Equilibrium Statistical Mechanics of Fermion Lattice Systems

We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the non-commutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle with a minimal assumption for the dynamics and without any explicit assumption on the potential. It holds also for spin lattice systems as well, yielding a vast improvement over known results. All formulations are in terms of a C*-dynamical systems for the Fermion (CAR) algebra with all or a part of the following assumptions: (I) The interaction is even with respect to the Fermion number. (Automatically satisfied when (IV) below is assumed.) (II) All strictly local elements of the algebra have the first time derivative. (III) The time derivatives in (II) determine the dynamics. (IV) The interaction is lattice translation invariant. A major technical tool is the conditional expectation from the total algebra onto the local subalgebra for any finite subset of the lattice, which induces a system of commuting squares. This technique overcomes the lack of tensor product structures for Fermion systems and even simplifies many known arguments for spin lattice systems.Comment: 103 pages, no figure. The Section 13 has become simpler and a problem in 14.1 is settled thanks to a referee. The format has been revised according to the suggestion of this and the other referee

Hypercontractivity on the qq-Araki-Woods algebras

Extending a work of Carlen and Lieb, Biane has obtained the optimal hypercontractivity of the $q$-Ornstein-Uhlenbeck semigroup on the $q$-deformation of the free group algebra. In this note, we look for an extension of this result to the type III situation, that is for the $q$-Araki-Woods algebras. We show that hypercontractivity from $L^p$ to $L^2$ can occur if and only if the generator of the deformation is bounded.Comment: 17 page

Asymmetrical generalization of length in the rat

Two groups of rats in Experiment 1 were required to escape from a square pool by swimming to 1 of 2 submerged platforms that were situated beside the centers of 2 opposite walls. To help rats find a platform, black panels of equal width were pasted to the middle of the walls that were adjacent to the platforms. The width of the 2 panels was 50 cm for Group 50, and 100 cm for Group 100. Test trials were then conducted in the same pool, but with the platforms removed and with a 50-cm panel on 1 wall and a 100-cm panel on the opposite wall. Group 50 expressed a stronger preference for the 100-cm than the 50-cm panel during the test, whereas Group 100 expressed a similar preference for both panels. Thus the degree of generalization from the short to the long panel was greater than from the long to the short panel. Experiments 2 and 3 pointed to the same conclusion. They were of a similar design to Experiment 1, except that the lengths of the panels for the 2 groups were 25 and 50 cm in Experiment 2, and 25 and 100 cm in Experiment 3. The results are explained by assuming the original training results in the walls without black panels entering into inhibitory associations. This inhibition is then assumed to generalize more to the short than the long test panels and thereby result in an asymmetry in the gradients of generalization between the different lengths

A sharpened nuclearity condition for massless fields

A recently proposed phase space condition which comprises information about the vacuum structure and timelike asymptotic behavior of physical states is verified in massless free field theory. There follow interesting conclusions about the momentum transfer of local operators in this model.Comment: 13 pages, LaTeX. As appeared in Letters in Mathematical Physic