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How many properties are required to measure full energy of a fundamental particle? I know $E=mc^2$, but what about charge, spin, etc? Which full equation would give me all parameters of any particle?

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The properties of a particle appear, and play a role in determining the energy, when the particle moves in some field. For instance, if it moves in an electrical field, the charge is a relevant property because in an electric field the particle is accelerated or decelerated and it also gets a potential energy. In a magnetic field is important also the spin (more exactly the intrinsic magnetic momentum of the particle. In the gravitational field, its mass is relevant.

So, you have to decide in which field, if any, moves your particle. The properties of the particle have to appear in the Hamiltonian of the motion.

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