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Neutrino oscillations: A relativistic example of a two-level system Elisabetta Sassaroli Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Department of Physics, Northeastern University, Boston, Massachusetts 02115 共Received 30 April 1998; accepted 3 March 1999兲 Neutrino flavor oscillations are discussed in terms of an explicit model. This model consists of two coupled Dirac equations with three parameters: the electron neutrino mass, the muon neutrino mass, and a coefficient which describes the possibility that the neutrino can flip flavor. The system is diagonalized to obtain the exact eigenvalues and eigenfunctions. The system is then quantized and the neutrino flavor wave functions are derived directly from the quantized fields. It is shown that neutrino flavor oscillation probabilities are recovered in a quantum field theory treatment only in the ultrarelativistic limit. © 1999 American Association of Physics Teachers. I. INTRODUCTION Neutrino flavor oscillations are a relativistic example of a two-level system. In his Lectures on Physics, Feynman1 describes many examples in which the approximation of a twostate system can be assumed. Some of his examples include the ammonia molecule, the hydrogen molecule, a spin 1/2 particle in a magnetic field, and oscillations of strangeness in the neutral K meson system. For example, the ammonia molecule (NH3) has the form of a pyramid with the nitrogen atom located above the plane of the three hydrogen atoms. Like any other, this molecule has an infinite number of states; however in the two-level system approximation, it is assumed that all the states remain fixed except for two: the nitrogen may be on one side of the plane of the hydrogen atom or on the other. The system can be described by the state vector 兩典, 兩 典 ⫽C 1 兩 1 典 ⫹C 2 兩 2 典 , 共1兲 where in the state 兩1典 the nitrogen is ‘‘up’’ and in the state 兩2典 the nitrogen is ‘‘down.’’ The coefficient C 1 ⫽ 具 1 兩 典 is the amplitude to be in state 兩1典 and C 2 ⫽ 具 2 兩 典 the amplitude to be in state 兩2典. The coefficients C 1,2 are obtained by diagonalizing the two coupled differential equations i dC 1 ⫽H 11C 1 ⫹H 12C 2 , dt dC 2 ⫽H 21C 1 ⫹H 22C 2 , i dt 共2兲 with H i j being the Hamiltonian matrix, which depends on the particular system which is studied. Feynman solved and discussed in great detail the set of equations given by Eq. 共2兲 for the ammonia molecule case and also applied these equations to the ammonia maser. In this paper we will to consider a relativistic generalization of the above set of two coupled differential equations to the neutrino flavor oscillation case. Neutrinos are relativistic noncharged particles of spin 1/2, which are produced in weak interaction processes.2,3 In the electroweak theory of Glashow, Salam, and Weinberg 共GSW兲, neutrinos are massless and they can exist in three different flavors: the electron, muon, and tau flavors. However as far as we know, there is no deep theoretical reason why neutrino masses, i.e., their rest energies, should be ex869 Am. J. Phys. 67 共10兲, October 1999 actly zero. Moreover neutrinos have masses in most extensions of the GSW theory. It is therefore extremely important to investigate experimentally and theoretically neutrino masses. The experimental investigation of the allowed direct and inverse  decay 共see, for example, Ref. 4 for more details兲 p→n⫹e ⫹ ⫹ e , ¯ e ⫹ p→n⫹e ⫹ , e ⫹n→p⫹e ⫺ , and forbidden processes e ⫹ p→n⫹e ⫹ , ¯ e ⫹n→p⫹e ⫺ , can be conveniently described by introducing an electron flavor quantum number. The electron e ⫺ and its neutrino ¯ e are assigned the value L e ⫽1, the positron e ⫹ and the antineutrino ¯ e are assumed to have L e ⫽⫺1, while all the remaining particles 共for example, the proton p and the neutron n in the above reactions兲 are assigned the value L e ⫽0. The electron flavor number is conserved in the allowed processes 共the total electron flavor number on the left-hand side of the reaction is equal to the total electron flavor number on the right-hand side兲, while the forbidden ones would violate this conservation law. Moreover the studies of processes such as, for example, ⫹→ ⫹⫹ , ⫺ →e ⫺ ⫹ ⫹¯ e , ¯ ⫹ p→n⫹ ⫹ , and the fact that processes of the type →e⫹ ␥ , ⫹ p→n⫹e ⫹ , are not observed experimentally lead to the introduction of the muon flavor quantum number L ⫽1 for muon ⫺ and its neutrino , L ⫽⫺1 is for to ⫹ and ¯ , and L ⫽0 for all the remaining particles. The tau leptons ( ⫺ , ⫹ , ,¯ ) are described with the help of the tau flavor number L . All the examples considered above satisfy the conservation of both lepton numbers L e and L separately 共the total lepton number on the left-hand side of the reaction is equal to the total lepton number on the right-hand side兲. Because © 1999 American Association of Physics Teachers 869 the origin of these quantum numbers is not clear, there is the possibility that they are only approximately conserved. If so, then a neutrino produced in a given flavor can transform into another flavor while propagating, as will be discussed in Sec. II. If this is the case, the phenomenon of neutrino flavor oscillations can arise in nature. Recent experiments strongly suggest the evidence of neutrino oscillations.5 More specifically, neutrino flavor oscillations can be a possible explanation of the atmospheric neutrino anomaly measured by three different experiments6–8 of the solar neutrino deficit observed by four different experiments,9–13 and of the evidence of neutrino masses obtained by the liquid scintillator nuclear detector 共LSND兲 experiment.14 Neutrinos are also considered to make a small contribution to the dark matter of the universe 共hot dark matter兲. We point out here that the properties of neutrinos 共electric charge zero, mass very small, maybe zero兲 make their detection very difficult. A typical interaction cross section between an electron neutrino and a nucleus at 1 MeV is of the order of 10⫺43 – 10⫺44 cm2. In this paper we are going to study neutrino oscillations as an example of a two-level system. The two coupled differential equations, which describe two-level systems, are valid in the rest frame of the system investigated. However, since neutrinos are relativistic particles of spin 1/2, it is important to write a relativistically invariant generalization of Eq. 共2兲. The simplest generalization of Eq. 共2兲 which is relativistically invariant and properly takes into account the neutrino spin is a system of two coupled Dirac equations with three parameters: the electron neutrino mass, the muon neutrino mass, and a coefficient which describes the possibility that the neutrino can flip its flavor, as will be discussed in Sec. III. 共For simplicity we are only investigating a two-flavor model兲. The system is diagonalized and the energy eigenvalues and eigenfunctions are obtained. These solutions represent an interesting new piece of relativistic quantum mechanics. Contrary to the nonrelativistic case where the diagonalization of the two coupled equations above gives directly the probability amplitudes for flipping from one state to the other, in a relativistic system, in order to deal properly with the states of negative energies we have to abandon the one-particle picture and adopt a many-particle formulation, e.g., quantum field theory. The solutions of the two-coupled Dirac equations are quantized according to the usual Jordan–Wigner anticommutation relations and the neutrino energy wave functions are obtained as matrix elements of the quantized Dirac fields, as described in Sec. IV. These wave functions describe neutrinos of given energy and in a state of mixed flavor at any space–time point. It is also shown that their sum describes neutrinos in a state of mixed flavor. Therefore it is not possible with Dirac fields to impose the boundary condition to have a given flavor for all the space points at a given time, let’s say at production. Hence, Dirac fields cannot properly describe neutrino oscillations, where it is assumed that the neutrino is produced in a state of given flavor and then subsequently oscillates in flavor. Neutrino oscillation probabilities can be recovered only if the following two approximations are made on the neutrino flavor wave function: 共i兲 the left-handed chiral component of the flavor wave function is considered as an observable wave function; 共ii兲 the ultrarelativistic approximation is assumed for the spinor component of the wave function. Approximation 共i兲 is due to the fact that neutrinos are 870 Am. J. Phys., Vol. 67, No. 10, October 1999 produced only through the weak interaction, which does not conserve parity. Approximation 共ii兲 appears to be related to the impossibility of simultaneously maintaining Lorentz invariance and obtaining standard neutrino oscillation probabilities in a way which is consistent with relativistic field theory. This problem is obviously a deep one and is associated with the possibility that neutrinos violate Lorentz invariance as well as the equivalence principle as discussed in Refs. 15–17. II. NEUTRINO OSCILLATIONS: STANDARD TREATMENT We will first review the standard quantum mechanical treatment of neutrino flavor oscillations. Suppose for example that a muon neutrino is produced in the reaction ⫹→ ⫹⫹ , and while it is propagating has a probability of flipping its flavor and becoming an electron neutrino, in the same way as it is possible for the nitrogen atom in the ammonia molecule to push its way through the three hydrogen atoms and flip to the other side, due to a quantum tunneling effect. The system can be described by a state vector 兩典 as a linear combination of the flavor eigenstates 兩 e 典 and 兩 典 , 兩 典 ⫽C e 兩 e 典 ⫹C 兩 典 , 兩典⫽ 共3兲 冉 冊 Ce , C 共4兲 with C e ⫽ 具 e 兩 典 , C ⫽ 具 兩 典 , and 兩 C e 兩 2 ⫹ 兩 C 兩 2 ⫽1. 共5兲 C e and C then become the amplitudes for detecting an electron neutrino and a muon neutrino, respectively. In analogy, in the neutral K meson system the oscillations occur between states of different strangeness 兩 K 0 典 (S⫽1) and 兩 K̄ 0 典 (S ⫽⫺1). To derive the time evolution of the coefficients C e (t) and C (t), the state vector 兩典 is written as a superposition of the energy 共mass兲 eigenstates 兩 I典 and 兩 II典 , 兩 典 ⫽C I兩 1 典 ⫹C II兩 2 典 , 兩典⫽ 共6兲 冉 冊 CI , C II 共7兲 with C 1 ⫽ 具 I兩 典 , C 2 ⫽ 具 II兩 典 , and 兩 C I兩 2 ⫹ 兩 C II兩 2 ⫽1, 共8兲 where C I and C II are the amplitudes for finding the neutrino in the energy states E 1 and E 2 , respectively. These coefficients evolve in time as C I共 t 兲 ⫽C I共 0 兲 e ⫺iE 1 t , C II共 t 兲 ⫽C II共 0 兲 e ⫺iE 2 t . 共9兲 Introducing the rotation matrix between flavor and mass eigenstates 冉 典典 冊 冉 cos共 兲 兩I ⫽ 兩 II sin共 兲 ⫺sin共 兲 cos共 兲 冊冉 冊 兩 e典 , 兩 典 共10兲 it is easy to see that the following relation between the energy and flavor amplitudes holds: Elisabetta Sassaroli 870 冉 冊冉 cos共 兲 C I共 t 兲 ⫽ C II共 t 兲 sin共 兲 ⫺sin共 兲 cos共 兲 冊冉 冊 C e共 t 兲 . C 共 t 兲 共11兲 Hence, the time evolution of the coefficients C e and C is given by 冉 冊冉 cos共 兲 C e共 t 兲 ⫽ C 共 t 兲 ⫺sin共 兲 sin共 兲 cos共 兲 冊冉 冊 C I共 0 兲 e ⫺iE 1 t . C II共 0 兲 e ⫺iE 2 t 共12兲 In Eq. 共12兲 the boundary condition must be imposed that we have only a given flavor at production. Suppose for example that at t⫽0 a muon neutrino is produced, i.e., C 共 0 兲 ⫽1, C e 共 0 兲 ⫽0. C 2 共 0 兲 ⫽cos共 兲 . 共14兲 C e 共 t 兲 ⫽sin共 兲 cos共 兲共 e ⫺iE 2 t ⫺e ⫺iE 1 t 兲 , 共15兲 ⫺iE 1 t 共16兲 C 共 t 兲 ⫽sin 共 兲 e ⫹cos 共 兲 e 2 ⫺iE 2 t . Space and therefore momentum is introduced by assuming in Eqs. 共15兲 and 共16兲 E 21 ⫽m 21 ⫹p 2 , E 22 ⫽m 22 ⫹p 2 , 共17兲 L⯝t. The probability of finding a given flavor is obtained by squaring Eqs. 共18兲 and 共19兲, 兩 C e 兩 2 ⫽sin2 共 2 兲 sin2 关共 E 2 ⫺E 1 兲 t/2兴 ⯝sin 共 2 兲 sin 2 2 冋 共 m 21 ⫺m 22 兲 L 4E 册 共18兲 , ⯝1⫺sin 共 2 兲 sin 2 冋 共 m 21 ⫺m 22 兲 L 4E 册 , 共19兲 with E⫽p. In order to compare with experimental data, the probabilities given by Eqs. 共18兲 and 共19兲 have to be averaged over the energy distribution of particles involved. For a more complete analysis of the phenomenology of neutrino flavor oscillations see, for example, Ref. 3. The assumption that the muon neutrino is created with a definite momentum p is only an approximation, as has been pointed out previously.18–21 It is in contradiction with fourmomentum conservation, for example for the reaction → . Each of the possible energy eigenstates has a somewhat different momentum pi . In the rest frame of the pion, energy conservation dictates that (i⫽1,2) M ⫽ 冑M 2 ⫹p2i ⫹ 冑m 2i ⫹p2i . 共20兲 III. DIAGONALIZATION OF THE TWO COUPLED DIRAC EQUATIONS In this section we will derive a relativistic generalization of the two coupled equations, given by Eq. 共2兲, in the context of the Dirac theory. The system of equations is diagonalized and the energy eigenvalues and eigenfunctions are derived. Suppose, for example, that a muon neutrino is produced with some small mass 共rest energy兲. The propagation of the 871 Am. J. Phys., Vol. 67, No. 10, October 1999 共21兲 e 共 x,t 兲 ⫽ 共 ␣ p̂⫹  m e 兲 e 共 x,t 兲 . t 共22兲 Now suppose that, due to a quantum tunneling effect, the muon neutrino has some probability of flipping its flavor and turning into an electron neutrino. If ␦ is the parameter that describes the flavor flipping possibility, then neutrino flavor oscillations can be described by the two coupled Dirac equations i e 共 x,t 兲 ⫽ 共 ␣ p̂⫹  m e 兲 e 共 x,t 兲 ⫹  ␦ 共 x,t 兲 , t 共23兲 i 共 x,t 兲 ⫽ 共 ␣ p̂⫹  m 兲 共 x,t 兲 ⫹  ␦ e 共 x,t 兲 . t 共24兲 We notice here that ␦ has the dimension of a mass. The details of the diagonalization of the system of equations are given in the Appendix; here we simply discuss the solutions. For a given momentum p⫽ 兩 p兩 the energy eigenvalues are 2 E 1,2⫽⫾ 冑p 2 ⫹m 1,2 , 共25兲 where m 1,2 are the ‘‘renormalized’’ masses 兩 C 兩 2 ⫽1⫺sin2 共 2 兲 sin2 关共 E 2 ⫺E 1 兲 t/2兴 2 共 x,t 兲 ⫽ 共 ␣ p̂⫹  m 兲 共 x,t 兲 , t with p̂⫽⫺iⵜ and ␣ and  are the Dirac matrices. If the muon neutrino does not undergo any flavor change, then it will move as a free particle according to Eq. 共21兲. In the same way, if a neutrino is produced as an electron neutrino and has a small mass, but it does not oscillate in flavor, its propagation in vacuum will be described by i The time evolution of the flavor amplitudes is obtained by substituting Eq. 共14兲 into Eq. 共12兲, 2 i 共13兲 From Eq. 共11兲 at t⫽0 we obtain C 1 共 0 兲 ⫽⫺sin共 兲 , free neutrino can be described by the Dirac equation 共for a review about the Dirac equation and its solutions see, for example, Ref. 22兲 m 1,2⫽ 21 关共 m e ⫹m 兲 ⫾R 兴 , R⫽ 冑共 m ⫺m e 兲 2 ⫹4 ␦ 2 . 共26兲 A system of two coupled Dirac equations possesses for a given value of the energy two eigenfunctions; one with spin up and one with spin down. If 1 (x,t) is the eigenfunction corresponding to the positive energy solution E 1 given by Eq. 共25兲, it is possible to show that it can be written in terms of a two-dimensional vector 冉 冊 1 Z 1⫽ 冑1⫹⌳ 2 ⌳ ⌳⫽ , m ⫺m e ⫹R 2␦ 共27兲 冑1⫹⌳ 2 times the solution of the Dirac equation of renormalized mass m 1 , momentum p, and energy E 1 , 1 共 x,t 兲 ⫽Z 1 1 1 冑V 冑2E 1 u 1 共 s,p兲 e ipx e ⫺iE 1 t , 共28兲 where s⫽1,2 is the spin index, u 1 (s,p) is the Dirac spinor of mass m 1 . The wave function 1 (x,t) has eight dimensions because the two-dimensional vector Z 1 multiplies a four-dimensional wave function. This last wave function describes a neutrino of given energy E 1 and spin 1/2. The vector Z 1 tells us that Elisabetta Sassaroli 871 in any location inside the volume V there is a probability equal to (1/1⫹⌳ 2 ) of finding the neutrino in the electron flavor and probability equal to (⌳ 2 /1⫹⌳ 2 ) of finding it in the muon flavor. Therefore 1 describes a neutrino of a given energy, but of mixed flavor at any space–time point. In the same way, the wave function corresponding to the other positive energy solution E 2 is 2 共 x,t 兲 ⫽Z 2 1 1 冑V 冑2E 2 u 2 共 s,p兲 e ipx e ⫺iE 2 t , 冉 冊 共29兲 where Z 2 is the vector ⌳ Z 2⫽ 冑1⫹⌳ 2 ⫺ 共30兲 1 and u 2 (s,p) is the Dirac spinor of renormalized mass m 2 . The wave function 2 describes a neutrino of given energy E 2 and mixed flavor at any space–time point. The solutions of negative energies are interpreted, as in the Dirac theory, as antiparticles of positive energy and are given by 1 1 共 x,t 兲 ⫽ 冉 冑V 冑2E 1,2 v 1,2共 s,p兲 e ipx e ⫹iE 1 t , The quantization procedure for the two coupled Dirac neutrino fields e and , defined by Eqs. 共23兲 and 共24兲, proceeds in the same way as for the case of Dirac theory. 共For a discussion of the quantization of the Dirac theory, see, for example, Ref. 22兲. We expand the neutrino field ˆ in terms of the energy eigenfunctions and found in Sec. III, ⫽ ˆ e 共 x,t 兲 ˆ 共 x,t 兲 冊 共32兲 where the operators b i and d i (i⫽1,2) satisfy the usual Jordan–Wigner anticommutation relations 兵 b i 共 s,p兲 ,b †j 共 s ⬘ ,p⬘ 兲 其 ⫽ ␦ i j ␦ pp ⬘ ␦ ss ⬘ , 兵 d i 共 s,p兲 ,d †j 共 s ⬘ ,p⬘ 兲 其 ⫽ ␦ i j ␦ pp ⬘ ␦ ss ⬘ . 共33兲 For a given value of the momentum p and spin s, there are four possible one-particle states, one for each energy value, given by Eq. 共25兲 d †1 共 s,p兲 兩 0 典 ⫽ 兩 ⫺1 ps 典 , 872 共36兲 兩 A 兩 2 ⫹ 兩 B 兩 2 ⫽1. 共37兲 The matrix element 具 0 兩 ˆ e 共 x,t 兲 兩 ⫹ 典 ⫽ e 共 x,t 兲 冋 1 1 ⫽ 冑V 冑1⫹⌳ 2 A u 2 共 s,p兲 ⫹B⌳ 冑2E 2 u 1 共 s,p兲 冑2E 1 e ⫺iE 1 t 册 e ⫺iE 2 t e ipx , 共38兲 gives the probability amplitude of finding a neutrino of momentum p and spin s at the space–time point (x,t) with the electron flavor. In the same way, the matrix element 具 0 兩 ˆ 共 x,t 兲 兩 ⫹ 典 ⫽ 共 x,t 兲 ⫽ 1 1 冑V 冑1⫹⌳ ⫺B 2 u 2 共 s,p兲 冑2E 2 冋 A⌳ u 1 共 s,p兲 冑2E 1 e ⫺iE 1 t 册 e ⫺iE 2 t e ipx , 共39兲 is the probability amplitude for the muon flavor. The coefficients A and B are determined through the initial boundary conditions. Suppose that at t⫽0, 兺p 兺s 兺i 关 b i共 s,p兲 i共 x,t 兲 ⫹d †i i共 x,t 兲兴 , b †1 共 s,p兲 兩 0 典 ⫽ 兩 1 ps 典 , 共35兲 where A and B satisfy the normalization condition 共31兲 IV. FIELD QUANTIZATION, ANTICOMMUTATION RELATIONS, AND FLAVOR WAVE FUNCTIONS 冉 冊 It is easy to see that the above wave function is the energy eigenfunction of energy E 1 defined in Eq. 共30兲. Similar considerations can be applied to the other states 兩 2 ps 典 , 兩 ⫺1 ps 典 , and 兩 ⫺2 ps 典 . These one-particle states represent states of mixed flavor at any given space–time point. When describing neutrino oscillations, we have to consider a superposition of energy states. A general state of positive charge, momentum p, and spin s is given by where v 1,2(s,p) are Dirac spinors of mass m 1 and m 2 , respectively. ˆ 共 x,t 兲 ⫽ 冊冉 e 共 x,t 兲 具 0 兩 ˆ e 共 x,t 兲 兩 1 ps 典 ⫽ . 共 x,t 兲 具 0 兩 ˆ 共 x,t 兲 兩 1 ps 典 兩 ⫹ 典 ⫽ 关 Ab †1 共 s,p兲 ⫹Bb †2 共 s,p兲兴 兩 0 典 , 冑1⫹⌳ 2 1,2共 x,t 兲 ⫽Z 1,2 The wave function associated with the one-particle state 兩 1 ps 典 is, for example, obtained as a matrix element of neutrino fields between the vacuum state and the one-particle state, b †2 共 s,p兲 兩 0 典 ⫽ 兩 2 ps 典 , d †2 共 s,p兲 兩 0 典 ⫽ 兩 ⫺2 ps 典 . Am. J. Phys., Vol. 67, No. 10, October 1999 共34兲 共 x,t⫽0 兲 ⫽0, 共40兲 i.e., we have only the electron flavor present. The other boundary condition is obtained from the normalization condition 冕 V d 3 x兩 e 共 x,t⫽0 兲 兩 2 ⫽1. 共41兲 However, the above boundary conditions cannot be applied in a consistent way to the flavor wave functions given by Eqs. 共38兲 and 共39兲 and at the same time satisfy the conservation of probability condition given by Eq. 共37兲. The following two approximations have to be made on the neutrino wave functions in order for us to be able to impose the boundary conditions given by Eqs. 共40兲 and 共41兲. Elisabetta Sassaroli 872 共a兲 The left-handed chiral components of the flavor wave functions are considered as observable wave functions. Mathematically this is equivalent to considering as observable wave functions23 eL 共 x,t 兲 ⫽ 共 1⫺ ␥ 5 兲 e 共 x,t 兲 , L 共 x,t 兲 ⫽ 共 1⫺ ␥ 5 兲 共 x,t 兲 , 共42兲 where e (x,t) and (x,t) are given by Eqs. 共38兲 and 共39兲 and ␥ 5 is defined by ␥ 5⫽ 冉 冊 0 1 1 0 共43兲 . 共b兲 The ultrarelativistic approximation, i.e., E⯝ p, is assumed in the spinor left-handed chiral components u L (s,p) of the flavor wave functions. By applying the approximations 共a兲 and 共b兲 one obtains the flavor neutrino wave functions eL 共 x,t 兲 ⫽ L 共 x,t 兲 ⫽ 冉 冊 1 1 ⫺iE t 共2兲 , 关 e 1 ⫹⌳ 2 e ⫺iE 2 兴 2 ⫺共2兲 冑V 1⫹⌳ & 共44兲 e ipx 冉 冊 ⌳ 1 ⫺iE t ⫺iE t 共 2 兲 . 关 e 1 ⫺e 2 兴 2 ⫺共2兲 冑V 1⫹⌳ & 共45兲 e ipx The probability densities of finding the electron and muon neutrino flavor are then given, respectively, by 冋 冉 冊 冉 冊 2⌳ 1 e 共 t 兲 ⫽ 1⫺ V 1⫹⌳ 2 共 t 兲 ⫽ 2⌳ 1 V 1⫹⌳ 2 2 2 sin2 册 共 E 2 ⫺E 1 兲 t sin2 , 2 共46兲 共 E 2 ⫺E 1 兲 t . 2 共47兲 The coefficient 关 2⌳/(1⫹⌳ 2 ) 兴 2 is equivalent to sin2(2) in Eqs. 共18兲 and 共19兲. Therefore Eqs. 共46兲 and 共47兲 are equivalent to the standard neutrino oscillation probabilities. It is important to discuss the meaning of both approximations. Approximation 共a兲 takes into account the fact that neutrinos are produced through the weak interaction, which does not conserve parity. Therefore neutrinos are created with negative helicity and antineutrinos with positive helicity, i.e., neutrinos are emitted with their spin polarized opposite to their direction of motion and antineutrinos have their spin polarized in the same direction. This statement is exactly true on the hypothesis that neutrinos are massless and it is certainly a very good approximation for very small neutrino masses. Approximation 共b兲 is related to the conservation of probability. The coefficients A and B, obtained by imposing the boundary conditions given by Eqs. 共40兲 and 共41兲 in the flavor wave functions eL (z,t), L (z,t), do not satisfy the normalization condition given by Eq. 共37兲. This condition is satisfied only if we assume that the terms of type p/(E ⫹m) in spinors u L are of order one, i.e., in the ultrarelativistic limit. Therefore only in the ultrarelativistic limit can we impose the condition of a given flavor at production without violating the condition that the sum of the probabilities must be one. Other authors,24,25 by using different approaches, have also found that the standard neutrino oscillations can be recovered only in the ultrarelativistic limit. 873 Am. J. Phys., Vol. 67, No. 10, October 1999 We believe that there are deep theoretical reasons for the impossibility of discussing in a consistent way flavor oscillations in field theory. In the literature, some hypotheses have been proposed in relation to this issue, especially in relation to the problem of CPT and Lorentz violations. For example, Coleman and Glashow have examined the assumption that Lorentz noninvariance leads to neutrino oscillations which are phenomenologically equivalent to those obtained by assuming that neutrinos violate the equivalence principle. A variety of different approaches have been considered in the literature to address the problem of CPT and Lorentz invariance violations, as discussed in Ref. 26. V. CONCLUSIONS We have investigated an explicit model of neutrino flavor oscillations in the framework of relativistic quantum mechanics and quantum field theory. This model, which is a relativistically invariant generalization of a two-level system, consists of two coupled Dirac equations. The system has been diagonalized and ‘‘second quantized’’ in order to deal properly with the states of negative energy. The neutrino wave functions are obtained as matrix elements of the quantized neutrino fields. These wave functions, however, describe neutrinos which are in a state of mixed flavor at any space–time point and only in the so-called ultrarelativistic limit do they describe the possibility of having a given flavor at production. ACKNOWLEDGMENTS This work is supported in part by funds provided by the U.S. Department of Energy 共D.O.E.兲 under cooperative research agreement No. DF-FC02-94ER40818. The author would like to thank Professor Alan H. Guth for his constructive criticism which led to a revision of the paper. She would also like to acknowledge fruitful discussions with Professor Kenneth Johnson and the MIT atomic and molecular interferometry group. APPENDIX In order to determine the energy eigenvalues and eigenfunctions of the system of equations 共23兲 and 共24兲 we consider the ansatz e ⫽ae ⫺i Px , 共A1兲 ⫺i Px 共A2兲 ⫽be , where P is the four-momentum P⫽(E,p), which is unknown and is to be determined so that the system of differential equations 共23兲 and 共24兲 is satisfied. The coefficients a and b are Dirac spinors, which can be written as a⫽ b⫽ 冉 冊 冉 冊 1 , 2 共A3兲 3 , 4 共A4兲 where 1,2 and 3,4 are two component spinors. Substituting Eqs. 共A1兲 and 共A2兲 into Eqs. 共23兲 and 共24兲, we obtain the system of linear homogeneous equations E 1 ⫽ •p 2 ⫹m e 1 ⫹ ␦ 3 , 共A5兲 E 2 ⫽ •p 1 ⫺m e 2 ⫺ ␦ 4 , Elisabetta Sassaroli 873 E 3 ⫽ •p 4 ⫹m 3 ⫹ ␦ 1 , u 2 共 s,p兲 ⫽ 冑E 2 ⫹m 2 共A6兲 E 4 ⫽ •p 3 ⫺m 4 ⫺ ␦ 2 , where are the Pauli matrices. The system of Eqs. 共A5兲 and 共A6兲 admits nontrivial solutions only if 共A7兲 ⌳ ⬘⫽ E 1,2⫽⫾ 冑p 共A8兲 with m 1,2 given by m 1,2⫽ 21 关共 m e ⫹m 兲 ⫾R 兴 , 共A9兲 R⫽ 冑共 m ⫺m e 兲 2 ⫹4 ␦ 2 . 共A10兲 Therefore, while in the free Dirac equation there are two energies 共one positive and one negative兲 for every possible value of the momentum p, for a system of two coupled Dirac equations, there are four possible values of the energy, two positive and two negative. This is due to the possibility of flavor oscillations. Also, because there is some chance that the neutrino can flip flavor, the rest energies of the electron and muon neutrino system are not simply m e and m but are given by Eq. 共A9兲. Corresponding to the positive energy solution E 1 ⫽ 冑m 21 ⫹p 2 , we have the following two solutions: 1 1 冑V 冑2E 1 1 共 s,p兲 e ipx e ⫺iE 1 t , 共A11兲 1 共 x,t 兲 ⫽ 1 冑1⫹⌳ 2 冉 冊 u 1 共 s,p兲 , ⌳u 1 共 s,p兲 共A12兲 and u 1 (s,p) is the Dirac spinor u 1 共 s,p兲 ⫽ 冑E 1 ⫹m 1 冉 冊 共s兲 p , 共s兲 E 1 ⫹m 1 3 共 s,p兲 ⫽ ⫽ m ⫺m e ⫹R . 2␦ 1 with 2 (s,p) given by 2 共 s,p兲 ⫽ 共A15兲 1 冑1⫹⌳ ⬘ 2 冉 共A16兲 冊 u 2 共 s,p兲 , ⌳ ⬘ u 2 共 s,p兲 and the Dirac spinor u 2 (s,p) is 874 Am. J. Phys., Vol. 67, No. 10, October 1999 冊 ⌳u 2 共 s,p兲 . ⫺u 2 共 s,p兲 共A19兲 1 冑V 冑2E 1 冉 1 冑1⫹⌳ ⬘ 1 冑1⫹⌳ 2 3 共 s,p兲 e ⫺ipx e ⫹iE 1 t , 2 冉 ⫺⌳ ⬘ v 1 共 s,p兲 v 1 共 s,p兲 冊 冊 v 1 共 s,p兲 , ⌳ v 1 共 s,p兲 冉 共A20兲 共A21兲 冊 p 共s兲 E 1 ⫹m 1 . 共s兲 共A22兲 For the energy eigenvalue ⫺E 2 we have the solution 2 共 x,t 兲 ⫽ 1 1 冑V 冑2E 2 with 4 (s,p) given by 共A13兲 2 共 s,p兲 e ipx e ⫺iE 2 t , 冑V 冑2E 2 1 1 冑1⫹⌳ 2 冉 and 共A14兲 1 冉 v 1 共 s,p兲 ⫽ 冑E 1 ⫹m 1 4 共 s,p兲 ⫽ For the other positive energy solution E 2 ⫽ 冑p 2 ⫹m 22 , we have 2 共 x,t 兲 ⫽ 冑1⫹⌳ 2 and v 2 共 s,p兲 ⫽ 冑E 2 ⫹m 1 with ⌳⫽ 1 with 3 (s,p) given by where s⫽1,2 is the spin index and 1 (s,p) is given by 1 共 s,p兲 ⫽ 共A18兲 Similarly, for the solutions of negative energies ⫺E 1 we have the eigenfunction: and 1 共 x,t 兲 ⫽ m ⫺m e ⫺R . 2␦ 2 共 s,p兲 ⫽ 2 ⫹m 1,2 , 共A17兲 We notice here that because ⌳⌳ ⬘ ⫽⫺1 we can write 2 (s,p) in terms of ⌳ as Solving Eq. 共A7兲, we obtain (p⫽ 兩 p兩 ) 2 冊 ⌳⬘ is defined as E 4 ⫺E 2 共 2p 2 ⫹2 ␦ 2 ⫹m 2e ⫹m 2 兲 ⫹p 4 ⫹ ␦ 4 ⫹p 2 共 2 ␦ 2 ⫹m 2e ⫹m 2 兲 ⫹m 2 m 2e ⫺2 ␦ 2 m e m ⫽0. 冉 共s兲 p . 共s兲 E 2 ⫹m 2 4 共 s,p兲 e ⫺ipx e ⫹iE 2 t , 共A23兲 冊 共A24兲 ⌳ v 2 共 s,p兲 , ⫺ v 2 共 s,p兲 冉 冊 p 共s兲 E 2 ⫹m 2 . 共s兲 共A25兲 1 R. 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Vitiello, ‘‘Quantum field theory of neutrino mixing,’’ Ann. Phys. 共N.Y.兲 244, 283–311 共1995兲; E. Alfinito, M. Blasone, A. Iorio, and G. Vitiello, ‘‘Squeezed neutrino oscillations in quantum field theory,’’ Phys. Lett. B 362, 91–96 共1995兲. 26 V. Alan Kostelecky, ‘‘The Status of CTP,’’ Talk presented at WEIN-98, Santa Fe, New Mexico, June 1998; hep-ph/9810365. ADJUSTING THE SPECTROMETER I sometimes regret that much of our modern apparatus, even for students, has all the interesting difficulties removed beforehand. If a student is going to work with a spectrometer, I think it is highly desirable that he should go through the process of adjusting the collimator, the telescope, and eye piece himself. It is desirable that he shall go through the process of getting the grating lines parallel with the axis of rotation. It is desirable that he shall know how to set the axis of the telescope perpendicular to the axis of rotation. Once the spectrometer is adjusted, all of the good of the experiment has been utilized. I do not think that the student learns much in the last act of measuring the wavelength of light. W. F. G. Swann, ‘‘The Teaching of Physics,’’ Am. J. Phys. 19共3兲, 182–187 共1951兲. 875 Am. J. Phys., Vol. 67, No. 10, October 1999 Elisabetta Sassaroli 875