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.     

The Madelung Picture as a Foundation of Geometric Quantum Theory

Despite its age, quantum theory still suffers from serious conceptual difficulties. To create clarity, mathematical physicists have been attempting to formulate quantum theory geometrically and to find a rigorous method of quantization, but this has not resolved the problem. In this article we argue that a quantum theory recursing to quantization algorithms is necessarily incomplete. To provide an alternative approach, we show that the Schrödinger equation is a consequence of three partial differential equations governing the time evolution of a given probability density. These equations, discovered by Madelung, naturally ground the Schrödinger theory in Newtonian mechanics and Kolmogorovian probability theory. A variety of far-reaching consequences for the projection postulate, the correspondence principle, the measurement problem, the uncertainty principle, and the modeling of particle creation and annihilation are immediate. We also give a speculative interpretation of the equations following Bohm, Vigier and Tsekov, by claiming that quantum mechanical behavior is possibly caused by gravitational background noise.     

Framework for possible unification of quantum and relativity theories

We put forward a framework, inspired by recent axiomatic and operational approaches to generalized quantum theories, wherein we investigate the possibility of unifying quantum and relativity theories. The framework concentrates on a detailed analysis of a general construction of reality that can be used in both quantum and relativity theories. By means of this construction of reality we clarify some well-known conceptual problems that stand in the way of a conceptual unification of quantum and relativity theories on a more profound physical level than the purely mathematical algebraic level on which unification attempts are generally investigated. More specifically we concentrate on the problem of what is physical reality in quantum and relativity theories.     

A categorical framework for quantum theory

Underlying any physical theory is a layer of conceptual frames. They connect the mathematical structures used in theoretical models with the phenomena, but they also constitute our fundamental assumptions about reality. Many of the discrepancies between quantum physics and classical physics (including Maxwell's electrodynamics and relativity) can be traced back to these categorical foundations. We argue that classical physics corresponds to the factual aspects of reality and requires a categorical framework which consists of four interdependent components: boolean logic, the linear‐sequential notion of time, the principle of sufficient reason, and the dichotomy between observer and observed. None of these can be dropped without affecting the others. However, quantum theory also addresses the “status nascendi” of facts, i.e., their coming into being. Therefore, quantum physics requires a different conceptual framework which will be elaborated in this article. It is shown that many of its components are already present in the standard formalisms of quantum physics, but in most cases they are highlighted not so much from a conceptual perspective but more from their mathematical structures. The categorical frame underlying quantum physics includes a profoundly different notion of time which encompasses a crucial role for the present. The article introduces the concept of a categorical apparatus (a framework of interdependent categories), explores the appropriate apparatus for classical and quantum theory, and elaborates in particular on the category of non‐sequential time and an extended present which seems to be relevant for a quantum theory of (space)‐time.     

Non-Kolmogorovian Probabilities and Quantum Technologies

In this work, we focus on the philosophical aspects and technical challenges that underlie the axiomatization of the non-Kolmogorovian probability framework, in connection with the problem of quantum contextuality. This fundamental feature of quantum theory has received a lot of attention recently, given that it might be connected to the speed-up of quantum computers—a phenomenon that is not fully understood. Although this problem has been extensively studied in the physics community, there are still many philosophical questions that should be properly formulated. We analyzed different problems from a conceptual standpoint using the non-Kolmogorovian probability approach as a technical tool.     

The Observer in the Quantum Experiment

A goal of most interpretations of quantum mechanics is to avoid the apparent intrusion of the observer into the measurement process. Such intrusion is usually seen to arise because observation somehow selects a single actuality from among the many possibilities represented by the wavefunction. The issue is typically treated in terms of the mathematical formulation of the quantum theory. We attempt to address a different manifestation of the quantum measurement problem in a theory-neutral manner. With a version of the two-slit experiment, we demonstrate that an enigma arises directly from the results of experiments. Assuming that no observable physical phenomena exist beyond those predicted by the theory, we argue that no interpretation of the quantum theory can avoid a measurement problem involving the observer.     

A Topos Perspective on the Kochen-Specker Theorem II. Conceptual Aspects and Classical Analogues

In a previous paper, we proposed assigning asthe value of a physical quantity in quantum theory acertain kind of set (a sieve) of quantities that arefunctions of the given quantity. The motivation was in part physical — such a valuationilluminates the Kochen–Specker theorem — andin part mathematical — the valuation arisesnaturally in the topos theory of presheaves. This paperdiscusses the conceptual aspects of this proposal. We also undertake two othertasks. First, we explain how the proposed valuationscould arise much more generally than just in quantumphysics; in particular, they arise as naturally in classical physics. Second, we give anothermotivation for such valuations (that applies equally toclassical and quantum physics). This arises fromapplying to propositions about the values of physical quantities some general axioms governingpartial truth for any kind of proposition.     

Quantum geometrodynamics: whence,whither?

Quantum geometrodynamics is canonical quantum gravity with the three-metric as the configuration variable. Its central equation is the Wheeler–DeWitt equation. Here I give an overview of the status of this approach. The issues discussed include the problem of time, the relation to the covariant theory, the semiclassical approximation as well as applications to black holes and cosmology. I conclude that quantum geometrodynamics is still a viable approach and provides insights into both the conceptual and technical aspects of quantum gravity.

These considerations reveal that the concepts of spacetime and time itself are not primary but secondary ideas in the structure of physical theory. These concepts are valid in the classical approximation. However, they have neither meaning nor application under circumstances when quantum-geometrodynamical effects become important. ...There is no spacetime, there is no time, there is no before, there is no after. The question what happens “next” is without meaning [1].

Dedicated to the memory of John Archibald Wheeler.     

The conceptual analysis (CA) method in theories of microchannels: Application to quantum theory. Part III. Idealizations. Hilbert space representation

We illustrate the application of the conceptual analysis (CA) method outlined in Part I by the example of quantum mechanics. In the present part the Hilbert space structure of conventional quantum mechanics is deduced as a consequence of postulates specifying further idealized concepts. A critical discussion of the idealizations of quantum mechanics is proposed. Quantum mechanics is characterized as a “statistically complete” theory and a simple and elegant formal recipe for the construction of the fundamental mathematical apparatus of quantum mechanics is formulated. Our analysis may also lead to a criticism of quantum mechanics as a “strongly idealized” theory. A critical analysis of the fundamental structure of quantum mechanics seems an indispensable and natural starting point for the construction of new theories. A major technical problem in a more general application of the CA method is the lack of mathematical representation theorems for more general algebraic structures.     

Quantum Theory and the Role of Mind in Nature

Orthodox Copenhagen quantum theory renounces the quest to understand the reality in which we are imbedded, and settles for practical rules describing connections between our observations. Many physicist have regarded this renunciation of our effort describe nature herself as premature, and John von Neumann reformulated quantum theory as a theory of an evolving objective universe interacting with human consciousness. This interaction is associated both in Copenhagen quantum theory and in von Neumann quantum theory with a sudden change that brings the objective physical state of a system in line with a subjectively felt psychical reality. The objective physical state is thereby converted from a material substrate to an informational and dispositional substrate that carries both the information incorporated into it by the psychical realities, and certain dispositions for the occurrence of future psychical realities. The present work examines and proposes solutions to two problems that have appeared to block the development of this conception of nature. The first problem is how to reconcile this theory with the principles of relativistic quantum field theory; the second problem is to understand whether, strictly within quantum theory, a person's mind can affect the activities of his brain, and if so how. Solving the first problem involves resolving a certain non-locality question. The proposed solution to the second problem is based on a postulated connection between effort, attention, and the quantum Zeno effect. This solution explains on the basic of quantum physics a large amount of heretofore unexplained data amassed by psychologists.     

Topological protection and quantum noiseless subsystems

Encoding and manipulation of quantum information by means of topological degrees of freedom provides a promising way to achieve natural fault tolerance that is built in at the physical level. We show that this topological approach to quantum information processing is a particular instance of the notion of computation in a noiseless quantum subsystem. The latter then provides the most general conceptual framework for stabilizing quantum information and for preserving quantum coherence in topological and geometric systems.     

Can von Neumann’s Theory Meet the Deutsch-Jozsa Algorithm?

The theoretical formalism of the implementation of the Deutsch-Jozsa algorithm relies on von Neumann’s theory. We try to investigate whether von Neumann’s theory meet our physical world. We derive a proposition concerning a quantum expectation value under the assumption of the existence of the orientation of reference frames in N spin-1/2 systems (1≤N<+∞). This assumption intuitively depictures our physical world. However, the quantum predictions within the formalism of von Neumann’s projective measurement violate the proposition with a magnitude that grows exponentially with the number of particles. Therefore, von Neumann’s theory cannot depicture our physical world with a violation factor that grows exponentially with the number of particles. Hence, von Neumann’s theory cannot meet the Deutsch-Jozsa algorithm. We propose the solution of the problem. Our solution is equivalent to changing Planck’s constant (?) to new constant (\(\hbar/\sqrt{2}\)). It may be that a new type of the quantum theory early approaches Newton’s theory in the macroscopic scale than the old quantum theory does so.     

Nonlinear classical theory of electromagnetism

A topological theory of electric charge is given. Einstein's criteria for the completion of classical electromagnetic theory are summarized and their relation to quantum theory and the principle of complementarity is indicated. The inhibiting effect that this principle has had on the development of physical thought is discussed. Developments in the theory of functions on nonlinear spaces provide the conceptual framework required for the completion of electromagnetism. The theory is based on an underlying field which is a continuous mapping of space-time into points on the two-sphere.     

Renormalization and asymptotic behavior in a finite quantum field theory with shadow states

In this paper we present a study of the renormalization problem in a finite quantum field theory with shadow states for a system of a physical scalar field interacting with a physical fermion field. In order to make the theory finite, two fermion shadow fields are introduced. We observe that the stability criterion of renormalization can not be satisfied simultaneously by both physical fields and shadow fields, if the finiteness of the theory is to be maintained. A physical interpretation of this result is given. Furthermore, we find that the effective complete propagators for large space-like momenta behave like free field propagators without the logarithmic factors observed in the non-abelian gauge theory.     

Quantization by parts,self-adjoint extensions,and a novel derivation of the Josephson equation in superconductivity

There has been a lot of interest in generalizing orthodox quantum mechanics to include POV measures as observables, namely as unsharp obserrables. Such POV measures are related to symmetric operators. We have argued recently that only maximal symmetric operators should describe observables.¹ This generalization to maximal symmetric operators has many physical applications. One application is in the area of quantization. We shall discuss a scheme, to he called quantization by parts,which can systematically deal with what may be called quantum circuits. As a specific application we shall present a novel derivation of the famous Josephson equation for the supercurrent through a Josephson junction in a superconducting circuit. An interesting effect emerges from our quantization scheme when applied to a superconducting Y-shape circuit configuration. We also propose an experimental test for this effect which is expected to shed light on some conceptual problems on the quantum nature of the condensate.