Some sources describe antimatter as just like normal matter, but "going backwards in time". What does that really mean? Is that a good analogy in general, and can it be made mathematically precise? Physically, how could something move backwards in time?
Quantum Field Theory – Is Antimatter Matter Going Backwards in Time? Understanding the Concepts
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Please bear in mind that I am an experimentalist, therefore I treat theoretical ideas and models as dependent on experimental observations and not vice versa.
Facts:
elementary particles have measurable quantum numbers. These quantum numbers define the particle.
elementary particles have mass
there exist elementary and composite (from elementary) particles that have the same mass as others but opposite charge, as the electron vs positron, proton vs antiproton, called antiparticles because:
there is a very high probability that when scattered against each other, the particle and antiparticle disappear and the energy appears in a plethora of other particles. This is called annihilation because charge disappears, and in general the opposite quantum numbers are "annihilated".
On the theoretical side these experimental observations are fitted beautifully by the Dirac equation if one considers the negative energy solutions to describe the antiparticles.
One thing we can be sure of is that measured antiparticles travel forward in time.
The popular theoretical interpretation of antiparticles being particles traveling backwards in time mainly comes from the Feynman diagrams. These are a brilliant mathematical tool for fitting and predicting measurements representing particle interactions as incoming lines and outgoing lines and in between lines representing virtual particles that carry the quantum numbers but are off mass shell.
Due to the CPT theorem, once a Feynman diagram is drawn, one can interpret the lines consistently according to CPT and will get the corresponding cross sections and probabilities for the change in the quantum numbers consistent with CPT (charge conjugation, parity and time reversal). Identifying a positron as a backwards-in-time electron is an elegant interpretation that in the Feynman diagrams exhibits the CPT symmetry they must obey.
What I am saying is, the statement "positrons are backward-going electrons" is a convenient and accurate mathematical representation for calculation purposes. "As if". There has not been any indication, not even a tiny one, that in nature (as we study it experimentally) anything goes backwards in time, as we define time in the laboratory.
Edit replying to comment by Nathaniel:
I'm curious: how would you expect empirical data from a backwards-travelling positron to differ from what we actually see?
In this bubble chamber picture we see the opposite to annihilation, the creation by a photon of an e+ e- pair. (This is an enlarged detail from the bubble chamber photo in the archives. The original web page with the letterings has disappeared, as of july2017)
The magnetic field that makes them go into helices is perpendicular to the plane of the photo. We identify the electron by the sign of its curvature as it leaves the vertex. The positron is the one going up to the left corner. We know it is not an electron that started its life before the vertex formed because as an electron/positron moves through the liquid it loses energy and the loss defines the time direction of the path. So the particle has to start at the vertex and end at the upper left, so it has the opposite curvature to the electron and it is a positron.
A Feynman diagram looks like a scattering in real space, or a pair production, but one cannot project the intricacies of the mathematics it represents onto real space. It is only the calculations of cross sections and probabilities that can be compared with measurements.
There is no massive fundamental/elementary "(anti)particle" (or better field) in our Universe that can move at the speed of light. The fastest known massive fundamental (anti)particles are ultra high energy astrophysical neutrinos which can move almost at speed of light but not exactly. The reason for this is that first of all they are the lightest known fundamental (anti)particles in Nature and are given high boost factors at some astrophysical sources as they're produced. So, there is no way for any massive "particle" in our Universe to travel yet faster than speed of light. The only (anti)particle that can travel at the speed of light (but not faster than that) is light itself, namely photon field. Since photon is its own anti particle, we can conclude that the only anti-matter travelling at speed of light is 'light.'
By the construction of Big Bang theory and creation of Universe from a singularity, the arrow of time has been always forwards in an expanding Universe. So, nothing in Nature can move opposite to that arrow (or backward in time as you called it.) The Feynman diagrams--which have been used to work out perturbation theories--make use of the time symmetry where as you mentioned you can replace an incoming electron with an outgoing positron and that has nothing to do with the reality of the processes being involved. They are just mathematical constructs.
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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.