I articulate and discuss a geometrical interpretation of Yang–Mills theory. Analogies and disanalogies between Yang–Mills theory and general relativity are also considered.

In this paper, we present a non-geometrodynamic quantum Yang–Mills theory of gravity based on the homogeneous Lorentz group within the general framework of the Poincare gauge theories. The obstacles of this treatment are that first, on the one hand, the gauge group that is available for this purpose is non-compact. On the other hand, Yang–Mills theories with non-compact groups are rarely healthy, and only a few instances exist in the literature. Second, it is not clear (...) how the direct observations of space–time waves can be explained when space–time has no dynamics. We show that the theory is unitary and is renormalizable to the one-loop perturbation. Although in our proposal, gravity is not associated with any elementary particle analogous to the graviton, classical helicity-two space–time waves are explained. Five essential exact solutions to the field equations of our proposal are presented as well. We also discuss a few experimental tests that can falsify the presented Yang–Mills theory. (shrink)

The elucidation of the gauge principle ``is the most pressing problem in current philosophy of physics" Redhead. This paper argues two points that contribute to this elucidation in the context of Yang-Mills theories. 1) Yang-Mills theories, including quantum electrodynamics, form a class. They should be interpreted together. To focus on electrodynamics is a mistake. 2) The essential role of gauge and BRST surplus is to provide a local theory that can be quantized and would be (...) equivalent to the quantization of the non-local reduced theory. (shrink)

We analyze the geometric foundations of classical Yang-Mills theory by studying the relationships between internal relativity, locality, global/local invariance, and background independence. We argue that internal relativity and background independence are the two independent defining principles of Yang-Mills theory. We show that local gauge invariance -heuristically implemented by means of the gauge argument- is a direct consequence of internal relativity. Finally, we analyze the conceptual meaning of BRST symmetry in terms of the invariance of (...) the gauge fixed theory under general local gauge transformations. (shrink)

Essay review of Gauging What’s Real: The Conceptual Foundations of Contemporary Gauge Theories R. Healey. Oxford University Press (2007). To be published in the Studies in History and Philosophy of Modern Physics, 39(3):687-693, 2008.

The discrete light-cone quantization (DLCQ) of a supersymmetric gauge theory in 1+1 dimensions is discussed, with particular attention given to the inclusion of the gauge zero mode. Interestingly, the notorious “zero-mode” problem is now tractable because of special supersymmetric cancellations. In particular, we show that anomalous zero-mode contributions to the currents are absent, in contrast to what is observed in the nonsupersymmetric case. An analysis of the vacuum structure is provided by deriving the effective quantum mechanical Hamiltonian of the (...) gauge zero mode. It is shown that the inclusion of the zero modes of the adjoint scalars and fermions is crucial for probing the phase properties of the vacua. We find that the ground-state energy is zero and thus consistent with unbroken supersymmetry and conclude that the light-cone Fock vacuum is unchanged with or without the presence of matter fields. (shrink)

In this paper I detail three major mathematical developments that led to the emergence of Yang–Mills theories as the foundation for the standard model of particle physics. In less than 10 years, work on renormalizability, the renormalization group, and lattice quantum field theory highlighted the utility of Yang–Mills type models of quantum field theory by connecting poorly understood candidate dynamical models to emerging experimental results. I use this historical case study to provide lessons for (...) theory construction in physics, and touch on issues related to renormalization group realism from a more historical perspective. In particular, I highlight the fact that much of the hard work in theory construction comes when trying to understand the consequences and representational capacities of a theoretical framework. (shrink)

Classical and quantum field theory provide not only realistic examples of extant notions of empirical equivalence, but also new notions of empirical equivalence, both modal and occurrent. A simple but modern gravitational case goes back to the 1890s, but there has been apparently total neglect of the simplest relativistic analog, with the result that an erroneous claim has taken root that Special Relativity could not have accommodated gravity even if there were no bending of light. The fairly recent acceptance (...) of nonzero neutrino masses shows that widely neglected possibilities for nonzero particle masses have sometimes been vindicated. In the electromagnetic case, there is permanent underdetermination at the classical and quantum levels between Maxwell's theory and the one-parameter family of Proca's electromagnetisms with massive photons, which approximate Maxwell's theory in the limit of zero photon mass. While Yang–Mills theories display similar approximate equivalence classically, quantization typically breaks this equivalence. A possible exception, including unified electroweak theory, might permit a mass term for the photons but not the Yang–Mills vector bosons. Underdetermination between massive and massless (Einstein) gravity even at the classical level is subject to contemporary controversy. (shrink)

The Yang-Mills mass gap problem has stood as one of the most profound unresolved questions in theoretical physics for nearly half a century. Despite extensive efforts, a rigorous mathematical proof of the mass gap's existence in Yang-Mills theory has remained elusive—until now. -/- The proof presented in this work represents not merely a technical achievement, but a fundamental paradigm shift in our understanding of the mathematical structures underlying physical reality. By rigorously demonstrating that the mass (...) gap arises as a direct consequence of the gauge-invariant constraint structure of Yang-Mills theory, we overturn decades of conventional assumptions and reveal a startling truth: what we perceive as physical laws are in fact mathematically unavoidable necessities emerging from the very consistency conditions required for a well-defined quantum field theory. (shrink)

We study the magnetic monopoles in non-Abelian gauge theories. The exact static, spherically symmetric solutions of the magnetic monopoles in both Yang-Mills and unified gauge theories are obtained. The energyE of the static system is calculable and it is either zero or infinite. The existence of the magnetic monopole solution is a consequence of symmetry rather than dynamics. We propose a new definition of the electromagnetic field tensor, which relates the static solution of gauge fields and the magnetic (...) monopole solution. Experimental implications are discussed. (shrink)

We relate personal encounters of three kinds with geometrical approaches in the development of a relativistic quantum field theory of the fundamental interactions—including interactions with Nathan Rosen. We characterize the geometrical structures involved and discuss the more recent attempts to develop a unified theory based on a Klein-Kaluza contraction of the eightfold extended supergravity.

The elucidation of the gauge principle ‘‘is the most pressing problem in current philosophy of physics’’ said Michael Redhead in 2003. This paper argues for two points that contribute to this elucidation in the context of Yang–Mills theories. (1) Yang–Mills theories, including quantum electrodynamics, form a class. They should be interpreted together. To focus on electrodynamics is potentially misleading. (2) The essential role of gauge and BRST symmetries is to provide a local field theory that (...) can be quantized and would be equivalent to the quantization of the non-local reduced theory. If this is correct, the gauge symmetry is significant, not so much because it implies ontological consequences, but because it allows us to quantize theories that we would not be able to quantize otherwise. Thus, in the context of Yang–Mills theories, it is essentially a pragmatic principle. This does not seem to be the case for the gauge symmetry in general relativity. (shrink)

Since it was first published, Quantum Field Theory in a Nutshell has quickly established itself as the most accessible and comprehensive introduction to this profound and deeply fascinating area of theoretical physics. Now in this fully revised and expanded edition, A. Zee covers the latest advances while providing a solid conceptual foundation for students to build on, making this the most up-to-date and modern textbook on quantum field theory available. -/- This expanded edition features several additional chapters, as (...) well as an entirely new section describing recent developments in quantum field theory such as gravitational waves, the helicity spinor formalism, on-shell gluon scattering, recursion relations for amplitudes with complex momenta, and the hidden connection between Yang-Mills theory and Einstein gravity. Zee also provides added exercises, explanations, and examples, as well as detailed appendices, solutions to selected exercises, and suggestions for further reading. (shrink)

In the philosophical literature on Yang-Mills theories, field formulations are taken to have more structure and to be local, while curve-based formulations are taken to have less structure and to be nonlocal. I formalize the notion of locality at issue and show that theories with less structure are nonlocal. However, the amount of structure had by some formulation is independent of whether it uses fields or curves. The relevant difference in structure is not a difference in set-theoretic structure. (...) Rather, it is a difference in the structure of the category of models of the theory. (shrink)

It is shown that, in some non-QCD theories, there are effects shared by QCD: (i) in SU(2) Yang–Mills theory containing a nonlinear spinor field, there is a mass gap; (ii) in SU(3) Proca–Higgs theory, there are flux tube solutions with a longitudinal electric field required for producing a force binding quarks; (iii) in non-Abelian Proca–Higgs theories, there exist flux tube solutions with a momentum directed along the tube axis and particlelike solutions with a nonvanishing total angular (...) momentum created by crossed color electric and magnetic fields; in QCD, such configurations may contribute to the proton spin. We discuss the conjecture that such non-QCD theories might be a consequence of approximate solution of an infinite set of Dyson–Schwinger equations describing the procedure of nonperturbative quantization. The phenomenon of dimensional transmutation for nonperturbative quantization and the analogy between nonperturbative quantization and turbulence modeling are also discussed. (shrink)

Science is a process, through which theoretical frameworks are developed, new phenomena defined and discovered, and properties of entities tested. The goal of this dissertation is to illustrate how high-energy physics exemplified the process of theory construction from the 1950s to 1970s, and the promising ways in which it can continue to do so today. The lessons learned from the case studies examined here can inform future physics, and may provide methodological clues as to the best way forward today. (...) I examine the discovery of parity nonconservation in weak interactions, the emergence of Yang-Mills theories as the foundation of the standard model, and contemporary precision testing of quantum electrodynamics. In each of these cases, I examine the details of the physicists’ practice to draw conclusions regarding the epistemology behind successful episodes of theory construction. I reconstruct the methodology of each episode in order to find generalizable lessons to apply to contemporary issues at the frontiers of the search for a theory of quantum gravity. In order to understand the many moving parts in each case study, I introduce a new terminology to distinguish the “parts” of a scientific discipline, inspired by the literature on scientific modelling. These terms—theoretical framework, dynamical model, phenomenological model, experiment, and mathematical tools—are meant to aid in investigating other quantitative scientific disciplines beyond high-energy physics. Ultimately, high-energy physics is at its best when various avenues of theoretical ideas are being pursued, spurring the development of new mathematical techniques to use as tools, and new ideas are quickly and vigorously tested experimentally. Proliferation of new ideas in response to theoretical developments is characteristic of the era of construction of the standard model, and is still ongoing in precision testing of quantum electrodynamics today. (shrink)

I review some recent work on applications of category theory to questions concerning theoretical structure and theoretical equivalence of classical field theories, including Newtonian gravitation, general relativity, and Yang-Mills theories.

Not all symmetries are on a par. For instance, within Newtonian mechanics, we seem to have a good grasp on the empirical significance of boosts, by applying it to subsystems. This is exemplified by the thought experiment known as Galileo’s ship: the inertial state of motion of a ship is immaterial to how events unfold in the cabin, but is registered in the values of relational quantities such as the distance and velocity of the ship relative to the shore. But (...) the significance of gauge symmetries seems less clear. For example, can gauge transformations in Yang-Mills theory—taken as mere descriptive redundancy—exhibit a similar relational empirical significance as the boosts of Galileo’s ship? This question has been debated in the last fifteen years in philosophy of physics. I will argue that the answer is ‘yes’, but only for a finite subset of gauge transformations, and under special conditions. Under those conditions, we can mathematically identify empirical significance with a failure of supervenience: the state of the Universe is not uniquely determined by the intrinsic state of its isolated subsystems. Empirical significance is therefore encoded in those relations between subsystems that stand apart from their intrinsic states. (shrink)

Rovelli’s “Why Gauge?” offers a parable to show that gauge-dependent quantities have a modal and relational physical significance. We subject the morals of this parable to philosophical scrutiny and argue that, while Rovelli’s main point stands, there are important disanalogies between his parable and Yang-Mills type gauge theory.

Some aspects of the Schrödinger equation in quantum field theory are considered in this article. The emphasis is on the Schrödinger functional equation for Yang-Mills theory, arising mainly out of Feynman's work on (2+1)-dimensional Yang-Mills theory, which he studied with a view to explaining the confinement of gluons. The author extended Feynman's work in two earlier papers, and the present article is partly a review of Feynman's and the author's work and some further (...) extension of the latter. The primary motivation of this article is to suggest that considering the Schrödinger functional equation in the context of Yang-Mills theory may contribute significantly to the solution of the confinement and related problems, an aspect which, in the author's opinion, has not received the attention it deserves. The relation of this problem with certain others such as those of quarks, superconductivity, and quantum gravity is considered briefly, together with certain basic aspects of the formalism that may be of interest in their own right, especially for the beginner. (shrink)

The equations of motion for a particle in a classical gauge field are derived from the invariance identities 2 and basic assumptions about the Lagrangian. They are found to be consistent with the equations of some other approaches to classical gauge-field theory, and are expressed in terms of a set of undetermined functions Eα. The functions Eα are found to satisfy a system of differential equations which has the same formal structure as a system of equations from Yang- (...) class='Hi'>Mills theory. 3 These results are obtained by a new method which applies techniques from the theory of functional equations to deduce the way in which the arguments of the Lagrangian must combine. The method constitutes an aid for obtaining the equations of motion when a non-gauge-invariant Lagrangian is chosen, and it is assumed that the equations of motion can be written in a gauge-invariant manner. (shrink)

We address a recent proposal concerning ‘surplus structure’ due to Nguyen et al.. We argue that the sense of ‘surplus structure’ captured by their formal criterion is importantly different from—and in a sense, opposite to—another sense of ‘surplus structure’ used by philosophers. We argue that minimizing structure in one sense is generally incompatible with minimizing structure in the other sense. We then show how these distinctions bear on Nguyen et al.’s arguments about Yang-Mills theory and on the (...) hole argument. (shrink)

A Lorentz-invariant gauge theory for massive fermions on R × S 3 spacetime is built up. Using the symmetry of S 3,we obtain Dirac-type equation and derive the expression of the fermionic propagator. Finally, starting from the SU(N) gauge-invariant Lagrangian, we obtain the set of Dirac-Yang-Mills equations on R × S 3 spacetime, pointing out major differences from the Minkowskian case.

The tetradic Lorentz-gauge invariant formulation of the SU(2) × U(1) theory in S3 × R space-time is presented and the general gauge covariant Dirac-Klein-Gordon-Maxwell-Yang-Mills equations are derived. A direct comparison of these equations to those of the SU(2) × U(1) gauge theory on Minkowskian background points out major differences effectively induced by the minimally coupling to S3 × R gravity.

Enormous and significant progress has been made in the important areas of entanglement, quantum computing and harnessing energy from the vacuum, which includes a sound theoretical basis, using the Einstein-Sachs theories to develop an anti-symmetric general relativity (AGR) approach to a higher topology O(3) electrodynamics. These developments also lead to the application of the Aharonov-Bohm effect and the Yang-Mills theory to the higher topology O(3) electrodynamics, as well as a deeper understanding and appreciation of these effects and (...) their impact on modern physics. The door is now open for the further development and unification of physics, including the gravitational, electromagnetic, weak, and strong forces, using the AGR approach. (shrink)

It has been shown that the orbits of motion for a wide class of non-relativistic Hamiltonian systems can be described as geodesic flows on a manifold and an associated dual by means of a conformal map. This method can be applied to a four dimensional manifold of orbits in spacetime associated with a relativistic system. We show that a relativistic Hamiltonian which generates Einstein geodesics, with the addition of a world scalar field, can be put into correspondence in this way (...) with another Hamiltonian with conformally modified metric. Such a construction could account for part of the requirements of Bekenstein for achieving the MOND theory of Milgrom in the post-Newtonian limit. The constraints on the MOND theory imposed by the galactic rotation curves, through this correspondence, would then imply constraints on the structure of the world scalar field. We then use the fact that a Hamiltonian with vector gauge fields results, through such a conformal map, in a Kaluza-Klein type theory, and indicate how the TeVeS structure of Bekenstein and Saunders can be put into this framework. We exhibit a class of infinitesimal gauge transformations on the gauge fields ${\mathcal{U}}_{\mu}(x)$ which preserve the Bekenstein-Sanders condition ${\mathcal{U}}_{\mu}{\mathcal{U}}^{\mu}=-1$ . The underlying quantum structure giving rise to these gauge fields is a Hilbert bundle, and the gauge transformations induce a non-commutative behavior to the fields, i.e. they become of Yang-Mills type. Working in the infinitesimal gauge neighborhood of the initial Abelian theory we show that in the Abelian limit the Yang-Mills field equations provide residual nonlinear terms which may avoid the caustic singularity found by Contaldi et al. (shrink)

Rovelli's "Why Gauge?" offers a parable to show that gauge-dependent quantities have a modal and relational physical significance. We subject the morals of this parable to philosophical scrutiny and argue that, while his overarching point stands, there are subtle disanalogies between his parable and our best gauge theories, e.g. Yang-Mills theory and General Relativity.

I consider two usages of the expression "gauge theory". On one, a gauge theory is a theory with excess structure; on the other, a gauge theory is any theory appropriately related to classical electromagnetism. I make precise one sense in which one formulation of electromagnetism, the paradigmatic gauge theory on both usages, may be understood to have excess structure, and then argue that gauge theories on the second usage, including Yang-Mills theory (...) and general relativity, do not generally have excess structure in this sense. (shrink)

Is change missing in Hamiltonian Einstein–Maxwell theory? Given the most common definition of observables, observables are constants of the motion and nonlocal. Unfortunately this definition also implies that the observables for massive electromagnetism with gauge freedom are inequivalent to those of massive electromagnetism without gauge freedom. The alternative Pons–Salisbury–Sundermeyer definition of observables, aiming for Hamiltonian–Lagrangian equivalence, uses the gauge generator G, a tuned sum of first-class constraints, rather than each first-class constraint separately, and implies equivalent observables for equivalent massive (...) electromagnetisms. For General Relativity, G generates 4-dimensional Lie derivatives for solutions. The Lie derivative compares different space-time points with the same coordinate value in different coordinate systems, like 1 a.m. summer time versus 1 a.m. standard time, so a vanishing Lie derivative implies constancy rather than covariance. Requiring equivalent observables for equivalent formulations of massive gravity confirms that G must generate the 4-dimensional Lie derivative for observables. These separate results indicate that observables are invariant under internal gauge symmetries but covariant under external gauge symmetries, but can this bifurcated definition work for mixed theories such as Einstein–Maxwell theory? Pons, Salisbury and Shepley have studied G for Einstein–Yang–Mills. For Einstein–Maxwell, both \ and \ are invariant under electromagnetic gauge transformations and covariant under 4-dimensional coordinate transformations. Using the bifurcated definition, these quantities count as observables, as one would expect on non-Hamiltonian grounds. (shrink)

The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of (...) Hilbert mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions. (shrink)

A classic result in the foundations of Yang-Mills theory, due to J. W. Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills Theory." Int. J. Th. Phys. 30, ], establishes that given a "generalized" holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such (...) that the holonomy map corresponds to the holonomies of that connection. Barrett also provided one sense in which this "recovery theorem" yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of "unique recovery" in Barrett's theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or "loop", formulation. (shrink)

Its is pointed out that the low-energy effective theory that describes the low-lying glueballs of the pure Yang-Mills theory sustains static classical stringlike solutions. We suggest that these objects can be identified with the QCD flux tubes and their energy per unit length with the string tension.

Like moths attracted to a bright light, philosophers are drawn to glitz. So in discussing the notions of ‘gauge’, ‘gauge freedom’, and ‘gauge theories’, they have tended to focus on examples such as Yang–Mills theories and on the mathematical apparatus of fibre bundles. But while Yang–Mills theories are crucial to modern elementary particle physics, they are only a special case of a much broader class of gauge theories. And while the fibre bundle apparatus turned out, in (...) retrospect, to be the right formalism to illuminate the structure of Yang–Mills theories, the strength of this apparatus is also its weakness: the fibre bundle formalism is very flexible and general, and, as such, fibre bundles can be seen lurking under, over, and around every bush. What is needed is an explanation of what the relevant bundle structure is and how it arises, especially for theories that are not initially formulated in fibre bundle language. Here I will describe an approach that grows out of the conviction that, at least for theories that can be written in Lagrangian/Hamiltonian form, gauge freedom arises precisely when there are Lagrangian/Hamiltonian constraints of an appropriate character. This conviction is shared, if only tacitly, by that segment of the physics community that works on constrained Hamiltonian systems. (shrink)

A gauge theory on R×S 3 topology is developed. It is a generalization to the previously obtained field theory on R×S 3 topology and in which equations of motion were obtained for a scalar particle, a spin one-half particle, the electromagnetic field of magnetic moments, and a Shrödinger-type equation, as compared to ordinary field equations defined on a Minkowskian manifold. The new gauge field equations are presented and compared to the ordinary Yang-Mills field equations, and the (...) mathematical and physical differences between them are discussed. (shrink)

Einstein's general relativity theory describes very well the gravitational phenomena in themacroscopic world. In themicroscopic domain of elementary particles, however, it does not exhibit gauge invariance or approximate Bjorken type scaling, properties which are believed to be indispensible for arenormalizable field theory. We argue that thelocal extension of space-time symmetries, such as of Lorentz and scale invariance, provides the clue for improvement. Eventually, this leads to aGL(4, R)-gauge approach to gravity in which the metric and the affine connection (...) acquire the status ofindependent fields. The Yang-Mills type field equations, the Noether identities, and conformal models of gravity are discussed within this framework. After symmetry breaking, Einstein's GR surfaces as an effective “low-energy” theory. (shrink)

We analyze the problem of constructing supersymmetric versions of gauge theories of particles and of gravity which have a closed supersymmetric algebra. Inparticular we present the basic no-go theorems that indicate that in four dimensions it is not possible to construct suitably extended supersymmetric versions of the above theories without drastic modification of the supersymmetric algebra. Two ways past the“N=3” barrier are discussed; that of central charges involved highly constrained versions which appearn difficult to quantize effectively, while the use of (...) light-cone variables seems to be the most promising. We give light-cone gauge versions of supersymmetric Yang-Mills theories for all extended cases of interest and briefly consider their ultraviolet divergence properties. (shrink)

This paper criticizes the traditional philosophical account of the quantization of gauge theories and offers an alternative. On the received view, gauge theories resist quantization because they feature distinct mathematical representatives of the same physical state of affairs. This resistance is overcome by a sequence of ad hoc modifications, justified in part by reference to semiclassical electrodynamics. Among other things, these modifications introduce "ghosts": particles with unphysical properties which do not appear in asymptotic states and which are said to be (...) purely a notational convenience. I argue that this sequence of modifications is unjustified and inadequate, making it a poor basis for the interpretation of ghosts. I then argue that gauge theories can be quantized by the same method as any other theory. On this account, ghosts are not purely notation: they are coordinates on the classical configuration space of the theory—specifically, on its gauge structure. This interpretation does not fall prey to the standard philosophical arguments against the significance of ghosts, due to Weingard. Weingard’s argumentative strategy, properly applied, in fact tells in favor of ghosts’ physical significance. (shrink)

This paper shows how the study of surpluses of structure is an interesting philosophical task. In particular I explore how local gauge symmetry in quantized Yang-Mills theories is the by-product of the specific dynamical structure of interaction. It is shown how in non relativistic quantum mechanics gauge symmetry corresponds to the freedom to locally define global features of gauge potentials. Also discussed is how in quantum field theory local gauge symmetry is replaced by BRST symmetry. This last (...) symmetry is apparently the result of the fact that we do not know how to define quantum Yang-Mills theories without unphysical gauge boson states. Since Yang-Mills theories describe successfully three of the four fundamental interactions the elucidation of this symmetry is a pressing philosophical question. (shrink)

In this paper, we will analyse the superloop space formalism for a four dimensional supersymmetric Yang–Mills theory in deformed superspace. We will deform the \ superspace by imposing imposing non-anticommutativity. This non-anticommutative deformation of the superspace will break half the supersymmetry of the original theory. So, this theory will have \ supersymmetry. We will analyse the superloop space duality for this deformed supersymmetric Yang–Mills theory using the \ superspace formalism. We will demonstrate (...) that the sources in the original theory will become monopoles in the dual theory, and the monopoles in the original theory will become sources in the dual theory. (shrink)

In the range of his intellectual interests and the profundity of his mathematical thought Hermann Weyl towered above his contemporaries, many of whom viewed him with awe. This volume, the most ambitious study to date of Weyl's singular contributions to mathematics, physics, and philosophy, looks at the man and his work from a variety of perspectives, though its gaze remains fairly steadily fixed on Weyl the geometer and space‐time theorist. Structurally, the book falls into two parts, described in the general (...) introduction by the editor: Part 1 contains four essays on particular aspects of Weyl's work, highlighting ideas he developed in various editions of his classic Raum‐Zeit‐Materie. Part 2 presents a lengthy study by Robert Coleman and Herbert Korté covering nearly the whole gamut of Weyl's mathematical research, an impressive feat. Both in the introduction as well as in footnotes to the articles Erhard Scholz's editorial voice chimes in discreetly, helping tie all five studies together.Coleman and Korté begin chronologically with Weyl's early work in analysis and the modern theory of Riemann surfaces before turning to differential geometry, unified field theory, and the space problem, a topic they use as a springboard for a discussion of their own recent work on the foundations of space‐time. They then take up Weyl's shift to group representation theory and its applications to quantum mechanics, ending with his much earlier research on the structure of the continuum. All of these topics are well handled, but the authors' own agendas coupled with their penchant for overlooking chronology in order to package Weyl's work into neat little bundles leave one feeling rather stranded and far removed from the sources of Weyl's inspiration. Moreover, the narrative style makes this part of the volume read like a technical appendix, albeit a most informative one. Readers who tackle Scholz's far more contextualized essay will be amply rewarded by comparing his views with the opinions set forth by Coleman and Korté in Part 2.Scholz gives a masterful account of Weyl's intellectual journeys from 1917 to 1925 in a study that serves as a fulcrum for the entire volume. Drawing on a number of recently published studies, including his own, on the interplay between mathematics and physics inspired by Einstein's theory of general relativity, Scholz describes how Weyl responded to this challenge by developing a truly infinitesimal space‐time geometry that generalized classical Riemannian geometry. Although unconvinced by Einstein's critique of his unified field theory, Weyl shifted his focus from this realm to the classical space problem, analyzed earlier with more primitive techniques by Hermann Helmholtz and Sophus Lie. In this connection, it should be mentioned that Thomas Hawkins has given a probing analysis of Weyl's related work on the representation of Lie groups in his tour‐de‐force work, Emergence of the Theory of Lie Groups . Scholz argues that Weyl's struggle to tame his modernized version of the space problem stemmed from a deep‐seated belief in his geometrical ideas, which in turn were nourished by philosophical musings. By demonstrating the closely related conceptual links that motivated Weyl's research in infinitesimal geometry, space‐time physics, and the foundations of mathematics, Scholz nicely illuminates the underlying fabric of epistemological concerns that occupied Weyl's attention during this fertile period.The three remaining essays in Part 1 focus on other aspects of Weyl's work in mathematical physics and cosmology. Skuli Sigurdsson's “Journeys in Spacetime” offers a broad interpretation of Weyl's career, one that emphasizes Weyl's sensitivity to cultural tensions as reflected in his philosophical roots, which combined phenomenology with facets of German idealism. Shaken by the annihilation of cultural values in Nazi Germany, Weyl became deeply aware of the gulf that separated his earlier life in Göttingen and Zurich from the one he took up at Princeton's Institute for Advanced Study in 1933. He tried to adapt, but felt out of place in an Anglo‐American scientific culture openly hostile toward metaphysics and speculative philosophy. Sigurdsson stresses these tensions, contrasting the introspective, creative individual against the backdrop of the collective in the age of the machine, but without spelling out which collective were most important for him. Wolfgang Pauli thought he knew and, like Einstein before him, he had no compunction about bluntly telling Weyl he was a mathematician, not a physicist.Pauli's opinions notwithstanding, Weyl did far more than just dabble around the mathematical edges of the new physics. If Coleman and Korté perhaps press their case for his visionary accomplishments too far, Norbert Straumann's essay “Ursprünge der Eichtheorien” suggests why Weyl's reputation among physicists has risen steadily ever since the advent of Yang‐Mills theory in the 1950s. In the course of describing Weyl's adaptation of his gauge transformation formalism to Dirac's electron theory, Straumann sheds considerable light on Pauli's role as self‐appointed watchman guarding the disciplinary boundary that separated theoretical physics from physical mathematics . He further suggests that disciplinary jealousy was a major reason why Pauli dismissed Weyl's two‐component formalism for spinors out of hand.In the realm of cosmology, on the other hand, Weyl's work has long since passed into the dustbins of history, as Hubert Goenner remarks in recounting a fascinating chapter in the infancy of space‐time physics. While doing so, Goenner shows how initially Weyl almost slavishly adopted what Einstein called Mach's principle, which asserts that the metric structure of space‐time is solely determined by the distribution of matter in the universe. This notion was quickly challenged by Willem De Sitter, who showed that Einstein's matter‐free field equations admitted a global solution with non‐zero constant curvature. Both Einstein and Weyl tried to argue that invisible masses must be present just over the “spatial horizon” of De Sitter's world in order to account for its curvature. Goenner meticulously analyzes the physical and mathematical issues at stake in this debate, stressing how Weyl gradually moved away from a strong physical interpretation to one in which mathematics models rather than physics models simply reveal natural phenomena. He argues further that Weyl's cosmological principle arose as the final expression of his search for a deeper physical meaning.Given the quality of these essays, it is regrettable that this book contains so little about Weyl's professional career, a weakness the editor could have redressed at least partially in his general introduction. This omission is all the more unfortunate given the dearth of readily accessible information about Weyl's life available elsewhere. For however mundane his outward existence may have been, the reader cannot be expected to appreciate the interplay between the world Weyl knew and his creative responses to it without fairly detailed knowledge of his biography. Shorn from these contexts, it becomes difficult to form a flesh‐and‐blood image of Weyl beyond the cliché‐ridden stereotype that sees him as a “heroic thinker in the grand German tradition.” While none of the authors falls into this trap, the collective impression they leave suggests a most enigmatic figure. Either Weyl the man tends to get lost in the shadows of his collected scientific output or he appears as a mystic loner, an outcast who abhorred the machine age in which he lived. Closer attention to the people in his life would no doubt produce a very different picture of the man and his interests. This major lacuna notwithstanding, the present volume will surely remain an indispensable resource for any future investigations of Weyl's staggering intellectual achievements. (shrink)

Generalizing the work of Einstein and Mayer, it is assumed that at each point of space-time there exists an N-dimensional linear vector space with N≥5. This space is decomposed into a four-dimensional tangent space and an (N - 4)-dimensional internal space. On the basis of geometric considerations, one arrives at a number of fields, the field equations being derived from a variational principle. Among the fields obtained there are the electromagnetic field, Yang-Mills gauge fields, and fields that can (...) be interpreted as describing matter. As a simple example, the case N=6 is considered. (shrink)

Successful applications of a conceptually novel setup of Quantum Field Theory, that accounts for all subtheories of the Standard Model (QED, Electroweak Interaction and Higgs, Yang–Mills and QCD) and beyond (Helicity 2), call for a perspective view in a broader conceptual context. The setting is “autonomous” in the sense of being intrinsically quantum. Its principles are: Hilbert space, Poincaré symmetry and causality. Its free quantum fields are obtained from Wigner’s unitary representations of the Poincaré group, with only (...) physical and observable degrees of freedom. A “quantization” of an “underlying” classical theory is not needed. It allows renormalizable perturbation theory with interactions whose detailed structure, and in some cases even the particle content, is predicted by internal consistency. The results confirm and extend observable predictions for the interactions of the Standard Model without assuming a “principle” of gauge invariance. (shrink)

Recently two pairs of authors have aimed to vindicate the longstanding "orthodox" or conventional claim that a first-class constraint generates a gauge transformation in typical gauge theories such as electromagnetism, Yang-Mills and General Relativity, in response to the Lagrangian-equivalent reforming tradition, in particular Pitts, _Annals of Physics_ 2014. Both pairs emphasize the coherence of the extended Hamiltonian formalism against what they take to be core ideas in Pitts 2014, but both overlook Pitts 2014's sensitivity to ways that one (...) might rescue the claim in question, including an additive redefinition of the electrostatic potential. Hence the bulk of the paper is best interpreted as arguing that the longstanding claim about separate first-class constraints is _either false or trivial_---de-Ockhamization (using more when less suffices by splitting one quantity into the sum of two) being trivial. Unfortunately section 9 of Pitts 2014, a primarily verbal argument that plays no role in other works, is refuted. Pooley and Wallace's inverse Legendre transformation to de-Ockhamized electromagnetism with an additively redefined electrostatic potential, however, opens the door to a precisely analogous calculation introducing a photon mass, which shows that a _second-class primary_ constraint generates a 'gauge transformation' in the exactly same sense---a _reductio ad absurdum_ of the orthodox claim that a first-class constraint generates a gauge transformation _and a second-class constraint does not_. A _reductio_ using massive electromagnetism was presaged in Pitts 2014. Gauge freedom by de-Ockhamization does not require any constraints at all, first-class or second-class, because any dynamical variable in any Lagrangian can be de-Ockhamized into exhibiting trivial additive artificial gauge freedom by splitting one quantity into the sum of two. Physically interesting gauge freedom, however, is typically generated by a tuned sum of first-class constraints, not each first-class constraint alone. (shrink)

The Higgs mechanism is an essential but elusive component of the Standard Model of particle physics. Without it Yang‐Mills gauge theories would have been little more than a warm‐up exercise in the attempt to quantize gravity rather than serving as the basis for the Standard Model. This article focuses on two problems related to the Higgs mechanism clearly posed in Earman’s recent papers (Earman 2003, 2004a, 2004b): what is the gauge‐invariant content of the Higgs mechanism, and what does (...) it mean to break a local gauge symmetry? (shrink)

The Higgs mechanism gives mass to Yang-Mills gauge bosons. According to the conventional wisdom, this happens through the spontaneous breaking of gauge symmetry. Yet, gauge symmetries merely reflect a redundancy in the state description and therefore the spontaneous breaking can not be an essential ingredient. Indeed, as already shown by Higgs and Kibble, the mechanism can be explained in terms of gauge invariant variables, without invoking spontaneous symmetry breaking. In this paper, we present a general discussion of such (...) gauge invariant treatments for the case of the Abelian Higgs model, in the context of classical field theory. We thereby distinguish between two different notions of gauge: one that takes all local transformations to be gauge and one that relates gauge to a failure of determinism. (shrink)

Professor Sir Roger Penrose's work, spanning fifty years of science, with over five thousand pages and more than three hundred papers, has been collected together for the first time and arranged chronologically over six volumes, each with an introduction from the author. Where relevant, individual papers also come with specific introductions or notes. Many important realizations concerning twistor theory occurred during the short period of this third volume, providing a new perspective on the way that mathematical features of the (...) complex geometry of twistor theory relate to actual physical fields. Following on from the nonlinear graviton construction, a twistor construction was found for (anti-)self-dual electromagnetism allowing the general (anti-)self-dual Yang-Mills field to be obtained. It became clear that some features of twistor contour integrals could be understood in terms of holomorphic sheaf cohomology. During this period, the Oxford research group founded the informal publication, Twistor Newsletter. This volume also contains the influential Weyl curvature hypothesis and new forms of Penrose tiles. (shrink)

The present special section brings together three articles that seek to elucidate the epistemological and ontological foundations of the Higgs mechanism, as well as the epistemic dynamics of the development of the models beyond the “standard model” of elementary particle physics.The standard model of elementary particle physics consists of two gauge theories; namely, the electroweak theory of the weak and electromagnetic interactions, and the theory of quantum chromo-dynamics which describes the strong interaction. This introduction will provide a brief (...) overview of the key issues concerning the foundations of the Higgs mechanism.Historically, the main motivation for the Higgs mechanism stemmed from the need to account for the non-zero masses of the intermediate vector bosons of the weak interaction within a gauge theory of the Yang-Mills type. One way to give mass to the IVBs, called “explicit symmetry breaking,” is to add mass terms directly to the Lagrangian of the gauge theor .. (shrink)

After a brief review of the field formulations and the relativistic non-instantaneous action-at-a-distance formulations of some well known classical theories, we study Rivacoba’s generalization of a theory with a linearly rising potential as a relativistic non-instantaneous action-at-a-distance theory. For this case we construct the corresponding field theory, which turns out to coincide with a model proposed by Kiskis to describe strong interactions. We construct the action functional for this field theory. Although this model belongs to the (...) class of Lagrangian theories with higher derivatives, it is shown that its action-at-a-distance counterpart has a much simpler form. A proposal to construct an action-at-a-distance description of Yang-Mills interactions is also presented. (shrink)

A critical review of gravitational wave theory is made. It is pointed out that the usual linear approach to the gravitational wave theory is neither conceptually consistent nor mathematically justified. Relying upon that analysis it is argued that—analogously to a Yang-Mills propagating field, which must be nonlinear to carry its gauge charge—a gravitational wave must necessarily be nonlinear to transport its own charge—that is, energy-momentum.