Non-perturbative

In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function

The function e^−1/x2. The MacLaurin series is identically zero, but the function is not.

${\displaystyle f(x)=e^{-1/x^{2}},}$

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,

${\displaystyle \Gamma ={\frac {(eE)^{2}}{4\pi ^{3}}}\mathrm {e} ^{-{\frac {\pi m^{2}}{eE}}}}$

which cannot be expanded in a Taylor series in the electric charge ${\displaystyle e}$, or the electric field strength ${\displaystyle E}$. Here ${\displaystyle m}$ is the mass of an electron and we have used units where ${\displaystyle c=\hbar =1}$.

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

• Lattice QCD
• Soliton
• Sphaleron
• Instanton
• BCFW recursion
• Operator product expansion
• Conformal bootstrap
• Loop quantum gravity
• Causal dynamical triangulation

References

1. Shifman, M. (2012). Advanced Topics in Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-19084-8.
2. 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.