October 23, 2008

Hopf Algebraic Renormalization

Posted by Urs Schreiber

The basic idea and starting point of Hopf algebra methods in renormalization of quantum field theories.

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 diagrams – to Laurent polynomials in a complex variable $z$ – called the (dimensional) regularization parameter$\mathrm{Amplitude}:\mathrm{CertainGraphs}\to \mathrm{LaurentPolynomials}$ 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 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 collection 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 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

$\varphi ={\varphi }_{+}\cdot \left({\varphi }_{-}{\right)}^{-1}\phantom{\rule{thinmathspace}{0ex}},$ with the product and the inverse on the right taken in the group $G$. Notice that by the assumption of holomorphicity ${\varphi }_{+}\left(0\right)$ is a well defined element of $G$.

Now: CK-Theorem:

a) If $G$ is the group of characters on any graded connected commutative Hopf algebra $H$$G=\mathrm{Hom}\left(H,ℂ\right)$ – then the Birkhoff decomposition always exists and is given by the formula ${\varphi }_{-}:\left(X\in H\right)↦\mathrm{Counit}\left(X\right)-\mathrm{PolePartOf}\left(\mathrm{Product}\left({\varphi }_{-}\otimes \varphi \right)\circ \left(1\otimes \left(1-\mathrm{Counit}\right)\right)\circ \mathrm{Coproduct}\left(X\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

b) 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.

c) The BPHZ renormalization-procedure for amplitudes is nothing but a) applied to the special case b).

Proof: A. Connes, D. Kreimer, Renormalization in quantum field theory I.

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 ${\underset{⏟}{{\varphi }_{+}}}_{\mathrm{renormalized}\mathrm{amplitude}}={\underset{⏟}{\varphi }}_{\mathrm{amplitude}}{\underset{⏟}{\cdot }}_{\mathrm{convolution}\mathrm{product}}{\underset{⏟}{{\varphi }_{-}}}_{\mathrm{counterterm}}\phantom{\rule{thinmathspace}{0ex}},$ 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

I learned most of what I know about this from Walter von Suijlekom, who I stayed with at the program at the Hausdorff Institute in Bonn over last summer.

Walter is thinking about the Hopf-algebraic formulation of BRST-BV methods in nonabelian gauge theory and gave a couple of talks to us about his work in Bonn.

In his nicely readable Renormalization of gauge fields using Hopf algebra 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 identies.

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 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. This is in his latest:

W. van Suijlekom: Representing Feynman graphs on BV-algebras.

Posted at October 23, 2008 5:52 PM UTC

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Re: Hopf Algebraic Renormalization

This is something that I never really understood.

AFAIU, the Hopf algebra way to compute a Feynman diagram consists of cutting it into two in all possible ways, and then reducing the calculation to the two subdiagrams, which are simpler. However, it seems to me that the procedure comes to a halt once the diagram is 1PI. Since the computation of 1PI diagrams is the hard part, how useful is this really?

Re: Hopf Algebraic Renormalization

AFAIU, the Hopf algebra way to compute a Feynman diagram consists of cutting it into two in all possible ways, and then reducing the calculation to the two subdiagrams, which are simpler

The Hopf algebra structure is a way to understand and equivalently reformulat the structure of the counterterm in the BPHZ regularization prescription. That involves looking at subgraphs of graphs in a recursive way. If you already know the latter procedure, then learning that this can be understood in terms of convolution products of characters on a Hopf alegbra of graphs may tell you little you didn’t know before as far as the mere computation of renormalized amplitudes is concerned.

how useful is this really?

I think the point of the Hopf-algebraic reformulation of the well-known BPHZ-renormalization scheme is that it may help to organize your thoughts about the latter and then see further than has been seen so far.

For instance, as I mentioned above, Dirk Kreimer claims (in Not so non-renormalizable gravity) to see previously missed patterns in perturbative quantum gravity this way, and Walter van Suijlekom shows (in Representing Feynman graphs on BV-algebras) how to clarify structures in perturbative nonabelian gauge theory, such as Dyson’s formula and the relation of Slavnov-Taylor identities to perturbative BRST-BV quantization.

I suppose it may be fair to say that at the moment it is not clear yet if the average pure QFT theorist interested in cranking out numbers profits from Hopf algebraic reformulations, but it seems clear that the method sheds light on the structure of perturbation theory as such.

Posted by: Urs Schreiber on October 24, 2008 10:15 AM | Permalink | Reply to this

Re: Hopf Algebraic Renormalization

As to the last comment:
although the average QFT theorist might not really need the Hopf algebra structure, do you think that a computer wanting to learn QFT might find it useful? (ie would it be useful for automated computations?)

Posted by: Simon on October 24, 2008 11:07 AM | Permalink | Reply to this

Re: Hopf Algebraic Renormalization

That was the point of papers like this. Is that the kind of calculation ‘average pure QFT theorists’ want to make?

Posted by: David Corfield on October 24, 2008 11:38 AM | Permalink | Reply to this

Re: Hopf Algebraic Renormalization

That was the point of papers like this. Is that the kind of calculation ‘average pure QFT theorists’ want to make?

Yes, thanks, good example. Let’s quote the abstract:

It is easy to sum chain-free self-energy rainbows, to obtain contributions to anomalous dimensions. It is also easy to resum rainbow-free self-energy chains. Taming the combinatoric explosion of all possible nestings and chainings of a primitive self-energy divergence is a much more demanding problem. We solve it in terms of the coproduct $\Delta$, antipode $S$, and grading operator $Y$ of the Hopf algebra of undecorated rooted trees. The vital operator is $S\star Y$, with a star product effected by $\Delta$. We perform 30-loop Padé-Borel resummation of 463 020 146 037 416 130 934 BPHZ subtractions in Yukawa theory, at spacetime dimension $d=4$, and in a trivalent scalar theory, at $d=6$, encountering residues of $S\star Y$ that involve primes with up to 60 digits. Even with a very large Yukawa coupling, $g=30$, the precision of resummation is remarkable; a 31-loop calculation suggests that it is of order ${10}^{-8}$.

#

Posted by: Urs Schreiber on October 24, 2008 11:49 AM | Permalink | Reply to this

Re: Hopf Algebraic Renormalization

We perform 30-loop Padé-Borel resummation of 463 020 146 037 416 130 934 BPHZ subtractions in Yukawa theory

by hand?

Posted by: jim stasheff on October 24, 2008 2:44 PM | Permalink | Reply to this

Re: Hopf Algebraic Renormalization

would it be useful for automated computations?

I wouldn’t be surprised, sure. For one, the formalism allows to easily explain to a computer theorist who knows nothing about QFT at all the algorithm which he should try to implement.

I mean: this is one of the big adavantegs of these reformulations of operational knowledge into a clear formalism. I see this a lot here in interaction between mathematicians and physicists: there are a bunch of formalizations to which the physicists say “looks like a weird way to say what we have been saying for decades”, but which is necessary to get mathematicians in the position of being able to contribute. And the same may be true for computers.

Posted by: Urs Schreiber on October 24, 2008 11:42 AM | Permalink | Reply to this

Re: Hopf Algebraic Renormalization

But a computer would still need to know the value of the 1PI diagrams (or the correct elementary diagrams that terminate the coproduct, if different). If there are infinitely many of these, you still need to compute a lot of Feynman integrals.

I think (but there are decades since I knew about this, so I might misremember) that you can make a Legendre transformation to the effective action, and then you only need to compute 1PI diagrams.

Re: Hopf Algebraic Renormalization

you still need to compute a lot of Feynman integrals.

That’s true. The Hopf alegbra formalism knows nothing about Feynman integrals. In fact it works just as well for any assignment of numbers(depending on a regularization parameter) to graphs that you can dream up.

The Hopf algebra technique only comes in at the next stage: after you know your Feynman rules and Feynman integrals, it tells you how to renormalize them.

Posted by: Urs Schreiber on October 24, 2008 1:26 PM | Permalink | Reply to this

Re: Hopf Algebraic Renormalization

By the way, could somebody remind me/teach me the way to make the following correct:

there is a monoidal category $\mathrm{TheoryOfGroups}$ which is the “theory of groups” in that monoidal functors from $\mathrm{TheoryOfGroups}$ to $\mathrm{Sets}$ are precisely groups on sets. $\mathrm{Groups}=\mathrm{Hom}\left(\mathrm{TheoryOfGroups},\mathrm{Sets}\right)\phantom{\rule{thinmathspace}{0ex}}.$ This also happens to be a PROP.

And models of $\mathrm{TheoryOfGroups}$ in vector spaces are precisely Hopf algebras

$\mathrm{HopfAlgebras}=\mathrm{Hom}\left(\mathrm{TheoryOfGroups},\mathrm{VectorSpaces}\right)\phantom{\rule{thinmathspace}{0ex}}.$

I suppose $\mathrm{TheoryOfGroups}$ is (almost?) self-dual in that it is equivalent to the category ${\mathrm{TheoryOfGroups}}^{\mathrm{op}}$ which explains why the contravariant functor $ℂ\left[-\right]:={\mathrm{Hom}}_{\mathrm{Sets}}\left(-,ℂ\right):\mathrm{Sets}\to \mathrm{VectorSpaces}$ sends a group $G$ to Hopf algebras in that we get $ℂ\left[G\right]:{\mathrm{TheoryOfGroups}}^{\mathrm{op}}\stackrel{{G}^{\mathrm{op}}}{\to }{\mathrm{Sets}}^{\mathrm{op}}\stackrel{ℂ\left[-\right]}{\to }\mathrm{VectorSpaces}\phantom{\rule{thinmathspace}{0ex}}.$

(Let’s assume all my sets are finite and all vector spaces of finite dimension.)

I suppose this is roughly what’s going on, but may be mistaken on details. Could you help me? I did look at this discussion over at the Secret Blogging Seminar, but it didn’t quite seem to contain the answer to all of the aspects I am asking about here.

Posted by: Urs Schreiber on October 24, 2008 12:17 PM | Permalink | Reply to this

Re: Hopf Algebraic Renormalization

I thought it was ${\mathrm{FreeGroups}}^{\mathrm{op}}$ which was equivalent to $\mathrm{TheoryOfGroups}$. If that’s wrong you can blame John:

you can reconstruct an algebraic theory from its category of free algebras in the simplest manner imaginable: just reversing the direction of all the morphisms!

Posted by: David Corfield on October 24, 2008 12:48 PM | Permalink | Reply to this

Re: Hopf Algebraic Renormalization

I thought it was ${\mathrm{FreeGroups}}^{\mathrm{op}}$ which was equivalent to $\mathrm{TheoryOfGroups}$.

Ah. Hm. So what, then, is the abstract reason that ${\mathrm{Hom}}_{\mathrm{Sets}}\left(-,ℂ\right)$ sends groups to Hopf algebras?

Posted by: Urs Schreiber on October 24, 2008 1:09 PM | Permalink | Reply to this

Re: Hopf Algebraic Renormalization

Urs wrote:

There is a monoidal category TheoryOfGroups which is the “theory of groups” in that monoidal functors from TheoryOfGroups to Sets are precisely groups on sets.

$\mathrm{Groups}=\mathrm{Hom}\left(\mathrm{TheoryOfGroups},\mathrm{Sets}\right).$

This would be true if you said ‘symmetric monoidal functors’; I’m scared to think about more general monoidal functors here.

This also happens to be a PROP.

People usually think of TheoryOfGroups is an algebraic theory (also known as ‘Lawvere theory’). In other words, it’s a category with finite products where all objects are powers of a single object. So, we can use the trick you sketch above to define groups in any category $C$ with finite products: we take product-preserving functors from TheoryOfGroups to $C$.

However, we can forget that TheoryOfGroups is an algebraic theory and think of it as a PROP. In other words, a symmetric monoidal category where all objects are tensor powers of a single object. This lets us look at things resembling groups in any symmetric monoidal category $C$: we take symmetric monoidal functors from TheoryOfGroups to $C$.

The forgetful 2-functor

$\mathrm{Algebraic}\mathrm{Theories}\to \mathrm{PROPs}$

is a rather subtle thing; I discuss it — and especially the example of TheoryOfGroups — in notes universal algebra notes, on pages 49–53. Everything I’m saying here can be found in those notes.

And models of TheoryOfGroups in vector spaces are precisely Hopf algebras

HopfAlgebras=Hom(TheoryOfGroups,VectorSpaces).

No, alas — they’re cocommutative Hopf algebras. This is one of the subtleties!

The point is that the diagonal map

$\Delta :\mathrm{TheoryOfGroups}\to \mathrm{TheoryofGroups}×\mathrm{TheoryOfGroups}$

is cocommutative, as it is in any category with finite products.

This is related to the fact that for a group $G$, the group algebra $ℂ\left[G\right]$ is always a cocommutative Hopf algebra.

I find it sad that I know no ‘automatic’ way to start with TheoryOfGroups and get the definition of Hopf algebra. Doing this would be like ‘quantizing’ the definition of group: starting with a definition of group that applies in any classical context (category with finite products), turning a crank, and getting a definition of ‘quantum group’ that applies in any quantum context (symmetric monoidal category).

The forgetful functor

$\mathrm{Algebraic}\mathrm{Theories}\to \mathrm{PROPs}$

is a kind of ‘quantization procedure’, but applied to the theory of groups it gives the PROP for cocommutative Hopf algebras, which is not what my heart desires.

I suppose $\mathrm{TheoryOfGroups}$ is (almost?) self-dual in that it is equivalent to the category ${\mathrm{TheoryOfGroups}}^{\mathrm{op}}$

No, ${\mathrm{TheoryofGroups}}^{\mathrm{op}}$ is equivalent to the category of finitely generated free groups.

The same thing works for any algebraic theory: this was one of the big theorems in Lawvere’s thesis.

David wrote:

I thought it was FreeGroups op which was equivalent to TheoryOfGroups. If that’s wrong you can blame John.

Okay, blame me: I said ‘free models’ when I should have said ‘finitely generated free models’. The point is that TheoryofGroups has objects $1,G,{G}^{2},\dots$ where $G$ is the ‘walking group’. These correspond to the free groups on $0,1,2\dots$ generators. So we only get finitely generated free groups.

I’ll fix “week136”.

Posted by: John Baez on October 24, 2008 3:34 PM | Permalink | Reply to this

Re: Hopf Algebraic Renormalization

John,

thanks for the explanation! This is very much appreciated.

I’ll try to get back to this when I have taken care of the other threads that are calling for my attention…

Posted by: Urs Schreiber on October 25, 2008 4:13 PM | Permalink | Reply to this

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