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Antiparticle

Corresponding to most kinds of particles, there is an associated antiparticle with the same mass and opposite electric charge. For example, the antiparticle of the electron is the positively charged antielectron, or positron, which is produced naturally in certain types of radioactive decay.

The laws of nature are very nearly symmetrical with respect to particles and antiparticles. For example, an antiproton and a positron can form an antihydrogen atom, which has almost exactly the same properties as a hydrogen atom. A physicist whose body was made of antimatter, doing experiments in a laboratory also made of antimatter, using chemicals and substances made of antiparticles, would find almost exactly the same results in all experiments. This leads to the question of why the formation of matter after the Big Bang resulted in a universe consisting almost entirely of matter, rather than being a half-and-half mixture of matter and antimatter. The discovery of CP violation helped to shed light on this problem by showing that this symmetry, originally thought to be perfect, was only approximate.

Particle-antiparticle pairs can annihilate each other, producing photons; since the charges of the particle and antiparticle are opposite, charge is conserved. For example, the antielectrons produced in natural radioactive decay quickly annihilate themselves with electrons, producing pairs of gamma rays.

Antiparticles are produced naturally in beta decay, and in the interaction of cosmic rays in the Earth's atmosphere. Because charge is conserved, it is not possible to create an antiparticle without either destroying a particle of the same charge (as in beta decay), or creating a particle of the opposite charge. The latter is seen in many processes in which both a particle and its antiparticle are created simultaneously, as in particle accelerators. This is the inverse of the particle-antiparticle annihilation process.

Although particles and their antiparticles have opposite charges, electrically neutral particles need not be identical to their antiparticles. The neutron, for example, is made out of quarks, the antineutron from antiquarks, and they are distinguishable from one another because neutrons and antineutrons annihilate each other upon contact. However, other neutral particles are their own antiparticles, such as photons, the hypothetical gravitons, and WIMPs. These are called Majorana particles and can annihilate with themselves.

Particle-antiparticle annihilation

An example of a virtual pion pair which influences the propagation of a kaon causing a neutral kaon to mix with the antikaon. This is an example of renormalization in quantum field theory— the field theory being necessary because the number of particles changes from one to two and back again.

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 in free space because it is impossible to conserve energy and momentum together in this process. However, in the Coulomb field of a nucleus the translational invariance is broken and single-photon annihilation may occur. [1] The reverse reaction (in free space, without an atomic nucleus) is also impossible for this reason. In quantum field theory this process is allowed only as an intermediate quantum state for times short enough that the violation of energy conservation can be accommodated 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 opens the way for neutral particle mixing through processes such as the one pictured here, which is a complicated example of mass renormalization.

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) operators. If |p,\sigma ,n  \rangle 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,\sigma,n \rangle\ =\ (-1)^{J-\sigma}\  |p,-\sigma,n^c \rangle ,

where nc denotes the charge conjugate state, i.e., 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,\sigma,n\rangle \ \propto \  |-p,-\sigma,n\rangle ,
CP\ |p,\sigma,n\rangle \ \propto \  |-p,\sigma,n^c\rangle ,
C\ |p,\sigma,n\rangle \ \propto \  |p,\sigma,n^c\rangle ,

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

\psi (x)=\sum_{k}u_k (x)a_k e^{-iE(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 ak 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=\sum_{k} E(k) a^\dagger_k a_k,\,

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\prime} = a^\dagger_k\ \mathrm{and}\  b^\dagger_{k\prime}=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

\psi(x)=\sum_{k_+} u_k (x)a_k  e^{-iE(k)t}+\sum_{k_-} u_k (x)b^\dagger _k e^{-iE(k)t},\,

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

H=\sum_{k_+} E_k a^\dagger _k a_k + \sum_{k_-}  |E(k)|b^\dagger_k b_k + E_0,\,

where E0 is an infinite negative constant. The vacuum state is defined as the state with no particle or antiparticle, i.e.,a_k |0\rangle=0 and b_k |0\rangle=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 ak and bk 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 negative energy modes of the electron field backward in time, Ernst Stueckelberg reached a pictorial understanding of the fact that the particle and antiparticle have equal mass m and spin J but opposite charges q. This allowed him to rewrite perturbation theory precisely in the form of diagrams. Richard Feynman later gave an independent systematic derivation of these diagrams from a particle formalism, and they are now called Feynman diagrams. Each line of a diagram represents a particle propagating either backward or forward in time. This technique is the most widespread method of computing amplitudes in quantum field theory today.

Since this picture was first developed by Ernst Stueckelberg, and acquired its modern form in Feynman's work, it is called the Feynman-Stueckelberg interpretation of antiparticles to honor both scientists.

 

ANTIPARTICLE

Corresponding to most kinds of particles, there is an associated antiparticle with the same mass and opposite electric charge. For example, the antiparticle of the electron is the positively charged antielectron, or positron, which is produced naturally in certain types of radioactive decay.

The laws of nature are very nearly symmetrical with respect to particles and antiparticles. For example, an antiproton and a positron can form an antihydrogen atom, which has almost exactly the same properties as a hydrogen atom. A physicist whose body was made of antimatter, doing experiments in a laboratory also made of antimatter, using chemicals and substances comprised of antiparticles, would find almost exactly the same results in all experiments. This leads to the question of why the formation of matter after the Big Bang resulted in a universe consisting almost entirely of matter, rather than being a half-and-half mixture of matter and antimatter. The discovery of CP violation helped to shed light on this problem by showing that this symmetry, originally thought to be perfect, was only approximate.

Particle-antiparticle pairs can annihilate each other, producing photons; since the charges of the particle and antiparticle are opposite, charge is conserved. For example, the antielectrons produced in natural radioactive decay quickly annihilate themselves with electrons, producing pairs of gamma rays.

Antiparticles are produced naturally in beta decay, and in the interaction of cosmic rays in the Earth's atmosphere. Because charge is conserved, it is not possible to create an antiparticle without either destroying a particle of the same charge (as in beta decay), or creating a particle of the opposite charge. The latter is seen in many processes in which both a particle and its antiparticle are created simultaneously, as in particle accelerators. This is the inverse of the particle-antiparticle annihilation process.

Although particles and their antiparticles have opposite charges, electrically neutral particles need not be identical to their antiparticles. The neutron, for example, is made out of quarks, the antineutron from antiquarks, and they are distinguishable from one another because neutrons and antineutrons annihilate each other upon contact. However, other neutral particles are their own antiparticles, such as photons, the hypothetical gravitons, and WIMPs. These are called Majorana particles and can annihilate with themselves.

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) operators. 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,\sigma,n>\ =\ (-1)^{J-\sigma}\ |p,-\sigma,n^c>,

where nc denotes the charge conjugate state, i.e., 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,\sigma,n>\ \propto \ |-p,-\sigma,n>,
CP\ |p,\sigma,n>\ \propto \ |-p,\sigma,n^c>,
C\ |p,\sigma,n>\ \propto \ |p,\sigma,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

\psi (x)=\sum_{k}u_k (x)a_k e^{-iE(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 ak 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=\sum_{k} E(k) a^\dagger_k a_k,\,

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\prime} = a^\dagger_k\ \mathrm{and}\ b^\dagger_{k\prime}=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

\psi(x)=\sum_{k_+} u_k (x)a_k e^{-iE(k)t}+\sum_{k_-} u_k (x)b^\dagger _k e^{-iE(k)t},\,

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

H=\sum_{k_+} E_k a^\dagger _k a_k + \sum_{k_-} |E(k)|b^\dagger_k b_k + E_0,\,

where E0 is an infinite negative constant. The vacuum state is defined as the state with no particle or antiparticle, i.e., a_k |0\rangle=0 and b_k |0\rangle=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 ak and bk 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 negative energy modes of the electron field backward in time, Ernst Stueckelberg reached a pictorial understanding of the fact that the particle and antiparticle have equal mass m and spin J but opposite charges q. This allowed him to rewrite perturbation theory precisely in the form of diagrams. Richard Feynman later gave an independent systematic derivation of these diagrams from a particle formalism, and they are now called Feynman diagrams. Each line of a diagram represents a particle propagating either backward or forward in time. This technique is the most widespread method of computing amplitudes in quantum field theory today.

Since this picture was first developed by Ernst Stueckelberg, and acquired its modern form in Feynman's work, it is called the Feynman-Stueckelberg interpretation of antiparticles to honor both scientists







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