[1] As an axiom, it offers a non-perturbative approach to quantum field theory.
One example is the vertex operator algebra, which has been used to construct two-dimensional conformal field theories.
Whether this result can be extended to QFT in general, thus resolving many of the difficulties of a perturbative approach, remains an open research question.
In practical calculations, such as those needed for scattering amplitudes in various collider experiments, the operator product expansion is used in QCD sum rules to combine results from both perturbative and non-perturbative (condensate) calculations.
[2] OPE Formulation and Application of Thirring Model are conceived by Kenneth G.
Heuristically, in quantum field theory the interest is in the physical observables represented by operators.
In conformal coordinate mappings, the radial ordering is instead more relevant.
Normal ordering of creation operators is useful when working in the second quantization formalism.
The non-normal-ordered terms can often be written as a commutator, and these have useful simplifying identities.
The radial ordering supplies the convergence of the expansion.
This result represents the expansion of two operators at two different points in the original coordinate system as an expansion around just one point in the space of displacements between points, with terms of the form: Related to this is that an operator on the complex plane is in general written as a function of
These are referred to as the holomorphic and anti-holomorphic parts respectively, as they are continuous and differentiable functions with finitely many singularities.
[1] In general, the operator product expansion may not separate into holomorphic and anti-holomorphic parts, especially if there are
Setting x free to live on a manifold, the operator product
In general, such rings do not possess enough structure to make meaningful statements; thus, one considers additional axioms to strengthen the system.
are required to be single-valued functions, rather than sections of some vector bundle.
The above can be viewed as a requirement that is imposed on a ring of functions; imposing this requirement on the fields of a conformal field theory is known as the conformal bootstrap.
It is currently hoped that operator product algebras can be used to axiomatize all of quantum field theory; they have successfully done so for the conformal field theories, and whether they can be used as a basis for non-perturbative QFT is an open research area.