Weak Interactions in a Background of a Uniform Magnetic Field. A Mathematical Model for the Inverse β Decay. I. ()
1. Introduction
A supernova is initiated by the collapse of a stellar core which leads to the formation of a protoneutron star which may be formed with strong magnetic fields typically of order 1016 Gauss. It turns out that the protoneutron star leads to the formation of a neutron star in a very short time during which almost all the gravitational binding energy of the protoneutron star is emmitted in neutrinos and antineutrinos of each type. Neutron stars have strong magnetic fields of order 1012 Gauss. Thus neutrinos interactions are of great importance because of their capacity to serve as mediators for the transport and loss of energy and the following processes, the so-called “Urca” ones or inverse β decays in Physics,
(1.1)
(1.2)
play an essential role in those phenomena and they are associated with the β decay
(1.3)
Here
(resp.
) is an electron (resp. a positron). p is a proton and n a neutron.
and
are the neutrino and the antineutrino associated with the electron.
See [1] [2] [3] [4] and references therein.
We only consider here high-energy neutrinos and antineutrinos which are indeed relativistic particles whose mass is zero or in anyway negligible.
Due to the large magnetic field strengths involved, it is quite fundamental to study the processes (1.1) and (1.2) in the presence of magnetic fields.
These realistic fields may be very complicated in their structure but we assume these fields to be locally uniform which is a very good hypothesis because the range of the weak interactions is very short. Our aim is to study the processes (1.1) and (1.2) in a background of a uniform magnetic field.
Throughout this work we restrict ourselves to the study of processes (1.1), the study of processes (1.2) and (1.3) would be quite similar. We choose the units such that
.
The advantage of a uniform magnetic field is that, in presence of this field, Dirac equation can be exactly solved. Using the generalized eigenfunctions of the Dirac equation and the canonical quantization we carefully define the quantized fields associated with the electrons, the positrons, the protons and the antiprotons in a uniform magnetic field.
For the neutrons and the neutrinos we define the corresponding quantized fields by using the helicity formalism for the free Dirac equation.
We then consider the Fock space for the electrons, the positrons, the protons, the antiprotons, the neutrons and the neutrinos.
In this paper we consider a mathematical model for the process (1.1) in a uniform magnetic field based on the Fermi’s Hamiltonian for the β decay. The physical interaction is a highly singular operator due to delta-distributions associated with the conservation of momenta and because of the ultraviolet divergences. In order to get a well defined Hamiltonian in the Fock space we have to substitute smoother kernels both for the delta-distributions and for dealing with the ultraviolet divergences. We then get a self-adjoint Hamiltonian with cutoffs in the Fock space when the kernels are square integrable.
We then study the essential spectrum of the Hamiltonian and prove the existence of a unique ground state with appropriate hypothesis on the kernels. The proof of the uniqueness of the ground state is a direct consequence of the proof of the existence of a ground state. The spectrum of the Hamiltonian is identical to its essential spectrum. Every result is obtained for a sufficiently small coupling constant. No infrared regularization is assumed. We adapt to our case the proofs given in [5] and [6] .
These results are new for the mathematical models in Quantum Field Theory with a uniform magnetic field.
The paper is organized as follows. In the next two sections, we quantize the Dirac fields for electrons, protons and their antiparticles in a uniform magnetic field. In the third section, we quantize the Dirac fields for free neutrons, neutrinos and their antiparticles in helicity formalism. The self-adjoint Hamil- tonian of the model is defined in the fourth section. We then study the essential spectrum and prove the existence of a unique ground state.
2. The Quantization of the Dirac Fields for the Electrons and the Protons in a Uniform Magnetic Field
In this paper we assume that the uniform classical background magnetic field in
is along the x3-direction of the coordinate axis. There are several choices of gauge vector potential giving rise to a magnetic field of magnitude
along the x3-direction. In this paper we choose the following vector potential
,
, where
(2.1)
Here
in
.
We recall that we neglect the anomalous magnetic moments of the particles of
.
The Dirac equation for a particle of
with mass
and charge e
in a uniform magnetic field of magnitude
along the x3-direction with the choice of the gauge (2.1) and by neglecting its anomalous magnetic moment is given by
(2.2)
acting in the Hilbert space
.
The scalar product in
is given by
We refer to [7] for a discussion of the Dirac operator.
Here
,
are the Dirac matrices in the standard form:
where
are the usual Pauli matrices.
By ( [7] , thm 4.3)
is essentially self-adjoint on
. The spectrum of
is equal to
(2.3)
The spectrum of
is absolutely continuous and its multiplicity is not uniform. There is a countable set of thresholds, denoted by S, where
(2.4)
with
. See [8] .
We consider a spectral representation of
based on a complete set of generalized eigenfunctions of the continuous spectrum of
. Those generalized eigenfunctions are well known. See [9] . In view of (2.1) we use the computation of the generalized eigenfunctions given by [10] and [11] . See also [4] and references therein.
Let
be the conjugate variables of
. By the Fourier transform in
we easily get
(2.5)
and
(2.6)
where
(2.7)
Here
is the
unit matrix.
is the reduced Dirac operator associated to
.
is essentially self-adjoint on
and has a pure point spectrum which is symmetrical with respect to the origin.
Set
(2.8)
The positive spectrum of
is the set of eigenvalues
and the negative spectrum is the set of eigenvalues
.
and
are simple eigenvalues and the multiplicity of
and
is equal to 2 for
.
Through out this work e will be the positive unit of charge taken to be equal to the proton charge.
We now give the eigenfunctions of
both for the electrons and for the protons. The eigenfunctions are labelled by
,
and
.
labels the nth Landau level.
are the eigenvalues of
. The electrons and the protons in all Landau levels with
can have different spin polarizations
. However in the lowest Landau state
the electrons can only have the spin orientation given by
and the protons can only have the spin orientation given by
.
2.1. Eigenfunctions of the Reduced Dirac Operator for the Electrons
We now compute the eigenfunctions of
with
where
is the mass of the electron.
and
will denote the eigenvalues of
for the electrons. We have
.
2.1.1. Eigenfunctions of the Electrons for Positive Eigenvalues
For
is of multiplicity two corresponding to
and
is multiplicity one corresponding to
.
Let
denote the eigenfunctions associated to
.
For
and
we have
(2.9)
where
(2.10)
Here
is the Hermite polynomial of order n and we define
(2.11)
For
and
we set
For
and
we have
(2.12)
Note that
(2.13)
where
is the adjoint in
.
2.1.2. Eigenfunctions of the Electrons for Negative Eigenvalues
For
is of multiplicity two corresponding to
and
is multiplicity one corresponding to
.
Let
denote the eigenfunctions associated with the eigenvalue
and with
.
For
and
we have
(2.14)
and for
we set
For
and
we have
(2.15)
Note that
(2.16)
where
is the adjoint in
.
The sets
and
of vectors in
form a orthonormal basis of
.
This yields for
in
(2.17)
where
.
Let
be the Fourier transform of
with respect to
and
:
We have
(2.18)
The complex coefficients
and
satisfy
(2.19)
2.2. Eigenfunctions of the Reduced Dirac Operator for the Protons
We now compute the eigenfunctions of
with
.
and
denote the eigenvalues of
for the proton. We have
.
2.2.1. Eigenfunctions of the Proton for Positive Eigenvalues
For
is of multiplicity two corresponding to
and
is of multiplicity one corresponding to
.
Let
denote the eigenfunctions associated with the eigen- value
and with
.
For
and
we have
(2.20)
where
(2.21)
For
and
we have
(2.22)
For
and
we set
Note that
where
is the adjoint in
.
2.2.2. Eigenfunctions of the Proton for Negative Eigenvalues
For
is of multiplicity two corresponding to
and
is of multiplicity one corresponding to
.
Let
denote the eigenfunctions associated with the eigen- value
and with
.
For
and
we have
(2.23)
For
and
we have
(2.24)
and for
and
we set
Note that
(2.25)
where
is the adjoint in
.
The sets
and
of vectors in
form an orthonormal basis of
.
This yields for
in
(2.26)
where
The complex coefficients
and
satisfy
(2.27)
We have
(2..28)
2.2.3. Eigenfunctions of the Positron for Positive Eigenvalues
The generalized eigenfunctions for the positron, denoted by
, are obtained from
by substituting the mass of the electron
for
. The associated eigenvalues are denoted by
with
.
2.2.4. Eigenfunctions of the Positron for Negative Eigenvalues
The generalized eigenfunctions for the positron, associated with the eigenvalues
and denoted by
, are obtained from
by substituting the mass of the electron
for
.
2.2.5. Eigenfunctions of the Antiproton for Positive Eigenvalues
The generalized eigenfunctions for the antiproton, denoted by
, are obtained from
by substituting the mass of the proton
for
. The associated eigenvalues are denoted by
with
.
2.2.6. Eigenfunctions of the Antiproton for Negative Eigenvalues
The generalized eigenfunctions for the antiproton, associated with the eigen- values
and denoted by
, are obtained from
by substituting the mass of the proton
for
.
2.3. Fock Spaces for Electrons, Positrons, Protons and Antiprotons in a Uniform Magnetic Field
It follows from Sections 2.1 and 2.2 that
are quantum variables for the electrons, the positrons, the protons and the antiprotons in a uniform magnetic field.
Let
be the quantum variables of a electron and of a positron and let
be the quantum variables of a proton and of an antiproton.
We set
for the configuration space for both the electrons, the positrons, the protons and the antiprotons.
is the Hilbert space associated to each species of fermions.
We have, by (2.17), (2.18), (2.19), (2.26), (2.27) and (2.28),
(2.29)
Let
and
denote the Fock spaces for the electrons and the posi- trons respectively and let
and
denote the Fock spaces for the protons and the antiprotons respectively.
We have
(2.30)
is the antisymmetric n-th tensor power of
.
is the vacuum state in
for
.
We shall use the notations
(2.31)
Set
.
(resp.
) are the annihilation (resp.creation) operators for the electron when
and for the proton when
if
.
(resp.
) are the annihilation (resp.creation) operators for the positron when
and for the antiproton when
if
.
The operators
and
fulfil the usual anticommutation relations (CAR)(see [12] ).
In addition, following the convention described in ( [12] , Section 4.1) and ( [12] , Section 4.2), we assume that the fermionic creation and annihilation operators of different species of particles anticommute (see [13] arXiv for explicit definitions). In our case this property will be verified by the creation and annihilation operators for the electrons, the protons, the neutrons, the neutrinos and their respective antiparticles.
Therefore the following anticommutation relations hold for
(2.32)
where
and
or
.
Recall that for
, the operators
(2.33)
are bounded operators on
and
for
and on
and
for
respectively satisfying
(2.34)
2.4. Quantized Dirac Fields for the Electrons and the Protons in a Uniform Magnetic Field
We now consider the canonical quantization of the two classical fields (2.17) and (2.26).
Recall that the charge conjugation operator
is given, for every
, by
(2.35)
Here * is the complex conjugation.
Let
be locally in the domain of
. We have
(2.36)
(2.36) shows that, by applying the charge conjugation (2.35) to a solution of the Dirac equation with a negative energy for some particle, we get a solution of the Dirac equation for the antiparticle with a positive energy.
Thus, by applying the charge conjugation (2.35) to (2.14), (2.15), (2.23) and (2.24) which are solutions of the Dirac equation for the electrons and protons with a negative energy, we obtain
(2.37)
The solutions of the right hand side of (2.37) are solutions of the Dirac equation for the positrons and antiprotons with a positive energy.
By (2.37) we set
(2.38)
By using (2.37) and (2.38) the symmetric of charge canonical quantization of the classical field (2.17) gives the following formal operator associated with the electron and denoted by
:
(2.39)
For a rigourous approach of the quantization see [22] .
We further note that
(2.40)
See [11] .
By (2.37) we now set
(2.41)
By using (2.37) and (2.41) the symmetric of charge canonical quantization of the classical field (2.26) gives the following formal operator associated to the proton and denoted by
:
(2.42)
We further note that
(2.43)
See [11] .
3. The Quantization of the Dirac Fields for the Neutrons and the Neutrinos in Helicity Formalism
As stated in the introduction we neglect the magnetic moment of the neutrons. Therefore neutrons and neutrinos are purely neutral particles without any electromagnetic interaction. We suppose that the neutrinos and antineutrinos are massless as in the Standard Model.
The quantized Dirac fields for free massive and massless particles of
are well-known.
In this work we use the helicity formalism, for free particles. See, for example, [7] [15] and [16] .
The helicity formalism for particles is associated with a spectral representation of the set of commuting self adjoint operators
.
are the
generators of space-translations and
is the helicity operator
where
and
with for
(3.1)
3.1. The Quantization of the Dirac Field for the Neutron in Helicity Formalism
The Dirac equation for the neutron of mass
is given by
(3.2)
acting in the Hilbert space
.
It follows from the Fourier transform that
(3.3)
where
(3.4)
Here
is the
unit matrix,
and
with
.
has two eigenvalues
and
where
The helicity, denoted by
, is given by
(3.5)
commutes with
and has two eigenvalues
and
.
Set (see ( [7] , Appendix. 1.F.] and [15] ) for
(3.6)
and
(3.7)
For
we set
and
We have
.
Let
(3.8)
The two eigenfunctions of the eigenvalue
associated with helicities
and
are denoted by
and are given by
(3.9)
We now turn to the eigenfunctions for the eigenvalue
.
The two eigenfunctions associated with the eigenvalue
and with helicities
and
are denoted by
and are given by
(3.10)
The four vectors
and
form an orthonormal basis of
.
and
is a complete set of generalized eigenfunctions of (3.2) with positive and negative eigenvalues
.
This yields for
in
(3.11)
with
(3.12)
3.1.1. Fock Space for the Neutrons
We recall that the neutron is not its own antiparticle.
Let
be the quantum variables of a neutron and an antineutron
where
is the momentum and
is the helicity. We set
for the configuration space of the neutron and the anti- neutron.
Let
and
denote the Fock spaces for the neutrons and the anti- neutrons respectively.
We have
(3.13)
is the antisymmetric n-th tensor power of
.
is the vacuum state in
for
.
In the sequel we shall use the notations
(3.14)
(resp.
) is the annihilation (resp.creation)operator for the neutron if
and for the antineutron if
.
The operators
and
fulfil the usual anticommutation relations (CAR) and they anticommute with
for
according to the convention described in ( [12] , Section 4.1). See [13] arXiv for explicit definitions.
Therefore the following anticommutation relations hold for
(3.15)
Recall that for
, the operators
(3.16)
are bounded operators on
and
satisfying
(3.17)
3.1.2. Quantized Dirac Field for the Neutron in Helicity Formalism
By (2.35) we get
(3.18)
Setting
(3.19)
and applying the canonical quantization we obtain the following quantized Dirac field for the neutron:
(3.20)
3.2. The Quantization of the Dirac Field for the Neutrino
Throughout this work we suppose that the neutrinos we consider are those associated with the electrons.
The Dirac equation for the neutrino is given by
(3.21)
acting in the Hilbert space
.
By (3.3) it follows from the Fourier transform that
(3.22)
where
(3.23)
has two eigenvalues
and
where
.
The helicity given by
commutes with
and has two eigenvalues
and
.
The two eigenfunctions of the eigenvalue
associated with helicities
and
are denoted by
. The two eigenfunctions of the eigenvalue
associated with helicities
and
are denoted by
. They are given by
(3.24)
The four vectors
and
form an orthonormal basis in
.
Turning now to the theory of neutrinos and antineutrinos (see [17] ) a neutrino has a helicity equal to
and a antineutrino a helicity equal to
. Neutrinos are left-handed and antineutrinos are right-handed.
is the eigen- function of a neutrino with a momentum
and an energy
.
is the eigenfunction of an antineutrino with a momentum
and an energy
.
Thus the classical field, denoted by
and associated with the neutrino and the antineutrino, is given by
(3.25)
with
3.2.1. Fock Space for the Neutrinos and the Antineutrinos
Let
be the quantum variables of a neutrino where
is the momentum and
is the helicity. In the case of the antineutrino we set
where
and
is the helicity.
is the Hilbert space of the states of the neutrinos and of the anti- neutrinos.
Let
and
denote the Fock spaces for the neutrinos and the anti- neutrinos respectively.
We have
(3.26)
is the antisymmetric n-th tensor power of
.
is the vacuum state in
for
.
In the sequel we shall use the notations
(3.27)
(resp.
) is the annihilation (resp.creation) operator for the neutrino and
(resp.
) is the annihilation (resp.creation) opera- tor for the antineutrino.
The operators
,
,
and
fulfil the usual anti- commutation relations (CAR) and they anticommute with
for
according the convention described in ( [12] , Section 4.1). See [13] arXiv for explicit definitions.
Therefore the following anticommutation relations hold for
(3.28)
Recall that for
, the operators
(3.29)
are bounded operators on
and
respectively satisfying
(3.30)
where
.
3.2.2. Quantized Dirac Field for the Neutrino
and
are generalized eigenfunctions of (3.21) with positive and negative eigenvalues
respectively.
By (2.35) we get
(3.31)
Setting
(3.32)
and applying the canonical quantization we obtain the following quantized Dirac field for the neutrino:
(3.33)
4. The Hamiltonian of the Model
The processes (1.1) and (1.2) are associated with the β decay of the neutron (see [3] [4] [17] and [18] ).
The β decay process can be described by the well known four-fermion effective Hamiltonian for the interaction in the Schrdinger representation:
(4.1)
Here
,
and
are the Dirac matrices in the standard representation.
and
are the quantized Dirac fields for p, n, e and
.
.
, where
is the Fermi coupling constant with
and
is the Cabbibo angle with
. Moreover
. See [19] .
The neutrino
is the neutrino associated to the electron and usually denoted by
in Physics.
From now on we restrict ourselves to the study of processes (1.1).
We recall that
.
4.1. The Free Hamiltonian
We set
(4.2)
We set
(4.3)
Let
(resp.
,
and
) be the Dirac Hamiltonian for the electron (resp.the proton, the neutron and the neutrino).
The quantization of
, denoted by
and acting on
, is given by
(4.4)
Likewise the quantization of
,
and
, denoted by
,
and
respectively,acting on
,
and
respectively, is given by
(4.5)
For each Fock space
, let
denote the set of vectors
for which each component
is smooth and has a compact support and
for all but finitely many (r). Then
is well-defined on the dense subset
and it is essentially self-adjoint on
. The self-adjoint extension will be denoted by the same symbol
with domain
.
The spectrum of
is given by
(4.6)
is a simple eigenvalue whose the associated eigenvector is the vacuum in
denoted by
.
is the absolutely continuous spectrum of
.
Likewise the spectra of
,
and
are given by
(4.7)
,
and
are the associated vacua in
,
and
respectively and are the associated eigenvectors of
,
and
respectively for the eigenvalue
.
The vacuum in
, denoted by
, is then given by
(4.8)
The free Hamiltonian for the model, denoted by
and acting on
, is now given by
(4.9)
is essentially self-adjoint on
.
Here
is the algebraic tensor product.
and
is the eigenvector associated with the simple eigenvalue
of
.
Let
be the set of the thresholds of
:
with
.
Likewise let
be the set of the thresholds of
:
with
.
Let
be the set of the thresholds of
:
Then
(4.10)
is the set of the thresholds of
.
4.2. The Interaction
By (4.1) let us now write down the formal interaction,denoted by
, involving the protons, the neutrons, the electrons and the neutrinos together with antiparticles in the Schrödinger representation for the process (1.1). We have
(4.11)
Set
(4.12)
After the integration with respect to
is given by
(4.13)
(4.14)
(4.15)
(4.16)
and
are responsible for the fact that the bare vacuum will not be an eigenvector of the total Hamiltonian as expected in Physics.
is formally symmetric.
In the Fock space
the interaction
is a highly singular operator due to the δ-distributions that occur in the
and because of the ultraviolet behaviour of the functions
and
.
In order to get well defined operators in
we have to substitute smoother kernels
,
, where
, both for the δ-distributions and the ultraviolet cutoffs.
We then obtain a new operator denoted by
and defined as follows in the Schrödinger representation.
(4.17)
with
(4.18)
(4.19)
(4.20)
(4.21)
Definition 4.1. The total Hamiltonian is
(4.22)
where g is a non-negative coupling constant.
The assumption that g is non-negative is made for simplicity but all the results below hold for
with
small enough.
We now give the hypothesis that the kernels
,
, and the coupling constant g have to satisfy in order to associate with the formal operator H a well defined self-adjoint operator in
.
Throughout this work we assume the following hypothesis
Hypothesis 4.2. For
we assume
(4.23)
Let
be the scalar product in
. We have
(4.24)
Set
(4.25)
We then have
Proposition 4.3. For every
we obtain
(4.26)
By (4.23), (4.24) and (4.25) the estimates (4.26) are examples of
estimates (see [20] ). The proof is similar to the one of ( [21] , Proposition 3.7) and details are omitted.
Let
be such that
(4.27)
We now have
Theorem 4.4. For any
such that
, H is a self-adjoint operator in
with domain
and is bounded from below. H is essentially self-adjoint on any core of
. Setting
we have for every
with
.
Here
is the spectrum of H and
is the essential spectrum of H.
Proof. By Proposition 4.2 and (4.27) the proof of the self-adjointness of H follows from the Kato-Rellich theorem.
We turn now to the essential spectrum. The result about the essential spectrum in the case of models involving bosons has been obtained by ( [14] , theorem 4.1) and [23] . In the case of models involving fermions the result has been obtained by [24] . In our case involving only massive fermions and massless neutrinos we use the proof given by [24] .
Thus we have to construct a Weyl sequence for H and
with
.
Let T be the self-adjoint multiplication operator in
defined by
. T is the spectral representation of
for the neutrinos
of helicity
in the configuration space
. See (3.27).
Every
belongs to the essential spectrum of T. Then there exists a Weyl sequence
for T and
such that
(4.28)
Let
(4.29)
In the following we identify
with its obvious extension to
.
An easy computation shows that, for every
,
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
(4.35)
(4.36)
(4.37)
Let
be the spectral measure of H. For any
the orthogonal projection
is different from zero because E belongs to
.
Let
such that
. We set
(4.38)
Let us chow that there exists a subsequence of
which is a Weyl sequence for H and
with
.
By Hypothesis 4.1, (4.30), (4.32), (4.34), (4.36) and the
estimates we get
(4.39)
Note that
(4.40)
We have for every
(4.41)
See [14] .
This yields
(4.42)
and
(4.43)
By (3.19) this yields for
(4.44)
Let
be an orthonormal basis of
and consider
(4.45)
where the indices can be assumed ordered
. Fock space vectors of this type form a basis of
(see [7] ). By ( [24] , Lemma 2.1) this yields for every
(4.46)
By (3.26) and Hypothesis 4.1 we have
(4.47)
It follows from (4.28), (4.38), (4.44), (4.46) and (4.47) that for every
(4.48)
This yields
(4.49)
In view of (4.49) there exists a subsequence
such that
(4.50)
Furthermore it follows from (4.46) that
.
The sequence
is a Weyl sequence for H and
with
.
In order to show that
we adapt the proof given in [5] and [6] . We omit the details.
This concludes the proof of theorem 4.4.
5. Existence of a Unique Ground State for the Hamiltonian H
Set
(5.1)
By (4.26) and (5.1) we get for every
(5.2)
In order to prove the existence of a ground state for the Hamiltonian H we shall make the following additional assumptions on the kernels
,
.
From now on
is the momentum of the neutrino with helicity
.
Hypothesis 5.1. There exists a constant
such that for
and
1)
2)
We have
Theorem 5.2. Assume that the kernels
and
,
, satisfy Hypothesis 4.1 and Hypothesis 5.1. Then there exists
such that H has a unique ground state for
.
In order to prove theorem 5.2 we first prove the existence of a spectral gap for some neutrino infrared cutoff Hamiltonians.
5.1. The Neutrino Infrared Cutoff Hamiltonians and the Existence of a Spectral Gap
Proof. Let us first define the neutrino infrared cutoff Hamiltonians.
For that purpose, let
with
on
and
on
. For
and
, we set
(5.3)
The operator
is the interaction given by (4.17) associated with the kernels
instead of
.
We then set
(5.4)
We now introduce
(5.5)
is the Fock space for the massless neutrino such that
.
We set
(5.6)
We have
(5.7)
We further set
(5.8)
In the following we identify
with its obvious extension to
.
We let
(5.9)
We identify
and
with their obvious extension to
and
respectively.
On
, we have
(5.10)
where
(resp.
) is the identity operator on
(resp.
).
Setting
(5.11)
we then get
(5.12)
and
(5.13)
On the other hand, for
such that
, we define the sequence
by
(5.14)
where
(5.15)
For
, we now introduce the neutrino infrared cutoff Hamiltonians on
by stting
(5.16)
We set, for
,
(5.17)
We introduce the neutrino infrared cutoff Hamiltonians on
by setting
(5.18)
We set, for
,
(5.19)
Note that
(5.20)
One easily shows that, for
,
(5.21)
See [5] [13] for a proof.
We now let
(5.22)
where
is the constant given in Hypothesis 5.2(2).
We further set,
(5.23)
(5.24)
and
(5.25)
Let
be such that
(5.26)
and let
(5.27)
Setting
(5.28)
and applying the same method as the one used for proving proposition 4.1 in [5] we finally get the existence of a spectral gap for
. We omit the details of the proof.
The proof of the following proposition is achieved.
Proposition 5.3. Suppose that the kernels
,
,
, satisfy Hypothesis 4.1 and Hypothesis 5.1(2). Then, for
,
is a simple eigenvalue of
for
, and
does not have spectrum in the interval
.
5.2. Proof of the Existence of a Ground State
Proof. In order to prove the existence of a ground state for H we adapt the proof of theorem 3.3 in [13] . By Proposition 5.3
has a unique ground state, denoted by
, in
such that
(5.29)
Therefore
has a unique normalized ground state in
, given by
, where
is the vacuum state in
,
(5.30)
Let
denote the interaction
. It follows from the pull-through formula that
(5.31)
where
(5.32)
(5.33)
Hence, by (5.30), (5.31), (5.32) and (5.33), we get
(5.34)
We further note that, for
,
(5.35)
where
The estimates (5.35) are examples of
estimates (see [20] ). The proof is similar to the one of ( [21] , Proposition 3.7) and details are omitted.
Let us estimate
. By (5.2) we get
(5.36)
and
(5.37)
By (5.21), we obtain
(5.38)
By (5.38)
is bounded uniformly with respect to n and
and by (5.34), (5.35) and (5.38) we get
(5.39)
uniformly with respect to n.
By Hypothesis 5.1(1) and (5.39) there exists a constant
such that
(5.40)
Since
, there exists a subsequence
, converging to
such
that
converges weakly to a state
. By adapting the proof of theorem 4.1 in [21] it follows from (5.40) that there exists
such that
and
for any
. Thus
is a ground state of H.
5.3. Uniqueness of a Ground State of the Hamiltonian H
Proof. The proof follows by adapting the one given in [6] . See also [25] .
In view of theorem 4.3 E is an eigenvalue of H with a finite multiplicity. Either E is a simple eigenvalue and the theorem is proved or its multiplicity is equal to
with
. Let us consider the second case. We wish to show by contradiction that E is a simple eigenvalue for g sufficiently small.
Let
be two vectors of the eigenspace of E. Each
with
is a ground state of H.
and
can be chosen such that
with
,
.
By (5.30) let
be a unique normalized ground state of
.
We have
(5.41)
where
is the spectral measure for the associated self-adjoint operator.
We have
(5.42)
We have to estimate
(5.43)
and
(5.44)
We first estimate (5.43).
By applying the same proof as the one used to get estimates (5.38), (5.39) and (5.40) with
instead of
we easily get
(5.45)
This yields
(5.46)
We now estimate (5.44)
Set
(5.47)
By proposition 5.3 we get
(5.48)
and
(5.49)
Note that
(5.50)
In view of (5.49) and of (5.50) we get
(5.51)
Hence
(5.52)
Here
has been introduced in proposition 5.3
Estimate of
. We have
(5.53)
is associated with the kernels
.
By adapting the proof of (5.2) to the estimate of
we finally get
(5.54)
where
(5.55)
Under Hypothesis 5.2(2) we get
(5.56)
This, together with (5.55), yields
(5.57)
where
.
Combing (5.41), (5.42), (5.46), (5.52) and (5.57) we finally get
(5.58)
Here
.
is a positive constant independent of g and it follows from (5.41) that, for g sufficiently small,
. This is a contradiction and
. This concludes the proof of theorem 5.2.
Acknowledgements
J.-C. G. acknowledges J.-M Barbaroux, J. Faupin and G. Hachem for helpful discussions.