For a good understanding of the physical concept of antiparticle and the closely related concept of charge, it is important to appreciate how these notions emerged. Therefore, we begin with a brief sketch of the associated history.
In the early days of quantum mechanics, the material world was thought to be built from three elementary particles, namely, the electron, the proton, and the neutron. The idea that there might be associated particles with the same mass and opposite charge (now called antiparticles) first arose from the characteristics of the one-particle Dirac equation. Writing it in Hamiltonian form, the resulting Dirac Hamiltonian has spectrum where m is the particle mass. The negative part of the spectrum was already considered unphysical by Paul Dirac himself: no such negative energies had been observed, and their presence would give rise, for example, to instability of the electron.
This physical inadequacy of the “first-quantized” description quickly led to the introduction of “second quantization,” as expressed in quantum field theory. In the quantum field theoretic version of the Dirac theory, the problem of unphysical negative energies is cured by a prescription that goes back to Dirac’s hole theory.
Specifically, Dirac postulated that all of the negative energy states of his equation are filled by a sea of unobservable particles. In his picture, annihilating such a negative energy particle with a given charge yields a hole in the sea, which should manifest itself as a new type of positive energy particle with the same mass, but opposite charge. This intuitive idea led Dirac to the prediction that a charged particle should have an oppositely charged partner (its antiparticle).
Ever since, this prediction has been confirmed by experiment, not only for all electrically charged particles, such as the electron and proton, but also for most of the electrically neutral particles, such as the neutron. In the latter case, one still speaks of the particle having a charge, whereas the remaining electrically neutral particles are identical to their antiparticles.
As already mentioned, in the second-quantized Dirac theory no negative energies occur. In the Dirac quantum field, the creation/annihilation operators of negative energy states are replaced by annihilation/creation operators of positive energy holes. This hole theory substitution therefore leads to a physical arena with an arbitrary number of particles and antiparticles with the same positive mass, now called Fock space. (A mathematically precise account can be found in the monograph by Thaller, 1992.)
As it soon turned out, the number of particles and antiparticles in a high-energy collision is not conserved, a phenomenon that can be naturally accommodated in the Fock spaces associated with interacting relativistic quantum field theories. The quantum field theory model that is the most comprehensive description of real-world elementary particle phenomena arose in the early 1970s. During the last few decades, this so-called Standard Model has been abundantly confirmed by experiment.
In spite of these successes, the problem of obtaining nonperturbative insights into the Standard Model remains daunting. This is an important reason why its classical version and various related, but far simpler, classical nonlinear field theories have been, and still are, widely studied. It is a striking and relatively recent finding that within this classical framework, there exist localized, smooth, finite-energy solutions with characteristics that are very reminiscent of particles. The most conspicuous examples in this respect are the soliton field theories, where there exist such particle-like solutions for any given particle number and where the particle numbers and their velocities are preserved in a scattering process.
To be sure, the latter soliton equations arose independently of particle physics. They involve a lower space-time dimension (mostly two), and they have applications in a great many areas that are far removed from high-energy physics. (An early survey that is still one of the best and most comprehensive can be found in Scott et al., 1973.)
Returning briefly to the latter area, there are also various equations, typically within a gauge-theoretic context, where particle-like solutions (instantons, monopoles, vortices, and so on) have been found. A closely related field is classical gravity, where various previously known solutions (such as the Schwarzschild and Kerr black holes) came to be viewed as particle-like solutions, an interpretation strengthened by the occurrence of “many-particle” generalizations.
It should be pointed out that within this nonlinear classical context, there are no explicit many-particle solutions where “creation” and “annihilation” occur. Indeed, it is not even clear how this would manifest itself on the classical level (as compared with the quantum level, where this is a clear-cut matter).
Focusing once again on the notions of charge and particles vs. antiparticles, it is an even more striking fact that these concepts are naturally present within some of the above-mentioned nonlinear field theory models with particle-like solutions. A prime example for gauge theories is given by instanton solutions, which are accompanied by anti-instanton solutions. Roughly speaking, these are distinguished by opposite generalized winding numbers, viewed as charges of a topological nature. More precisely, these localized solutions minimize the energy in certain homotopy classes. This means they are stable under small perturbations.
Turning to soliton field theories, it should be mentioned at the outset that for most soliton equations (for instance, for their most well-known representative, the KdV equation), there exists only one type of soliton, hence no notion of charge. The sine-Gordon field theory is a paradigm for theories where more than one type of soliton occurs. We proceed by using it to exemplify the notions of particle, antiparticle, and charge at the classical level.
The sine-Gordon equation
(1)
can be obtained as the Euler-Lagrange equation associated with the Lagrangian
(2)
The corresponding energy functional reads
(3)
Obviously, the constant solutions
(4)
yield E=0. These are the so-called vacuum solutions.
There exist, however, two distinct classes of nonconstant, time-independent, finite-energy solutions, namely,
(5)
They connect the two vacuum solutions and for x→±∞. The functions have winding numbers ±1 and may be viewed as generators of the homotopy group π1(S1)=Z. They are interpreted as one-particle and one-antiparticle solutions (soliton and antisoliton) of the sine-Gordon equation, having charges ±1. (By Lorentz invariance, they can be boosted to constant velocity υ, with |υ| smaller than the speed of light, which is 1 for the units chosen in (1).)
The analogy with electrical charge is strengthened by the existence of soliton-antisoliton bound states, the so-called breathers. More generally, there exist solutions with N+ solitons, N− antisolitons, and N0 breathers, where N+, N−, N0 are arbitrary integers. These particle numbers and the velocities are conserved in the collision, the nonlinear interaction showing up only in factorized position shifts.
From the viewpoint of elementary particle physics, these findings are regarded as stepping stones for a better understanding of the associated quantum field theory. In particular, the existence of particle-like solutions stabilized by charges of a topological nature is believed to signal the existence of a corresponding stable quantum particle. A lucid survey in which this scenario is expounded is Coleman (1977).
Returning to the sine-Gordon example (also discussed in Coleman, 1977), the above scenario has not only been confirmed, but considerably enlarged: at the quantum level, the sine-Gordon theory yields a model of interacting solitons and antisolitons, whose scattering preserves particle numbers and other characteristics (such as the set of velocities), yielding an explicitly known factorized S-matrix. Thus, the classical picture is essentially preserved under quantization (cf. the review by Zamolodchikov & Zamolodchikov (1979)). It should be stressed that this absence of particle creation and annihilation is highly nongeneric for relativistic quantum field theories; it occurs for only completely integrable soliton type field theories in two space-time dimensions.
The phenomenon of oppositely charged particles and antiparticles with an attractive interaction between them has also come up in a setting quite different from the above field-theoretic one. Consider the classical Hamiltonian
(6)
on the phase space
(7)
By contrast to the above partial differential equations, most of which are wave equations at face value, the Hamiltonian equations resulting from (6) have, from the outset, a point particle interpretation. To be specific, they describe N nonrelativistic particles on the line with a repulsive pair interaction.
The special character of this Hamiltonian is already manifest from the scattering to which it leads: the particle velocities are conserved and the position shifts are factorized, just as for solitons. Indeed, Hamiltonian (6) is one version of the completely integrable nonrelativistic Calogero-Moser models. (These are surveyed at the classical and quantum levels in Olshanetsky & Perelomov (1981) and Olshanetsky & Perelomov (1983), respectively.) As pointed out first by Calogero, the substitution xk→xk+iπ, k=1,…, N−, has the effect of turning the repulsive 1/sinh2 interaction between particles 1,…, N− and particles N−+1,…, N into an attractive −1/cosh2 interaction. Thus, one obtains an integrable system that can be viewed as describing N− antiparticles and N+=N−N− particles in interaction.
The nonrelativistic Calogero-Moser systems have been generalized to a relativistic setting, both on the classical and on the quantum levels. Again, there are versions describing particles and antiparticles, with repulsion between like charges and attraction between opposite charges. These versions have been studied at the classical level in Ruijsenaars (1994), where the intimate connection to the sine-Gordon solutions with arbitrary numbers of solitons, antisolitons, and breathers is also detailed. More generally, Ruijsenaars (2001) is a recent survey of this relation between the solitons and antisolitons of the sine-Gordon field theory and the point particles and antiparticles of certain relativistic Calogero-Moser systems, covering both classical and quantum aspects.
Coleman, S. 1977. Classical lumps and their quantum descendants. In New Phenomena in Subnuclear Physics, Proceedings Erice 1975, edited by A.Zichichi, New York: Plenum, pp. 297–421
Olshanetsky, M.A. & Perelomov, A.M. 1981. Classical integrable systems related to Lie algebras. Physics Reports, 71:313–400
Olshanetsky, M.A. & Perelomov, A.M. 1983. Quantum integrable systems related to Lie algebras. Physics Reports, 94:313–404
Ruijsenaars, S.N.M. 1994. Action-angle maps and scattering theory for some finite-dimensional integrable systems. II. Solitons, antisolitons, and their bound states. Publications RIMS Kyoto University, 30:865–1008
Ruijsenaars, S.N.M. 2001. Sine-Gordon solitons vs. relativistic Calogero-Moser particles. In Integrable Structures of Exactly Solvable Two-dimensional Models of Quantum Field Theory, edited by S.Pakuliak & G.von Gehlen Delft: Kluwer, pp. 273–292
Scott, A.C., Chu, F.Y.F. & McLaughlin, D.W. 1973. The soliton: a new concept in applied science. Proceedings of the IEEE, 61:1443–1483
Thaller, B. 1992. The Dirac Equation, Berlin: Springer
Zamolodchikov, A.B. & Zamolodchikov, Al.B. 1979. Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Annals of Physics (NY), 120:253–291
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where m is the particle mass. The negative part of the spectrum was already considered unphysical by Paul Dirac himself: no such negative energies had been observed, and their presence would give rise, for example, to instability of the electron.
and 
for x→±∞. The functions 
have winding numbers ±1 and may be viewed as generators of the homotopy group π1(S1)=Z. They are interpreted as one-particle and one-antiparticle solutions (soliton and antisoliton) of the sine-Gordon equation, having charges ±1. (By Lorentz invariance, they can be boosted to constant velocity υ, with |υ| smaller than the speed of light, which is 1 for the units chosen in (1).)
has the effect of turning the repulsive 1/sinh2 interaction between particles 1,…, N− and particles N−+1,…, N into an attractive −1/cosh2 interaction. Thus, one obtains an integrable system that can be viewed as describing N− antiparticles and N+=N−N− particles in interaction.
