Hello,
for my bachelor's thesis I need to understand the Hopf Algebra of Feynman Diagrams. As I have only litte knowledge in Algebra by now I wanted to ask where I could start and what preknowledge I should have in order to understand this topic. Can you perhaps recommend a textbook or some other reference regarding Hopf Algebra or Quantum Groups suitable for my need?
Thank you in advance.
Hopf Algebra for Physicists – Mathematical Physics Reference
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Most of this is standard theory of path integrals known to mathematical physicists so I will try to address all of your questions.
First let me say that the hypothesis you list for the action $S$ to make the path integral well defined, ie that $S=Q+V$ where $Q$ is quadratic and non-degenerate and $V$ is bounded are extremely restrictive. One should think of $V$ as defining the potential energy for interactions of the physical system and while it certainly true that one expects this to be bounded below, there are very few physical systems where this is also bounded above (this is also true for interesting mathematical applications...). Essentially requiring that the potential be bounded implies that the asymptotic behavior of $S$ in the configuration space is totally controlled by the quadratic piece. Since path integrals with quadratic actions are trivial to define and evaluate, it is not really that surprising or interesting that by bounding the potential one can make the integral well behaved.
Next you ask if anyone has studied the question of when an action $S$ gives rise to a well defined path integral: $ \int \mathcal{D}f \ e^{-S[f(x)]}$
The answer of course is yes. The people who come to mind first are Glimm and Jaffe who have made whole careers studying this issue. In all cases of interest $S$ is an integral $S=\int L$ where the integral is over your spacetime manifold $M$ (in the simplest case $\mathbb{R}^{n}$) and the problem is to constrain $L$. The problem remains unsolved but nevertheless there are some existence proofs. The basic example is a scalar field theory, ie we are trying to integrate over a space of maps $\phi: M \rightarrow \mathbb{R}$. We take an $L$ of the form:
$ L = -\phi\Delta \phi +P(\phi)$
Where in the above $\Delta$ is the Laplacian, and $P$ a polynomial. The main nontrivial result is then that if $M$ is three dimensional, and $P$ is bounded below with degree less than seven then the functional integral exists rigorously. Extending this analysis to the case where $M$ has dimension four is a major unsolved problem.
Moving on to your next point, you ask about another approach to path integrals called perturbation theory. The typical example here is when the action is of the form $S= Q+\lambda V$ where $Q$ is quadratic, $V$ is not, and $\lambda$ is a parameter. We attempt a series expansion in $\lambda$. The first thing to say here, and this is very important, is that in doing this expansion I am not attempting to define the functional integral by its series expansion, rather I am attempting to approximate it by a series. Let me give an example of the difference. Consider the following function $f(\lambda)$:
$f(\lambda)=\int_{-\infty}^{\infty}dx \ e^{-x^{2}-\lambda x^{4}}$
The function $f$ is manifestly non-analytic in $\lambda$ at $\lambda=0$. Indeed if $\lambda<0$ the integral diverges, while if $\lambda \geq 0$ the integral converges. Nevertheless we can still be rash and attempt to define a series expansion of $f$ in powers of $\lambda$ by expanding the exponential and then interchanging the order of summation and integration (illegal to be sure!). We arrive at a formal series:
$s(\lambda)=\sum_{n=0}^{\infty}\frac{\lambda^{n}}{n!}\int_{-\infty}^{\infty}dx \ e^{-x^{2}}(-x^{4})^{n}$
Of course this series diverges. However this expansion was not in vain. $s(\lambda)$ is a basic example of an asymptotic series. For small $\lambda$ truncating the series at finite order less than $\frac{1}{\lambda^{2}}$ gives an excellent approximation to the function $f(\lambda)$
Returning to the example of Feynman integrals, the first point is that the perturbation expansion in $\lambda$ is an asymptotic series not a Taylor series. Thus just as for $s(\lambda)$ it is misguided to ask if the series converges...we already know that it does not! A better question is to ask for which actions $S$ this approximation scheme of perturbation theory itself exists. On this issue there is a complete and rigorous answer worked out by mathematical physicists in the late 70s and 80s called renormalization theory. A good reference is the book by Collins "Renormalization." Connes and Kreimer have not added new results here; rather they have given modern proofs of these results using Hopf algebras etc.
Finally I will hopefully answer some of your questions about Chern-Simons theory. The basic point is that Chern-Simons theory is a topological field theory. This means that it suffers from none of the difficulties of usual path integrals. In particular all quantities we want to compute can be reduced to finite dimensional integrals which are of course well defined. Of course since we lack an independent definition of the Feynman integral over the space of connections, the argument demonstrating that it reduces to a finite dimensional integral is purely formal. However we can simply take the finite dimensional integrals as the definition of the theory. A good expository account of this work can be found in the recent paper of Beasley "Localization for Wilson Loops in Chern-Simons Theory."
Overall I would say that by far the currently most developed approach to studying path integrals rigorously is that of discretization. One approximates spacetime by a lattice of points and the path integral by a regular integral at each lattice site. The hard step is to prove that the limit as the lattice spacing $ a $ goes to zero, the so-called continuum limit, exists. This is a very hard analysis problem. Glimm and Jaffe succeeded in using this method to construct the examples I mentioned above, but their arguments appear limited. Schematically when we take the limit of zero lattice size we also need to take a limit of our action, in other words the action should be a function of $ a$. We now write $S(a)=Q+\lambda V+H(a,\lambda)$ Where as usual $Q$ is quadratic $V$ is not an $\lambda$ is a parameter. Our original action is $S=Q +\lambda V$
The question is then can we find an $H(a,\lambda)$ such that a suitable $a\rightarrow 0$ limit exists? A priori one could try any $H$ however the arguments of Glimm and Jaffe are limited to the case where $H$ is polynomial in $\lambda$. Physically this means that the theory is very insensitive to short distance effects, in other words one could modify the interactions slightly at short distances and one would find essentially the same long distance physics. It seems that new methods are needed to generalize to a larger class of continuum limits.
I will try to contribute a partial answer. First I want to comment on the Lindstrom-Gessel-Viennot determinant coming from quantum mechanics stuff, in physics this is known as the Slater determinant, giving the formula for the wavefunction of a multi-fermionic system. This gives a nice picture to think of LGV as yet another instance of the Boson-Fermion correspondence. In the boson case, one allows all paths and obtains the total number as the permanent of the LGV matrix (this is obvious from the definition of the permanent), and in the fermion case one gets a system with states counted by the determinant of the LGV matrix. Of course the non-intersecting part comes into play because fermions in addition satisfy the Pauli exclusion principle, therefore they cannot occupy the same site at the same time.
Now, LGV and related results have an interesting history even in fields other than combinatorics. Fisher, in "Walks Walls, Wetting and Melting" 1984, considered the vicious walkers model in statistical mechanics, which considers mutually avoiding directed lattice paths. From this perspective it is interesting to look at some configurations which aren't solved by the usual LGV theorem, for instance when the paths are allowed to intersect at vertices but not edges, or when two paths are allowed to intersect in at most 2 consecutive vertices (the terminology for this classification is $n$-friendly walkers, see here). Viennot and others considered such variants after the relation between the combinatorics of lattice paths and statistical mechanics was established, it turns out that some of these models also have determinantal formulas associated to them. One main article is "From the Bethe Ansatz to the Gessel-Viennot Theorem" by R. Brak, J. W. Essam, and A. L. Owczarek, the point here is that LGV related results can be proven using transfer matrix methods as well, which is a powerful point of view in light of the models I mentioned above where the usual LGV fails (i.e. outside of the six vertex model).
Now if you need something more rigorous relating LGV matrices to fermion models, this can be done, but it doesn't seem to have been written nicely anywhere. Sometimes this is mentioned in the literature in the case of graphs like $\mathbb Z^2$, see for example "Domino tilings and the six-vertex model at its free fermion point" by P.L. Ferrari and H. Spohn, but I believe there should be a more general setting to talk about this. If you take the point of view that Greg Kuperberg mentioned in his answer to this previous MO question, that Kasteleyn-Percus matrices are essentially equivalent to LGV matrices, then I believe there is more literature on interpreting these as models of Majorana fermions living on the graph. The article I'm thinking of is "Dimer Models, Free Fermions and Super Quantum Mechanics" by Dijkgraaf, Orlando and Reffert.
As a last note, I wanted to say that I don't fully understand your motivation to want to identify every occurrence of the determinant with a LGV (or Kasteleyn-Percus) context, given that even within graph theory there are families of objects (even paths or random walks, as mentioned above) which are counted by determinants of a different sort of flavor. As to the question about non-commutative weights to LGV, I can't offer any insight, except perhaps to suggest looking at previous work on non-commutative extensions of the LGV theorem, such as the extension proved in "Noncommutative Schur Functions and their Applications" by Fomin and Greene (available from Fomin's website). But this is probably not very useful since even in their case the ring is almost commutative.
With regard to your question on rank and signature, there is a lot one could explore. As a start, if you look at the Laplacian matrix associated to a graph, then the rank has a clear combinatorial meaning, it measures the number of connected components of the graph (well, the maximum number of edges in an acyclic spanning subgraph, to be more precise), but there is not much to say about the signature since all eigenvalues of the Laplacian are nonnegative. When it comes to the adjacency matrix of the graph, the connection to combinatorial quantities gets even weaker (but it is an area of research), for example the rank of the adjacency matrix was conjectured to have something to do with the chromatic number of the graph, but the relation can not be so nice as was shown by Alon and Seymour. There has been some work on interpreting the signature of the adjacency matrix as well, with some early articles by Torgasev (for example here), but it is of similar nature. One can compute rank and signature easily by linear algebra methods and therefore hopes to use these to bound graph-theoretic quatities, but apparently not the other way around.
Best Answer
Dominique Manchon's lecture notes, which are very well-known amongst people working on Connes--Kreimer renormalisation, offer exactly the sort of detailed, accessible introduction to Hopf algebras and Connes--Kreimer renormalisation that you're looking for. However, you should first be thoroughly comfortable with abstract linear algebra and with the basics of ring and module theory, and you should be familiar with the basic language of category theory and of representation theory. Roughly speaking, if you can follow along with Chapter 1 of the notes, you should have the bare minimum needed.