Equivalent Theories and Changing Hamiltonian Observables in General Relativity

Change and local spatial variation are missing in Hamiltonian general relativity according to the most common definition of observables as having 0 Poisson bracket with all first-class constraints. But other definitions of observables have been proposed. In pursuit of Hamiltonian–Lagrangian equivalence, Pons, Salisbury and Sundermeyer use the Anderson–Bergmann–Castellani gauge generator G, a tuned sum of first-class constraints. Kucha? waived the 0 Poisson bracket condition for the Hamiltonian constraint to achieve changing observables. A systematic combination of the two reforms might use the gauge generator but permit non-zero Lie derivative Poisson brackets for the external gauge symmetry of General Relativity. Fortunately one can test definitions of observables by calculation using two formulations of a theory, one without gauge freedom and one with gauge freedom. The formulations, being empirically equivalent, must have equivalent observables. For de Broglie-Proca non-gauge massive electromagnetism, all constraints are second-class, so everything is observable. Demanding equivalent observables from gauge Stueckelberg–Utiyama electromagnetism, one finds that the usual definition fails while the Pons–Salisbury–Sundermeyer definition with G succeeds. This definition does not readily yield change in GR, however. Should GR’s external gauge freedom of general relativity share with internal gauge symmetries the 0 Poisson bracket (invariance), or is covariance (a transformation rule) sufficient? A graviton mass breaks the gauge symmetry (general covariance), but it can be restored by parametrization with clock fields. By requiring equivalent observables, one can test whether observables should have 0 or the Lie derivative as the Poisson bracket with the gauge generator G. The latter definition is vindicated by calculation. While this conclusion has been reported previously, here the calculation is given in some detail.     

Perturbative Solutions of the Extended Constraint Equations in General Relativity

The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface in an asymptotically simple space-time satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-linear system of partial differential equations for the induced metric, the second fundamental form and two other tensorial quantities defined on , and are equivalent to the usual constraint equations that satisfies as a space-like hypersurface in a space-time satisfying Einstein’s vacuum equation. This article develops a method for finding perturbative, asymptotically flat solutions of the extended constraint equations in a neighbourhood of the flat solution on Euclidean space. This method is fundamentally different from the ‘classical’ method of Lichnerowicz and York that is used to solve the usual constraint equations.     

Crossing relations derived from (Extended) Relativity

Recently, Special Relativity has been straightforwardly extended to Superluminal inertial frames and faster-than-light objects. The Extended Relativity theory not only allowed building up a self-consistent classical theory of tachyons, but reveals itself useful also for the understanding of standard (subluminal) physics, i.e. of usual particles. In this paper, it is shown that Extended Relativity allows: (i) deriving the usual Crossing Relations of elementary particle (high-energy) physics; and (ii) deriving the CPT-covariance theorem as a particular case of G-covariance (i.e., covariance under the new group of Generalised Lorentz transformations, both subluminal and Superluminal).In this framework, the Analyticity postulate is unnecessary: it is better substituted by the G-covariance requirement.Moreover, new crossing-type relations are predicted on the basis of mere Extended Relativity. They may well serve as a test for relativistic covariance of force fields like strong interactions and, particularly, weak interactions, and possible new interaction fields (whicha priori are not relativistically covariant).     

Stochastically and Intrinsically Extended Non-relativistic Quantum Particles

Stochastically and intrinsically extended non relativistic quantum particles are described by combining the ideas of a stochastic quantum theory and a quantum functional theory. The former relates the extension to imperfect real measurements while the latter considers it as intrinsic. Physical states, Positive-Operator-Valued measures connected to measurement, and propagators are given and discussed. The stochastic theory is sufficient when the bilocal field describing the particle has a product form.     

Hamiltonian Anomalies from Extended Field Theories

We develop a proposal by Freed to see anomalous field theories as relative field theories, namely field theories taking value in a field theory in one dimension higher, the anomaly field theory. We show that when the anomaly field theory is extended down to codimension 2, familiar facts about Hamiltonian anomalies can be naturally recovered, such as the fact that the anomalous symmetry group admits only a projective representation on the Hilbert space, or that the latter is really an abelian bundle gerbe over the moduli space. We include in the discussion the case of non-invertible anomaly field theories, which is relevant to six-dimensional (2, 0) superconformal theories. In this case, we show that the Hamiltonian anomaly is characterized by a degree 2 non-abelian group cohomology class, associated to the non-abelian gerbe playing the role of the state space of the anomalous theory. We construct Dai-Freed theories, governing the anomalies of chiral fermionic theories, and Wess-Zumino theories, governing the anomalies of Wess-Zumino terms and self-dual field theories, as extended field theories down to codimension 2.     

High-Speed Scattering of Charged and Uncharged Particles in General Relativity

After a brief consideration of the high-speed scattering of two point charges we thoroughly discuss high-speed scattering for a charged particle by a fixed mass and of two uncharged particles of comparable masses. We use perturbation technique over Minkowski spacetime in the de Donder gauge and solve the field equations and the resulting equations of motion (which take the reaction of the particles' quasistatic self-field into account) by iteration. The obtained energy-momentum conservation laws allow the computation of second-order corrections for the scattering angle and the cross section. The asymptotic structure of the far-field indicates synchrotron radiation (electromagnetic and gravitational, respectively) which causes an energy loss whose reaction on the motion is briefly considered in the low-velocity limit including bound motion. (For neutral particles this is a third-order effect).     

Recovering the Effective Cosmological Constant in Extended Gravity Theories

In the framework of extended gravity theories, we discuss the meaning of a time-dependent cosmological constant and give a set of conditions to recover an asymptotic de Sitter behaviour for a class of cosmological models independently of initial data. To this purpose we introduce a time-dependent (effective) quantity which asymptotically becomes the true cosmological constant. We will deal with scalar-tensor, fourth and higher than fourth-order theories.     

On the Principle of Relativity

In a recent article [1] M.A. Oliver argues there is a conflict between Einstein's Special Theory of Relativity (STR) and Cosmology. In ascertaining this conflict (see below), Oliver finds allies in Bergmann [2] and Bondi [3]. To resolve this conflict, he proposes to restore the classical (mechanical) concepts of space and time [1, p.666] and an absolute rest-frame. I shall devote a few words (1) to the Principle of Relativity and (2) to the notion of cosmic time in cosmology; this enables me (3) to argue that the alleged conflict between STR and Cosmology is based on a misunderstanding of the Principle of Relativity. (4) Finally I take a critical look at Oliver's allies.     

On the Long Time Behavior of Infinitely Extended Systems of Particles Interacting via Kac Potentials

We analyze the long time behavior of an infinitely extended system of particles in one dimension, evolving according to the Newton laws and interacting via a non-negative superstable Kac potential (x)=(x), (0, 1]. We first prove that the velocity of a particle grows at most linearly in time, with rate of order . We next study the motion of a fast particle interacting with a background of slow particles, and we prove that its velocity remains almost unchanged for a very long time (at least proportional to ^–1 times the velocity itself). Finally we shortly discuss the so called Vlasov limit, when time and space are scaled by a factor .     

Quantum General Relativity and the Meaning of (R + R2) Theories

It is argued that, due to the cut-off lengths arising in Quantum General Relativity, R2 corrections of Einstein's theory cannot be interpreted as quantum corrections.     

On Parametrized General Relativity

A physical framework has been proposed which describes manifestly covariant relativistic evolution using a scalar time . Studies in electromagnetism, measurement, and the nature of time have demonstrated that in this framework, electromagnetism must be formulated in terms of -dependent fields. Such an electromagnetic theory has been developed. Gravitation must also use of -dependent fields, but many references do not take the metric's dependence on fully into account. Others differ markedly from general relativity in their formulation. In contrast, this paper outlines steps towards a -dependent classical intrinsic formulation of gravitation, patterned after general relativity, which we call parametrized general relativity (PGR). Given the existence of a preferred foliation, the Hamiltonian constraint is removed. We find that some nonmetricity in the connection is allowed, unlike in general relativity. Conditions on the allowable nonmetricity are found. Consideration of the initial value problem confirms that the metric signature should normally be O(3, 2) rather than O(4, 1). Following the lead of earlier works, we argue that concatenation (integration over ) is unnecessary for relating parametrized physics to experience, and propose an alternative to it. Finally, we compare and contrast PGR with other relevant gravitational theories.     

Extended U(1) Conformal Field Theories and Zk-Parafermions

A constructive approach is developed for studying local chiral algebras generated by a pair of oppositely charged fields Ψ (z, ±g) such that the operator product expansion (OPE) of Ψ(z₁, g) Ψ(z₂, −g) involves a U(1) current. The main tool in the study is the factorization property of the charged fields (exhibited in [PT 2, 3]) for Virasoro central charge c < 1 into U(1)-vertex operators tensored with ZAMOLODCHIKOV-FATEEV [ZF1] (generalized) Z_k-parafermions. The case Δ₂ = 4 (Δ₁ − 1), where Δ_v = Δ_k−v(Δ₀ = 0) are the conformaldimensions of the parafermionic currents, is studied in detail. For Δ_v = 2v(1 − v/k) the theory is related to GEPNER'S [Ge] Z₂ [so (k)] parafermions and the corresponding quantum field theoretic (QFT) representations of the chiral algebra are displayed. The Coulomb gas method of [CR] is further developed to include an explicit construction of the basic parafermionic current Ψ of weight Δ = Δ₁. The characters of the positive energy representations of the local chiral algebra are written as sums of products of Kac's string functions and classical θ-functions.     

Renormalization and Operator Product Expansion in Theories with Massless Particles

Renormalization procedure in theories including massless particles is presented. With the help of counterterm formalizm the operator product expansion for arbitrary composite fields is derived. The coefficient functions are explicitly expressed in terms of certain Green's functions.     

Theories of Variable Mass Particles and Low Energy Nuclear Phenomena

Variable particle masses have sometimes been invoked to explain observed anomalies in low energy nuclear reactions (LENR). Such behavior has never been observed directly, and is not considered possible in theoretical nuclear physics. Nevertheless, there are covariant off-mass-shell theories of relativistic particle dynamics, based on works by Fock, Stueckelberg, Feynman, Greenberger, Horwitz, and others. We review some of these and we also consider virtual particles that arise in conventional Feynman diagrams in relativistic field theories. Effective Lagrangian models incorporating variable mass particle theories might be useful in describing anomalous nuclear reactions by combining mass shifts together with resonant tunneling and other effects. A detailed model for resonant fusion in a deuterium molecule with off-shell deuterons and electrons is presented as an example. Experimental means of observing such off-shell behavior directly, if it exists, is proposed and described. Brief explanations for elemental transmutation and formation of micro-craters are also given, and an alternative mechanism for the mass shift in the Widom–Larsen theory is presented. If variable mass theories were to find experimental support from LENR, then they would undoubtedly have important implications for the foundations of quantum mechanics, and practical applications may arise.     

Conformal aspects of the Palatini approach in Extended Theories of Gravity

The debate on the physical relevance of conformal transformations can be faced by taking the Palatini approach into account gravitational theories. We show that conformal transformations are not only a mathematical tool to disentangle gravitational and matter degrees of freedom (passing from the Jordan frame to the Einstein frame) but they acquire a physical meaning considering the bi-metric structure of Palatini approach which allows to distinguish between spacetime structure and geodesic structure. These facts are relevant at least at cosmological scales, while at small scale (i.e. in the spacetime regions relevant for observations) the conformal factor is slowly varying and its effects are not relevant. Examples of higher-order and non-minimally coupled theories are worked out and relevant cosmological solutions in Einstein frame and Jordan frame are discussed showing that also the interpretation of cosmological observations can drastically change depending on the adopted frame.     

On the Geodesic Hypothesis in General Relativity

In this paper, we give a rigorous derivation of Einstein’s geodesic hypothesis in general relativity. We use small material bodies ${\phi^\epsilon}$ governed by the nonlinear Klein–Gordon equations to approximate the test particle. Given a vacuum spacetime ${([0, T]\times\mathbb{R}^3, h)}$ , we consider the initial value problem for the Einstein-scalar field system. For all sufficiently small ε and δ ≤ ε ^q , q > 1, where δ, ε are the amplitude and size of the particle, we show the existence of the solution ${([0, T]\times\mathbb{R}^3, g, \phi^\epsilon)}$ to the Einstein-scalar field system with the property that the energy of the particle ${\phi^\epsilon}$ is concentrated along a timelike geodesic. Moreover, the gravitational field produced by ${\phi^\epsilon}$ is negligibly small in C ¹, that is, the spacetime metric g is C ¹ close to the given vacuum metric h. These results generalize those obtained by Stuart in (Ann Sci École Norm Sup (4) 37(2):312–362, 2004, J Math Pures Appl (9) 83(5):541–587, 2004).     

On the Foundation of the Principle of Relativity

The relation of the special and the general principle of relativity to the principle of covariance, the principle of equivalence and Mach's principle, is discussed. In particular, the connection between Lorentz covariance and the special principle of relativity is illustrated by giving Lorentz covariant formulations of laws that violate the special principle of relativity: Ohm's law and what we call Aristotle's first and second laws. An Aristotelian universe in which all motion is relative to absolute space is considered. The first law: a free particle is at rest. The second law: force is proportional to velocity. Ohm's law: the current density is proportional to the electrical field strength. Neither of these laws fulfills the principle of relativity. The examples illustrate, in the context of Lorentz covariance and special relativity, Kretschmann's critique of founding Einstein's general principle of relativity on the principle of general covariance. A modification of the principle of covariance is suggested, which may serve as a restricted criterium for a physical law to satisfy Einstein's general principle of relativity. Other objections that have been raised to the validity of Einstein's general principle of relativity are based upon the preferred state of inertial frames in the general, as well as in the special theory, the existence of tidal effects in true gravitational fields, doubts as to the validity of Mach's principle, whether electromagnetic phenomena obey the principle, and, finally, the anisotropy of the cosmic background radiation. These objections are reviewed and discussed.