References

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.

Introduction This chapter develops the ideas of classical field theory in the context of special relativity. We use a scalar field and the electromagnetic field as examples of classical fields. The discussion of scalar field theory will allow us to understand concepts that are unique to field theory in a somewhat simpler context than electromagnetism; it will also be useful later on in the study of topics such as inflation, quantum field theory in curved spacetime, etc. As regards electromagnetism, we concentrate on those topics that will have direct relevance in the development of similar ideas in gravity (gauge invariance, Hamilton–Jacobi theory for particle motion, radiation and radiation reaction, etc.). The ideas developed here will be used in the next chapter to understand why a field theory of gravity – developed along similar lines – runs into difficulties. The concept of an action principle for a field will be extensively used in Chapter 6 in the context of gravity. Other topics will prove to be valuable in studying the effect of gravity on different physical systems. External fields of force In non-relativistic mechanics, the effect of an external force field on a particle can be incorporated by adding to the Lagrangian the term − V(t, x) , thereby adding to the action the integral of − V dt . Such a modification is, however, not Lorentz invariant and hence cannot be used in a relativistic theory.

Two basic geometric approaches to the modern theory of gauge fields are analyzed and compared. The first approach is an extension of the Kaluza-Klein unified theory of gravity and electromagnetism (1921). The second approach generalizes Cartan's formulation (1925) of Riemannian geometry and GR which now is transformed to fiber bundle theory. The goal of this paper is to show that the above-mentioned geometric approaches to the classical gauge field theory are nonequivalent and lead to different forms of quantum gauge field theory. Bibliography:12 titles.

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.

A classical nonrelativistic effective field theory for a real Lorentz-scalar\nfield $\\phi$ is most conveniently formulated in terms of a complex scalar field\n$\\psi$. There have been two derivations of effective Lagrangians for the\ncomplex field $\\psi$ in which terms in the effective potential were determined\nto order $(\\psi^* \\psi)^4$. We point out an error in each of the effective\nLagrangians. After correcting the errors, we demonstrate the equivalence of the\ntwo effective Lagrangians by verifying that they both reproduce $T$-matrix\nelements of the relativistic real scalar field theory and by also constructing\na redefinition of the complex field $\\psi$ that transforms terms in one\neffective Lagrangian into the corresponding terms of the other.\n

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.

Quantum states can be understood as either (i) describing quantum systems or (ii) representing observes' knowledge about quantum systems. These different meanings are shown to imply different transformation properties in relativistic field theories. The rules for the reduction of quantum states and the transformation properties of quantum states under Lorentz transformations are derived for case (ii). The results obtained are applied to a quantum system recently presented and analyzed by Aharonov and Albert. It is shown that the present results, combined with Aharonov and Albert's, amount to a proof of Bohr's view that quantum states represent observes' knowledge about quantum systems.

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

called the special theory of relativity, no doubt one of the boldest and most interesting theories ever designed.' Generations of physicists, mathematicians and philosophers were inspired by it. Bitter battles were fought over its principles and deductions which clashed with many an honourable idea backed up by honourable philosophical and scientific traditions. Yet all the knocks and bites from left and right seem to have left Einstein's theory unshaken. Today the battle-grounds of yesterday look rather deserted. The critic is likely to be silenced in the face of the theory's spectacular success. In the following I shall try to throw some new light on one or two old problems which Einstein's theory gives rise to. Section z is concerned with the obstacles we are faced with when we try to combine Einstein's special theory of relativity with a causal account of relativistic phenomena. The argument does not show, of course, that Einstein's theory is false, but it points to a possible incompleteness and to certain limitations inherent in the special theory of relativity. In section 3 a substratum theory of relativity is outlined which seems to be as simple and coherent as Einstein's special theory of relativity; the Lorentz transformations are deduced from two basic principles without making use of the principle of relativity. Finally, in section 4, the two theories are compared and some general conclusions are drawn. But let us first have a short look at the ideas underlying Einstein's theory. To begin with, the special theory of relativity is a transformation theory. This is its most fundamental aspect. Suppose we know the space and time co-ordinates of an event particle e (event with negligible spatiotemporal extension) in an inertial frame of reference F; then the theory

This paper is a sequel to earlier ones, (1984), in which, on the one hand, classical field theories were described on a curved Newtonian spacetime, and, on the other hand, the Newtonian gravitation theory was formulated on a five-dimensional spacetime with a metric of signature (++++-) and a covariantly constant vector field. Here the authors show that Lagrangians for matter fields are easily formulated on this extended spacetime from simple invariance arguments and that stress-energy tensors can be derived from them in the usual manner so that four-dimensional spacetime expressions are obtained that are consistent in the relativistic as well as the Newtonian case. In the former the theory is equivalent to general relativity. When the magnitude of the distinguished vector field vanishes equations for the (covariant) Newtonian limit follow. They demonstrate this here explicitly in the case of the Klein-Gordon/Schrodinger and the Dirac field and its covariant non-relativistic analogue, the Levy-Leblond field. In especially the latter example the covariant Newtonian theory simplifies dramatically in this five-dimensional form.

Quantum field theory forms the present theoretical framework for our understanding of the fundamental interactions of particle physics. This up-dated and expanded text examines gauge theories and their symmetries with an emphasis on their physical and technical aspects. Beginning with a new chapter giving a systematic introduction to classical field theories and a short discussion of their canonical quantization and the discrete symmetries C, P and T, the book provides a brief exposition of perturbation theory, the renormalization programme, and the use of the renormalization group equation. It then explores topics of current research interest including chiral symmetry and its breaking, anomalies, and low energy effective lagrangians and some basics of supersymmetry. A chapter on basics of the electroweak theory is now included. Professor Pokorski, a distinguished theoretical physicist, has presented here a self-contained text for graduate courses in physics; the only prerequisite is some grounding in quantum field theory.

Effective Lagrangians with non-linear coupling schemes are applied for the relativistic description of the nuclear many-body problem. The most widely used models of this type is Relativistic Mean Field (RMF) theory for closed shell nuclei and Relativistic Hartree-Bogoliubov (RHB) theory for open-shell nuclei with pairing correlations. We discuss the basic ideas of this concept and recent applications for nuclei far from the valley of stability.

A rigorous theory of variational principles in quantum mechanics is formulated. This result is obtained by generalizing the standard theory of the inverse problem of the variational caleulus, according to new concepts of differentiability on complex Banach spaces. Our theory provides sound mathematical foundations to all known variational principles of quantum mechanics and a powerful tool to find new ones. As a simple application, we derive rigorously two variational principles for the time-dependent Schrödinger equation: the conventional one, requiring both an initial and a final condition, and a new one, requiring an initial condition only.

A new gauge field theory of a continuum with dislocations and disclinations