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# Antiparticle

For each kind of particle, there is an associated antiparticle with the same mass but opposite electromagnetic, weak, and strong charges, as well as spin. Some particles, notably photons, have no distinct antiparticle, or, put in another way, are identical to their antiparticle. Such particles are called real neutral particles, in contrast with, say, neutrons or neutral kaons, which are not identical to their antiparticles. Each quantum number of a real neutral particle is identical with its antiparticle's one.

Particle-antiparticle pairs can arise from the interactions of other particles, and can annihilate one another to produce other particles. The annihilation products (and possible means of pair production) depend on the interactions of the particles involved. For example, an electron and positron (or antielectron) will tend to annihilate to photons, the quanta of the electromagnetic field, since their interactions are primarily electromagnetic. Proton-antiproton pairs, interacting through the strong nuclear force, tend to annihilate to collections of mesons and their own antiparticles, mostly various types of pion. In either case, however, increasing the energy of the collision (as in a particle accelerator) can lead to the production of more exotic products as the necessary energy becomes available, and this process is an important tool in particle physics.

Antiparticles are produced by certain nuclear reactions (most notably positive beta decay), and by sufficiently energetic particle collisions, including naturally occurring cosmic ray collisions. Antimatter is a collection of antiparticles, in particular antiprotons, antineutrons and positrons 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 by Carl D. Anderson in a cloud chamber experiment. Dirac's prediction stemmed from the existence of negative energy states, which in a relativistic universe cannot be discarded a priori. To keep all interacting electrons in his theory from eventually falling into negative energy states, Dirac posited that these extra states must all be filled already. In that case, a negative-energy 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.

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## History

### Experiment

In 1932, Carl D. Anderson found positrons created by cosmic-ray collisions in a cloud chamber, in which moving electrons (or positrons) leave behind trails as they move through the gas. The electric charge-to-mass ratio of a particle can be measured by observing the curling of its cloud-chamber track in a magnetic field. Originally, positrons, because of the direction that their paths curled, were mistaken for electrons travelling in the opposite direction.

The antiproton and antineutron were found by Emilio Segrè and Owen Chamberlain in 1955. Since then the antiparticles of many other subatomic particles have been created in particle accelerator experiments. In recent years, complete atoms of antimatter have been assembled out of antiprotons and positrons, collected in electromagnetic traps.

### Hole theory

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 a particle and its antiparticle annihilate, the energy released is the sum of mc2 of the two particles (or more accurately, the sum of the $\sqrt{p^2c^2 + m^2c^4}$ of the particles). If antimatter 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.

Dirac developed "hole theory" 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. Dirac's own relativistic wave equation for electrons seemed to imply that both negative- and positive-energy electrons really existed. That being the case, since a collection of electrons would tend to radiate energy in their interactions, there seemed to be nothing to stop every electron in the universe from emitting enough energy to fall into a negative energy state. To prevent this from happening in his theory (which would obviously be at odds with the world we see around us), Dirac invoked the Pauli exclusion principle applying to all fermions such as the electron; this prevents any two electrons from occupying exactly the same quantum state. He proposed a "sea" of negative-energy electrons that would fill the universe, already occupying all of the lower energy states so that no other electrons could fall into them.

However, given the right combination of energy and momentum, from, for example, a collision of two photons, one of these particles could be lifted out of the Dirac 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 — the absence of a negative-energy electron, which would itself, according to the mathematics, act exactly like a positive-energy electron, only with a positive charge. If an electron were to hit this hole, a negative-energy state would become available to it, and the electron could emit enough energy in the form of photons to send it into that lower state, disappearing into the sea of negative-energy electrons.

In the lab, this would appear as a pair of photons colliding and transforming 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 antimatter is completely consistent with what has been observed in the laboratory, and theoretically, the anti-particle should exhibit a "normal" gravitational force. This description is also similar to the behavior of charge carriers in a semiconductor, for which the hole theory is still considered valid.

However, nobody (including Dirac) was very satisfied with the idea that the universe was completely filled with a sea of negative-energy electrons. For one thing, it was hard to see how the corresponding infinite sea of negative charge could be made to make sense. For another, bosons also have antiparticles, but since they do not obey the Pauli exclusion principle, hole theory doesn't work for them.

### Antiparticles in quantum field theory

The root of the trouble lay in Dirac's treatment of his wave theory for the electron as a way of reconciling special relativity with the single-particle quantum mechanics represented by the Schrödinger equation. Eventually the Dirac theory would became incorporated into quantum electrodynamics (QED), the first successful quantum field theory; the hole theory was abandoned and the situation became far less paradoxical.

In Shin'ichiro Tomonaga's and Julian Schwinger's formulation, the entity described by Dirac's wave equation is not the wave function of a single electron, but, rather, a quantum field constructed of operators that create and destroy both electrons and positrons. Dirac's positive-energy solutions to the wave equation are associated with operators that destroy electrons (by convention) and create positrons; his negative-energy solutions are associated with the creation of electrons and the destruction of positrons. The particles themselves occur only in positive-energy states, though they can seem to have negative energy in the intermediate virtual states used in the bookkeeping of perturbation theory.

Richard Feynman's formulation of QED treated positrons in a seemingly very different (and very strange) but actually equivalent manner: he thought of the Dirac wave equation as describing the wave function of a particle, but allowed the particle to travel backwards in time. Feynman then interpreted the negative-energy solutions as positive-energy electrons moving backwards in time, in which case they would act much like forward-in-time particles with the opposite charge—antiparticles. Two particles of the same charge travelling in different directions through time could attract electromagnetically.

Annihilation and pair production then appear to be nothing but a generalization of scattering, as can be seen in the following example. Say you have an electron, travelling forward through time, and (according to Feynman's picture) 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 backward in time, which sends it hurling forward through time once again.

 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.

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 is travelling off to the right while the positron moves to the left, colliding with a regular electron at T5 and releasing energy. The whole reaction appears to be the scattering of an electron and a photon, with an intermediate state consisting of a pair of electrons and a virtual positron.

Feynman made this picture precise with his formalism of Feynman diagrams, in which particle and antiparticle interactions are visualized as a set of paths in space-time. Many would argue that the paths shouldn't be taken too literally: these diagrams just represent terms in a perturbation theory approximation to quantum field theory. However, Feynman himself seems to have thought of them much more literally as representing particles zigzagging back and forth through space-time in a grand path integral encompassing all possibilities.

## Properties of antiparticles

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 $i \hbar \nabla$ to $-i \hbar \nabla$. 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 (which determines the time evolution of the system) commutes with CPT, (CPT)-1H(CPT)=H, that is, all of known physics possesses 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, $H=\sqrt{{(p-A(x))^2 c^2 + m^2 c^4}}+V(x)$, 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 $H=-\sqrt{{(p-A(x))^2 c^2 + m^2 c^4}}+V(x)$, then apply CPT. The potential(both A and V) is negated by the charge conjugation, giving $H=-\sqrt{{(p+A(x))^2 c^2 + m^2 c^4}}-V(x)$. The kinetic momentum is negated by the parity, and the potentials are replaced with their mappings under a parity transformation, giving $H=-\sqrt{{(-p-A(-x))^2 c^2 + m^2 c^4}}-V(-x)$. The time reversal negates the total momentum and the hamiltonian, giving $H=\sqrt{{(p-A(-x))^2 c^2 + m^2 c^4}}+V(-x)$.

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 ψ(x) with $\int a(y) \psi(y-x) \, dx$ 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 [Jx,Jy] = iJz which transforms under complex conjugation to [Jx,Jy] = - iJz, 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 intrinsic 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):

particle antiparticle
mass m m
spin quantum number s s
electric charge q -q
color charge {r,g,b} {−r,−g,−b}
intrinsic parity eiφ eiφ