The Poincaré algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states
Confluentes Mathematici, Tome 4 (2012) no. 4.

By introducing an unconventional realization of the Poincaré algebra alt1 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 alt1 of in this setting. Finally, we construct canonical Appell systems, coherent states and Leibniz function for alt1 as a tool for bosonic quantization.

Publié le :
DOI : 10.1142/S1793744212500065

Malte Henkel 1 ; René Schott 1 ; Stoimen Stoimenov 1 ; Jérémie Unterberger 1

1
@article{CML_2012__4_4_A2_0,
     author = {Malte Henkel and Ren\'e Schott and Stoimen Stoimenov and J\'er\'emie Unterberger},
     title = {The {Poincar\'e} algebra in the context of ageing systems: {Lie} structure, representations, {Appell} systems and coherent states},
     journal = {Confluentes Mathematici},
     publisher = {World Scientific Publishing Co Pte Ltd},
     volume = {4},
     number = {4},
     year = {2012},
     doi = {10.1142/S1793744212500065},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744212500065/}
}
TY  - JOUR
AU  - Malte Henkel
AU  - René Schott
AU  - Stoimen Stoimenov
AU  - Jérémie Unterberger
TI  - The Poincaré algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states
JO  - Confluentes Mathematici
PY  - 2012
VL  - 4
IS  - 4
PB  - World Scientific Publishing Co Pte Ltd
UR  - https://cml.centre-mersenne.org/articles/10.1142/S1793744212500065/
DO  - 10.1142/S1793744212500065
LA  - en
ID  - CML_2012__4_4_A2_0
ER  - 
%0 Journal Article
%A Malte Henkel
%A René Schott
%A Stoimen Stoimenov
%A Jérémie Unterberger
%T The Poincaré algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states
%J Confluentes Mathematici
%D 2012
%V 4
%N 4
%I World Scientific Publishing Co Pte Ltd
%U https://cml.centre-mersenne.org/articles/10.1142/S1793744212500065/
%R 10.1142/S1793744212500065
%G en
%F CML_2012__4_4_A2_0
Malte Henkel; René Schott; Stoimen Stoimenov; Jérémie Unterberger. The Poincaré algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states. Confluentes Mathematici, Tome 4 (2012) no. 4. doi : 10.1142/S1793744212500065. https://cml.centre-mersenne.org/articles/10.1142/S1793744212500065/

[1] F. Avram and M. S. Taqqu, Noncentral limit theorems and Appell polynomials, Ann. Probab. 15 (1987) 767–775.

[2] A. O. Barut, Conformal group → Schrödinger group → Dynamical group – the maxi- mal kinematical group of the massive Schrödinger particle, Helv. Phys. Acta 46 (1973) 496–503.

[3] F. Baumann, S. Stoimenov and M. Henkel, Local scale-invariances in the bosonic contact and pair-contact processes, J. Phys. A Math. Gen. 39 (2006) 4095–4118.

[4] C. D. Boyer, R. T. Sharp and P. Winternitz, Symmetry-breaking interactions for the time dependent Schrödinger equation, J. Math. Phys. 17 (1976) 1439–1451.

[5] A. J. Bray, Theory of phase-ordering kinetics, Adv. Phys. 43 (1994) 357–459.

[6] G. Burdet, M. Perrin and P. Sorba, About the non-relativistic structure of the con- formal algebra, Comm. Math. Phys. 34 (1973) 85–90.

[7] R. Cherniha and M. Henkel, The exotic conformal Galilei algebra and nonlinear partial differential equations, J. Math. Anal. Appl. 369 (2010) 120–132.

[8] L. Clavelli and P. Ramond, Group-theoretical construction of dual amplitudes, Phys. Rev. D 3 (1971) 988–990.

[9] L. F. Cugliandolo, Dynamics of glassy systems, in Slow Relaxation and non- equilibrium Dynamics in Condensed Matter, Les Houches Session 77, July 2002, eds. J.-L. Barrat, J. Dalibard, J. Kurchan, M. V. Feigel’man (Springer, 2003).

[10] C. Duval and P. A. Horv´athy, Non-relativistic conformal symmetries and Newton- Cartan structures, J. Phys. A: Math. Theor. 42 (2009) 465206.

[11] P. Feinsilver, Y. Kocik and R. Schott, Representations of the Schrödinger algebra and appell systems, Prog. Phys. 52 (2004) 343–359.

[12] P. Feinsilver and R. Schott, Algebraic Structures and Operator Calculus, Vol. 1: Rep- resentations and Probability Theory (Kluwer, 1993).

[13] P. Feinsilver and R. Schott, Algebraic Structures and Operator Calculus, Vol. 3: Rep- resentations of Lie Groups (Kluwer, 1996).

[14] P. Feinsilver and R. Schott, Appell systems on Lie groups, J. Th. Probab. 5 (1992) 251–281.

[15] D. B. Fuks, Cohohomology of infinite-dimensional Lie algebras, in Monographs in Contemporary Soviet Mathematics, Consultants Bureau (Springer, 1986).

[16] L. Guieu and C. Roger, L’algèbre et le groupe de Virasoro: Aspects géométriques et algébriques, généralisations, Publications CRM (Montreal Univ. Press, 2007).

[17] P. Havas and J. Plebanski, Conformal extensions of the Galilei group and their relation to the Schrödinger group, J. Math. Phys. 19 (1978) 482–488.

[18] M. Henkel, Schrödinger-invariance and strongly anisotropic critical systems, J. Stat. Phys. 75 (1994) 1023–1061.

[19] M. Henkel, Phenomenology of local scale-invariance: From conformal invariance to dynamical scaling, Nucl. Phys. B 641 (2002) 405–486.

[20] M. Henkel and J. Unterberger, Schrödinger-invariance and space-time symmetries, Nucl. Phys. B 660 (2003) 407–435.

[21] M. Henkel and J. Unterberger, Supersymmetric extensions of Schrödinger-invariance, Nucl. Phys. B 746 (2006) 155–201.

[22] M. Henkel, R. Schott, S. Stoimenov and J. Unterberger, On the dynamical symmetric algebra of ageing: Lie structure, representations and Appell systems, Quantum Probab. White Noise, 20 (2007) 233–240.

[23] Eds. M. Henkel, M. Pleimling and R. Sanctuary, Ageing and the Glass Transition, Lecture Notes in Physics, Vol. 716 (Springer-Verlag, 2007).

[24] M. Henkel and M. Pleimling, Non-equilibrium Phase Transitions Vol. 2: Ageing and Dynamical Scaling Far from Equilibrium (Springer, 2010).

[25] M. Henkel and S. Stoimenov, On non-local representations of the ageing algebra, Nucl. Phys. B 847 (2011) 612–627.

[26] C. G. Jacobi, Vorlesungen über Dynamik (1842/43), 4. Vorlesung, in Gesammelte Werke, eds. A Clebsch und E. Lottner (Akademie der Wissenschaften, 1866/1884).

[27] S. Lie, Über die Integration durch bestimmte Integrale von einer Klasse linearer par- tieller Differentialgleichungen, Arch. Math. Naturvidenskab. 6 (1881) 328.

[28] J. Lukierski, P. C. Stichel and W. J. Zakrewski, Exotic galilean conformal symmetry and its dynamical realisations, Phys. Lett. A 357 (2006) 1–5; Accelaration-extended galilean symmetries with central charges and their dynamical realizations, Phys. Lett. B 650 (2007) 203–207.

[29] U. Niederer, The maximal kinematical invariance group of the free Schrödinger equa- tion, Helv. Phys. Acta 45 (1972) 802–810.

[30] A. Picone and M. Henkel, Local scale-invariance and ageing in noisy systems, Nucl. Phys. B 688 (2004) 217–265.

[31] C. Roger and J. Unterberger, The Schrödinger–Virasoro Lie group and algebra: From geometry to representation theory, Ann. Inst. H. Poincaré 7 (2006) 1477–1529.

[32] S. Stoimenov and M. Henkel, Dynamical symmetries of semi-linear Schrödinger and diffusion equations, Nucl. Phys. B 723 (2005) 205–233.

[33] J. Unterberger and C. Roger, The Schrödinger–Virasoro Algebra (Springer, 2011).

[34] M. Virasoro, Subsidiary conditions and ghosts in dual-resonance models, Phys. Rev. D 1 (1970) 2933–2936.

Cité par Sources :