Editing Antiparticle (section)

== 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 (physics)|field]] without mixing the annihilation and creation operators by writing

::<math>\psi (x)=\sum_{k}u_k (x)a_k e^{-iE(k)t},\,</math>

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<sub>k</sub>'' denotes the corresponding annihilation operators. Of course, since we are dealing with [[fermion]]s, we have to have the operators satisfy canonical anti-commutation relations. However, if one now writes down the [[Hamiltonian (quantum mechanics)|Hamiltonian]]

::<math>H=\sum_{k} E(k) a^\dagger_k a_k,\,</math>

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

::<math>b_{k\prime} = a^\dagger_k\ \mathrm{and}\ b^\dagger_{k\prime}=a_k,\,</math>

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

::<math>\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},\,</math>

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

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

where ''E<sub>0</sub>'' is an infinite negative constant. The [[vacuum state]] is defined as the state with no particle or antiparticle, ''i.e.'', <math>a_k |0\rangle=0</math> and <math>b_k |0\rangle=0</math>. Then the energy of the vacuum is exactly ''E<sub>0</sub>''. 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 theory|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 <!--'[[Feynman–Stueckelberg interpretation]]', '[[Feynman-Stueckelberg interpretation]]', '[[Stueckelberg–Feynman interpretation]]', '[[Stueckelberg-Feynman interpretation]]', '[[Feynman–Stückelberg interpretation]]', '[[Feynman-Stückelberg interpretation]]', '[[Stückelberg–Feynman interpretation]]', and '[[Stückelberg-Feynman interpretation]]' redirect here-->===

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 (quantum mechanics)|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 diagram]]s. 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.<ref>{{cite book|author=Griffiths, D.J. |author-link=David J. Griffiths |page=61 |year=2008 |title=Introduction to Elementary Particles |edition=2nd |publisher=[[John Wiley & Sons]] |isbn=978-3-527-40601-2|quote=}}</ref> This technique is the most widespread method of computing [[probability amplitude|amplitudes]] in [[quantum field theory]] today.

Since this picture was first developed by Stückelberg,<ref>Stückelberg, Ernst (1941), "La signification du temps propre en mécanique ondulatoire." ''Helv. Phys. Acta'' '''14''', pp. 322–323.</ref> and acquired its modern form in Feynman's work,<ref>{{cite journal|first=Richard P.|last=Feynman|title=Space-time approach to non-relativistic quantum mechanics|journal= [[Reviews of Modern Physics]]|volume= 20|pages= 367–387|year=1948|doi=10.1103/RevModPhys.20.367|bibcode = 1948RvMP...20..367F|issue=2 |url=https://authors.library.caltech.edu/47756/1/FEYrmp48.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://authors.library.caltech.edu/47756/1/FEYrmp48.pdf |archive-date=2022-10-09 |url-status=live}}</ref> it is called the '''Feynman–Stückelberg interpretation'''<!--boldface per WP:R#PLA--> of antiparticles to honor both scientists.