Foundational Problems in and Methodological Lessons from Quantum Field Theory

Rom Harre & Harvey Brown both of the University of Oxford: Introduction I. QUANTUM FIELD THEORY AS AN OBJECT OF PHILOSOPHICAL STUDY: TWO VIEWS: Michael Redhead, University of Cambridge: A philosopher looks at quantum field theory James T. Cushing, University of Notre Dame: Foundational problems in and methodological lessons from quantum field theory II. THE PROBLEMS OF VIRTUAL PARTICLES AND RENORMALIZATION: Robert Weingard, Rutgers University: Virtual particles and the interpretation of quantum field theory Rom Harre: Parsing the amplitudes Paul Teller, University of Illinois: Three problems of renormalization III. COVARIANCE PRINCIPLES IN QUANTUM FIELD THEORY: Gordon N. Fleming, Pennsylvania State University: Hyperplane-dependent quantized fields and Lorentz invariance Tian-yu Cao, Trinity College, Cambridge: Gauge theory and the geometrization of fundamental physics IV. MATHEMATICAL FOUNDATIONS OF QUANTUM FIELD THEORY: Ray F. Streater, King's College, London: Why should anyone want to axiomatize quantum field theory? Simon Saunders, Wolfson College, Oxford: The algebraic notation of quantum field theory.

Holism and nonseparability by analogy

In theoretical physics, important new results have often been found by realising that two different concepts are related to each other at a deep and fundamental level. Examples of such relations are dualities which relate two seemingly different quantum theories to each other by stating that the theories are in fact equivalent. In particular, the Hilbert spaces and the dynamics of the two theories agree. From a mathematical point of view, this means that the theories are identical. However, from a physical point of view, their descriptions may differ, for instance there may be different Lagrangians for the two theories. The duality examples mentioned in box 5.1 either relate quantum field theories together, or they relate string theories together. The Anti-de Sitter/Conformal Field Theory correspondence (AdS/CFT), however, is a new type of duality which relates a quantum field theory on flat spacetime to a string theory. This is particularly remarkable since string theory is a very promising candidate for a consistent theory of quantum gravity. Naively, quantum field theory on flat spacetime does not appear to be a theory of quantum gravity. However, the AdS/CFT correspondence, being a duality, implies that the two theories are equivalent. This explains why many scientists think that the AdS/CFT correspondence, discovered by Maldacena in 1997, is one of the most exciting discoveries in modern theoretical physics in the last two decades. Moreover, the AdS/CFT correspondence is an important realisation of the holographic principle . This principle states that in a gravitational theory, the number of degrees of freedom in a given volume V scales as the surface area ∂ V of that volume, as described in box 5.2. The theory of quantum gravity involved in the AdS/CFT correspondence is defined on a manifold of the form AdS × X , where AdS is the Anti-de Sitter space and X is a compact space. The quantum field theory may be thought of as being defined on the conformal boundary of this Anti-de Sitter space.

In this work, we develop a new technique for the numerical study of quantum field theory. The procedure, borrowed from non-relativistic quantum mechanics, is that of finding the eigenvalues of a finite Hamiltonian matrix. The matrix is created by evaluating the matrix elements of the Hamiltonian operator on a finite basis of states. The eigenvalues and eigenvectors of the finite dimensional matrix become an accurate approximation to those of the physical system as the finite basis of states is extended to become more complete. We study a model of scalars coupled to fermions in 0+1 dimensions as a simple field theory to consider in the course of developing the technique. We find in the course of studying this model a change of basis which diagonalizes the Hamiltonian in the large coupling limit. The importance of this transformation is that it can be generalized to higher dimensional field theories involving a trilinear coupling between a Bose and a Fermi field. Having developed the numerical and analytical techniques, we consider a Fermi field coupled to a Bose field in 1+1 dimensions with the Yukawa coupling λΨΨΨ. We extend the large coupling limit basis of the 0+1 dimensional model to this case using a Bogoliubov transformation on the fermions. Although we do not use this basis in the numerical work due to its complexity, it provides a handle on the behavior of the system in the large coupling limit. In this model we consider the effects of renormalization and the generation of bound states.

An imaginary-time method was developed for calculating the probability of particle transmission through smooth barriers variable with time. Within the imaginary-time method, the tunneling process is described by using classical equations of motion written in terms of an imaginary time (t → it), while the probability of tunneling is determined by the imaginary part of the action functional, this imaginary part being calculated along the subbarrier particle trajectory. The fundamentals of the imaginary-time method are surveyed, and its applications in the theory of atomic-state ionization under the effect of constant electric and magnetic fields that have various configurations, in the field of intense monochromatic laser radiation and of an ultrashort electromagnetic pulse, in the process of Lorentz ionization of atoms and ions during their motion in a strong magnetic field, etc., are outlined. The applications of the imaginary-time method in relativistic cases—for example, in the theory of ionization of levels of multiply charged ions whose binding energy is commensurate with the electron rest energy—and in quantum field theory (Schwinger effect, which consists in the production of electron-positron pairs from a vacuum by a superstrong external field) are briefly described. Particular attention is given to methodological issues and details of the imaginary-time method that are of importance in solving specific physics problems, but which are usually skipped in original publications.

Quantum field theory is the traditional solution to the problems inherent in melding quantum mechanics with special relativity. However, it has also long been known that an alternative first-quantized formulation can be given for relativistic quantum mechanics, based on the parametrized paths of particles in spacetime. Because time is treated similarly to the three space coordinates, rather than as an evolution parameter, such a spacetime approach has proved particularly useful in the study of quantum gravity and cosmology. This paper shows how a spacetime path formalism can be considered to arise naturally from the fundamental principles of the Born probability rule, superposition, and Poincaré invariance. The resulting formalism can be seen as a foundation for a number of previous parametrized approaches in the literature, relating, in particular, “off-shell” theories to traditional on-shell quantum field theory. It reproduces the results of perturbative quantum field theory for free and interacting particles, but provides intriguing possibilities for a natural program for regularization and renormalization. Further, an important consequence of the formalism is that a clear probabilistic interpretation can be maintained throughout, with a natural reduction to nonrelativistic quantum mechanics.

The aim of relativistic quantum mechanics is to describe the finer details of the structure of atoms and molecules, where relativistic effects become nonnegligible. It is a sort of intermediate realm, between the familiar nonrelativistic quantum mechanics and fully relativistic quantum field theory, and thus it lacks the simplicity and elegance of both. Yet it is a necessary tool, mostly for quantum chemists. Pilkuhn's book offers to this audience an up-to-date survey of these methods, which is quite welcome since most previous textbooks are at least ten years old. The point of view of the author is to start immediately in the relativistic domain, following the lead of Maxwell's equations rather than classical mechanics, and thus to treat the nonrelativistic version as an approximation. Thus Chapter 1 takes off from Maxwell's equations (in the noncovariant Coulomb gauge) and gradually derives the basic aspects of Quantum Mechanics in a rather pedestrian way (states and observables, Hilbert space, operators, quantum measurement, scattering,. Chapter 2 starts with the Lorentz transformations, then continues with the Pauli spin equation and the Dirac equation and some of their applications (notably the hydrogen atom). Chapter 3 is entitled `Quantum fields and particles', but falls short of treating quantum field theory properly: only creation/annihilation operators are considered, for a particle in a box. The emphasis is on two-electron states (the Pauli principle, the Foldy--Wouthuysen elimination of small components of Dirac spinors, Breit projection operators. Chapter 4 is devoted to scattering theory and the description of relativistic bound states. Chapter 5, finally, covers hyperfine interactions and radiative corrections. As we said above, relativistic quantum mechanics is by nature limited in scope and rather inelegant and Pilkuhn's book is no exception. The notation is often heavy (mostly noncovariant) and the mathematical level rather low. The central topic is the description of atoms and molecules, including relativistic effects. The author fulfils this program in a reasonable way and offers a valuable tool to the targeted audience. I am not overly enthusiastic about the end result, but I might be prejudiced. Clearly, going further would require the full power of quantum field theory, but this is clearly beyond the scope of the book.

The relationship between physics and phenomenology is a blind spot of current literature in a twofold sense. First, while many phenomenologists work on the interface to psychology, the cognitive sciences, biology, artificial intelligence, etc., only very few engage in phenomenological reflections on physics. This is puzzling since, as we shall see below, Husserl and other classical phenomenologists have been quite clear on the significance of such an undertaking, stressing the central role physics plays among the natural sciences. Secondly, while philosophy of physics has emerged as an important sub-discipline of philosophy in the twentieth century, there is hardly any work that addresses the role of phenomenology for the development of modern physics or the philosophical interpretation of physical theories (with notable exceptions that will be discussed below). This is puzzling since, as we shall see below, the father of modern field theory, Hermann Weyl, was inspired by phenomenological considerations when formulating his gauge principle, and there is a long (although usually unnoticed) history of phenomenological motifs in interpretations of quantum mechanics. Section 15.1 addresses the question of how to interpret physical theories and how to estimate the relationship between mathematical models and physical reality. In Sect. 15.2, we shed light on the perspectivist features of Husserl’s philosophy of science and discuss how it relates to current versions of scientific perspectivism. Section 15.3 addresses how phenomenological ideas have influenced the development and interpretation of early modern physics. In Sect. 15.4, we discuss observer-dependence in modern physics and highlight phenomenological motifs in Wheeler’s conception of “observer-participancy.” Finally, Sect. 15.5 sheds light on phenomenological motifs in a novel interpretation of quantum mechanics, namely QBism, and emphasizes similarities between phenomenology and the program of reconstructing quantum mechanics, a program that currently enjoys popularity in the field of quantum foundations. We shall see that the worldviews delivered by phenomenology and physics might be closer than is commonly assumed.

The mathematical universe of the quantum topos, which is formulated on the basis of classical Boolean snapshots, delivers a neo-realist description of quantum mechanics that preserves realism. The main contribution of this article is developing formal objectivity in physical theories beyond quantum mechanics in the topos-theory approach. It will be shown that neo-realist responses to non-perturbative structures of quantum field theory do not preserve realism. In this regard, the method of Feynman graphons is applied to reframe the task of describing objectivity in quantum field theory in terms of replacing the standard Hilbert-space/operator-algebra ontology with a new context category built from a certain family of topological Hopf subalgebras of the topological Hopf algebra of renormalization as algebraic/combinatorial data tied to non-perturbative structures. This topological-Hopf-algebra ontology, which is independent of instrumentalist probabilities, enables us to reconstruct gauge field theories on the basis of the mathematical universe of the non-perturbative topos. The non-Boolean logic of the non-perturbative topos cannot be recovered by classical Boolean snapshots, which is in contrast to the quantum-topos reformulation of quantum mechanics. The article formulates a universal version of the non-perturbative topos to show that quantum field theory is a globally and locally neo-realist theory which can be reconstructed independent of the standard Hilbert-space/operator-algebra ontology. Formal objectivity of the universal non-perturbative topos offers a new route to build objective semantics for non-perturbative structures.

The origin of the observed acceleration of the expansion of the universe is a\nmajor problem of modern cosmology and theoretical physics. Simple estimations\nof the contribution of vacuum to the density energy of the universe in quantum\nfield theory are known to lead to catastrophic large values compared to\nobservations. Such a contribution is therefore generally not regarded as a\nviable source for the acceleration of the expansion. In this letter we propose\nthat the vacuum contribution actually provides a small positive value to the\ndensity energy of the universe. The underlying mechanism is a manifestation of\nthe quantum nature of the gravitational field, through a Casimir-like effect\nfrom an additional compact dimension of space. A key ingredient is to assume\nthat only modes with wavelength shorter than the Hubble length contribute to\nthe vacuum. Such a contribution gives a positive energy density, has a Lorentz\ninvariant equation of state in the usual 4D spacetime and hence can be\ninterpreted as a cosmological constant. Its value agrees with observations for\na radius of a 5th extra dimension given by $35\\,\\mu$m. This implies a\nmodification of the gravitational inverse square law around this scale, close\nbut below existing limits from experiments testing gravity at short range.\n

In the development of quantum mechanics in the 1920s, both matrix mechanics (developed by Born, Heisenberg and Jordon) and wave mechanics (developed by Schrödinger) prevailed. These early attempts corresponded to the quantum mechanics of particles. Matrix mechanics was found to lead directly to the Schrödinger equation, and the Schrödinger equation could be used to derive the alternative problem for matrix mechanics. Later emphasis lay on the development of the dynamics of fields, where the classical field equations were quantized (see, for example, Weinberg). Today, quantum field theory is one of the most successful physical theories ever developed. The symmetry between particle and wave mechanics is exploited herein. One of the important properties of quantum mechanics is that it is linear, leading to some confusion about how to treat the problem of nonlinear classical field equations. In the present paper we address the case of classical nonlinear soliton equations which are exactly integrable in terms of the periodic/quasiperiodic inverse scattering transform. This means that all physical spectral solutions of the soliton equations can be computed exactly for these specific boundary conditions. Unfortunately, such solutions are highly nonlinear, leading to difficulties in solving the associated quantum mechanical problems. Here we find a strategy for developing the quantum mechanical solutions for soliton dynamics. To address this difficulty, we apply a recently derived result for soliton equations, i.e., that all solutions can be written as quasiperiodic Fourier series. This means that soliton equations, in spite of their nonlinear solutions, are perfectly linearizable with quasiperiodic boundary conditions, the topic of finite gap theory, i.e., the inverse scattering transform with periodic/quasiperiodic boundary conditions. We then invoke the result that soliton equations are Hamiltonian, and we are able to show that the generalized coordinates and momenta also have quasiperiodic Fourier series, a generalized linear superposition law, which is valid in the case of nonlinear, integrable classical dynamics and is here extended to quantum mechanics. Hamiltonian dynamics with the quasiperiodicity of inverse scattering theory thus leads to matrix mechanics. This completes the main theme of our paper, i.e., that classical, nonlinear soliton field equations, linearizable with quasiperiodic Fourier series, can always be quantized in terms of matrix mechanics. Thus, the solitons and their nonlinear interactions are given an explicit description in quantum mechanics. Future work will be formulated in terms of the associated Schrödinger equation.

BackgroundQuaternions have emerged as powerful tools in higher-dimensional quantum mechanics as they provide homogeneous four-dimensional structure in quantum field theories, offer compact representations, and incorporate spin naturally. Quantum field theories then lead to the unification of fundamental interactions so the use of quaternion becomes necessary when we are dealing with higher-dimensional theories. On the other hand, supersymmetry is the theory of bosons and fermions and is an essential constituent of grand unified theories. The use of quaternion in supersymmetric field theories provides an excellent framework for higher-dimensional unification theories.ResultA complete theory for supersymmetric quaternionic quantum mechanics has been constructed for N = 1, 2, 4 supersymmetry in terms of one, two, and four supercharges and Hamiltonians, respectively. It has been shown that N = 4 SUSY is the quaternionic extension of the N = 2 complex SUSY and N = 1 real SUSY; also spin is the natural outcome of using quaternion units. Pauli and Dirac Hamiltonian and their relationship have also been obtained in quaternion space. It has been shown that quaternionic quantum mechanics are superior to ordinary and complex quantum mechanics because in the quaternion framework we do not need three different theories for N = 1,2,4 SQM but a single theory only.ConclusionsIt has been concluded that N = 1 real SUSY is equal to N = 2 complex SUSY which in turn is equal to N = 4 quaternion SUSY so one can arrive at higher-dimensional quantum field theories starting from lower-dimensional quantum theories. Higher-dimensional quaternion field theories are suitable for nonphotonic light cone particles which are not allowed in complex QFT, also noncommutative nature of quaternion gives an extra degree of freedom and may provide the possibility of some new particle, dark matter, or new phenomenon. Though quaternions provide an excellent framework in higher-dimensional field theories, there are certain challenges due to their noncommutativity as calculations become tedious where large terms are involved. Keeping in view the noble features of quaternion, we expect some development to get a better understanding of N = 8 supergravity, maximal supergravity (D = 11 − n), and maximal supersymmetry theories (N = 10) in terms of quaternion operators.

This is an introductory level review of recent applications of resurgent trans-series and PicardLefschetz theory to quantum mechanics and quantum field theory. Resurgence connects local perturbative data with global topological structure. In quantum mechanical systems, this program provides a constructive relation between different saddles. For example, in certain cases it has been shown that all information around the instanton saddle is encoded in perturbation theory around the perturbative saddle. In quantum field theory, such as sigma models compactified on a circle, neutral bions provide a semi-classical interpretation of the elusive IR-renormalon, and fractional kink instantons lead to the non-perturbatively induced gap, of order of the strong scale. In the path integral formulation of quantum mechanics, saddles must be found by solving the holomorphic Newton’s equation in the inverted (holomorphized) potential. Some saddles are complex, multi-valued, and even singular, but of finite action, and their inclusion is strictly necessary to prevent inconsistencies. The multi-valued saddles enter either via resurgent cancellations, or their phase is tied with a hidden topological angle. We emphasize the importance of the destructive/constructive interference effects between equally dominant saddles in the Lefschetz thimble decomposition. This is especially important in the context of the sign problem.

It is demonstrated how quantum mechanics is generated by stochastic momentum kicks from the force carriers, transmitting the fundamental interactions between the point particles. The picture is consistent with quantum field theory and points out that the force carriers are the only quantum particles. Since the latter are waves in the coordinate space, they are responsible for the wavy character of quantum mechanics.

This two-tier conference signals a new phase of philosophers' interest in quantum field theory, which has been growing noticeably in the last few years. However, some prominent physicists have shown their deep suspicions against ignorant philosophers' intrusion into their profession, and have expressed their hostility quite openly. In the philosophy community, some prominent philosophers of physics have also expressed their suspicions against the rationale of moving away from the profound foundational problems raised by Einstein and Bohr, Bohm and Bell, such as those concerning the nature of space-time and measurement, possibility and implications of hidden variables and nonlocality, and stepping into the technical complexity of quantum field theory, which is only an application of quantum mechanics in general without intrinsically distinct philosophical questions to be explored. In order to dispel these suspicions, it is desirable to highlight certain aspects of quantum field theory which require philosophical reflections and deserve further investigations. This discussion intends to suggest that philosophers can learn many important lessons from quantum field theory, and may be of some help in clarifying its conceptual foundations. At this stage of crisis that quantum field theory is experiencing now, the clarification may contribute to the radical transformation of our basic concepts in theoretical physics, which is necessary for a happy resolution of the crisis and the emergence of a new promising fundamental physical theory.