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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.
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166:, 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,
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In physics, such functions arise for phenomena which are impossible to understand by perturbation theory, at any finite order. In
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264:{\displaystyle \Gamma ={\frac {(eE)^{2}}{4\pi ^{3}}}\mathrm {e} ^{-{\frac {\pi m^{2}}{eE}}}}
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Schwinger, Julian (1951-06-01). "On Gauge
Invariance and Vacuum Polarization".
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which cannot be expanded in a Taylor series in the electric charge
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are examples. A concrete, physical example is given by the
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is the mass of an electron and we have used units where
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Functions that can't be described by perturbation theory
131:= 0. Every coefficient of the Taylor expansion around
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135:= 0 is exactly zero, but the function is non-zero if
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485:(5). American Physical Society (APS): 664–679.
50:or process is one that cannot be described by
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31:is identically zero, but the function is not.
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454:Advanced Topics in Quantum Field Theory
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113:{\displaystyle f(x)=e^{-1/x^{2}},}
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456:. Cambridge University Press.
429:Causal dynamical triangulation
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123:which does not equal its own
54:. An example is the function
589:. You can help Knowledge by
532:. You can help Knowledge by
7:
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148:'t Hooft–Polyakov monopoles
127:in any neighborhood around
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414:Operator product expansion
359:{\displaystyle c=\hbar =1}
585:-related article is a
528:-related article is a
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327:{\displaystyle m}
307:{\displaystyle E}
287:{\displaystyle e}
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23:The function
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152:domain walls
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526:mathematics
450:Shifman, M.
389:Lattice QCD
37:mathematics
630:Categories
436:References
160:instantons
156:flux tubes
499:0031-899X
404:Instanton
399:Sphaleron
348:ℏ
236:π
230:−
211:π
177:Γ
85:−
452:(2012).
382:See also
48:function
394:Soliton
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41:physics
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