# Quantization and evaluating a complex function on multiple Riemann sheets

+ 1 like - 0 dislike
174 views

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 ?

asked Mar 3, 2014
edited May 14, 2015

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

## 1 Answer

+ 2 like - 0 dislike

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.

## Your answer

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ys$\varnothing$csOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.