Geometric quantization of thermodynamics, phase transitions, and asymptotes at critical points
If \(\psi\) is a one-form on the manifold \(Q\), we may consider it as a mapping from \(Q\) to \(T^*Q\) which satisfies the equation \(\psi^* \vartheta = \vartheta\), where \(\vartheta \) is the canonical one-form on the cotangent bundle \(T^*Q\). This allows to establish a natural one-to-one correspondence between the Lagrangian submanifolds of \(T^*Q\) and the closed one-forms on \(Q\). If the one-form corresponding to the Lagrangian submanifold is of the form \(dU\), \(U\) is called a generating function for the submanifold. Here the author considers nine-dimensional configurational manifold of intensive thermodynamic parameters, the temperature \(T\), the pressure \(P\), the electric field strength \(\vec E\), the magnetic field strength \(\vec H\) and the electrostatic potential \(\varphi\). Together with the corresponding momenta which are chosen to be the extensive specific quantities -- the entropy \(s\), the volume \(V\), the polarization \(\vec p\), the magnitization \(\vec \mu\) and the charge \(e\), they form an eighteen-dimensional phase space \(\mathbb{R}^{18}\). The one-form \(dU=T ds-P ds+\vec E.d\vec p+\vec H.d\vec\mu+\varphi de\) defines a new thermodynamic quantity and a Lagrangian submanifold \(\Lambda\) in \(\mathbb{R}^{18}\). With \(\Lambda\) one can associate the so-called ``canonical tunnel operator'' \(K_{\Lambda, \hbar}\). Specification of the canonical tunnel operator defines axiomatically the geometric quantization of thermodynamics. This line is not traced out further but after taking the classical limit, an integral equation is derived from which the local equations of state follow.
scientific article; zbMATH DE number 830732