Can apparent superluminal neutrino speeds be explained as a quantum weak measurement?

Probably not. PACS numbers: 03.65.Ta, 03.65.Xp, 14.60.Pq If recent measurements [1] suggesting that neutrinos travel faster than light survive scrutiny, the question of their theoretical interpretation will arise. Here we discuss the possibility that the apparent superluminality is a quantum interference effect, that can be interpreted as a weak measurement [2–5]. Although the available numbers strongly indicate that this explanation is not correct, we consider the idea worth exploring and reporting—also because it might suggest interesting experiments, for example on electron neutrinos, about which relatively little is known. Similar suggestions, though not interpreted as a weak measurement [6, 7] or not accompanied by numerical estimates [6, 8], have been proposed independently. The idea, following analogous theory and experiment [9] involving light in a birefringent optical fibre, is based on the fact that the vacuum is birefringent for neutrinos. We consider the initial choice of neutrino flavour as a preselected polarization state, together with a spatially localized initial wavepacket. Since a given flavour is a superposition of mass eigenstates, which travel at different speeds, the polarization state will change during propagation, evolving into a superposition of flavours. The detection procedure postselects a polarization state, and this distorts the wavepacket and can shift its centre of mass from that expected from the mean of the neutrino velocities corresponding to the different masses. This shift can be large enough to correspond to an apparent superluminal velocity (though not one that violates relativistic causality: it cannot be employed to send signals). Large shifts, corresponding to states arriving at the detector that are nearly orthogonal to the polarization being detected, are precisely of the type considered in weak measurement theory. It seems that only muon and tau neutrino flavours are involved in the experiment, and we denote these by 1 (muon) and 2 (tau). The initial beam, with ultrarelativistic central momentum p̄, is almost pure muon, which can be represented as a superposition, with mixing angle θ , of mass states |+〉 and |−〉, with m+ > m–: | 0 (x)〉 = (cos θ |+〉 + sin θ |−〉) exp ( i p̄x ) f (x) . (1) 1751-8113/11/492001+05$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA 1 J. Phys. A: Math. Theor. 44 (2011) 492001 Fast Track Communication Here, f (x) represents the envelope of the initial wavepacket, normalized and centred on x = 0. For convenience, and with no effect on the final results (7) and (8), we take this as a Gaussian of width W , that is, f (x) = 1 π1/4 √ W exp ( − x 2 2W 2 ) . (2) The two mass states evolve with different phases and group velocities, so that the state arriving at the detector after travelling for time t is | (x, t)〉 = exp ( i p̄x ) [ cos θ exp ( −i tE+ ) |+〉 f (x − tv+) + sin θ exp ( −i tE− ) |−〉 f (x − tv−) ] . (3) This is an approximation, neglecting the spreading and distortion [10] of the individual packets—both negligible in the present case. E± and v± are the energies and group velocities of the two mass states, and we write E± = Ē ± 2 E, v± = v̄ ± 2 v, x = v̄t + ξ, (4) in which the new coordinate ξ measures deviation from the centre of the wavepacket expected by assuming it travels with the mean velocity. In the experiment, the detector postselects the muon flavour [1], so the final spatial wavepacket is F (ξ , t) = N exp ( i p̄x ) (cos θ 〈+| + sin θ 〈−|) | (ξ + v̄t)〉 = N exp ( i ( p̄x − Ēt) ) [ cos θ exp ( −i t E 2 ) f ( ξ − 1 2 t v ) + sin θ exp ( +i t E 2 ) f ( ξ + 1 2 t v )] , (5) where N is a normalization constant. Thus the shift in the measured final position of the wavepacket is found, after a short calculation, to be ξ̄ = ∫ ∞ −∞ dξ ξ ∣∣F (ξ , t)2∣∣ = 1 2 t v cos 2θ 1 − 2 sin 2θ ( 1 − cos ( t E ) exp (− (t v )2 4W 2 )) . (6) In the prefactor, t v is the relative shift of the two mass wavepackets, expected from the difference of their group velocities. This is multiplied by a factor representing the influence of the measurement, that is, of the preselection and postselection and the evolution. If t v is small compared with the width of the packet, as we will see that it certainly is in the neutrino case, the shift simplifies to ξ̄ = 1 2 t v cos 2θ 1 − sin 2θ sin ( t E 2 ) , (7) which involves neither the width nor the shape of the wavepacket. When interpreted as an effective velocity shift, that is, veff = ξ̄ t = 1 2 v cos 2θ 1 − sin 2θ sin ( t E 2 ) , (8) this is the same as the result found in [6].