This presentation will address the relationship between operators for raising and lowering quantum states and for creating and annihilating fermion states on the one hand and Dirac-like field equations on the other. Clifford algebra and geometric calculus offer insights into this relationship. We examine the derivation of Dirac-like field equations for fermions in selected geometric algebras and the characteristics of the solutions for each. The first-order, Dirac-like field equations are found by factoring the second-order Klein-Gordon equation in the selected Clifford algebras $C\!\ell_{p,q}$. In previous work, this author focused on raising and lowering operators for electroweak fermion states expressed as Fock space ladders in $C\!\ell_{4,1}$, including an algebraic vacuum state, but unconnected to field equations. The present work seeks to merge the operators and field equations as a quantum field theory.