Antiparticle

Conservation of baryon number

In 1938 Ernst Stueckelberg proposed conservation of baryon number as a law governing elementary particle reactions. All baryons, like protons and neutrons, are assigned baryon number +1 while their antiparticle counterparts are assigned -1. For example, the decay of the neutron, $${\displaystyle n\to p^{+}+e^{-}+{\overline{\nu }}_e}$$ conserves baryon number because the neutron on the left side and the proton on the right side both have baryon number +1 while other particles are not baryons. Since protons are the lowest energy baryon, no decay reaction can yield another baryon: the conservation law predicts the observed stability of the proton.

Elementary antiparticles

Composite antiparticles

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⁺
→ γγ (the two-photon annihilation of an electron-positron pair) are 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. 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

Quantum states of a particle and an antiparticle are interchanged by the combined application of charge conjugation ${\displaystyle C}$, parity ${\displaystyle P}$ and time reversal ${\displaystyle T}$. ${\displaystyle C}$ and ${\displaystyle P}$ are linear, unitary operators, ${\displaystyle T}$ is antilinear and antiunitary, ${\displaystyle ⟨\mathrm{\Psi }|T\mathrm{\Phi }⟩=⟨\mathrm{\Phi }|T^{-1}\mathrm{\Psi }⟩}$. If ${\displaystyle |p,\sigma ,n⟩}$ denotes the quantum state of a particle ${\displaystyle n}$ with momentum ${\displaystyle p}$ and spin ${\displaystyle J}$ whose component in the z-direction is ${\displaystyle \sigma }$, then one has

${\displaystyle CPT|p,\sigma ,n⟩=(-1{)}^{J-\sigma }|p,-\sigma ,n^c⟩,}$

where ${\displaystyle n^c}$ denotes the charge conjugate state, that is, the antiparticle. In particular a massive particle and its antiparticle transform under the same irreducible representation of the Poincaré group which means the antiparticle has the same mass and the same spin.

If ${\displaystyle C}$, ${\displaystyle P}$ and ${\displaystyle T}$ can be defined separately on the particles and antiparticles, then

${\displaystyle T|p,\sigma ,n⟩\propto |-p,-\sigma ,n⟩,}$

${\displaystyle CP|p,\sigma ,n⟩\propto |-p,\sigma ,n^c⟩,}$

${\displaystyle C|p,\sigma ,n⟩\propto |p,\sigma ,n^c⟩,}$

where the proportionality sign indicates that there might be a phase on the right hand side.

As ${\displaystyle CPT}$ anticommutes with the charges, ${\displaystyle CPTQ=-QCPT}$, particle and antiparticle have 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

${\displaystyle \psi (x)=\sum _k{u}_k(x)a_k{e}^{-iE(k)t},}$

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 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

${\displaystyle H=\sum _{k}E(k)a_k^{†}{a}_k,}$

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

${\displaystyle b_{k\mathrm{\prime }}=a_k^{†}\mathrm{a}\mathrm{n}\mathrm{d}{b}_{k\mathrm{\prime }}^{†}=a_k,}$

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

${\displaystyle \psi (x)=\sum _{k_{+}}{u}_k(x)a_k{e}^{-iE(k)t}+\sum _{k_{-}}{u}_k(x)b_k^{†}{e}^{-iE(k)t},}$

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

${\displaystyle H=\sum _{k_{+}}{E}_k{a}_k^{†}{a}_k+\sum _{k_{-}}|E(k)|b_k^{†}{b}_k+E_0,}$

where E₀ is an infinite negative constant. The vacuum state is defined as the state with no particle or antiparticle, i.e., ${\displaystyle a_k|0⟩=0}$ and ${\displaystyle b_k|0⟩=0}$. 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_k and b_k 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.

Feynman–Stückelberg interpretation

By considering the propagation of the negative energy modes of the electron field backward in time, Ernst Stückelberg 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. In Feynman diagrams, anti-particles are shown traveling backwards in time relative to normal matter, and vice versa. This technique is the most widespread method of computing amplitudes in quantum field theory today.

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