The present paper is an attempt to generalize the canonical quantization procedure for the Dirac field to curved spaces. Thereby the metric field acts as an unquantized back-ground field which is not influenced by the quantized Dirac field. After the investigation of a 3-dimensional eigenvalue problem the equations of motion are derived for the field operators in the Heisenberg picture. Finally we consider special Riemannian spaces admitting the Killing vectors.
The influence of a time-dependent gravitational field (Robertson–Walker metric with spherical 3-space) on a quantized scalar field conformally coupled to the geometry is studied for two different cases: prescribed background field and reaction on the metric. In the latter model, the expectation values of the stress-energy tensor are the only source terms in the Einstein equations. The equation of state contains the limiting regimes of matter-dominated (\(P=0\)) and radiation-dominated (\(P =\frac {1}{3}\varepsilon \)) dynamics.
The Källén–Lehmann spectral function for the product \(\sigma _{\mu _1}\dots \sigma _{\mu _n}{\mit \Delta }^+(x;a) \sigma _{\nu _1}\dots \sigma _{\nu _m}{\mit \Delta }^+(x;b)\) is calculated. It is shown that the spectral function can be obtained without explicit calculations of the integral by means of algebraic methods.
In this note we apply the results of our previous paper to the investigation of spectral representation of one-loop diagram with arbitrary spin for a class of derivative couplings.
It is shown that the dual model given by Dorren, Rittenberg and Yaffe, similar to the dual Pokorski–Satz model, leads to the Gribov–Morrison rule for the parent states. In the Pokorski–Satz model this parity rule does not apply to the daughters. In the model considered by us the first daughter under \(N_{\gamma }\)-trajectory satisfies the Gribov–Morrison rule.
Isospin-zero and isospin-one clusters are constructed out of pions and used to describe final states with the correct value of the total isospin in high energy proton–proton collisions. General formulas for averages and correlation functions are given. A model for uncorrelated production and decay of clusters is studied in detail and shown to give good agreement with experiment.
Absolute values of the real part of the pp, \(\overline {\rm p}\)p forward scattering amplitude and the slope of the differential cross section have been determined on the basis of data on the elastic differential cross section outside the region of the Coulomb interference. The method is based on the optimal use of analyticity of the scattering amplitude in the cos \(\theta \) plane and leads to significantly larger absolute values of the real part and slope in comparison with those obtained by means of the Bethe formula on the basis of data on differential cross sections in the region of the Coulomb interference.
The excitation functions for the \(^{24}\)Mg(\(\alpha \),p)\(^{27}\)Al and \(^{24}\)Mg(\(\alpha \),\(\alpha \))\(^{24}\)Mg reactions leading to several excited states of \(^{27}\)Al and \(^{24}\)Mg, respectively, were measured at \(\theta _{\rm lab}=40^{\circ }\), 80\(^{\circ }\)32’, 144\(^{\circ }\)19’, and 175\(^{\circ }\) over an \(\alpha \)-energy range of 23.05–28.55 MeV, in \(\sim 200\) keV steps. Statistical analysis of these excitation functions was performed. The direct interaction contribution \(y_{\rm D}\) to the reactions studied was obtained from auto-correlation coefficients. Cross-correlation coefficients calculated between different reaction channels did not indicate the existence of resonances common to all investigated channels. The final state spin dependence of the cross-sections for the \(^{24}\)Mg(\(\alpha \),p)\(^{27}\)Al reaction was also examined. It is in a good agreement with the predictions of the statistical model of compound nucleus reactions.
