I have long been interested in physics as (working toward) a description of absolute truth and, as a consequence, have had a number of discussions with people with religious and metaphysical beliefs far from mine own. These conversations often turn toward ontology, the study of existence vs. non-existence. It seems to me that the fundamental ontology in physics is characterization of mass/energy.
For example, when textbooks work through solving basic quantum mechanics problems, they describe the case where the wave equation equals 0 as the "trivial" case because it means that the particle doesn't exist. If a particle has a non-zero wave function, then it also has an energy (even if not entirely well defined) described by the Hamiltonian operator operating on that wave function. Similarly, the creation and annihilation operators represent the "creation" or "annihilation" of a particle by adding or removing a unit of mass/energy to the system. Even within a Newtonian system, for an object to have an effect it must have either mass and/or energy.
Granted, field theories appear to be an exception in the sense that the magnetic, electric, induction, displacement, etc. fields are often treated as real entities that do not have any intrinsic energy. Nonetheless, texts often explicitly warn students not to be concerned with the "reality" of these fields but to use them as a descriptive tool. Modern theories that incorporate virtual and exchange particles do away with these fields and suggest a description more in line with the mass/energy paradigm.
Recognizing that there is not universal consensus on the point, in what sense does the philosophical foundation of physics imply that existence is contingent upon the "posession" of mass/energy?
Equivalently, does it make sense to say that something may exist without mass or energy?