The goal of this book is to present classical mechanics, quantum mechanics, and statistical mechanics in an almost completely algebraic setting, thereby introducing mathematicians, physicists, and engineers to the ideas relating classical and quantum mechanics with Lie algebras and Lie groups. The book emphasizes the closeness of classical and quantum mechanics, and the material is selected in a way to make this closeness as apparent as possible. 
Much of the material covered here is not part of standard textbook treatments of classical or quantum mechanics (or is only superficially treated there). For physics students who want to get a broader view of the subject, this book may therefore serve as a useful complement to standard treatments of quantum mechanics. 
Almost without exception, this book is about precise concepts and exact results in classical mechanics, quantum mechanics, and statistical mechanics. The structural properties of mechanics are discussed independent of computational techniques for obtaining quantitatively correct numbers from the assumptions made. The standard approximation machinery for calculating from first principles explicit thermodynamic properties of materials, or explicit cross sections for high energy experiments can be found in many textbooks and is not repeated here.

/pdf/classical-and-quantum-mechanics-via-lie-algebras-53t2e4uwfs.pdf

Classical and quantum mechanics via Lie algebras

This paper gives a concise, mathematically rigorous description of phenomenological equilibrium thermodynamics for single-phase systems in the absence of chemical reactions and external forces. 
The present approach is similar to that of Callen, who introduces in his well-known thermodynamics book the basic concepts by means of a few postulates from which everything else follows. His setting is modified to match the more fundamental approach based on statistical mechanics. Thermodynamic stability is derived from kinematical properties of states outside equilibrium by rigorous mathematical arguments, superseding Callen's informal arguments that depend on a dynamical assumption close to equilibrium. 
From the formulas provided, it is an easy step to go to various examples and applications discussed in standard textbooks such as Callen or Reichl. A full discussion of global equilibrium would also involve the equilibrium treatment of multiple phases and chemical reactions. Since their discussion offers no new aspects compared with traditional textbook treatments, they are not treated here.

/pdf/phenomenological-thermodynamics-in-a-nutshell-c1y7vv7sda.pdf

Phenomenological thermodynamics in a nutshell

On the basis of new, concise foundations, this paper establishes the four laws of thermodynamics, the Maxwell relations, and the stability requirements for response func- tions, in a form applicable to global (homogeneous), local (hydrodynamic) and microlocal (kinetic) equilibrium. The present, self-contained treatment needs very little formal machinery and stays very close to the formulas as they are applied by the practicing physicist, chemist, or engineer. From a few basic assumptions, the full structure of phenomenological thermodynamics and of classical and quantum statistical mechanics is recovered. Care has been taken to keep the foundations free of subjective aspects (which traditionally creep in through information or probability). One might describe the paper as a uniform treatment of the nondynamical part of classical and quantum statistical mechanics "with- out statistics" (i.e., suitable for the definite descriptions of single objects) and "without mechanics" (i.e., independent of microscopic assumptions). When enriched by the tradi- tional examples and applications, this paper may serve as the basis for a course on thermal physics.

On the foundations of thermodynamics

Über G Falks 'einfachere' Definition des Begriffs der Entropie

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Über G Falks 'einfachere' Definition des Begriffs der Entropie