Wick's lemma



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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 observablesphysics terminologymaths terminology
1)supercommutative productAA:A 1A 2:\phantom{AA} :A_1 A_2:
normal ordered product
AAA 1A 2\phantom{AA} A_1 \cdot A_2
pointwise product of functionals
2)non-commutative product
(deformation induced by Poisson bracket)
AAA 1A 2\phantom{AA} A_1 A_2
operator product
AAA 1 HA 2\phantom{AA} A_1 \star_H A_2
star product for Wightman propagator
3)AAT(A 1A 2)\phantom{AA} T(A_1 A_2)
time-ordered product
AAA 1 FA 2\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 Δ H\Delta_H
A 1 HA 2= (()())exp((Δ H) ab(x,y)δδΦ a(x)δδΦ b(y))(A 1A 2) \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 Δ F\Delta_F
A 1 FA 2= (()())exp((Δ F) ab(x,y)δδΦ a(x)δδΦ b(y))(A 1A 2) \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).


Let 𝒲\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:

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

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

(PolyObs(E,L) mc, Δ H)Wick's lemma(End(FockSpace),) \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)


Original articles include

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

and Dütsch 18, theorem 2.17

Textbook accounts include

See also

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)

Last revised on December 23, 2017 at 13:45:19. See the history of this page for a list of all contributions to it.