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

A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity

We revisit and prove some convexity inequalities for trace functions conjectured in the earlier part I. The main functional considered is \Phi_{p,q}(A_1,A_2,...,A_m) = (trace((\sum_{j=1}^m A_j^p)^{q/p}))^{1/q} for m positive definite operators A_j. In part I we only considered the case q=1 and proved the concavity of \Phi_{p,1} for 0 < p \leq 1 and the convexity for p=2. We conjectured the convexity of \Phi_{p,1} for 1< p < 2. Here we not only settle the unresolved case of joint convexity for 1 \leq p \leq 2, we are also able to include the parameter q\geq 1 and still retain the convexity. Among other things this leads to a definition of an L^q(L^p) norm for operators when 1 \leq p \leq 2 and a Minkowski inequality for operators on a tensor product of three Hilbert spaces -- which leads to another proof of strong subadditivity of entropy. We also prove convexity/concavity properties of some other, related functionals.Comment: Proof of a conjecture in math/0701352. Revised version replaces earlier draft. 18 pages, late

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

The H\"older Inequality for KMS States

We prove a H\"older inequality for KMS States, which generalises a well-known trace-inequality. Our results are based on the theory of non-commutative $L_p$-spaces.Comment: 10 page

Lightfront holography and area density of entropy associated with localization on wedge-horizons

It is shown that a suitably formulated algebraic lightfront holography, in which the lightfront is viewed as the linear extension of the upper causal horizon of a wedge region, is capable of overcoming the shortcomings of the old lightfront quantization. The absence of transverse vacuum fluctuations which this formalism reveals, is responsible for an area (edge of the wedge) -rearrangement of degrees of freedom which in turn leads to the notion of area density of entropy for a ``split localization''. This area proportionality of horizon associated entropy has to be compared to the volume dependence of ordinary heat bath entropy. The desired limit, in which the split distance vanishes and the localization on the horizon becomes sharp, can at most yield a relative area density which measures the ratio of area densities for different quantum matter. In order to obtain a normalized area density one needs the unknown analog of a second fundamental law of thermodynamics for thermalization caused by vacuum fluctuation through localization on causal horizons. This is similar to the role of the classical Gibbs form of that law which relates Bekenstein's classical area formula with the Hawking quantum mechanism for thermalization from black holes. PACS: 11.10.-z, 11.30.-j, 11.55.-mComment: The last two sections have been modified. This is the form in which the paper will be published in IJP

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

Inequalities for quantum skew information

We study quantum information inequalities and show that the basic inequality between the quantum variance and the metric adjusted skew information generates all the multi-operator matrix inequalities or Robertson type determinant inequalities studied by a number of authors. We introduce an order relation on the set of functions representing quantum Fisher information that renders the set into a lattice with an involution. This order structure generates new inequalities for the metric adjusted skew informations. In particular, the Wigner-Yanase skew information is the maximal skew information with respect to this order structure in the set of Wigner-Yanase-Dyson skew informations. Key words and phrases: Quantum covariance, metric adjusted skew information, Robertson-type uncertainty principle, operator monotone function, Wigner-Yanase-Dyson skew information

Joint system quantum descriptions arising from local quantumness

Bipartite correlations generated by non-signalling physical systems that admit a finite-dimensional local quantum description cannot exceed the quantum limits, i.e., they can always be interpreted as distant measurements of a bipartite quantum state. Here we consider the effect of dropping the assumption of finite dimensionality. Remarkably, we find that the same result holds provided that we relax the tensor structure of space-like separated measurements to mere commutativity. We argue why an extension of this result to tensor representations seems unlikely

A Characterization of right coideals of quotient type and its application to classification of Poisson boundaries

Let $G$ be a co-amenable compact quantum group. We show that a right coideal of $G$ is of quotient type if and only if it is the range of a conditional expectation preserving the Haar state and is globally invariant under the left action of the dual discrete quantum group. We apply this result to theory of Poisson boundaries introduced by Izumi for discrete quantum groups and generalize a work of Izumi-Neshveyev-Tuset on $SU_q(N)$ for co-amenable compact quantum groups with the commutative fusion rules. More precisely, we prove that the Poisson integral is an isomorphism between the Poisson boundary and the right coideal of quotient type by maximal quantum subgroup of Kac type. In particular, the Poisson boundary and the quantum flag manifold are isomorphic for any q-deformed classical compact Lie group.Comment: 28 pages, Remark 4.9 adde