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Matterwave

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1) We have that there are 4 internal states of a spin 1/2 particle, 2 of which correspond to particle with two possible spin orientations, and 2 of which correspond to its antiparticle with 2 possible spin orientations. Does that mean we can have a particle which is a linear combination of particle+antiparticle states? This in my mind would be analogous to a particle which is a linear combination of spin up and spin down for example. If not, then why not? But if so, what is the physical meaning of an particle in a superposition of, for example, electron+positron states? By charge conservation, it seems this should be forbidden. But then again when we first learned QM, we found immediately that strict Energy conservation appears to be broken for linear combinations of different energy states, but that the expectation value of energy is conserved. Is there some analogous "the expectation value of charge is conserved" statement here?

2) Can we transform a particle into its antiparticle via a Lorentz transformation? In my mind, this would be analogous to transforming a spin up along z particle to a spin up along x particle by rotating my x axis into the z axis. If not, then why not?

3) Since the Lorentz boosts cannot be represented by unitary group of transformations, to construct a probability current and density which remains normalized under a Lorentz boost, we cannot use ##u^\dagger \gamma^\mu u## but instead we use ##\overline{u}\gamma^\mu u## where ##\bar{u}=u^\dagger\gamma^0##. Is this basically modifying the metric on our space from the usual metric to a modified metric where I have to multiply by a gamma matrix in the middle of all inner products? This would seem to, in turn, change our correspondences of kets to bras. The physical state which is dual to ##u## is now ##\overline{u}## and not ##u^\dagger##?