The Double Slit Experiment With Polarizers

The double slit experiment provides a standard way of demonstrating how quantum mechanics works. We consider modifying the standard arrangement so that a photon beam incident upon the double slit encounters a polarizer in front of either one or both of the slits.Comment: 6 page

Renormalization Mass Scale and Scheme Dependence in the Perturbative Contribution to Inclusive Semileptonic bb Decays

We examine the perturbative calculation of the inclusive semi-leptonic decay rate $\Gamma$ for the $b$-quark, using mass-independent renormalization. To finite order of perturbation theory the series for $\Gamma$ will depend on the unphysical renormalization scale parameter $\mu$ and on the particular choice of mass-independent renormalization scheme; these dependencies will only be removed after summing the series to all orders. In this paper we show that all explicit $\mu$-dependence of $\Gamma$, through powers of ln$(\mu)$, can be summed by using the renormalization group equation. We then find that this explicit $\mu$-dependence can be combined together with the implicit $\mu$-dependence of $\Gamma$ (through powers of both the running coupling $a(\mu)$ and the running $b$-quark mass $m(\mu)$) to yield a $\mu$-independent perturbative expansion for $\Gamma$ in terms of $a(\mu)$ and $m(\mu)$ both evaluated at a renormalization scheme independent mass scale $I\!\!M$ which is fixed in terms of either the "$\overline{MS}$ mass" $\overline{m}_b$ of the $b$ quark or its pole mass $m_{pole}$. At finite order the resulting perturbative expansion retains a degree of arbitrariness associated with the particular choice of mass-independent renormalization scheme. We use the coefficients $c_i$ and $g_i$ of the perturbative expansions of the renormalization group functions $\beta(a)$ and $\gamma(a)$, associated with $a(\mu)$ and $m(\mu)$ respectively, to characterize the remaining renormalization scheme arbitrariness of $\Gamma$. We further show that all terms in the expansion of $\Gamma$ can be written in terms of the $c_i$ and $g_i$ coefficients and a set of renormalization scheme independent parameters $\tau_i$.Comment: 26 pages, 4 figures, typo correcte

Ionization Parameter as a Diagnostic of Radiation and Wind Pressures in H II Regions and Starburst Galaxies

The ionization parameter U is potentially useful for measuring radiation pressure feedback from massive star clusters, as it reflects the radiation-to-gas-pressure ratio and is readily derived from mid-infrared line ratios. We consider several effects which determine the apparent value of U in HII regions and galaxies. An upper limit is set by the compression of gas by radiation pressure. The pressure from stellar winds and the presence of neutral clumps both reduce U for a given radiation intensity. The most intensely irradiated regions are selectively dimmed by internal dust absorption of ionizing photons, inducing observational bias on galactic scales. We explore these effects analytically and numerically, and use them to interpret previous observational results. We find that radiation confinement sets the upper limit log_10 U = -1 seen in individual regions. Unresolved starbursts display a maximum value of ~ -2.3. While lower, this is also consistent with a large portion of their HII regions being radiation dominated, given the different technique used to interpret unresolved regions, and given the bias caused by dust absorption. We infer that many individual, strongly illuminated regions cannot be dominated by stellar winds, and that even when averaged on galactic scales, shocked wind pressures cannot be large compared to radiation pressure. Therefore, most HII regions cannot be adiabatic wind bubbles. Our models imply a metallicity dependence in the physical structure and dust attenuation of radiation-dominated regions, both of which should vary strongly across a critical metallicity of about one-twentieth solar.Comment: 19 pages, 14 figures, accepted for publication in Ap

Gauge Model With Extended Field Transformations in Euclidean Space

An SO(4) gauge invariant model with extended field transformations is examined in four dimensional Euclidean space. The gauge field is $(A^\mu)^{\alpha\beta} = 1/2 t^{\mu\nu\lambda} (M^{\nu\lambda})^{\alpha\beta}$ where $M^{\nu\lambda}$ are the SO(4) generators in the fundamental representation. The SO(4) gauge indices also participate in the Euclidean space SO(4) transformations giving the extended field transformations. We provide the decomposition of the reducible field $t^{\mu\nu\lambda}$ in terms of fields irreducible under SO(4). The SO(4) gauge transformations for the irreducible fields mix fields of different spin. Reducible matter fields are introduced in the form of a Dirac field in the fundamental representation of the gauge group and its decomposition in terms of irreducible fields is also provided. The approach is shown to be applicable also to SO(5) gauge models in five dimensional Euclidean space.Comment: 31 pages, Plain LaTe

Summation of Higher Order Effects using the Renormalization Group Equation

The renormalization group (RG) is known to provide information about radiative corrections beyond the order in perturbation theory to which one has calculated explicitly. We first demonstrate the effect of the renormalization scheme used on these higher order effects determined by the RG. Particular attention is payed to the relationship between bare and renormalized quantities. Application of the method of characteristics to the RG equation to determine higher order effects is discussed, and is used to examine the free energy in thermal field theory, the relationship between the bare and renormalized coupling and the effective potential in massless scalar electrodynamics