An electromagnetic wave (like a propagating photon) is known to carry it's electric and magnetic field-vectors perpendicular and each depending on the differential change of the other thus "creating" each other and therefore appearing in-phase and reaching their minima/maxima together. I'm interested to know whether there was any uncertainty as a principle discussed in the underlying principles of the Maxwell equations, like $\Delta E \Delta B \geq \hbar$. I appreciate links, hints and answers.
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One can see the consistency with the Heisenberg Uncertainty Principle by the definition of wavelength and frequency of the electromagnetic wave:
$\lambda\nu/c=1$ where c is the velocity of light
Multiplying both sides by $h$ and considering $\lambda$ as $\delta(x)$ and $p=h\nu/c$ for a photon,
$$\lambda h\nu/c \sim h$$
$$\delta(x)\delta(p)\sim h$$
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According to this Letter to the Editor from 1933, written by G. Lemaitre at MIT (or at that time University of Louvain) https://journals.aps.org/pr/abstract/10.1103/PhysRev.43.148 https://doi-org.ezproxy.library.ubc.ca/10.1103/PhysRev.43.148
$\Delta E_x\Delta H_z = hc/l^4$
"refers to the mean measures of two perpendicular components $E_x$, $H_z$ of the electric and magnetic fields in a cube of side $l$. It must be understood as referring to the time-mean values of this field during the time $l/c$."
It has only 2 citing articles, and likely became out-dated shortly after its publish date. For instance, one article which cites it in 2002,
https://doi-org.ezproxy.library.ubc.ca/10.1016/S1355-2198(02)00033-3
Claims: "But it is, in fact, not obvious that Lemaıtre in 1934 was unaware of the vacuum energy arising in quantum field theory. For instance he discusses Heisenberg uncertainty relations for the electromagnetic field in a short article from 1933 (Lemaıtre, 1933) in connection with the then newly formulated quantum principles for the electromagnetic field."
Since electric and magnetic fields are not conserved quantities like energy and momentum, they are not typically used in Heisenberg's uncertainty principle. More often position and momentum or energy and time are used as Anna V. suggests. However, variance relations exist for the ground-states of displacement fields $D$ and $B$ in a dielectric (1991): https://doi.org/10.1103/PhysRevA.43.467
$\Delta D\Delta B = \frac{1}{2}\hbar\omega$
The right-hand-side of the above equation is the "zero-point-energy" or "vacuum energy" for photons mentioned above. It occurs at a temperature of 0K because of Heisenberg's uncertainty relation.
