# 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:

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

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 (xy)}}$

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

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 .