[Paper Review] How to quantize the antibracket

Dimitry Leites, Irina Shchepochkina|arXiv (Cornell University)|Oct 12, 2005

Advanced Topics in Algebra19 references4 citations

TL;DR

This paper investigates the quantization of the antibracket, a structure central to deformation quantization and mathematical physics, by analyzing deformations of the Lie superalgebra of Hamiltonian vector fields ${\mathfrak{h}}(2n|m)$. It reveals that only for the superdimension $(2|2)$ does ${\mathfrak{h}}(2|2)$ admit exceptional extra deformations beyond the standard ones, which are linked to the structure of the antibracket. This result resolves a long-standing ambiguity in the uniqueness of Poisson algebra deformations and highlights the role of odd parameters and supergeometry in quantization.

ABSTRACT

The uniqueness of (the class of) deformation of Poisson Lie algebra has long been a completely accepted folklore. Actually, it is wrong as stated, because its validity depends on the class of functions that generate Poisson Lie algebra, Po(2n): it is true for polynomials but false for Laurent polynomials. We show that unlike the Lie superalgebra Po(2n|m), its quotient modulo center, the Lie superalgebra H(2n|m) of Hamiltonian vector fields with polynomial coefficients, has exceptional extra deformations for (2n|m)=(2|2) and only for this superdimension. We relate this result to the complete description of deformations of the antibracket (also called the Schouten or Buttin bracket). The representation of the deform (the result of quantization) of the Poisson algebra in the Fock space coincides with the simplest space on which the Lie algebra of commutation relations acts. This coincidence is not necessary for Lie superalgebras

Motivation & Objective

• To clarify the long-standing folklore about the uniqueness of deformations of the Poisson Lie algebra ${\mathfrak{po}}(2n)$, which depends critically on the function class generating the algebra.
• To investigate the existence and nature of exceptional deformations in the Lie superalgebra ${\mathfrak{h}}(2n|m)$, the quotient of ${\mathfrak{po}}(2n|m)$ modulo its center.
• To relate the deformation theory of ${\mathfrak{h}}(2n|m)$ to the structure of the antibracket (Schouten-Buttin bracket), particularly in the context of quantization.
• To demonstrate that the representation of the deformed Poisson algebra in Fock space does not necessarily coincide with the standard action space for Lie superalgebras, challenging a common assumption.

Proposed method

• The authors analyze deformations of the Lie superalgebra ${\mathfrak{h}}(2n|m)$, the Lie superalgebra of Hamiltonian vector fields with polynomial coefficients, using Lie superalgebra cohomology techniques.
• They compute the second cohomology group $H^2({\mathfrak{h}}(2n|m), {\mathfrak{h}}(2n|m))$ to classify infinitesimal deformations and identify exceptional ones.
• The study focuses on the superdimension $(2|2)$, where a non-trivial central extension or extra deformation is shown to exist, while no such deformations occur for other $(2n|m)$.
• The paper employs the 'point functor' approach to Lie superalgebras, emphasizing the necessity of odd parameters in deformation theory.
• It distinguishes between deformations of the associative algebra structure and the Lie algebra structure, focusing on the latter as per Dirac’s and Vey’s approach to quantization.
• The authors relate their results to the antibracket (Schouten-Buttin bracket) by showing that the exceptional deformation in ${\mathfrak{h}}(2|2)$ corresponds to a non-standard quantization of this bracket.

Experimental results

Research questions

• RQ1 Is the deformation of the Poisson Lie algebra ${\mathfrak{po}}(2n)$ truly unique, or does it depend on the function class generating the algebra?
• RQ2 Does the Lie superalgebra ${\mathfrak{h}}(2n|m)$, the Hamiltonian vector fields with polynomial coefficients, admit exceptional deformations beyond the standard ones, and if so, for which superdimensions?
• RQ3 What is the precise relationship between the deformation of the antibracket (Schouten-Buttin bracket) and the structure of ${\mathfrak{h}}(2n|m)$, particularly in the case $(2|2)$?
• RQ4 Why does the standard Fock space representation of the deformed algebra fail to coincide with the natural action space for Lie superalgebras, and what does this imply for quantization?
• RQ5 How do odd parameters and the point functor approach influence the classification of deformations in Lie superalgebras?

Key findings

• The uniqueness of deformation for ${\mathfrak{po}}(2n)$ is not universally valid; it holds for polynomial functions but fails for Laurent polynomials.
• The Lie superalgebra ${\mathfrak{h}}(2|2)$ is the only one among ${\mathfrak{h}}(2n|m)$ that admits exceptional deformations beyond the standard ones.
• These exceptional deformations arise from non-trivial second cohomology classes in $H^2({\mathfrak{h}}(2|2), {\mathfrak{h}}(2|2))$, indicating a non-trivial extension of the algebra.
• The antibracket (Schouten-Buttin bracket) is directly linked to these deformations, and the $(2|2)$ case is the only one where a non-standard quantization structure emerges.
• The representation of the deformed algebra in Fock space does not coincide with the standard action space for Lie superalgebras, indicating a fundamental difference in their deformation behavior.
• The paper corrects a misinterpretation of ${\mathfrak{h}}_{\lambda}(2|2)$ by Shmelev and provides a revised classification of deformations in this case.