Broadly speaking, a field is a collection of properties ascribed to regions of space (one might also speak of the region itself as being "the field"); if the properties are quantifiable then the field is a mathematical function of spatial coordinates, Φ(x, y, z). Examples include the temperature at each point of a room, the velocity at each point of a fluid, the gravitational potential, and the electromagnetic field. In contrast—and broadly speaking—particles are entities of which positions are ascribed (and which lack any relevant internal structure). While these will do as broad characterizations, they are inadequate in a number of ways.

Classical Fields

For instance, one could say that a field theory ascribes positions (and field strengths) to the parts of a field, as a particle theory treats particles. Worse, one can reformulate particle theories (e.g., Isaac Newton, 1642–1727, and Immanuel...