Clifford geometric parameterization of inequivalent vacua (Q2746983)

The author proposes a method, called geometric, to parameterize inequivalent vacuum by dynamical data. Introducing a notion of quantum Clifford algebras with arbitrary bilinear forms, isomorphic algebras are, then, distinguished by different filtrations and induced gradings. The idea of a vacuum is introduced as the unique algebraic projection on the base field embedded in the Clifford algebra, which is equivalent to the term vacuum in axiomatic quantum field theory and the GNS construction in \(C^*\)-algebras. NEWLINENEWLINENEWLINEIt is shown that this approach is equivalent to the usual picture where one product is fixed but the existence of a variety of GNS states is employed. As the authors stresses, the geometric approach implies the striking novelty of the fact that dynamical data determine uniquely the vacuum without positivity required. The usual concept of a statistical quantum state can be generalized to geometric meaningful situations which are now non-statistical and non-definite. An application to physics is provided by a \(U(2)\)-symmetry producing a gap equation which governs a phase transition.