First steps in classical field theory

2021 ◽  
pp. 435-448
Author(s):  
Andrew M. Steane

Classical field theory, as it is applied to the most simple scalar, vector and spinor fields in flat spacetime, is described. The Klein-Gordan, Weyl and Dirac equations are obtained, and some features of their solutions are discussed. The Yukawa potential, the plane wave solutions, and the conserved currents are obtained. Spinors are introduced, both through physical pictures (flagpole and flag) and algebraic defintions (complex vectors). The relationship between spinors and four-vectors is given, and related to the Lie groups SU(2) and SO(3). The Dirac spinor is introduced.

2021 ◽  
pp. 24-34
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras

The purpose of this chapter is to recall the principles of Lagrangian and Hamiltonian classical mechanics. Many results are presented without detailed proofs. We obtain the Euler–Lagrange equations of motion, and show the equivalence with Hamilton’s equations. We derive Noether’s theorem and show the connection between symmetries and conservation laws. These principles are extended to a system with an infinite number of degrees of freedom, i.e. a classical field theory. The invariance under a Lie group of transformations implies the existence of conserved currents. The corresponding charges generate, through the Poisson brackets, the infinitesimal transformations of the fields as well as the Lie algebra of the group.


1974 ◽  
Vol 53 ◽  
pp. 27-46
Author(s):  
H. A. Bethe

An equation of state is developed for densities from nuclear density (3 x 1014 g cm−3) to about 1016 g cm−3. The repulsive interaction between baryons dominates and empirical arguments for its existence are given. This interaction is attributed to vector meson exchange, and is derived from classical field theory whereupon a Yukawa potential results. The potential actually assumed is a modification of the Reid potential. Arguments are given that the baryons will not form a crystal lattice. The actual calculations were done using Pandharipande's method. The particles present at high density certainly include nucleons, Λ and Σ. The presence of Δ is questionable but that of π is likely. Results are given for the concentration of various species. With the more likely assumption about interactions, the concentration of each permissible species of particle is about equal at ϱ = 1016 g cm−3. The relation between energy and density is nearly independent of the assumptions on the species permitted and the energy is about 3 GeV particle−1 at ϱ = 1016 g cm−3. The relation between pressure and energy density is given, which yields a sound velocity equal to c at a few times 1015 g cm−3. Results for the structure of neutron stars are given. The maximum mass is about 2 solar masses and the maximum moment of inertia 1045 g cm2.


2013 ◽  
Vol 22 (12) ◽  
pp. 1342018 ◽  
Author(s):  
SHINJI MUKOYAMA ◽  
JEAN-PHILIPPE UZAN

The Lorentzian metric structure allows one to implement the relativistic notion of causality in any field theory and to define a notion of time dimension. We propose that at the microscopic level the metric is Riemannian and that the Lorentzian structure, usually thought as fundamental, is in fact an effective property, that emerges in some regions of a 4-dimensional space with a positive definite metric. We argue that a decent classical field theory for scalars, vectors and spinors in flat spacetime can be constructed, and that gravity can be included under the form of a covariant Galileon theory instead of general relativity.


1994 ◽  
Vol 06 (05a) ◽  
pp. 1071-1083 ◽  
Author(s):  
MOSHÉ FLATO ◽  
JACQUES SIMON ◽  
ERIK TAFLIN

In this article we present an announcement of results concerning: a) A solution to the Cauchy problem for the M-D equations, namely global existence, for small initial data at t = 0, of solutions for the M-D equations. b) Arguments from which asymptotic completeness for the M-D equations follows. c) Cohomological interpretation of the results in the spirit of nonlinear representation theory and its connection to the infrared tail of the electron in M-D classical field theory. The full detailed results will be published elsewhere.


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