimensional.

  14. 14.

    In this subsection, it would seem overly pedantic to insist on the name Morse phase or index for wave functions or propagators, reserving the name Maslov index for what appears in traces of propagators; as a compromise we speak here of Morse alias Maslov phases for multibranch wave functions or propagators.

  15. 15.

    For simplicity, we specialize to two-freedom systems from hereon; there is only a single Lyapunov exponent λ p for the pth orbit then.

  16. 16.

    As already pointed out in the footnote accompanying the definition (5.20.3) of the two-point correlator, two averages are needed to produce at least a piecewise constant function with step heights, upwards or downwards, of order \(\frac {1}{N}\). One of these is over the center energy as discussed here; the second, not worked into the formulae here in order to save space can be over a small window for the offset variable e.

  17. 17.

    Some authors prefer to absorb the Maslov phase factor in the stability amplitude which then becomes complex.

  18. 18.

    Keating and Müller [58] give reasons for which one can hope that the “exponentially small” corrections \(\propto \int _{T_H/2}^\infty dT(\ldots )\) even vanish identically.

  19. 19.

    We must apologize for using the tilde on one of these two symbols which in contrast to everywhere else in this book does not mean transposition.

  20. 20.

    Grassmann algebra and Grassmann integrals will accompany us in the remainder of this chapter. The reader may want to refresh versatility with these techniques by a glance at Sect. 5.13.

  21. 21.

    The matrices U, V have appeared in Sects. 6.6, 6.9, and 8.8 under the names A, D.

  22. 22.

    Transposition of a supermatrix involves interchanging the indices of the matrix elements and afterwards flipping the sign of the lower left elements in each 2 × 2 block.

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Authors and Affiliations

  1. Universität Duisburg-Essen, Essen, Germany

    Fritz Haake

  2. School of Mathematical Science, University of Nottingham, Nottingham, UK

    Sven Gnutzmann

  3. Center for Theoretical Physics, Polish Academy of Sciences, Warszawa, Poland

    Marek Kuś

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Haake, F., Gnutzmann, S., Kuś, M. (2018). Semiclassical Roles for Classical Orbits. In: Quantum Signatures of Chaos. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-97580-1_10

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