The theoretical prediction that did not fit a wrong experimental finding and was probably right

The superluminal group velocity difference of neutrinos that was NOT measured by Opera

I take advantage of the Physics Nobel Prize 2015 rewarding the discovery of neutrino oscillations to shine a different light on a famous experiment that made big headlines in september 2011 (and gave me opportunity to start this blog!)

we discuss the possibility that the apparent superluminality is a quantum interference effect, that can be interpreted as a weak measurement [2, 3, 4, 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... The initial beam, with ultrarelativistic central momentum p, is almost pure muon, which can be represented as a superposition, with mixing angle θ, of mass states |+> and |- >, with m₊ > m_- ... The two mass states evolve with different phases and group velocities 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_±=E±1/2ΔE, v_±=v±1/2Δv, x=vt+ξ... 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]... thus the shift in the measured final position of the wavepacket [can be interpreted]... as an effective velocity shift, that is  

[where the prefactor, tΔv is the relative shift of the two mass wavepackets, expected from the difference of their group velocities (it is small compared with the width of the packet ... in the neutrino case). The main factor represents the influence of the measurement-that is of the pre- and postselection and the evolution]... The possibility of superluminal velocity measurement arises because the amplification factor in (8) can be arbitrarily large if sin²2θ and sin²(tΔE/2ℏ) are close to unity, corresponding to near-orthogonality of |pre> and |post>. 

For neutrinos with momentum p, ... the group velocity [difference Δv is given by -ΔE/p]. Thus Δv<0, so, in order for the apparent velocity to be superluminal, Δveff in (8) must be positive; this can be accommodated by making cos2θ negative. 

Note also that v₊ and v_-- are less than c if both neutrino masses are nonzero, so the individual mass eigenstate wavepackets move with subluminal group velocities; any superluminal velocity arising from (8) is a consequence of pulse distortion ... associated with the postselection, i.e. considering only arriving muon neutrinos. In the more conventional superluminal wave scenario [10], group velocities faster than light, and the pulse distortions that enable them to occur, are associated with propagation of frequencies near resonance, for which there is absorption, i.e. non-unitary propagation. That is also true in the optical polarization experiments [9] and in the neutrino situation considered here, with the difference that the nonunitarity, which gives rise to the superluminal velocity, is not continuous during propagation but arises from the sudden projection onto the postselected state.

In the [Opera] experiment, the energies of the neutrinos varied over a wide range, with an average of cp = 28.1GeV. For the difference in the squared masses, with electron neutrinos neglected and m₊ and m_- identified with the standard m₂ and m₃, a measured value [13] is m₊²c⁴-m_-²c⁴≃2.43×10^-3eV². This gives

Δv/c=-1.5×10^-24.                  (16) 

 The apparent velocity measured in the experiment [1] was (1+2.5×10^-5)c . Comparison with the quantum velocity shift Δv_eff in (8) would require knowlege of m₊ and m_-, not just their squared difference, and the individual masses are not known. But even on the most optimistic assumption, that m_-=0, it is immediately clear that it is unrealistic to imagine that the quantum amplification factor in (8) can bridge the gap of 19 orders of magnitude between (16) and the measured superluminal velocity.

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

M. V. Berry, N. Brunner, S. Popescu, P. Shukla

(Submitted on 13 Oct 2011 (v1), last revised 14 Nov 2011 (this version, v2))

Remark: for the anecdote the abstract of this article by the distinguished mathematical physicist Michael Berry and his collaborators might be the shortest one ever written since it answered laconically to the question asked in the title : "probably not". And time has proved that it was right...

A superluminal group velocity of photons that was effectively measured
While the theoretical prediction from the last paragraph has not been tested by the Opera experiment and will stay quite hard to test empirically given the smallness of the effect, the physics behind it is pretty sound and falsifiable in other contexts. I think the article below is a nice illustration:

The physics of light propagation is a very timely topic because of its relevance for both classical [1] and quantum [2] communication. Two kind of velocities are usually introduced to describe the propagation of a wave in a medium with dispersion ω( k): the phase velocity v_ph=ωk and the group velocity v_g=∂ω/∂k . Both of these velocities can exceed the speed of light in vacuum c in suitable cases [3]; hence, neither can describe the speed at which the information carried by a pulse propagates in the medium. Indeed, since the seminal work of Sommerfeld, extended and completed by Brillouin [4], it is known that information travels at the signal velocity, defined as the speed of the front of a square pulse. This velocity cannot exceed c [5]. The fact that no modification of the group velocity can increase the speed at which information is transmitted has been directly demonstrated in a recent experiment [6]. Superluminal (or even negative) and, on the other extreme, exceedingly small group velocities, have been observed in several media [7]. In this letter we report observation of both superluminal and delayed pulse propagation in a tabletop experiment that involves only a highly birefringent optical fiber and other standard telecom devices. 

Before describing our setup, it is useful to understand in some more detail the mechanism through which anomalous group velocities can be obtained. For a light pulse sharply peaked in frequency, the speed of the center-of-mass is the group velocity v_g of the medium for the central frequency [3]. In the absence of anomalous light propagation, the local refractive index of the medium is n_f , supposed independent on frequency for the region of interest. The free propagation simply yields v_g=L/t_f where L is the length of the medium and t_f =n_f L/c is the free propagation time. One way to allow fast- and slow-light amounts to modify the properties of the medium in such a way that it becomes opaque for all but the fastest (slowest) frequency components. The center-of-mass of the outgoing pulse appears then at a time t = t_f+<t>, with <t> the mean time of arrival once the free propagation has been subtracted; obviously <t><0 for fast-light, <t>>0 for slow-light. If the deformation of the pulse is weak, the group velocity is still the speed of the center-of-mass, now given by  

                          v_g=Lt_f+<t>.                                                                         (1) 

This can become either very large and even negative (<t>→−∞) or very small (<t>→∞) — although in these limiting situations the pulse is usually strongly distorted, so that our reasoning breaks down.

 

Direct measurement of superluminal group velocity and of signal velocity in an optical fiber 

Nicolas Brunner, Valerio Scarani, Mark Wegmuller, Matthieu Legre, Nicolas Gisin

(Submitted on 20 Jul 2004 (v1), last revised 10 Jan 2005 (this version, v2))

Neutrino oscillations : experiment validated and awarded the 2015 Physics Nobel Prize ...

... but theory is still under [discuss]{construct}ion

Neutrino physics is one of the most interesting and vividly discussed topics in high-energy physics today. Especially the question whether the neutrinos can oscillate or not (i.e. different neutrinos can change into each other) gave rise to a huge number of experiments to actually observe these oscillations. At least since the results from the Super Kamiokande ... and the SNO experiment ... are published, it is widely believed that neutrino oscillations are an experimentally verified fact. However, the first hint has already been found in 1964 when the Homestake experiment ... discovered the solar neutrino problem. That is, the number of measured electron neutrinos from the sun is by a factor of 2-3 less than the number of neutrinos predicted by the standard solar model (SSM). 

Since within the standard model (SM) of particle physics the neutrinos are massless, and consequently cannot oscillate, their measurement shows that new physics beyond the SM exists. And indeed nowadays the experiments on neutrino oscillations are important to measure the unknown parameters of the SM and its minimal extensions. In particular, these unknown parameters are the neutrino masses and the entries in the neutrino mixing matrix. 

From all the measurements made to discover neutrino oscillations one should think that the theory behind [them] is well established and understood. But surprisingly this is not the case. The first who mentioned the idea of neutrino oscillations, though he assumed neutrino-antineutrino oscillations, was Pontecorvo in 1957 [Pon57, Pon58]. A few years later Maki, Nakagawa and Saka were the first to consider oscillations between the electron and the muon neutrino [MNS62]. Then it took around 20 years before Kayser in 1981 showed that the up to that point used plane-wave approximation cannot hold for oscillating neutrinos and he proposed a wave packet treatment [Kay81], which then has again not been discussed for around 10 years. In the early 90s the discussion on the theoretical description of neutrino oscillations finally started with several seminal papers. First, Giunti, Kim and Lee explicitly calculated the oscillation probability for the neutrinos in a wave packet model [GKL91] and then showed that the state vectors used for the quantum mechanical description are, in general, ill-defined [GKL92]. In 1993 they published together with Lee a calculation of the probability in a quantum field theoretical framework without using state vectors for the neutrinos [GKLL93]. And finally, in 1995 Blasone and Vitiello showed that the description of mixed particles in quantum field theory (QFT) yields unexpected problems for the interpretation of neutrinos as particles. By only using exact—without perturbation—QFT methods they calculated an oscillation probability which differs significantly from the other results [BV95]. All these different approaches are even today still under discussion, but however under the assumption of relativistic neutrinos which have tiny mass squared differences, all approaches give the same result. Thus, the theoretical discussion on the right description of the neutrinos does not spoil the experimental results, because today we are only able to measure ultra-relativistic neutrinos whose energy is at least a few orders of magnitude higher than their mass. 

Diploma Thesis On Theories of Neutrino Oscillations (Summary and Characterisation of the Problematic Aspects) Daniel Kruppke September 2007

A quantum field theory for flavor states ...

The study of mixing of fields of different masses in the context of Quantum Field Theory (QFT) has produced recently very interesting and in some sense unexpected results ... The story begins in 1995 when in Ref.[1], it was proved the unitary inequivalence of the Hilbert spaces for (fermion) fields with definite flavor on one side and those (free fields) with definite mass, on the other. The proof was then generalized to any number of fermion generations [7] and to bosonic fields [2, 5]. This result strikes with the common sense of Quantum Mechanics (QM), where one has only one Hilbert space at hand: the inconsistencies that arise there have generated much controversy and it was also claimed that it is impossible to construct an Hilbert space for flavor states [16] (see however Ref.[6] for a criticism of that argument). In fact, not only the flavor Hilbert space can be consistently defined [1], but it also provide a tool for the calculation of flavor oscillation formulas in QFT ..., which exhibit corrections with respect to the usual QM ones [20, 21]. From a more general point of view, the above results show that mixing is an “example of non-perturbative physics which can be exactly solved”, as stated in Ref.[13]. Indeed, the flavor Hilbert space is a space for particles which are not on-shell and this situation is analogous to that one encounters when quantizing fields at finite temperature [22] or in a curved background [23]. In the derivation of the oscillation formulas by use of the flavor Hilbert space, both for bosons and for fermions, a central role is played by the flavor charges [9] and indeed it was found that these operators satisfy very specific physical requirements [6, 8]. 

Mixing and oscillations of neutral particles in Quantum Field Theory

Massimo Blasone, Jonathan S. Palmer

(Submitted on 23 May 2003 (v1), last revised 10 Jun 2003 (this version, v2))

... with an unfinished taste

Blasone and Vitiello (BV) have attempted to construct a Fock space for neutrino flavor states [4]... Giunti conclude that “the Fock spaces of flavor neutrinos are ingenuous mathematical constructs without physical relevance” [3]. 

... there is another issue that plagues the scheme in [5]. The problem is that the neutrino flavor vacuum defined in [5] is time-dependent and hence Lorentz invariance is manifestly broken. Recently, BV and collaborators attempted to tackle this issue by proposing neutrino mixing as a consequence of neutrino interactions with an external non-abelian gauge field [7]. Under this framework, the Lorentz violation of the neutrino flavor vacuum can be attributed to the presence of a fixed external field which specifies a preferred direction in spacetime. However, at the moment, there is not a single sign of such a non-abelian gauge field in neutrino experiments. They proposed that this scheme can be tested in the tritium decay, but again the indefinite mass mνα becomes an observable quantity. Also, given the current stringent bounds on Lorentz violations [8], it is unclear whether this scheme will survive... 

In this article, we first gave a detailed review on the current status of the understanding about the neutrino flavor states. At the end of the review, we were led to conclude that it is currently unclear how to construct a consistent and physically relevant Fock space of neutrino flavor states. We proceeded to prove that if one insists on second-quantizing the neutrino flavor fields and thereby constructing the flavor states, then they are approximately well-defined only when neutrinos are ultra-relativistic or the mass differences are negligible compared to energy...   

However, we showed that one can consistently describe weak interactions by only neutrino mass eigenstates. At the same time, we argued that the second quantization of neutrino flavor fields generally lacks physical relevance because their masses are indefinite. Thus, neutrino flavor states lose their physical significance and they should simply be interpreted as definitions to denote specific linear combinations of mass eigenstates involved in weak interactions. Under this interpretation, there is no physical motivation to construct the Fock space of neutrino flavor states from the first principles of quantum field theory. 

On Neutrino Flavor States

Chiu Man Ho

(Submitted on 16 Sep 2012 (v1), last revised 26 Nov 2012 (this version, v2))