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Authors
Kazuo Fujikawa
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We propose an idea of the constrained Feynman amplitude for the scattering of the charged lepton and the virtual W-boson, $l_{beta} + W_{rho} rightarrow l_{alpha} + W_{lambda}$, from which the conventional Pontecorvo oscillation formula of relativistic neutrinos is readily obtained using plane waves for all the particles involved. In a path integral picture, the neutrino propagates forward in time between the production and detection vertices, which are constrained respectively on the 3-dimensional spacelike hypersurfaces separated by a macroscopic positive time $tau$. The covariant Feynman amplitude is formally recovered if one sums over all possible values of $tau$ (including negative $tau$).
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We investigate potential quantum nonlinear corrections to Diracs equation through its sub-leading effect on neutrino oscillation probabilities. Working in the plane-wave approximation and in the $mu-tau$ sector, we explore various classes of nonlinearities, with or without an accompanying Lorentz violation. The parameters in our models are first delimited by current experimental data before they are used to estimate corrections to oscillation probabilities. We find that only a small subset of the considered nonlinearities have the potential to be relevant at higher energies and thus possibly detectable in future experiments. A falsifiable prediction of our models is an energy dependent effective mass-squared, generically involving fractional powers of the energy.
In this paper, we have constructed the Feynman path integral method for non-paraxial optics. This is done by using the mathematical analogy between a non-paraxial optical system and the generalized Schrodinger equation deformed by the existence a minimal measurable length. Using this analogy, we investigated the consequences of a minimal length in this optical system. This path integral has been used to obtain instanton solution for such a optical systems. Moreover, the Berry phase of this optical system has been investigated. These results may disclose a new way to use the path integral approach in optics. Furthermore, as such system with an intrinsic minimal length have been studied in quantum gravity, the ultra-focused optical pluses can be used as an optical analog of quantum gravity.
Neutrino mixing and oscillations in quantum field theory framework had been studied before, which shew that the Fock space of flavor states is unitarily inequivalent to that of mass states (inequivalent vacua model). A paradox emerges when we use these neutrino weak states to calculate the amplitude of $W$ boson decay. The branching ratio of W(+) -> e(+) + nu_mu to W(+) -> e(+) + nu_e is approximately at the order of O({m_i^2}/{k^2}). The existence of flavor changing currents contradicts to the Hamiltonian we started from, and the usual knowledge about weak processes. Also, negative energy neutrinos (or violating the principle of energy conservation) appear in this framework. We discuss possible reasons for the appearance of this paradox.
The resolvent of supersymmetric Dirac Hamiltonian is studied in detail. Due to supersymmetry the squared Dirac Hamiltonian becomes block-diagonal whose elements are in essence non-relativistic Schrodinger-type Hamiltonians. This enables us to find a Feynman-type path-integral representation of the resulting Greens functions. In addition, we are also able to express the spectral properties of the supersymmetric Dirac Hamiltonian in terms of those of the non-relativistic Schrodinger Hamiltonians. The methods are explicitly applied to the free Dirac Hamiltonian, the so-called Dirac oscillator and a generalization of it. The general approach is applicable to systems with good and broken supersymmetry.
The historical discovery of neutrino oscillations using solar and atmospheric neutrinos, and subsequent accelerator and reactor studies, has brought neutrino physics to the precision era. We note that CP effects in oscillation phenomena could be difficult to extract in the presence of unitarity violation. As a result upcoming dedicated leptonic CP violation studies should take into account the non-unitarity of the lepton mixing matrix. Restricting non-unitarity will shed light on the seesaw scale, and thereby guide us towards the new physics responsible for neutrino mass generation.
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