The ongoing quantization of the four fundamental forces of nature represents one of the most fruitful grounds for cross-pollination between physics and mathematics. While remaining vastly open, substantial progress has been made in the last decades: the expression of all basic physical theories in terms of geometry, specifically as gauge theories. This is accomplished by the recognition of the strong, weak, and electromagnetic fields as Yang-Mills (gauge) fields, and by the re-writing of general relativity in terms of gauge connection variables.

The method of canonical quantization offers several advantages in treating gauge theories: the gauge fields themselves are the basic variables, while gauge constraints promote to quantum operators whose commutation relations reflect the classical Poisson brackets.

In this thesis I construct a zero-energy ground state for canonically quantized Yang-Mills theory, for a particular ("nonlinear normal") factor ordering of the Hamiltonian operator. The inspiration for this project is to find an alternative to the Chern-Simons and Kodama states. These are closely related ground state solutions for (respectively) quantum Yang-Mills theory and quantum gravity with a positive cosmological constant. Objections to the Chem-Simons and Kodama states come from, among other arguments, their apparent lack of well-defined decay "at infinity." The ground state I have constructed, as the exponentiation of a strictly non-positive functional, manifestly enjoys good decay properties. In addition, I have constructed a similar ground state for scalar ϕ⁴ theory. The construction of these ground states represents a generalization to quantum field theories of work done by my thesis advisor V. Moncrief, in collaboration with M. Ryan, for quantum mechanical situations.

Gauge, rotation, and translation invariance are directly verifiable for the nonlinear normal ordered Yang-Mills ground state; invariance under boosts remains as a question for future work. The analogous state for the abelian case (free Maxwell theory) enjoys full Poincare invariance.