Quantization and evaluating a complex function on multiple Riemann sheets

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In Lecture 3 of his mathematical physics course (starting at about 44:30), Carl Bender mentions that the evaluation of a complex function on multiple Riemann sheets can be used to describe the quantization of the laws of nature we observe.

How is this way of thinking about quantization connected to the usual methods of quantization applied in physics, such as for example covariant quantization, light-cone quantization, or BRST quantization ?

edited May 14, 2015

Could you please specify at which minute in the video? (Done in the edited version)

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He talks about a square matrix H(z) depending on a complex variable. He says that the eigenvalues of $H(z)$ form a multi-sheeted Riemann surface (the multiset of $E$ with $\det (E-H(z))=0$ for some $z$), and explained that the spectrum has discrete (quantized) level for fixed $z$ although the surface smoothly connects all eigenvalues.

This is just a fact of linear algebra over the complex numbers. The only connection to quantum mechanics is that for real $z$ (if the coefficients are real analytic, as in his 2-dimensional examples) the matrix can be viewed as a Hamiltonian acting on a finite-dimensional Hilbert space.

Calling this quantization is just an illustration, not a hard fact; it cannot be used to quantize anything.

answered May 14, 2015 by (14,009 points)

Would this count as some form of analytical continuation of the (Hamiltonian) equation to the complex numbers to connect the discrete set of solutions?

@conformal_gk: In some sense yes, as the Hamiltonian is Hermitian only for real $z$. However, in the above scenario, $z$ can be a fairly arbitrary parameter in the expression of $H$, which means that upon varying $z$ one doesn't get useful information about a fixed Hamiltonian.

The analytically most meaningful complex deformations are of very special kind, related to the often possible analytic continuation of the resolvent $G(E)=(E-H)^{-1}$ for a fixed Hamiltonian. The ''right'' deformation here is a deformation of the inner product via so-called complex scaling, which allows one to see resonances as isolated poles rather than as narrow peaks in the spectral density of the continuous spectrum, by rotating the branch cuts of the continuous spectrum along the positive real line into another, complex direction.