If you put a photon detector near a magnet (with the magnetic field static in time), is there some probability that the photon detector will detect a photon?

Does QFT not predict that a photon could be detected? If we have a uniform static classical $\vec{B}$ field in the $z$ direction, e.g. $A_{classical}^{\mu}=(0,-By,0,0)$ and we couple electrons to this classical field by adding the term $H_{interaction}=\bar{\psi}\gamma_{\mu}\psi A^{\mu}_{classical}$ to the QED Lagrangian, then there are then Feynman diagrams which seem to indicate that the classical field can produce real photons. (e.g. the Feynman diagram where the classical source emits a photon which fluctuates into an $e^{+}e^{-}$ bubble with a real photon emitted off of the bubble).


Here is an argument based on spacetime symmetry and the fact that U(1) theory is abelian (therefore photons don't interact with each other) to show that the probability is zero.

Suppose we start from a uniform magnetic field in the $z$ direction. Since there is no preferred direction in the $xy$-plane, the final state photon must be emitted along the $z$ direction. The final state magnetic field must be uniform along the $z$ direction as well, due to the same argument.

Therefore if the probability is non-zero, a uniform magnetic field will continue to decay spontaneously until its strength approaches zero (as we approach $T=\infty$), and produce a collection of photons. Therefore the end state at $T=\infty$ will simply be a collection of photons, ie. $a^\dagger(\omega_1)\cdots a^\dagger(\omega_n)|0\rangle$.

Now QED is time symmetric. Therefore we can start from the final collection of photons and evolve backwards in time. However since in U(1) theories gauge particles do not interact with each other, we simply have a collection of free particles --- the interaction Hamiltonian is zero, and the initial state must be a collection of free particles as well. QED.

EDIT: Before anybody starts babbling about "non-linear" optics and "four-photon scattering", let me point out that the current experimentally accessible field intensity is well below the Schwinger limit [1], at which point we expect new physics to come out. Our current experimentally verified theory contains only linear photon interactions. As theorists tend to get carried away with formalism we should always remind ourselves that physics is ultimately an experimental science, and any theoretical formalism is ultimately an effective theory.

[1] https://en.wikipedia.org/wiki/Schwinger_limit

  • $\begingroup$ There is no radiation field so nothing for a photon detector to detect. A static magnetic field cannot produce a radiative transition. $\endgroup$
    – my2cts
    Aug 14 '19 at 22:11
  • $\begingroup$ @my2cts Exactly (at least classically). I believe the intention of the original poster is to ask whether quantum corrections can make this happen. Classically a static field cannot radiate (in both Maxwell E&M and Einstein gravity). OP was trying to argue that if field strength were strong enough quantum effects can pull a virtual pair out of the vacuum and therefore make static field radiate. This has never been seen in experiments, and neither does any of the "non-linear" corrections to the vacuum Maxwell equation due to quantum effects, as the limit is well above what we can achieve .... $\endgroup$ Aug 16 '19 at 3:29
  • $\begingroup$ @my2cts experimentally right now. When we can finally see this in experiments we will revise our theory --- at the Schwinger limit there will probably be new physics unforeseen by current theory anyway. So for now my answer would be no: static field cannot radiate. This is how physics is supposed to work, namely that we as theorists should produce theories that actually describe our physical universe as experimentalists have measured. $\endgroup$ Aug 16 '19 at 3:30
  • $\begingroup$ My parenthetical edit was made because OP and I discussed this off-line, and the impression I got from that conversation seem to suggest that OP rejects the notion that all our physics theories are effective theories constructed out of experimental results, and regards mathematical formalism as absolute truth that exists above and dictates our physical experiences. My comment is that this represents a fundamental misunderstanding of the scientific method and is derived from an ontology reminiscent of the kind of medieval scholasticism that has been rejected by modern science. $\endgroup$ Aug 16 '19 at 3:47

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