Download Neutrino oscillations: A relativistic example of a two-level system Elisabetta Sassaroli

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
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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
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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.
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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
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Elisabetta Sassaroli
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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.
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875