Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Poincare algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states

Malte Henkel, Rene Schott, Stoimen Stoimenov, Jeremie Unterberger

TL;DR

The paper investigates ageing systems through an unconventional realization of the Poincaré-algebra analogue \(\mathfrak{p}_3\) in the form of \(\mathfrak{alt}_1\), framing it as a dynamical symmetry of non-equilibrium dynamics and enabling the computation of two-time correlators within a Galilei-invariant deterministic sector. It shows that \(\mathfrak{alt}_1\) embeds in an infinite-dimensional algebra \(\mathcal{W}\) and relates to Virasoro/SV structures, while constructing Casimir operators, explicit matrix and dual representations, and invariant differential operators. The authors develop Wick products and Appell systems for \(\mathfrak{alt}_1\), then build coherent states and the Leibniz function to realize a bosonic quantization scheme and reveal how the Lie algebra is reconstructed from Leibniz data. These results provide a robust algebraic framework for analysing ageing phenomena and suggest avenues for stochastic processes and representation-theoretic methods in non-equilibrium physics. Overall, the work links dynamical symmetries, infinite-dimensional extensions, and Appell/coherent-state techniques to a tractable, representation-theoretic approach to ageing dynamics.

Abstract

By introducing an unconventional realization of the Poincare algebra alt_1 of special relativity as conformal transformations, we show how it may occur as a dynamical symmetry algebra for ageing systems in non-equilibrium statistical physics and give some applications, such as the computation of two-time correlators. We also discuss infinite-dimensional extensions of alt_1 in this setting. Finally, we construct canonical Appell systems, coherent states and Leibniz functions for alt_1 as a tool for bosonic quantization.

The Poincare algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states

TL;DR

The paper investigates ageing systems through an unconventional realization of the Poincaré-algebra analogue in the form of , framing it as a dynamical symmetry of non-equilibrium dynamics and enabling the computation of two-time correlators within a Galilei-invariant deterministic sector. It shows that embeds in an infinite-dimensional algebra and relates to Virasoro/SV structures, while constructing Casimir operators, explicit matrix and dual representations, and invariant differential operators. The authors develop Wick products and Appell systems for , then build coherent states and the Leibniz function to realize a bosonic quantization scheme and reveal how the Lie algebra is reconstructed from Leibniz data. These results provide a robust algebraic framework for analysing ageing phenomena and suggest avenues for stochastic processes and representation-theoretic methods in non-equilibrium physics. Overall, the work links dynamical symmetries, infinite-dimensional extensions, and Appell/coherent-state techniques to a tractable, representation-theoretic approach to ageing dynamics.

Abstract

By introducing an unconventional realization of the Poincare algebra alt_1 of special relativity as conformal transformations, we show how it may occur as a dynamical symmetry algebra for ageing systems in non-equilibrium statistical physics and give some applications, such as the computation of two-time correlators. We also discuss infinite-dimensional extensions of alt_1 in this setting. Finally, we construct canonical Appell systems, coherent states and Leibniz functions for alt_1 as a tool for bosonic quantization.

Paper Structure

This paper contains 14 sections, 7 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A Brief Perspective on the Algebra $\mathfrak{alt}_1$
  3. The abstract Lie algebra $\mathfrak{alt}_1$
  4. Central extensions: an introduction
  5. Infinite-dimensional extension of $\mathfrak{alt}_1$
  6. Some results on representations of ${\cal W}$
  7. Casimir Operators and Matrix Representations for $\mathfrak{alt}_1$.
  8. Casimir operators
  9. Matrix representations
  10. Cartan Decomposition and Dual Representations
  11. Wick Products and Appell Polynomials
  12. Appell Systems of the Algebra $\mathfrak{alt}_1$
  13. Coherent States and Leibniz Function
  14. Concluding Remarks

Key Result

Proposition 2.1

The following Lie algebra isomorphisms hold true. First, where $\varepsilon$ is a 'Grassmann' variable. Second, where ${\mathfrak{p}}_3 \cong {\mathfrak{so}}(2,1)\ltimes\mathbb{R}^3$ is the relativistic Poincaré algebra in (2+1)-dimensions.

Figures (1)

  • Figure 1: (a) Root diagram of the complex Lie algebra $B_2$ and the identification of the generators (\ref{['eq:zetaGen']}) of the complexified conformal Lie algebra $(\mathfrak{conf}_3)_{\mathbb{C}}\supset(\mathfrak{sch}_1)_{\mathbb{C}}$. The double circle in the centre denotes the Cartan subalgebra. The generators belonging to the three non-isomorphic parabolic subalgebras Henk03 are indicated by the full points, namely (b) $\widetilde{\mathfrak{sch}}_1$, (c) $\widetilde{\mathfrak{age}}_1$ and (d) $\widetilde{\mathfrak{alt}}_1$.

Theorems & Definitions (13)

  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.1
  • Proposition 2.3
  • Remark 3.1
  • Lemma 4.1
  • Example 5.1
  • Example 5.2
  • Remark 5.1
  • Proposition 6.1
  • ...and 3 more