Geometric BRST quantization,I: Prequantization

This is the first part of a two-part paper dedicated to the definition of BRST quantization in the framework of geometric quantization. After recognizing prequantization as a manifestation of the Poisson module structure of the sections of the prequantum line bundle, we define BRST prequantization and show that it is the homological analog of the symplectic reduction of prequantum data. We define a prequantum BRST cohomology theory and interpret it in terms of geometric objects. We then show that all Poisson structures correspond under homological reduction. This allows to prove, in the BRST context, that prequantization and reduction commute.     

Geometric Prequantization for the Binary Black Holes

In this letter, the effective-one-body Hamiltonian of two spinning black hole considered and prequantization operators obtained by using the closed 2-form. It is indeed an application of prequantization method in a given physical system. Our results may be considered as mathematical tool and is useful to obtain the wave function.     

Local transformations and KMS states

A unified approach to the DLR and KMS conditions is presented emphasizing local properties.     

KMS States and Conformal Measures

From a non-constant holomorphic map on a connected Riemann surface we construct an étale second countable locally compact Hausdorff groupoid whose associated groupoid C*-algebra admits a one-parameter group of automorphisms with the property that its KMS states correspond to conformal measures in the sense of Sullivan. In this way certain quadratic polynomials give rise to quantum statistical models with a phase transition arising from spontaneous symmetry breaking.     

Condition of Prequantization to Two—Dimensional Manifolds with Boundary

The Weil‘s integrality condition of prequantization is generalized to two-dimensional phase space with boundaries.It is shown that in the prequantization condition a term related to the symplectic potential on the boundary appears.The necessity of the generalized condition is proved by analyzing the isolated singularities of the Hermitian bundle while the sufficiency of the condition is proved via geometric construction on the space of equivalence class.     

Stability properties and the KMS condition

First we derive stability properties of KMS states and subsequently we derive the KMS condition from stability properties. New results include a convergent perturbation expansion for perturbed KMS states in terms of appropriate truncated functions and stability properties of ground states. Finally we extend the results of Haag, Kastler, Trych-Pohlmeyer by proving that stable states ofL ¹-asymptotically abelian systems which satisfy a weak three point cluster property are automatically KMS states. This last theorem gives an almost complete characterization of KMS states, ofL ¹-asymptotic abelian systems, by stability and cluster properties (a slight discrepancy can occur for infinite temperature states).Supported during this research by the Norwegian Research Council for Science and Humanities     

Classical KMS condition and Tomita-Takesaki theory

Relationships between the classical KMS condition and the time evolution for classical systems are discussed.     

Quantum detailed balance and KMS condition

A definition of detailed balance for quantum dynamical semigroups is given, and its close connection with the KMS condition is investigated.A fellowship from the Italian Ministry of Public Education is acknowledgedA fellowship from the Italian National Research Council is acknowledged     

Linear response theory and the KMS condition

The response, relaxation and correlation functions are defined for any vector state ε of a von Neumann algebra \(\mathfrak{M}\) , acting on a Hilbert space ?, satisfying the KMS-condition. An operator representation of these functions is given on a particular Hilbert space . With this technique we prove the existence of the static admittance and the relaxation function. Finally we generalize the fluctuation-dissipation theorem and other relations between the above mentionned functions to infinite systems.     

Partial Dynamical Systems and the KMS Condition

 Given a countably infinite 0–1 matrix A without identically zero rows, let 𝒪 _A be the Cuntz–Krieger algebra recently introduced by the authors and 𝒯 _A be the Toeplitz extension of 𝒪 _A , once the latter is seen as a Cuntz–Pimsner algebra, as recently shown by Szymański. We study the KMS equilibrium states of C ^* -dynamical systems based on 𝒪 _A and 𝒯 _A , with dynamics satisfying for the canonical generating partial isometries s _x and arbitrary real numbers N _x > 1. The KMS_β states on both 𝒪 _A and 𝒯 _A are completely characterized for certain values of the inverse temperature β, according to the position of β relative to three critical values, defined to be the abscissa of convergence of certain Dirichlet series associated to A and the N(x). Our results for 𝒪 _A are derived from those for 𝒯 _A by virtue of the former being a covariant quotient of the latter. When the matrix A is finite, these results give theorems of Olesen and Pedersen for 𝒪 _n and of Enomoto, Fujii and Watatani for 𝒪 _A as particular cases. Received: 21 November 2000 / Accepted: 31 May 2002 Published online: 22 November 2002 RID="*" ID="*" Partially supported by CNP_q RID="**, ***"     

A spectral characterization of KMS states

Let be a state on aC*-dynamical system . For each of the following properties of : (1) is -K MS with respect to for some given , 0<+, (2) is either a KMS state or a ground state, necessary and sufficient conditions are given involving only the spectral subspaces of associated with . The results provide a new insight in the concept of passivity, introduced by W. Pusz and S. L. Woronowicz.Aangesteld navorser N.F.W.O., Belgium, on leave from Katholieke Universiteit Leuven. Research partially supported by N.A.T.O.     

KMS states for the Weyl algebra

We consider the possible automorphism groups for the Weyl algebra overR, resp.T, and classify those for which KMS states, unique or not unique, exist.     

On the existence of KMS states

A necessary and sufficient condition for the existence of KMS states is formulated in terms of a certain left ideal in the algebraA ⁰ A.     

Relaxation,amplification, and the KMS conditions

We formulate the dynamics of an oscillator, $\text{gS}$, harmonically coupled to an infinite quasifree bose system Σ, the initial states of Σ and Σ being mutually uncorrelated and the initial state of Σ being quasifree, primary, and stationary. It is shown that, subject to certain regularity conditions, the oscillator $\text{gS}$ will either relax towards a terminal state, whose value is independent of the initial state of the oscillator, or else will become amplified, according to whether the energy quantum for $\text{gS}$ is positive or negative. In the former case, the weak-coupling limit for the terminal state of $\text{gS}$ is a Gibbs state, corresponding to a temperature which takes the same value for all $\text{Σ-}\text{gS}$ couplings of the prescribed class, if and only if the initial state of Σ satisfies the KMS conditions for that temperature. We argue that this result provides an operational justification for identifying equilibrium conditions with KMS conditions, at least for the class of stationary states of Σ that we consider.     

Cyclic cocycles from graded KMS functionals

Each graded KMS functional of aZ/2-gradedC*-algebra with respect to a supersymmetric one-parameter automorphism group gives rise to a cyclic cocycle.     

Passive states and KMS states for general quantum systems

We characterize equilibrium states of quantum systems by a condition of passivity suggested by the second principle of thermodynamics. Ground states and -KMS states for all inverse temperatures 0 are completely passive. We prove that these states are the only completely passive ones. For the special case of states describing pure phases, assuming the passivity we reproduce the results of Haag et al.     

Closed 2-Form of 2D Black Holes from Geometric Prequantization Method

In this paper, we consider effective Hamiltonian of 2D dilatonic black hole adding Axion field and calculate closed 2-form by using geometric prequantization method. It yields to the Schrödinger equation which may be solved to obtain wave function. We obtained a condition on the cosmological constant to obtain appropriate Hamilton equation of motions.