# How does a magnetic monopole break time reversal symmetry?

I read in the article on the magnetic monopole in the German Wikipedia that the path of an electrically charged particle in the field of a magnetic monopole breaks time reversal symmetry. This means, that under the time reversal transformation ($t \mapsto -t$) the interaction between electric and magnetic charge is not invariant.

I don't know whether it is possible to give a perhaps simple argument to illustrate this? Or to give a reference to an explanation?

The English Wikipedia article on magnetic monopoles has the following equation for the 'extended' Lorentz-Force of a magnetic field on a electrically and magnetically charged particle:

$$\vec{F}=q_{\mathrm e}\left(\vec{E}+\vec{v}\times\vec{B}\right) + q_{\mathrm m}\left(\vec{B}-\vec{v}\times\frac{\vec{E}}{c^2}\right)$$

Under time reversal ($t$ is replaced by $-t$), some of these quantities change sign ('odd', annotated with a $-$ sign below), some don't ('even', annotated with a $+$ sign below):

$$\underbrace{\vec{F}}_+ = \underbrace{q_{\mathrm e}}_+ \left( \underbrace{\vec{E}}_+ + \underbrace{\vec{v}}_- \times \underbrace{\vec{B}}_- \right) + \underbrace{q_{\mathrm m}}_? \left( \underbrace{\vec{B}}_- - \underbrace{\vec{v}}_- \times \underbrace{\frac{\vec{E}}{c^2}}_+ \right)$$

You'll notice that the first term in parentheses is 'even' under time reversal and so is the entire first part up to the plus sign between the parentheses. This is the usual Lorentz force in the absence of magnetic monopoles.

The second term in parentheses is however odd under time reversal (changes sign). In order not to break the behaviour of the force $\vec{F}$ under time reversal, $q_{\mathrm m}$ would have to have 'odd' parity under time reversal, i.e. would have to change sign. Such a quantity is also called a pseudoscalar under time reversal (as opposed to the electric charge which is a scalar under time reversal).

While mathematically speaking it is possible to keep invariance under time reversal, a (magnetic) charge which flips sign under time reversal would be very a counter-intuitive object...

On the other hand, if magnetic monopoles exist and their charge does not flip sign under time reversal, time reversal symmetry would be broken.

Remark: In the text above, $\vec{B}$ was assumed to flip sign under time reversal. This can e.g. be derived from Faraday's law of induction (and exploiting the fact that $\vec{E}$ is even under time reversal).

Intuitively, this can also be understood with currents generating magnetic fields where the currents (of electric charges) would flip their direction under time reversal (like a velocity) and the moving electric charges don't flip sign.

Decomposing a general magnetic field into a 'current' and 'monopole' component:

$$\vec{B}_\mathrm{tot} = \vec{B}_\mathrm{currents} + \vec{B}_\mathrm{monopoles}$$

one would also conclude that in order to keep invariance under time reversal, both parts on the right side must transform the same way, i.e. the monopole field must flip its sign. A monopole field however looks as follows:

$$\vec{B}(\vec{x})_\mathrm{monopole} = K \cdot q_{\mathrm m} \cdot \dfrac{\vec{x}}{|\vec{x}|} \cdot \dfrac{1}{|\vec{x}|^2}$$

which only can flip its sign under time reversal if $q_{\mathrm m}$ does.