Operator product development

In quantum field theory , operator product development (OPE) is a method for representing a product of two operators at different spacetime points as a series of operators at a single spacetime point. The coefficients of these operators are functions of the difference between the two spacetime points and are called Wilson coefficients according to Kenneth Wilson . Mathematically, the operator product development is:

${\displaystyle \lim _{x\to y}{\mathcal {O}}_{1}(x){\mathcal {O}}_{2}(y)=\sum _{n}C^{(n)}(x-y){\mathcal {O}}^{(n)}(y)}$

The operators contain the information about the physics on short distances, the coefficients the information about the physics on large length scales. ${\displaystyle {\mathcal {O}}^{(n)}}$ ${\displaystyle C^{(n)}}$

Example: Fermi interaction

An application example for an operator product development is the (historical) Fermi interaction which represents an operator product development for two charged Dirac currents and a massive photon propagator. Then: ${\displaystyle j^{\mu }(x)={\bar {\psi }}(x)\gamma ^{\mu }\psi (x)}$ ${\displaystyle D_{\mu \nu }(x,y)=\int \mathrm {d} ^{4}p{\frac {\mathrm {i} g_{\mu \nu }}{\square +m_{W}^{2}}}e^{-\mathrm {i} p(x-y)}}$

${\displaystyle j^{\mu }(x)D_{\mu \nu }j^{\nu }(y)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{m_{W}^{2n}}}j^{\mu }(y)\square ^{n}j_{\mu }(y)}$

In the Fermi interaction, this series is broken off after the first summand, because the Fermi theory of the weak interaction was established for . ${\displaystyle \square j=p^{2}j\ll m_{W}^{2}j}$

literature

• Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, New York 2014, ISBN 978-1-107-03473-0 .