Physics of charges in relativistic quantum field theory

We study the properties of the "formal" charge operator Q = d3 x j0(x). It is well known that some of the properties usually associated with the "formal" charge operator are, in fact, not satisfied by Q itself. However, in the case of an exact symmetry, an operator can be defined in terms of the matrix elements of Q which does satisfy the expected properties. For broken symmetries no such well-defined spacelike charge operator exists. In this paper we attempt to understand the problems associated with Q intuitively. We suggest several Gedanken experiments which accentuate the basic problems and, at the same time, give the related phenomena a sound, physically understandable basis.