Izumi Ojima

Local Covariant Operator Formalism of Non-Abelian Gauge Theories and Quark Confinement Problem

A manifestly covariant and local canonical operator formalism of non-Abelian gauge theories is presented in its full detail. This formalism, applicable to Yang-Mills theories as well as to gravity, not only provides us a transparent understanding in the scattering theoretical aspects, but also makes it possible to discuss other important problems directly related to the (Heisenberg) operators and the state vectors: As for the former, the physical S-matrix unitarity is proved quite generally on the basis of the representation of the algebra of the BRS charge, and asymptotic field analysis is explicitly performed for some examples. As for the latter, the problems of observables and the well-definedness of charge operators are discussed and clear results are obtained, where the locality and covariance of the formalism are indispensable. Observables are shown to be invariant under the BRS transformation as well as the unbroken global gauge groups. By analyzing the structure of “Maxwell” equations in YM theories, the converse of the Higgs theorem is found to hold. This turns out to lead to a remarkably simple criterion of quark confinement in QCD. The present formalism is found useful also for the U(1) problem and the charge universality proof in the Weinberg-Salam model. General theory of indefinite metric quantum fields is developed to some extent.

Manifestly covariant canonical formulation of Yang-Mills theories physical state subsidiary conditions and physical S-matrix unitarity

Abstract A satisfactory canonical formulation of Yang-Mills theories is presented in a manifestly covariant manner. The subsidiary conditions for physical states are given as Q B |phys> = Q C |phys> = 0 in terms of the two conserved charges Q B and Q C . These conditions assure the physical S -matrix unitarity.

Thermodynamic Properties of Non-equilibrium States in Quantum Field Theory

Abstract Within the framework of relativistic quantum field theory, a novel method is established which allows for distinguishing non-equilibrium states admitting locally a thermodynamic interpretation. The basic idea is to compare these states with global equilibrium states (KMS states) by means of local thermal observables. With the help of such observables, the states can be ordered into classes of increasing local thermal stability. Moreover, it is possible to identify states exhibiting certain specific thermal properties of interest, such as a definite local temperature or entropy density. The method is illustrated in a simple model describing the spatio-temporal evolution of a “big heat bang.”

Subsidiary conditions and physical S-matrix unitarity in indefinite-metric quantum gravitation theory

Abstract The quantum theory of gravitational fields is formulated in a manifestly Lorentz covariant manner in the framework of indefinite-metric quantum field theory. The physical state subspace is defined by the two subsidiary conditions Q B |phys〉 = Q c |phys〉 = 0, where the conserved charges Q B and Q c are the generators of the BRS transformation ond of the Faddeev-Popov ghost scale transformation, respectively. By clarifying the metric structures of the Fock space of asymptotic fields with help of the Ward-Takahashi identities, the physical S -matrix unitarity is established in just the same way as in the canonical theory of Yang-Mills fields.

Entropy production and its positivity in nonlinear response theory of quantum dynamical systems

A formulation ofentropy production is given with the aid of relative entropy in the nonlinear response theory of a quantum dynamical system. It allows a natural interpretation of the quantity in terms of familiar thermodynamic notions, such as force and response current conjugate to it, without sacrificing the full nonlinearity in the perturbing force. For the understanding ofdissipativity aspositive entropy production, the stationarity of states and coarse graining of time scale turn out to be essential, which are implemented by some time averaging procedures involvingalmost periodic external forces. Finally, it is shown that the obtained result reduces, in the linear response regime, to the power dissipation appearing in the well-known fluctuation-dissipation relation.

Lorentz invariance vs. temperature in QFT

The Lorentz invariance of a relativistic QFT is shown to be broken spontaneously at finite temperature with a zero-energy Goldstone spectrum which does not necessarily carry a one-particle structure. The Lorentz behaviour of the temperature is determined and the relation between Gibbs states in different Lorentz frames is clarified.

Micro-Macro Duality in Quantum Physics

Micro-Macro Duality means here the universal mutual relations between the microscopic quantum world and various macroscopic classical levels, which can be formulated mathematically as categorical adjunctions. It underlies a unified scheme for generalized sectors based upon selection criteria proposed by myself in 2003 to control different branches of physics from a unified viewpoint, which has played essential roles in extending the Doplicher-Haag-Roberts superselection theory to various situations with spontaneously as well as explicitly broken symmetries. Along this line of thought, the state correlations between a system and a measuring apparatus necessary for measurements can canonically be formulated within the context of group duality; the obtained measurement scheme is not restricted to the quantum mechanical situations with finite number of particles but can safely be applied to quantum field theory with infinite degrees of freedom whose local subalgebras are given by type III von Neumann algebras.

A Unified Scheme for Generalized Sectors Based on Selection Criteria: Order Parameters of Symmetries and of Thermality and Physical Meanings of Adjunctions

A unified scheme for treating generalized superselection sectors is proposed on the basis of the notion of selection criteria to characterize states of relevance to each specific domain in quantum physics, ranging from the relativistic quantum fields in the vacuum situations with unbroken and spontaneously broken internal symmetries, through equilibrium and non-equilibrium states to some basic aspects in measurement processes. This is achieved by the help of c → q and q → c channels: the former determines the states to be selected and to be parameterized by the order parameters, and the latter provides the physical interpretations of selected states in terms of order parameters. This formulation extends the traditional range of applicability of the Doplicher-Roberts construction method for recovering the field algebra and the gauge group (of the first kind) from the data of group invariant observables to the situations with spontaneous symmetry breakdown: in use of the machinery proposed, the physical and mathematical meaning of basic structural ingredients associated with the spontaneously broken symmetry are re-examined, such as the degenerate vacua parameterized by the variables belonging to the relevant homogeneous space, the Goldstone modes and condensates, etc. The geometrical meaning of the space of order parameters is naturally understood in relation with the adjunction as the classifying space of a sector structure. As further examples of applications, some basic notions arising in the mathematical framework of quantum theory are reformulated and examined in connection with control theory.

Entropy production and nonequilibrium stationarity in quantum dynamical systems. Physical meaning of van Hove limit

With aid of the so-called dilation method, a concise formula is obtained for the entropy production in the algebraic formulation of quantum dynamical systems. In this framework, the initial ergodic state of an external force system plays a pivotal role in generating dissipativity as a conditional expectation. The physical meaning of van Hove limit is clarified through the scale-changing transformation to control transitions between microscopic and macroscopic levels. It plays a crucial role in realizing the macroscopic stationarity in the presence of microscopic fluctuations as well as in the transition from non-Markovian (groupoid) dynamics to Markovian dissipative processes of state changes. The extension of the formalism to cases with spatial and internal inhomogeneity is indicated in the light of the groupoid dynamical systems and noncommutative integration theory.

Observables and quark confinement in the covariant canonical formalism of Yang-Mills theory

The theorem stating the absence of localized charged physical states in QED is extended to QCD in the manifestly covariant canonical formalism. The correct relation between the Gupta-Bleuler condition and the coherent photon states associated to the charged sectors is clarified in the case of QED, which secures the existence of electrons as non-localized charged physical states.