Validity and failure of some entropy inequalities for CAR systems

Basic properties of von Neumann entropy such as the triangle inequality and what we call MONO-SSA are studied for CAR systems. We show that both inequalities hold for any even state. We construct a certain class of noneven states giving counter examples of those inequalities. It is not always possible to extend a set of prepared states on disjoint regions to some joint state on the whole region for CAR systems. However, for every even state, we have its `symmetric purification' by which the validity of those inequalities is shown. Some (realized) noneven states have peculiar state correlations among subsystems and induce the failure of those inequalities.Comment: 14 pages, latex, to appear in JMP. Some typos are correcte

Non-formality of the odd dimensional framed little balls operads

We prove that the chain operad of the framed little balls (or disks) operad is not formal as a non-symmetric operad over the rationals if the dimension of their balls is odd and greater than 4.Comment: 10 pages, presentation improved, errors collected, references adde

Rational homotopy theory and differential graded category

We propose a generalization of Sullivan's de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan's theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.Comment: 28pages, revised version, title changed, to appear in JPA

On fermion grading symmetry

We consider the univalence superselection rule. One would say perhaps ``There is no indication in nature to invalidate this rule. Fermions do not condensate!'' To explain our motivation, let us recall the correspondence of fermion systems and Pauli systems by the Jordan-Wigner transformation. For a finite lattice, fermion grading symmetry corresponds to the Pauli grading. For an infinite lattice, the Pauli-grading can be spontaneously broken e.g. for the XY-model. What is the status of the fermion grading? Nature tells that fermion grading symmmetry cannot be broken for any physical model. But it seems that its rigorous support is needed.Comment: A revised versio

On quasi-free dynamics on the resolvent algebra

The resolvent algebra is a new C*-algebra of the canonical commutation relations of a boson field given by Buchholz-Grundling. We study analytic properties of quasi-free dynamics on the resolvent algebra. Subsequently we consider a supersymmetric quasi-free dynamics on the graded C*-algebra made of a Clifford (fermion) algebra and a resolvent (boson) algebra. We establish an infinitesimal supersymmetry formula upon the GNS Hilbert space for any regular state satisfying some mild requirement which is standard in quantum field theory. We assert that the supersymmetric dynamics is given as a C*-dynamics.Comment: This paper has been withdrawn by the author due to its being based on some unjustified assumptions. Also there are mathematically incorrect arguments that stem from the assumption