The function e−1/x2. The MacLaurin series is identically zero, but the function is not.
which does not equal its own Taylor series in any neighborhood around x = 0. Every coefficient of the Taylor expansion around x = 0 is exactly zero, but the function is non-zero if x ≠ 0.
In physics, such functions arise for phenomena which are impossible to understand by perturbation theory, at any finite order. In quantum field theory, 't Hooft–Polyakov monopoles, domain walls, flux tubes, and instantons are examples.[1] A concrete, physical example is given by the Schwinger effect,[2] whereby a strong electric field may spontaneously decay into electron-positron pairs. For not too strong fields, the rate per unit volume of this process is given by,
which cannot be expanded in a Taylor series in the electric charge , or the electric field strength . Here is the mass of an electron and we have used units where .
In theoretical physics, a non-perturbative solution is one that cannot be described in terms of perturbations about some simple background, such as empty space. For this reason, non-perturbative solutions and theories yield insights into areas and subjects that perturbative methods cannot reveal.
perturbative, mathematics, physics, perturbative, function, process, that, cannot, described, perturbation, theory, example, functionthe, function, maclaurin, series, identically, zero, function, displaystyle, which, does, equal, taylor, series, neighborhood, . In mathematics and physics a non perturbative function or process is one that cannot be described by perturbation theory An example is the functionThe function e 1 x2 The MacLaurin series is identically zero but the function is not f x e 1 x 2 displaystyle f x e 1 x 2 which does not equal its own Taylor series in any neighborhood around x 0 Every coefficient of the Taylor expansion around x 0 is exactly zero but the function is non zero if x 0 In physics such functions arise for phenomena which are impossible to understand by perturbation theory at any finite order In quantum field theory t Hooft Polyakov monopoles domain walls flux tubes and instantons are examples 1 A concrete physical example is given by the Schwinger effect 2 whereby a strong electric field may spontaneously decay into electron positron pairs For not too strong fields the rate per unit volume of this process is given by G e E 2 4 p 3 e p m 2 e E displaystyle Gamma frac eE 2 4 pi 3 mathrm e frac pi m 2 eE which cannot be expanded in a Taylor series in the electric charge e displaystyle e or the electric field strength E displaystyle E Here m displaystyle m is the mass of an electron and we have used units where c ℏ 1 displaystyle c hbar 1 In theoretical physics a non perturbative solution is one that cannot be described in terms of perturbations about some simple background such as empty space For this reason non perturbative solutions and theories yield insights into areas and subjects that perturbative methods cannot reveal See also editLattice QCD Soliton Sphaleron Instanton BCFW recursion Operator product expansion Conformal bootstrap Loop quantum gravity Causal dynamical triangulationReferences edit Shifman M 2012 Advanced Topics in Quantum Field Theory Cambridge University Press ISBN 978 0 521 19084 8 Schwinger Julian 1951 06 01 On Gauge Invariance and Vacuum Polarization Physical Review 82 5 American Physical Society APS 664 679 doi 10 1103 physrev 82 664 ISSN 0031 899X nbsp This mathematics related article is a stub You can help Wikipedia by expanding it vte nbsp This quantum mechanics related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Non perturbative amp oldid 1203500266, wikipedia, wiki, book, books, library,
