SYMPLECTIC PATHOLOGY
Copyright policy )SYMPLECTIC PATHOLOGY
Let \(Z\) be a real vector space with algebraic dual \(Z^*\) and on the direct sum \(V_z= Z\oplus Z^*\) define \(\Omega_z (x\oplus \xi, y\oplus \eta)= \xi(y)- \eta (x)\), for \(x,y\in Z\) and \(\xi, \eta\in Z^*\). The author shows here that the real symplectic vector space \((V_z, \Omega_z)\) given by infinite-dimensional \(Z\) has a variety of pathological properties. That is, it does not admit unitary structure and moderating norm. His result is expressed as follows from the viewpoint of the quantum field theory (QFT): The minimal \(C^*\)-Weyl algebra \({\mathfrak A} (V_z, \Omega_z)\) by infinite-dimensional \(Z\) admits neither Fock states nor more generally quasi-free states. Whether this result has real significance for QFT is a question.
- University of Florida United States
Applications of functional analysis in quantum physics, Geometry of classical groups, quasi-free states, real symplectic vector space, Fock states, Geometry and quantization, symplectic methods, algebraic dual, quantum field theory, minimal \(C^*\)-Weyl algebra
Applications of functional analysis in quantum physics, Geometry of classical groups, quasi-free states, real symplectic vector space, Fock states, Geometry and quantization, symplectic methods, algebraic dual, quantum field theory, minimal \(C^*\)-Weyl algebra
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