Problem Setting

When we solve both the KG equation and the Dirac equation, there is the need to introduce negative energy solutions in order to have a complete set of states. There are then two types of solutions to the Dirac equation:

$psi_1 (s,x) = u(s,p)e^{-i(Et-vec{p}.vec{x})}$

$psi_2 (s,x) = v(s,p)e^{i(Et-vec{p}.vec{x})}$

where E > 0

When acting with the energy operator $ifrac{partial}{partial t}$ we get

$ifrac{partial}{partial t}psi_1 = +E psi_1$

$ifrac{partial}{partial t}psi_2 = -E psi_2$

When faced with these negative energy solutions, Mark thomson says this in his "Modern Particle Physics" book:

"The modern interpretation of the negative energy solutions, due to Stückelberg
and Feynman, was developed in the context of quantum field theory. The $E < 0$
solutions are interpreted as negative energy particles which propagate backwards
in time. These negative energy particle solutions correspond to physical positive
energy antiparticle states with opposite charge, which propagate forwards in time."

Given this, my question relies on the next sentence:

"Since the time dependence of the wavefunction, $e^{−iEt}$, is unchanged under
the simultaneous transformation $E→−E$ and $t→−t$ these two pictures are mathematically
equivalent"

Which I agree with, but it seems to have nothing to do with what we have obtained when solving the Dirac equation.

Question

It seems that interpreting negative energy solutions going backwards in time as positive energy solutions (antiparticles) going forwards in time leads to a contradiction.
If the antiparticles are physical states with positive energy, then their time dependence should be of the form:

$e^{-iEt}$ where E > 0 and time is running forwards as expected from the usual Schrödinger time evolution.

However the antiparticle time evolution is given by $e^{iEt}$ where $E > 0$. This seems to be incompatible with any physical picture because either:

1. It has negative energy and is going forwards in time $e^{-i(-E)t}$
2. It has positive energy and is going backwards in time $e^{-iE(-t)}$

Neither of these interpretations looks compatible with physical reality and to emphasize that i also quote Thomson’s book before this:

"Apart from possessing different charges, antiparticles behave very much like particles;
they propagate forwards in time from the point of production, ionise the gas in
tracking detectors, produce the same electromagnetic showers in the calorimeters
of large collider particle detectors, and undergo many of the same interactions as
particles"