Particle-antiparticle pairs arise from pure energy and will annihilate one another to give pure energy, usually in the form of photons. Antiparticles are produced by nuclear reactions and cosmic rays. Antimatter is a collection of antiparticles, in particular antiprotons, antineutrons and positrons (anti-electrons) in a similar composition as matter.
The existence of antiparticles was predicted by Dirac a few years before the first one, the antielectron or positron, was found. The idea stemmed from the existence of negative energy states, which in a relativistic universe can not be discarded a priori. Since electrons normally seek the lowest possible energy state, Dirac posited that these extra states must all be filled with what are called virtual particles. In that case, a virtual particle could be promoted to a positive energy state, creating a real particle and leaving a hole that would behave exactly the same, but with opposite charge.
Experimentation with anti-matter goes back quite some time to the cloud chambers, in which moving electrons (or positrons) leave behind trails as they move through the gas. Originally, positrons, because of the electromagnetic forces acting on them, were mistaken for electrons travelling in the opposite direction.
It is tempting to sometimes think of antimatter as consisting of negative energy, or possibly even having negative mass. However, this cannot be the case. When matter and anti-matter collide, the energy released is the sum of mc2 of the two particles (or more accurately, the sum of the √(p2c2 + m2c4) of the particles). If anti-matter had negative energy, the energy released from the two colliding particles would equal 0, since the positive and negative energies would cancel each other out.
Originally, the idea of anti-matter came to Dirac when, among other things he looked at the true form of the equation E = mc2, which is actually E2 = p2c2 + m2c4, and realized that the "2" sign means that the equation for energy can have two solutions, a negative energy solution and a positive energy solution. That being the case, since electrons always seek to fill the lowest possible energy state, there seemed to be nothing to stop every electron in the universe from emitting enough enery to fall into a negative energy state. To correct this he propposed a "sea of negative electrons" that would fill the universe, occupying all of the lower energy states. However, given sufficient energy, from for example, a photon, one of these particles could be lifted out of the sea of negative energy to become a positive energy particle. But when lifted out, it would leave behind a hole in the sea of negative energy - a space of zero energy, which would itself, according to the mathematics, act exactly like an electron, but with a positive charge. If an electron were to hit this area of zero energy a lower negative energy state would become available and the electron would emit enough energy to send it into that lower state, dissappearing into the sea of negative electrons.
In the lab, this would appear as a photon travelling along and suddenly splitting into an electron and positron. The positron then would hit another electron (or possibly, the same one), and energy would be released as the two particles annihilate each other.
This theory of anti-matter is completely consistent with what has been
observed in the laboratory, and theoretically, the anti-particle should
exhibit a "normal" gravitational force.
However, nobody (including Dirac) was very satisfied with the idea that the universe was completely filled with a sea of negative electrons, particularly because bosons have antiparticles, though hole theory doesn't work for them. Richard Feynman, however, shortly afterwards, showed that negative energy forward in time and positive energy backwards in time solutions are not allowed to the energy equation. A negative energy running backwards in time would appear to be exactly the same as a positive energy particle running forward in time except for its polarization, which would cause two particles of the same charge travelling in different directions through time to attract electromagnetically.
So, how does this fit into the idea of anti-matter?
Say you have an electron, travelling forward through time, and it emits a photon with enough energy and in the right direction to send it hurling back in time. It continues along for a while, then emits another photon, which sends it hurling forward through time once again.
To us, observing this reaction travelling only forward in time, at T1 we see a photon split up into two particles, a positron and an electron. The electron travelling off to the right while the positron moves to the left, colliding with a regular electron at T5 and releasing energy.
In "reality" the electron starting at the left end, moves forward in time until T5 when it emits enough energy to slip into a negative energy state. It can only do so if the photon is emitted in a way that will send it back in time, however, which is a very low probability. It continues backwards in time with negative energy but since it is travelling backwards in time the negative energy, from the viewpoint of something moving forwards in time, appears to be acting like a positive energy particle.
A particle's wave function can be changed to that of its antiparticle by applying the charge conjugation, parity, and time reversal(which, contrary to the name, involves complex conjugation in addition to replacing t with -t) operations.
The charge conjugation operator has no effect on the momentum. The parity operation negates the momentum, since -∂/∂x=∂/∂(-x). The time reversal operator also negates the momentum, because the momentum operator is changed from to . Thus the net effect of the CPT operation leaves the momentum unchanged.
The energy is unchanged by the parity and charge conjugation operators. The time reversal operator also leaves the energy unchanged as shown. Complex conjugation negates the energy(see above argument for momentum). Replacing t with -t also negates the energy(see above argument for momentum under parity transformations). Thus the net effect of time reversal leaves momentum unchanged, as was to be shown. Since energy is invariant under the C, P, and T operators, it is invariant under CPT transformations as well.
Because the hamiltonian commutes with CPT, (CPT)-1H(CPT)=H, that is the Hamiltonian posseses CPT symmetry. Since the momentum has the same magnitude after the CPT operation, this implies that the mass does as well, so a particle and its antiparticle must have the same mass, as will be shown.
For the particle, , where A is the potential momentum(such as magnetic potential times electric charge). For the antiparticle we choose the negative energy solution for a particle, ie , then apply CPT. The potential(both A and V) is negated by the charge conjugation, giving . The kinetic momentum is negated by the parity, and the potentials are replaced with their mappings under a parity transformation, giving . The time reversal negates the total momentum and the hamiltonian, giving .
Due to the fact that for a spherically symmetric potential, which is required for this argument(without it parity will have a more complicated form which replaces with for some a(x)), potential momentum is an even function of position and potential energy is an odd function of position, which gives it the same form as that of the particle, in terms of the particles potential. Since the momentum is the same, the mass must be the same as well in order for the hamiltonian to be invariant.
Obviously orbital angular momentum is negated, since r X p transforms into (-r) X (p), where X is the cross product and r instead of x is the position(since we are now considering multiple dimensions). Total angular momentum must also be negated as seen from the commutation relations such as which transforms under complex conjugation to , etc. Spin is thus also negated(note the spin quantum number is the same as it expresses the magnitude alone).
Charges, such as electric charge, and color charge are negated because the corresponding potentials are negated. Electric current densities(along with other current densities) may similarly be seen to be negated.
The intristic parity is unchanged, since C, P, and T all commute, thus [P,CPT]=0.
Since the hamiltonian and the energy are CPT invariant, if the original particle
was possible, the result is possible as well. Thus there is no reason applying the CPT operation to a particle will not produce another particle(this is the modern theoretical argument for the existence of antiparticles).
In summary here are the properties of antiparticles(only those that distinguish particle species are considered here):
History
t5 ----*------/
t4 ----/\\----/
t3 ---/--\\--/
t2 --/----\\/
t1 -/-----*
The Y axis is time and the X axis is position. The "*" are places where photons are emitted, the "/" and "\\" trace out the path of the particle, from left to right, and the "-" designate a specific point in time, labeled as t1, t2, t3, t4, and t5.Properties of antiparticles
| particle | antiparticle | |||
|---|---|---|---|---|
| mass | m | m | ||
| spin quantum number | s | s | ||
| electric charge | q | -q | ||
| color charge | {r,g,b} | {-r,-g,-b} | ||
| intristic parity |