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renormalization

Idea

One way to make sense of the path integral used for quantization of classical field theory to quantum field theory is to define it as a resummation of the the Feynman perturbation series of Feynman diagrams.

But for a generic action functional this prescription produces in each term of this series an ill-defined diverging term. Renormalization is a process of adjusting the action functional such that the classical field theory remains the same, but while at the same time adding interactions whose effect is to counteract these divergences to yield a series that is termwise finite. The terms for these extra interactions in the action functional are therefore called counterterms.

A well-developed method for renormalization, described in more detail below, expresses each of the terms in the perturbation series as a Laurent series in a certain parameter which is divergent at the parameter value of interest. In this description the process of renormalization corresponds to discarding the poles of this series.

The precise way in which this Laurent series is constructed has recently been understood to have an elegant desciption in terms of Hopf algebra and is now known as Hopf algebraic renormalization .

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

(…)

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 graph?s – to Laurent polynomials in a complex variable $z$ – called the (dimensional) regularization parameter –

$$\mathrm{Amplitude}:\mathrm{CertainGraphs}\to \mathrm{LaurentPolynomials}$$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 $\varphi :S^1\to G$ in a complex group $G$ is a continuation of the loop to

• a holomorphic function ${\varphi }_{+}$ on the standard disk inside the circle;

• a holomorphic function ${\varphi }_{-}$ on the complement of this disk in the projective complex plane

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

  $$\varphi ={\varphi }_{+}\cdot ({\varphi }_{-}{)}^{-1},$$\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 ${\varphi }_{+}(0)$ 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 $H$

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

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

   $${\varphi }_{-}:(X\in H)\mapsto \mathrm{Counit}(X)-\mathrm{PolePartOf}(\mathrm{Product}({\varphi }_{-}\otimes \varphi )\circ (1\otimes (1-\mathrm{Counit}))\circ \mathrm{Coproduct}(X)).$$\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, $H=\mathrm{Graphs}$, 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

• Alain Connes, Dirk Kreimer, Renormalization in quantum field theory I (arXiv)

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 ${\varphi }_{+}$ of an amplitude $\varphi$ amounts to using the above formula to compute the counterterm ${\varphi }_{-}$ and then evaluating the right hand side of

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

In his nicely readable

• 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

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

where

1. the fields $\varphi$ are sections of a graded field bundle $E$ on which $Q$ is a differential, $⟨-,-⟩$ a compatible antibracket pairing such that $(E,Q,⟨⟩)$ 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 Chern-Simons theory.

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

1. The setup

2. Operator (heat) kernels and propagators

3. The renormalization group operator

4. The path integral

5. Renormalized action

6. Renormalization

The setup

Definition

A free field theory $(E,⟨,-,-⟩.Q)$:

kinematics:

• A smooth manifold $X$ (“spacetime”/”worldvolume”);

• a $\mathbb{Z}$-graded complex vector bundle $E\to X$ (the “field bundle” containing also in general antifields and ghosts);

• equipped with a bundle homomorphism (the “antibracket density”)

  $$⟨-,-{⟩}_{\mathrm{loc}}:E\times E\to {\mathrm{Dens}}_X$$\langle -,-\rangle_{loc} \;\colon\; E \times E \to Dens_X

  from the fiberwise tensor product of $E$ with itself to the compex density bundle which is fiberwise

  • non-degenerate

  • anti-symmetric

  • of degree -1

Write ${ℰ}_c≔{\Gamma }_{\mathrm{cp}}(E)$ for the space of sections of the field bundle of compact support. Write

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

for the induced pairing on sections

$$⟨\varphi ,\psi ⟩={\int }_{x\in X}⟨\varphi (x),\psi (x){⟩}_{\mathrm{loc}}.$$\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 $E\stackrel{\simeq }{\to }{E}^{*}\otimes {\mathrm{Dens}}_X$ and we write

$$E^{!}≔E^{*}\otimes {\mathrm{Dens}}_X.$$E^! \coloneqq E^* \otimes Dens_X \,.

dynamics

• A differential operator on sections of the field bundle

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

  of degree 1 such that

  1. $(ℰ,Q)$ is an elliptic complex;

  2. $Q$ is self-adjoint with respect to $⟨-,-⟩$ in that for all fields $\varphi ,\psi \in {ℰ}_c$ of homogeneous degree we have $⟨\varphi ,Q\psi ⟩=(-1{)}^{\mid \varphi \mid }⟨Q\varphi ,\psi ⟩$.

From this data we obtain:

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

  $$S:\varphi \mapsto {\int }_X⟨\varphi ,Q\varphi ⟩.$$S \;\colon\; \phi \mapsto \int_X \langle \phi, Q \phi\rangle \,.

• The classical BV-complex is the symmetric algebra $\mathrm{Sym}{ℰ}^{!}$ of sections of $E^{!}$ equipped with the induced action of the differential $Q$ and the pairing

  $$\{\alpha ,\beta \}≔{\int }_{x\in X}⟨\alpha (x),\beta (x)⟩.$$\{\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 $Q^{\mathrm{GF}}$ such that

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

is a generalized Laplace operator.

Operator (heat) kernels and propagators

Definition

For $K=\sum K\prime \otimes K″\in ℰ\otimes ℰ$ the corresponding convolution operator $K\star :ℰ\to ℰ$ is

$$K\star e=(-1{)}^{\mid e\mid }\sum K\prime \otimes ⟨K″,e⟩.$$K \star e = (-1)^{\vert e\vert} \sum K' \otimes \langle K'', e\rangle \,.

Definition

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

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

Proposition

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

Definition

For $\varphi =\sum \varphi \prime \otimes \varphi ″\in {\mathrm{Sym}}^2ℰ$ write

$${\partial }_{\varphi }≔\frac{1}{2}\sum {\partial }_{\varphi ″}{\partial }_{\varphi \prime }.$$\partial_{\phi} \coloneqq \frac{1}{2} \sum \partial_{\phi''} \partial_{\phi'} \,.

This is (Costello 07, p. 32).

Proposition

$$[{\partial }_{\varphi },Q]={\partial }_{Q\varphi }.$$[\partial_\phi, Q] = \partial_{Q \phi} \,.

The renormalization group operator

Definition

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

$$P(ϵ,T)≔{\int }_{ϵ}^T(Q^{\mathrm{GF}}\otimes 1)K_{t}dt\in ℰ\otimes ℰ.$$P(\epsilon, T) \coloneqq \int_{\epsilon}^T (Q^{GF} \otimes 1) K_t \; d t \;\;\;\;\; \in \mathcal{E} \otimes \mathcal{E}\,.

(Costello 07, p. 33)

Definition

Write

$$\Gamma (P(ϵ,T),S)≔\hslash \log \left(\exp \left(\hslash {\partial }_{P(ϵ,T)}\right)\exp (I/\hslash )\right)\in 𝒪(ℰ,ℂ[[\hslash ]]),$$\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.

(Costello 07, def. 6.6.1)

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

$$\hslash \log {\int }_{x\in (\mathrm{Im}{Q}^{\mathrm{GF}}{)}_{ℝ}}\exp (\frac{1}{2}⟨x,Qx⟩+I(x+a)d\mu )=\Gamma (P(0,\mathrm{\infty }),I)(a)$$\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

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

does not in general exist. Renormalization is the process of adding $\hslash$-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 $S:\varphi \mapsto \frac{1}{2}⟨\varphi ,Q\mathrm{phi}⟩+I(\varphi )$, a renormalization is a power series

$$I^R(\hslash ,ϵ)=I(\hslash )-\sum _{\genfrac{}{}{0ex}{}{i>0}{k\ge k}}{\hslash }^i{I}_{i,k}^{\mathrm{CT}}(ϵ)$$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

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

by def. 8, exists.

The $I_{i,k}^{\mathrm{CT}}$ are called the counterterms.

Renormalization

Definition

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

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

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

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

Theorem

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

• each counterterm is in the chosen ${𝒜}_{>0}$ as a function of $ϵ$;

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

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

Definition

Given a renormalization $I^R$, write for all $T\in {ℝ}_{ht0}$

$${\Gamma }^R(P(0,T),I)≔\underset{ϵ\to 0}{\to }\Gamma (P(ϵ,R),I-I^{\mathrm{CT}}(ϵ)).$$\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

The renormalization group flow?

$${\Gamma }^R(P(0,T\prime ),I)=\Gamma (P(T,T\prime ),{\Gamma }^R(P(0,T),I))$$\Gamma^R(P(0,T'), I) = \Gamma(P(T,T'), \; \Gamma^R(P(0,T), I ))

holds.

(Costello 07, lemma 9.0.6).

Related concepts

• renormalization scheme
• perturbation theory
• BV quantization
• gauge fixing, Gribov ambiguity

References

Traditional physics accounts

BPHZ renormalization was introduced in particular in

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

A review is in

• R. E. Borcherds, Renormalization and quantum field theory, arxiv/1008.0129

In BV formalism

Discussion in the context of BV-BRST formalism is in

• Kevin Costello, Renormalization and Effective Field Theory AMS (2011) (publisher webpage)

building on

• Kevin Costello, Renormalisation and the Batalin-Vilkovisky formalism (arXiv:0706.1533)

See also at factorization algebra of observables

In AQFT

A discussion of renormalization in the context of AQFT, as well as a detailed comparison of major renormalization schemes is in

• Romeo Brunetti, Klaus Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds Commun.Math.Phys.208:623-661 (2000) (arXiv)

• Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen, Perturbative Algebraic Quantum Field Theory and the Renormalization Groups Adv. Theor. Math. Physics 13 (2009), 1541-1599 (arXiv:0901.2038)

Hopf algebraic BPHZ formulation

• 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

• Christian Brouder, Quantum field theory meets Hopf algebra (arxiv:hep-th/0611153)

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

• D. J. Broadhurst, Dirk Kreimer, Renormalization automated by Hopf algebra (arXiv:hep-th/9810087)

See also

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

Operadic description

• Jean-Louis Loday, Nikolay M. Nikolov, Operadic construction of the renormalization group (arXiv:1202.1206)

Relations to motives, polylogarithms, positivity

• Alexander B. Goncharov, Marcus Spradlin, C. Vergu, Anastasia Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys.Rev.Lett.105:151605,2010 arxiv/1006.5703

• Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, Jaroslav Trnka, Scattering amplitudes and the positive Grassmannian, arxiv/1212.5605

• Spencer Bloch?, Hélène Esnault, Dirk Kreimer, On motives associated to graph polynomials, Commun.Math.Phys. 267 (2006) 181-225 math.AG/0510011 doi