### M Theory Lesson 218

The Secret Blogging Seminar discusses an astonishing mathematical paper on the quantum Fourier transform and geometric quantisation, the process of canonically associating a quantum system to a classical phase space.

In quantum computation, or M theory, one is used to confusing the discrete and quantum Fourier transforms, because both points and states are really just elements of a finite set and both transforms have basically the same definition. This paper focuses on the idea (due to Deligne) of a canonical (in a heavily category theoretic sense) normalisation factor for the transform. This factor is not simply $\frac{1}{\sqrt{n}}$, as commonly used by physicists concerned with unitarity, but rather a Gaussian term. For our favourite case of $n = 3$, this factor would be

$\sum_{k = 0,1,2} \frac{1}{\psi (- \frac{1}{2} k^{2})}$

where $\psi$ is the character that defines the Fourier transform. That is, it naturally involves sixth roots of unity. So the Gaussian $\frac{1}{2}$ is responsible for the introduction of twice as many roots, which occurred for instance in the modular relation associated to $B_{3}$. This normalisation is also similar to the one occurring in the choice of mass scale for Koide operators, which rely on the square roots of the masses.

In quantum computation, or M theory, one is used to confusing the discrete and quantum Fourier transforms, because both points and states are really just elements of a finite set and both transforms have basically the same definition. This paper focuses on the idea (due to Deligne) of a canonical (in a heavily category theoretic sense) normalisation factor for the transform. This factor is not simply $\frac{1}{\sqrt{n}}$, as commonly used by physicists concerned with unitarity, but rather a Gaussian term. For our favourite case of $n = 3$, this factor would be

$\sum_{k = 0,1,2} \frac{1}{\psi (- \frac{1}{2} k^{2})}$

where $\psi$ is the character that defines the Fourier transform. That is, it naturally involves sixth roots of unity. So the Gaussian $\frac{1}{2}$ is responsible for the introduction of twice as many roots, which occurred for instance in the modular relation associated to $B_{3}$. This normalisation is also similar to the one occurring in the choice of mass scale for Koide operators, which rely on the square roots of the masses.

## 3 Comments:

So what is the classical phase space associated with the quantum space given by the MUBs of the Pauli algebra? (That we're working with, I suppose.)

Carl, although Geometric Quantization in the mathematical sense had a big influence on the direction I took 10 yrs ago, physically I'm not sure that the complex number case is directly interesting (except as a model of Coecke type categorical axioms). This paper is interesting for its use of

finite fields, which I am always imagining in the background when we talk about MUBs.So, having rephrased it, you ask an interesting question: what would I think of as an analogue of classical phase space for MUB problems? The answer is that from a categorical point of view, the bases themselves are in some sense

classicalin the context of the quantum measurement. That is, the classical structure does not exist independently of the quantum one. This is also basically the result of this paper, although on a much higher level.Kea, A friend passes on this link, which probably needs a unifersity account to read: Heisenberg groups—the fundamental ingredient to describe information, its transmission and quantization"

For a singularity free gradient field in an open set of an oriented Euclidean space of dimension three we define a natural principal bundle out of an immanent complex line bundle. The fibres of this bundle encode information. The elements of both bundles are called internal variables. Several other natural bundles are associated with the principal bundle and, in turn, determine the vector field. Two examples are given and it is shown that for a constant vector field circular polarized waves with values in the principal bundle are associated with the vector field. These waves transmit information encoded in internal variables and, moreover, determine a Schrödinger representation. On U(2) a relation between spin representations and Schrödinger representations is established. The link between the spin ½ model and the Schrödinger representations yields a connection between a microscopic and a macroscopic viewpoint. Quantization and its link to information is derived out of the Schrödinger representation.

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