# Contents

## Idea

The construction of a perturbative quantum field theory from a given local Lagrangian density (rigorously via causal perturbation theory) involves ambiguities associated with the detailed nature of the quantum processes at point interactions. What is called renormalization is making a choice of fixing these ambiguities to produce a perturbative quantum field theory (Wightman et al. 76).

In causal perturbation theory the renormalization ambiguities are understood as the freedom of extending the time-ordered product of operator-valued distributions to the interaction points, in the sense of extension of distributions. The notorious “infinities that plague quantum field theory” arise only if this extension is not handled correctly and has to be fixed. Hence what historically is called “renormalization” could from a mathematical point of view just be called “normalization” (a point made vividly for instance in Scharf 95, Scharf 01).

The main theorem of perturbative renormalization theory states that any two choices of such (re-)normalizations are uniquely related by a re-definition of the interaction Lagrangian density which introduces further point interactions of higher order (“counter terms”),

The extension of distributions of the time-ordered product may naturally be organized via graphs, the Feynman graphs (Garcia-Bondia & Lazzarini 00, Keller 10, chapter IV), and hence the renormalized perturbative S-matrix defining the perturbative quantum field theory is expressed as a formal power series in renormalized Feynman graphs, the Feynman perturbation series (Keller 10 (IV.12)).

Historically the Feynman perturbation series was motivated from intuition about the would-be path integral, and this is still a popular point of view, albeit its lack of rigorous formulation. One may understand the axiomatics on the S-matrix in causal perturbation theory as defining the result of the path integral without actually doing an integration over field configurations.

But while path integral quantization for perturbative quantum field theory remains elusive, it has been shown that the (re-)normalized perturbative quantum field theory thus constructed via causal perturbation theory is, at least under favorable circumstances, equivalently the (Fedosov) formal deformation quantization of the covariant phase space induced by the given interacting Lagrangian density (Collini 16). This identifies the (re-)normalization freedom with the usual freedom in choosing formal deformation quantization.

This also suggest that the construction of the full non-perturbative quantum field theory ought to be given by a strict deformation quantization of the covariant phase space. But presently no example of such for non-trivial interaction in spacetime dimension $\geq 4$ is known. In particular the phenomenologically interesting case of a complete construction of interacting field theories on 4-dimensional spacetimes is presently unknown. For the case of Yang-Mills theory this open problem is one of the “Millenium Problems” (see at quantization of Yang-Mills theory).

## Definitions

There are different procedures of renormalization

1. Stückelberg-Bogoliubov-Epstein-Glaser renormalizationenormalization)

2. BPHZ and Hopf-Algebraic renormalization

3. Of theories in BV-CS form

### Stückelberg-Bogoliubov-Epstein-Glaser renormalization

Based on Stückelberg-Peterman-53, Bogoliubov-Shirkov 76, developed by Epstein-Glaser 73. Generalized to quantum field theory on curved spacetimes in Brunetti-Fredenhagen 99.

### BPHZ and Hopf-algebraic renormalization

#### The phenomenon

In the study of perturbative quantum field theory one is concerned with functions – called amplitudes – that take a collection of graphs – called Feynman graphs – to Laurent polynomials in a complex variable $z$ – called the (dimensional) regularization parameter

$Amplitude : CertainGraphs \to LaurentPolynomials$

and wishes to extract a “meaningful” finite component when evaluated at vanishing regularization parameter $z = 0$.

A prescription – called renormalization scheme – for adding to a given amplitude in a certain recursive fashion further terms – called counterterms – such that the resulting modified amplitude – called the renormalized amplitude – is finite at $z=0$ was once given by physicists and is called the BPHZ-procedure .

This procedure justifies itself mainly through the remarkable fact that the numbers obtained from it match certain numbers measured in particle accelerators to fantastic accuracy.

#### Its combinatorial Hopf-algebraic interpretation

The combinatorial Hopf algebraic approach to perturbative quantum field theory, see for instance

• Hector Figueroa, Jose M. Gracia-Bondia, Combinatorial Hopf algebras in quantum field theory I (arXiv),

starts with the observation that the BPHZ-procedure can be understood

• by noticing that there is secretly a natural group structure on the collection of amplitudes;

• which is induced from the fact that there is secretly a natural Hopf algebra structure on the vector space whose basis consists of graphs;

• and with respect to which the BPHZ-procedure is simply the Birkhoff decomposition of group valued functions on the circle into a divergent and a finite part.

The Hopf algebra structure on the vector space whose basis consists of graphs can be understood most conceptually in terms of pre-Lie algebras.

#### The Connes-Kreimer theorem

A Birkhoff decomposition of a loop $\phi : S^1 \to G$ in a complex group $G$ is a continuation of the loop to

• a holomorphic function $\phi_+$ on the standard disk inside the circle;

• a holomorphic function $\phi_-$ on the complement of this disk in the projective complex plane

• such that on the unit circle the original loop is reproduced as

$\phi = \phi_+ \cdot (\phi_-)^{-1} \,,$

with the product and the inverse on the right taken in the group $G$.

Notice that by the assumption of holomorphicity $\phi_+(0)$ is a well defined element of $G$.

###### Theorem

(Connes-Kreimer)

1. If $G$ is the group of characters on any graded connected commutative Hopf algebra $H$

$G = Hom(H,\mathbb{C})$

then the Birkhoff decomposition always exists and is given by the formula

$\phi_- : (X \in H) \mapsto Counit(X) - PolePartOf( Product(\phi_- \otimes \phi) \circ (1 \otimes (1 - Counit)) \circ Coproduct (X) ) \,.$
2. There is naturally the structure of a Hopf algebra, $H = Graphs$, on the graphs considered in quantum field theory. As an algebra this is the free commutative algebra on the “1-particle irreducible graphs”. Hence QFT amplitudes can be regarded as characters on this Hopf algebra.

3. The BPHZ renormalization-procedure for amplitudes is nothing but the first item applied to the special case of the second item.

###### Proof

The proof is given in

#### The Hopf-algebra perspective on QFT

This result first of all makes Hopf algebra an organizational principle for (re-)expressing familiar operations in quantum field theory.

Computing the renormalization $\phi_+$ of an amplitude $\phi$ amounts to using the above formula to compute the counterterm $\phi_-$ and then evaluating the right hand side of

$\underbrace{\phi_+}_{renormalized amplitude} = \underbrace{\phi}_{amplitude} \underbrace{\cdot}_{convolution product} \underbrace{\phi_-}_{counterterm} \,,$

where the product is the group product on characters, hence the convolution product of characters.

Every elegant reformulation has in it the potential of going beyond mere reformulation by allowing to see structures invisible in a less natural formulation. For instance Dirk Kreimer claims that the Hopf algebra language allows him to see patterns in perturbative quantum gravity previously missed.

#### Gauge theory and BV-BRST with Hopf algebra

Walter von Suijlekom is thinking about the Hopf-algebraic formulation of BRST-BV methods in nonabelian gauge theory

• Walter von Suijlekom?, Renormalization of gauge fields using Hopf algebra, (arXiv)

he reviews the central idea: the BRST formulation of Yang-Mills theory manifests itself at the level of the resulting bare i.e. unnormalized amplitudes in certain relations satisfied by these, the Slavnov-Taylor identities .

Renormalization of gauge theories is consistent only if these relations are still respected by renormalized amplitudes, too. We can reformulate this in terms of Hopf algebra now:

the relations between amplitudes to be preserved under renormalization must define a Hopf ideal in the Hopf algebra of graphs.

Walter von Suijlekom proves this to be the case for Slavnov-Taylor in his theorem 9 on p. 12

As a payoff, he obtains a very transparent way to prove the generalization of Dyson’s formula to nonabelian gauge theory, which expresses renormalized Green’s functions in terms of unrenormalized Green’s functions “at bare coupling”. This is his corollary 12 on p. 13.

In the context of BRST-BV quantization these statements are subsumed, he says, by the structure encoded in the Hopf ideal which corresponds to imposing the BV-master equation. See also (Suijlekom).

### Of theories in BV-CS form

In (Costello 07) a comparatively simple renormalization procedure is given that applies to theories that are given by action functionals which can be given in the form

$S(\phi) = \langle \phi , Q \phi \rangle + I(\phi)$

where

1. the fields $\phi$ are sections of a graded field bundle $E$ on which $Q$ is a differential, $\langle -,-\rangle$ a compatible antibracket pairing such that $(E,Q, \langle \rangle)$ is a free field theory (as discussed there) in BV-BRST formalism;

2. $I$ is an interaction that is at least cubic.

These are action functionals that are well adapted to BV-BRST formalism and for which there is a quantization to a factorization algebra of observables.

Most of the fundamental theories in physics are of this form, notably Yang-Mills theory. In particular also all theories of infinity-Chern-Simons theory-type coming from binary invariant polynomials are perturbatively of this form, notably ordinary 3d Chern-Simons theory.

For a discussion of just the simple special case of 3d CS see (Costello 11, chapter 5.4 and 5.14).

For comparison of the following with other renormalization schemes, see at (Costello 07, section 1.7).

#### The setup

###### Definition

A free field theory $(E, \langle, -,-\rangle. Q)$:

Write $\mathcal{E}_c \coloneqq \Gamma_{cp}(E)$ for the space of sections of the field bundle of compact support. Write

$\langle -,-\rangle \;\colon\; \mathcal{E}_c \otimes \mathcal{E}_c \to \mathbb{C}$

for the induced pairing on sections

$\langle \phi, \psi\rangle = \int_{x \in X} \langle \phi(x), \psi(x)\rangle_{loc} \,.$

The paring being non-degenerate means that we have an isomorphism $E \stackrel{\simeq}{\to} E^* \otimes Dens_X$ and we write

$E^! \coloneqq E^* \otimes Dens_X \,.$

dynamics

• A differential operator on sections of the field bundle

$Q \;\colon\; \mathcal{E} \to \mathcal{C}$

of degree 1 such that

1. $(\mathcal{E}, Q)$ is an elliptic complex;

2. $Q$ is self-adjoint with respect to $\langle -,-\rangle$ in that for all fields $\phi,\psi \in \mathcal{E}_c$ of homogeneous degree we have $\langle \phi , Q \psi\rangle = (-1)^{{\vert \phi\vert}} \langle Q \phi, \psi\rangle$.

From this data we obtain:

• The action functional $S \colon \mathcal{E}_c \to \mathbb{C}$ of this corresponding free field theory is

$S \;\colon\; \phi \mapsto \int_X \langle \phi, Q \phi\rangle \,.$
• The classical BV-complex is the symmetric algebra $Sym \mathcal{E}^!$ of sections of $E^!$ equipped with the induced action of the differential $Q$ and the pairing

$\{\alpha,\beta\} \coloneqq \int_{x \in X} \langle \alpha(x), \beta(x)\rangle \,.$
###### Definition

An interaction term $I \in \cdots$

(…)

###### Definition

A gauge fixing operator $Q^{GF}$ such that

$H \coloneqq [Q, Q^{GF}]$

#### Operator (heat) kernels and propagators

###### Definition

For $K = \sum K' \otimes K'' \in \mathcal{E} \otimes \mathcal{E}$ the corresponding convolution operator $K \star \colon \mathcal{E} \to \mathcal{E}$ is

$K \star e = (-1)^{\vert e\vert} \sum K' \otimes \langle K'', e\rangle \,.$
###### Definition

For $H \colon \mathcal{E} \to \mathcal{E}$ a linear operator, a heat kernel for it is a function $K_{(-)} \colon \mathbb{R}_{\gt 0} \to \mathcal{E}\otimes \mathcal{E}$ such that for each $t \in \mathbb{R}_{\gt 0}$ the convolution with $K_t$, def. 4, reproduces the exponential of $t\cdotH$:

$K_t \star = \exp(-t H) \;\;\;\;\; \forall t \in \mathbb{R}_{\gt 0} \,.$
###### Proposition

For $H$ a generalized Laplace operator such as the $[Q,Q^{GF}]$ of def. 3 there is a unique heat kernel which is moreover a smooth function of $t$.

###### Definition

For $\phi = \sum \phi' \otimes \phi'' \in Sym^2 \mathcal{E}$ write

$\partial_{\phi} \coloneqq \frac{1}{2} \sum \partial_{\phi''} \partial_{\phi'} \,.$

This is (Costello 07, p. 32).

###### Proposition
$[\partial_\phi, Q] = \partial_{Q \phi} \,.$

#### The renormalization group operator

###### Definition

For $Q^{GF}$ the gauge fixing operator of def. 3, and $K_t$ the heat kernel of the corresponding generalized Laplace operator $H = [Q, Q^{GF}]$ by prop. 1, write for $\epsilon, T \in \mathbb{R}_{\gt 0}$

$P(\epsilon, T) \coloneqq \int_{\epsilon}^T (Q^{GF} \otimes 1) K_t \; d t \;\;\;\;\; \in \mathcal{E} \otimes \mathcal{E}\,.$
###### Definition

Write

$\Gamma(P(\epsilon,T), S) \coloneqq \hbar log \left( \exp\left(\hbar \partial_{P(\epsilon,T)}\right) \exp\left(I/\hbar\right) \right) \;\; \in \mathcal{O}(\mathcal{E}, \mathbb{C}[ [ \hbar ] ]) \,,$

where $\partial_{\cdots}$ is given by def. 6.

#### The path integral

###### Proposition

If $X = *$ is the point, then the path integral over the action functional exists as an ordinary integral and is equal to

$\hbar log \int_{x \in (Im Q^{GF})_{\mathbb{R}}} \exp\left( \tfrac{1}{2} \langle x, Q x\rangle + I(x + a) d \mu \right) = \Gamma(P(0,\infty), I)(a)$

This is (Costello 07, lemma 6.6.2).

###### Remark

For $X$ of positive dimension, the limit

$\underset{\epsilon \to 0}{\lim} \Gamma(P(\epsilon,\infty), I)(a)$

does not in general exist. Renormalization is the process of adding $\hbar$-corrections to the action – the counterterms – such as to make it exist after all. In this case we may regard the limit, by prop. 3, as the definition of the path integral.

#### Renormalized action

###### Definition

Given the action functional $S \colon \phi \mapsto \frac{1}{2}\langle \phi, Q phi\rangle + I(\phi)$, a renormalization is a power series

$I^R(\hbar, \epsilon) = I(\hbar) - \underset{{i \gt 0} \atop { k \geq k}}{\sum} \hbar^i I^{CT}_{i,k}(\epsilon)$

such that the limit

$\underset{\epsilon \to 0}{\lim} \Gamma(P(\epsilon,T), I^R(\epsilon)) \,,$

by def. 8, exists.

The $I^{CT}_{i,k}$ are called the counterterms.

#### Renormalization

###### Definition

A renormalization scheme is a decomposition of functions $\mathcal{A}$ on $(0,1)$, as a vector space, into a direct sum

$\mathcal{A} \simeq \mathcal{A}_{\geq 0} \oplus \mathcal{A}_{\gt 0}$

such that the functions $f \in \mathcal{A}_{\geq 0}$ are non-singular in that $\underset{\epsilon \to 0}{\lim} f(\epsilon)$ exists.

Hence this is a choice of picking the singularities in functions that are not necessarily defined at $\epsilon = 0$.

###### Theorem

Given any choice of renormalization scheme, def. 10, there exists a unique choice of counterterms $\{I^{CT}_{k,i}\}$, def. 9 such that

• each counterterm is in the chosen $\mathcal{A}_{\gt 0}$ as a function of $\epsilon$;

• for $k \gt 0$ the germ of a counterterm at $x \in X$ depends only on the germ of the given field theory data $(E, Q, Q^{GF})$ at that point.

This is (Costello 07, theorem B, p. 38).

###### Definition

Given a renormalization $I^{R}$, write for all $T \in \mathbb{R}_{\ht 0}$

$\Gamma^R(P(0,T), I) \coloneqq \underset{\epsilon \to 0}{\to} \Gamma(P(\epsilon, R), I - I^{CT}(\epsilon)) \,.$
###### Remark

We think of $\Gamma^R(P(0,T), I)$ as the renormalized effective action of the original action at scale? $T$.

###### Proposition
$\Gamma^R(P(0,T'), I) = \Gamma(P(T,T'), \; \Gamma^R(P(0,T), I ))$

holds.

## References

After the original informal suggestions by Tomonaga, Julian Schwinger, Richard Feynman and Freeman Dyson,

• Freeman Dyson, The raditation theories of Tomonaga, Schwinger and Feynman, Phys. Rev. 75, 486, 1949 (pdf)

the mathematics of renormalization was finally understood and summarized in the 1975 Erice Majorana School:

• G. Velo and Arthur Wightman (eds.) Renormalization Theory Proceedings of the 1975 Erice summer school, NATO ASI Series C 23, D. Reidel, Dordrecht, 1976

which included causal perturbation theory, BPHZ renormalization, proof of the forest formula? and the BRST complex method for gauge theory.

Little advancement happened until the identification of Hopf algebra structure in the forest formula? due to

• Dirk Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2 , 303 (1998) (q-alg/9707029)

This finally triggered the formulation of causal perturbation theory in terms of Feynman diagrams in

Thus the original “dark art” of the construction of perturbative quantum field theory via renormalization finds a complete rigorous formulation.

In

it is shown that (at least under favorable circumstances) the construction of perturbative quantum field theory via causal perturbation theory is equivalently Fedosov deformation quantization of the given interacting Lagrangian density, thus identifying the renormalization freedom with the Freedom in choosing a deformation quantization.

### Review

Reviews include

(in the context of effective field theory)

### Stückelberg-Bogoliubov-Epstein-Glaser renormalization

• Stückelberg, E. C. G., and Peterman, A., La normalisation des constants dans la theorie des quanta Helv. Phys. Acta 26, 499 (1953); and earlier references therein

• Bogoliubov, N. N., and Shirkov, D. V., Introduction to the Theory of Quantized Fiels, New York: John Wiley and Sons, 1976, 3rd edition

• Henri Epstein, Vladimir Glaser, The Role of locality in perturbation theory, Annales Poincaré Phys. Theor. A 19 (1973) 211.

Exposition includes

A discussion of Epstein-Glate renormalization in the context of locally covariant perturbative quantum field theory is in

For more see at perturbation theory – In AQFT.

Applications to the renoamrliaztion of Yang-Mills theory on curved background spacetimes is accomplished in

### BPHZ Renormalization

BPHZ renormalization was introduced in particular in

• K. Hepp.: Théorie de la Renormalisation Lect. Notes in Phys. Springer (1969)

Review includes

The original articles on this are

• Alain Connes, Dirk Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000), no. 1, 249–273, hep-th/9912092, MR2002f:81070, doi, II. The $\beta$-function, diffeomorphisms and the renormalization group, Comm. Math. Phys. 216 (2001), no. 1, 215–241; hep-th/0003188, MR2002f:81071, doi

An introduction and review to the Hopf-algebraic description of renormalization is in

A textbook treatment is

• Dirk Kreimer, Knots and Fenyman diagrams , Cambridge Lecture Notes in Physics. 13. Cambridge: Cambridge University Press.
• Joseph C. Varilly, The interface of noncommutative geometry and physics, hep-th/0206007

Some heavywheight automated computations using this formalism are discussed in

• W. van Suijlekom, Representing Feynman graphs on BV-algebras , (arXiv)

### In BV formalism

Discussion in the context of BV-BRST formalism is in

building on

A vaguely related approach earlier appeared in