Relativistic Covariance Research Articles

Covariant version of the Pauli Hamiltonian, spin-induced noncommutativity, Thomas precession, and the precession of spin

We show that there is a manifestly covariant version of the Pauli Hamiltonian with equations of motion quadratic on spin and field strength. Relativistic covariance inevitably leads to noncommutative positions: classical brackets of the position variables are proportional to the spin. It is the spin-induced noncommutativity that is responsible for transforming the covariant Hamiltonian into the Pauli Hamiltonian, without any appeal to the Thomas precession formula. The Pauli theory can be thought to be $1/c^2$ approximation of the covariant theory written in special variables. These observations clarify the long standing question on the discrepancy between the covariant and Pauli Hamiltonians. We also discuss the transformational properties of spin axis in the passage from laboratory to comoving and instantaneous frames, and reveal the role of Thomas spin-vector in the covariant scheme.

Quantum Reality, Perspectivalism and Covariance

Paul Busch has emphasized on various occasions the importance for physics of going beyond a merely instrumentalist view of quantum mechanics. Even if we cannot be sure that any particular realist interpretation describes the world as it actually is, the investigation of possible realist interpretations helps us to develop new physical ideas and better intuitions about the nature of physical objects at the micro level. In this spirit, Paul Busch himself pioneered the concept of “unsharp quantum reality”, according to which there is an objective non-classical indeterminacy—a lack of sharpness—in the properties of individual quantum systems. We concur with Busch’s motivation for investigating realist interpretations of quantum mechanics and with his willingness to move away from classical intuitions. In this article we try to take some further steps on this road. In particular, we pay attention to a number of prima facie implausible and counter-intuitive aspects of realist interpretations of unitary quantum mechanics. We shall argue that from a realist viewpoint, quantum contextuality naturally leads to “perspectivalism” with respect to properties of spatially extended quantum systems, and that this perspectivalism is important for making relativistic covariance possible.

Basic observables for the accelerated electron and its field

We revisit in the framework of the classical theory the problem of the accelerated motion of an electron, taking into account the effect of the radiation emission. We present results for the momentum and energy of the electromagnetic field of an accelerated electron for a spatial region excluding a vicinity of the electron and a procedure to compensate their singularities in the limit of the point electron. From them we infer expressions for the observables momentum and energy of the electron. They lead, together with those corresponding to the emitted radiation, to an equation of motion of the electron that coincides, in the case of an external electromagnetic field, with the Lorentz-Dirac equation. Based on the results for the linear momentum and using the same calculation method, we obtain the equations corresponding to the angular momentum of the electron and its field. While the formalism used is not wholly manifestly covariant, arguments based on relativistic covariance are invoked and used at appropriate places, where they play an essential role.

Relativistic covariance of Ohm's law

The derivation of Lorentz-covariant generalizations of Ohm's law has been a long-term issue in theoretical physics with deep implications for the study of relativistic effects in optical and atomic physics. In this article, we propose an alternative route to this problem, which is motivated by the tremendous progress in first-principles materials physics in general and ab initio electronic structure theory in particular. We start from the most general, Lorentz-covariant first-order response law, which is written in terms of the fundamental response tensor [Formula: see text] relating induced four-currents to external four-potentials. By showing the equivalence of this description to Ohm's law, we prove the validity of Ohm's law in every inertial frame. We further use the universal relation between [Formula: see text] and the microscopic conductivity tensor [Formula: see text] to derive a fully relativistic transformation law for the latter, which includes all effects of anisotropy and relativistic retardation. In the special case of a constant, scalar conductivity, this transformation law can be used to rederive a standard textbook generalization of Ohm's law.

Exact quantisation of the relativistic Hopfield model

We investigate the quantisation in the Heisenberg representation of a relativistically covariant version of the Hopfield model for dielectric media, which entails the interaction of the quantum electromagnetic field with the matter dipole fields, represented by a mesoscopic polarisation field. A full quantisation of the model is provided in a covariant gauge, with the aim of maintaining explicit relativistic covariance. Breaking of the Lorentz invariance due to the intrinsic presence in the model of a preferred reference frame is also taken into account. Relativistic covariance forces us to deal with the unphysical (scalar and longitudinal) components of the fields, furthermore it introduces, in a more tricky form, the well-known dipole ghost of standard QED in a covariant gauge. In order to correctly dispose of this contribution, we implement a generalised Lautrup trick. Furthermore, causality and the relation of the model with the Wightman axioms are also discussed.

Electromagnetic gauge-freedom and work

We argue that the definition of the thermodynamic work done on a charged particle by a time-dependent electromagnetic field (EMF) is an open problem, because the particle’s Hamiltonian is not gauge-invariant. The solution of this problem demands accounting for the source of the field. Hence we focus on the work done by a heavy body (source) on a lighter particle when the interaction between them is electromagnetic and relativistic. The work can be defined via the gauge-invariant kinetic energy of the source. We uncover a formulation of the first law (or the generalized work-energy theorem) which is derived from relativistic dynamics, has definite validity conditions, and relates the work to the particle’s Hamiltonian in the Lorenz gauge. Thereby the thermodynamic work also relates to the mechanic work done by the Lorentz force acting on the source. The formulation of the first law is based on a specific separation of the overall energy into those of the source, particle and EMF. This separation is deduced from a consistent energy–momentum tensor. Hence it holds relativistic covariance and causality.

Symmetric energy-momentum tensor: The Abraham form and the explicitly covariant formula

We compare the known in literature, explicitly covariant 4-dimensional formula for the symmetric energy-momentum tensor of electromagnetic field in a medium and the energy-momentum tensor derived by Abraham in the 3-dimensional vector form. It is shown that these two objects coincide only on the physical configuration space Γ¯, formed by the field vectors and the velocity of the medium, which satisfy the Minkowski constitutive relations. It should be emphasized that the 3-dimensional vector formulae for the components of the energy-momentum tensor were obtained by Abraham only on Γ¯, and the task of their extension to the whole unconditional configuration space Γ was not posed. In order to accomplish the comparison noted above, we derive the covariant formula a new by another method, namely, by generalizing the Abraham reasoning. The comparison conducted enables one to treat the explicitly covariant formula as a unique consistent extension of the Abraham formulae to the whole configuration space Γ. Thus the question concerning the relativistic covariance of the original 3-dimensional Abraham formulae defined on Γ¯ is solved positively. We discuss in detail the relativistic covariance of the 3-dimensional vector formulae for individual components of the 4-dimensional tensors in electrodynamics which is manifested in the form-invariance of these formulae under Lorentz transformations.

Terminality Implies No-signalling ...and Much More Than That

A ‘process theory’ is any theory of systems and processes which admits sequential and parallel composition. ‘Terminality’ unifies normalisation for pure states, trace-preservation for positive superoperators, and adding up to identity of positive operators in quantum theory, and it moreover generalises this to arbitrary process theories. We show that terminality and no-signalling coincide in any process theory, provided one makes causal structure explicit. In fact, making causal structure explicit is necessary to even make sense of no-signalling in process theories. We conclude that because of its much simpler mathematical form, terminality should be taken to be a more fundamental notion than no-signalling. We also point out that terminality imposes many other nice features upon process theories, for example, it even imposes relativistic covariance.

The nonlinear dirac equation in Bose–Einstein condensates: vortex solutions and spectra in a weak harmonic trap

We analyze the vortex solution space of the -dimensional nonlinear Dirac equation for bosons in a honeycomb optical lattice at length scales much larger than the lattice spacing. Dirac point relativistic covariance combined with s-wave scattering for bosons leads to a large number of vortex solutions characterized by different functional forms for the internal spin and overall phase of the order parameter. We present a detailed derivation of these solutions which include skyrmions, half-quantum vortices, Mermin–Ho and Anderson–Toulouse vortices for vortex winding For we obtain topological as well as non-topological solutions defined by the asymptotic radial dependence. For arbitrary values of ℓ the non-topological solutions include bright ring-vortices which explicitly demonstrate the confining effects of the Dirac operator. We arrive at solutions through an asymptotic Bessel series, algebraic closed-forms, and using standard numerical shooting methods. By including a harmonic potential to simulate a finite trap we compute the discrete spectra associated with radially quantized modes. We demonstrate the continuous spectral mapping between the vortex and free particle limits for all of our solutions.

Classical gauged massless Rarita-Schwinger fields

We show that, in contrast to known results in the massive case, a minimally gauged massless Rarita-Schwinger field yields consistent classical and quantum theories, with a generalized fermionic gauge invariance. To simplify the algebra, we study a two-component left chiral reduction of the massless theory. We formulate the classical theory in both Lagrangian and Hamiltonian form for a general non-Abelian gauging, and analyze the constraints and the Rarita-Schwinger gauge invariance of the action. An explicit wave front calculation for Abelian gauge fields shows that wave-like modes do not propagate with superluminal velocities. The quantized case is studied in gauge covariant radiation gauge and $\Psi_0=0$ gauge for the Rarita-Schwinger field, by both functional integral and Dirac bracket methods. In $\Psi_0=0$ gauge, the constraints have the form needed to apply the Faddeev-Popov method for deriving a functional integral. The Dirac bracket approach in $\Psi_0=0$ gauge yields consistent Hamilton equations of motion, and in covariant radiation gauge leads to anticommutation relations with the correct positivity properties. We discuss relativistic covariance of the anticommutation relations, and of Rarita-Schwinger scattering from an Abelian potential. We note that fermionic gauge transformations are a canonical transformation, but further details of the transformation between different fermionic gauges are left as an open problem.

Free Quantum Field Theory from Quantum Cellular Automata

After leading to a new axiomatic derivation of quantum theory, the new informational paradigm is entering the domain of quantum field theory, suggesting a quantum automata framework that can be regarded as an extension of quantum field theory to including an hypothetical Planck scale, and with the usual quantum field theory recovered in the relativistic limit of small wave-vectors. Being derived from simple principles (linearity, unitarity, locality, homogeneity, isotropy, and minimality of dimension), the automata theory is quantum ab-initio, and does not assume Lorentz covariance and mechanical notions. Being discrete it can describe localized states and measurements (unmanageable by quantum field theory), solving all the issues plaguing field theory originated from the continuum. These features make the theory an ideal framework for quantum gravity, with relativistic covariance and space-time emergent solely from the interactions, and not assumed a priori. The paper presents a synthetic derivation of the automata theory, showing how from the principles lead to a description in terms of a quantum automaton over a Cayley graph of a group. Restricting to Abelian groups we show how the automata recover the Weyl, Dirac and Maxwell dynamics in the relativistic limit. We conclude with some new routes about the more general scenario of non-Abelian Cayley graphs.

Elementary particles, the concept of mass, and emergent spacetime

It is argued that the problem of space quantization should be considered in close connection with the problem of mass quantization. First, the nonlocality of quantum physics suggests that if spacetime emerges from the underlying quantum layer, this emergence should occur simultaneously at all distance and momentum scales, and not just at the Planck scale. Second, the spectrum of elementary particles provides us with a lot of important information, experimentally inaccessible at the Planck scale, that could be crucial in unravelling the mechanism of emergence. Accordingly, we start with a brief review of some fundamental issues appearing both in the spectroscopy of excited baryons and in connection with the concept of quark mass. It is pointed out that experiment suggests the inadequacy of the description of baryonic interior in terms of ordinary spacetime background. Thus, it is argued that one should be able to learn about the emergence of space from the studies of the quark/hadron transition. The problem of mass is then discussed from the point of view of nonrelativistic phase space and its Clifford algebra, which proved promising in the past. Connection with the Harari-Shupe explanation of the pattern of a single Standard Model generation is briefly reviewed and a proposal for the reintroduction of relativistic covariance into the phase-space scheme is presented.

Lagrangian for Frenkel electron and position’s non-commutativity due to spin

We construct a relativistic spinning-particle Lagrangian where spin is considered as a composite quantity constructed on the base of a non-Grassmann vector-like variable. The variational problem guarantees both a fixed value of the spin and the Frenkel condition on the spin-tensor. The Frenkel condition inevitably leads to relativistic corrections of the Poisson algebra of the position variables: their classical brackets became noncommutative. We construct the relativistic quantum mechanics in the canonical formalism (in the physical-time parametrization) and in the covariant formalism (in an arbitrary parametrization). We show how state vectors and operators of the covariant formulation can be used to compute the mean values of physical operators in the canonical formalism, thus proving its relativistic covariance. We establish relations between the Frenkel electron and positive-energy sector of the Dirac equation. Various candidates for the position and spin operators of an electron acquire clear meaning and interpretation in the Lagrangian model of the Frenkel electron. Our results argue in favor of Pryce’s (d)-type operators as the spin and position operators of Dirac theory. This implies that the effects of non-commutativity could be expected already at the Compton wavelength. We also present the manifestly covariant form of the spin and position operators of the Dirac equation.

Combined open shell Hartree-Fock theory of atomic and molecular systems constructed from noncharged scalar particles

By the use of condition of relativistic covariance, Dirac group theory, Clifford algebra and complete orthonormal sets of w ða � Þ -self-frictional exponential type orbitals (w ða � Þ -SFETOs) introduced by the author in standard convention, the Hartree-Fock (HF) theory is suggested for multideterminantal single configuration states with any number of open shells of atoms and molecules constructed from the Standard Model-Fermi (SM-F) particles with m 6 0;s ¼ 0; and e ¼ 0 defined in the Standard Model of particle physics. It is shown that the origin of stability of these systems is the quantum damping or self-frictional forces produced by the SM-F particle itself. As an application, we have presented the periodic table for the SM-F atomic elements using Pauli principle of spinless noncharged identical SM-F particles. V C 2014 Physics Essays Publication .( http://dx.doi.org/10.4006/0836-1398-27.3.351) Resume´: Par l'utilisation de la condition de covariance relativiste, theorie des groupes de Dirac, algebre de Clifford et des ensembles complets de orthonormeesw ða � Þ -auto-friction orbitales de type exponentielle (w ða � Þ

Where do bosons actually belong?

We explore a variety of reasons for considering su(1, 1) instead of the customary h(1) as the natural unifying frame for characterizing boson systems. Resorting to the Lie–Hopf structure of these algebras, that shows how the Bose–Einstein statistics for identical bosons is correctly given in the su(1, 1) framework, we prove that quantization of Maxwell’s equations leads to su(1, 1), relativistic covariance being naturally recognized as an internal symmetry of this dynamical algebra. Moreover su(1, 1) rather than h(1) coordinates are associated to circularly polarized electromagnetic waves. As for interacting bosons, the su(1, 1) formulation of the Jaynes–Cummings model is discussed, showing its advantages over h(1).

Probing the neutron skin thickness in collective modes of excitation

Nuclear collective motion provides valuable constraint on the size of neutron-skin thickness and the properties of nuclear matter symmetry energy. By employing relativistic nuclear energy density functional (RNEDF) and covariance analysis related to $\chi^2$ fitting of the model parameters, relevant observables are identified for dipole excitations, which strongly correlate with the neutron-skin thickness $(r_{np})$, symmetry energy at saturation density $(J)$ and slope of the symmetry energy $(L)$. Using the RNEDF framework and experimental data on pygmy dipole strength ($^{68}$Ni, $^{132}$Sn, $^{208}$Pb) and dipole polarizability ($^{208}$Pb), it is shown how the values of $J$, and $L$, and $r_{np}$ are constrained. The isotopic dependence of moments associated to dipole excitations in $^{116-136}$Sn shows that the low-energy dipole strength and polarizability in neutron-rich nuclei display strong sensitivity to the symmetry energy parameter $J$, more pronounced than in isotopes with moderate neutron-to-proton number ratios.

Causal Categories: Relativistically Interacting Processes

A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This paper is concerned with the encoding of a fixed causal structure within a symmetric monoidal category: causal dependencies will correspond to topological connectedness in the graphical language. We show that correlations, either classical or quantum, force terminality of the tensor unit. We also show that well-definedness of the concept of a global state forces the monoidal product to be only partially defined, which in turn results in a relativistic covariance theorem. Except for these assumptions, at no stage do we assume anything more than purely compositional symmetric-monoidal categorical structure. We cast these two structural results in terms of a mathematical entity, which we call a causal category. We provide methods of constructing causal categories, and we study the consequences of these methods for the general framework of categorical quantum mechanics.

Dynamic coarse-graining approach to quantum field theory

We build quantum field theory on the thermodynamic master equation for dissipative quantum systems. The vacuum is represented by a thermodynamic equilibrium state in the low-temperature limit. All regularization is consistently provided by a friction mechanism; with decreasing friction parameter, only degrees of freedom on shorter and shorter length scales are damped out of a quantum field theory. No divergent integrals need to be manipulated. Renormalization occurs as a tool to refine perturbation expansions, not to remove divergences. Relativistic covariance is recovered in the final results. We illustrate the proposed thermodynamic approach to quantum fields for the ${\ensuremath{\varphi}}^{4}$ theory by calculating the propagator and the $\ensuremath{\beta}$ function, and we offer some suggestions on its application to gauge theories.

MAKING NONLOCAL REALITY COMPATIBLE WITH RELATIVITY

It is often argued that hypothetic nonlocal reality responsible for nonlocal quantum correlations between entangled particles cannot be consistent with relativity. I review the most frequent arguments of that sort, explain how they can all be circumvented, and present an explicit Bohmian model of nonlocal reality (compatible with quantum phenomena) that fully obeys the principle of relativistic covariance and does not involve a preferred Lorentz frame.

Relativistic Covariance of the 2-Nucleon Contact Interactions

Relativistic covariance restricts the number of two-nucleon contact operators. We show that these constraints can be implemented starting from a complete set of relativistically invariant contact operators up to order Q in the parity-violating sector and order Q 2 in the parity-conserving one.