Quantum Theory of Polymers

Jean Marie André, Jean‐Luc Brédas, J. Delhalle +3 more

Dejan I. Raković, M. Dugić, M.B. Plavšić

Alejandro Corichi, Tatjana Vukašinac, José Antonio Zapata

A rather nonstandard quantum representation of the canonical commutation relations of quantum mechanics systems, known as the polymer representation, has gained some attention in recent years, due to its possible relation with Planck scale physics. In particular, this approach has been followed in a symmetric sector of loop quantum gravity known as loop quantum cosmology. Here we explore different aspects of the relation between the ordinary Schr\"odinger theory and the polymer description. The paper has two parts. In the first one, we derive the polymer quantum mechanics starting from the ordinary Schr\"odinger theory and show that the polymer description arises as an appropriate limit. In the second part we consider the continuum limit of this theory, namely, the reverse process in which one starts from the discrete theory and tries to recover back the ordinary Schr\"odinger quantum mechanics. We consider several examples of interest, including the harmonic oscillator, the free particle, and a simple cosmological model.

Russell B. Thompson

Abstract The Feynman quantum-classical isomorphism between classical statistical mechanics in 3+1 dimensions and quantum statistical mechanics in 3 dimensions is used to connect classical polymer self-consistent field theory with quantum time-dependent density functional theory. This allows the theorems of density functional theory to relate non-relativistic quantum mechanics to a classical statistical mechanical derivation of polymer self-consistent field theory for ring polymers in a 4 dimensional thermal-space. One dynamic postulate is added to two static postulates which allows for a description of quantum physics from a 5 dimensional thermal-space-time ensemble perspective. A connection with aspects of classical field theory can be made in the classical limit.

Bertrand I. Halperin, Alexander Sevrin

Aleksandar M. Spasic, Jyh-Ping Hsu

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 II. The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 A. The Born-Oppenheimer Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . 218 B. The Molecules Conformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 C. Levinthal’s Paradox: A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 D. Levinthal’s Paradox: Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 III. New Approach: The Clue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 IV. The Fundamentals of the Decoherence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 A. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 B. Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 C. The Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 D. The Environment-Induced Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 E. The Physical Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 V. New Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 A. The Conformation Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 B. Nonstationary State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 C. The Relaxation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 D. The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 VI. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 VII. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 In this chapter, we describe the problem of polymer conformational stability and transitions in the framework of the so-called quantum decoherence theory. We propose a rather qualitative scenario yet bearing generality in the context of the quantum decoherence theory, enabling us to reproduce both, existence and stability of the polymers conformations, and the short time scales for the quantum-mechanical processes resulting effectively in the conformational transitions. The proposed model is qualitative yet providing us with the possibility to overcome the main obstacle in resolving the problem of (semi-)classically unreasonably long time necessary for the change of conformation of the polymers in a solution.