Topic

# Power series

About: Power series is a(n) research topic. Over the lifetime, 7169 publication(s) have been published within this topic receiving 118970 citation(s).

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Abstract: The probability of a configuration is given in classical theory by the Boltzmann formula exp [— V/hT] where V is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of h. The formula is developed for this correction by means of a probability function and the result discussed.

5,808 citations

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Abstract: A systematic study is made of the non-perturbative effects in quantum chromodynamics. The basic object is the two-point functions of various currents. At large Euclidean momenta q the non-perturbative contributions induce a series in (μ2/q2) where μ is some typical hadronic mass. The terms of this series are shown to be of two distinct types. The first few of them are connected with vacuum fluctuations of large size, and can be consistently accounted for within the Wilson operator expansion. On the other hand, in high orders small-size fluctuations show up and the high-order terms do not reduce (generally speaking) to the vacuum-to-vacuum matrix elements of local operators. This signals the breakdown of the operator expansion. The corresponding critical dimension is found. We propose a Borel improvement of the power series. On one hand, it makes the two-point functions less sensitive to high-order terms, and on the other hand, it transforms the standard dispersion representation into a certain integral representation with exponential weight functions. As a result we obtain a set of the sum rules for the observable spectral densities which correlate the resonance properties to a few vacuum-to-vacuum matrix elements. As the last bid to specify the sum rules we estimate the matrix elements involved and elaborate several techniques for this purpose.

3,635 citations

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Abstract: It is shown how certain thermodynamic functions, and also the radial distribution function, can be expressed in terms of the potential energy distribution in a fluid. A miscellany of results is derived from this unified point of view. (i) With g(r) the radial distribution function and Φ(r) the pair potential, it is shown that g exp (Φ/kT) may be written as a Fourier integral, or as a power series in r2 the terms of which alternate in sign. (ii) A potential‐energy distribution which is independent of the temperature implies an equation of state which is a generalization of a number of well‐known approximations. (iii) The grand partition function of the one‐dimensional lattice gas is obtained from thermodynamic arguments without evaluating a sum over states. (iv) If in a two‐dimensional honeycomb (three‐coordinates) lattice gas fr(r=0, 1, 2, 3) is the fraction of all the empty sites which at equilibrium are neighbored by exactly r filled sites, then at the critical density the values of all four of the f's ...

2,369 citations

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Abstract: The renormalized propagation functions DFC and SFC for photons and electrons, respectively, are investigated for momenta much greater than the mass of the electron. It is found that in this region the individual terms of the perturbation series to all orders in the coupling constant take on very simple asymptotic forms. An attempt to sum the entire series is only partially successful. It is found that the series satisfy certain functional equations by virtue of the renormalizability of the theory. If photon self-energy parts are omitted from the series, so that D_(FC)=D_F, then S_(FC) has the asymptotic form A[p^2m^2]^n[iγ⋅p]^(−1), where A=A(e_1^2) and n=n(e_1^2). When all diagrams are included, less specific results are found. One conclusion is that the shape of the charge distribution surrounding a test charge in the vacuum does not, at small distances, depend on the coupling constant except through a scale factor. The behavior of the propagation functions for large momenta is related to the magnitude of the renormalization constants in the theory. Thus it is shown that the unrenormalized coupling constant e_0^2/4πℏc, which appears in perturbation theory as a power series in the renormalized coupling constant e_1^2/4πℏc with divergent coefficients, may behave in either of two ways: (a) It may really be infinite as perturbation theory indicates; (b) It may be a finite number independent of e_1^2/4πℏc.

1,068 citations

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Abstract: For a system with degenerate energies, the power series expansions of the $S$-matrix elements may become singular. An elementary theorem in quantum mechanics is proved which shows that under certain general conditions such singularities do not appear in the power series expansions of the transition probabilities, provided these are averaged over an appropriate ensemble of degenerate states. Application of this theorem leads to the cancellations of mass singularities and infrared divergences in quantum electrodynamics. The question of whether a charged particle can have zero mass is studied.

1,024 citations