nLab
interaction vertex redefinition

Contents

Context

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

In perturbative quantum field theory an (adiabatically switch) interaction action functional is a local observable of the form

gS int+jALocObs(E BV-BRST)[[,g,j]]g,j. g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle \,.

In the Feynman perturbation series this is interpreted as the label of vertices in multigraphs underlying Feynman diagrams which encode field interactions at that point.

In the process of (re-)normalization of perturbative QFT one considers redefinitions of such “interaction vertices” by terms of higher order in the coupling constant gg ad source field jj:

𝒵:gS int+jAgS int+jA+S counter𝒪(g 2,j 2,gj). \mathcal{Z} \;\colon\; g S_{int} + j A \;\mapsto\; g S_{int} + j A + \underset{ \mathcal{O}(g^2, j^2, g j) }{ \underbrace{ S_{counter} }} \,.

In the context of effective QFT via UV cutoffs, the correction term S counterS_{counter} is called a counterterm.

Under mild assumptions, such interaction vertex redefinitions form a group, called the Stückelberg-Petermann renormalization group. See there for more.

Definition

Definition

(perturbative interaction vertex redefinition)

Let (E BV-BRST,L,Δ H)(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H) be a gauge fixed free field vacuum (this def.).

A perturbative interaction vertex redefinition (or just vertex redefinition, for short) is an endofunction

𝒵:LocObs(E BV-BRST)[[,g,j]]g,jLocObs(E BV-BRST)[[,g,j]]g,j \mathcal{Z} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle \longrightarrow LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle

on local observables with formal parameters adjoined (this def.) such that there exists a sequence {Z k} k\{Z_k\}_{k \in \mathbb{N}} of continuous linear functionals, symmetric in their arguments, of the form

(LocObs(E BV-BRST)[[,g,j]]g,j) [[,g,j]] kLocObs(E BV-BRST)[[,g,j]]g,j \left( {\, \atop \,} LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle {\, \atop \,} \right)^{\otimes^k_{\mathbb{C}[ [ \hbar, g, j] ]}} \longrightarrow LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle

such that for all gS int+jALocObs(E BV-BRST)[[,g,j]]g,jg S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle the following conditions hold:

  1. (perturbation)

    1. Z 0(gS int+jA)=0Z_0(g S_{int + j A}) = 0

    2. Z 1(gS int+jA)=gS int+jAZ_1(g S_{int} + j A) = g S_{int} + j A

    3. and

      𝒵(gS int+jA) =Zexp (gS int+jA) k1k!Z k(gS int+jA,,gS int+jAkargs) \begin{aligned} \mathcal{Z}(g S_{int} + j A) & = Z \exp_\otimes( g S_{int} + j A ) \\ & \coloneqq \underset{k \in \mathbb{N}}{\sum} \frac{1}{k!} Z_k( \underset{ k \, \text{args} }{ \underbrace{ g S_{int} + j A , \cdots, g S_{int} + j A } } ) \end{aligned}
  2. (field independence) The local observable 𝒵(gS int+jA)\mathcal{Z}(g S_{int} + j A) depends on the field histories only through its argument gS int+jAg S_{int} + j A , hence by the chain rule:

    (1)δδΦ a(x)𝒵(gS int+jA)=𝒵 gS int+jA(δδΦ a(x)(gS int+jA)) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \mathcal{Z}(g S_{int} + j A) \;=\; \mathcal{Z}'_{g S_{int} + j A} \left( \frac{\delta}{\delta \mathbf{\Phi}^a(x)} (g S_{int} + j A) \right)

Properties

The following proposition should be compared to the axiom of causal additivity of the S-matrix scheme (this equation):

Proposition

(local additivity of vertex redefinitions)

Let (E BV-BRST,L,Δ H)(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H) be a gauge fixed free field vacuum (this def.) and let 𝒵\mathcal{Z} be a vertex redefinition (def. ).

Then for all local observables O 0,O 1,O 2LocObs(E BV-BRST)[[,g,j]]g,jO_0, O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g, j\rangle with spacetime support denoted supp(O i)Σsupp(O_i) \subset \Sigma (this def.) we have

  1. (local additivity)

    (supp(O 1)supp(O 2)=) AA𝒵(O 0+O 1+O 2)=𝒵(O 0+O 1)𝒵(O 0)+𝒵(O 0+O 2). \begin{aligned} & \left( supp(O_1) \cap supp(O_2) = \emptyset \right) \\ & \Rightarrow \phantom{AA} \mathcal{Z}( O_0 + O_1 + O_2) = \mathcal{Z}( O_0 + O_1 ) - \mathcal{Z}(O_0) + \mathcal{Z}(O_0 + O_2) \end{aligned} \,.
  2. (preservation of spacetime support)

    supp(𝒵(O 0+O 1)𝒵(O 0))supp(O 1) supp \left( {\, \atop \,} \mathcal{Z}(O_0 + O_1) - \mathcal{Z}(O_0) {\, \atop \,} \right) \;\subset\; supp(O_1)

    hence in particular

    supp(𝒵(O 1))=supp(O 1) supp \left( {\, \atop \,} \mathcal{Z}(O_1) {\, \atop \,} \right) = supp(O_1)

(Dütsch 18, exercise 3.98)

Proof

Under the inclusion

LocObs(E BV-BRST)PolyObs(E BV-BRST) LocObs(E_{\text{BV-BRST}}) \hookrightarrow PolyObs(E_{\text{BV-BRST}})

of local observables into polynomial observables we may think of each Z kZ_k as a generalized function, as for time-ordered products in this remark.

Hence if

O j=Σj Σ (L j) O_j = \underset{\Sigma}{\int} j^\infty_\Sigma( \mathbf{L}_j )

is the transgression of a Lagrangian density L\mathbf{L} we get

Z k((O 1+O 2+O 3),,(O 1+O 2+O 3))=j 1,,j k{0,1,2}Σ kZ(L j 1(x 1),,L j k(x k)). Z_k( (O_1 + O_2 + O_3) , \cdots , (O_1 + O_2 + O_3) ) = \underset{ j_1, \cdots, j_k \in \{0,1,2\} }{\sum} \underset{\Sigma^{k}}{\int} Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) ) \,.

Now by definition Z k()Z_k(\cdots) is in the subspace of local observables, i.e. those polynomial observables whose coefficient distributions are supported on the diagonal, which means that

δδΦ a(x)δδΦ b(y)Z k()=0AAforAAxy \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \frac{\delta}{\delta \mathbf{\Phi}^b(y)} Z_{k}(\cdots) = 0 \phantom{AA} \text{for} \phantom{AA} x \neq y

Together with the axiom “field independence” (1) this means that the support of these generalized functions in the integrand here must be on the diagonal, where x 1==x kx_1 = \cdots = x_k.

By the assumption that the spacetime supports of O 1O_1 and O 2O_2 are disjoint, this means that only the summands with j 1,,j k{0,1}j_1, \cdots, j_k \in \{0,1\} and those with j 1,,j k{0,2}j_1, \cdots, j_k \in \{0,2\} contribute to the above sum. Removing the overcounting of those summands where all j 1,,j k{0}j_1, \cdots, j_k \in \{0\} we get

Z k((O 1+O 2+O 3),,(O 1+O 2+O 3)) =j 1,,j k{0,1}Σ kZ(L j 1(x 1),,L j k(x k)) =j 1,,j k{0}Σ kZ(L j 1(x 1),,L j k(x k)) =j 1,,j k{0,2}Σ kZ(L j 1(x 1),,L j k(x k)) =Z k((O 0+O 1),,(O 0+O 1))Z k(O 0,,O 0)+Z k((O 0+O 2),,(O 0+O 2)). \begin{aligned} & Z_k\left( {\, \atop \,} (O_1 + O_2 + O_3) , \cdots , (O_1 + O_2 + O_3) {\, \atop \,} \right) \\ & = \underset{ j_1, \cdots, j_k \in \{0,1\} }{\sum} \underset{\Sigma^{k}}{\int} Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) ) \\ & \phantom{=} - \underset{ j_1, \cdots, j_k \in \{0\} }{\sum} \underset{\Sigma^{k}}{\int} Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) ) \\ & \phantom{=} - \underset{ j_1, \cdots, j_k \in \{0,2\} }{\sum} \underset{\Sigma^{k}}{\int} Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) ) \\ & = Z_k\left( {\, \atop \,} (O_0 + O_1), \cdots, (O_0 + O_1) {\, \atop \,}\right) - Z_k\left( {\, \atop \,} O_0, \cdots, O_0 {\, \atop \,} \right) + Z_k\left( {\, \atop \,} (O_0 + O_2), \cdots, (O_0 + O_2) {\, \atop \,} \right) \end{aligned} \,.

This directly implies the claim.

References

The original discussion is due to

in the context of the Stückelberg-Petermann renormalization group.

For more see the references there and those at main theorem of perturbative renormalization.

Last revised on January 31, 2018 at 09:51:41. See the history of this page for a list of all contributions to it.