Is anti-matter matter going backwards in time?
To the best of my knowledge, most physicists don't believe that antimatter is actually matter moving backwards in time. It's not even entirely clear what would it really mean to move backwards in time, from the popular viewpoint.
If I'm remembering correctly, this idea all comes from a story that probably originated with Richard Feynman. At the time, one of the big puzzles of physics was why all instances of a particular elementary particle (all electrons, for example) are apparently identical. Feynman had a very hand-wavy idea that all electrons could in fact be the same electron, just bouncing back and forth between the beginning of time and the end. As far as I know, that idea never developed into anything mathematically grounded, but it did inspire Feynman and others to calculate what the properties of an electron moving backwards in time would be, in a certain precise sense that emerges from quantum field theory. What they came up with was a particle that matched the known properties of the positron.
Just to give you a rough idea of what it means for a particle to "move backwards in time" in the technical sense: in quantum field theory, particles carry with them amounts of various conserved quantities as they move. These quantities may include energy, momentum, electric charge, "flavor," and others. As the particles move, these conserved quantities produce "currents," which have a direction based on the motion and sign of the conserved quantity. If you apply the time reversal operator (which is a purely mathematical concept, not something that actually reverses time), you reverse the direction of the current flow, which is equivalent to reversing the sign of the conserved quantity, thus (roughly speaking) turning the particle into its antiparticle.
For example, consider electric current: it arises from the movement of electric charge, and the direction of the current is a product of the direction of motion of the charge and the sign of the charge.
$$\vec{I} = q\vec{v}$$
Positive charge moving left ($+q\times -v$) is equivalent to negative charge moving right ($-q\times +v$). If you have a current of electrons moving to the right, and you apply the time reversal operator, it converts the rightward velocity to leftward velocity ($-q\times -v$). But you would get the exact same result by instead converting the electrons into positrons and letting them continue to move to the right ($+q\times +v$); either way, you wind up with the net positive charge flow moving to the right.
By the way, optional reading if you're interested: there is a very basic (though hard to prove) theorem in quantum field theory, the TCP theorem, that says that if you apply the three operations of time reversal, charge conjugation (switch particles and antiparticles), and parity inversion (mirroring space), the result should be exactly equivalent to what you started with. We know from experimental data that, under certain exotic circumstances, the combination of charge conjugation and parity inversion does not leave all physical processes unchanged, which means that the same must be true of time reversal: physics is not time-reversal invariant. Of course, since we can't actually reverse time, we can't test in exactly what manner this is true.
Antimatter is in every precise meaningful sense matter moving backward in time. The notion of "moving backward in time" is nonsensical in a Hamiltonian formulation, because the whole description can only go forward in time. That's the definition of what the Hamiltonian does--- it takes you forward in time a little bit. So if you formulate quantum mechanics the Hamiltonian way, this idea is difficult to understand (still it can be done--- Stueckelberg discovered this connection before the path integral, when field Hamiltonians were the only tool).
But in Feynman's particle path-integral picture, when you parametrize particles by their worldline proper time, and you renounce a global causal picture in favor of particles splitting and joining, the particle trajectories are consistent with relativity, but only if the trajectories include back-in-time trajectories, where coordinate time ticks in the opposite sense to proper time.
Looked at in the Hamiltonian formalism, the coordinate time is the only notion of time. So those paths where the proper time ticks in the reverse direction look like a different type of particle, and these are the antiparticles.
Sometimes there is an idenification, so that a particle is its own antiparticle.
Precise consequence: CPT theorem
The "C" operator changes all particles to antiparticles, the P operator reflects all spatial directions, and the T operator reflects all motions (and does so by doing complex conjugation). It is important to understand that T is an operator on physical states, it does not abstractly flip time, it concretely flips all momenta and angular momenta (a spinning disk is spinning the other way), so that things are going backwards. The parity operator flips all directions, but not angular momenta.
The CPT theorem says that any process involving matter happens exactly the same when done in reverse motion, in a mirror, to antimatter.
The CPT operator is never the identity, aside from the case of a real scalar field. CPT acting on an electron produces a positron state, for example. CPT acting on a photon produces a photon going in the same direction with opposite polarization (if P is chosen to reflect all spatial coordinate axes, this is a bad convention outside of 3+1 dimensions).
This theorem is proved by noting that a CPT operator corresponds to a rotation by 180 degrees in the Euclidean theory, as described on Wikipedia.
Precise consequence: crossing
Any amplitude involving particles A(k_1,k_2,...,k_n) is analytic in the incoming and outgoing momenta, aside from pole and cut singularities caused by producing intermediate states. In tree-level perturbation theory, these amplitudes are analytic except when creating physical particles, where you find poles. So the scattering amplitudes make sense for any complex value of the momenta, since going around poles is not a problem.
In terms of mandelstam variables for 2-2 scattering, s,t,u (s is the CM energy, t is the momentum transfer and u the other momentum transfer, to the other created particle), the amplitude is an analytic function of s and t. The regions where the particles are on the mass shell are given by mandelstam plot, and there are three different regions, corresponding to A+B goes to C+D , Cbar + B goes to Abar+ D, and A + Dbar goes to C+Bbar. These three regimes are described by the exact same function of s,t,u, in three disconnected regions.
In starker terms, if you start with pure particle scattering, and analyticaly continue the amplitudes with particles with incoming momentum k's (with positive energy) to negative k's, you find the amplitude for the antiparticle process. The antiparticle amplitude is uniquely determined by the analytic contination of the particle amplitude for the energy-momentum reversed.
This corresponds to taking the outgoing particle with positive energy and momentum, and flipping the energy and momentum to negative values, so that it goes out the other way with negative energy. If you identify the lines in Feynman diagrams with particle trajectories, this region of the amplitude gives the contribution of paths that go back in time.
So crossing is the other precise statement of "Antimatter is matter going back in time".
Causal pictures
The notion of going back in time is acausal, meaning it is excluded automatically in a Hamiltonian formulation. For this reason, it took a long time for this approach to be appreciated and accepted. Stueckelberg proposed this interpretation of antiparticles in the late 1930s, but Feynman's presentation made it stick.
In Feynman diagrams, the future is not determined from the past by stepping forward timestep by timestep, it is determined by tracing particle paths proper-time by proper-time. The diagram formalism therefore is philosophically very different from the Hamiltonian field theory formalism, so much so Feynman was somewhat disappointed that they were equivalent.
They are not as easily equivalent when you go to string theory, because string theory is an S-matrix theory formulated entirely in Feynman language, not in Hamiltonian language. The Hamiltonian formulation of strings requires a special slicing of space time, and even then, it is less clear and elegant than the Feynman formulation, which is just as acausal and strange. The strings backtrack in time just like particles do, since they reproduce point particles at infinite tension.
If you philosophically dislike acausal formalisms, you can say (in field theory) that the Hamiltonian formalism is fundamental, and that you believe in crossing and CPT, and then you don't have to talk about going back in time. Since crossing and CPT are the precise manifestations of the statement that antimatter is matter going back in time, you really aren't saying anything different, except philosophically. But the philosophy motivates crossing and CPT.
This refers to Feynman's 1949 theory.
See http://www.upscale.utoronto.ca/PVB/Harrison/AntiMatter/AntiMatter.htmllink text
From there: "Feynman's Theory of Antimatter
In 1949 Richard Feynman devised another theory of antimatter.
The spacetime diagram for pair production and annihilation appears to the right. An electron is travelling along from the lower right, interacts with some light energy and starts travelling backwards in time. An electron travelling backwards in time is what we call a positron. In the diagram, the electron travelling backwards in time interacts with some other light energy and starts travelling forwards in time again. Note that throughout, there is only one electron.
A friend of mine finds the image of an electron travelling backwards in time, interpreted by us as a positron, to be scary.
Feynman in his original paper proposing this theory wrote:
"It is as though a bombardier flying low over a road suddenly sees three roads and it is only when two of them come together and disappear again that he realizes that he has simply passed over a long switchback in a single road." (Physical Review 76, (1949), 749.)
Note that Feynman's theory is yet another echo of the fact, noted above, that a negatively charged object moving from left to right in a magnetic field has the same curvature as a positive object moving from right to left.
Feynman's theory is mathematically equivalent to Dirac's, although the interpretations are quite different. Which formalism a physicist uses when dealing with antimatter is usually a matter of which form has the simplest structure for the particular problem being solved.
Note that in Feynman's theory, there is no pair production or annihilation. Instead the electron is just interacting with electromagnetic radiation, i.e. light. Thus the whole process is just another aspect of the fact that accelerating electric charges radiate electric and magnetic fields; here the radiation process is sufficiently violent to reverse the direction of the electron's travel in time.
Nambu commented on Feynman's theory in 1950:
"The time itself loses sense as the indicator of the development of phenomena; there are particles which flow down as well as up the stream of time; the eventual creation and annihilation of pairs that may occur now and then is no creation or annihilation, but only a change of direction of moving particles, from past to future, or from future to past." (Progress in Theoretical Physics 5, (1950) 82).
About Formally Equivalent Descriptions ...."
Then you mix in another very interesting problem, namely the origin of the apparent matter antimatter asymmetry in the observable universe (observed absence of annihilation radiation except in special circumstances) and point out that it may be related to a very very hard problem indeed, namely the origin of time asymmetry. One problem at a time, please. Maybe separate questions, but the answers will likely be more or less over your head since, to the extent that they are even partially understood, they are still being figured out.