How does physics bear upon metaphysics; and why did Plato hold that philosophy cannot be written down?

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.

We study the equilibration properties of classical integrable field theories at a finite energy density, with a time evolution that starts from initial conditions far from equilibrium. These classical field theories may be regarded as quantum field theories in the regime of high occupation numbers. This observation permits to recover the classical quantities from the quantum ones by taking a proper limit. In particular, the time averages of the classical theories can be expressed in terms of a suitable version of the LeClair–Mussardo formula relative to the generalized Gibbs ensemble. For the purposes of handling time averages, our approach provides a solution of the problem of the infinite gap solutions of the inverse scattering method.

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

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.

We analyze the field radiation due to a general source in the inertial and accelerated frames in classical massless scalar field theory in Minkowski spacetime. The classical particle numbers in these frames are defined and further justified in the context of quantum mechanics. The simple relation between the two expressions shows how some aspects of the Fulling-Davies-Unruh thermal bath are present in classical field theory. Finally we show how our results and an operator algebra can be used to obtain an operator expression for the Fulling-Davies-Unruh effect. We also comment on the expected analogue of our results for the Schwarzschild black hole.

The classical approximation provides a non-perturbative approach to time-dependent problems in finite temperature field theory. We study the divergences in hot classical field theory perturbatively. At one-loop, we show that the linear divergences are completely determined by the classical equivalent of the hard thermal loops in hot quantum field theories, and that logarithmic divergences are absent. To deal with higher-loop diagrams, we present a general argument that the superficial degree of divergence of classical vertex functions decreases by one with each additional loop: one-loop contributions are superficially linearly divergent, two-loop contributions are superficially logarithmically divergent, and three- and higher-loop contributions are superficially finite. We verify this for two-loop SU(N) self-energy diagrams in Feynman and Coulomb gauges. We argue that hot, classical scalar field theory may be completely renormalized by local (mass) counterterms, and discuss renormalization of SU(N) gauge theories.

W. Fulton, Enumerative geometry (with notes by Alastair Craw): Enumerative geometry (with notes by Alastair Craw) Bibliography A. Bertram, Computing Gromov-Witten invariants with algebraic geometry: Introduction and motivation Localization $J$-functions An alternative to WDVV Bibliography D. S. Freed, Classical field theory and supersymmetry: Introduction Classical mechanics Lagrangian field theory and symmetries Classical bosonic theories on Minkowski spacetime Fermions and the supersymmetric particle Free theories, quantization, and approximation Supersymmetric field theories Supersymmetric $\sigma$-models Bibliography J. W. Morgan, Introduction to supermanifolds: Introduction to supermanifolds Bibliography C. V. Johnson, Notes on introductory general relativity: Notes on introductory general relativity Bibliography.

We derive a generic identity which holds for the metric (i.e. variational) energy–momentum tensor under any field transformation in any generally covariant classical Lagrangian field theory. The identity determines the conditions under which a symmetry of the Lagrangian is also a symmetry of the energy–momentum tensor. It turns out that the stress tensor acquires the symmetry if the Lagrangian has the symmetry in a generic curved spacetime. In this sense, a field theory in flat spacetime is not self-contained. When the identity is applied to the gauge invariant spin-2 field in Minkowski space, we obtain an alternative and direct derivation of a known no-go theorem: a linear gauge invariant spin-2 field, which is dynamically equivalent to linearized general relativity, cannot have a gauge invariant metric energy–momentum tensor. This implies that attempts to define the notion of gravitational energy density in terms of the metric energy–momentum tensor in a field-theoretical formulation of gravity must fail.

It is known that, in quantum field theory, localized operations, e.g., given by unitary operators in local observable algebras, may lead to non-causal, or superluminal, state changes within their localization region. In this article, it is shown that, both in quantum field theory as well as in classical relativistic field theory, there are localized operations which correspond to “instantaneous” spatial rotations (leaving the localization region invariant) leading to superluminal effects within the localization region. This shows that “impossible measurement scenarios” which have been investigated in the literature, and which rely on the presence of localized operations that feature superluminal effects within their localization region, do not only occur in quantum field theory, but also in classical field theory.

In this scientific preface to the first issue of International Journal of Geometric Methods in Modern Physics1, we briefly survey some peculiarities of geometric techniques in quantummodels. Contemporary quantum theory meets an explosion of different types of quantization. Some of them (geometric quantization, deformation quantization, noncommutative geometry, topological field theory etc.) speak the language of geometry, algebraic and differential topology. We do not pretend for any comprehensive analysis of these quantization techniques, but aims to formulate and illustrate their main peculiarities. As in any survey, a selection of topics has to be done, and we apologize in advance if some relevant works are omitted. Geometry of classical mechanics and field theory is mainly differential geometry of finitedimensional smooth manifolds, fiber bundles and Lie groups. The key point why geometry plays a prominent role in classical field theory lies in the fact that it enables one to deal with invariantly defined objects. Gauge theory has shown clearly that this is a basic physical principle. At first, a pseudo-Riemannian metric has been identified to a gravitational field in the framework of Einstein’s General Relativity. In 60-70th, one has observed that connections on a principal bundle provide the mathematical model of classical gauge potentials [1-3]. Furthermore, since the characteristic classes of principal bundles are expressed in terms of the gauge strengths, one can also describe the topological phenomena in classical gauge models [4]. Spontaneous symmetry breaking and Higgs fields have been explained in terms of reduced G-structures [5]. A gravitational field seen as a pseudo-Riemannian metric exemplifies such a Higgs field [6]. In a general setting, differential geometry of smooth fiber bundles gives the adequate mathematical formulation of classical field theory, where fields are represented by sections of fiber bundles and their dynamics is phrased in terms of jet manifolds [7]. Autonomous classical mechanics speaks the geometric language of symplectic and Poisson Web: http://www.worldscinet.com/ijgmmp/ijgmmp.shtml

Axions and axion-like particles are bosonic quantum fields. They are often assumed to follow classical field equations due to their high degeneracy in the phase space. In this work, we explore the disparity between classical and quantum field treatments in the context of density and velocity fields of axions. Once the initial density and velocity field are specified, the evolution of the axion fluid is unique in the classical field treatment. However, in the quantum field treatment, there are many quantum states consistent with the given initial density and velocity field. We show that evolutions of the density perturbations for these quantum states are not necessarily identical and, in general, differ from the unique classical evolution. To illustrate the underlying physics, we consider a system of large number of bosons in a one-dimensional box, moving under the gravitational potential of a heavy static point-mass. We ignore the self-interactions between the bosons here. Starting with homogeneous number density and zero velocity field, we determine the density perturbations in the linear regime in both quantum and classical field theories. We find that classical and quantum evolutions are identical in the linear regime if only one single-particle state is occupied by all the bosons and the self-interaction is absent. If more than one single-particle states are occupied, the density perturbations in quantum evolutions differ from the classical prediction after a certain time which depends upon the parameters of the system.

xTras: A field-theory inspired xAct package for mathematica

A Hamilton-Jacobi formalism of classical relativistic field theory is developed. Both "time-independent" and "time-dependent" formulations are given, and the relation between them is discussed. In the former, the constants of the motion are identified with the "new" field variables, whereas in the latter they are the values of the fields on a suitable spacelike surface. The explicit introduction of a Hamiltonian density is avoided. As an illustration of the respective procedures, the classical Dirac and Klein-Gordon free fields are solved explicitly. A perturbation method is formulated for the case of fields in interaction. The metric tensor is not treated as a field quantity.

We show that various relativistic potential models (all sharing exact relativistic two-body kinematics and a common nonrelativistic limit) can be distinguished by agreement or disagreement with relativistic corrections produced by classical field theory. We find that the only one of these models whose relativisic corrections duplicate those of classical field theory is the minimal Todorov equation. Conversely, we derive the Todorov equation from the semirelativistic dynamics of classical field theory, thus exposing the classical field-theoretic origins of its characteristic minimal potential structures and dependences on effective one-body variables.

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.