Theory of Neutrino Detection -- Flavor Oscillations and Weak Values
TTheory of Neutrino Detection - Flavor Oscillationsand Weak Values
Yago P. Porto-Silva ∗ , Marcos C. de Oliveira † ∗ † Instituto de Física “Gleb Wataghin”, Universidade Estadual de Campinas, Campinas, SP,Brazil ∗ Max-Planck-Institut für Kernphysik, 69117 Heidelberg, GermanyE-mail: ∗ yporto@ifi.unicamp.br, † marcos@ifi.unicamp.br We show that, in the relativistic limit, the quantum theory of neutrino oscil-lations can be described through the theory of weak measurements with pre-and post-selection. The weak nature of neutrino detection allows simultaneousdetermination of flavor and energy without problems related to the collapse ofthe wavefunction. Together with post-selection, a non-trivial quantum inter-ference emerges, allowing one to describe a flavor neutrino as one single par-ticle, despite its superposition of masses. We write down the flavor equationof motion and calculate the flavor oscillation probability by showing preciselyhow a single neutrino interferes with itself.
Introduction
Neutrino oscillations is an experimentally established phenomenon by which neutrinos undergoflavor transformations periodically as they propagate large enough distances [1, 2, 3]. It isdescribed by a simple quantum mechanical model in which the flavor states are not eigenstatesof the propagation Hamiltonian, but some linear combination of them, and, as a consequence,the flavor content changes with time and distance [4, 5, 6, 7, 8].Most of the literature treats the eigenstates of propagation as plane waves: states with def-inite mass, energy, and momentum. Besides, assumptions as same energies or same momentafor all the propagation eigenstates, also known as mass eigenstates, are usually made. Allthese assumptions are unphysical in the sense that they can violate coherence of mass eigen-states, energy-momentum conservation and are unable to describe space-time localized pro-cesses as neutrino production and detection [9, 10, 11]. To overcome these difficulties manyauthors have proposed treatments based on neutrino quantum mechanical (QM) wave-packets1 a r X i v : . [ qu a n t - ph ] F e b
12, 13, 14, 15, 16, 17, 18, 19, 20] and quantum field theory [21, 22, 23, 24, 25, 26, 27, 28, 29].Here, we explore neutrino oscillations at the level of QM wave-packets.In this paper, we shall work with Gaussian wave-packets . The uncertainties in energyand momentum of a massive neutrino wave-packet come from the approximate conservation ofmean energies and momenta of all particles in the production and detection processes. Whenthese uncertainties are large enough so that one cannot, even in principle, resolve the masses,the produced and detected states can be written as a coherent superposition of mass eigen-states. When neutrinos are detected, their flavor is revealed by the charged leptons producedin the interaction, and their energy and momentum can, in principle, be reconstructed by mea-suring energies and momenta of all other particles involved in the detection process. Even ifall this information is inaccessible to the experimentalist, it is available to the particles in thedetection, and this, by itself, configures a measurement. However, from QM, two incompatible observables are being measured at the same time in this detection process: flavor and energy-momentum .The most critical consequence of measuring two incompatible observables at the same timeis that they randomly mess up information about each other, being manifestly complementary.Therefore, in the same way that, in general, measuring momentum degrades the informationabout the position of quantum particles, the measurement of energy-momentum of the neutrinoshould disturb previous information about flavor, in particular, the flavor transitions during thepropagation, and it would have been impossible to study neutrino oscillations. The reasonwhy the energy-momentum measurement does not prevent neutrino oscillations is, again, thelarge uncertainties in energy and momentum in the detection process. Indeed, this type ofmeasurement, with large uncertainties, is an example of what is called in the literature weakmeasurements [31, 32, 33] and their main feature is to disturb very little the quantum state ofthe system, not degrading the information about complementary observables . Therefore, thesame condition that allows neutrinos to be produced coherently also warrant flavor, energy, andmomentum to be measured simultaneously by the detection particles.Based on weak measurements plus post-selection [34, 35, 36, 37, 38, 39], i.e. , the fact thatusually only one flavor type is measured in the end, we develop the theory of neutrino detectionand derive a relativistic quantum mechanics theory of particles described by a superpositionof masses. We notice that flavor neutrinos obey Klein-Gordon continuity equation (ignoringspin) with momentum and energy described by weak values . This is a remarkable connectionbetween two completely independent developments in 20th-century quantum mechanics. Fromthis formalism, the oscillation probability can be calculated without the conceptual issues usu- See [28] for a discussion on the conditions under which a Gaussian envelope is a good approximation for anarbitrary wave-packet. Incompatible observables do not share the same set of eigenstates: the flavor eigenstates are certainly not thesame as the energy-momentum eigenstates [30]. The concept of weak measurements has nothing to do with the concept of weak interactions in the StandardModel. Weak values are the measured values when weak measurements with pre- and post-selection are performed.
Neutrino wave-packets
In this section, we review the standard wave-packet formalism of neutrino oscillations in onedimension . We use natural units ( (cid:126) = c = 1 ) throughout the paper.Consider a process at (average) coordinates ( t = 0 , x = 0) that produces a neutrino offlavor α which propagates and is detected at ( T, L ) with flavor β . Using a normalized Gaussianenvelope, we can write the one-particle state of the neutrino produced at the origin: (cid:12)(cid:12) ν Pα (0 , (cid:11) ≡ (cid:12)(cid:12) ν Pα (cid:11) = (cid:88) a U ∗ αa (cid:12)(cid:12) ν Pa (cid:11) = (cid:88) a U ∗ αa (cid:90) dp √ π (cid:112) E a ( p ) φ P ( p − p a ) | ν a ( p ) (cid:105) , (1)with E a ( p ) = (cid:112) p + m a and (cid:90) dp | φ P ( p − p a ) | = 1 → φ P ( p − p a ) = 1(2 πσ pP ) e − ( p − pa )24 σ pP . (2)Here the flavor eigenstate (cid:12)(cid:12) ν Pα (cid:11) is a superposition of mass eigenstate wave-packets, (cid:12)(cid:12) ν Pa (cid:11) , withmass m a , weightened by the complex-conjugated PMNS matrix elements, U ∗ αa . The mass eigen-states themselves are a superposition of energy and momentum eigenstates | ν a ( p ) (cid:105) . In the x -space, ν a wave function, at time t , is given by [43]: (cid:12)(cid:12) ν Pa ( x, t ) (cid:11) = (cid:90) dp √ π (cid:112) E a ( p ) φ P ( p − p a ) e − iE a ( p ) t e ipx . (3)The average momenta and momentum uncertainties of different mass eigenstates, p a and σ pP , respectively, are determined by the kinematics and by the properties of the particles in-volved in the production ( P ) process. We assume all mass eigenstates are extremely relativistic, p a >> m a , so that we can approximate their average energies by [43], (cid:15) a ≈ E + ξ m a E , (4) This is a good approximation for cases in which the distance between neutrino source and detector is largecompared to their size [28].
3n which E is the energy determined by the kinematics of the production process if neutrinomasses are neglected and ξ E = ∂(cid:15) a ∂m a (cid:12)(cid:12)(cid:12)(cid:12) m a =0 (5)is the coefficient of the first-order term if one expands (cid:15) a = (cid:112) p a + m a around m a = 0 . Thecorresponding momenta are p a ≈ E − (1 − ξ ) m a E . (6)For a given process, ξ can be calculated from energy-momentum conservation up to order m a E .The effective momentum-space uncertainty of the produced neutrino wave-packets σ pP is σ pP ∼ min { δ pP , δ eP } (7)where δ pP and δ eP are, respectively, the momentum and energy uncertainties in the productionprocess. In configuration-space, σ xP = σ pP .The detection process, in the standard formalism, is considered by propagating the ket in (1)from the origin to ( T, L ) and then projecting it on the state (cid:12)(cid:12) ν Dβ (cid:11) = (cid:88) a U ∗ βa (cid:90) dp √ π (cid:112) E a ( p ) φ D ( p − p a ) | ν a ( p ) (cid:105) , (8)with (cid:90) dp | φ D ( p − p a ) | = 1 → φ D ( p − p a ) = 1(2 πσ pD ) e − ( p − pa )24 σ pD , (9)which takes into account the effective momentum-space uncertainty σ pD of the detection ( D )wave-packet, related to δ pD and δ eD in a similar way to (7), σ pD ∼ min { δ pD , δ eD } , (10)and σ xD = σ pD . The average momentum p a seen in the detection process is determined by thekinematics of the production process . Notice that (1) and (8) are normalized independently.Now, we compute A αβ ( L, T ) = (cid:10) ν Dβ (cid:12)(cid:12) e − i H T + i p L (cid:12)(cid:12) ν Pα (0 , (cid:11) = (cid:10) ν Dβ (cid:12)(cid:12) ν Pα ( L, T ) (cid:11) , (11) i.e. , the amplitude of probability of detection of neutrinos in state (cid:12)(cid:12) ν Dβ (cid:11) when they weregenerated in state (cid:12)(cid:12) ν Pα (0 , (cid:11) after traveling the distance L during the time interval T , being This constrain can be relaxed, see [28] the Hamiltonian and p the momentum operators. Using the condition (cid:104) ν a ( p ) | ν a ( p (cid:48) ) (cid:105) =(2 π )2 E a ( p ) δ ( p − p (cid:48) ) : A αβ ( L, T ) = 1(4 π σ pP σ pD ) (cid:88) a U ∗ αa U βa (cid:90) dp e − ( p − pa )24 σ p e − iE a ( p ) T + ipL . (12)Let us consider sharply peaked Gaussian functions in momentum space , with average mo-mentum and energy given by (4) and (6). In this context, the relativistic dispersion relation canbe approximated by E a ( p ) ≈ (cid:15) a + v a ( p − p a ) , with v a = ∂E a ( p ) ∂p (cid:12)(cid:12)(cid:12)(cid:12) p = p a = p a (cid:15) a ≈ − m a E . (13)Thus, A αβ ( L, T ) = (cid:115) σ xP σ xD σ x (cid:88) a U ∗ αa U βa exp (cid:18) − i(cid:15) a T + ip a L − ( L − v a T ) σ x (cid:19) . (14)In (14), σ x is the effective size of the detection region , that takes into account space and timeintervals in which the neutrino and all particles in the detection process are overlapped. Sincethe neutrino that reached the detection process carries information about the production process, σ x takes into account features of both production and detection, in a similar way to σ p - theeffective resolution with which the detection process can measure momentum: σ x = σ xP + σ xD and σ p = 1 σ pP + 1 σ pD . (15)Both are related by σ x σ p = .Squaring the amplitude in (14) and integrating out the T dependence, we obtain P αβ ( L ) = 2 σ xP σ xD σ x (cid:88) a,b U ∗ αa U βa U αb U ∗ βb e i (1 − ξ ) ∆ m ab E L × (cid:90) dT exp (cid:20) − ( L − v a T ) + ( L − v b T ) σ x (cid:21) e − iξ ∆ m ab E T . (16)After integration, we substitute the expression in (13) for the velocity of relativistic mass eigen-states in the exponents, preserving terms of first order in m a E (or one order higher if first ordervanishes) and find P αβ ( L ) = 2 √ πσ xP σ xD σ x (cid:88) a,b (cid:115) v a + v b U ∗ αa U βa U αb U ∗ βb e − i ∆ m abL E e − (cid:0) LLabcoh (cid:1) e − (∆ (cid:15)ab )28 σ e , (17) Dispersion due to different phase velocities is negligible [27]. In the literature σ x is most commonly referred as the size of the wave-packet. Here we want to emphasize thatit is related to the momentum resolution in the detection process. ∆ p ab << σ p , with p , p and p the mean momenta of the massive neutrino wave-packetsgiven in (6).with ∆ (cid:15) ab = ξ ∆ m ab L E , and L abcoh = 4 √ E | ∆ m ab | σ x , (18)where σ e ≈ ( v a + v b ) σ p .The probability in (17) is not normalized, and its magnitude is manifestly dependent on thesizes of produced and detected wave-packets and their overlap. Indeed, (cid:88) β P αβ ( L ) = 2 √ πσ xP σ xD σ x (cid:88) a | U αa | v a ≈ √ πσ xP σ xD σ x . (19)In addition, it is not dimensionless but has unit of length. Fixing it, therefore, is not a matter onlyof a constant factor, its calculation is conceptually incorrect. This is called the normalizationproblem and, to get rid of it, unitarity has to be imposed. This is rather unsatisfactory and asymptom that the formalism has consistency problems [20, 28].The discussion about the physical meaning of the exponentials (17) can be found in manyreferences [13, 15, 20, 27, 43]. Here we highlight: • The exponential e − (cid:0) LLabcoh (cid:1) defines the coherence length, L abcoh , that is the effective distanceafter which mass eigenstates ν a and ν b lose coherence due to separation of their wave-packets. For L << L abcoh wave-packet separation is negligible. • The term e − (∆ (cid:15)ab )28 σ e defines the conditions under which neutrinos are produced and de-tected coherently. In the limit, ∆ (cid:15) ab << σ e , (Coherence Condition) (20)the conditions for coherent production and detection of the mass eigenstates ν a and ν b are set (see fig. 1). In the relativistic regime and in one dimension, (20) is equivalent to ∆ p ab << σ p [19].Therefore, L << L abcoh and ∆ E ab << σ e are usually referred as the conditions for theobservability of neutrino oscillations. 6 eak measurements and weak values In this section, we formalize the concept of quantum measurement in the von Neumann regimeand use it to distinguish between the strong (great disturbance, wavefunction collapse) and weakmeasurements (very little disturbance) [31, 32, 33, 34, 35, 36]. von Neumann measurements
In the von Neumann measurement model [44, 45], the measuring device (or pointer) is a sec-ondary quantum system with canonical variables x D and p D satisfying [ x D , p D ] = i , in naturalunits. The Hamiltonian that describes the interaction between the system and the device usuallycouples the observable of interest, that we call A , with some of the canonical variables, x D , forexample, so that the change in the conjugate variable, p D , reveals information about A . Theinteraction Hamiltonian can be written as H int = − δ ( t − t ) Ax D , (21)where the delta function assures the interaction to happen for times only on the vicinity of t while, at any other instant, the system evolves freely. To illustrate how the pointer variable, p D in our example, acquires information about A , we compute its evolution at times close to t inthe Heisenberg picture: ddt p D ( t ) = i [ H int , p D ( t )] = − iδ ( t − t ) A ( t )[ x D ( t ) , p D ( t )] = δ ( t − t ) A ( t ) , (22)then, p D ( t > t ) − p D ( t < t ) = (cid:90) dtδ ( t − t ) A ( t ) = A ( t ) . (23)Therefore, the change in the pointer immediately after t gives the information about the statusof the observer of interest at t . Statistics of the pointer variable
Consider an ensemble defined by a system prepared in state | ψ i (cid:105) and measuring device in state | φ (cid:105) . A system ensemble prepared in a specific initial state defines a preselected ensemble. Weknow that the effect of the measurement on the device is to change the status of its pointervariable p D proportionally to the system observable of interest A according to (23). Startingfrom the initial state of the system plus measuring device, | Ω i (cid:105) = | ψ i (cid:105) | φ (cid:105) , we find that theimpact of the measurement on this state is given by, | Ω f (cid:105) = e − i (cid:82) H int dt | Ω i (cid:105) = e i Ax D | ψ i (cid:105) | φ (cid:105) . (24)7igure 2: Illustration of a pointer wavefunction that can resolve the spectrum of eigenvaluesof the system observable A in a strong measurement, σ p << ∆ a ij . The amplitude of thedistributions are proportional to the probability amplitude of the system to be in the state | a i (cid:105) .In the limiting case of σ p → , one recovers Born rule. Detector and system are fully entangledimmediately after the measurement.Projecting (24) into the pointer variable space (cid:104) p D | Ω f (cid:105) = (cid:88) a | a (cid:105) (cid:104) a | ψ i (cid:105) (cid:104) p D | e ia x D | φ (cid:105) = (cid:88) a | a (cid:105) (cid:104) a | ψ i (cid:105) φ ( p D − a ) , (25)in which {| a (cid:105)} are the eigenvectors of operator A , and φ ( p D − a ) = (cid:104) p D − a | φ (cid:105) is the shifted(by a ) wavefunction, φ ( p D ) , due to the action of the translation operator e ia x D . The probabilitydistribution of the pointer apparatus state after the measurement is given by the absolute squareof (25) P f ( p D ) = | (cid:104) p D | Ω f (cid:105) | = (cid:88) a | (cid:104) a | ψ i (cid:105) | | φ ( p D − a ) | . (26)Remark that for the probability interpretation to hold we need a normalized pointer wavefunc-tion, (cid:104) φ | φ (cid:105) = (cid:90) dp D | φ ( p D ) | = 1 . (27)What we call strong or weak measurement depends very much on the spread of the apparatuswavefunction in the p D -space, σ p , relative to the separation, ∆ a ij , of the eigenvalues, { a i } , ofthe system. If the pointer can resolve the spectrum, in other words, if σ p << ∆ a ij , (Strong measurement) (28)it is called strong measurement, and is pictorially represented in fig. 2.In the opposite limit, σ p >> ∆ a ij (Weak measurement) (29)we say that the system is weakly measured by the apparatus, see fig. 3. It is common to model the pointer with a Gaussian wavefunction. cannot resolve the spectrum of eigenvaluesof the system observable A in a weak measurement, σ p << ∆ a ij . The amplitude of the distribu-tions are proportional to the probability amplitude of the system to be in the state | a i (cid:105) . Becauseof the poor resolution, detector and system are not fully entangled and the system wavefunctionis very little disturbed after the measurement.Weak measurements were first proposed as a way in which one can extract average stateinformation without fully collapsing the system [45]. In fact, due to the large uncertainty σ p in the apparatus wavefunction, after the measurement, the state of the apparatus is not stronglycorrelated or entangled with any of the states {| a i (cid:105)} of the system. This can be seen graphicallyin fig 3. For comparison, observe how the states of the system and apparatus are fully correlatedafter a strong measurement (resembling Born rule) in fig. 2.A useful way of thinking about weak measurements is that the eigenvalues of A are so closethat the effect of the translation operator on the pointer is very small. More precisely, supposeour pointer just measured a (went from zero to a by means of e ia x D ) in fig. 3, to move itscenter to a we operate with: e i ( a − a ) x D = 1 + i ∆ a x D −
12 (∆ a ) x D + ... (30)with ∆ a = a − a . However, due to (29), (cid:104) φ | (∆ a ) x D | φ (cid:105) = (∆ a ) σ x ≈ (∆ a ) σ p << . (31)Hence, in case the measurement is weak, it is enough to use the expansion in (30) up to firstorder.Although strong and weak measurements are conceptually different, they are quantitativelyequivalent with respect to expectation values [46]. In other words, (cid:104) Ω f | p D | Ω f (cid:105) = (cid:104) ψ i | A | ψ i (cid:105) , (32)independently of σ p . Pre- and post-selected ensembles
If the expectation values of the observables of the system are the same independently of themeasurement type (see (32)), then one can judge unnecessary to talk explicitly about the weak9ature of the neutrino energy-momentum measurement. The problem is that, as we are goingto see in next section, neutrino oscillation measurements are, in general, made in pre- and post-selected ensembles and the results of these measurements are not expectation values, but weakvalues , a concept introduced in 1988 by Aharanov, Albert and Vaidman (AAV) [34, 37, 38, 39].To understand the concept of weak value, suppose that after the measurement describedby the Hamiltonian in (21) in the system ensemble Ω i , we focus only on the measurementoutcomes from the system subensemble that ended up in some specific state | ψ f (cid:105) . We name thissubensemble Ω if . We can find the statistics of the apparatus pointer variable by taking (24) andapplying to it the system conditional final state, | Ω if (cid:105) = (cid:104) ψ f | Ω f (cid:105) = (cid:104) ψ f | e − i (cid:82) H int dt | Ω i (cid:105) = (cid:104) ψ f | e i Ax D | ψ i (cid:105) | φ (cid:105) . (33)Now, we expand the exponential inside (33), | Ω if (cid:105) ≈ (cid:104) ψ f | ψ i (cid:105) (1 + iA w x D − A w x D + ... ) | φ (cid:105) , (34)where A nw is called the n-th order weak value of A , A nw ≡ (cid:104) ψ f | A n | ψ i (cid:105) (cid:104) ψ f | ψ i (cid:105) . We use the hypothesis ofweak measurements to argue that, for the apparatus, the action of the Hamiltonian in (21) is justa small perturbation and truncate the expansion at first order in x D , (cid:104) p D | Ω if (cid:105) ≈ (cid:104) ψ f | ψ i (cid:105) (cid:104) p D | iA w x D | φ (cid:105) , (35)using the approximation iA w x D ≈ e iA w x D , which acts as a translation operator in p D -space (cid:104) p D | iA w x D | φ (cid:105) ≈ (cid:104) p D − A w | φ (cid:105) . (36)Assuming for the state of the pointer before the measurement, (cid:104) p D | φ (cid:105) = φ ( p D ) ∝ e − p D σ p , with σ x σ p = 12 , (37)and that A w is a complex number, A w = Re A w + i Im A w , then, the probability distribution forthe apparatus after the measurement is | (cid:104) p D − Re A w − i Im A w | φ (cid:105) | ∝ e (Im Aw )22 σ p e − ( pD − Re Aw )22 σ p ≈ | φ ( p D − Re A w ) | , (38)which is approximately the initial probablity distribution translated by Re A w in p D -space.Therefore, instead of moving the pointer to some eigenvalue, as in (26), a weak measurementwith pre- and post-selection move the pointer to real part of the observable weak value, see fig.4. We usually think in eigenvalues as the only possible answers to single quantum measure-ments, but what this section teaches us is that with enough uncertainty in the detection and10igure 4: Illustration of the resulting pointer wavefunction (green), with mean given by the realpart of the weak value, Re A w , when, together with weak measurements, there is post-selectionof the state | ψ f (cid:105) . Its amplitude is proportional to (cid:104) ψ f | ψ i (cid:105) . On purpose, Re A w is shown outsidethe range of eigenvalues a i , to highlight one of the most interesting properties of weak values.post-selection, an entirely new type of answer appears: the weak value, A w . The weak valueis considered a property of a single system under pre- and post-selection, revealed by a singlemeasurement [47]. All the physical consequences of the interactions of the system under suchcircumstances depend on A w . Weak values can lie beyond the range of eigenvalues of A , so-called anomalous weak values, as illustrated in fig. 4, and, at its core, is a complicated quantuminterference effect that is mathematically described by the concept of superoscillations [48, 49].Observe in fig. 4 that, due to the uncertainty in the detection process, the eigenfunctionsof the observable A look like wave-packets with mean values a i . Under general pre- and post-selection, i.e. initial and final states are not restricted to be eigenvectors of observable A -an effective wave-packet emerges with the mean value given by Re A w . What comes in thefollowing section can be already anticipated: if the total momentum uncertainty in the neutrinodetection is large enough (see fig. 1) and we post-select a given flavor (see (8)), then a kind of“flavor wave-packet” emerges with mean momentum given by a weak value . Neutrino oscillations and the weak regime
In this section we aim to construct a quantum theory of neutrino detection. Start by interpretingthe detection region (see discussion before (15)), as an apparatus (or pointer) that will measureneutrino energy and momentum with uncertainties σ e and σ p , respectively. In the relativisticone-dimensional case, it is redundant to talk in terms of momentum and energy; then, in thefollowing, we refer to momentum measurement. In the pointer interpretation , (20) must beunderstood in the same sense as (29): σ p >> ∆ p ab . (Weak measurement) (39)As an apparatus, the detection region has conjugate variables p D and x D obeying [ x D , p D ] = i . In p D -space, its wavefunction is given by the combination of the production and detection The same happens for energy. : (cid:104) p D | φ (cid:105) = φ ( p D ) = φ P ( p D ) φ D ( p D ) ∝ e − p D σ pP e − p D σ pD = e − p D σ p , (40)where we used (15). Thus, we model the detection region as a Gaussian pointer with resolution σ p , as in previous section. Here, σ p is the momentum resolution in neutrino detection, accordingto (15).We construct the Hamiltonian coupling the neutrino momentum, p , to the pointer conjugatevariable, x D as H int ( t ) = − δ ( t − T ) px D , (41)where T is the average time of detection. According to this Hamiltonian, in analogy with (23),after measurement (assuming initial value of p D is zero): p D ( t > T ) = p ( t = T ) . (42)In case the measurement is made for a neutrino mass eigenstate ν a , described by (1), we have,in Heisenberg picture (initial state (cid:12)(cid:12) ν Pa (cid:11) | φ (cid:105) ), (cid:10) ν Pa (cid:12)(cid:12) (cid:104) φ | p D ( t > T ) | φ (cid:105) (cid:12)(cid:12) ν Pa (cid:11) = (cid:104) φ | φ (cid:105) (cid:10) ν Pa (cid:12)(cid:12) p ( t = T ) (cid:12)(cid:12) ν Pa (cid:11) = p a , (43)with | φ (cid:105) the (normalized) state of the detection region. In other words, the detection regionmomentum distribution after the measurement ( t > T ) is clustered around p D = p a , given in(6), as expected. This is just telling us that the detection process behaves as if a wave-packetwith mean momentum p a and uncertainty σ p just arrived. In our example, if p a is known, thecorresponding energy, (cid:15) a , is also known.Next subsection is devoted to the most general case of coherent detection of several masseigenstates with post-selection (detection of a specific flavor). Weak values naturally appear. Neutrino oscillations with pre- and post-selection
In the case the neutrino is preselected in the state (cid:12)(cid:12) ν Pα (cid:11) , evolves freely to (cid:12)(cid:12) ν Pα ( L, T ) (cid:11) , until beingdetected and, consequently, post-selected in the state (cid:12)(cid:12) ν Dβ (cid:11) , we can write for the initial state | Ω α (cid:105) = (cid:12)(cid:12) ν Pα ( L, T ) (cid:11) | φ (cid:105) . (44)In analogy with (33), | Ω αβ ( L, T ) (cid:105) = (cid:10) ν Dβ | Ω α (cid:11) = (cid:10) ν Dβ (cid:12)(cid:12) e − i (cid:82) H int dt (cid:12)(cid:12) ν Pα ( L, T ) (cid:11) | φ (cid:105) = (cid:10) ν Dβ (cid:12)(cid:12) e i px D (cid:12)(cid:12) ν Pα ( L, T ) (cid:11) | φ (cid:105) . (45) Actually, it is just after the measurement, p D → p D − p a , that φ P and φ D will be equal to the Gaussianevelopes in (2) and (9). Notice that we are not imposing any kind of new interaction in the detection process, H int , here, is just anartifact of calculation. | ν β (cid:105) constitutes post-selection. As usual, the resulting wave-packet (green) hasmean given by Re p αβw and we interpret it as the wave-packet of the detected ν β or, generically,the “flavor wave-packet”. Its amplitude is proportional to (cid:104) ν β | ν α ( L, T ) (cid:105) .Using the weak measurement hypothesis (39), | Ω αβ ( L, T ) (cid:105) ≈ (cid:10) ν Dβ | ν Pα ( L, T ) (cid:11) (cid:18) ip αβw x D (cid:19) | φ (cid:105)≈ (cid:10) ν Dβ | ν Pα ( L, T ) (cid:11) (cid:12)(cid:12) φ ( p D − Re { p αβw } ) (cid:11) , (46)where p αβw is also a function of L and T , given by p αβw ( L, T ) = (cid:10) ν Dβ (cid:12)(cid:12) p (cid:12)(cid:12) ν Pα ( L, T ) (cid:11)(cid:10) ν Dβ | ν Pα ( L, T ) (cid:11) . (47)Hence, the neutrino momentum measured by the particles in the detection region at averagecoordinates ( T, L ) is given by p D = Re { p αβw ( L, T ) } . Analogously, the energy is the real part of (cid:15) αβw ( L, T ) = (cid:10) ν Dβ (cid:12)(cid:12) H (cid:12)(cid:12) ν Pα ( L, T ) (cid:11)(cid:10) ν Dβ | ν Pα ( L, T ) (cid:11) . (48)Notice that the flavor neutrino behaves as a single particle wave-packet with average energiesand momenta Re (cid:15) αβw and Re p αβw at ( T, L ) , see fig. 5, in the same sense the massive wave-packets have averages (cid:15) a and p a . As explained after (38), the detection process effectivelysees a “flavor wave-packet” as a consequence of large uncertainties and post-selection. In this specific context, flavor neutrinos can be considered particles with their own wave-packets. Normalization and probability current
In this subsection, we work with one massive neutrino ν a , mass m a , and explain how to write itswavefunction, probability density and current satisfying the pointer interpretation. Any othermassive particles, such as electrons or muons, would have the same treatment. In the nextsubsection we mix the massive neutrinos and find an analogous treatment for flavor neutrinos.13qs. (27) and (40) imply that for our interpretation of detection region as a pointer, produc-tion and detection Gaussian envelopes should not be normalized separately but in a correlatedmanner, (cid:90) dp | φ ( p − p a ) | = (cid:90) dp | φ P ( p − p a ) | | φ D ( p − p a ) | = 1 , (49)and therefore φ ( p − p a ) = 1(2 πσ p ) e ( p − pa )24 σ p . (50)This has a simple interpretation – the final momentum distribution, φ , represents the detectionof one particle independently of how exactly φ P and φ D overlap . The coordinate space wave-function at time T and position L for this particle is, in analogy with (3), the space-time integralof φ ( p − p a ) : A a ( L, T ) = (cid:10) ν Da (cid:12)(cid:12) ν Pa ( L, T ) (cid:11) = (cid:90) dp √ π (cid:112) E a ( p ) φ ( p − p a ) e − iE a ( p ) T + ipL . (51)Hence, instead of propagating the produced state to the detected state, we bring produced anddetected states together and propagate them as one particle . For sharply peaked wave-packets, A a ( L, T ) ≈ (cid:90) dp √ π √ (cid:15) a φ ( p − p a ) e − i(cid:15) a T e − iv a ( p − p a ) T e ipL = 1 √ π √ (cid:15) a (cid:18) πσ x (cid:19) exp (cid:18) − i(cid:15) a T + ip a L − ( L − v a T ) σ x (cid:19) . (52)Relativistic particles with defined masses, such as ν a , obey Klein-Gordon equation . From(52), the Klein-Gordon current for an arbitrary particle produced as (cid:12)(cid:12) ν Pa (cid:11) and detected as (cid:12)(cid:12) ν Da (cid:11) after propagating a distance L during some time T is given by J a ( L, T ) = 2 p a | A a ( L, T ) | . (53)Together with the probability density, ρ a ( L, T ) = 2 (cid:15) a | A a ( L, T ) | , J a ( L, T ) satisfies the Klein-Gordon continuity equation, ∂∂T ρ a ( L, T ) + ∂∂L J a ( L, T ) = 0 . (54) This is a feature that every measurement formalism in QM must take care of. Think of the double-slitexperiment, there is a particle propagating as a wave, this wave partially reflects in the wall, partially goes throughone slit or the other, but when we get to measure the particle on the other side of the wall, it is the entire particleon the spot. It is a sharp "click" of the detector, and the intensity of the click does not depend on factors such ashow the wave overlaps with the slit. These factors become important in the many-particle cases, for predicting therate of clicks at a specific location. We write A a = (cid:10) ν Da (cid:12)(cid:12) ν Pa ( L, T ) (cid:11) to keep track that φ ( p − p a ) = φ P ( p − p a ) φ D ( p − p a ) , therefore the integralis a “correlated” inner product. In a sense, this is a time symmetric formulation [50, 51, 52]. As matter of fact, ν a obeys Dirac equation. Klein-Gordon density and currents are approximations of theirrespective Dirac counterparts when spinor degrees of freedom are ignored. The calculations in the following canbe reproduced without ignoring the spinors by using Gordon decomposition [53]. A a ( L, T ) is normalized in (51), however, is so that there is only one massive neutrino ν a in the wholespace at any given time: (cid:90) dL ρ a ( L, T ) = 1 . (55)Therefore, A a ( L, T ) satisfies two constraints, that we summarize:1. For the pointer interpretation of the detection process, we need a correlated normalizationof produced and detected envelopes in (49).2. For the probability interpretation of the neutrino wavefunction in a relativistic theory, weimpose that the number of neutrinos in all space at any given time is one, (55).Note that, with such a convention, ρ a has dimension of 1/ length as it should be in a one-particletheory and J a ∝ time . Therefore, integrating J a for the whole time of the experiment shouldgive us the probability that, after the detection or "click", the detector will register a particle ofindex a : P a ( L ) = (cid:90) dT J a ( L, T ) = 1 . (56)For the case of particles with superposition of masses, as flavor neutrinos, it is not straight-forward to find the corresponding (53) and (54) [54, 55, 56, 57]. But, in the next section, weuse the interpretation of “flavor wave-packets”, that comes from weak measurement and post-selection, to guess the form of their probability current. Presumably, after time integration, itmight give us the expression for the flavor oscillation probability. Neutrino oscillation Probability
Taking seriously the idea that the flavor neutrinos can be considered particles described by the“flavor wave-packets”, we guess the form of their probability current by analogy with eq. 53: J αβ ( L, T ) = 2 Re { p αβw }| A αβ ( L, T ) | , (57)with A αβ ( L, T ) = (cid:80) a U ∗ αa U βa A a ( L, T ) and Re { p αβw } in place of the average momentum of thewave-packet. It can be shown that J αβ obeys a continuity equation of the form: (cid:88) β (cid:18) ∂∂T ρ αβ ( L, T ) + ∂∂L J αβ ( L, T ) (cid:19) = 0 , (58)with ρ αβ ( L, T ) = 2 Re { (cid:15) αβw }| A αβ ( L, T ) | . Indeed, (57) and (58) can be derived from manipu-lating Klein-Gordon equation without ever referring to weak measurements. Thus, weak valuesspontaneously appear and highlight the underlying weak regime in the physics of mixed parti-cles . Different from (53), (57) describes a probability that is not conserved, in general, due toflavor transformations. This is the content of a future paper. P αβ ( (cid:126)L ) = (cid:90) dT (cid:90) S dA J αβ ( (cid:126)L, T ) . (59)Since we are working in just one dimension, this integral simplifies to P αβ ( L ) = (cid:90) dT J αβ ( L, T ) . (60)This is equivalent to (56) in the context of mixed particles, it gives the probability that thedetector will register index β after the "click". Substituting (57) into (60), we have P αβ ( L ) = (cid:90) dT { p αβw }| A αβ ( L, T ) | = 2 Re (cid:88) a,b U ∗ αa U βa U αb U ∗ βb (cid:90) dT (cid:10) ν Pb ( L, T ) | ν Db (cid:11) (cid:10) ν Da (cid:12)(cid:12) p (cid:12)(cid:12) ν Pa ( L, T ) (cid:11) . (61)Now, (cid:10) ν Da (cid:12)(cid:12) (cid:12)(cid:12) ν Pa ( L, T ) (cid:11) ≈ A a ( L, T ) as given by (52). For the second term in (61), (cid:10) ν Da (cid:12)(cid:12) p (cid:12)(cid:12) ν Pa ( L, T ) (cid:11) ≈ (cid:90) dp √ π √ (cid:15) a p φ ( p − p a ) e − i(cid:15) a T e − iv a ( p − p a ) T e ipL = (cid:18) p a + 2 i L − v a T σ x (cid:19) √ π √ (cid:15) a (cid:18) πσ x (cid:19) × exp (cid:18) − i(cid:15) a T + ip a L − ( L − v a T ) σ x (cid:19) , (62)or (cid:10) ν Da (cid:12)(cid:12) p (cid:12)(cid:12) ν Pa ( L, T ) (cid:11) ≈ (cid:18) p a + 2 i L − v a T σ x (cid:19) A a ( L, T ) . (63)Back to (61), we have P αβ ( L ) = 1 π (cid:18) πσ x (cid:19) Re (cid:26) (cid:88) a,b U ∗ αa U βa U αb U ∗ βb p a √ (cid:15) a √ (cid:15) b e i (1 − ξ ) ∆ m ab E L × (cid:90) dT exp (cid:20) − ( L − v a T ) + ( L − v b T ) σ x (cid:21) e − iξ ∆ m ab E T (cid:27) , (64)in which relations from (4) and (6) have been used . After integration, P αβ ( L ) ≈ Re (cid:26) (cid:88) a,b U ∗ αa U βa U αb U ∗ βb (cid:115) v a + v b p a √ (cid:15) a √ (cid:15) b e − i ∆ m abL E e − (cid:0) LLabcoh (cid:1) e − (∆ (cid:15)ab )28 σ e (cid:27) . (65) We neglect the integral involving the term i L − v a T σ x . P αβ is dimensionless, (cid:80) β P αβ ( L ) = 1 and there is no normalization problem(see (19)). When the dependence of p a , (cid:15) a and v a on the index a is negligible, (cid:115) v a + v b p a √ (cid:15) a √ (cid:15) b ≈ . (66)Moreover, the result of the summation in (65) is real, and therefore P αβ ( L ) ≈ (cid:88) a,b U ∗ αa U βa U αb U ∗ βb e − i ∆ m abL E e − (cid:0) LLabcoh (cid:1) e − (∆ (cid:15)ab )28 σ e . (67)Hence, the probability that the detector will register index β depends on the distance L betweenthe source and the detector as well as the energy E , computed for the case of massless neutrinos(see (4)). This formula, which is the basis of all neutrino oscillations phenomenology, has anoscillatory behavior, and its frequency is proportional to the squared mass differences, ∆ m ab .Here we derived it in a different route, appealing to the features of the detection process and themost basic principles of QM: the complementarity, through the uncertainty principle and themeasurement postulate. It is not exaggerated to say that the present theory addresses each ofthe issues that a complete quantum theory should present.Notice that, in case L >> L coh or ∆ (cid:15) ab >> σ e , the mass eigenstates can be resolved and themeasurement is strong by definition, implying the detection would collapse the wavefunctionof the single flavor neutrino to one of its mass eigenstates. In such a situation, the interferenceof the neutrino with itself is inaccessible, and the detection probability should be described bya statistical mixture of the mass eigenstates destroying the oscillatory pattern of (67).It is important to say that, in particle physics, the quantum one particle treatment is not ingeneral applicable. Elementary interactions involve the creation and annihilation of particles,and quantum field theory must be applied. In special cases of ultra-relativistic neutrinos inthe laboratory frame, with sharply peaked momentum distributions and scattering amplitudesthat are insensitive to the absolute masses, m a , the many contributions to the detection ratecan be factored, and a definition of oscillation probability makes sense [27, 28, 43]. In suchsituations, that happens to be the one of most practical interest, neutrino flavor states are usefulapproximations, and the theory of neutrino oscillations presents us with all the richness of QM. Conclusion
This work revolves around the idea that, in the relativistic limit, the features of neutrino de-tection in oscillation experiments are well described by the theory of weak measurements withpre- and post-selection. From this simple observation, everything else follows.On the one hand, the weak nature of the phenomenon enables us to reconcile the fact thatenergy-momentum and flavor are measured simultaneously during the neutrino detection even17hough, in the quantum mechanical model of neutrino oscillations, they are incompatible ob-servables. On the other hand, in analogy with the concept of weak values, we can describeflavor neutrinos as single particles with their own wave-packets. The “flavor wave-packet” is,then, the consequence of a highly non-trivial quantum interference effect that happens due toquantum uncertainties and post-selection. In mathematical physics, this interference effect iscalled superoscillations .With the wave-packet description of flavor neutrinos, we can treat them as single particlesin despite their superposition of masses. As relativistic particles, they obey a specific type ofKlein-Gordon continuity equation (by ignoring spin) and, therefore, they have an associatedprobability current. From the probability current, it is straightforward to calculate the time-independent flavor oscillation probability. The connection between the Klein-Gordon equationof motion and weak-values – two completely independent developments in quantum mechanics– is one of the main results of this paper.The previous quantum mechanical treatments of neutrinos oscillations are unsatisfactorydue to problems that come either from the use of plane waves or wave-packet treatments thatdo not address some of basic principles deep enough. The treatment in this present paper tellsthe narrative, step by step, of how a single neutrino interferes with itself.
Acknowledgements
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de NívelSuperior - Brasil (CAPES) - Finance Code 001. YPPS acknowledges support from FAPESPfunding Grants No. 2014/19164- 453 6, No. 2017/05515-0 and No. 2019/22961-9. MCOalso acknowledges support from CNPq. YPPS is thankful to O. L. G. Peres, M. E. Chaves,A. Y. Smirnov and E. Akhmedov for enlightening discussions on wave-packets and neutrinooscillations.
References [1] Y. Fukuda et al. Evidence for oscillation of atmospheric neutrinos.
Phys. Rev. Lett. ,81:1562–1567, 1998.[2] Q. R. Ahmad et al. Direct evidence for neutrino flavor transformation from neutral currentinteractions in the Sudbury Neutrino Observatory.
Phys. Rev. Lett. , 89:011301, 2002.[3] T. Araki et al. Measurement of neutrino oscillation with KamLAND: Evidence of spectraldistortion.
Phys. Rev. Lett. , 94:081801, 2005.[4] B. Pontecorvo. Mesonium and anti-mesonium.
Sov. Phys. JETP , 6:429, 1957. [Zh. Eksp.Teor. Fiz.33,549(1957)]. 185] V. N. Gribov and B. Pontecorvo. Neutrino astronomy and lepton charge.
Phys. Lett. ,28B:493, 1969.[6] Shalom Eliezer and Arthur R. Swift. Experimental Consequences of electron Neutrino-Muon-neutrino Mixing in Neutrino Beams.
Nucl. Phys. , B105:45–51, 1976.[7] Harald Fritzsch and Peter Minkowski. Vector-Like Weak Currents, Massive Neutrinos,and Neutrino Beam Oscillations.
Phys. Lett. , 62B:72–76, 1976.[8] Samoil M. Bilenky and B. Pontecorvo. Quark-Lepton Analogy and Neutrino Oscillations.
Phys. Lett. , 61B:248, 1976. [,248(1975)].[9] Boris Kayser. On the Quantum Mechanics of Neutrino Oscillation.
Phys. Rev. , D24:110,1981.[10] R. G. Winter. NEUTRINO OSCILLATION KINEMATICS.
Lett. Nuovo Cim. , 30:101–104, 1981.[11] Carlo Giunti and Chung W. Kim. Quantum mechanics of neutrino oscillations.
Found.Phys. Lett. , 14(3):213–229, 2001.[12] S. Nussinov. Solar Neutrinos and Neutrino Mixing.
Phys. Lett. , 63B:201–203, 1976.[13] C. Giunti, C. W. Kim, and U. W. Lee. When do neutrinos really oscillate?: Quantummechanics of neutrino oscillations.
Phys. Rev. , D44:3635–3640, 1991.[14] J. Rich. Quantum mechanics of neutrino oscillations.
Phys. Rev. D , 48:4318–4325, Nov1993.[15] C. Giunti and C. W. Kim. Coherence of neutrino oscillations in the wave packet approach.
Phys. Rev. , D58:017301, 1998.[16] Ken Kiers, Shmuel Nussinov, and Nathan Weiss. Coherence effects in neutrino oscilla-tions.
Phys. Rev. , D53:537–547, 1996.[17] Ken Kiers and Nathan Weiss. Neutrino oscillations in a model with a source and detector.
Phys. Rev. , D57:3091–3105, 1998.[18] Evgeny Kh. Akhmedov and Alexei Yu. Smirnov. Paradoxes of neutrino oscillations.
Phys.Atom. Nucl. , 72:1363–1381, 2009.[19] Evgeny Akhmedov. Do non-relativistic neutrinos oscillate?
JHEP , 07:070, 2017.[20] Evgeny Akhmedov. Quantum mechanics aspects and subtleties of neutrino oscillations. In
International Conference on History of the Neutrino: 1930-2018 Paris, France, September5-7, 2018 , 2019. 1921] I. Yu. Kobzarev, B. V. Martemyanov, L. B. Okun, and M. G. Shchepkin. Sum Rules forNeutrino Oscillations.
Sov. J. Nucl. Phys. , 35:708, 1982. [Yad. Fiz.35,1210(1982)].[22] C. Giunti, C. W. Kim, J. A. Lee, and U. W. Lee. On the treatment of neutrino oscillationswithout resort to weak eigenstates.
Phys. Rev. , D48:4310–4317, 1993.[23] M. Blasone and Giuseppe Vitiello. Quantum field theory of fermion mixing.
Annals Phys. ,244:283–311, 1995. [Erratum: Annals Phys.249,363(1996)].[24] W. Grimus and P. Stockinger. Real oscillations of virtual neutrinos.
Phys. Rev. , D54:3414–3419, 1996.[25] Mikael Beuthe. Oscillations of neutrinos and mesons in quantum field theory.
Phys. Rept. ,375:105–218, 2003.[26] M. Beuthe. Towards a unique formula for neutrino oscillations in vacuum.
Phys. Rev. ,D66:013003, 2002.[27] C. Giunti. Neutrino wave packets in quantum field theory.
JHEP , 11:017, 2002.[28] Evgeny Kh. Akhmedov and Joachim Kopp. Neutrino Oscillations: Quantum Mechanicsvs. Quantum Field Theory.
JHEP , 04:008, 2010. [Erratum: JHEP10,052(2013)].[29] Andrew Kobach, Aneesh V. Manohar, and John McGreevy. Neutrino Oscillation Mea-surements Computed in Quantum Field Theory.
Phys. Lett. , B783:59–75, 2018.[30] Jun John Sakurai.
Modern quantum mechanics; rev. ed.
Addison-Wesley, Reading, MA,1994.[31] E. Arthurs and J. L. Kelly. B.s.t.j. briefs: On the simultaneous measurement of a pair ofconjugate observables.
The Bell System Technical Journal , 44(4):725–729, April 1965.[32] Stan Gudder. Non-disturbance for fuzzy quantum measurements.
Fuzzy Sets and Systems ,155(1):18 – 25, 2005. Measures and conditioning.[33] Christopher A. Fuchs and Asher Peres. Quantum-state disturbance versus infor-mation gain: Uncertainty relations for quantum information.
Physical Review A ,53(4):2038–2045, Apr 1996.[34] Yakir Aharonov, David Z. Albert, and Lev Vaidman. How the result of a measurementof a component of the spin of a spin-1/2 particle can turn out to be 100.
Phys. Rev. Lett. ,60:1351–1354, Apr 1988.[35] Yakir Aharonov and Lev Vaidman. Properties of a quantum system during the time intervalbetween two measurements.
Phys. Rev. A , 41:11–20, Jan 1990.2036] Lars M Johansen. What is the value of an observable between pre- and postselection?
Physics Letters A , 322(5-6):298–300, Mar 2004.[37] Justin Dressel, Mehul Malik, Filippo M. Miatto, Andrew N. Jordan, and Robert W. Boyd.Colloquium: Understanding quantum weak values: Basics and applications.
Reviews ofModern Physics , 86(1):307–316, Mar 2014.[38] Yutaka Shikano. Theory of "weak value" and quantum mechanical measurements, 2011.[39] Lupei Qin, Wei Feng, and Xin-Qi Li. Simple understanding of quantum weak values.
Scientific Reports , 6(1), Feb 2016.[40] M. V. Berry, N. Brunner, S. Popescu, and P. Shukla. Can apparent superluminal neutrinospeeds be explained as a quantum weak measurement?
J. Phys. , A44:492001, 2011.[41] Shogo Tanimura. Apparent Superluminal Muon-neutrino Velocity as a Manifestation ofWeak Value. 2011.[42] H. Minakata and A. Yu. Smirnov. Neutrino Velocity and Neutrino Oscillations.
Phys.Rev. , D85:113006, 2012.[43] Carlo Giunti and Chung W. Kim.
Fundamentals of Neutrino Physics and Astrophysics .Oxford University Press, mar 2007.[44] John Von Neumann and Nicholas A Wheeler.
Mathematical foundations of quantum me-chanics; New ed.
Princeton University Press, Princeton, NJ, Mar 2018.[45] Yakir Aharonov and Daniel Rohrlich. Quantum paradoxes: Quantum theory for the per-plexed.
Quantum Paradoxes: Quantum Theory for the Perplexed, by Yakir Aharonov,Daniel Rohrlich, pp. 299. ISBN 3-527-40391-4. Wiley-VCH , September 2003. , 09 2003.[46] Yakir Aharonov and Alonso Botero. Quantum averages of weak values.
Physical ReviewA , 72(5), Nov 2005.[47] Lev Vaidman, Alon Ben-Israel, Jan Dziewior, Lukas Knips, Mira Weißl, Jasmin Meinecke,Christian Schwemmer, Ran Ber, and Harald Weinfurter. Weak value beyond conditionalexpectation value of the pointer readings.
Physical Review A , 96(3), Sep 2017.[48] Y Aharonov, F Colombo, I Sabadini, D C Struppa, and J Tollaksen. Some mathemati-cal properties of superoscillations.
Journal of Physics A: Mathematical and Theoretical ,44(36):365304, aug 2011.[49] M V Berry and Pragya Shukla. Pointer supershifts and superoscillations in weak measure-ments.
Journal of Physics A: Mathematical and Theoretical , 45(1):015301, nov 2011.2150] Yakir Aharonov, Peter G. Bergmann, and Joel L. Lebowitz. Time symmetry in the quan-tum process of measurement.
Phys. Rev. , 134:B1410–B1416, Jun 1964.[51] B. Reznik and Y. Aharonov. Time-symmetric formulation of quantum mechanics.
Physi-cal Review A , 52(4):2538–2550, Oct 1995.[52] Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen. A time-symmetric formulation ofquantum mechanics.
Physics Today , 63:27–, 11 2010.[53] J.J. Sakurai.
Advanced Quantum Mechanics . Addison-Wesley Series in AdvancedPhysics. Addison-Wesley, 1987.[54] Marek Zralek. From kaons to neutrinos: Quantum mechanics of particle oscillations.
ActaPhys. Polon. , B29:3925–3956, 1998.[55] B. Ancochea, A. Bramon, R. Munoz-Tapia, and M. Nowakowski. Space dependent prob-abilities for K0 - anti-K0 oscillations.
Phys. Lett. , B389:149–156, 1996.[56] Massimo Blasone, Petr Jizba, and Giuseppe Vitiello. Currents and charges for mixedfields.
Phys. Lett. , B517:471–475, 2001.[57] Massimo Blasone, Paulo Pires Pacheco, and Hok Wan Chan Tseung. Neutrino oscillationsfrom relativistic flavor currents.