Definition of Charge Please give me a definition of charge. A textbook I have defines it to be - "A fundamental property of matter, which determines whether the matter will receive an electric flow or liberate one" I don't find this convincing enough. Any help would be appreciated.
 A: In quantum field theory, particles are simply excitations of fields. And interactions are determined by symmetry in an extremely elegant way, see gauge principle. Symmetry is the central concept in fundamental physics. Except that it determines the interaction, it can be also used to classify particles. For instance the spin of particles is characterized by the representation of Lorenz Group $SO(3,1)$ which is locally isomorphic to $SU(2)\otimes SU(2)$. Also the charge comes from the symmetry. Specifically, charge $Q$ actually is the generator of a symmetry since the conservative charge is related to the symmetry by the well kown Noether theorem. For example, considering a free complex scalar field theory
$$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi*-\frac{1}{2}m^2\phi\phi*.$$
The Lagrangian is invariant under transformation 
$$\phi\to e^{i\alpha}\phi,$$
which is called $U(1)$ symmetry. Let's define $\phi\to e^{iQ\alpha}\phi$, then we have 
$$e^{-iQ\alpha}\phi e^{iQ\alpha}=e^{i\alpha}\phi,$$
which is followed by $[Q,\phi]=-\phi$. Note that we have treated $\phi$ as a field operator as in QFT. When dealing with the state space of field $\phi$ (in Fock space), we also obtain the charge for every particle.
