A Nontrivial Lattice Scalar Quantum Field Model

Abstract

We introduce a new Euclidean quantum model with zero-spin scaled scalar fields on a spacetime hypercubic lattice, which we refer to as the hybrid model. The model coupling parameters are the hopping parameter ${\kappa }^2>0$ and the field interaction coupling $\alpha >0$. Using the technique of lattice scaled fields, with a noncanonical scaling that corresponds to a partial renormalization (wave function renormalization), we show that the partition function of this model satisfies both thermodynamic and ultraviolet stability bounds, with constants which are uniform in the lattice volume ${\Lambda }_s$ and the lattice spacing $a\in (0,1]$. As a consequence, the model has a finite free energy per degree of freedom in the thermodynamic limit ${\Lambda }_s\nearrow \infty$, at least in the subsequential sense. Subsequently, we can also take the ultraviolet or continuum limit $a\searrow 0$. Besides the finiteness property of the free energy, for sufficiently small values of the model coupling parameters ${\kappa }^2$ and $\alpha$, we show the model satisfies Osterwalder-Schrader positivity and construct the underlying physical quantum mechanical Hilbert space, and, using a polymer expansion, we show correlations also remain bounded in the thermodynamic limit. Finally, we perturbatively compute an infinite volume truncated four-point correlation showing it is nonzero. Noting that, for fixed $a$, perturbation series in ${\kappa }^2$ and $\alpha$ converges, these results are sufficient to show that our hybrid model makes sense as a lattice quantum field theory in spacetime dimension $d=3,4$, with possible extensions to other higher values of $d$.

22 pages