Superposition of Electron and Positron Particle States

Question

Let $b_k^\dagger ,b_k$ represent the creation and annihilation operators for an electron in state $k$. Let $d_j^\dagger ,d_j$ represent the same for a positron in state $j$. And let $|0\rangle$ represent the vacuum.

Is it possible to have a state described by $ \left( b_k^\dagger + re^{i\theta} d_k^\dagger \right)|0\rangle $? I include the $re^{i\theta}$ for generality.

How do I interpret such a state? If I make measurements of the number of particles, the energy, the momentum, the charge, etc... what would I observe?

The question of how many particles is easy. The answer is 1. (On that note, can we have superpositions of states with differing number of particles?)

What the energy and momentum are depends on what the labels $k$ and $j$ mean.

But what about charge? What would I measure for charge? If we enclosed the system in a box that measured the electric field, what would we get for $\oint{\vec{E}}\cdot d\vec{A}$?

Thanks!

Answer 1

The Vacuum sector Hilbert space is generated by the action of $U(1)$ invariant operators $\overline{\hat\psi(x)}\Gamma\hat\psi(x)$ (and their Fourier transforms, where $\Gamma$ is an arbitrary Dirac matrix) on the Lorentz and translation invariant vacuum vector, which does not allow the state you ask about to be constructed because $U(1)$ invariant operators do not change the charge. One can also discuss this in terms of superselection rules, which is, however, an alternative presentation of $U(1)$ (or other) invariance of observables. [Strictly speaking, note that $\hat\psi(x)$ and its Fourier transform are operator-valued distributions, not operators, which introduces issues that anyone who might be concerned about such details can fill in.]

The question of how many particles there are in a given state is best answered in aggregate: what is the aggregate number of electrons minus the number of positrons, which for the vacuum sector is zero. One can create other Hilbert spaces, with different aggregate numbers of electrons/positrons, which can then be used to construct mixed states, but not superpositions.

There is, however, a lot else that could be said.

Answer 2

Yes, you can make a physical system for which the Hilbert space contains states which are superpositions of electrons and positrons.

Mathematically, this is because the Hilbert space depends on the boundary conditions at spatial infinity, and these boundary conditions may be themselves be quantum superpositions. (Note, this generalizes rather than contradicts Peter Morgan's answer. He is considering the vacuum sector, which means trivial boundary conditions at infinity.)

Gedankenexperimentally, you trap electron-positron pairs in separate jars and then use a binary observable derived from an atomic system to decide which to mail off to Alpha Centauri. The Hilbert space which describes your remaining jar will be in a superposition of states.