Le Bellac Research Articles

Forty years of Galilean Electromagnetism (1973–2013)

We review Galilean Electromagnetism since the 1973 seminal paper of Jean-Marc Levy-Leblond and Michel Le Bellac and we explain for the first time all the historical experiments of Rowland, Vasilescu Karpen, Roentgen, Eichenwald, Wilson, Wilson and Wilson, which were previously interpreted in a Special Relativistic framework by showing the uselessness of the latter for setups involving slow motions of a part of the apparatus. Galilean Electromagnetism is not an alternative to Special Relavity but is precisely its low-velocity limit in Classical Electromagnetism.

Non-relativistic limits of Maxwell’s equations

In 1973, Le Bellac and Lévy-Leblond (Nuovo Cimento B 14 217–234) discovered that Maxwell’s equations possess two non-relativistic Galilei-covariant limits, corresponding to |E| ≫ c|B| (electric limit) or |E| ≪ c|B| (magnetic limit). Here, we provide a systematic, yet simple, derivation of these two limits based on a dimensionless form of Maxwell’s equations and an expansion of the electric and magnetic fields in a power series of some small parameters. Using this procedure, all previously known results are recovered in a natural and unambiguous way. Some further extensions are also proposed.

On some applications of Galilean electrodynamics of moving bodies

We discuss the seminal article by Le Bellac and Lévy-Leblond in which they identified two Galilean limits (called “electric” and “magnetic” limits) of electromagnetism and their implications. Recent work has shed new light on the choice of gauge conditions in classical electromagnetism. We show that the recourse to potentials is compelling in order to demonstrate the existence of both (electric and magnetic) limits. We revisit some nonrelativistic systems and related experiments, in the light of these limits, in quantum mechanics, superconductivity, and the electrodynamics of continuous media. Much of the current technology where waves are not taken into account can be described in a coherent fashion by the two limits of Galilean electromagnetism instead of an inconsistent mixture of these limits.

On the electrodynamics of moving bodies at low velocities

We discuss an article by Le Bellac and Lévy-Leblond in which they have identified two Galilean limits of electromagnetism (1973 Nuovo Cimento B 14 217–33). We use their results to point out some confusion in the literature, and in the teaching of special relativity and electromagnetism. For instance, it is not widely recognized that there exist two well-defined non-relativistic limits, so that researchers and teachers are likely to utilize an incoherent mixture of both. Recent works have shed new light on the choice of gauge conditions in classical electromagnetism. We retrieve the results of Le Bellac and Lévy-Leblond first by examining orders of magnitudes and then with a Lorentz-like manifestly covariant approach to Galilean covariance based on a five-dimensional Minkowski manifold. We emphasize the Riemann–Lorenz approach based on the vector and scalar potentials as opposed to the Heaviside–Hertz formulation in terms of electromagnetic fields.

Nonrelativistic Wave Equations with Gauge Fields

We illustrate a metric formulation of Galilean invariance by constructing wave equations with gauge fields. It consists of expressing nonrelativistic equations in a covariant form, but with a five-dimensional Riemannian manifold. First we use the tensorial expressions of electromagnetism to obtain the two Galilean limits of electromagnetism found previously by Le Bellac and Levy-Leblond. Then we examine the nonrelativistic version of the linear Dirac wave equation. With an Abelian gauge field we find, in a weak field approximation, the Pauli equation as well as the spin—orbit interaction and a part reminiscent of the Darwin term. We also propose a generalized model involving the interaction of the Dirac field with a non-Abelian gauge field; the SU(2) Hamiltonian is given as an example.

The Galilean covariance of quantum mechanics in the case of external fields

Textbook treatments of the Galilean covariance of the time-dependent Schrödinger equation for a spinless particle seem invariably to cover the case of a free particle or one in the presence of a scalar potential. The principal objective of this paper is to examine the situation in the case of arbitrary forces, including the velocity-dependent variety resulting from a vector potential. To this end, we revisit the 1964 theorem of Jauch which purports to determine the most general form of the Hamiltonian consistent with “Galilean-invariance,” and argue that the proof is less than compelling. We then show systematically that the Schrödinger equation in the case of a Jauch-type Hamiltonian is Galilean covariant, so long as the vector and scalar potentials transform in a certain way. These transformations, which to our knowledge have appeared very rarely in the literature on quantum mechanics, correspond in the case of electrodynamical forces to the “magnetic” nonrelativistic limit of Maxwell’s equations in the sense of Le Bellac and Lévy-Leblond (1973). Finally, this Galilean covariant theory sheds light on Feynman’s “proof” of Maxwell’s equations, as reported by Dyson in 1990.

Comment on “Infrared and pinching singularities in out of equilibrium QCD plasmas”

Analyzing the dilepton production from out of equilibrium quark-gluon plasma, Le Bellac and Mabilat have recently pointed out that, in the reaction rate, the cancellation of mass (collinear) singularities takes place only in physical gauges, and not in covariant gauges. They then have estimated the contribution involving pinching singularities. After giving a general argument for the gauge independence of the production rate, we explicitly confirm the gauge independence of the mass-singular part. The contribution involving pinching singularities develops mass singularities, which is also gauge dependent. This “additional” contribution to the singular part is responsible for the gauge independence of the “total” singular part.

Comment on “Infrared and pinching singularities in out of equilibrium QCD plasmas”

Analyzing the dilepton production from out of equilibrium quark-gluon plasma, Le Bellac and Mabilat have recently pointed out that, in the reaction rate, the cancellation of mass (collinear) singularities takes place only in physical gauges, and not in covariant gauges. They then have estimated the contribution involving pinching singularities. After giving a general argument for the gauge independence of the production rate, we explicitly confirm the gauge independence of the mass-singular part. The contribution involving pinching singularities develops mass singularities, which is also gauge dependent. This `` additional'' contribution to the singular part is responsible for the gauge independence of the `` total'' singular part. We give a sufficient condition, under which cancellation of mass singularities takes place.

Celestial mechanics, conformal structures, and gravitational waves.

The equations of motion for $N$ non-relativistic particles attracting according to Newton's law are shown to correspond to the equations for null geodesics in a $(3N+2)$-dimensional Lorentzian, Ricci-flat, spacetime with a covariantly constant null vector. Such a spacetime admits a Bargmann structure and corresponds physically to a generalized pp-wave. Bargmann electromagnetism in five dimensions comprises the two Galilean electro-magnetic theories (Le Bellac and L\'evy-Leblond). At the quantum level, the $N$-body Schr\"odinger equation retains the form of a massless wave equation. We exploit the conformal symmetries of such spacetimes to discuss some properties of the Newtonian $N$-body problem: homographic solutions, the virial theorem, Kepler's third law, the Lagrange-Laplace-Runge-Lenz vector arising from three conformal Killing 2-tensors, and motions under inverse square law forces with a gravitational constant $G(t)$ varying inversely as time (Dirac). The latter problem is reduced to one with time independent forces for a rescaled position vector and a new time variable; this transformation (Vinti and Lynden-Bell) arises from a conformal transformation preserving the Ricci-flatness (Brinkmann). A Ricci-flat metric representing $N$ non-relativistic gravitational dyons is also pointed out. Our results for general time-dependent $G(t)$ are applicable to the motion of point particles in an expanding universe. Finally we extend these results to the quantum regime.

Pomeron cuts and inclusive reactions

The contributions of Pomeron cuts to inclusive cross sections in the central region are studied within the framework of the Reggeon calculus. The two-particle correlation function contains a term independent of the rapidities of the observed particles and only weakly dependent on the total rapidity $Y$. The integrated correlation functions ${f}_{n}$ grow as ${Y}^{n\ensuremath{-}1}$ and satisfy Le Bellac's inequality automatically. The multiplicity distribution contains multiple peaks at very high energy.

On correlations in inclusive reactions

We consider the nature of correlations in inclusive hadronic reactions. Assuming the existence of hadronic scaling, we study the constraints imposed by energy-momentum conservation, and deduce that they seem not to lead to correlations of long range in rapidity. We show that short-range correlations must obey various restrictions in order to guarantee consistency with conservation laws. Following Le Bellac and Wilson, we investigate the nature of correlations when diffractive processes are present, and show that these processes lead to long-range correlations. In Mueller’s Regge analysis of inclusive reactions these would correspond to failures of the Pomeranchuk singularity to factorize. Lower bounds are established for the average correlation between two particles of similar type when there is nonshrinking diffraction. The existence of correlations between particles of different types cannot be established unless specific additional assumptions are made. The implications of our results for experiment are discussed.

Dispersive Approach to Current Algebra

The current-algebra scheme, not including equal-time commutators of two space components, is formulated entirely in terms of covariant dispersion relations (i.e., over the laboratory energy for fixed mass) where the number of subtractions is chosen according to the usual Regge picture. To do this, one expresses the equal-time commutators and retarded commutators directly in terms of the covariant dispersion integrals, using an identity between covariant and noncovariant dispersion integrals of Lorentz-invariant distributions which is proved here, by means of the Jost-Lehmann representation, from the locality of the currents. This last result is a generalization of previous works of Schroer and Stichel and of Le Bellac and the author. In the scheme discussed here, the equal-time commutators and retarded commutators are rigorously defined as a consequence of the physical high-energy behavior, contrary to the usual approach where they are only formally written as product of the commutator with $\ensuremath{\delta}({x}_{0})$ and $\ensuremath{\theta}({x}_{0})$, respectively. In fact, we show that the properties usually derived formally (as, e.g., by partial integration) are rigorously true in our scheme. The quantities which in the usual approach are not well defined, and/or are model-dependent---such as the Schwinger term, the seagull term, and the equal-time commutator of the current and the divergence---are given in this approach by the subtraction constants of the dispersion relations introduced. They are shown to exhibit the properties which usually are more or less assumed, or only obtained in models; e.g., we prove that the divergence of the seagull term is equal to the Schwinger term. Only matrix elements averaged over spin and/or relative momenta, with the vacuum matrix element subtracted, are considered. No explicit value of the equal-time commutator is assumed at the beginning, so as to clearly show the respective roles of the internal symmetry group and of the analyticity and high-energy behavior. This paper also completes a previous work of the author on the infinite-momentum limit based on the same approach.

Factorisation, kinematic factors of the Regge residue function and conspiracy

When helicity amplitudes are Reggeised, the kinematic singularities and zeros of the helicity amplitudes are not generally compatible with the factorisation theorem for the Regge residue function. We find the simplest (but not unique) class of solutions for the extra zeros which must be added to give factorisation. The residue function for a t-channel helicity amplitude may be written βΛμ( t= = β Λμ(t), where WΛμ( t) is a kinematic function. To satisfy the factorisation theorem we must be able to write WΛμ( t)= VΛ( t) Vμ( t). Our main result is that near t = 0 the vertex function VΛ( t) is proportional to t q ̂ where, for unequal masses, q ̂ = −1 2 α+ 1 2 |M−|Λ|| and, for equal masses (each with spin s), q ̂ = 1 4 [1–(−1) Λ+M] if M ⩾ 2s and q ̂ = 1 4 [1–([t-1) Λ+2s+ 1 2 if M ≤ 2s , where M is a non-negative parameter (integer for boson trajectories, half-integer for fermion trajectories), identified with a quantum number characterising the Toller pole from which the Regge pole at complex angular momentum α arises. The general results for VΛ( t) at thresholds and pseudothresholds, as well as at t = 0, are given. As a subsidiary result we show that the analytic structure of helicity amplitudes at thresholds, and at pseudothresholds for unequal masses, follow from non-relativistic arguments without study of the Trueman and Wick crossing relations. The results are applied, as an example, to the pion conspiracy problem, and some earlier results of Le Bellac are rederived.