In relativistic quantum physics, a theoretical concept of the photon is formulated as the quantum of the (electro)magnetic field. How is working that concept in the particular case of static magnetic field?
Unless the magnetic field is time-varying, it will not generate electrical field. Therefore, electromagnetic waves and hence the photon will never come into existence due to static magnetic field alone.
If I have correctly understood you, within relativistic quantum physics (quantum electrodynamics) there is no concept "constant magnetic field"?
There is notion of electromagnetic field, if one has to study constant magnetic field in space-time alone, there is already classical electrodynamics.
What is a quantitative sign on which we can say that the ensemble of photons (which are quantum objects) has turned into a classical static magnetic field?
A static magnetic field is a vacuum configuration. Some care is required in specifying boundary conditions, but it can't be distinguished from a configurations of no photons-which is what a vacuum means.
No: time varying electric fields inevitably produce (time varying) magnetic fields, (that's from the displacement term of the Maxwell-Ampère equation and Faraday's law, with Gauss' law and the statement that the magnetic field doesn't have point-like sources being the constraints) and these solutions are superpositions of electromagnetic waves, that are coherent superpositions of real photons, whose energy is related to their momentum by E = |p|c.
A static magnetic field is another solution of Maxwell's equations, that's a vacuum solution, since E=n hbar ω (with n=number of photons and ω the frequency) so a static solution has ω->0, thus |p|->0, therefore E->0.
The Maxwell equations are the classical equations and their solutions are classical fields. They don't have anything to do with virtual photons, that describe electric and magnteic fields that are not solutions of Maxwell's equations.
A static field has fixed external charges and steady currents as sources-and it should be recalled that photons don't carry electric (or magnetic) charge themselves. So a static field doesn't affect the charges or currents that are its source-else it wouldn't remain static.
An experimental method is proposed for the measurement of the photon's fundamental magnetic field operator B̂Π. It is shown that the quantum field and semi-classical descriptions of the optical Zeeman effect produce a different pattern of splitting in a 1S→1P transition in an atom. The quantum field operator B̂Π of the photon produces a pattern of three lines, each displaced from the original 1S→1P line, while its classical equivalent BΠ produces a pattern akin to conventional Zeeman splitting with a static magnetic field;
doi:10.1016/0921-4526(92)90582-D
Transition lines *mean* non-zero frequency, which, in turn, *means* that the fields aren't static: they have a harmonic time dependence, defined by the frequency of the transition line. Transition lines are thus completely irrelevant when discussing static fields.
In QED, because of freedom given by gauge invariance (see eq. (5.1.6) in [1] - Akhiezer A.I., Berestetsky V.B." Quantum Electrodynamics"), only real photons with cross polarization (/lambda=1,2) make a contribution to energy of the field whereas a "scalar" photon (/lambda=3) and a photon with "longitudinal" polarization (/lambda=4) don't give a contribution to energy.
The condition of vacuum of the field is a state where the number of cross polarized photons (/lambda=1,2) is equal to zero, but always there are the "scalar" and " longitudinal" (/lambda=3,4) photons.
However, wave function of the "scalar" and" longitudinal" (/lambda=3,4) photons can't be normalized (see below eq. (5.2.1) in [1]).
Moreover, in QED we have a freedom of choice of vacuum (for example, eq. (5.2.4) and above of eq. (6.1.1) in [1]) that results in total absence of "scalar" and" longitudinal" (/lambda=3,4) photons in a condition of vacuum.
As in such ambiguous conditions, proceeding from a vacuum state unambiguously and authentically to calculate the physical mechanism of formation of a constant magnetic field which is equal, for example, 3 Tesla, 8 Tesla, 81 Tesla, 325 Tesla ….?
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