Classical Field Theory

Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which most closely approximates quantum electrodynamics. Indeed, in inertial frames, there is a general connection between classical field theories with classical zero-point radiation and quantum field theories. However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Quantum field theory applies the canonical quantization procedure (depending on the local time coordinate) to a mirror-walled box, and, in general, each non-inertial coordinate frame has its own vacuum state. In complete contrast, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; in empty space, the correlation functions depend upon only the geodesic separations (and their coordinate derivatives) between the spacetime points. It makes no difference whether a box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same correlation function except for small end-point (Casimir) corrections. Thus in classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. It is also noted that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. Based upon the classical analysis, it is suggested that the claimed heating effects of acceleration through the vacuum may not exist in nature.

Classical Field Theory is a graduate level introduction to modern classical field theory, including electromagnetism and general relativity and describes various classical methods for fields with n...

A common choice of configuration space for classical field theory is an infinite-dimensional vector space of functions or tensor fields on space or spacetime, the elements of which are called fields. Here we relate our treatment of infinite-dimensional Hamiltonian systems discussed in §2.1 to classical Lagrangian and Hamiltonian field theory and then give examples. Classical field theory is a large subject with many aspects not covered here; we treat only a few topics that are basic to subsequent developments; see Chapters 6 and 7 for additional information and references.KeywordsPoisson BracketTravel Wave SolutionSymplectic FormSchrodinger EquationClassical Field TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present the fundamentals of a prequantum model with hidden variables of classical field type—prequantum classical statistical field theory (PCSFT). In some sense, this is the comeback of classical field theory, but with an extension to massive particles. All quantum averages (including correlations of entangled systems) can be represented as classical field averages and correlations. In the literature, entanglement of biparticle systems (in particular, biphotons) has been represented in the PCSFT framework. In this paper, we consider triparticle entanglement. It is well known that in conventional quantum mechanics, the cases of biparticle and triparticle entanglement are quite different. Roughly speaking, triparticle entanglement is not reducible to biparticle entanglement. The PCSFT approach to entanglement (as a correlation of classical signals) reflects this quantum situation. The PCSFT representations of correlations in biparticle and triparticle systems differ essentially. In particular, from the mathematical viewpoint the latter is fundamentally more complicated than the former.

Einstein equations, Yang-Mills equations and classical field theory as compatibility conditions of linear partial differential operators

FROM FERMAT TO GAUSS. Fermat, Euler and Quadratic Reciprocity. Lagrange, Legendre and Quadratic Forms. Gauss, Composition and Genera. Cubic and Biquadratic Reciprocity. CLASS FIELD THEORY. The Hilbert Class Field and p = x 2 + ny 2 . The Hilbert Class Field and Genus Theory. Orders in Imaginary Quadratic Fields. Class Fields Theory and the Cebotarev Density Theorem. Ring Class Field and p = x 2 + ny 2 . COMPLEX MULTIPLICATION. Elliptic Functions and Complex Multiplication. Modular Functions and Ring Class Fields. Modular Functions and Singular j--Invariants. The Class Equation. Ellpitic Curves. References. Index.

A geometrical theorem on the asymptotic space-time properties of conservation laws in a classical field theory

On-shell constructibility is redefining our understanding of perturbative quantum field theory. The tree-level S-matrix of constructible theories is completely determined by a set of recurrence relations and a reduced number of scattering amplitudes. In this paper, we revisit the on-shell constructibility of gravitational theories making use of new results on soft theorems and recurrence relations. We show that using a double complex shift and an all-line soft deformation allows us to relax the technical conditions for constructibility, in order to include more general propagators and higher-derivative interactions that prevent using conventional Britto-Cachazo-Feng-Witten (BCFW) shifts. From this result we extract a set of criteria that guarantee that a given gravitational action has the same tree-level S-matrix in Minkowski spacetime as general relativity, which implies the equivalence at all orders in perturbation theory between these classical field theories on asymptotically flat spacetimes. As a corollary we deduce that the scattering amplitudes of general relativity and unimodular gravity are the same for an arbitrary number of external particles (as long as the S-matrix of the latter is unitary), thus extending previous works that were able to deal only with n=4 and n=5 amplitudes.

Our treatment of integrable systems in the first three chapters made essential use of the Hamiltonian formalism both in finite and infinite dimensional settings. In the next two chapters we shall concentrate on classical field theory, where the covariant formulation requires the Lagrangian formalism. It is assumed that the reader has covered the Lagrangian treatment of classical mechanics and classical field theory at the basic level [102, 187]. The aim of this chapter is not to provide a crash course in these subjects, but rather to introduce less standard aspects.

Part I of this book is introductory in nature. Its purpose is to motivate our geometric approach to supersymmetric field theory. We show how geometric structures arise in classical and quantum field theories on quite general grounds. In Chapter 1 we consider the basic geometric structures which hold independently of supersymmetry. In Chapter 2 we specialize to the supersymmetric case (rigid and local) where more elegant structures emerge. Not being part of the technical body of the book, these chapters are rather elementary and sketchy. However, we show how dualities, modularity, and other stringy patterns are universal features of field theory.

AE 1. AE (377). 2. AE AE - (AE ) (379). 2.1. lf R P . 2.2. AE. 2.3. AE. 2.4. AE lf R P - . 2.5. AE AE AE . 2.6. . 2.7. - . 2.8. "" AE . 2.9. - AE AE . 2.10. "" . -. 2.11. AE . 3. AE AE (387). 3.1. AE AE -AE. 3.2. AE AE AE -AE . 4. AE (390). 4.1. . 4.2. AE . 4.3. , AEAE. 4.4. AE lf R P AEAE. 4.5. AE -. 4.6. . 4.7. AE -AE -. 4.8. AE lf R P AEAE. AE AE AE AE. 5. (395). 5.1. , .AE. 5.2. - . 5.3. AE . 5.4. AE "" -. 5.5. . . 6. - (399). 6.1. Q- . 6.2. Q- . 6.3. Q- - AE. 6.4. Q-. 6.5. Q-. 6.6. AE Q-. 7. (403). (404). (404). (404).

This paper describes affine Toda field theory which is a theory of r scalar fields in two-dimensional Minkowski space-time, where r is the rank of a compact semi-simple Lie algebra g. The classical field theory is determined by the lagrangian density L = 1/2 {partial derivative}{sub {rho}}{phi}{sup a}{partial derivative}{sup {mu}}{phi}{sup a} {minus} V({phi}) where V({phi}) = m{sup 2}/{beta}{sup 2} {Sigma}{sub 0}{sup r}n{sub i}e{sup {beta}{alpha}{sub i} {center dot} {phi}}. m and {beta} are real, classically unimportant constants, {alpha}{sub i} i = 1, . . . ,r are the simple roots of the Lie algebra g, and {alpha}{sub 0} = {Sigma}{sub 1}{sup 4} n{alpha}{sub i} is a linear combination of the simple roots; it corresponds to the extra spot on an extended Dynkin diagram for g. A reasonable question to ask is whether the classical integrability survives into the quantum field theory and, if so, what is the spectrum and to what extent is it possible to calculate explicitly quantities of interest such as S-matrices and form factors. The recent discoveries leave no doubt that these relatively simple models have much structure and their study (even in the {beta}{sup 2} {gt} 0 regime) will be informative. In this short review, the ADEmore » series of Lie algebras will be singled out for special attention.« less

In many formulations of Hamiltonian perturbation theory a Green’s function becomes undefined when some parameter is allowed to vanish. Here various examples are discussed to illustrate this phenomenon, and it is shown that they are all realizations of a general theorem. The cases considered are examples in classical mechanics, quantum mechanics, electrodynamics and field theory. The prime object is to illustrate the unity of the examples and thus to make the application of the procedure to field theory models of current interest more transparent. One example that we refer to is the skyrmion model.

The geometric framework for the Hamilton-Jacobi theory developed in the studies of Cariñena et al. [Int. J. Geom. Methods Mod. Phys. 3(7), 1417–1458 (2006)], Cariñena et al. [Int. J. Geom. Methods Mod. Phys. 13(2), 1650017 (2015)], and de León et al. [Variations, Geometry and Physics (Nova Science Publishers, New York, 2009)] is extended for multisymplectic first-order classical field theories. The Hamilton-Jacobi problem is stated for the Lagrangian and the Hamiltonian formalisms of these theories as a particular case of a more general problem, and the classical Hamilton-Jacobi equation for field theories is recovered from this geometrical setting. Particular and complete solutions to these problems are defined and characterized in several equivalent ways in both formalisms, and the equivalence between them is proved. The use of distributions in jet bundles that represent the solutions to the field equations is the fundamental tool in this formulation. Some examples are analyzed and, in particular, the Hamilton-Jacobi equation for non-autonomous mechanical systems is obtained as a special case of our results.

The physical meaning and the geometrical interpretation of causality implementation in classical field theories are discussed. Causality in field theory are kinematical constraints dynamically implemented via solutions of the field equations, but in a limit of zero-distance from the field sources part of these constraints carries a dynamical content that explains old problems of classical electro-dynamics with deep implications to the nature of physical interactions.