Chirality as a Classical Observable in Algebraic Quantum Mechanics

Abstract

Ordinary Hilbert-space quantum mechanics leads to a wrong prediction for the ground state of chiral molecules such as alanine. This does not mean that quantum mechanics is incorrect but only that it is not applied properly. A detailed analysis shows that chirality corresponds to a classical observable (a superselection rule) which is generated by the environment, i.e. by the influence of an infinite system. For both, classical observables and infinite systems, Hilbert space quantum mechanics is inappropriate and has to be replaced by algebraic quantum mechanics.

Two models for chirality are discussed:

• The spin-boson model, where the single (eventually chiral) molecule is described by a two-level system. The infinitely many bosons of the model mimick the radiation field (the environment) which is inseparably coupled to the molecule.

• The Ising model with a transverse field, which is built up of infinitely many spins representing, e.g., an infinite crystal.

Chiral KMS-states (thermodynamic states) arise only in the latter model. It is shown that this result fits nicely into a more subtle discussion of the different notions of states and their interpretation in algebraic quantum mechanics. For single individual molecules chirality may only be described on the level of pure states of the system. The possibility of a phase transition in the spin-boson model is discussed.