nLab Wick's lemma

Idea

Wick’s lemma is a combinatorial formula for the product in associative algebras that appear in the context of free field theory.

From the point of view of path integral quantization, Wick’s lemma is about the moments of Gaussian probability distributions. See at Feynman diagram for more on this.

From the point of view of causal perturbation theory Wick’s lemma expresses the Moyal deformation quantization of a free field theory (Wick algebras) in terms of operator products on.

	free field algebra of quantum observables	physics terminology	maths terminology
1)	supercommutative product	$\phantom{AA} :A_1 A_2:$ normal ordered product	$\phantom{AA} A_1 \cdot A_2$ pointwise product of functionals
2)	non-commutative product (deformation induced by Poisson bracket)	$\phantom{AA} A_1 A_2$ operator product	$\phantom{AA} A_1 \star_H A_2$ star product for Wightman propagator
3)		$\phantom{AA} T(A_1 A_2)$ time-ordered product	$\phantom{AA} A_1 \star_F A_2$ star product for Feynman propagator
	perturbative expansion of 2) via 1)	Wick's lemma	Moyal product for Wightman propagator $\Delta_H$ $\begin{aligned} & A_1 \star_H A_2 = \\ & ((-)\cdot (-)) \circ \exp \left( \hbar \int (\Delta_H)^{ab}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \right)(A_1 \otimes A_2) \end{aligned}$
	perturbative expansion of 3) via 1)	Feynman diagrams	Moyal product for Feynman propagator $\Delta_F$ $\begin{aligned} & A_1 \star_F A_2 = \\ & ((-)\cdot (-)) \circ \exp \left( \hbar \int (\Delta_F)^{ab}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \right)(A_1 \otimes A_2) \end{aligned}$

From the point of view of BV-quantization Wick’s lemma arises as a consequence of the homological perturbation lemma (Gwilliam, section 2.3).

Statement

Let $\mathcal{W}$ be the Wick algebra of the free scalar field, hence the space of microcausal observables with product the star product induced by the Wightman propagator:

$$\mathcal{W} \;\coloneqq\; \left( PolyObs(E,\mathbf{L})_{mc}, \star_{\Delta_H} \right)$$

Then the evident map from $\mathcal{W}$ to linear operators on the Fock space equipped with their operator product $\circ$ is an associative algebra isomorphism onto its image:

$$\left( PolyObs(E,\mathbf{L})_{mc}, \star_{\Delta_H} \right) \underoverset{\simeq}{\text{Wick's lemma}}{\hookrightarrow} \left( End(FockSpace), \circ \right)$$

(Dütsch 18, theorem 2.17, following Dütsch-Fredenhagen 00,, pages 10-11, Dütsch-Fredenhagen 01)

Related concepts

• perturbation theory

• Feynman diagram

References

Original articles include

• Gian-Carlo Wick, The evaluation of the collision matrix, Phys. Rev. 80, 268-272 (1950)

• Klaus Hepp, Théorie de la Renormalisation Lect. Notes in Physics, Springer 1969

• Romeo Brunetti, Klaus Fredenhagen, Rainer Verch, theorem 2.4 in The generally covariant locality principle – A new paradigm for local quantum physics, Commun.Math.Phys.237:31-68, 2003 (arXiv:math-ph/0112041)

The interpretation as an algebra isomorphism to the star product with respect to the Wightman propagator is made explicit in

• Michael Dütsch, Klaus Fredenhagen, pages 10-11 of Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion, Commun.Math.Phys. 219 (2001) 5-30 (arXiv:hep-th/0001129)

• Michael Dütsch, Klaus Fredenhagen, Perturbative algebraic quantum field theory and deformation quantization, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) (arXiv:hep-th/0101079)

and Dütsch 18, theorem 2.17

Textbook accounts include

• Günter Scharf, theorem 3.1 in Finite Quantum Electrodynamics – The Causal Approach, Berlin: Springer-Verlag, 1995, 2nd edition

• Michael Dütsch, E.26 - E.30 in From classical field theory to perturbative quantum field theory, 2018

See also

• Wikipedia, Wick’s theorem

Discussion of Wick’s lemma as a consequence of the homological perturbation lemma for BV-complexes is in

• Owen Gwilliam, section 2.3 Factorization algebras and free field theories PhD thesis (pdf)