Multiple zeta values in deformation quantization

Oxford Mathematician Erik Panzer talks about his and colleagues' work on devising an algorithm to compute Kontsevich's star-product formula explicitly, solving a problem open for more than 20 years.

"The transition from classical mechanics to quantum mechanics is marked by the introduction of non-commutativity. For example, let us consider the case of a particle moving on the real line.

From commutative classical mechanics...

Classically, the state of the particle is described by its position $x$ and its momentum $p$. These coordinates parametrize the phase space, which is the tangent space $M=T^1 \mathbb{R} \cong \mathbb{R}^2$. One can view $x,p\colon M \rightarrow \mathbb{R}$ as smooth functions on the phase space, and the set $A=C^{\infty}(M)$ of all smooth functions on the phase space is an algebra with respect to the (commutative) multiplication of functions, e.g. $x\cdot p = p \cdot x$. The dynamics of the system is determined by a Hamiltonian $\mathcal{H} \in A$, which dictates the time evolution of a state according to \begin{equation*} x'(t) = \{x(t), \mathcal{H}\} \quad\text{and}\quad p'(t) = \{p(t), \mathcal{H}\}, \qquad\qquad(1) \end{equation*} where the Poisson bracket on the phase space is given by \begin{equation*} \{\cdot,\cdot\}\colon M\times M \longrightarrow M, \qquad \{f,g\} = \frac{\partial f}{\partial x} \frac{\partial g}{\partial p} - \frac{\partial f}{\partial p} \frac{\partial g}{\partial x}. \end{equation*}

...to non-commutative quantum mechanics.

In the quantum world, the state is described by a wave function $\psi$ that lives in a Hilbert space $L^2(\mathbb{R})$ of square-integrable functions on $\mathbb{R}$. Position and momentum now are described by operators $\hat{x},\hat{p}$ that act on this Hilbert space, namely \begin{equation*} \hat{x} \psi(x) = x \cdot \psi(x) \quad\text{and}\quad \hat{p} \psi(x) = -\mathrm{i}\hbar\frac{\partial}{\partial x} \psi(x) \end{equation*} where $\hbar$ is the (very small) reduced Planck constant. Note that these operators on the Hilbert space do not commute, $\hat{x}\hat{p} \neq \hat{p}\hat{x}$, and the precise commutator turns out to be \begin{equation*} [\hat{x},\hat{p}] = \hat{x}\hat{p}-\hat{p}\hat{x} = \mathrm{i}\hbar = \mathrm{i}\hbar \{x,p\}. \end{equation*} The observable quantity associated to an operator $\hat{f}$ is its expection value $\langle {\psi}| \hat{f} |\psi \rangle$, which is common notation in physics for the scalar product of $\psi$ with $\hat{f}\psi$. The time evolution of such an observable is determined by the Hamiltonian operator $\hat{\mathcal{H}}$ in a way very similar to (1): \begin{equation*} \mathrm{i}\hbar \frac{\partial \langle \psi(t) | \hat{f}| \psi(t)\rangle}{\partial t} = \langle \psi(t)| \big[\hat{f},\hat{\mathcal{H}} \big] |\psi(t)\rangle. \qquad\qquad(2) \end{equation*}

Deformation quantization...

Let us now generalise the example: For each classical quantity $f\in A=C^{\infty}(M)$, there should be a quantum analogue $\hat{f}$ that acts on some Hilbert space. We can think of this quantization $f\mapsto \hat{f}$ as an embedding of the commutative algebra $A$ into a non-commutative algebra of operators. Comparing (1) and (2), we see that quantization identifies the commutator $[\cdot,\cdot]$ with $i\hbar$ times the Poisson bracket $\{\cdot,\cdot\}$. The goal of deformation quantization is to describe this non-commutative structure algebraically on $A$, without reference to the Hilbert space. Namely, can we find a non-commutative product $f\star g$ of smooth functions such that \begin{equation*} \hat{f} \hat{g} = \widehat{f\star g} \quad\text{?} \end{equation*} Since the non-commutativity comes in only at order $\hbar$, it follows from the above that we should have \begin{equation*} f \star g = f\cdot g + \frac{\mathrm{i}\hbar}{2} \{f,g\} + \mathcal{O}(\hbar^2), \qquad\qquad(3)\end{equation*} i.e. the star-product is commutative to leading order (and therefore called a deformation of the commutative product), and the first order correction in $\hbar$ is determined by the Poisson structure.

...has a universal solution. 

Maxim Kontsevich proved in 1997 that such quantizations always exist: We can allow for an arbitrary smooth manifold $M$ with a Poisson structure - that means a Lie bracket $\{\cdot,\cdot\}$ on $A=C^{\infty}(M)$ which acts in both slots as a derivation (Leibniz rule).

Theorem (Kontsevich). Given an arbitrary Poisson manifold $(M,\{\cdot,\cdot\})$, there does indeed exist an associative product $\star$ on the algebra $A[[\hbar]]$ of formal power series in $\hbar$, such that equation (3) holds.

In fact, Kontsevich gives an explicit formula for this star-product in the form \begin{equation*} f \star g = f \cdot g + \sum_{n=1}^{\infty} \frac{(\mathrm{i}\hbar)^n}{n!} \sum_{\Gamma \in G_n} c(\Gamma) \cdot B_{\Gamma}(f,g), \end{equation*} where at each order $n$ in $\hbar$, the sum is over a finite set $G_n$ of certain directed graphs, like

and so on. The term $B_{\Gamma}(f,g)$ is a bidifferential operator acting on $f$ and $g$, which is defined in terms of the Poisson structure. It can be written down very easily directly in terms of the graph. The remaining ingredients in the formula are the real numbers $c(\Gamma) \in \mathbb{R}$. These are universal constants, which means that they do not depend on the Poisson structure or $f$ or $g$. Once these constants are known, the star-product for any given Poisson structure can be written down explicitly.

What are the weights?

These weights are defined as integrals over configurations of points in the upper half-plane $\mathbb{H}$. The simplest example is the unique graph in $G_1$, which gives (exercise!)

This is precisely the factor $1/2$ in front of the Poisson bracket in (3). But for higher orders, these integrals become much more complicated, and until very recently it was not known how they could be computed. Due to this problem, the star product was explicitly known only up to order $\hbar^{\leq 3}$.

It was conjectured, however, that the weights should be expressible in terms of multiple zeta values, which are sums of the form
\begin{equation*}
\zeta(n_1,\ldots,n_d)
=
\sum_{1\leq k_1<\cdots<k_d} \frac{1}{k_1^{n_1} \cdots k_d^{n_d}} \in \mathbb{R},
\end{equation*}
indexed by integers $n_1,\ldots,n_d$. They generalize the Riemann zeta function $\zeta(n)$, and they play an important role in the theory of periods and motives. In particular, there is a (conjectural) Galois theory for multiple zeta values.

In recent work with Peter Banks and Brent Pym, we developed an algorithm that can evaluate the weight integrals $c(\Gamma)$ for arbitrary graphs $\Gamma$:

Theorem (Banks, Panzer, Pym). For an arbitrary graph $\Gamma\in G_n$, the number $c(\Gamma)$ is a rational linear combination of multiple zeta values, normalized by $(2\mathrm{i}\pi)^{n_1+\ldots+n_d}$.

We calculated and tabulated the weights of all 50821 graphs that appear in $G_n$ for $n\leq 6$, which gives the star-product for all Poisson structures up to order $\hbar^{\leq 6}$. This allows us to study the quantizations in explicit examples and compare them to other quantization methods. Furthermore, our result shows that the Galois group of multiple zeta values acts on the space of star-products.

This research started off with a project by Peter Banks over the summer 2016, when Brent Pym was still in Oxford. Since then we have developed our proof and implemented all steps in computer programs, which are starproducts (publicly available} and automatize the computation of the star product and Kontsevich weights. The underlying techniques make heavy use of the theory of multiple polylogarithms on the moduli space $\mathfrak{M}_{0,n}$ of marked genus 0 curves, a generalization of Stokes' theorem to manifolds with corners and a theory of single-valued integration, influenced by work of Francis Brown (Oxford) and Oliver Schnetz (Erlangen-Nürnberg)."

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