Geometry, Particles and Fields

Abstract

Library of Congress Cataloging-in-Publication Data Felsager, Bj0rn.Geometry, particles, and fields I Bj0rn Felsager. p. cm. -(Graduate texts in contemporary physics) lncludes index.

Solvent Stokes’ Shifts Revisited: Application and Comparison of Thompson−Schweizer−Chandler−Song−Marcus Theories with Ooshika−Bakshiev−Lippert Theories

“…The gradient of ϕ (except for the part involving κ) is then given by the gradient of the time-coordinate part Φ alone (if k 0 = ω/c) and is essentially the electric field of a dipole – the Laplacian of which vanishes outside of the source. The ‘time’ term then makes up for the ‘source’ term in the Gaussian approximation to the Landau Ginzburg model. Then, from the Klein-Gordon equation the mass term is provided only by the decaying term κ, as we expected from the similarity of the screened electrostatic (Debye-like) model with that of a massive, spin-zero boson, or : $${\mathrm{normalm}}^2{\mathrm{normalc}}^2∕{\hslash }^2=\kappa +(1∕{\mathrm{c}}^2{{\tau }^2}_{\text{diel}})+2{\mathrm{ik}}_0(\kappa -1∕\mathrm{normalc}{\tau }_{\text{diel}})$$ The Hamiltonian density (54) for this model is given by : $$H=(1∕2)[{(1∕\mathrm{c}\mathrm{d}\varphi ∕\mathrm{dt})}^2+{(\nabla \varphi )}^2+(m^2{c}^2∕{\hslash }^2){\varphi }^2]=m^2{c}^2∕{\hslash }^2{\varphi }^2+(1∕2){\mathrm{boldE}}^2$$ This density gives the energy of the system at a particular time when integrated spatially. Given a time dependent ∇ϕ (see below), we can evaluate the energy difference at two times by the trivial device of writing: $$\Delta \mathrm{W}\equiv {\mathrm{normalW}}_{\text{emmision}}-{\mathrm{normalW}}_{\text{excitation}}=\int {\int }_{\mathrm{italicin}}^{\mathrm{italicout}}\mathrm{dH}/\mathrm{dt}$$…”

“…A massive vector boson is a known model from Quantum Field Theory(53,54). It also, except for the‘time’ term, resembles the Gaussian approximation to the Landau-Ginzburg model for order parameters in phase transitions(49).…”

“…We expect, as with the massive boson model, and as in the Debye model of solution electrostatics, to obtain a function of the dipole-order parameter to be of the form ∼ κ( μ I /r) exp(−κr) . If we take the Klein Gordon model as a starting point(53,54), $${\partial }_{\mu }{\partial }^{\mu }\varphi =-(m^2{c}^2∕{\hslash }^2)\varphi =(-1∕c^2){\mathrm{normald}}^2\varphi ∕{\mathrm{dt}}^2+{\nabla }^2\varphi$$ where ϕ is our ‘pseudoparticle’ wavefunction which we wish to roughly characterize as an ‘order parameter’ for the system. One possible solution for ϕ is identical with the solution for the vector potential of a radiating dipole (55) together with the electrostatic potential of the dipole, or the four-vector (Φ, A r ) = ([ μ • r /r 3 ], iω μ /r exp(−i(ωt+ k 0 • r )), where the characteristic energy of the ‘lattice’ vibration is hω.…”

“…The k 0 wave vector might be that of a librational mode of the solvent lattice for dipolar liquids, for example. Moreover, the ϕ 4 model of the Landau-Ginzburg treatment(49) (a well-known variation of the Klein-Gordon equation as well(54)) is a good first approximation to a dissociating Morse-potential-like dipolaron theory. Taking account of these asymptotes we ought to transform ϕ => ϕαρg(r)S D ϕ ⇒ ϕα ′ ρsol g(r)S D and ∂ μ => (αρg(r)) −1/2 ∂ μ . ∂ μ ⇒ ( α ′ ρsol g(r)) 1/2 ∂ μ We then see that the expected ‘energy density’ of the “dipolaron’ is αρg(r)S D 2 μ I 2 /r 6 α ′ ρsol g(r)S D 2 μ I 2 /r 6 which when integrated over space becomes −1/3 αρg(r)S D 2 μ I 2 /a 3 − α ′ ρsol g(r)S D 2 μ I 2 /a 3 and if we take the difference between zero time and infinity we obtain: $${\mathrm{normalE}}_{\mathrm{solv}}\cong 1∕3{\alpha }^{\prime }{\rho }_{\mathrm{italicsol}}\mathrm{g}\left(\mathrm{normalr}\right){{\mu }_{\mathrm{I}}}^2∕a^3[normal{S}_D\left({\epsilon }_0\right)-normal{S}_D\left(n^2\right)]+{\alpha }^{\prime }{\rho }_{\mathrm{sol}}\mathrm{g}\left(\mathrm{normalr}\right){{\mu }_{\mathrm{I}}}^2{{\mathrm{k}}_0}^2({\kappa }_{(\mathrm{normalt}=\infty )}-{\kappa }_{(\mathrm{normalt}=0)})$$ for which, if we had used the Clausius-Mossotti version of S D (which Pollock et al57 specifically reject ) S D = (ε − 1)/(ε+2), we would obtain a formula very similar to eq(1a) except for some additional terms.…”

Nonlinear Gravitational Waves as Dark Energy in Warped Spacetimes

“…In the next order 4 T (2) tt there will be terms proportional with higher orders of (n 2 − n 1 +P). These higher order terms will be suppressed by the warp factor, so the vortex will not become unstable as is the case when one breakup the vortex string in multiple flux [21].…”