# Hopf algebra and quantum fields

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Quantum field theory aims at an understanding of one-particle irreducible (1PI) Green functions. Such Green functions provide inverse propagators and vertex functions. Their knowledge determines full Green functions and all scattering amplitudes of the theory.

It is the task of this review to illuminate the underlying Hopf algebraic structure of 1PI Green functions. In practice, computation of 1PI Green functions proceeds by the computation of multiloop feynman integrals. We will emphasize the underlying Hopf algebra behind the renormalization of such perturbative contributions to physical amplitudes, and will also exhibit conceptual aspects beyond perturbation theory.

## Free QFT, interacting QFT

We classify 1PI Green functions $$G^{r}$$ by the amplitude $$r$$ to which they contribute. We let $$|r|$$ be the number of external asymptotic states to be specified for that amplitude.

We assume this number is greater or equal to two.

In the latter case, the Green function $$G^{r,|r|=2}$$ provides an inverse propagator, if more than two legs are attached to each other, we agree to speak of a vertex function.

To distinguish between the types of amplitudes $$r$$ explicit quantum numbers (spin, momenta, masses) for the asymptotic states have to be assigned as necessary.

### Edges vs propagators

Edges are defined from free quantum field theory. The representation theory of the Poincar'e group immediately teaches us that covariances for free fields come as fermionic or bosonic propagators. Those propagators are the inverse of the Fourier transforms of the Dirac and Klein--Gordon operators, for example$P_0(k,m):=\frac{1}{k^2-m^2+i\eta}$

for the propagator of a spin zero boson of mass $$m\ ,$$

and

$$P_{1/2}(k,m):=\frac{k\!\!\!/+m}{k^2-m^2+i\eta}$$

for a spin-half fermion of similar mass.

Here, the $$i\eta,\;0<\eta\ll 1$$ notation serves as a reminder that these free propagators respect causality. For the inverse propagators we write $$P^{-1}_0(k,m):=k^2-m^2$$ and $$P^{-1}_{1/2}(k,m):=k\!\!\!/-m.$$ Propagators serve as edges in Feynman graphs. To discuss 1PI graphs, we need next to define vertices.

### Vertices vs interaction

Vertices describe local interactions. They correspond to monomials in the Lagrangian formed from products of fields and their derivatives. Again, representation theory guarantees that in a chosen dimension of spacetime, there is at most a finite number of local interactions which are renormalizable.

For example, if we formulate the quantum field theory of a single spin-zero massive field $$\phi$$ over Minkowski space (four-dimensional and flat), we obtain the propagator $$P_0$$ above for the free theory. As the Klein-Gordon operator is quadratic in field derivatives, the field itself has unit scaling weight so that the monomial $$\phi\Box\phi$$ has weight four and hence a quartic interaction $$g\phi^4/4!$$ can be added which has corresponding weight four for a dimensionless coupling $$g$$ and leads to a renormalizable interacting field theory.

### Renormalizability

Renormalizability is a statement about selfsimilarity. To see how this comes about, let us first remind ourselves that a Feynman graph to be one-particle irreducible (1PI) if and only if it is connected after removal of any one of its internal edges.

Let us next assign a weight $$w=-1$$ to each fermionic propagator $$P_{1/2}\ ,$$ and a weight $$w=-2$$ to each bosonic $$P_0\ ,$$ hence $$w=+1$$ to an inverse fermionic propagator, and $$w=+2$$ to an inverse bosonic propagator.

A vertex $$v$$ to which $$n_f$$ fermionic lines and $$n_b$$ bosonic lines couple, has weight $$w(v)=-n_f(D-1)/2-n_b(D-2)/2+D\ .$$ Here, $$D$$ is the dimension of spacetime, often set to four. For a 1PI graph $$\Gamma\ ,$$ let

$$\mathbf{\omega}_D(\Gamma):=D|\Gamma|+\sum_{\mathrm{weights}\;w}w,$$

where the sum is over the weights of all vertices and internal edges of $$\Gamma\ ,$$ and $$|\Gamma|$$ is the lowest Betti number of the graph, the number of independent cycles in it (the loop number, in physics).

For each such graph $$\Gamma\ ,$$ we let $$\mathrm{res}(\Gamma)$$ be the graph obtained by shrinking all its internal edges to zero length. We call it the graphical residue.

$$\mathrm{res}(\Gamma)$$ is then a vertex $$v(\Gamma)$$ connected to the external edges of the graph. If the number of the external edges is two, this corresponds to an inverse propagator. See the examples below.

We call a field theory renormalizable in $$D$$ dimensions, if

$$w(\mathrm{res}(\Gamma))=\mathbf{\omega}_D(\Gamma)$$

holds for all 1PI graphs $$\Gamma$$ in the theory.

Example: Consider the graph

Figure 1: A graph in quantum electrodynamics.

It has two internal fermionic propagators of weight -1. The vertices which couple a fermion pair to a boson have weight $$-(D-1)-(D-2)/2+D$$ which vanishes in four dimensions, and hence

$$\omega_4=2=w(P_{1}^{-1})$$ for this graph which has one loop, with $$\mathrm{res}(\Gamma)$$ an inverse photon propagator, which has weight +2 as the photon is a spin-one boson.

## The Hopf algebra

### Basics on Hopf algebras

We are mainly concerned with Hopf algebras $$H = \oplus_{n=0}^\infty H_n\ .$$ The graded pieces $$H_n$$ are finite dimensional vectorspaces over $$\mathbb{Q}$$ say. As an algebra our Hopf algebras are free commutative algebras with generators given by some suitable countable set. They are connected Hopf algebras, with $$\mathbb{Q}1=H_0.$$ The counit $$e:H\to\mathbb{Q}$$ annihilates any element in the augmentation ideal $$\oplus_{n=1}^\infty H_n,$$ and we let $$P$$ be the projector into this ideal. If we write $$\Delta(h)=\sum h^{\prime}\otimes h^{\prime\prime}$$ for the coproduct in Sweedler's notation, we have the antipode recursively as $$S(h)=-\sum S(h^\prime)P(h^{\prime\prime}).$$

### Graphs

We will base our Hopf algebras on graphs as generators: as algebras, they all are free commutative algebras on generators indexed by countable sets of Feynman graphs. A grading will always be provided by the fundamental Betti number: the number of independent cycles in the graph. Truncating at any finite order in that grading corresponds to a Taylor expansion of Green functions in a suitable parameter up to that grading.

Let $$\Gamma$$ be a 1PI Feynman graph. Then, $$\Delta(\Gamma)=\Gamma\otimes 1+1\otimes\Gamma+\sum_{\gamma\subset\Gamma}\gamma\otimes\Gamma/\gamma$$ gives a coproduct $$\Delta$$ for the free commutative algebra over graphs. Here, $$\gamma=\cup_i\gamma_i$$ is a disjoint union of graphs $$\gamma_i$$ such that each $$\gamma_i$$ is a proper superficially divergent subgraph of $$\Gamma\ .$$ The sum is over all such unions $$\gamma\ .$$

This gives a Hopf algebra structure to Feynman graphs, with the empty graph furnishing the unit, a counit which projects to scalars, $$e(\Gamma)=0, \Gamma\in Aug(H),$$

and


an antipode $$S(\Gamma)=-\Gamma-\sum_{\gamma\subset\Gamma}S(\gamma)\Gamma/\gamma.$$ Example: Consider the graph

Figure 2: A Feynman graph $$\Gamma\ .$$

It has two subgraphs,

Figure 3: $$\gamma_1$$
Figure 4: $$\gamma_2$$
.

We hence have three contributions in the above sum$\gamma\in\{\gamma_1,\gamma_2,\gamma_1\gamma_2\}.$ The cographs $$\Gamma/\gamma$$ are

Figure 5: $$\Gamma/\gamma_1$$
Figure 6: $$\Gamma/\gamma_2$$
Figure 7: $$\Gamma/(\gamma_1\gamma_2)$$

Let us compute the coproduct$\Delta(\Gamma)=\Gamma\otimes 1 + 1 \otimes \Gamma +\gamma_1\otimes \Gamma/\gamma_1+\gamma_2\otimes\Gamma/\gamma_2+ \gamma_1\gamma_2\otimes\Gamma/(\gamma_1\gamma_2).$ Similarly, the antipode in this example is $$S(\Gamma)=-\Gamma+\gamma_1\;(\Gamma/\gamma_1)+\gamma_2\;(\Gamma/\gamma_2)-\gamma_1\gamma_2\;(\Gamma/(\gamma_1\gamma_2)).$$

### Flags

To find the connection between graphs and words we turn to flags. To each graph $$\Gamma$$ we can assign a finite number of flags $$F_i(\Gamma)\ .$$ Each such flag is a sequence $$\gamma_1\subset\cdots \subset \Gamma$$ such that the quotient of a graph in the sequence by its predecessor is a primitive element in the Hopf algebra. It is implied that the first element in the sequence is primitive. We let $$f_i^\Gamma\in \mathbb{N}$$ be the length of the flag $$F(\Gamma_i).\ .$$

Now consider the graph $$\Gamma$$ above. It allows for two flags$F_1(\Gamma)=\gamma_1\subset(\gamma_1\cup\gamma_2)\subset\Gamma$

and

$$F_1(\Gamma)=\gamma_2\subset(\gamma_1\cup\gamma_2)\subset\Gamma.$$

There is a natural Hopf algebra structure on flags given as

$$\Delta_{F}(F(\Gamma))=1\otimes F(\Gamma)+F(\Gamma)\otimes 1+\sum_{i=1}^{f^\Gamma-1}[\gamma_1\subset\cdots\subset\gamma_{i}]\otimes \gamma_{i+1}\subset\cdots\subset\Gamma,$$ where we identify $$\gamma_{f^\Gamma}=\Gamma.$$

There is also a natural shuffle algebra structure on flags which is apparent if we write a flag as a sequence $$p_1|p_2|\cdots|p_{f^\Gamma}$$ of primitives $$p_i:=\gamma_{i+1}/\gamma_i.$$

$$\gamma_1\subset\cdots\subset \Gamma\to p_1|p_2|\cdots|p_{f^\Gamma},$$ with $$\gamma_1=p_1$$ and $$p_{f^\Gamma}=\Gamma/\gamma_{f^\Gamma-1}\ .$$ This shuffle algebra plays a remarkable role in undertanding the Feynman rules discussed below.

## Perturbative aspects

### Perturbative renormalization

#### Regularization in physics

Here, we will concentrate on regularization of perturbative quantum field theory by complex regulators. The most prominent one is dimensional regularization....

#### renormalized and unrenormalized Feynman rules

Perturbative renormalization starts with unrenormalized Feynman rules. We regard them as maps $$\Phi:H\to V$$into a ring $$V$$ such that $$\Phi(h_1h_2)=\Phi(h_1)\Phi(h_2).$$ More details will be given in a detailed discussion. Furthermore, we introduce an endomorphism $$R:V\to V$$ which is a Rota--Baxter map$R(v_1v_2)+R(v_1)R(v_2)=R(v_1R(v_2))+R(R(v_1)v_2).$ With these ingredients, the transition from unrenormalized Feynman rules $$\Phi$$ to local counterterms $$S_R:H\to V$$ and renormalized Feynman rules $$\Phi_R$$ becomes algebraic for the counterterm

$$S_R(\Gamma)=-R(\Phi(\Gamma)+\sum_{\gamma\subset\Gamma}S_R(\gamma)\Phi(\Gamma/\gamma))$$

and the renormalized Feynman rules

$$\Phi_R(\Gamma)=m_V\circ(S_R\otimes\Phi)\circ\Delta(\Gamma)\equiv S_R\star\Phi(\Gamma).$$ All three maps $$\Phi,S_R,\Phi_R\in \mathrm{Spec}(H).$$

### The renormalization group and the Dynkin operator

For a flag $$F=\subset_i^\Gamma\gamma_i$$ as above, we have a canonical primitive element

$$p_F=[S\star Y](F)=|F|F-\sum_{j=1}^{f^\Gamma-1}S(F_{1,i})|F_{F_{i+1,f^\Gamma}}|F_{i+1,f^\Gamma}.$$

Let us now write for a kinematical renormaliztion scheme

$$\Phi_R(\Gamma)=\sum_{j=1}^{\mathrm{aug}(\Gamma)}c_j^\Gamma(\Theta) \ln^j s$$

as before.

Then,

$$c_j^\Gamma(\Theta)=\frac{1}{j!}m^{j-1}\circ [c_1\otimes\cdots\otimes c_1]\Delta^{j-1}(\Gamma).$$

### Sub Hopf algebras

There are dinstiguished series of graphs which appear in physics: for a given 1PI Green function determines a sum over all 1PI graphs with the corresponding external leg structure$X(g)=1+\sum_{\mathbf{res}(\Gamma)=r}g^{|\Gamma|}\frac{\Gamma}{\mathrm{Aut}(\Gamma)}.$

This series in $$g$$ has coefficients in the Hopf algebra, as at any finite loop order $$|\Gamma|\ ,$$ only a finite number of graphs exist which contribute to the chosen amplitude $$r\ .$$ It is hence well-defined as a formal series $$\in H[[g]]\ .$$

Writing it as a sum over homogenous components $$c_k^r:= \sum_{\mathbf{res}(\Gamma)=r,|\Gamma|=k}g^{|\Gamma|}\frac{\Gamma}{\mathrm{Aut}(\Gamma)},$$ reveals remarkable properties.

In particular, we can regard the coefficients $$c_k^r\in H$$ as generators of a sub-Hopf algebra upon realizing that coproduct ,math>\Delta[/itex] preserves the free commutative sub-algebra on these generators.

### Hopf (co-)Ideals

Consider a subset $$I$$ of generators of $$H\ .$$ It corresponds to an ideal also denoted by $$I$$ of $$H$$ in an obvious manner.

It might happen that $$I$$ is also a coideal$\Delta(I)\subset I\otimes H+H\otimes I.$

We can then sensibly speak about the Hopf algebra $$H/I\ .$$ This elementary construction is of utter importance in physics.

### Ward Identities

Let us find the (co-)ideal which gives us the Ward--Takahashi identities.

## Beyond perturbation theory

We saw above how to replace a sum over all graphs by a sum over primitive elements. This has consequences.

### Dyson Schwinger equations

Let us set them up for amplitudes which do need renormalization, $$\forall\;r\in\mathcal{R}\ :$$

$$X^r(g)= 1\pm \sum_{k=1}^\infty g^k B_+^{r;k}(X^r(g)Q^{k}(g)).$$

Here, the positive sign is taken if $$r$$ is a scattering amplitude and the negative sign if $$r$$ is the inverse of an amplitude of propagation. Furthermore,

$$Q(g):=\frac{X^v}{\prod_{e\in \mathrm{Ext}(r)}\sqrt{X^e(g)}}.$$

Here, we assume that $$X^v$$ is the sole independent scattering amplitude of the theory needing renormalization, possibly after dividing by a suitable (co-)ideal.

Finally, the maps $$B_+^{r;k}$$ are Hochschild one-cocycles of the theory such that the series $$X^r(g)$$ appear as fixpoints of this system for all amplitudes needing renormalization.

Applications of the Feynman rules turns this into a set of integral equations which are the Dyson--Schwinger equations of QFT where under the Feynman rules $$B_+^{r;k}(1)$$ maps to the $$k$$-loop integral kernel of the amplitude $$r\ ,$$ sometimes dubbed a skeleton.

We now assume that the renormalization group equations hold. For a !PI Green function, we identify a suitable kinematical parameter $$s$$ say, made out of Lorentz scalars (masses $$m$$ and scalar products $$q_i\cdot q_j$$ of external momenta) which is in general position (vanishes only when all external momenta are identically zero). We assume is has the technical dimension of a squared mass, so that the Green function depends on this parameter $$s\ ,$$ a reference scale from renormalization $$\mu^2\ ,$$ and generalized angles $$\Theta_i\in\{m^2/s,\cdots,q_i\cdot q_j/s\}\ .$$

$$G^r=G^r(\ln s/\mu^2,\Theta_i,g)=1\pm \sum_{j=1}^\infty \gamma_j^r(g;\Theta_i)\ln^j\frac{s}{\mu^2}.$$

The renormalization group equations then demands relations between the coefficient functions $$\gamma_j^r.$$

Combining these relations with the analytic form of the DSE leads to a system of ordinary differential equations for the anomalous dimensions of field theory.

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