 Journal of Modern Physics, 2015, 6, 1942-1949 Published Online October 2015 in SciRes. http://www.scirp.org/journal/jmp http://dx.doi.org/10.4236/jmp.2015.613200 How to cite this paper: Goryachev, B.I. (2015) The Model of Neutrino Vacuum Flavour Oscillations and Quantum Mechanics. Journal of Modern Physics, 6, 1942-1949. http://dx.d oi.org/10. 4236/jm p .2015. 613200 The Model of Neutrino Vacuum Flavour Oscillations and Quantum Mechanics Boris I. Goryachev Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia Email: bigor@srd.sinp. msu.ru Received 28 July 2015; accepted 26 October 2015; published 29 October 2015 Copyright © 2015 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract It is shown that the model of vacuum flavour oscillations is in disagreement with quantum me- chanics theorems and postulates. Features of the model are analyzed. It is noted that apart from the number of mixed mass states neutrino oscillations is forbidden by Fock-Krylov theorem. A possible reason of oscillation model inadequacy is discussed. Keywords Maximal State Mixing, Fock-Krylov Theorem, Superposition Principle, Superselection Rule 1. Introduction For a number of decad es searc h for neutrino oscillations is the basic direction in the field of massive neutrino expe- rimental phy sics. Idea of these oscillations caused by transitions was advanced on the analogy with oscillations [1] and later was exte nded to oscillati ons of neutrino with differe nt flavours [2] [3]. This type of oscillations is of vital importance for the attempts to explain solar neutrino deficit, and almost a ll ex- periments in this field are directed to the search for flavour neutrino oscillations. Hereinafter precisely these oscilla- tions will be regarded as neutrino os cillations. Neutrino oscillations hypothesis is based on the assumption that flavour neutrino states being weak interac- tions Hamiltonian eige nstates, are not ei genstates of mass operator and can be obtained by m ixing the latter: , (1) where U-unitary m ixing m atrix. The number of mixed mass states in (1) is equal to the number of interaction states. Usually, two variants: 2f- and  B. I. Goryachev 3f-oscillations, are used for the experimental data analysis. The latter variant takes into account the full number of possible flavours (three). As is known mixing (1) results in the transitions from one flavour to another, so observa- tion of neutrino oscillati ons would mean flavour le pton numbers noncons ervation. In spite of prevailing hopefulness in the evaluation of the possibilities of the neutrino oscillation model, a number of questions related to such eval uation still remai n. Foremost two of them shoul d be marked out: 1) Why do none of “direct” experiments, which do not search for neutrino oscillations, discovers nonconservation of the lepton flavour num ber? 2) Why is the mixing in qua rk sector significantly smaller than the suppos ed mixing of neutrino stat es? In the majority of the studies parameters of the neutrino oscillations models are evaluated according to the existed experimental data. This way is associated with considerable difficulties. In particular, the events, qualified by expe- rimenters as desired ones , can be simulat e d by bac kground proc e s ses. Therefore, the problem of studying of the neu- trino oscillations model basing on the quantum mechanics theorems, not taking into account experimental data, is of particular interest. The present report concerns precisely this problem . The analysis below is based on the theorems true for isolated systems. Therefore, conclusions obtained in the present study concern vac uum neutrino oscillati ons. Effects of the interaction with matter [4] [5] are out of the questi on. As usually, neutrinos are considered as stable particles. Section 2 includes some specific characteristics of the vacuum neutrino oscillations model (in 2f- and 3f-variants), which lead to the limits for the selection of the m odel parameters numbe r. Consistency of the oscillat ion m odel and Fock-Krylov the orem [6] along with specific c haracters of the mi ssing of neutrino mass states are analyzed in Section 3. Appendix includes the notations for the standard mixing matrix ele- ments, used in the present report due to their spelling conve nience. 2. Specific Characteristics of the Vacuum Neutrino Oscillations Model 2.1. 2f-Oscillations In the case of two fla vours, we will designat e inte raction st ate, gene rated i n the source at t = 0, as , and the state missing from the initial neutrino flux at t = 0 as . In respect to the solar neutrino , and muon neutrino can occur as . For the long-baseline accelerator experiments, the following identification is possible: and . Then, acc ording to (1) and agre ed notations (see Appendix) (2a) , (2b ) where θ is mixing angle. Mass states can be expressed in the foll owing manner (herei nafter ħ = c = 1) (3) where -ei genstate of the moment um operator wi th ei genvalue pi, and εi-total energy of state. Mass states are the solutions of the wave equation and are described by plane waves. If neutrino moves, for instance, along z-a xis, t hat: () 22 ~ exp ii ii ipzpm t ν −+ (4) As applied to the experiments on the search for neutrino oscillations t value means time of neutrino transit from the source t o the detector. As usually, wave functions are normalized in the appropriate volume and system of these functions is orthonormal. So if pi are different, during calculation of matrix elements cross terms formed by not coincident mass states will become zer o, and it will lead to zero oscill ations. As is known these oscillations appe ar due to inte rference of different mass states in (1) (for instance, see [7]) and restriction (5)  B. I. Goryachev is the necessary criterion for t he neutrino vacuum oscillati ons model . Selecti on of C-const ant in (3) allows to provide “correct” initial conditions c oncerning intensities of the fluxes of neut rino with different flavours. We will express the norms Nf of and states and means of physical quantities F for these states according to the rules (6) and , (7) where is an operator of the given quantity F. For Nf (t) the following formulas are true: ()()( ) 2 1sin 2cos f Nt Ct ω =±Θ , (8) where plus and minus signs are attributed to a- and b-states, respectively. In the case of relativistic neutrino (9) According to (5), (7) and (8) mean val ue of momentum is given by the expression , (10) that come out from (5). For the problem at hand momentum is an integral of motion. For the calculation of the mean values of energy (for free particles ) condition (5), which leads to the appearance of cross terms, is responsible for com plex value of energy (11) for any time t (except , etc.; these values are exclude d from consideration up to Secti on 3). For relativistic neutrino ( )()() 22 221 1212 01cos 21sin2cos 22 2r mm mm EReEpt p θ θω − +− ≡=+ ±± (1 2) and ( ) ( ) ( )()( ) 1 22 12 sin 2sin1sin 2cos 4rr Im Emmtt p θω θω − =−± (13) For plus-m inus signs the upper and l ower signs are attributed to a- and b-states , respe ctivel y. The sam e note is true for the formulas (15), (19), (20), (22)-(24) along with (26) and (28). Complex energy embodies the continuity t he energy E distribution f or the flavour states at hand (1). We will show that value is e qual to the half -width Γf (in terms of spectroscopy) of states, which is caused by transitions between a- and b-states. Let’s analyze the number of flavour states, proportional to Nf in (8). Then according to definiti on from [8] } { d Γnumber of transitions per unit timed f ff N Nt = ≡ (14) Indeed, taking into consi deration (8), (9) and (14) we obtain ( ) ( ) ( ) ( )()( ) 1 22 12 sin 2 Γddsin1sin 2cos 2 ff frr NtNm mtt p θω θω − ==−± (15) and comparing (15) and (13), we ha ve (16)  B. I. Goryachev The width of the state Γf i ndicates the measure of unc ertainty for the value of energy (mass) of the flavour ne utrino. Energy defines “center of gravity” of the energy distribution of neutrino and can be identified with the mean energy of this distribution. Hereinafter we’ll consider mean energy of the flavour neutrino as E0 value, tak- ing into account that for the problem at hand these neut rinos are emitted with the same momentum p. For the oscillation model Γf width also oscillates, because as can be seen from (8) and (14) during different time periods t each state and is mostly either “filled” or “cleaned”. Usually during parametrization of the experimental data within the frames of the oscillation model and 2f-variant two parameters are found-mixing angle θ and value . However, such a number of pa rameters are redundant. Indeed, formula (12) shows that E0 varies with time, while quantum mechanics canons require mean energy to be conserved in the isolated system. Fixing of E0 is possible by setting , which means that condition of max- imal mixing of neutrino mass states is realized. In the case of maximal mixi ng formulas (8), (12) and (13) a re simpli fied in the following ma nner: ( )() 2 1 cos fr Nt Ct ω = ± , (17) (18) and ( )( ) 22 1 12 sin1 cos 4 rr mm Im Ett p ωω − − = ± . (19) Besides the maxim al mixing, in orde r t o obtain the “c orrect” i nitial conditi ons (20) according to (17) it is necessary to set (21) Conditions (20) mean, that at t = 0 only a-states should be “filled”. Due to relation (5) the present problem can be analyzed in the in trinsic frame of r eference of neutr ino (p = 0). Thereby relations (8), (12) and (13) lead to the fol- lowing: ( )() ( ) 20 1sin2cos,12 , f Nt CtC θω =±= (22) ( )()( ) ( ) 1 12 12 00 cos 21sin2s 22 ,co mm mm Et θ θω − +− =+± (23) ( ) ( ) ( ) ( ) 1 12 00 sin 2sin1sin 2co,s 2 mm Im Ett θω θω − − = ± (24) where (25) It should be stressed that in any f rame of axes the vac uum oscillations m odel should enforce two conditions: 1) Conservation of the mean energy of the free particles (neutrino), which leads to the necessity for the maximal mixing in (1); 2) Initial conditions, which mean that only one type of ne utrino should oc cur in frames of flavour s tates at t = 0. Fulfillment of the condition 2) is taken ou t in 2f-variant b y means of s electio n of normalization constant (21) and by meeting 1) requirement. Taking into account 1) and 2) in the intrinsic frames of reference of neutrino we obtain ( ) ( ) 20 1cos,12 , f Nt CtC ω =±= (26) (27)  B. I. Goryachev ( )( ) 1 12 00 sinc s 2.1o mm Im Ett ωω − − = ± (28) As seen from (17) and (26) mee ting the requir ements 1) and 2) leads to the time dependen ce of Nf(t), coinciden t with harmonic oscillations. So the model of the vacuum flavor oscillations is associated with availability of some neutrinos “ inner clock”, which define the rate of transitions from one flavour state into another. Corresponding cyc- lic frequen cies ω in different moving frames of reference should be related by Lorentz transfor mations, and it lea ds to the relation: , (2 9) where γ is Lorentz factor of relativisti c neutr i no. Acc ording to (9) and (25) this relation is realized, if the value (m1 + m2)/2, cor respondent to t he m aximal m ixing m ode , is t aken as e ffective mas s of acti ve ne utrino. In interpretation of experimental data within the frame of oscillation model one usually calculates probabilities of the transitions and , where t is time of the neutrino flight from the source to the detector. But in the experiment it is possible to measure only the detection probability for neutrino with given flavour. Under this approach it is necessary to know flavour composition of neutrino, generated by the source at t = 0 . Usual ly this composition is given “ma nually” on the assumption wit h lepton numbers’ conservat ion. In the present report, the norms of state vectors Nf(t) which contain all necessary information for comparison with the data of the experiments mentioned above are calculated. It is easy to show that under realization of 1) and 2) re- quirements the re sults, obtained by bot h approa ches, are t he same . 2.2. 3f-Oscillations In the most of studies 2f-variant is used for parametrization of the experimental data within the frames of the oscilla- tion model. Section 2.2 contains analysis of the problem, to what extent maximal mixing of mass states (i.e. ua1 = ua2 = ua3) allows t o enforce natural physical requi reme nts 1) and 2) in 3f-variant. We’ll calculate in the neutrino intr insic fra me of reference. Matrix (A.2) will be used as mixing matrix (see Appendix). The followi ng values can be selected: 2 233 1 3,23,1 2,S CSC === = (30) which lead to the “correct ” initial conditions ( )( )( ) 01, 00, 00. ab c NNN= == (31) These conditions do not depend on the selection of S1 (and C1, respectiv ely), if values ( 30) are fixed. So the form of the m ixing mat rix in 3f-variant can vary essentially. We’ll give two examples of these matrices, associated to the maximal mixing of mass states. 1) , (32a) (32b) (32c) 2) , (33a) ()( ) 1 23 31 2331 2313 b bb u uu=−+ =−= (33b) ( )() 12 3 31 2331 2313 cc c uu u=− =−+= (33c) Matrices (32) and (33) result in the same expression for the norm Na(t), ( )( ) () ( ) 212 1323 2 1 coscoscos 3 a NtCm mtm mtmmt = +−+−+− (34)  B. I. Goryachev , (35) where, as is known, only two values of three (mi − mk) are ind epen den t. I t w as alr ead y s tre s sed in (31), that Na (0) = 1. Really, it runs from (34) and (35). Here expressions for Nb (0) and Nc (0) are discarded. Further, for the maximal mixi ng we obtain the follow ing relation (for state) ( ) ()() ( ) ( )() ( ) ()() 123 123 012 1323 3 312213123 1 12 1323 1coscos cos 3 33 cos coscoscos 2 1 coscoscos 3 mmm mmm Em mtm mtmmt mmm mtmmmtmmmt mmt mmt mmt − ++ ++ =+−+ −+− − −+−+− ×+−+−+− (36) At t = 0 (37) and it corresponds to the value (27) in the case of 2f-variant. The structure of the formula (36) for E0 shows, t hat this value is not a motion i ntegral and varies wit h increasing of t (time of the particles’ flight from the sourc e to the detector). So, 3f-variant in the maximal mixing mode does not allow simultaneous reali zation of 1) and 2) requirements. Expression for the imaginary com ponent of the state’s energy in 3f-variant is as follows: () () ()()() () ( ) ()( ) 1212 13132323 1 12 1323 1sin sinsin 3 2 1 coscoscos 3 ImEmm mmtmm mmtmm mmt mmt mmtmmt − =−−−+−− +−− ×+−+−+− (38) As is seen from (38) at t = 0 the value . Most often for parametrization of the experimental data mixing matrix of 3f-variant, reduced to the corresponding matrix of 2f-variant, is used (see Appendi x). As it was shown in Section 2.1, in this case the maxim al mixing of mass states allows the realizati on of both requirement s 1) and 2). 3. The Neutrino Oscillations Model and the Quantum Theory Canons Performed in the previous section analysis is of methodological character. The problem of to what extent the model of the vacuum neutrino oscillations is reasonable within the as regards to the quantum mechanics, is the subject of the pres ent sectio n. Here it is conven ient to ap ply the theorem by Fock-Krylov [6] (see also [9]). According to this theorem for arbitrary (incl uding quasi-stationary) state of the isolated system the foll owing relation is t rue: ( )()() 2 exp d aa LtiEt WEE= − ∫ (39) where Wa (E) is distribution func ti on for e ne rgy E in state at t = 0, and La (t) is probability that by the time t the system is still at the state. Function Wa (E) is obviously normalized (40) For the problem at hand the sta tes and are the sam e. Vanishing of the spectroscopic width of the state as time tends to t = 0 (see relations (19), (28), (38)) is the distinctive feature of the oscillati on model. As a conseque nce the following relati on is true: , (41) here is the Dirac delta -function. From (41) it follows that (42)  B. I. Goryachev Formu la (42) holds throughout t. In other words, initial state does not vary with time. It means that transi- tions into other states with flavours and, respecti vely, flavour oscilla tions are not avail able. The following property can be taken as the crucial point of the oscillation model: mixing states are the states of Hermitean Hamiltonians, but the resulting state, in general, corresponds to non-Hermitean Hamiltonian. Es- sential points of the problem of neutrino oscillations are convenient to be considered within the intrinsic frame of reference of neutrino. Therefore model’s difficulty mentioned above can be qualified as paradox of “mixing by mass”, which can be hardly referred to any physical meaning. The importance of using of Hermitean Hamiltonians in quantum mechanics is well known. In particular, the authors of [10] note that in the field of radioactive decay and scattering of the atomic nuclei and particles all theorems on an expansion of arbitrary function in eigenfunctions, which form a complete system, belong to the set of wave functions for the real eigenvalues of energy E. Functions of state, corresponding to complex values of E, are excluded from this system of eigenfunctions. We can ignore the imaginary component of the energy E for long-lived particles and nuclei, but within the frames of the problem of neutrino oscillations this component is essential, because it defines the rate of flavour oscillations. One can guess that the mentioned paradox is caused by using of the states superposition principle (see (1)) . It is known, that superpos ition of the states with diffe rent values of total electric ch arge, for instance, are physical- ly unrealized. In the field of quantum theor y it is embodied in the rules of superselection [11]. In the field of classical physics the electric charge and the mass of the particle act as charges for electromag- netic and gravitational inter a ctions, respectively. Besides, for the stable elementary particles, to which neutrino belongs, group-theoretical approach (within the frames of quantum-mechanical Poincare group concept) predicts particular value of the elementary particle mass. In view of the above no tes the model, assuming coincidence of the interaction states and neutrino mass states, has a major appeal in comparison to the oscillation model due to the clearance of the former from the paradox of “mixing by mass”. Within the frames of this model lepton numbers of neutrino conserve. 4. Conclusions The model of vacuum flavour oscillations is not in agreement with the theorems and postulates of quantum mechan- ics. Even within the frames of two flavour s “c orrect” initia l c onditions a nd requi reme nts of neutrino energy t o be, mo- tion integral is true only in case of ma ximal mixing of ma ss states. Without regard to the numbe r of such states (i.e. to i), neutrino oscillations are forbi dden by Fock-Krylov theorem. Although mixing states are described by ordinary plane waves, Hamiltonian of the resulting state is non-Hermi- tean. This paradox ca n’t be e xplaine d within the frames of quant um mec ha nics ca nons. B ut it can be al s o resolved as neutrino oscillations, if the rules of superselecti on are taken into ac count. Acknowledgements The author appreciates greatly A. V. Grigoriev and S. I. Svertilov for the fruitful discussion of the article. References [1] Pontecorvo, B. (1957) Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 34, 247-249, [1958, Soviet Physics—JETP, 7, 172-173]. [2] Maki, Z., Nakagawa, M. and Sakata, S. (1962) Progress of Theoretical Physics, 28, 870-880. http://dx.doi.org/10.1143/PTP.28.870 [3] Gribov, V. and Pontecorvo, B. (1969) Physics Letters B, 28 493-496. http://dx.doi.org/10.1016/0370-2693(69)90525-5 [4] Wolfenstein, L. (1978) Physical Review D, 17, 2369. http://dx.doi.org/10.1103/PhysRevD.17.2369 [5] Mikheev, S. and Smirnov, A. (1985) Yadernaya Fizika, 42, 1441-1448, [Soviet Journal of Nuclear Physics, 42, 913- 917]. [6] Fock, V. and Krylov, S. (1947) Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 17, 93-107. [7] Kayser, B. (1981) Physical Review D, 24 110. http://dx.doi.org/10.1103/PhysRevD.24.110  B. I. Goryachev [8] Blatt, J. and Weisskopf, V. (1952) Theoretical Nuclear Physics. Wiley & Sons Inc., New York, London. [9] Davydov, A. (1965) Quantum Mechanics. Pergamon Press, New York. [10] Baz, A., Zel’dovich, Ya. and Perelomov, A. (1969) Scattering, Reactions and Decay, in Nonrelativistic Quantum Me- chanics. Jerusalem. [11] Wick, G., Wigner , E. and Wightman, A. (1952) Physical Review, 88, 101-105; (1970) Physical Review D, 1, 3267- 3269. Appendix Usually within 3f-variant mixing matrix U with foll owing designations is used: 1131221312313 aaa ucc ucsus=== (A.1.1) 123 1223 13 12223 1223 13 12323 13b bb ucs ssc ucc sssusc=−−= −= (A .1.2) 1231223 13 122231223 13 12323 13cc c usscscusccss ucc= −=−−= (A .1.3) Here, symbols s and c mean the sine and c osine functions, respecti vely. Matrix (A.1) corresponds t o the case of CP-invariance conservati on. In this article , more conve nient and compact expre ssions for mixing m atrix elements are used: 1 232233 2aaa ucc ucs us== = (A.2.1) 113 123213 123312b bb ucssscuccsss usc=−−= −= (A.2.2) 113123213 123312cc c usscscusccss ucc= −=−−= (A.2.3) Matrix (A.2) can be reduced to the mixing matrix of 2f-variant, if we set (i.e. and ) and ( and ). In this case, only one mixing angle remains (see formulas 2a and 2b).
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