nLab
time-ordered product

Context

Algebraic Quantum Field Theory

(, , )

Concepts

: , , ,

    • ,

      • ,

  • ,

,

    • , ,

  • ,

    • /

    • ,

      • ,

  • ,

    • ,

  • ,

  • ,

  • ,

    • ,

    • ,

    • /

    • ,

Theorems

States and observables

Operator algebra

    • ,

Local QFT

  • ()

Perturbative QFT

Contents

Idea

In relativistic perturbative quantum field theory, the time-ordered product is a product on suitably well-behave observables which re-orders its arguments according to the causal ordering of their spacetime supports befor multiplying with the Wick algebra product.

(Analogously reverse causal ordering this is called the reverse-time ordered or anti-time ordered prouct.)

For example for point-evaluation field observables and distinct events x,yΣx,y \in \Sigma the time-ordered product is defined by

T(Φ a(x)Φ b(y)){Φ a(x)Φ a(y) | xnot in the past ofy Φ a(y)Φ a(x) | otherwise T(\mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y)) \;\coloneqq\; \left\{ \array{ \mathbf{\Phi}^a(x) \mathbf{\Phi}^a(y) &\vert& x\, \text{not in the past of}\ y \\ \mathbf{\Phi}^a(y) \mathbf{\Phi}^a(x) &\vert& \text{otherwise} } \right.

This may be understood as arising from the causal additivity-axiom of the perturbative S-matrix. It generalizes the 1-dimensional time-ordering (path ordering) of the Dyson series in quantum mechanics.

More precisely, the time-ordere product is a commutative algebra-structure on the microcausal polynomial observables of a free Lagrangian field theory equipped with a vacuum state (Hadamard state) which on regular polynomial observables given on the regular polynomial observables by the star product which is induced (via this def.) by the Feynman propagator and which is extended from there, in the sense of extensions of distributions, to all microcausal polynomial observables. (This extension is the “renormalization” of the time-ordered product).

Definition

On regular polynomial observables

Definition

(time-ordered product on regular polynomial observables)

Let (E,L)(E,\mathbf{L}) be a free Lagrangian field theory over a Lorentzian spacetime and with Green-hyperbolic Euler-Lagrange differential equations; write Δ S=Δ +Δ \Delta_S = \Delta_+ - \Delta_- for the induced causal propagator. Let moreover Δ H=i2Δ S+H\Delta_H = \tfrac{i}{2}\Delta_S + H be a compatible Wightman propagator and write Δ F=i2(Δ ++Δ )+H\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H for the induced Feynman propagator.

Then the time-ordered product on the space of off-shell regular polynomial observable PolyObs(E) regPolyObs(E)_{reg} is the star product induced by the Feynman propagator (via this prop.):

PolyObs(E) reg[[]]PolyObs(E) reg[[]] PolyObs(E) reg[[]] (A 1,A 2) A 1 FA 2 \array{ PolyObs(E)_{reg}[ [\hbar] ] \otimes PolyObs(E)_{reg}[ [\hbar] ] &\overset{}{\longrightarrow}& PolyObs(E)_{reg}[ [\hbar] ] \\ (A_1, A_2) &\mapsto& \phantom{\coloneqq} A_1 \star_F A_2 }

hence

A 1 FA 2(()())exp(Σ×ΣΔ F ab(x,y)δδΦ a(x)δδΦ b(y)dvol Σ(x)dvol Σ(y)) A_1 \star_F A_2 \; \coloneqq \; ((-)\cdot(-)) \circ \exp\left( \underset{\Sigma \times \Sigma}{\int} \Delta_F^{a b}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \, dvol_\Sigma(x) \, dvol_\Sigma(y) \right)

(Notice that this does not descend to the on-shell observables, since the Feynman propagator is not a solution to the homogeneous equations of motion.)

Proposition

(time-ordered product is indeed causally ordered Wick algebra product)

Let (E,L)(E,\mathbf{L}) be a free Lagrangian field theory over a Lorentzian spacetime and with Green-hyperbolic Euler-Lagrange differential equations; write Δ S=Δ +Δ \Delta_S = \Delta_+ - \Delta_- for the induced causal propagator. Let moreover Δ H=i2Δ S+H\Delta_H = \tfrac{i}{2}\Delta_S + H be a compatible Wightman propagator and write Δ F=i2(Δ ++Δ )+H\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H for the induced Feynman propagator.

Then the time-ordered product on regular polynomial observables (def. ) is indeed a time-ordering of the Wick algebra product H\star_H in that for all pairs of regular polynomial observables

A 1,A 2PolyObs(E) reg[[]] A_1, A_2 \in PolyObs(E)_{reg}[ [\hbar] ]

with disjoint spacetime support we have

T(A 1A 2)={A 1 HA 2 | supp(A 1)supp(A 2) A 2 HA 1 | supp(A 2)supp(A 2). T(A_1 A_2) \;=\; \left\{ \array{ A_1 \star_H A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2) \\ A_2 \star_H A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2) } \right. \,.

Here S 1S 2S_1 {\vee\!\!\!\wedge} S_2 is the causal order relation (“S 1S_1 does not intersect the past cone of S 2S_2”). Beware that for general pairs (S 1,S2)(S_1, S-2) of subsets neither S 1S 2S_1 {\vee\!\!\!\wedge} S_2 nor S 2S 1S_2 {\vee\!\!\!\wedge} S_1.

Proof

Recall the following facts:

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

    supp(Δ ±)V¯ ± supp(\Delta_{\pm}) \subset \overline{V}^{\pm}
  2. they turn into each other under exchange of their arguments (this cor.):

    Δ ±(y,x)=Δ (x,y). \Delta_\pm(y,x) = \Delta_{\mp}(x,y) \,.
  3. the real part HH 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) H(x,y) = H(y,x)

Using this we compute as follows:

A 1Δ FA 2 =A 1i2(Δ ++Δ )+HA 2 ={A 1i2Δ ++HA 2 | supp(A 1)supp(A 2) A 1i2Δ +HA 2 | supp(A 2)supp(A 2) ={A 1i2Δ ++HA 2 | supp(A 1)supp(A 2) A 2i2Δ ++HA 1 | supp(A 2)supp(A 2) ={A 1i2(Δ +Δ )+HA 2 | supp(A 1)supp(A 2) A 2i2(Δ +Δ )+HA 1 | supp(A 2)supp(A 2) ={A 1Δ HA 2 | supp(A 1)supp(A 2) A 2Δ HA 1 | supp(A 2)supp(A 2) \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

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

given by

𝒯Aexp(12ΣΔ F(x,y) abδ 2δΦ a(x)δΦ b(y))A \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

T(A 1A 2) A 1 FA 2 =𝒯(𝒯 1(A 1)𝒯 1(A 2)) \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

PolyObs(E) reg[[]]PolyObs(E) reg[[]] ()() PolyObs(E) reg[[]] 𝒯𝒯 𝒯 PolyObs(E) reg[[]]PolyObs(E) reg[[]] () F() PolyObs(E) reg[[]] \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] ] }

(Brunetti-Dütsch-Fredenhagen 09, (12)-(13), Fredenhagen-Rejzner 11b, (14))

Proof

Since the Feynman propagator is symmetric (this prop.), the statement is a special case of this prop.).

Example

(time-ordered exponential of regular polynomial observables)

Let

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

be a regular polynomial observables of degree zero, and write

exp(V)=1+V+12!VV+13!VVV+ \exp(V) = 1 + V + \tfrac{1}{2!} V \cdot V + \tfrac{1}{3!} V \cdot V \cdot V + \cdots

for the exponential of VV with respect to the pointwise product.

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

exp 𝒯(V) 1+V+12V FV+13!V FV FV+ =𝒯exp()𝒯 1(V) \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}

On local observables

The time-ordered product on regular polynomial observables from prop. extends to a product on polynomial local observables, then taking values in microcausal observables:

T:PolyLocObs(E) n[[]]PolyObs(E) mc[[]]. T \;\colon\; PolyLocObs(E)^{\otimes_n}[ [\hbar] ] \longrightarrow PolyObs(E)_{mc}[ [\hbar] ] \,.

This extension is not unique. A choice of such an extension, satisfying some evident compatibility conditions, is a choice of renormalization scheme for the given perturbative quantum field theory. Every such choice corresponds to a choice of perturbative S-matrix for the theory. This construction is called causal perturbation theory.

product in \,\, induces
( )
()

\,

terminology terminology
1)AA:A 1A 2:\phantom{AA} :A_1 A_2:
AAA 1A 2\phantom{AA} A_1 \cdot A_2
pointwise product of functionals
2)
( induced by )
AAA 1A 2\phantom{AA} A_1 A_2
AAA 1 HA 2\phantom{AA} A_1 \star_H A_2
for
3)AAT(A 1A 2)\phantom{AA} T(A_1 A_2)
AAA 1 FA 2\phantom{AA} A_1 \star_F A_2
for

of 2) via 1)

for 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}

of 3) via 1)

for 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}

References

See also the references at S-matrix

The equivalence of the time-ordered product on regular observables to the point-wise product was maybe first highlighted in

and then further amplified in

Last revised on August 1, 2018 at 08:17:55. See the history of this page for a list of all contributions to it.