Self-consistent Green’s function formalism for acoustic and light scattering. Part 3: Unitarity and symmetry

This paper completes two previous papers in which we have developed the self-consistent Green’s function formalism for acoustic and light scattering. It is concerned with the unitarity and symmetry properties of the interaction and far field scattering operator of this formalism. We will show that these are primarily mathematical properties, and that the principles of energy conservation and reciprocity, which express our physical experience, can be modeled by these mathematical properties. For this we have to distinguish two experimental configurations, and only one of these configurations will allow us to relate energy conservation to unitarity. Closely related to this are questions concerning the definition and measurability of the scattering quantities and the importance of the optical and generalized optical theorem. These questions will be also discussed from the point of view of the self-consistent Green’s function formalism.     

The Weil-B.R.S. algebra of a Lie algebra and the anomalous terms in gauge theory

In this paper, we explain the computation we made in collaboration with M. Talon and C.M. Viallet of anomalous terms in gauge theory [1], [2], [3]. We relate our constructions to standard mathematical constructions. The paper is self-contained in the sense that all mathematical concepts and results we use are explained.     

Generalized states in SFT

The search for analytic solutions in open string fields theory à la Witten often meets with singular expressions, which need an adequate mathematical formalism to be interpreted. In this paper we discuss this problem and propose a way to resolve the related ambiguities. Our claim is that a correct interpretation requires a formalism similar to distribution theory in functional analysis. To this end we concretely construct a locally convex space of test string states together with the dual space of functionals. We show that the above suspicious expressions can be identified with well defined elements of the dual.     

A mathematical interpretation of Dirac's formalism part a: Dirac bases in trajectory spaces

In this paper the notion of Dirac basis will be introduced. It is the continuous pendant of the discrete basis for Hilbert spaces. The introduction of this new notion is closely related to the theory of generalized functions. Here De Graaf's theory will be employed. It is based on the triplet $\text{S}{}_{\mathrm{<mtext>X,</mtext><mtext>A</mtext>}}\text{?X?}\text{T}{}_{\mathrm{<mtext>X,</mtext><mtext>A</mtext>}}$ where X is a Hilbert space. In a well specified way any member of $\text{T}{}_{\mathrm{<mtext>X,</mtext><mtext>A</mtext>}}$ can be expanded with respect to a Dirac basis. Both the introduction of Dirac bases and a new interpretation of Dirac's bracket notion will lead to a mathematical rigorization of various aspects of Dirac's formalism for quantum mechanics. This rigorization goes much beyond earlier proposals.     

Singular reduction of implicit Hamiltonian systems

This paper develops the theory of singular reduction for implicit Hamiltonian systems admitting a symmetry Lie group. The reduction is performed at a singular value of the momentum map. This leads to a singular reduced topological space which is not a smooth manifold. A topological Dirac structure on this space is defined in terms of a generalized Poisson bracket and a vector space of derivations, both being defined on a set of smooth functions. A corresponding Hamiltonian formalism is described. It is shown that solutions of the original system descend to solutions of the reduced system. Finally, if the generalized Poisson bracket is nondegenerate, then the singular reduced space can be decomposed into a set of smooth manifolds called pieces. The singular reduced system restricts to a regular reduced implicit Hamiltonian system on each of these pieces. The results in this paper naturally extend the singular reduction theory as previously developed for symplectic or Poisson Hamiltonian systems.     

Reply to “On the Morse oscillator with a kinetic coupling” by Fernández

Having analyzed Fernández's approach for the Morse oscillator with a kinetic coupling [Phys. Lett. A 229 (1997) 262] and contrasting it with our original paper [Phys. Lett. A 213 (1996) 226], we point out that a new representation is implicitly used by Fernández. The dynamic phase difference of two wave functions obtained via two different approaches is due to different representations adopted in these two papers respectively.     

The Nambu mechanics as a class of singular generalized dynamical formalisms

In a recent work Nambu has proposed ac-number dynamical formalism which can allow an odd numbern of canonical variables. Naturally associated to this new mechanics there exists ann-linear bracket whose study opens interesting possibilities. The purpose of this work is to show that besides this bracket another one which is bilinear and in fact a Lie bracket can also be associated with the Nambu mechanics. For anyn, however, this bracket is singular. In a sense previously used by the present author, this result exhibits the Nambu mechanics as an interesting class of singular generalized dynamical formalisms irrespective of the number of phase space variables. Reasons are given suggesting that such singular formalisms would be, within our context, the only ones capable of describing classical analogues of generalized quantum systems.     

Quantum theory of collective motions and an application to theory of extended hadrons

It is shown that the theory of collective motions can be reduced to a gauge problem in Dirac's generalized Hamiltonian formalism. As an example, we study the φ⁴ theory in two-dimensional space-time by using our formalism.     

Quantum nonlocality for two-mode correlated states based on generalized pseudospin operators

In this Letter, we investigate the quantum nonlocality of two-mode correlated states. We find that the pseudospin formalism [Z.B. Chen, J.W. Pan, G. Hou, Y.D. Zhang, Phys. Rev. Lett. 88 (2002) 040406] generally fails to depict the nonlocality of the states when the photon number difference between the two modes is odd. The formalism is then generalized such that the nonlocality of a two-mode correlated state can be well revealed without regard to the difference. Later we consider the nonlocality of the two-mode intelligent SU(1,1) states in the generalized formalism and compare our results with the entanglement of the corresponding states.     

Conditional expectations,conditional distributions,anda posteriori ensembles in generalized probability theory

A general probabilistic framework containing the essential mathematical structure of any statistical physical theory is reviewed and enlarged to enable the generalization of some concepts of classical probability theory. In particular, generalized conditional probabilities of effects and conditional distributions of observables are introduced and their interpretation is discussed in terms of successive measurements. The existence of generalized conditional distributions is proved, and the relation to M. Ozawa'sa posteriori states is investigated. Examples concerning classical as well as quantum probability are discussed.     

The generalized zero-mode supersymmetry scheme and the confluent algorithm

We show the relationship between the mathematical framework used in recent papers by Rosu et al. (2014) [1–3] and the second-order confluent supersymmetric quantum mechanics. In addition, we point out several immediate generalizations of the approach taken in the latter references. Furthermore, it is shown how to apply the generalized scheme to the Dirac and to the Fokker–Planck equation.     

The conceptual analysis (CA) method in theories of microchannels: Application to quantum theory. Part II. Idealizations. “Perfect measurements”

The application of the conceptual analysis (CA) method outlined in Part I is illustrated on the example of quantum mechanics. In Part II, we deduce the complete-lattice structure in quantum mechanics from postulates specifying the idealizations that are accepted in the theory. The idealized abstract concepts are introduced by means of a topological extension of the basic structure (obtained in Part I) in accord with the “approximation principle”; the relevant topologies are not arbitrarily chosen; they are fixed by the choice of the idealizations. There is a typical topological asymmetry in the mathematical scheme. Convexity or linear structures do not play any role in the mathematical methods of this approach. The essential concept in Part II is the idealization of “perfect measurement” suggested by our conceptual analysis in Part I. The Hilbert-space representation will be deduced in Part III. In our papers, we keep to the tenet: The mathematical scheme of a physical theory must be rigorously formulated. However, for physics, mathematics is only a nice and useful tool; it is not purpose.     

Particle creation,classicality and related issues in quantum field theory: I. Formalism and toy models

The quantum theory of a harmonic oscillator with a time dependent frequency arises in several important physical problems, especially in the study of quantum field theory in an external background. While the mathematics of this system is straightforward, several conceptual issues arise in such a study. We present a general formalism to address some of the conceptual issues like the emergence of classicality, definition of particle content, back reaction etc. In particular, we parameterize the wave function in terms of a complex number (which we call excitation parameter) and express all physically relevant quantities in terms it. Many of the notions—like those of particle number density, effective Lagrangian etc., which are usually defined using asymptotic in–out states—are generalized as time-dependent concepts and we show that these generalized definitions lead to useful and reasonable results. Having developed the general formalism we apply it to several examples. Exact analytic expressions are found for a particular toy model and approximate analytic solutions are obtained in the extreme cases of adiabatic and highly non-adiabatic evolution. We then work out the exact results numerically for a variety of models and compare them with the analytic results and approximations. The formalism is useful in addressing the question of emergence of classicality of the quantum state, its relation to particle production and to clarify several conceptual issues related to this. In Paper II which is a sequel to this, the formalism will be applied to analyze the corresponding issues in the context of quantum field theory in background cosmological models and electric fields.