Yong Moon Park

Massless quantum sine–Gordon equation in two space–time dimensions: Correlation inequalities and infinite volume limit

We prove new correlation inequalities for the massive and massless quantum sine–Gordon equations. These results are then used to construct infinite volume limit theory for the massless (S–G)2 model that satisfies the Osterwalder–Schrader axioms. As consequences, infinite volume limit theories for the classical, neutral, statistical mechanical systems with two‐body Coulomb potentials and for the massive Thirring model exist.

Nonequilibrium dynamics of infinite particle systems with infinite range interactions

We discuss the existence and uniqueness of nonequilibrium dynamics of infinitely many particles interacting via superstable pair interactions in one and two dimensions. The interaction is allowed to be of infinite range and singular at the origin. Under suitable regularity conditions on the interaction potential, we show that if the potential decreases polynomially as the distance between interacting two particles increases, then the tempered solution to the system of Hamiltonian equations exists. Moreover, if the potential satisfies further that either it has a subexponential decreasing rate or it is everywhere two-times continuously differentiable, then we show that the tempered solution is unique. The results extend those of Dobrushin and Fritz obtained for finite range interactions.

Dirichlet forms and symmetric Markovian semigroups on CCR algebras with respect to quasi-free states

Employing the construction method of Dirichlet forms on standard forms of von Neumann algebras developed in Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2000, Vol. 3, No. 1, pp. 1–14 (Ref. 1), we construct Dirichlet forms and associated symmetric Markovian semigroups on CCR algebras with respect to quasi-free states. More precisely, let A(h0) be the CCR algebra over a complex separable pre-Hilbert space h0 and let ω be a quasi-free state on A(h0). For any normalized admissible function f and complete orthonormal system (CONS) {gn}⊂h0, we construct a Dirichlet form and corresponding symmetric Markovian semigroup on the natural standard form associated to the GNS representation of (A(h0),ω). It turns out that the form is independent of admissible function f and CONS {gn} chosen. By analyzing the spectrum of the generator (Dirichlet operator) of the semigroup, we show that the semigroup is ergodic and tends to the equilibrium exponentially fast.

Improvement of uncertainty relations for mixed states

We study a possible improvement of uncertainty relations. The Heisenberg uncertainty relation employs commutator of a pair of conjugate observables to set the limit of quantum measurement of the observables. The Schrodinger uncertainty relation improves the Heisenberg uncertainty relation by adding the correlation in terms of anti-commutator. However both relations are insensitive whether the state used is pure or mixed. We improve the uncertainty relations by introducing additional terms which measure the mixtureness of the state. For the momentum and position operators as conjugate observables and for the thermal state of quantum harmonic oscillator, it turns out that the equalities in the improved uncertainty relations hold.

Construction of a family of quantum Ornstein–Uhlenbeck semigroups

For a given quasi-free state on the CCR algebra over one dimensional Hilbert space, a family of Markovian semigroups which leave the quasi-free state invariant is constructed by means of noncommutative elliptic operators and Dirichlet forms on von Neumann algebras. The generators (Dirichlet operators) of the semigroups are analyzed and the spectrums together with eigenspaces are found. When restricted to a maximal Abelian subalgebra, the semigroups are reduced to a unique Markovian semigroup of classical Ornstein–Uhlenbeck process.

FEYNMAN–KAC REPRESENTATION AND MARKOV PROPERTY OF SEMIGROUPS GENERATED BY NONCOMMUTATIVE ELLIPTIC OPERATORS

We construct Markovian semigroups generated by noncommutative elliptic operators on a von Neumann algebra. We first introduce a general framework of quantum Feynman–Kac formula in terms of unitary evolutions and multipliers, and then apply the general result to our problem. We construct multipliers which are determined by operator valued stochastic differential equations and satisfy cocycle relations with respect to a randomized flow on a von Neumann algebra. Our works are greatly influenced by that of Lindsay and Sinha.10 In order to ensure Markov property of the semigroups, we modify the method employed by Lindsay and Sinha.10

Dirichlet forms and Dirichlet operators for infinite particle systems: Essential self-adjointness

We study the Dirichlet forms and the associated Dirichlet operators for Gibbs measures on infinite particle configuration space. The Dirichlet forms are defined to be “gradient-type” forms by introducing a measurable field of rigged Hilbert spaces on the configuration space. Under mild conditions on the interaction including singular potentials, we show that the pre-Dirichlet operator is symmetric and that the closure of the pre-Dirichlet form satisfies the Markovian property. When the interaction is three times differentiable and decreasing sub-exponentially, we show that the Dirichlet operator is essentially self-adjoint on a domain consisting of bounded smooth local functions.

The λφ43 Euclidean quantum field theory in a periodic box

The ultraviolet cutoff (the lattice cutoff) normalized Schwinger functions converge as the ultraviolet cutoff (the lattice cutoff) is removed. The limit Schwinger functions are the moments of the normalized physical measure. As a consequence of the lattice approximation, the Lee–Yang theorem and various correlation inequalities hold for the λφ43 field theory in a periodic box.

Ergodic property of Markovian semigroups on standard forms of von Neumann algebras

We give sufficient conditions for ergodicity of the Markovian semigroups associated to Dirichlet forms on standard forms of von Neumann algebras constructed by the method proposed by Park. We apply our result to show that the diffusion type Markovian semigroups for quantum spin systems are ergodic in the region of high temperatures where the uniqueness of the KMS state holds.

DIRICHLET FORMS AND SYMMETRIC MARKOVIAN SEMIGROUPS ON ℤ2-GRADED VON NEUMANN ALGEBRAS

We extend the construction of Dirichlet forms and symmetric Markovian semigroups on standard forms of von Neumann algebras given in [1] to the case of ℤ2-graded von Neumann algebras. As an application of the extension, we construct symmetric Markovian semigroups on CAR algebras with respect to gauge invariant quasi-free states and also investigate detailed properties such as ergodicity of the semigroups.