Antiparticle

Corresponding to each kind of particle, there is an associated antiparticle with the same mass and spin but with many other quantum numbers flipped in sign. Some particles, such as the photon, are identical to their antiparticle; such particles must have no electric charge, but not all charge neutral particles are of this kind. The laws of nature were thought to be symmetric between particles and antiparticles until CP violation experiments found that time-reversal symmetry is violated in nature. The observed excess of baryons over anti-baryons in the universe is one the primary unsolved problems in cosmology.

Particle-antiparticle pairs can annihilate each other if they are in appropriate quantum states. They can also be produced in various processes. These processes are used in today's particle accelerators to create new particles and to test theories of particle physics. High energy processes in nature can create antiparticles. These are visible in cosmic rays and in certain nuclear reactions. The word antimatter properly refers to (elementary) antiparticles, composite antiparticles made with them (such as antihydrogen) and to larger assemblies of either.

Properties of antiparticles

Quantum states of a particle and an antiparticle can be interchanged by applying the charge conjugation (C), parity (P), and time reversal (T). If |p,σ,n> denotes the quantum state of a particle (n) with momentum p, spin J whose component in the z-direction is σ, then one has

CPT |p,σ,n>  =  (-1)^J-σ |p,-σ,n^c>,

where n^c denotes the charge conjugate state, ie, the antiparticle. This behaviour under CPT is the same as the statement that the particle and its antiparticle lie in the same irreducible representation of the Poincare group. Properties of antiparticles can be related to those of particles through this. If T is a good symmetry of the dynamics, then

T |p,σ,n>  α  |-p,-σ,n>

CP |p,σ,n>  α  |-p,σ,n^c>

C |p,σ,n>  α  |p,σ,n^c>,

where the proportionality sign indicates that there might be a phase on the right hand side. In other words, particle and antiparticle must have

• the same mass m
• the same spin state J
• opposite electric charges q and -q.

Quantum field theory

This section draws upon the ideas, language and notation of canonical quantization of a quantum field theory.

One may try to quantize an electron field without mixing the annihilation and creation operators by writing

ψ(x)  =  ∑_k u_k(x) a_k e^{-i E(k)t},

where we use the symbol k to denote the quantum numbers p and σ of the previous section and the sign of the energy, E(k), and a_k denotes the corresponding annihilation operators. Of course, since we are dealing with fermions, we have to have the operators satisfy canonical anti-commutation relations. However, if one now writes down the Hamiltonian

H  =  ∑_k E(k) a⁺_k a_k,</b>

then one sees immediately that the expectation value of H need not be positive. This is because E(k) can have any sign whatsoever, and the combination of creation and annihilation operators has expectation value 1 or 0.

So one has to introduce the charge conjugate antiparticle field, with its own creation and annihilation operators satisfying the relations

b_k'  =  a⁺_k and b⁺_k'  =  a_k

where k' has the same p, and opposite σ and sign of the energy. Then one can rewrite the field in the form

ψ(x)  =  ∑_k(+) u_k(x) a_k e^{-i E(k)t}  +  ∑_k(-) u_k(x) b⁺_k e^{-i E(k)t},

where the first sum is over positive energy states and the second over those of negative energy. The energy becomes

H  =  ∑_k(+) E(k) a⁺_k a_k  +  ∑_k(-) |E(k)| b⁺_k b_k  +  E₀,

where E₀ is an infinite negative constant. The vacuum state is defined as the state with no particle or antiparticle, ie, a_k |0> = 0 and b_k |0> = 0. Then the energy of the vacuum is exactly E₀. Since all energies are measured relative to the vacuum, H is positive definite. Analysis of the properties of a</sub>k</sub> and b</sub>k</sub> shows that one is the annihilation operator for particles and the other for antiparticles. This is the case of a fermion.

This approach is due to Vladimir Fock, Wendell Furry and Robert Oppenheimer. If one quantizes a real scalar field, then one finds that there is only one kind of annihilation operator; therefore real scalar fields describe neutral bosons. Since complex scalar fields admit two different kinds of annihilation operators, which are related by conjugation, such fields describe charged bosons.

The Feynman-Stueckelberg interpretation

By considering the propagation of the positive energy half of the electron field backward in time, Richard Feynman showed that causality is violated unless one allows some particles to travel faster than light. However, if particles are allowed to do that, then from the point of view of another inertial observer it would look like it was travelling backward in time with the opposite charge. Hence Feynman reached a pictorial understanding of the fact that the particle and antiparticle have equal mass m and spin J but opposite charges. This allowed him to rewrite perturbation theory precisely in the form of diagrams, called Feynman diagrams, of particles propagating back and forth in time. This technique now is the most widespread method of computing amplitudes in quantum field theory.

This picture was independently developed by Ernst Stueckelberg, and has been called the Feynman-Stueckelberg interpretation of antiparticles.

History

Experiment

In 1932, soon after the prediction of positrons by Dirac, Carl D. Anderson found that cosmic-ray collisions produced these particles in a cloud chamber— a particle detector 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

... the development of quantum field theory made the interpretation of antiparticles as holes unnecessary, even though unfortunately it lingers on in many textbooks.  —  Steven Weinberg in The quantum theory of fields, Vol I, p 14, ISBN 0521550017

Solutions of the Dirac equation contained negative energy quantum states. As a result, an electron could always radiate energy and fall into a negative energy state. Even worse, it could keep radiating infinite amount of energy because there were infinitely negative energy states available. To prevent this unphysical situation from happening, Dirac proposed that a "sea" of negative-energy electrons fills the universe, already occupying all of the lower energy states so that, due to the Pauli exclusion principle no other electron could fall into them. Sometimes, however, one of these negative energy particles could be lifted out of this Dirac sea to become a positive energy particle. But when lifted out, it would leave behind a hole in the sea which would act exactly like a positive energy electron with a reversed charge. These he interpreted as the proton, and called his paper of 1930 A theory of electrons and protons.

Dirac was aware of the problem that his picture implied an infinite negative charge for the universe. Dirac tried to argue that we would perceive this as the normal state of zero charge. Another difficulty was the difference in masses of the electron and the proton. Dirac tried to argue that this was due to the electromagnetic interactions with the sea, until Hermann Weyl proved that hole theory was completey symmetric between negative and positive charges. Dirac also predicted a reaction e + p → γ + γ (electron and proton annihilate to give two photons). Robert Oppenheimer and Igor Tamm proved that this would cause ordinary matter to disappear too fast. A year later, in 1931, Dirac modified his theory and postulated the positron, anew particle of the same mass as the electron. The discovery of this particle the next year removed the last two objections to his theory.

However, the problem of infinite charge of the universe remains. Also, as we now know, bosons also have antiparticles, but since they do not obey the Pauli exclusion principle, hole theory doesn't work for them. An unified interpretation of antiparticles is now available in quantum field theory, which solves both these problems.

Particle-antiparticle annihilation

If a particle and antiparticle are in the appropriate quantum states, then they can annihilate each other and produce other particles. Reactions such as e⁺  +  e^-  →  γ  +  γ (the two-photon annihilation of an electron-positron pair) is an example. The single-photon annihilation of an electron-positron pair, e⁺  +  e^-  →  γ cannot occur because it is impossible to conserve energy and momentum together in this process. The reverse reaction is also impossible for this reason. However, in quantum field theory this process is allowed as an intermediate quantum state for times short enough that the violation of energy conservation can be accomodated by the uncertainty principle. This opens the way for virtual pair production or annihilation in which a one particle quantum state may fluctuate into a two particle state and back. These processes are important in the vacuum state and renormalization of a quantum field theory. It also open the way for neutral particle mixing through processes such as the one pictured here: which is a complicated example of mass renormalization.

See also

• Parity, charge conjugation and time reversal symmetry.
• CP violations and the baryon asymmetry of the universe.
• Quantum field theory and the list of particles

References and external links

• The quantum theory of fields (Vol 1), by Steven Weinberg (http://www.amazon.com/exec/obidos/ASIN/0521550017/qid=1119099936/sr=2-1/ref=pd_bbs_b_2_1/102-7631552-5785747), (Cambridge University Press, 1995) [ISBN 0521550017].ca:Antipartícula

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