A Hierarchical Finite Element Method for Quantum Field Theory
Abstract.
We study a model of scalar quantum field theory in which spacetime is a discrete set of points obtained by repeatedly subdividing a triangle into three triangles at the centroid. By integrating out the field variable at the centroid we get a renormalized action on the original triangle. The exact renormalization map between the angles of the triangles is obtained as well. A fixed point of this map happens to be the cotangent formula of Finite Element Method which approximates the Laplacian in two dimensions.
Arnab Kar
Department of Physics and Astronomy,
University of Rochester,
Rochester, New York 14627, USA.
 PACS numbers:

02.70.Dh, 41.20.Cv, 11.25.Hf
1. Introduction
The most successful regularization method in understanding nonperturbative Quantum Field Theory (QFT) is the lattice method,[1, 2] which replaces spacetime by a periodically arranged finite set of points. Numerical simulations based on this are becoming increasingly accurate, therefore any attempt at a mathematical formulation of quantum field theory must build on this success and aim to improve upon it.
The classical analogue of the problem would be the solution of Partial Differential Equations (PDEs). In the early days a lattice with identically shaped fundamental regions was used in numerical solutions of PDEs. Later it was realized that using meshes adapted to the boundary conditions makes more economical use of computing resources by adding more points where the field varies rapidly and fewer where it varies slowly. The Finite Element Method[3, 4] was developed in the seventies: it allows fundamental cells to have different shapes and sizes and use sophisticated interpolation methods to model the field in the interior of each cell. Some of the mathematical ideas were anticipated by Whitney[5] in his work in topology and extended by Patodi[6]. The Whitney elements have provided a basis for a discrete formulation of geometry. This Discrete Differential Geometry is useful not only to solve PDEs, but also to model shapes for use in computer graphics[7, 8].
The analogue in Quantum Field Theory is to replace the periodic lattice with a mesh that contains different length scales. This approach has been looked at by groups in the past and met with varying degrees of success. The first approach in this direction was by Christ, Friedberg and Lee[9] (except they proposed to average over all locations of lattice points as a way to restore rotation invariance, which did not turn out to be helpful). There is also some early work by Bender, Guralnik and Sharp[10]. Patodi’s FEM to solve the eigenvalue problem for Laplacians was not noticed by physicists at this time. Since then much of the work on Lattice Gauge Theories has been computational along with some analytic work[11].
We propose to adapt existing methods of QFT and develop new Finite Element Methods to understand the essential problem from Wilson’s point of view: how to integrate out some variables and get an effective theory for the remaining degrees of freedom (for a recent review, see the volume[12]). The simplest case is the one dimensional lattice (the set of integers), which has a natural subdivision into even and odd numbered elements. By integrating out the odd sites and leaving only the even sites we are left with an identical lattice with a different separation between field points. Unfortunately there is no simple procedure to extend this into higher dimensions.
A natural idea would be to divide space into triangles (simplices in higher dimensions) and to fit them together to form larger ones, allowing us to integrate out the interior vertices and obtain an effective large scale theory. An advantage of our regularization method is that the renormalization map can be calculated exactly. The transformation between the angles of the triangles from subsequent generations is obtained at each stage of the subdivision. The first examples[13] we constructed this way ignored the shape (information contained in the angles) of the triangles. The Finite Element Method used by engineers leads to a “cotangent formula”[14]. It approximates the Laplacian in two dimensions on one hand and also happens to be a fixed point[15] of the renormalization dynamics. We determine this dynamics explicitly.
We expect this fixed point to be a continuum limit on a fractal, analogous to the Bethe lattice for which the renormalization group can be exactly calculated. Such QFTs can serve as approximations to theories on Euclidean spaces. Or perhaps at short distances, spacetime really is not Euclidean.
If generalized to the Ising model, nonlinear sigma models or to four dimensional field theories, we could get interesting examples of Discrete Conformal Field Theory[16]. In our approach we do not average over triangulations. Such an average has been proposed as an approach to quantum gravity[17] and as a way to restore translation invariance[9].
2. The Cotangent Formula
In the early days of computational engineering, Duffin[14] derived a formula for the discrete approximation for the energy of an electrostatic field on a planar domain. In this Finite Element Method the plane is divided into triangles where the field is specified at each vertex and the energy of the field is the sum of contributions from each triangle. An approximation for the energy of a triangle is obtained by linear interpolation of the field to the interior.
Suppose the vertices correspond to field values . Each point in the interior of the triangle divides it into three subtriangles with vertices , and respectively.
If the ratio of the area of a subtriangle opposite to to the larger triangle is
then
We can use the pair as coordinates instead of the cartesian components of . The linear interpolation of the field values to the point is then
The energy of the interpolated field inside a triangle on calculation turns out to be
where are the cotangents of the angles at the vertices.
Proof.
Define the vectors along the sides of the triangle (see Fig. 1),
Using for as coordinates,
Then the metric tensor of the plane in these coordinates has as components the dot products of the sides:
Also, is twice the area of the triangle. The cotangents are
Then,
and
Thus, using ,
This can be rewritten as
as claimed. ∎
3. The Geometry of Triangles
The space of similarity classes of triangles (with marked vertices) is a hyperboloid[18]. This can be understood in several ways. A pair of sides of a triangle forms a basis, thus the space of marked triangles may be identified with : this group acts transitively and without a fixed point on the space of bases. Quotienting by rotation, scaling and reflection around a side gives
which is a hyperboloid. This argument generalizes to dimensions: the similarity classes of marked simplices is .
An equivalent point of view is that is the space of symmetric tensors of determinant one: a pair of sides of the triangles define a symmetric tensor through their inner products. By scaling we can choose this symmetric tensor to have determinant one. It is clear that acts on the space of such tensors transitively, with as the isotropy group at one point. Again this generalizes to dimensions.
A more explicit point of view will be useful in what follows. A similarity class of marked triangles is determined by the angles at the vertices (or, for convenience, the cotangents of the angles). Since the angles of a triangle add up to , the cotangents satisfy
(1) 
This can be written as
Since has signature (, this is the equation for a timelike hypersurface in Minkowski space . Setting
the “cotangent identity” (1) becomes the equation for a hyperboloid
The quantity is the ratio of the sum of squares of the sides to the area of the triangle. It is a minimum for an equilateral triangle and becomes large for a flat triangle (one with small area or large perimeter).
So far we discussed triangles with marked vertices but we should also consider invariant transformations of the vertices. The group of permutations of vertices is generated by the cyclic permutation
and the interchange of a pair of vertices
These permutations act on the cotangents through the matrices
We can also parametrize by the complex number
By a translation, we can choose the first vertex and by a rotation and scaling we may choose . is then the coordinate of the remaining vertex.
Then the permutation of the vertices becomes
By a reflection around the side , we can choose ; equivalently . Note that is the reflection around the perpendicular from vertex to the opposite side of the triangle.
3.1. Subdivision of a triangle
We can subdivide a triangle into three subtriangles of equal area by connecting the centroid to the vertices by straight lines. (If we subdivide at some other interior point, we get similar results).
To see this, choose a coordinate system with so that Then,
By dropping a perpendicular from to the side we get
The remaining angle is given by solving the cotangent formula:
We can thus express the cotangents of this subtriangle as where
Note that
since the cotangent identity is preserved. Thus subdivisions are represented by Lorentz transformations in . Note the symmetry under the interchange of and :
The cotangents of the remaining subtriangles are given by cyclic permutations and (see Fig. 3). In this convention, the central angle is listed first.
In the complex parametrization , the subdivision corresponds to
which is the complex coordinate of the centroid when . Recall that in this parametrization Clearly, both and map the upper half plane to itself.
4. Renormalization Dynamics
Consider a Gaussian scalar field with values at the vertices of a triangle with cotangents . The most general quadratic form for the discrete approximation to the action will be
The coefficients are functions of the cotangents satisfying the symmetry
For example, the cotangent formula corresponds to the choice
If we subdivide the triangle and associate a field at the central vertex, the action will be the sum of contributions from each triangle.
where
The effective action after integrating out the central field variable is given by
where is a normalization constant.
On comparing coefficient of , we get
On comparing coefficient of , we get
Using we can verify that as needed for symmetry. The denominator
is invariant under and hence, under all permutations.
The semigroup generated by the map on the space of pairs of functions on the hyperboloid is the renormalization dynamics (“renormalization group”). This explicit example should help understand such dynamics. For example, is there is a notion of entropy that increases monotonically? Its fixed points correspond to some sort of continuum limit (which could be fractals[21, 22]).
5. Fixed Points
An obvious fixed point consists of constant . This corresponds to the “Apollonian subdivisions” considered in an earlier paper[13].
We now show that the cotangent formula of the FEM
is also a fixed point[15] of the above dynamics. It is not hard to verify that
so that .
Similarly,
from which follows.
This fixed point describes some sort of continuum limit of two dimensional scalar field theory. As in the examples of Ref. [13] it is likely to be a fractal of dimension less than two; but we have not been able to determine this dimension yet. An extension of this method to higher dimensions and to gauge theories would be interesting. We hope to return to these issues in the future.
6. Acknowledgement
We thank Abdelmalek Abdesselam, Abhishek Agarwal, Alex Iosevich and V. Parameswaran Nair for discussions related to this work. We also thank Bianca Dittrich for bringing Ref. [15] to our attention.
Footnotes
 arnabkar@pas.rochester.edu
 Also at the Department of Mathematics
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