time-ordered product in nLab

Context

Algebraic Quantum Field Theory

• Idea
• Definition
• On regular polynomial observables
• On local observables
• Properties
• Relation to path integral
• Related concepts
• References

Proof

Recall the following facts:

1. the advanced and retarded propagators $\Delta_{\pm}$ by definition are supported in the future cone/past cone, respectively

   $$supp(\Delta_{\pm}) \subset \overline{V}^{\pm}$$

2. they turn into each other under exchange of their arguments (this cor.):

   $$\Delta_\pm(y,x) = \Delta_{\mp}(x,y) \,.$$

3. the real part $H$ of the Feynman propagator, which by definition is the real part of the Wightman propagator is symmetric (by definition or else by this prop.):

   $$H(x,y) = H(y,x)$$

Using this we compute as follows:

$$\begin{aligned} A_1 \underset{\Delta_{F}}{\star} A_2 \; & = A_1 \underset{\tfrac{i}{2}(\Delta_+ + \Delta_-) + H}{\star} A_2 \\ & = \left\{ \array{ A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_1 \underset{\tfrac{i}{2}\Delta_- + H}{\star} A_2 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \\ & = \left\{ \array{ A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \\ & = \left\{ \array{ A_1 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \\ & = \left\{ \array{ A_1 \underset{\Delta_H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \underset{\Delta_H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \end{aligned}$$

Proposition

(time-ordered product on regular polynomial observables isomorphic to pointwise product)

The time-ordered product on regular polynomial observables (def. ) is isomorphism to the pointwise product of observables (this def.) via the linear isomorphism

$$\mathcal{T} \;\colon\; PolyObs(E)_{reg}[ [\hbar] ] \longrightarrow PolyObs(E)_{reg}[ [\hbar] ]$$

given by

$$\mathcal{T}A \;\coloneqq\; \exp\left( \tfrac{1}{2} \hbar \underset{\Sigma}{\int} \Delta_F(x,y)^{a b} \frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)} \right) A$$

in that

$$\begin{aligned} T(A_1 A_2) & \coloneqq A_1 \star_{F} A_2 \\ & = \mathcal{T}( \mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2) ) \end{aligned}$$

hence

$$\array{ PolyObs(E)_{reg}[ [\hbar] ] \otimes PolyObs(E)_{reg}[ [\hbar] ] &\overset{(-)\cdot (-)}{\longrightarrow}& PolyObs(E)_{reg}[ [\hbar] ] \\ {}^{\mathllap{\mathcal{T} \otimes \mathcal{T}}}_\simeq\Big\downarrow && \downarrow^{\mathrlap{\mathcal{T}}}_\simeq \\ PolyObs(E)_{reg}[ [\hbar] ] \otimes PolyObs(E)_{reg}[ [\hbar] ] &\overset{(-) \star_F (-)}{\longrightarrow}& PolyObs(E)_{reg}[ [\hbar] ] }$$

Example

(time-ordered exponential of regular polynomial observables)

Let

$$V \in PolyObs_{reg, deg = 0}[ [ \hbar ] ]$$

be a regular polynomial observables of degree zero, and write

$$\exp(V) = 1 + V + \tfrac{1}{2!} V \cdot V + \tfrac{1}{3!} V \cdot V \cdot V + \cdots$$

for the exponential of $V$ with respect to the pointwise product.

Then the exponential $\exp_{\mathcal{T}}(V)$ of $V$ with respect to the time-ordered product $\star_F$ (def. ) is equal to the conjugation of the exponential with respect to the pointwise product by the time-ordering isomorphism $\mathcal{T}$ from prop. :

$$\begin{aligned} \exp_{\mathcal{T}}(V) & \coloneqq 1 + V + \tfrac{1}{2} V \star_F V + \tfrac{1}{3!} V \star_F V \star_F V + \cdots \\ & = \mathcal{T} \circ \exp(-) \circ \mathcal{T}^{-1}(V) \end{aligned}$$