Abstract
In this work, we first present a detailed analysis of temperature-time duality in the 3D Ising model, by inspecting the resemblance between the density operator in quantum statistical mechanics and the evolution operator in quantum field theory, with the mapping β = (k B T)−1 → it. We point out that in systems like the 3D Ising model, for the nontrivial topological contributions, the time necessary for the time averaging must be infinite, being comparable with or even much larger than the time of measurement of the physical quantity of interest. The time averaging is equivalent to the temperature averaging. The phase transitions in the parametric plane (β, it) are discussed, and a singularity (a second-order phase transition) is found to occur at the critical time t c , corresponding to the critical point β c (i.e, T c ). It is necessary to use the 4-fold integral form for the partition function for the 3D Ising model. The time is needed to construct the (3 + 1)D framework for the quaternionic sequence of Jordan algebras, in order to employ the Jordan-von Neumann-Wigner procedure. We then turn to discuss quite briefly temperature-time duality in quantum-chemical many-electron theory. We find that one can use the known one-dimensional differential equation for the Slater sum S(x, β) to write a corresponding form for the diagonal element of the Feynman propagator, again with the mapping β → it.
Similar content being viewed by others
References
Zhang Z.D.: Phil. Mag. 87, 5309 (2007)
Ławrynowicz J., Marchiafava S., Niemczynowicz A.: Adv. Appl. Clifford Algebra 20, 733 (2010)
J. Ławrynowicz, S. Marchiafava, M. Nowak-Kȩpczyk, Trends Differential Geometry, Complex Analysis and Mathemtical Physics, in Proceedings of the 9th International Workshop on Complex Structures, Integrability and Vector Fields, Sofia, Bulgaria, 25–29 August 2008, ed. by K. Sekigawa, V.S. Gerdjikov, S. Dimiev (World Scientific, Singapore), pp. 156–166 doi:10.1142/9789814277723_0018
Lawrynowicz J.: Act. Phys. Superf. 11, 101 (2009)
March N.H.: Electron Density Theory of Atoms and Molecules. Academic, New York (1992)
Parr R.G., Yang W.: Density Functional Theory of Atoms and Molecules. Oxford University Press, New York (1989)
March N.H., Howard I.A.: Phys. Status Solidi B. 237, 265 (2003)
Zhang Z.D.: Philosphy Mag. 88, 3097 (2008)
Das A.: Field Theory, a Path Integral Approach. World Scientific, Singapore (1993)
Francesco P.D., Mathieu P., Sénéchal D.: Conformal Field Theory. Springer, New York (1996)
Binney J.J., Dowrick N.J., Fisher A.J., Newman M.E.J.: The Theory of Critical Phenomena, An Introduction to the Renormalization Group. Clarendon Press, Oxford (1992)
Jordan P., von Neumann J., Wigner E.: On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35, 29–64 (1934)
Zhang Z.D., March N.H.: Phase Trans. 84, 299 (2011)
Z.D. Zhang, Phil. Mag. 89, 765 (2009). See also 0812.0194v6 for more detailed discussions, specially, these on singularities at/near infinite temperature.
Jones W., March N.H.: Theoretical Solid–State Physics. Dover, New York (1985)
Howard I.A., March N.H., Nieto L.M.: Phys. Rev. A 66, 054501 (2002)
De Marco B., Jin D.S.: Science 285, 1703 (1999)
Sondheimer E.H., Wilson A.H.: Proc. R. Soc. Lond. Ser. A 210, 173 (1951)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, Z.D., March, N.H. Temperature-time duality exemplified by Ising magnets and quantum-chemical many electron theory. J Math Chem 49, 1283–1290 (2011). https://doi.org/10.1007/s10910-011-9820-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-011-9820-9