A free particle is described by its kinematic properties: energy, momentum, & angular momentum. Unlike the previous properties, energy & momentum take continuous values, but they are related by the mass: Nonrelativistically,

conservation | symmetry |
---|---|

energy | time translation |

momentum | space translation |

angular momentum | rotation |

*The relativistic quadratic equation has a 2nd solution, E ≈ - m - p⃗²/2m (antiparticle), but then K = E - m ≈ - 2m → - ∞ in the nonrelativistic limit.

The analogs to the particle kinematic properties are

particle | field |
---|---|

m | κ |

E (or K) | i∂/∂t |

p⃗ | -i∇⃗ |

angular momentum | index |

Energy & momentum can be expressed in terms of numbers, rather than (differential) operators, by Fourier transformation:

φ(x⃗,t) ~ ∫dk⃗ dω exp(ik⃗∙x⃗-iωt) φ̃(k⃗,ω)

Except for scalar fields, the wave equations tend to be more complicated. In particular, the concept of a "gauge field" must be introduced. The result is that a free field must describe an irreducible representation of the Poincaré group (including space-time translations, rotations, & Lorentz boosts).

Fields also carry energy, momentum, & angular momentum, in the usual sense of how they appear in conservation laws & symmetry principles, but their form does not seem to resemble that of particles. In particular, they are defined as spatial integrals of local functions of the fields (with a few derivatives). There we find similar problems: Avoiding the introduction of ε₀ (& μ₀), the Coulomb force between electrons looks like

On the other hand, the Yang-Mills action for a self-interacting field looks like

Note that, if we use consistent dimensions for energy & momentum, & thus the action, both for electron particle + Coulomb & for Yang-Mills, then the dimensionless quantities for the couplings are e²/ℏ for the former & g²ℏ for the latter. Thus the "classical" limit ℏ → 0 is strong coupling for the former & weak coupling for the latter. This relates to the fact that classical particles & classical fields are different limits of quantum theory. (Introducing interactions with particles @ the same time as self-interactions for fields makes the limit ℏ → 0 poorly defined.)

But the waves really appear in the Fourier expansion of a quantum mechanical wave function, not a field. This is much closer, but the full identification between particles & fields requires quantum field theory.

In practice, one usually works with the Fourier transform version of the Taylor expansion in fields (i.e., just start in momentum space),

Not only is this construction conceptually simpler (working completely with functions instead of operators), but in the relativistic case it's manifestly Lorentz covariant @ each step. Canonical quantization, on the other hand, must take a detour into a Hamiltonian analysis & restore Lorentz invariance @ the end, a procedure that can become so complicated that *no one ever uses it for the Standard Model*. Yet many professors still torture their students by pretending canonical quantization of field theory is more "pedagogical", just like the way they teach freshmen physics as if it were still the 19th century.