nLab operator product expansion

Idea

In quantum field theory the term operator product expansion (OPE) refers to Taylor expansion of products in the algebra of quantum observables of field observables $\mathbf{\Phi}^a(x)$ in the form

(1)$$\mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y) \;\sim\; C^{a b}{}_c(x) \mathbf{\Phi}^c(y) \,.$$

Here “$\sim$” means something like “as $x$ gets close to $y$ the expectation values in the given vacuum state of observables proportional to the left hand side approach those of observables proportional to the right hand side”, schematically:

$$\left\langle \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y) A \right\rangle \underset{x \to y}{\longrightarrow} C^{a b}{}_c(x) \left\langle \mathbf{\Phi}^c(y) A \right\rangle$$

If the algebra of observables is represented via operator-valued distributions acting on some Hilbert space, as commonly assumed, then this becomes an expansion of (distributional) operator products, whence the name operator product expansion (Wilson 69, Wilson-Zimmermann 72).

For conformal field theory and specifically for 2d CFT the operator product expansion is well understood, is neatly captured by the concept of vertex operator algebras. Via the conformal bootstrap the equation (1) defines or at least constrains the conformal field theory (Ferrara-Grillo-Gatto 73, Polyakov 74, Belavin-Polyakov-Zamolodchikov 84).

Attempts to similarly use the concept of the operator product expansion to define more general perturbative quantum field theory (and possibly non-perturbative quantum field theory) are discussed in (Hollands-Olbermann 09, Holland-Hollands 13). The concept of factorization algebras, rooted as it is in that of vertex operator algebras, may be thought of as meaning to axiomatize operator product expansions in the generality of Euclidean (Wick rotated) field theory.

Apparently Kontsevich 10 suggested that perturbative QFT should be thought of as the deformation theory controlled by a certain L-infinity algebra which is constructed from OPEs.

See also

• Vertex operator algebra

References

The general concept is due to

• Kenneth Wilson, Nonlagrangian models of current algebra, Phys. Rev., 179:1499–1512, 1969.

• Kenneth Wilson, Wolfhart Zimmermann, Operator product expansions and composite field operators in the general framework of quantum field theory, Commun. Math. Phys., 24:87–106, 1972

and is further developed in

• Stefan Hollands, Heiner Olbermann, Perturbative Quantum Field Theory via Vertex Algebras, J. Math. Phys. 50:112304, 2009 (arXiv:0906.5313)

• Jan Holland, Stefan Hollands, Operator product expansion algebra, J. Math. Phys. 54, 072302 (2013) (arXiv:1205.4904)

Historical recollections regarding non-perturbative effects in QCD (hadrodynamics) via operator product expansion:

• Mikhail Shifman, OPE-based Methods in Nonperturbative QCD, Eur. Phys. J. C (2022) [arXiv:2208.10600]

Development for QFT on curved spacetimes:

• Stefan Hollands, Robert Wald, Axiomatic quantum field theory in curved spacetime, Commun. Math. Phys. 293:85-125, 2010 (arXiv:0803.2003)

• Stefan Hollands, Robert M. Wald, The Operator Product Expansion in Quantum Field Theory [arXiv:2312.01096]

Review:

• Operator Product Expansion

• Stefan Hollands, Robert Wald, The Operator Product Expansion in Quantum Field Theory, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2312.01096]

The relevance of the concept of OPEs to conformal field theory (conformal bootstrap) was noticed in

• S. Ferrara, A. F. Grillo, and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161–188

• Alexander Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23–42

• Alexander Belavin, Alexander Polyakov, and Alexander Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B241 (1984) 333–380

There was also a talk

• Maxim Kontsevich, talk at Renormalization: algebraic, geometric and probabilistic aspects, Institut Camille Jordan (Lyon) June 16 - 18, 2010

On operator product expansions in 2D conformal field theories (via holography as Koszul duality):

• Kevin Costello, Natalie Paquette, On the associativity of one-loop corrections to the celestial OPE (arXiv:2204.05301)

• Natalie Paquette, 4d physics from 2d chiral correlators (slides, video)