Lawrence J. Landau

Stochastic processes associated with KMS states

Abstract A general mathematical framework is presented for the connection between quantum statistical mechanics and periodic stochastic processes. “Stochastically positive KMS systems,” “periodic OS-positive stochastic processes,” “periodic positive semigroup structures” are defined and shown to be equivalent. The objects of the Tomita-Takesaki theory are explicitly exhibited in terms of the associated stochastic process and the statements of the theory are proven. A (two-sided) Markov property for the stochastic process is related to a cyclicity property of the KMS state. Perturbations are constructed by the Feynman-Kac-Nelson formula. The general framework is applied to KMS states given by density matrices.

Construction of a unique self-adjoint generator for a symmetric local semigroup

Abstract A definition is given of a symmetric local semigroup of (unbounded) operators P(t) (0 ⩽ t ⩽ T for some T > 0) on a Hilbert space H , such that P(t) is eventually densely defined as t → 0. It is shown that there exists a unique (unbounded below) self-adjoint operator H on H such that P(t) is a restriction of e−tH. As an application it is proven that H0 + V is essentially self-adjoint, where e−tH0 is an Lp-contractive semigroup and V is multiplication by a real measurable function such that V ∈ L2 + e and e−δV ∈ L1 for some e, δ > 0.

On the Bose-Einstein condensation of an ideal gas

A mathematically precise treatment is given of the well-known Bose-Einstein condensation of an ideal gas in the grand canonical ensemble at fixed density. The method works equally well for any of the standard boundary conditions and it is shown that the finite volume activity converges and that in three dimensions condensation occurs for Dirichlet, Neumann, periodic, and repulsive walls.

On the uniqueness of the equilibrium state for plane rotators

We study the classical statistical mechanics of the plane rotator, and show that there is a unique translation invariant equilibrium state in zero external field, if there is no spontaneous magnetization. Moreover, this state is then extremal in the equilibrium states. In particular there is a unique phase for the two dimensional rotator, and a unique phase for the three dimensional rotator above the critical temperature. It is also shown that in a sufficiently large external field the Lee-Yang theorem implies uniqueness of the equilibrium state.

Energy gap, clustering, and the Goldstone theorem in statistical mechanics

We prove a Goldstone-type theorem for a wide class of lattice and continuum quantum systems, both for the ground state and at nonzero temperature. For the ground state (T=0) spontaneous breakdown of a continuous symmetry implies no energy gap. For nonzero temperature, spontaneous symmetry breakdown implies slow clustering (noL1 clustering). The methods apply also to nonzero-temperature classical systems.

From the Euclidean group to the Poincaré group via Osterwalder-Schrader positivity

Given a continuous representation of the Euclidean group inn+1 dimensions, together with a covariant system of subspaces, which satisfies Osterwalder-Schrader positivity, we construct a continuous unitary representation of the orthochronous Poincaré group inn+1 dimensions satisfying the spectral condition. A similar result holds for the covering groups of the Euclidean and Poincaré group.

Supersymmetry and the Parisi-Sourlas Dimensional Reduction: A Rigorous Proof

Functional integrals that are formally related to the average correlation functions of a classical field theory in the presence of random external sources are given a rigorous meaning. Their dimensional reduction to the Schwinger functions of the corresponding quantum field theory in two fewer dimensions is proven. This is done by reexpressing those functional integrals as expectations of a supersymmetric field theory. The Parisi-Sourlas dimensional reduction of a supersymmetric field theory to a usual quantum field theory in two fewer dimensions is proven.

On the absence of spontaneous breakdown of continuous symmetry for equilibrium states in two dimensions

Using the Bogoliubov inequality, we extend previously known results concerning the absence of continuous symmetry breakdown for equilibrium states of certain quantum and classical lattice, and continuum systems in two space dimensions.

On the non-classical structure of the vacuum

Abstract Given any three causally separated space-time regions there are observables associated with these regions which do not possess a joint distribution in the vacuum state. This result extends to any state with the Reeh-Schlieder property. This non-classical structure of the vacuum implies, due to the vacuum asymptotic condition, the genericity of non-classical structure for all density matrices in the vacuum sector.

Absence of symmetry breakdown and uniqueness of the vacuum for multicomponent field theories

Correlation inequalities are used to show that the two component λ(φ2)2 model (with HD, D, HP, P boundary conditions) has a unique vacuum if the field does not develop a non-zero expectation value. It follows by a generalized Coleman theorem that in two space-time dimensions the vacuum is unique for all values of the coupling constant. In three space-time dimensions the vacuum is unique below the critical coupling constant.For then-componentP(|φ|2)2+μφ1 model, absence of continuous symmetry breaking, as μ goes to zero, is proven for all states which are translation invariant, satisfy the spectral condition, and are weak* limit points of finite volume states satisfyingNlocτ and higher order estimates.