Modal Field Theory and Quasi-sparse Eigenvector Diagonalization

Of the many approaches to non-perturbative quantum field theory, we can identify two general computational strategies. The first is the method of Monte Carlo. The main advantages of this approach is that it can treat many higher dimensional field theories, requires relatively little storage, and can be performed with massively parallel computers. The other strategy is the method of explicit diagonalization. The strong points of this method are that it is immune to the fermion sign problem, can handle complex-valued actions, and yields direct information about the spectrum and eigenstate wavefunctions. Modal field theory is a simple Hamiltonian framework which can accomodate either computational strategy. The first step is to approximate field theory as a finite-dimensional quantum mechanical system. The approximation is generated by decomposing field configurations into free wave modes and has been explored using both spherical partial waves 1 and periodic box modes . From there we can analyze the properties of the reduced system using Monte Carlo, diagonalization, or some other computational method. In this short review we present two different approaches using the modal field formalism which address the example of φ theory in 1 + 1 dimensions. We begin with a summary of the Monte Carlo calculation of the critical behavior of φ theory in Euclidean space. We then discuss the computational challenges of a diagonalization-based approach and conclude with a calculation of the lowest energy states of the theory using a diagonalization technique known as the quasi-sparse eigenvector method.