Open Access Library Journal
Vol.04 No.12(2017), Article ID:81163,35 pages
10.4236/oalib.1104142
Weak Interactions in a Background of a Uniform Magnetic Field. A Mathematical Model for the Inverse β Decay. I.
Jean-Claude Guillot
Centre de Mathématiques Appliquées, École Polytechnique-C.N.R.S, Palaiseau Cedex, France
Copyright © 2017 by author and Open Access Library Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: November 10, 2017; Accepted: December 16, 2017; Published: December 19, 2017
ABSTRACT
In this paper we consider a mathematical model for the inverse β decay in a uniform magnetic field. With this model we associate a Hamiltonian with cutoffs in an appropriate Fock space. No infrared regularization is assumed. The Hamiltonian is self-adjoint and has a unique ground state. We study the essential spectrum and determine the spectrum. The coupling constant is supposed sufficiently small.
Subject Areas:
Functional Analysis, Particle Physics
Keywords:
Beta Decay, Uniform Magnetic Field, Weak Interactions, Spectral Theory
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.
Cite this paper
Guillot, J.-C. (2017) Weak Interactions in a Background of a Uniform Magnetic Field. A Mathe- matical Model for the Inverse β Decay. I. Open Access Library Journal, 4: e4142. https://doi.org/10.4236/oalib.1104142
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