# nLab Wick algebra -- table

free field algebra of quantum observablesphysics terminologymaths 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}

Last revised on January 18, 2018 at 15:50:43. See the history of this page for a list of all contributions to it.