# Particles as a limit of classical field theory

A common academic exercise has been to show that classical mechanics is a limit of quantum mechanics, usually by putting $\hbar \rightarrow 0$. Similarly is it possible to show that a limit to field theory will lead to a particle ?
May be by considering a initial field configuration such that the "lumpiness" of the particle is there, and then as it evolves it maintains that. Clearly such an initial field configuration will continue to evolve as it is in a free field theory as two different modes don't talk to each other, but what about an interacting model ? Are there any instances in the literature which attempts to establish this connection ?

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The quantum version of this classical question: physics.stackexchange.com/q/26960/2451 –  Qmechanic Oct 1 '12 at 15:32
–  Arnold Neumaier Nov 23 '12 at 11:40

The classical limit of a boson quantum field is a classical field. Fermion quantum fields do not have a classical limit. See Linde's book on particle physics and inflationary cosmology (chapter 2) http://arxiv.org/abs/hep-th/0503203 for a reference.

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 Coherent states are special states concentrated in a small region of phase space, hence having a reasonably well-defined position and momentum, resembling a classical particle. They usually retain their lumpy nature for a time of the order of $O(\hbar^{-1})$, which becomes infinitely large in the classical limit. However, in a field theory such as QED, the coherent states related to photons turn in the classical limit into modes of the classical electromagnetic field.

[Original statement, not addressing the lumpiness:]

In quantum mechanics, if you have a relativistic quantum field theory and consider the sector corresponding to the eigenvalue 1 of a number operator, you end up with a 1-particle theory. No limit is necessary to do so.

Once one has more than one particle, one has to account of all infinitely many of them, due to possible pair creation. However, if there is a mass gap (i.e., if there is a smallest positive mass), a low energy approximation will reduce the number of particles to at most a finite number.

Starting with classical mechanics, you can take general relativity coupled to matter fields. There, astrophysicists often consider the approximation where all planets, all stars, or even all galaxies are points. In one such approximation one gets the traditional Newtonian description of matter in the solar system by a coupled system of ODEs, where the sun, all planets and large asteroids, some moons and some artificial satellites are treated as point particles.

However, it is not a clearly defined limit, rather a clearly defined approximation. On the other hand, $\hbar$ is a constant of Nature, hence the so-called classical limit is also just an approximation in which one neglects all powers of $\hbar$.

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In the first place, classical mechanics is not recovered in the $\hbar \rightarrow 0$ limit of quantum mechanics Check Classical limit of quantum mechanics for details.

What particles do you consider? For instance, in classical electrodynamics the electron is not associated to any field but is introduced from the relativistic mechanics of classical particles, whereas the electromagnetic field is not described in particle terms (photons only arise in quantum field theory). Similar remarks apply to a field theory of gravitation.

What we can do is just the inverse of what you ask. We can derive classical field theory as a limit from a theory of particles. This was shown by Feynman and Wheeler for electrodynamics. They start with a classical theory with only particles and next obtain the fields as functionals of particle paths. This theory has several advantages over the old field theory, because there is no self-interaction divergences and other nuisances. The original paper is Classical Electrodynamics in Terms of Direct Interparticle Action

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