Manifest Lorentz Covariance Research Articles

A Lorentz-covariant interacting electron–photon system in one space dimension

A Lorenz-covariant system of wave equations is formulated for a quantum-mechanical two-body system in one space dimension, comprised of one electron and one photon. Manifest Lorentz covariance is achieved using Dirac's formalism of multi-time wave functions, i.e., wave functions $\Psi(\mathbf{x}_{{ph}},\mathbf{x}_{{el}})$ where $\mathbf{x}_{{el}},\mathbf{x}_{{ph}}$ are the generic spacetime events of the electron and photon, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifold $\{\mathbf{x}_{{el}}=\mathbf{x}_{{ph}}\}$, compatible with particle current conservation. The corresponding initial-boundary-value problem is proved to be well-posed. Electron and photon trajectories are shown to exist globally in a Hypersurface Bohm--Dirac theory, for typical particle initial conditions. Also presented are the results of some numerical experiments which illustrate Compton scattering as well as a new phenomenon: photon capture and release by the electron.

Canonical analysis with no second-class constraints of BF gravity with Immirzi parameter

In this paper we revisit the canonical analysis of $BF$ gravity with the Immirzi parameter and a cosmological constant. By examining the constraint on the $B$ field, we realize that the analysis can be performed in a Lorentz-covariant fashion while utterly avoiding the introduction of second-class constraints during the whole process. Finally, we make contact with the description of the phase space of first-order general relativity in terms of canonical variables with manifest Lorentz covariance subject to first-class constraints only recently introduced.

Covariant Hamiltonian for gravity coupled top-forms

We review the covariant canonical formalism initiated by D'Adda, Nelson and Regge in 1985, and extend it to include a definition of form-Poisson brackets (FPB) for geometric theories coupled to $p$-forms, gauging free differential algebras. The form-Legendre transformation and the form-Hamilton equations are derived from a $d$-form Lagrangian with $p$-form dynamical fields $\phi$. Momenta are defined as derivatives of the Lagrangian with respect to the "velocities" $d\phi$ and no preferred time direction is used. Action invariance under infinitesimal form-canonical transformations can be studied in this framework, and a generalized Noether theorem is derived, both for global and local symmetries. We apply the formalism to vielbein gravity in $d=3$ and $d=4$. In the $d=3$ theory we can define form-Dirac brackets, and use an algorithmic procedure to construct the canonical generators for local Lorentz rotations and diffeomorphisms. In $d=4$ the canonical analysis is carried out using FPB, since the definition of form-Dirac brackets is problematic. Lorentz generators are constructed, while diffeomorphisms are generated by the Lie derivative. A "doubly covariant" hamiltonian formalism is presented, allowing to maintain manifest Lorentz covariance at every stage of the Legendre transformation. The idea is to take curvatures as "velocities" in the definition of momenta.

Relativistic Classical Mechanics and Electrodynamics

This book presents classical relativistic mechanics and electrodynamics in the Feynman-Stueckelberg event-oriented framework formalized by Horwitz and Piron. The full apparatus of classical analytical mechanics is generalized to relativistic form by replacing Galilean covariance with manifest Lorentz covariance and introducing a coordinate-independent parameter ���� to play the role of Newton's universal and monotonically advancing time. Fundamental physics is described by the ����-evolution of a system point through an unconstrained 8D phase space, with mass a dynamical quantity conserved under particular interactions. Classical gauge invariance leads to an electrodynamics derived from five ����-dependent potentials described by 5D pre-Maxwell field equations. Events trace out worldlines as ���� advances monotonically, inducing pre-Maxwell fields by their motions, and moving under the influence of these fields. The dynamics are governed canonically by a scalar Hamiltonian that generates evolution of a 4D block universe defined at ���� to an infinitesimally close 4D block universe defined at ����+��������. This electrodynamics, and its extension to curved space and non-Abelian gauge symmetry, is well-posed and integrable, providing a clear resolution to grandfather paradoxes. Examples include classical Coulomb scattering, electrostatics, plane waves, radiation from a simple antenna, classical pair production, classical CPT, and dynamical solutions in weak field gravitation. This classical framework will be of interest to workers in quantum theory and general relativity, as well as those interested in the classical foundations of gauge theory.

Quantum tomography for collider physics: illustrations with lepton-pair production

Quantum tomography is a method to experimentally extract all that is observable about a quantum mechanical system. We introduce quantum tomography to collider physics with the illustration of the angular distribution of lepton pairs. The tomographic method bypasses much of the field-theoretic formalism to concentrate on what can be observed with experimental data. We provide a practical, experimentally driven guide to model-independent analysis using density matrices at every step. Comparison with traditional methods of analyzing angular correlations of inclusive reactions finds many advantages in the tomographic method, which include manifest Lorentz covariance, direct incorporation of positivity constraints, exhaustively complete polarization information, and new invariants free from frame conventions. For example, experimental data can determine the entanglement entropy of the production process. We give reproducible numerical examples and provide a supplemental standalone computer code that implements the procedure. We also highlight a property of complex positivity that guarantees in a least-squares type fit that a local minimum of a chi ^{2} statistic will be a global minimum: There are no isolated local minima. This property with an automated implementation of positivity promises to mitigate issues relating to multiple minima and convention dependence that have been problematic in previous work on angular distributions.

Berry Curvature and Four-Dimensional Monopoles in the Relativistic Chiral Kinetic Equation

We derive a relativistic chiral kinetic equation with manifest Lorentz covariance from Wigner functions of spin-1/2 massless fermions in a constant background electromagnetic field. It contains vorticity terms and a four-dimensional Euclidean Berry monopole which gives an axial anomaly. By integrating out the zeroth component of the 4-momentum p, we reproduce the previous three-dimensional results derived from the Hamiltonian approach, together with the newly derived vorticity terms. The phase space continuity equation has an anomalous source term proportional to the product of electric and magnetic fields (FσρF[over ˜]σρ∼EσBσ). This provides a unified interpretation of the chiral magnetic and vortical effects, chiral anomaly, Berry curvature, and the Berry monopole in the framework of Wigner functions.

Spontaneously broken Lorentz symmetry for Hamiltonian gravity

In Ashtekar's Hamiltonian formulation of general relativity, and in loop quantum gravity, Lorentz covariance is a subtle issue that has been strongly debated. Maintaining manifest Lorentz covariance seems to require introducing either complex-valued fields, presenting a significant obstacle to quantization, or additional (usually second class) constraints whose solution renders the resulting phase space variables harder to interpret in a spacetime picture. After reviewing the sources of difficulty, we present a Lorentz covariant, real formulation in which second class constraints never arise. Rather than a foliation of spacetime, we use a gauge field y, interpreted as a field of observers, to break the SO(3,1) symmetry down to a subgroup SO(3)_y. This symmetry breaking plays a role analogous to that in MacDowell-Mansouri gravity, which is based on Cartan geometry, leading us to a picture of gravity as 'Cartan geometrodynamics.' We study both Lorentz gauge transformations and transformations of the observer field to show that the apparent breaking of SO(3,1) to SO(3) is not in conflict with Lorentz covariance.

Chiral bosons in noncommutative spacetime

Underlying a general noncommutative algebra with both noncommutative coordinates and noncommutative momenta in a (1+1)-dimensional spacetime, a chiral boson Lagrangian with manifest Lorentz covariance is proposed by linearly imposing a generalized self-duality condition on a noncommutative generalization of massless real scalar fields. A significant property uncovered for noncommutative chiral bosons is that the left- and right-moving chiral scalars cannot be distinguished from each other, which originates from the noncommutativity of coordinates and momenta. An interesting result is that Dirac's method can be consistently applied to the constrained system whose Lagrangian explicitly contains space and time. The self-duality of the noncommutative chiral boson action does not exist.

A new first class algebra, homological perturbation and extension of pure spinor formalism for superstring

Based on a novel first class algebra, we develop an extension of the pure spinor (PS) formalism of Berkovits, in which the PS constraints are removed. By using the homological perturbation theory in an essential way, the BRST-like charge $Q$ of the conventional PS formalism is promoted to a bona fide nilpotent charge $\hat{Q}$, the cohomology of which is equivalent to the constrained cohomology of $Q$. This construction requires only a minimum number (five) of additional fermionic ghost-antighost pairs and the vertex operators for the massless modes of open string are obtained in a systematic way. Furthermore, we present a simple composite "$b$-ghost" field $B(z)$ which realizes the important relation $T(z) = \{\hat{Q}, B(z)\} $, with $T(z)$ the Virasoro operator, and apply it to facilitate the construction of the integrated vertex. The present formalism utilizes U(5) parametrization and the manifest Lorentz covariance is yet to be achieved.

REDUCED PHASE SPACE QUANTIZATION OF MASSIVE VECTOR THEORY

We quantize massive vector theory in such a way that it has a well-defined massless limit. In contrast to the approach by Stückelberg where ghost fields are introduced to maintain manifest Lorentz covariance, we use reduced phase space quantization with nonlocal dynamical variables which in the massless limit smoothly turn into the photons, and check explicitly that the Poincaré algebra is fulfilled. In contrast to conventional covariant quantization our approach leads to a propagator which has no singularity in the massless limit and is well behaved for large momenta. For massive QED, where the vector field is coupled to a conserved fermion current, the quantum theory of the nonlocal vector fields is shown to be equivalent to that of the standard local vector fields. An inequivalent theory, however, is obtained when the reduced nonlocal massive vector field is coupled to a nonconserved classical current.

Supergravity in 10 + 2 dimensions as consistent background for superstring

We present a consistent theory of N=1 supergravity in twelve-dimensions with the signature (10,2). Even though the formulation uses two null vectors violating the manifest Lorentz covariance, all the superspace Bianchi identities are satisfied. After a simple dimensional reduction to ten-dimensions, this theory reproduces the N=1 supergravity in ten-dimensions, supporting the consistency of the system. We also show that our supergravity can be the consistent backgrounds for heterotic or type-I superstring in Green-Schwarz formulation, by confirming the kappa-invariance of the total action. This theory is supposed to be the purely N=1 supergravity sector for the field theory limit of the recently predicted F-theory in twelve-dimensions.

Solving the constraints in off-shell linearized N = 8 supergravity

The linearized N = 8 supergravity obtained from dimensional reduction by Legendre transformation requires the solution of differential constraints for off-shell closure of the algebra. We solve these constraints, thereby arriving at an unconstrained theory for which, however, either the spin content is not preserved, or manifest Lorentz covariance is lost.

Extrapolation to the continuum limit from the cluster expansion on the lattice

In this paper the problem of obtaining the continuum-limit theory (which has manifest Lorentz covariance) from the lattice approximation for a weak-coupling as well as for a strong-coupling regime is studied. We use the lattice cluster expansion of the kinetic term around the static ultralocal approximation and we study a theory without infinite renormalization problems,i.e. the anharmonic oscillator. We find that, in the weak-coupling case, the lattice approximation gives us the correct continuum limit by simply taking the limit for zero lattice spacing; in the strong-coupling regime one needs a suitable analytic continuation.

A finite formulation of quantum electrodynamics

A recent finite formulation of field theory is applied to quantum electrodynamics. The basic assumptions of the formulation are Bogoliubov causality and TCP invariance of the S-operator; interpolating fields are never used.Ultraviolet divergences never appear in the calculations, and there is manifest Lorentz covariance at every stage.In particular, the calculation of the proper third-order vertex is carried through.

A finite formulation of perturbative field theory

A formalism is set up for obtaining the perturbative S-operator for an arbitrary field theory, "renormalizable" or "nonrenormalizable". Ultraviolet divergences never appear. The basic assumptions of the theory are Bogoliubov causality and TCP invariance of the S-operator. Interpolating fields are never used.In this formalism, calculations are based on a uniqueness theorem, rather than on an explicit prescription for the coefficients in the normal expansion of the S-operator. However, there is manifest Lorentz covariance at each stage of the calculations. Boundary conditions are considered in some detail.Example calculations using the present formalism are carried out for the [Formula: see text] theory and for the Fermi interaction involving the muon, electron, and neutrinos. Third- and even fourth-order perturbative results can be obtained without great difficulty, at least for cubic interactions.

Spin and Statistics with an Electromagnetic Field

Because of the impossibility of simultaneously satisfying the requirements of manifest Lorentz covariance and positive-definite (finite) norm in the Hilbert space, no simple proof of the connection between spin and statistics with an electromagnetic field has been given. This note is to point out that it is indeed not necessary to have manifest Lorentz covariance in the full 3 + 1 space to show such a connection. Using the axial gauge A3 = 0, we have succeeded in constructing a simple straightforward proof.

Quantization of the gravitational field

Present quantum theories of the gravitational field generally work in ‘flat space’. The original attempt at quantization was made by Gupta (1952) and carried out by him to first order. He started with the Lagrangian of the classical theory and applied the normal methods of quantization to it, treating the g ab as ordinary variables which have no connexion with the metric. Owing to the ambiguities in the ordering of the factors, such a programme cannot be carried through exactly. However, it may be that one can write down a Lagrangian to any order in the gravitational constant and obtain consistent equations of motion up to that order. Quantization in flat space can only be regarded as a provisional solution of the problem for several reasons. One would like to be able to formulate the equations of a theory exactly, even though approximations have to be made in their solution. This cannot be done or, at any rate, has not yet been done, in the flat space gravitation theory. Also, one has to introduce an indefinite metric and unphysical states, just as in quantizing electrodynamics with the Lorentz gauge. Both these disadvantages would be overcome if Arnowitt, Deser & Misner (1960) succeed in their programme of finding canonical variables for the quantum theory. At the moment a solution does not appear to be in sight, and in any case it would yield a theory which completely lacked manifest Lorentz co-variance.