It's very misleading to say an antiparticle is "going backward in time". The opposite is the case!
Antiparticles where predicted by Dirac in a very complicated and peculiar way. The space-time structure of special relativity (Minkowski space) implies the possible wave equations, which were taken as candidates for equation for single-particle wave functions for relativistic particles in analogy to the Schrödinger equation in non-relativistic quantum theory. In the latter case, of course, one has to use the space-time structure of Newtonian physics (Galilei space, if you wish to give it a name). Then the Born interpretation of the wave function, i.e., the definition of a conserved probability current with a positive definite probability distribution is straight forward. Also for the free-particle case the Hamiltonian has only positive eigenvalues, i.e., there's always a stable ground state.
In relativistic physics, the restriction to positive energies leads to very complicated "non-local" solutions, which also do not obey microcausality, i.e., although the wave equation is relativistically covariant there's signal propagation faster than light. Thus you have to admit solutions with negative frequencies, but if interpreted in a naive single-particle way this implies negative energies, and the Hamiltonian is not bounded from below, and there's no stable ground state. Dirac's idea now was to declare all negative-energy states occupied ("Dirac sea") and this state to represent the "vacuum" (which is a contradiction). Now, switching on interactions, enables transitions of a particle in the Dirac sea to the positive-energy states, leaving behind a hole, which behaves as an oppositely charged particle with the same mass as the particles. Such a hole can be reinterpreted as representing a such defined anti-particle with positive energy moving in the opposite direction.
This whole construct is not only pretty inconsistent but also very complicated to deal with. However, it leads to the correct predictions (at least for QED). It's also inconsistent from a fundamental point of view, because you start with a single-particle description and end up with infinitely many particles in the Dirac sea. Such a system should, however, be described as a many-body system to begin with. The above discribed particle-antiparticle pair-creation process also implies that particle number is not conserved (but charge is). The most convenient way to describe such a system is quantum field theory with an empty vacuum and creation and annihilation operators with respect to a well-defined single-particle basis (for free particles these are the momentum-polarization eigenstates). Then the only trick needed to reinterpret the negative-frequency modes as positive-energy excitations is that you have to put a creation operator in the mode expansion of the field operator and flip three-momentum. For the positive-frequency modes you put an annihilation operator as in non-relativistic many-body theory (where you have only annihilation operators in that expansion). This superposition of negative and positive-frequency modes, together with the canonical equal-time commutatation/anticommutation relations also leads to a microcausal local quantum system with a stable ground state.
Together with this assumptions, you can derive the correct QFT's systematically from the group theory of the Poincare group, implying the spin-statistics and CPT theorem, both of which are confirmed by all observations so far.