# Does this physical situation distinguish whether you are viewing it a mirror?

The weak interaction's lack of $P$-symmetry is often explained by saying that "the amplitudes for processes involving the weak interaction are different from the amplitudes for the same processes reflected in a mirror." So if you observe a process involving the weak interaction, you could tell whether you were observing it directly or observing its mirror reflection.

The textbook example if this asymmetry is the fact that in the rest frame of a particle that decays and produces a neutrino, the resulting neutrino is always left-helical (its spin is antiparallel to its velocity in this frame). But the mirror-reflected version of a left-helical neutrino is a right-helical neutrino, because momentum is odd under P but spin in even. Thus if you see left-helical neutrino produced in the rest frame of a decay process, you are viewing the process directly, while if you see a right-helical neutrino produced, you are viewing the process in a mirror.

This example always bothered me, because the spin of a neutrino is a pseudovector, and pseudovectors are not directly measurable. Quantum mechanical spin isn't a physical rotation of a rigid body, so you can't tell the spin of a particle just by "looking at it" - you need to perform an experiment which involves a cross product, which converts the spin back into a "true" (polar) vector or scalar. (For example, the magnetic field is not directly measurable - its only measurable effect is its deflection of a charged particle ${\bf F} = q\, {\bf v} \times {\bf B}$, which is a true vector. As rob's very nice answer to Is this an example of Parity violation? notes, "whenever you compute observables ... you always wind up using the right-hand rule an even number of times.")

Factoring in this wrinkle, you need to more careful than just cavalierly asserting that the spin flips direction when viewed in a mirror, because you need to specify the measuring device which is converting the spin into a true vector/scalar, and you also need to consider what happens when you reflect the measuring device in the mirror. But after some searching, I was unable to find any example of a physical process that literally looks different when viewed in a mirror. So I tried to think of a minimal experiment where you really could tell if you were viewing it directly or in a mirror only from measurable (i.e. true vector or scalar) quantities, without having to magically intuit the direction of particle spins. I think I've come up with one - you can send a neutrino through a loop of electric current:

This is basically a modification of the usual Stern-Gerlach spin measurement for spins parallel to the direction of velocity. The red particle is a neutrino (not an antineutrino) initially moving with velocity $v_0$, which was produced by the decay of a particle at rest in this frame. The black ring is an electric current flow, and the blue curves are the induced magnetic field lines. (I deliberately haven't labeled their orientation, since it would be reversed under mirror-reflection.) Since we are in the rest frame of the process that produced the neutrino, its left-chirality implies left-helicity in this frame, and so its magnetic dipole moment points to the left. (You may not be used to thinking of neutrinos as having magnetic dipole moments, but their very small mass gives them a very weak moment: http://nucla.physics.ucla.edu/sites/default/files/NeutrinoMagneticMoment_2012Nov8.pdf.)

Consider Setup A, where the electric current flows as indicated in the diagram, so the magnetic field lines point to the right. ${\bf \mu \cdot B}$ is strongly negative inside the ring and more weakly negative away from it, so the spatially varying magnetic field induces an outward-pointing force on the neutrino via its magnetic dipole moment. The neutrino will therefore slow down as it enters the ring and then speed back up to $v_0$ as it exits it.

Now consider Setup B, where the current instead flows the other way around the ring, so the magnetic field points to the left. The neutrino's left-helicity now implies that ${\bf \mu \cdot B}$ is strongly positive inside the ring and more weakly positive away from it, so the magnetic fields would induce an inward force on the neutrino. The neutrino would therefore speed up as it enters the ring and then slow back down to $v_0$ as it exits it.

Now consider looking at Setup B in a horizontal mirror that reflects everything vertically. In the mirror, the electric current would appear to travel in the direction indicated in the diagram. In fact, initially looking at Setup B in the mirror would be indistinguishable from looking at Setup A directly - both would look exactly like the diagram. But of course, the mirror-reflected version of Setup B would still show the neutrino first speeding up then slowing down. So we finally have an unambiguous distinction: if we see an experiment that initially corresponds to the diagrammed setup and the neutrino slows down inside the ring, then we are looking (at Setup A) directly. If instead the neutrino speeds up inside the ring, then we are looking (at Setup B) in a mirror.

(This experiment would be impossible to perform in practice, because the neutrino masses are so tiny that their magnetic dipole moments are unmeasurable and we can't measure the amount that their speed differs from that of light, so we can't measure if they ever speed up or slow down. An example of a more realistic setup would be one where a $\pi^+$ pion at rest decays into a neutrino and an antimuon. You could more feasibly perform this experiment on the antimuon, which is much more massive and sedate, then use the fact that the antimuon's and neutrino's spins must be antialigned (since the pion has spin 0) to get essentially the same results as described above.)

I think that a similar setup could work to distinguish a system from its CP-inverse, although you would need to observe many different neutrinos passing through the ring, and there would only be a slight asymmetry in the fraction of neutrinos speeding up vs. slowing down.

Can anyone see a flaw in my reasoning? Or think of a simple physical setup that can distinguish simply by looking at it whether you're viewing it in a mirror?