# Is the electric charge associate with quantum systems counterfactual definite?

Is the electric charge associated with quantum systems counterfactual definite? Counterfactual definiteness refers to the inability to ''speak'' meaningfully (in quantum systems) of the definiteness of the results of a measurement that has not been performed yet. It is a stand usually taken as a result of the violation of Bell's inequality by quantum mechanics. In other words you cannot associate definite values to a dynamical variable associated with a quantum system that has not yet been measured.

However, unlike the dynamical variables like position, momentum and kinetic energy, electric charge does not have an operator (am I right?). And there is no quantum indeterminacy associated with the measurement of charge. So, presumably we can talk of the charge of a quantum system independent of measurement. Is it not therefore counterfactual definite?

## 1 Answer

As far as I can understand the ''easy'' solution for this question is to inspect the properties of the charge operator $Q$. So yes. There is indeed a charge operator.

Usually when constructing a charge it is representative of an intrinsic symmetry (usually the gauge symmetry) of our system and disregarding nuances of our construction we know that the dynamics of our system is independent of transformations on the intrinsic structure $\phi\rightarrow e^{iq}\phi$, we have a continuous symmetry and therefore also a conserved charge $Q$. If our charge is constructed as such and is conserved it commutes with the Hamiltonian $[H,Q]=0$, so when creating charge eigenstates we know that these will not evolve with time and will be stationary (for a conserved charge it commutes with all observables), or rather: Any initial measurement of the system will not change the result of subsequent measurements of charge. As such there is no need for a superposition of charge states since the overall charge is conserved and no measurement of observables will subsequently generate a superposition.

This is from my own intuition of charge as a part of intrinsic symmetries and energy as a part of an extrinsic symmetry (skipping much nuance).

However there is also a possible discussion using superselection rules, I am however not qualified to comment on that aspect. Hopefully I managed to answer your question in some manner.