Showing papers in "Physical Review D in 1972"

Can gravitation have a finite range

Abstract: No acceptable tensor gravitational theory with arbitrarily long but finite range exists. In linear approximation, the infinite-range limit is a scalar-tensor mixture implying an effective matter-matter coupling different from the strictly infinite-range prediction and contradicted by experiment. Compensation of the scalar requires the admixture of a ghost scalar coupling. In the massive version of the full Einstein theory, (a) there are necessarily six rather than the five tensor degrees of freedom, (b) the energy has no lower bound, (c) the infinite-range limit seems not to exist at all, and (d) lowest-order forces are the same as in the massive linearized theory.

Atomic coherent states in quantum optics

Abstract: The central problem of Quantum Optics (laser theory, super-radiance, resonant propagation, etc.) is the description of the interaction between N atoms and an electromagnetic field confined in a cavity of finite volume. A suitable model Hamiltonian for this problem is the following one (ℏ = 1) $$\begin{array}{*{20}{l}} {{\rm{H = }}\sum\limits_{\rm{k}} {{{\rm{\omega }}_{\rm{k}}}} {\rm{a}}_{\rm{k}}^{\rm{ + }}{{\rm{a}}_{\rm{k}}}{\rm{ + }}\frac{{{{\rm{\omega }}_{\rm{o}}}}}{{\rm{2}}}\sum\limits_{{\rm{i = 1}}}^{\rm{N}} {{{\rm{S}}_{{\rm{3}}\left( {\rm{i}} \right)}}} }\\ {{\rm{ + }}\sum\limits_{{\rm{k,i}}} {{{\rm{g}}_{\rm{k}}}\left( {{{\rm{a}}_{\rm{k}}}{\rm{S}}_{\rm{i}}^{\rm{ + }}{{\rm{e}}^{{\rm{i \bullet }}{{{\rm{}}}_{\rm{i}}}}}{\rm{ + a}}_{\rm{k}}^{\rm{ + }}{\rm{S}}_{\rm{i}}^{\rm{ - }}{{\rm{e}}^{{\rm{ - i \bullet }}{{{\rm{}}}_{\rm{i}}}}}} \right)} {\rm{,}}} \end{array}$$ where ak, a k + are Bose operators describing the k-th field mode and S i ± ,S3i are Pauli operators describing the atom located at position x i as a two-level system.

Scale and conformal transformations in galilean-covariant field theory

Abstract: It is demonstrated that there exists the possibility of defining scale and conformal transformations in such a way that these constitute exact invariance operations of the Schr\"odinger equation. Unlike the relativistic case there is only a single conformal transformation and the usual eleven-parameter extended Galilei group is consequently enlarged to a thirteen-parameter group. The generalization to the case of fields of arbitrary spin is carried out within the framework of minimal-component theories whose interactions respect scale and conformal invariance. One finds that the bare-internal-energy term can be used to break these additional invariance operations in much the same way as the mass term in special relativity. The generators and conservation laws associated with all space-time symmetries of minimal-component Galilean-invariant field theories are derived, it being shown that, in analogy to the relativistic case, the operators which appear in these equations can be redefined so as to allow the formulation of scale and conformal invariance entirely in terms of those operator densities relevant to the transformations of the Galilei group.

Nonexistence of baryon number for static black holes

Abstract: Wheeler has conjectured that black holes should have no well-defined baryon number, and that as a result the law of conservation of baryons should be transcended in black-hole physics. We show here that a static black hole cannot have any exterior classical scalar or massive vector fields. We consider the modifications that would arise from a quantum-theoretical treatment, and we conclude that such a black hole cannot interact with the exterior world via virtual mesons such as the $\ensuremath{\pi}$ and $\ensuremath{\rho}$. Because of this we find no way for external measurements to assign unambiguously a baryon number to such a black hole in agreement with Wheeler's prediction.

Calculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light Beam

Abstract: Wiegrefe, Fedorov, Costa de Beauregard, and Schilling have discussed the transverse energy flux existing in total reflection of an elliptically polarized light beam, the latter two proposing formulas for the transverse shift of the reflected beam. We have calculated the transverse shift by an energy-flux-conservation argument similar to Kristoffel's and to Renard's in their deduction of the longitudinal Goos-H\"anchen shift, thus obtaining a formula different from those of the previous authors. We have also tested experimentally the existence of the transverse shift, in the optimal case of circular polarization and quasilimit total reflection, by using two slightly different multiplying procedures. Our measurements definitely vindicate our own formula for the transverse shift against both Costa de Beauregard's and Schilling's. The relevance of our results in connection with noncollinearity of velocity and momentum of the spinning photon inside the evanescent wave is very briefly discussed.

Generalization of the Einstein Theory

Abstract: The assumption that ${{T}^{\ensuremath{ u}}}_{\ensuremath{\mu};\ensuremath{ u}}=0$ in curved space-time is questioned. Field equations are given which are consistent with the assumption ${{T}^{\ensuremath{ u}}}_{\ensuremath{\mu};\ensuremath{ u}}=\ensuremath{\lambda}{R}_{,\ensuremath{\mu}}$, and which reduce to the Einstein equations when $\ensuremath{\lambda}=0$. The equations are equivalent to the Einstein equations in empty space-time, but differ from them in the presence of matter. Applications to cosmology, stellar structure, and collapsing objects are suggested.

Nonspherical Perturbations of Relativistic Gravitational Collapse. II. Integer-Spin, Zero-Rest-Mass Fields

Abstract: A nearly spherical star collapses through its gravitational radius. Nonspherical perturbations exist in its density, pressure, electromagnetic field, and gravitational field, and in other (hypothetical) zero-rest-mass, integer-spin fields coupled to sources in the stars. Paper I analyzed the evolution of scalar-field and gravitational-field perturbations. This paper treats fields of arbitrary integer spin and zero rest mass, using the Newman-Penrose tetrad formalism. The analysis of each multipole ($\mathrm{order}=l$) of each field ($\mathrm{spin}=s$) is reduced to the study of a two-dimensional wave equation, with a "curvature potential" that differs little from one field to another. The analysis of this wave equation for the scalar case ($s=0$) carries over completely to fields of arbitrary spin $s$. In particular, any radiatable multipole ($l\ensuremath{\ge}s$) gets radiated away completely in the late stages of collapse; if the multipole is static prior to the onset of collapse, it will die out as ${t}^{\ensuremath{-}(2l+2)}$ at late times. Nonradiatable multipoles ($lls$) are conserved. This paper also treats gravitational perturbations in the Newman-Penrose framework, and supplies some technical details missing in the gravitational-perturbation analysis of Paper I.

Effect of anomalies on quasi-renormalizable theories.

Abstract: Apparently nonrenormalizable field theories, such as the new models for weak interactions, can become renormalizable when gauge invariance of the second kind is present. However, the anomaly associated with the axial-vector current may destroy this gauge invariance in perturbation theory, even though it is present in the Lagrangian. When this happens the theory remains nonrenormalizable. Nevertheless it is possible, by enlarging the theory, to remove the anomaly at the expense of introducing additional fermion fields, which correspond to as-yet-unobserved particles.

Generalized renormalizable gauge formulation of spontaneously broken gauge theories.

Abstract: The spontaneously broken gauge theory is formulated in the generalized renormalizable gauge (${R}_{\ensuremath{\xi}}$ gauge). A parameter $\ensuremath{\xi}$ can be adjusted to include existing gauges, $U$ gauge, $R$ gauge, and 't Hooft-Feynman gauge as special cases. Three applications of the ${R}_{\ensuremath{\xi}}$-gauge formulation are given. First we compute the weak correction to the muon magnetic moment unambiguously in the existing models for leptons. Secondly, we discuss the large-momentum-transfer limit of the Pauli magnetic form factor of the muon. Finally, we discuss the static charge of the neutrino, and show that an appropriate regularization makes it vanish.

Bound Geodesics in the Kerr Metric

Abstract: The bound geodesics (orbits) of a particle in the Kerr metric are examined. (By "bound" we signify that the particle ranges over a finite interval of radius, neither being captured by the black hole nor escaping to infinity.) All orbits either remain in the equatorial plane or cross it repeatedly. A point where a nonequatorial orbit intersects the equatorial plane is called a node. The nodes of a spherical (i.e., constant radius) orbit are dragged in the sense of the spin of the black hole. A spherical orbit near the one-way membrane traces out a helix-like path lying on a sphere enclosing the black hole.

Nonexistence of Baryon Number for Black Holes. II

Abstract: In a previous paper we showed that a static (nonrotating) black hole cannot be endowed with exterior scalar-meson or massive vector-meson fields. Here we show that the same is true for massive spin-2 meson fields. We also extend the above results to the case of a rotating stationary black hole. We conclude from our results that a black hole in its final (static or stationary) state cannot interact with the exterior world via the strong interactions which are mediated by meson fields such as the $\ensuremath{\pi}$ (scalar), $\ensuremath{\rho}$ (vector), and $f$ (spin-2). A direct consequence of this is the impossibility of determining the baryon number of the black hole by means of exterior measurements alone. This results in the transcendence of the law of baryon-number conservation as originally predicted by Wheeler. All the above conclusions hold both in general relativity and in the Brans-Dicke theory. We also show that the result of Hawking, that stationary black holes are identical in general relativity and in Brans-Dicke theory, holds even when the effects of the strong interactions (of the stellar material out of which the black hole was formed) are taken into account. Our final conclusion is that in both theories the Israel-Carter conjecture that "all stationary single-black-hole exteriors are of the Kerr-Newman type" holds even when the strong interactions are "turned on."

Gravitational spin interaction

Abstract: The gravitational spin interaction is investigated by studying the deviation from geodesic motion of spinning test bodies. The force on a spinning test body at rest in the exterior field of an arbitrary stationary, rotating source is evaluated and found to be given by$\stackrel{\ensuremath{\rightarrow}}{\mathrm{F}}=\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{ abla}}\left(\frac{\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}+3(\stackrel{^}{r}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}})(\stackrel{^}{r}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}})}{{r}^{3}}\right)+O(\frac{1}{{r}^{5}})$ where $\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}$ is the spin of the test body, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}$ is the angular momentum of the source of the gravitational field, and geometrized units $G=c=1$ are used. Thus, the gravitational spin-spin force has the same form as the force between two dipoles in electromagnetism except that its sign is opposite, i.e., "north pole" attracts "north pole" in gravitational spin-spin interaction. The gravitational spin-spin torque, previously investigated by Schiff, Mashhoon, and Wilkins, also has opposite sign to the corresponding electromagnetic dipole-dipole torque. The sign of the spin-spin force agrees with that predicted by Hawking on the basis of the fact that less energy can be extracted from colliding black holes if their spins are parallel rather than antiparallel. Furthermore, it is shown that the spin-interaction energy quantitatively accounts for the angular momentum dependence in Hawking's formula for the upper limit for energy released from colliding black holes. The gravitational spin-orbit force is also investigated, and it is found to differ in form from the corresponding electromagnetic spin-orbit force.

Spectrum generating algebra and no ghost theorem for the dual model

Abstract: The dual model is generally factorized using Lorentz oscillators ${a}_{n}^{\ensuremath{\mu}}$ with ghost (or negativenorm) states arising from the indefinite metric ($[{a}_{n}^{0}, {a}_{n}^{0\ifmmode\dagger\else\textdagger\fi{}}]=\ensuremath{-}1$). Here all ghost states are proven to decouple for unit Regge intercept (${\ensuremath{\alpha}}_{0}=1$) as a consequence of the Virasoro gauges (${L}_{n}$). By reformulating vertices in light-cone variables and exploiting the local commutators (for ${Q}^{\ensuremath{\mu}}$, ${P}^{\ensuremath{\mu}}$) on the Koba-Nielson circle, the spectrum-generating algebra (${A}_{n}^{i}$, ${A}_{n}^{(+)}$) is found that commutes with all the gauges ${L}_{n}$. All physical states are explicitly constructed. The noghost theorem follows from the remarkable isomorphism of the transverse generators ${A}_{n}^{i}$ ($i=1, 2$) of Del Giudice, Di Vecchia, and Fubini to the original oscillators $\sqrt{n}{a}_{n}^{i}$, $[{A}_{n}^{i}, {A}_{m}^{j}]=n{\ensuremath{\delta}}_{\mathrm{ij}}{\ensuremath{\delta}}_{n+m,0}$, and the isomorphism (up to $c$ numbers) of the longitudinal generators ${A}_{n}^{(+)}$ with the conformal group generators ${L}_{l}$, $[{A}_{n}^{(+)}, {A}_{m}^{(+)}]=(n\ensuremath{-}m){A}_{n+m}^{(+)}+2{n}^{3}{\ensuremath{\delta}}_{n+m,0}$. Increasing the number of spatial oscillators (${a}_{n}^{i}$, $i=1, \dots{}, D\ensuremath{-}1$), one observes a critical dimension $D=26$. For $Dg26$ ghosts appear, for $Dl26$ there are no ghosts, and ${A}_{1}^{(+)}$ gives the null states postulated by Brower and Thorn. But for $D=26$, all ${A}_{n}^{(+)}$ correspond to null states, so that the second-order Pomeranchukon is precisely a Regge pole (${\ensuremath{\alpha}}_{P}=\frac{1}{2}{\ensuremath{\alpha}}^{\ensuremath{'}}s+2$) as proposed by Lovelace.

Gauge Theories Without Anomalies

Abstract: Theories with spontaneously broken gauge symmetry but without triangle anomalies can yield renormalizable models of weak and electromagnetic interactions. We identify and discuss the class of anomaly-free gauge theories.

Perturbation Lagrangian Theory for Scalar Fields-Ward-Takahashi Identity and Current Algebra

Abstract: Under the assumption that the time-ordered products are given by Feynman rules with Bogoliubov's renormalization scheme, it is shown that the operator forms of Euler-Lagrange equations of motion and Noether's theorem for a wide class of perturbation Lagrangian theories for scalar fields are valid. Ward-Takahashi identities for currents in Noether's theorem are also proved without recourse to equal-time commutators, and current-algebra results follow naturally in a subclass of Lagrangian field theories. Covariant Schwinger terms are present in these identities and their nature is determined.

Spontaneously Broken Gauge Symmetries. I. Preliminaries

Abstract: This is the first of a series of papers addressed to the renormalizability question of spontaneously broken gauge theories. We give a brief outline of the motivation for such an investigation and describe the manner in which the renormalizability of such theories will be proved in the sequel. Put briefly, we will show that in an appropriate gauge, ultraviolet divergences of a spontaneously broken gauge theory are removed completely by the gauge-invariant counterterms in the Lagrangian which would make the Greon's functions of the corresponding unbroken gauge theory finite, that the $S$ matrix computed in this gauge is unitary, and that the $S$ matrix is independent of the gauge chosen. In this paper, the renormalizability question of the unbroken gauge theory is considered. We derive the Ward-Takahashi identities of the theory. We discuss several ways of regulating divergent Feynman integrals of the theory without destroying gauge invariance. Infrared divergences are avoided by the device of intermediate renormalization, wherein we choose as subtraction points some points where external momenta are Euclidean. This suffices to establish that the Bogoliubov-Parasiuk-Hepp renormalization will give renormalized Green's functions which satisfy the Ward-Takahashi identities. The existence of finite, renormalized Green's functions satisfying the Ward-Takahashi identities provides us with the means of proving the renormalizability of the spontaneously broken symmetry case. The Ward-Takahashi identities were previously derived for the gauge bosons by Slavnov. We present here a new derivation. The discussions on regularization methods and intermediate renormalization procedure and the renormalization conditions for matter fields, we believe, are new contributions of the present paper.

Centrifugal-barrier effects in resonance partial decay widths, shapes, and production amplitudes

Abstract: (1) We review the experimental status of identifiable centrifugal-barrier effects in: (i) shapes of resonances, (ii) accounting for deviations in SU(2) and SU(3) predictions for ratios between partial decay widths, and (iii) accounting for the dependence on spin of the elastic widths of resonances lying on the same Regge trajectory. (2) We present a contour plot from which the "kinematic partial width" (centrifugal-barrier penetration factor times two-body phase space) of any resonance into any decay channel may be obtained immediately, given the orbital angular momentum, semiclassical impact parameter, and an assumed strong-interaction radius. Using this technique we display, as an example, the kinematically preferred decays of the high-spin mesons on the leading meson trajectory (assumed linear). (3) We obtain an approximate lower bound on the strong-interaction radius on the basis of general considerations. (4) We show that, for high-spin resonances, it can be expected that the mass enhancements in different decay channels associated with these resonances may be shifted by a half-width or so with respect to each other, because of different centrifugal-barrier effects in the different channels. (5) We also show that production cross sections of high-spin particles on leading linear trajectories by either formation or peripheral processes will be substantially suppressed by centrifugal-barrier effects. (6) Finally, we observe that the production cross sections and decay widths of mesons on the leading trajectory of the Veneziano model can be understood almost entirely on the basis of kinematical considerations if the radius of the region in which the decay particles interact strongly grows linearly with the mass of the decaying resonance.

Light-Cone Behavior of Perturbation Theory

Abstract: A technique introduced by Symanzik is used to derive a series of equations obeyed order by order in perturbation theory by the structure functions ${W}_{1}$ and $\ensuremath{ u}{W}_{2}$ entering the cross section for inelastic electron scattering. These equations relate the ${q}^{2}$, $\ensuremath{ u}$, and coupling-constant dependence of ${W}_{1}$ and $\ensuremath{ u}{W}_{2}$ in a manner reminiscent of the renormalization-group results of Gell-Mann and Low. The equations are used to compute the leading logarithmic contribution to $\ensuremath{ u}{W}_{2}$ in a theory of fermions coupled to pseudoscalar particles and a theory of fermions coupled to vector particles.

Weak-Interaction Corrections to the Muon Magnetic Moment and to Muonic-Atom Energy Levels

Abstract: The weak contributions to the anomalous magnetic moment of the muon are calculated in a proposed theory of the weak and electromagnetic interactions. The result is finite and of order $\ensuremath{\Delta}{g}_{\ensuremath{\mu}}\ensuremath{\approx}{10}^{\ensuremath{-}8}$, too small to be measured at present. The present agreement between theoretical and experimental values of the muon magnetic moment does not even rule out the possibility that the scalar meson required by this theory has a very small mass. If this mass were sufficiently small, then the scalar-meson field would produce shifts of the order of a hundred parts per million in muonic-atom energy levels.

Stellar energy-loss rates in a convergent theory of weak and electromagnetic interactions.

Abstract: The stellar energy-loss rates due to the production of neutrino pairs are calculated in Weinberg's theory of electromagnetic and weak interactions. The ratio of the total rate +p g3 calculated here to the rate calculated in the ordinary theory of weak interactions is 10 where the uncertainty comes entirely from the lack of knowledge of the 5'-meson mass. The ratio of the experimental rate to the rate calculated in the ordinary theory is 10 ' . Thus Weinberg's theory gives numbers well within the experimental limits for all values of the 8\"mass.

Effects of a Neutral Intermediate Boson in Semileptonic Processes

Abstract: The observable effects of a neutral intermediate vector boson in semileptonic processes are considered in the context of a proposed renormalizable theory of the weak and electromagnetic interactions. With strange particles neglected, this theory allows the calculation of neutral-current form factors in terms of charged-current form factors and electromagnetic form factors. The results are neither confirmed nor refuted by present data. One possible method of incorporating the strange particles is briefly discussed.

Hadronic Matter at High Density

Abstract: The density of hardron levels $\ensuremath{\rho}(m)$ has been derived from the statistical bootstrap model to have the asymptotic form $\ensuremath{\rho}(m)\ensuremath{\sim}c{m}^{a}{e}^{\mathrm{bm}}$ with $a\ensuremath{\le}\ensuremath{-}\frac{5}{2}$. We study the properties of ultradense hadronic matter characterized by level densities of this type. If $al\ensuremath{-}\frac{5}{2}$ (as is preferred by the bootstrap model), we show that hadronic matter may be thermodynamically unstable at high densities. This instability is characterized by macroscopic fluctuations of the energy density and the existence of a negative specific heat. Implications of the instability are discussed with reference to high-energy collisions and astrophysics.

Inelastic Electron-Proton Scattering at Large Momentum Transfers and the Inelastic Structure Functions of the Proton

Abstract: Differential cross sections for electrons scattered inelastically from hydrogen have been measured at 18\ifmmode^\circ\else\textdegree\fi{}, 26\ifmmode^\circ\else\textdegree\fi{}, and 34\ifmmode^\circ\else\textdegree\fi{}. The range of incident energy was 4.5 to 18 GeV, and the range of four-momentum transfer squared was 1.5 to 21 ${(\frac{\mathrm{GeV}}{c})}^{2}$. With the use of these data in conjunction with previously measured data at 6\ifmmode^\circ\else\textdegree\fi{} and 10\ifmmode^\circ\else\textdegree\fi{}, the contributions from the longitudinal and transverse components of the exchanged photon have been separately determined. The values of the ratio of the photoabsorption cross sections $\frac{{\ensuremath{\sigma}}_{S}}{{\ensuremath{\sigma}}_{T}}$ are found to lie in the range 0 to 0.5. The question of scaling of $2{M}_{p}{W}_{1}$ and $\ensuremath{ u}{W}_{2}$ as a function of $\ensuremath{\omega}$ is discussed, and scaling is verified for a large kinematic range. Also, a new scaling variable which reduces to $\ensuremath{\omega}$ in the Bjorken limit is introduced which extends the scaling region. The behavior of ${\ensuremath{\sigma}}_{T}$ and ${\ensuremath{\sigma}}_{S}$ is also discussed as a function of $\ensuremath{ u}$ and ${q}^{2}$. Various weighted sum rules of $\ensuremath{ u}{W}_{2}$ are evaluated.

Solution of the scalar wave equation in a kerr background by separation of variables

Abstract: The effect of the Kerr gravitational field on wave phenomena is explored by examining the inhomogeneous wave equation for a scalar massive field in a Kerr background geometry. The equation is separated in Boyer-Lindquist coordinates. The angular functions are spheroidal harmonics, and the radial equation is reduced to a one-dimensional Schr\"odinger equation with an effective potential.

Efimov's effect: a new pathology of three-particle systems. ii.

Abstract: By studying the eigenvalue spectrum of the Faddeev kernel in a certain singular limit, we give an independent proof of an effect recently deduced by Efimov: When three identical particles interact via short-range pairwise potentials, the number of three-body bound states grows without limit when the pairwise scattering length $a$ becomes large. [The number of bound states is then roughly $(\frac{1}{\ensuremath{\pi}})\mathrm{ln}(\ensuremath{\Lambda}|a|)$, where $\ensuremath{\Lambda}$ is a momentum cutoff]. We extend our proof to the case where only two particles are identical and show that Efimov's effect persists in the special limiting cases with two heavy and one light particle, and with two light and one heavy particle.

Nonmeasurability of the quantum numbers of a black hole

Abstract: We consider neutral-meson (scalar and vector), electromagnetic, and neutrino quantum fields interacting with classical sources on a Schwarzschild background. The meson fields decouple from the source when it reaches $r=2m$. This means that the coupling constant is effectively set equal to zero at the horizon. Such a phenomenon does not happen for an electromagnetic field. The sharp difference between vector mesons of any mass and photons suggests that charged black holes can exist only if the photon mass is exactly zero. Neutrinos do not decouple from the source either, but the remaining coupling is given by an unobservable phase factor. These results provide further evidence in favor of Wheeler's conjecture that the only measurable quantum numbers of a black hole are mass, charge, and angular momentum and that, consequently, the laws of conservation of baryon and lepton number are "transcended" in black-hole physics.

Spontaneously Broken Gauge Symmetries. II. Perturbation Theory and Renormalization

Abstract: The second paper in this series is devoted to the formulation of a renormalizable perturbation theory of Higgs phenomena (spontaneously broken gauge theories). In Sec. II, we reformulate the renormalization prescription for massless Yang-Mills theories in terms of gauge-invariant renormalization counterterms in the action. Section III gives a group-theoretic discussion of Higgs phenomena. We discuss the possibility that an asymmetric vacuum is stable, and show how the symmetry of the physical vacuum determines the mass spectrum of the gauge bosons. We show further that in a special gauge ($U$ gauge), all unphysical fields can be eliminated. Section IV discusses the quantization of a spontaneously broken gauge theory in the $R$ gauge, where, as we show in Sec. V, Green's functions are made finite by the renormalization counterterms of the symmetric theory (in which the gauge invariance is not spontaneously broken). The $R$-gauge formulation makes use of redundant fields for the sake of renormalizability. Section VI is a discussion of the low-energy limits of propagators in the $R$-gauge formulation. In Sec. VII we show that the particles associated with redundant fields peculiar to the $R$-gauge formulation are unphysical, i.e., they do not contribute to the sum over intermediate states.

Massless, Euclidean Quantum Electrodynamics on the 5-Dimensional Unit Hypersphere

Abstract: We show that the Feynman rules for vacuum-polarization calculations and the equations of motion in massless, Euclidean quantum electrodynamics can be transcribed, by means of a stereographic mapping, to the surface of the 5-dimensional unit hypersphere. The resulting formalism is closely related to the Feynman rules, which we also develop, for mass-less electrodynamics in the conformally covariant O(5, 1) language. The hyperspherical formulation has a number of apparent advantages over conventional Feynman rules in Euclidean space: It is manifestly infrared-finite, and it may permit the development of approximation methods based on a semiclassical approximation for angular momenta on the hypersphere. The finite-electron-mass, Minkowski-space generalization of our results gives a simple formulation of electrodynamics in (4, 1) de Sitter space.