[Paper Review] Non-Locality of Equivariant Star Products on $T*(RP^n)$

Ranee Brylinski|ArXiv.org|Oct 27, 2000

Advanced Topics in Algebra2 references3 citations

TL;DR

This paper investigates the non-locality of equivariant star products on the cotangent bundle of real projective space $T^*(\mathbb{RP}^n)$, deriving explicit formulas for the coefficients $C_p^\lambda(\cdot,\cdot)$ of the star product $\star_\lambda$ when $\lambda = \frac{1}{2}$ and $n$ is odd. It shows these coefficients involve non-differential operators through inverses of Euler vector field terms, and expresses them via new $SL_{n+1}(\mathbb{C})$-invariant bidifferential operators $Z_p$ on $\mathbb{CP}^n$, proving that $C_p^\lambda$ fails to be bidifferential except for $\lambda = \frac{1}{2}$, $p=1$, due to the presence of $(E + r)^{-1}$ terms.

ABSTRACT

Lecomte and Ovsienko constructed $SL_{n+1}(R)$-equivariant quantization maps $Q_λ$ for symbols of differential operators on $λ$-densities on $\RP^n$. We derive some formulas for the associated graded equivariant star products $star_λ$ on the symbol algebra $Pol(T*\RP^n)$. These give some measure of the failure of locality. Our main result expresses (for $n$ odd) the coefficients $C_p$ of $star_λ$ when $λ=\half$ in terms of some new $SL_{n+1}(C)$-invariant algebraic bidifferential operators $Z_p$ on $T*\CP^n$ and the operators $(E+\frac{n}{2}\pm s)^{-1}$ where $E$ is the fiberwise Euler vector field and $s\in\{1,2,...,[\frac{p}{2}]\}$.

Motivation & Objective

• To analyze the structure of equivariant star products $\star_\lambda$ on the symbol algebra $\mathrm{Pol}(T^*\mathbb{RP}^n)$ arising from Lecomte-Ovsienko quantization maps.
• To understand the failure of locality in the coefficients $C_p^\lambda(\cdot,\cdot)$ of these star products, particularly the non-bidifferential nature of $C_p^\lambda$ for $\lambda = \frac{1}{2}$ and $p \geq 2$.
• To construct and characterize new $SL_{n+1}(\mathbb{C})$-invariant algebraic bidifferential operators $Z_p$ on $T^*\mathbb{CP}^n$ that encode the non-local structure of $C_p^\lambda$ for $\lambda = \frac{1}{2}$ and odd $n$.
• To establish a precise formula expressing $C_p^\lambda(\phi,\psi)$ in terms of $Z_p$ and inverse Euler operators $(E + \frac{n}{2} \pm s)^{-1}$, revealing the algebraic origin of non-locality.

Proposed method

• Derives the star product $\star_\lambda$ from the Lecomte-Ovsienko $SL_{n+1}(\mathbb{R})$-equivariant quantization maps $\mathcal{Q}_\lambda$, using their formulas from [L-O, §4.5] to compute the coefficients $C_p^\lambda(\phi,\psi)$.
• Introduces the operator $E$, the fiberwise Euler vector field on $T^*\mathbb{RP}^n$, and analyzes the appearance of its inverse $E + r$ in the coefficients, which causes non-locality.
• Constructs $Z_p(\phi,\psi)$ as a new $SL_{n+1}(\mathbb{C})$-invariant bidifferential operator on $T^*\mathbb{CP}^n$ via analytic continuation and complexification of real Darboux coordinates.
• Uses the relation $C_p^\lambda(\phi,\psi) = \frac{1}{\prod_{s=1}^{[p/2]} (E' + s)(E' - s + p - d)} Z_p(S_p^{-1}\phi, S_p^{-1}\psi)$ for $\lambda = \frac{1}{2}$, $n$ odd, to express $C_p^\lambda$ in terms of $Z_p$ and inverse Euler operators.
• Applies the $SL_{n+1}(\mathbb{C})$-invariance of $Z_p$ to extend the operators algebraically to $T^*\mathbb{CP}^n$, ensuring global well-definedness.
• Employs the momentum functions $\mu^x$ and their quantization to derive explicit formulas for $C_2(\mu^x, \psi)$, showing it involves $1/(E'(E'+1))$ and a fourth-order differential operator $L^x$, proving non-locality.

Experimental results

Research questions

• RQ1 Why do the coefficients $C_p^\lambda(\cdot,\cdot)$ of the equivariant star product $\star_\lambda$ fail to be bidifferential for $\lambda = \frac{1}{2}$ and $p \geq 2$?
• RQ2 What is the algebraic structure of the non-local terms in $C_p^\lambda(\phi,\psi)$, and how can they be systematically described?
• RQ3 Can the non-local coefficients $C_p^\lambda(\phi,\psi)$ for $\lambda = \frac{1}{2}$ and odd $n$ be expressed in terms of $SL_{n+1}(\mathbb{C})$-invariant bidifferential operators on $\mathbb{CP}^n$?
• RQ4 How do the inverse Euler operators $(E + r)^{-1}$ contribute to the non-locality of $C_p^\lambda$, and what is their role in the star product structure?
• RQ5 Is there a canonical way to construct the bidifferential operators $Z_p$ that encode the non-local part of $C_p^\lambda$ for $\lambda = \frac{1}{2}$, and what is their total degree?

Key findings

• The coefficients $C_p^\lambda(\cdot,\cdot)$ of the star product $\star_\lambda$ are not bidifferential for $\lambda = \frac{1}{2}$ and $p \geq 2$, due to the presence of inverse Euler operators $(E + r)^{-1}$, which are non-local.
• For $\lambda = \frac{1}{2}$ and odd $n$, the coefficients $C_p^\lambda(\phi,\psi)$ are expressed as $C_p^\lambda(\phi,\psi) = \frac{1}{\prod_{s=1}^{[p/2]} (E' + s)(E' - s + p - d)} Z_p(S_p^{-1}\phi, S_p^{-1}\psi)$, where $Z_p$ is a new $SL_{n+1}(\mathbb{C})$-invariant bidifferential operator of total degree $-p$.
• The operator $Z_2(\phi,\psi)$ is explicitly computed and shown to be a fourth-order differential operator; for $\phi = \mu^x$, $C_2(\mu^x, \psi) = \frac{1}{E'(E'+1)} L^x(\psi)$ with $L^x$ of order 4 and no left factors of the form $E' + c$ for $n \geq 2$, proving non-locality.
• The operator $L^x$ extends uniquely to an algebraic differential operator on $T^*\mathbb{CP}^n$, and its order is 4 for $n \geq 2$, with a different form for $n=1$, where it has order 3.
• The formula $C_p^\lambda(\phi,\psi) = \frac{1}{\prod_{i=1}^{[p/2]} (E' + i)(E' - i + p - d)} L_p^\phi(\psi)$ holds for $\phi \in \mathcal{A}^d$, with $L_p^\phi$ a differential operator, and $L_p^\phi$ is algebraic if $\phi \in \mathcal{R}$, the algebra generated by momentum functions.
• The construction of $Z_p$ via complexification and $SL_{n+1}(\mathbb{C})$-invariance ensures that $Z_p$ is not a power of the Poisson tensor, as its total order is too large for that interpretation.