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Short Setup

I am curious about the the mechanics of fields, whether electromagnetic, gravitational, etc. So as a specific example in order to simplify (hopefully) how to ask this question, consider pair production or annihilation, specifically interactions between electrons and positrons.

My specific curiosity is in how we handle the "sudden" creation or loss of a source and how the source then creates its associated field. My naive guess is that the loss of a source (e.g., annihilation) might be easier to handle, but let's focus on creation of a source for now.

Questions

The specific example of pair production interested me because I was wondering:

  1. How does the field arise from the sources?
    • Meaning, is it delayed until after the source fully "condenses" from the Higgs field, for instance?
    • Or does the field slowly "turn on" as the source "condenses"?
      • Perhaps I should have asked whether it we know if it takes time for sources to form (I assume it does). If so, can we describe this from first principles or is it empirically based?
    • Am I asking that in a very uncomfortable manner? If so, I would appreciate any suggestions. For instance, is there a better word than "condense"? I was thinking about water condensation as a very loose analogy to the purpose of the Higgs field in the formation of massive objects.
  2. Why can they be produced without immediately recombining?
    • That is, how "close" are the two sources spatially after formation and what energies are needed to overcome the associated potential?
  3. Does the field emanate/propagate as if from a point source?
    • I ask because I am trying to determine whether this is a simple Gauss's law issue (i.e., far away it's just like a point source) or if the field emanates only from the "surface" of the source (e.g., proton).
    • If it starts from "within" the source as the source forms, do we know how to handle this mathematically?
      • My concern arises from the "similar" problem of bubble formation, where the bubble starts with zero radius which would require an infinite energy. Perhaps the analogy is not appropriate?

Purpose

I ask these questions because it struck me this morning, while driving to work, that I did not know how and when a field "turned on." I started to think about pair production and then quickly became confused (not difficult to do) and thought many of the quantum whisperers on this site would be much better at explaining these nuances than any of my failed rummaging through Wikipedia or scholarly articles and old text books.

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Quantum fields cannot be turned on or off. The field itself exists for all time and space. It is possible to excite various modes of a quantum field at various spacetime points. These field excitations are interpreted as particles. When no excitations are present (i.e. no particles are present) the quantum field is in the vacuum state. Particles do not act as a source for the fields; they are the excitations of the fields themselves. In an interacting theory, such as Quantum Electrodynamics, electrons, positrons and photons are allowed to interact. Contrary to the macroscopic picture, however, electrons do not act as a source of the electromagnetic field. Instead, the electron-positron spinor field interacts with the photon vector field.

Recall from the quantum harmonic oscillator the creation and annihilation operators $a^{\dagger}$ and $a$. The creation operator $a^{\dagger}$ has the property that when it acts on the $n$-th energy eigenstate, it yields the ($n+1$)-th energy eigenstate: \begin{equation} a^{\dagger}|{n}\rangle = \sqrt{n+1}|n+1\rangle, \end{equation} while the annihilation operator $a$ has the property that it lowers energy eigenstates: \begin{equation} a|n\rangle = \sqrt{n}|n-1\rangle. \end{equation} The vacuum state $|0\rangle$ is defined so that \begin{equation} a|0\rangle =0. \end{equation} In the canonical picture, quantum fields are defined in terms of creation and annihilation operators: \begin{equation} \phi(\vec{x},t)=\int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}} \left[a_p(t)e^{-ipx}+a^{\dagger}_p(t)e^{ipx}\right], \end{equation} where \begin{equation} a_p^{\dagger}|0\rangle=\frac{1}{\sqrt{2\omega_p}}|p\rangle, \end{equation} with $\omega_p = \sqrt{\vec{p}^2+m^2}$. In other words, the quantum field $\phi(\vec{x})$ is an operator that acts on the vacuum and creates a particle at position $\vec{x}$.

Fundamental particles are modeled as point particles but this does not imply a "simple Gauss's Law" situation. The fact that Quantum Electrodynamics is mediated by the photon does not imply that there is any analogy between the behavior of macroscopic electric and magnetic fields and the quantized photons mediating the force. One striking difference is the fact that the Coulomb potential is not exactly $1/r$; higher order quantum effects contribute to logarithmic corrections to the strength of the electromagnetic force.

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  • $\begingroup$ So the particles are a product of the fields and they do not act as the source? $\endgroup$ – honeste_vivere Nov 24 '15 at 1:27
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Nicest question since a long time. Your argumentation is excellent. You are the only one who give a comment to my last question - about the measured maximum distance of the electron influence.

I suppose here, that electric fields are finite. That follows immediately if one agree that the electric field is quantized. A quanta has to have a finite energy and that is the reason that the electric field could not spread out to infinity.

In pair production a photon with its electric field and magnetic field gets converted into the electric fields of an electron and a positron and into their magnetic dipole moments. Not to forget the intrinsic spins of the particles.

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