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The WKB approximation provides the correct exponential decay of eigenstates inside classically forbidden regions if one allows classical momenta to be imaginary. The typical example is a double well potential or a finite-height barrier. However, WKB theory (or its grown-up sibling) is a semiclassical approach to solve the time-independent Schroedinger equation, not it's time-dependent version. The application of semiclassical reasoning in the time-domain is mostly due to van Vleck, Maslov and Gutzwiller, resulting in the primitive semiclassical propagator.

The inclusion of complex trajectories in semiclassical calculations is still an active research area, as it would in principle allow for the semiclassical modelling of time-dependent tunneling. However, if one considers the Feynman path integral formulation of quantum mechanics (from which the semiclassical propagator can also be obtained), the approach is to usually interpret the integral as a "sum over all paths". This includes superluminal, discontinuous, etc, but if one evaluates the path integral by steepest descent, the requirement that the phase fulfills a variational principle pops up. Then people are comfortable and say "oh it's just the classical action :-)", and the trajectories selected are just the classical ones.

Now here's my problem: The phase in the full path integral reduces to the standard classical action if and only if the trajectories are assumed real in the path integral itself, unless someone can prove that a filter appears when evaluating by steepest descent that projects all complex trajectories in real space. As for myself, I cannot prove this, and I think it is a false statement. Thus, if the Feynman path integral involves complex trajectories from the start, then the classical equations that emerge in the semiclassical limit are complex as well. Is this reasoning correct? It leads to an even more embarrassing lack of connection between quantum and classical mechanics in the semiclassical limit (as if the current state-of-the-art were not already quite bad).

N.B. A complex trajectory arising as a solution to the complexified Hamiltonian equations can fulfill real boundary conditions and be complex-valued outside the endpoints. This is due to complexification doubling the dimension of phase space, such that the complex momenta and positions do not have to be [fully] related by Cauchy-Riemann conditions. This is well-explained here.

Qmechanic
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QuantumBrick
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1 Answers1

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  1. The target space for a path integral $\int\!{\cal D}\phi ~e^{\frac{i}{\hbar}S}$ could be a complex manifold.

    Example: The coherent state path integral, cf. e.g. this Phys.SE post.

  2. But say that the target space is a real manifold and the corresponding classical field theory is manifestly real. In other words, all the virtual paths/histories in the path integral are assumed to be real. Even in that case the method of steepest descent (MSD) may yield complex integration contours.

    Example: This happens already for oscillatory Gaussian integrals, cf. e.g. this Phys.SE post.

    Of course one may object, and say that this is just caused by the imaginary unit $i$ in the Boltzmann factor $e^{\frac{i}{\hbar}S}$.

  3. Let us therefore consider a manifestly real Euclidean path integral $\int\!{\cal D}\phi ~e^{-\frac{1}{\hbar}S}$ instead. Even in that case there may be stationary points in the complex plan that supports complex integration contours for the MSD.

Qmechanic
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