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Wikipedia: "electron electric dipole moment would imply a violation of both parity invariance and time reversal invariance" ---

Yes, it is true that the electron electric dipole moment violate parity invariance.

BUT How does the electron electric dipole moment violate time reversal invariance?

TEH
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zeta
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2 Answers2

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If the electron has an electric dipole moment $p$, then it must satisfy $p = kS$ where $S$ is the spin and $k$ is a scalar constant. But spin is time-reversal-odd, so this would make the electron electric dipole moment time-reversal-odd too, which is different from every other electric dipole moment, which is time-reversal-even. This would imply that T fails to be a symmetry of the laws of electromagnetism.

Brian Bi
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  • Let us write electric dipole moment by definition as $\vec{P}_e= q \vec{d}$. How do you see T violation at all? T makes $q$ and $\vec{d}$ invariant? – zeta Jan 21 '24 at 23:31
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    @zeta Electrons are point particles; they don't have physical charge separation like you imagine, so $P_e = qd$ doesn't apply to them. What is necessarily true is what I said: $P_e = kS$. The answers to this question explain: https://physics.stackexchange.com/q/367731/25572 – Brian Bi Jan 22 '24 at 00:49
  • @BrianBi It is more accurate to say that the upper limit of the electron radius is 10$^{-22} m. https://en.m.wikipedia.org/wiki/Electron#:~:text=Observation%20of%20a%20single%20electron,to%20be%2010%E2%88%9222%20meters, if we believe the reference. – my2cts Jan 22 '24 at 09:17
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Time-reversal is tricky, and a lot of logic for conserved quantities do not apply to it (for example just because the Hamiltonian respects time-reversal does not mean that there is a conserved "time-reversal" quantum number). There are hand wavy arguments, but the most general argument is through Kramers' Theorem (https://en.wikipedia.org/wiki/Kramers%27_theorem). This states that for a half-integer spin system with a Hamiltonian that is symmetric under time-reversal, every energy eigenstate has another eigenstate with the same energy related by time-reversal [1].

The addition of an external electric field to a system with electric dipole moment $\vec{d}$ will remove the degeneracy between the $+m$ and $-m$ states (which are related by time-reversal). Therefore the Hamiltonian describing a half-integer spin system which has an electric-dipole moment interaction cannot commute with the time-reversal operator.

Addition of an external $\vec{B}$-field will lift the degeneracy between the spin projections, and therefore the Hamiltonian for this half-integer spin system does not respect time-reversal invariance. This is to be expected though, as if we considered the full system under time-reversal the $\vec{B}$-field would flip sign as well. This is not true for an $\vec{E}$-field which would not change sign under time-reversal. Therefore an electric dipole moment in a half-integer spin system requires that the system itself breaks time-reversal symmetry.

This argument applies no matter how complicated the $\vec{E}$-field configuration is (so long as it's static). It also applies to gravitational dipole moments [2].

References:

  1. Sakurai, J.J., Napolitano, J. "Modern Quantum Mechanics", 2nd edition (2017)
  2. Weinberg, S. "The Quantum Theory of Fields, Volume 1 Foundations" (1995)
TEH
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