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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).
There are different procedures of renormalization
Stückelberg-Bogoliubov-Epstein-Glaser renormalizationenormalization)
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.
See at causal perturbation theory and locally covariant perturbative quantum field theory.
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 –
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.
The combinatorial Hopf algebraic approach to perturbative quantum field theory, see for instance
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.
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
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 $G$.
(Connes-Kreimer)
If $G$ is the group of characters on any graded connected commutative Hopf algebra $H$
then the Birkhoff decomposition always exists and is given by the formula
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.
The BPHZ renormalization-procedure for amplitudes is nothing but the first item applied to the special case of the second item.
The proof is given in
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
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.
Walter von Suijlekom is thinking about the Hopf-algebraic formulation of BRST-BV methods in nonabelian gauge theory
In his nicely readable
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).
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
where
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;
$I$ 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).
A free field theory $(E, \langle, -,-\rangle. Q)$:
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”)
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 $\mathcal{E}_c \coloneqq \Gamma_{cp}(E)$ for the space of sections of the field bundle of compact support. Write
for the induced pairing on sections
The paring being non-degenerate means that we have an isomorphism $E \stackrel{\simeq}{\to} E^* \otimes Dens_X$ and we write
A differential operator on sections of the field bundle
of degree 1 such that
$(\mathcal{E}, Q)$ is an elliptic complex;
$Q$ 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
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
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
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 $t\cdotH$:
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 $t$.
For $\phi = \sum \phi' \otimes \phi'' \in Sym^2 \mathcal{E}$ write
This is (Costello 07, p. 32).
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 $\epsilon, T \in \mathbb{R}_{\gt 0}$
Write
where $\partial_{\cdots}$ is given by def. 6.
If $X = *$ is the point, then the path integral over the action functional exists as an ordinary integral and is equal to
This is (Costello 07, lemma 6.6.2).
For $X$ of positive dimension, the limit
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.
Given the action functional $S \colon \phi \mapsto \frac{1}{2}\langle \phi, Q phi\rangle + I(\phi)$, a renormalization is a power series
such that the limit
by def. 8, exists.
The $I^{CT}_{i,k}$ are called the counterterms.
A renormalization scheme is a decomposition of functions $\mathcal{A}$ on $(0,1)$, as a vector space, into a direct sum
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 $\epsilon = 0$.
Given any choice of renormalization scheme, def. 10, there exists a unique choice of counterterms $\{I^{CT}_{k,i}\}$, def. 9 such that
This is (Costello 07, theorem B, p. 38).
Given a renormalization $I^{R}$, write for all $T \in \mathbb{R}_{\ht 0}$
We think of $\Gamma^R(P(0,T), I)$ as the renormalized effective action of the original action at scale? $T$.
The renormalization group flow
holds.
After the original informal suggestions by Tomonaga, Julian Schwinger, Richard Feynman and Freeman Dyson,
the mathematics of renormalization was finally understood and summarized in the 1975 Erice Majorana School:
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
This finally triggered the formulation of causal perturbation theory in terms of Feynman diagrams in
J.M. Gracia-Bondia, S. Lazzarini, Connes-Kreimer-Epstein-Glaser Renormalization (arXiv:hep-th/0006106)
Kai Keller, chapter IV of Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization, PhD thesis (arXxiv:1006.2148)
Michael Dütsch, Klaus Fredenhagen, Kai Keller, Katarzyna Rejzner, Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization, J. Math. Phy. 55(12), 122303 (2014) (arXiv:1311.5424)
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.
Reviews include
G. Peter Lepage, What is Renormalization?, talk 1989 (arXiv:hep-ph/0506330)
Steven Weinberg, Effective Field Theory, Past and Future (arXiv:0908.1964)
(in the context of effective field theory)
R. E. Borcherds, Renormalization and quantum field theory, (arxiv/1008.0129)
Arnold Neumaier, Renormalizatin without infinities – an elementary tutorial (pdf)
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
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)
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 was introduced in particular in
Review includes
Klaus Sibold, Bogoliubov-Parasiuk-Hepp-Zimmermann renormalization scheme, Scholarpedia, 5(5):7306. doi:10.4249/scholarpedia.7306
Alain Connes, Matilde Marcolli Noncommutative Geometry, Quantum Fields and Motives
The original articles on this are
An introduction and review to the Hopf-algebraic description of renormalization is in
A textbook treatment is
Some heavywheight automated computations using this formalism are discussed in
See also
Discussion in the context of BV-BRST formalism is in
building on
See also at factorization algebra of observables.
A vaguely related approach earlier appeared in
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