Group-theoretical foundations of classical mechanics: The Lagrangian gauge … (original) (raw)

Classical and Quantum Theory with a New Symmetry

arXiv (Cornell University), 2009

A formal symmetry between generalized coordinates and momenta is postulated to formulate classical and quantum theories of a particle coupled to an Abelian gauge field. It is shown that the symmetry (a) requires the field to have dynamic degrees of freedom and to be a connection in a non-flat space-time manifold, and (b) leads to a quantum theory free of the measurement problem. It is speculated that gravitomagnetism could be a possible source of the gauge field.

The General-Covariant and Gauge-Invariant Theory of Quantum Particles in Classical Backgrounds

International Journal of Modern Physics D, 2003

A new approach to the concept of particles and their production in quantum field theory is developed. A local operator describing the current of particle density is constructed for scalar and spinor fields in arbitrary gravitational and electromagnetic backgrounds. This enables one to describe particles in a local, general-covariant and gauge-invariant way. However, the current depends on the choice of a 2-point function. There is a choice that leads to the local non-conservation of the current in a gravitational or an electromagnetic background, which describes local particle production consistent with the usual global description based on the Bogoliubov transformation. The most natural choice based on the Green function calculated using the Schwinger–DeWitt method leads to the local conservation of the current, provided that interactions with quantum fields are absent. Interactions with quantum fields lead to the local non-conservation of the current which describes local particle...

Towards the Unity of Classical Physics. With Some Implications for Quantum Physics

AIP Conference Proceedings, 2007

The unity of classical mechanics and electromagnetism is proposed to be established through putting both on equal footing. The special-relativistic equations of motion for the particles and fields, the Maxwell-Lorentz force, and the Yukawa potential are derived exploiting Newton's and Euler's (stationary-)state descriptions, Newton's decomposition of forces into body-and position-dependent factors, and Helmholtz's analysis of the relationships between forces and energies. For instance, the magnetic Maxwell-Lorentz force is a special case of the Lipschitz force being a general class of forces that leave the kinetic energy constant. Adding this force to Newton's force of gravity leads to self-standing fields representing the mediating agent of interaction sought by Newton. Thus, equal footing is realized through a foundation on common principles, but not through a reduction to mechanical models.

Classical Physics, Fundamental Principles, and Lagrangian Approach

Undergraduate Lecture Notes in Physics, 2017

In this chapter it is shown how some of the basic equations of classical physics behave with respect to the requirements of the fundamental principles we have discussed earlier. In particular the examples have been chosen with the aim of naturally ferrying our reasoning toward the realm of non-Newtonian physics. These exercises therefore want to show: 1. How, contrary to what is commonly expected, in some cases even the Euclidean (i.e., "geometrical") covariance can hardly be regarded as self-evident 2. How the covariance, and therefore the fundamental principles from which it derives, can be used as a "minimum requirement" to guess some properties of the physical laws' equations, but also as a way to explore extensions or different versions of an existing theory 3. That classical electromagnetism is incompatible with the principle of relativity in its Galilean form, whose consequences will lead us to special relativity. It has to be stressed that in this chapter, when necessary, x k are intended as vector components dx k , and not just as coordinates. 4.1 Equations of Motion and Newtonian Gravitational Force Newton's second law of dynamics The very first case to start with is Newton's law of dynamics F = ma. This is a very simple one because the force on the left-hand side of the equation is a vector in the Euclidean and Galilean sense by definition. In other words, we require that for the transformations of the Galilean covariance group (i.e., translations, rotations, and Galilean boosts) and for any transformation of the coordinate system it isF = F.

On the gauge structure of classical mechanics

Transport Theory and Statistical Physics, 2000

A self consistent gauge theory of Classical Lagrangian Mechanics, based on the introduction of the bundle of affine scalars over the configuration manifold is proposed. In the resulting setup , the "Lagrangian" L is replaced by a section of a suitable principal fiber bundle over the velocity space, called the lagrangian bundle, while the associated Poincaré-Cartan 2-form is recognized as the curvature 2-form of a connection induced by L on a second "co-lagrangian" principal bundle. A parallel construction leads to the identification of a hamiltonian and a co-hamiltonian bundle over the phase space. An analysis of the properties of these spaces provides an intrinsic geometrical characterization of the Legendre transformation, thus allowing a systematic translation of the hamiltonian formalism into the newer scheme.

A Simple Introduction to Particle Physics I: Foundations and the Standard Model

This is the first of a series of papers in which we present a brief introduction to the relevant mathematical and physical ideas that form the foundation of Particle Physics, including Group Theory, Relativistic Quantum Mechanics, Quantum Field Theory and Interactions, Abelian and Non-Abelian Gauge Theory, and the SU(3)xSU(2)xU(1) Gauge Theory that describes our universe apart from gravity. These notes are not intended to be a comprehensive introduction to any of the ideas contained in them. Among the glaring omissions are CPT theorems, evaluations of Feynman Diagrams, Renormalization, and Anomalies. The topics were chosen according to the authors preferences and agenda. These notes are intended for a student who has completed the standard undergraduate physics and mathematics courses. Furthermore, these notes should not and will not in any way take the place of the related courses, but rather provide a primer for detailed courses in QFT, Gauge Theory, String Theory, etc., which will fill in the many gaps left by this paper.

A Simple Introduction to Particle Physics Part II

arXiv: High Energy Physics - Theory, 2009

This is the second in a series of papers intended to provide a basic overview of some of the major ideas in particle physics. Part I [arXiv:0810.3328] was primarily an algebraic exposition of gauge theories. We developed the group theoretic tools needed to understand the basic construction of gauge theory, as well as the physical concepts and tools to understand the structure of the Standard Model of Particle Physics as a gauge theory. In this paper (and the paper to follow), we continue our emphasis on gauge theories, but we do so with a more geometrical approach. We will conclude this paper with a brief discussion of general relativity, and save more advanced topics (including fibre bundles, characteristic classes, etc.) for the next paper in the series. We wish to reiterate that these notes are not intended to be a comprehensive introduction to any of the ideas contained in them. Their purpose is to introduce the "forest" rather than the "trees". The primary e...

Towards the Unity of Classical Physics

2009

The unity of classical mechanics and electromagnetism is proposed to be established through putting both on equal footing. The special-relativistic equations of motion for the particles and fields, the Maxwell-Lorentz force, and the Yukawa potential are derived exploiting Newton’s and Euler’s (stationary-)state descriptions, Newton’s decomposition of forces into bodyand position-dependent factors, and Helmholtz’s analysis of the relationships between forces and energies. For instance, the magnetic MaxwellLorentz force is a special case of the Lipschitz force being a general class of forces that leave the kinetic energy constant. Adding this force to Newton’s force of gravity leads to self-standing fields representing the mediating agent of interaction sought by Newton. Thus, equal footing is realized through a foundation on common principles, but not through a reduction to mechanical models.