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Register a new userPath Integral for Non-Paraxial Optics
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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.
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The well-known (1+1D) nonlinear Schrodinger equation (NSE) governs the propagation of narrow-band pulses in optical fibers and others one-dimensional structures. For exploration the evolution of broad-band optical pulses (femtosecond and attosecond) it is necessary to use the more general nonlinear amplitude equation (GNAE) which differs from NSE with two additional non-paraxial terms. That is way, it is important to make clear the difference between the solutions of these two equations. We found a new analytical soliton solution of GNAE and compare it with the well-known NSE one. It is shown that for the fundamental soliton the main difference between the two solutions is in their phases. It appears that, this changes significantly the evolution of optical pulses in multisoliton regime of propagation and admits a behavior different from that of the higher-order NSE solitons.
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.
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 $tau$).
Hawkings seminal discovery of black hole evaporation was based on the semi-classical, perturbative method. Whether black hole evaporation may result in the loss of information remains undetermined. The solution to this paradox would most likely rely on the knowledge of the end-life of the evaporation, which evidently must be in the non-perturbative regime. Here we reinterpret the Hawking radiation as the tunneling of instantons, which is inherently non-perturbative. For definitiveness, we invoke the picture of shell-anti-shell pair production and show that it is equivalent to that of instanton tunneling. We find that such a shell pair production picture can help to elucidate firewalls and ER=EPR conjectures that attempt to solve the information paradox, and may be able to address the end-life issue toward an ultimate resolution.
Path integral solutions are obtained for the the PT-/non-PT-Symmetric and non-Hermitian Morse Potential. Energy eigenvalues and the corresponding wave functions are obtained.
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