Does anyone know how the method of logarithmic accuracy works and what do I have to know about it (as far as applied Mathematics is concerned)? Any references, examples or guidelines would be appreciated.

Thank you

  • $\begingroup$ Where have you seen it? A Google search didn't turn up anything that looked right. $\endgroup$ Apr 16 '13 at 20:51
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    $\begingroup$ I have seen in a Quantum Mechanics class, in which the professor used the method to solve an integral with infrared divergence. However, I did not ask the question in the physics exchange because it appeared to be a pure mathematical question that could have been answered better by a mathematician. $\endgroup$
    – Heber
    Apr 16 '13 at 21:05
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    $\begingroup$ I am a physicist and I have no idea what you mean by "infrared divergence." $\endgroup$
    – Ron Gordon
    Apr 16 '13 at 21:21

A quantity is said to be calculated “to logarithmic accuracy” when in an expansion like $$ F(\varepsilon) = c \sum_{n=2}^{\infty} A^n \sum_{m=2}^{n} a_{n,m} \ln^{n-m}(\varepsilon/E) ~, $$ only the coefficients $a_{n,m}$ associated with the most divergent terms are calculated exactly while the coefficients for $m>2$ are determined by an approximation procedure.

ref: Duke, C. B., and S. D. Silverstein, “Does Logarithmic Accuracy Uniquely Define the Low‐Temperature Properties of Dilute Magnetic Alloy Systems?”, J. Appl. Phys. 39 (1968) 708, http://dx.doi.org/10.1063/1.2163592 .

As a side remark, infra-red divergence, i.e., divergence of calculated physical quantities at low energy/momentum/temperature scales, is a widely known (and perhaps, ubiquitous) issue in current-day condensed matter physics, and has many important consequences.

In models of condensed matter physics, ultraviolet divergences present not a problem but a nuisance. Their presence indicates that the continuum description is incomplete, i.e. the behaviour of long-wavelength excitations depends on shorter length scales. ... . Infrared divergences are more interesting. Their appearance is always an indication of an incorrectly chosen reference ground state. For example, if electrons in a metal attract, it is wrong to approximate them as free particles; the real ground state is a superconducting condensate of electron pairs. This ground state is orthogonal to the ground state of the non-interacting electron gas and therefore is unreachable by a perturbation expansion. There is no universal recipe for how to choose a correct ground state. Sometimes it is relatively easy (as for ordinary superconductors), sometimes it is not easy at all.

[Tsvelik, A. M. Quantum field theory in condensed matter physics. Cambridge University Press (2006), p. 48]


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