In a previous note [1] we showed that it is possible to derive and modify the relativistic Feynman propagator of a free (forceless) particle (fermion spin 12 , boson spin 0 and 1) on the basis of Lévy stochastic processes [2]. We adopt here the space-time relativistic approach of Feynman’s propagators (for bosons and fermions) instead of the canonical Lagrangian-Hamiltonian quantized field theory. The rationale for this choice is that for the development of our basic ideas the former alternative is better suited to exhibit the connection between the propagator of quantum mechanics and the underlying Lévy processes. More precisely, the relativistic Feynman propagators are here linked to a dynamical theory based on a particular Lévy process: a point, already discussed in a previous paper [3], which is here analyzed thoroughly with the purpose of deducing its consequences for the basic interactions among the fundamental constituents, namely quarks, leptons, gluons, photons and so on. A stochastic process X(t), t ≥ 0 on a probability space (Ω,F ,P) is a Lévy process if X(0) = 0, P-qo, if it has independent and stationary increments, and if it is stochastically continuous. To simplify the notation, in this introduction we will restrict ourselves only to one-dimensional processes, but the three-dimensional extension is straightforward and will be adopted in the subsequent sections. It is well known [2, 4, 5] that all its laws are infinitely divisible, but we will be mainly interested in the non stable (and in particular non Gaussian) case. In this case the characteristic functions of the process increments are [φ(u)] where φ is infinitely divisible, but not stable, and τ is a time scale. The transition probability density of a particle moving from the space-time point 1 to 2 then is

Starting from the relation between the kinetic energy of a free Levy-Schroedinger particle and the logarithmic characteristic of the underlying stochastic process, we show that it is possible to get a precise relation between renormalizable field theories and a specific Levy process. This subsequently leads to a particular cut-off in the perturbative diagrams and can produce a phenomenological mass spectrum that allows an interpretation of quarks and leptons distributed in the three families of the standard model.

In continuation of a previous paper a close connection between Feynman propagators and a particular L\'evy stochastic process is established. The approach can be easily applied to the Standard Model SU_C(3)xSU_L(2)xU(1) providing qualitative interesting results. Quantitative results, compatible with experimental data, are obtained in the case of neutrinos.

We introduce a modification in the relativistic hamiltonian in such a way that (1) the relativistic Schrödinger equations can always be based on an underlying Lévy process, (2) several families of particles with different rest masses can be selected, and finally (3) the corresponding Feynman diagrams are convergent when we have at least three different masses.

The general expression of the Stern-Gerlach force is deduced for a relativistic spin-1/2 particle which travels inside a time varying magnetic field. This result was obtained either by means of two Lorentz boosts or starting from Dirac's equation. Then, the utilization of this interaction for attaining the spin states separation is reconsidered in a new example using a new radio-frequency arrangement.

We analyze the extension of the well known relation between Brownian motion and the Schrödinger equation to the family of the Lévy processes. We consider a Lévy–Schrödinger equation where the usual kinetic energy operator–the Laplacian–is generalized by means of a selfadjoint, pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Lévy–Khintchin formula shows then how to write down this operator in an integro-differential form. When the underlying Lévy process is stable we recover as a particular case the fractional Schrödinger equation. A few examples are finally given and we find that there are physically relevant models–such as a form of the relativistic Schrödinger equation–that are in the domain of the non stable Lévy–Schrödinger equations.

Use of the Stern-Gerlach force for attaining the spin- states separation of a particle beam is reconsidered in a new method where the magnetic moments are made to precess, at variance with a previously considered case where the magnetic moment conserves its direction in space.

In beam dynamics a detailed formalism based on Nelson's stochastic mechanics is presented for the description of betatron and synchrotron oscillations in both undamped and damped cases. The propagator formulation is used. The requirement of a comparison with more conventional (classical) approaches is emphasized.

We present the design, simulations and measurements of a prototype passive RF cavity polarimeter, with a transversely polarized electron beam interacting with the longitudinal gradient of the transverse component of magnetic field in the TE011 mode of the cavity. Signal levels at the MIT-Bates storage ring may approach a microwatt, permitting fast and accurate polarization measurement.

A non-conventional theoretical approach for high-density beams in accelerator and storage rings is proposed and used to calculate the losses in the main ring of the HIDIF proposal.

We show, either quantum mechanically or classically, that the variation of the effective mass induced in a charged particle by the presence of an ultra-strong electromagnetic field may lead to observable consequences. In particular, we discuss how this effect may operate on electrons close to the surface of a magnetar, a neutron star with a super-strong magnetic field of the order of $10^{11}$ Tesla.

A formulation of the Quantum-like theory based on Feynman's propagator is presented for particle beams. Applications to betatronic oscillations of LHC and HIDIF show encouranging predictions.

A formulation of the Quantum-like theory based on Feynman's propagator is presented for particle beams. Applications to betatronic oscillations of LHC and HIDIF show encouranging predictions.

The time varying relativistic Stern‐Gerlach force, which acts over a charged particle endowed with a magnetic moment, is deduced from the Dirac Hamiltonian finding its coincidence with the classical expression. Possible drawbacks related to the Heisenberg uncertainty principle are discussed.

The relativistic Stern‐Gerlach interaction is here considered as a tool for obtaining the spin state separation of an unpolarized (anti)proton beam circulating in a ring. Drawbacks, such as spin precessions within the TE rf cavity, spurious kicks due to the transverse electric field and, worst of all, filamentation in the longitudinal phase plane are analyzed. Possible remedies are proposed and their feasibility is discussed.

The Stern-Gerlach interaction, between a moving charged particle endowed with a magnetic moment and a radio-frequency e.m. field, is studied by means of a semi-classical approach. Theoretical results are presented, and a possible experimental check of this theory is discussed.

An interpretation of the formation of halo in accelerators based on quantum-like theory by a diffraction model is given in terms of the transversal beam motion. Physical implications of the longitudinal dynamics are also examined.