We consider a Hamiltonian with cutoffs describing the weak decay
of spin 1 massive bosons into the full family of leptons. The Hamiltonian is
a self-adjoint operator in an appropriate Fock space with a unique ground
state. We prove a Mourre estimate and a limiting absorption principle above
the ground state energy and below the first threshold for a sufficiently small
coupling constant. As a corollary, we prove the absence of eigenvalues and absolute
continuity of the energy spectrum in the same spectral interval.
1. Introduction
In this article, we consider a mathematical model of the weak interaction as patterned according to the Standard Model in Quantum Field Theory (see [1, 2]). We choose the example of the weak decay of the intermediate vector bosons into the full family of leptons.
The mathematical framework involves fermionic Fock spaces for the leptons and bosonic Fock spaces for the vector bosons. The interaction is described in terms of annihilation and creation operators together with kernels which are square integrable with respect to momenta. The total Hamiltonian, which is the sum of the free energy of the particles and antiparticles and of the interaction, is a self-adjoint operator in the Fock space for the leptons and the vector bosons and it has an unique ground state in the Fock space for a sufficiently small coupling constant.
The weak interaction is one of the four fundamental interactions known up to now. But the weak interaction is the only one which does not generate bound states. As it is well known, it is not the case for the strong, electromagnetic, and gravitational interactions. Thus we are expecting that the spectrum of the Hamiltonian associated with every model of weak decays is absolutely continuous above the energy of the ground state, and this article is a first step towards a proof of such a statement. Moreover a scattering theory has to be established for every such Hamiltonian.
In this paper we establish a Mourre estimate and a limiting absorption principle for any spectral interval above the energy of the ground state and below the mass of the electron for a small coupling constant.
Our study of the spectral analysis of the total Hamiltonian is based on the conjugate operator method with a self-adjoint conjugate operator. The methods used in this article are taken largely from [3, 4] and are based on [5, 6]. Some of the results of this article have been announced in [7].
For other applications of the conjugate operator method see [8–19].
For related results about models in Quantum Field Theory see [20, 21] in the case of the Quantum Electrodynamics and [22] in the case of the weak interaction.
The paper is organized as follows. In Section 2, we give a precise definition of the model we consider. In Section 3, we state our main results and in the following sections, together with the appendix, detailed proofs of the results are given.
2. The Model
The weak decay of the intermediate bosons and involves the full family of leptons together with the bosons themselves, according to the Standard Model (see [1, formula (4.139)] and [2]).
The full family of leptons involves the electron and the positron , together with the associated neutrino and antineutrino , the muons and together with the associated neutrino and antineutrino and the tau leptons and together with the associated neutrino and antineutrino .
It follows from the Standard Model that neutrinos and antineutrinos are massless particles. Neutrinos are left handed, that is, neutrinos have helicity and antineutrinos are right handed, that is, antineutrinos have helicity .
In what follows, the mathematical model for the weak decay of the vector bosons and that we propose is based on the Standard Model, but we adopt a slightly more general point of view because we suppose that neutrinos and antineutrinos are both massless particles with helicity . We recover the physical situation as a particular case. We could also consider a model with massive neutrinos and antineutrinos built upon the Standard Model with neutrino mixing [23].
Let us sketch how we define a mathematical model for the weak decay of the vector bosons into the full family of leptons.
The energy of the free leptons and bosons is a self-adjoint operator in the corresponding Fock space (see below), and the main problem is associated with the interaction between the bosons and the leptons. Let us consider only the interaction between the bosons and the electrons, the positrons, and the corresponding neutrinos and antineutrinos. Other cases are strictly similar. In the Schrödinger representation the interaction is given by (see [1, page 159, equation (4.139)] and [2, page 308, equation (21.3.20)])
where , and are the Dirac matrices and and are the Dirac fields for , , and .
We have
Here where is the mass of the electron, and and are the normalized solutions to the Dirac equation (see [1, Appendix]).
The operators and (resp., and ) are the annihilation and creation operators for the electrons (resp., the positrons) satisfying the anticommutation relations (see below).
Similarly we define and by substituting the operators and for and with . The operators and (resp., and ) are the annihilation and creation operators for the neutrinos associated with the electrons (resp., the antineutrinos).
For the fields we have (see [24, Section 5.3])
Here where is the mass of the bosons . is the antiparticule of . The operators and (resp., and ) are the annihilation and creation operators for the bosons (resp., ) satisfying the canonical commutation relations. The vectors are the polarizations of the massive spin 1 bosons (see [24, Section 5.2]).
The interaction (2.1) is a formal operator and, in order to get a well-defined operator in the Fock space, one way is to adapt what Glimm and Jaffe have done in the case of the Yukawa Hamiltonian (see [25]). For that sake, we have to introduce a spatial cutoff such that , together with momentum cutoffs and for the Dirac fields and the fields, respectively.
Thus when one develops the interaction with respect to products of creation and annihilation operators, one gets a finite sum of terms associated with kernels of the form
where is the Fourier transform of . These kernels are square integrable.
In what follows, we consider a model involving terms of the above form but with more general square integrable kernels.
We follow the convention described in [24, Section 4.1] that we quote: “The state-vector will be taken to be symmetric under interchange of any bosons with each other, or any bosons with any fermions, and antisymmetric with respect to interchange of any two fermions with each other, in all cases, whether the particles are of the same species or not.’’ Thus, as it follows from [24, Section 4.2], fermionic creation and annihilation operators of different species of leptons will always anticommute.
Concerning our notations, from now on, denotes each species of leptons. denotes the electron the positron and the neutrinos , . denotes the muons , and the neutrinos and , and denotes the tau-leptons and the neutrinos and .
Let be the quantum variables of a massive lepton, where and is the spin polarization of particles and antiparticles. Let be the quantum variables of a massless lepton where and is the helicity of particles and antiparticles, and, finally, let be the quantum variables of the spin bosons and where and is the polarization of the vector bosons (see [24, Section 5]). We set for the leptons and for the bosons. Thus is the Hilbert space of each lepton and is the Hilbert space of each boson. The scalar product in , is defined by
Here
The Hilbert space for the weak decay of the vector bosons and is the Fock space for leptons and bosons that we now describe.
Let be any separable Hilbert space. Let (resp., ) denote the antisymmetric (resp., symmetric) th tensor power of . The fermionic (resp., bosonic) Fock space over , denoted by (resp., , is the direct sum
where . The state denotes the vacuum state in and in .
For every , is the fermionic Fock space for the corresponding species of leptons including the massive particle and antiparticle together with the associated neutrino and antineutrino, that is,
We have
with
Here (resp., ) is the number of massive fermionic particle (resp., antiparticles) and (resp., ) is the number of neutrinos (resp., antineutrinos). The vector is the associated vacuum state. The fermionic Fock space denoted by for the leptons is then
and is the vacuum state.
The bosonic Fock space for the vector bosons and , denoted by , is then
We have
where . Here (resp., ) is the number of bosons (resp., ). The vector is the corresponding vacuum.
The Fock space for the weak decay of the vector bosons and , denoted by , is thus
and is the vacuum state.
For every let denote the set of smooth vectors for which has a compact support and for all but finitely many . Let
Here is the algebraic tensor product.
Let denote the set of smooth vectors for which has a compact support and for all but finitely many .
Let
The set is dense in .
Let be a self-adjoint operator in such that is a core for . Its extension to is, by definition, the closure in of the operator with domain when , of the operator with domain when , and of the operator with domain when . Here is the operator identity on .
The extension of to is a self-adjoint operator for which is a core and it can be extended to . The extension of to is, by definition, the closure in of the operator with domain , where is the extension of to . The extension of to is a self-adjoint operator for which is a core.
Let be a self-adjoint operator in for which is a core. The extension of the self-adjoint operator is, by definition, the closure in of the operator with domain when , of the operator with domain when , and of the operator with domain when . The extension of to is a self-adjoint operator for which is a core.
We now define the creation and annihilation operators.
For each , (resp., ) is the fermionic annihilation (resp., fermionic creation) operator for the corresponding species of massive particle when and for the corresponding species of massive antiparticle when . The operators and are defined as usually (see, e.g., [20, 26]; see also the detailed definitions in [27]).
Similarly, for each , (resp., ) is the fermionic annihilation (resp., fermionic creation) operator for the corresponding species of neutrino when and for the corresponding species of antineutrino when . The operators and are defined in a standard way, but with the additional requirements that for any , , and , the operators and anticommutes, where stands either for a or for no symbol (see the detailed definitions in [27]).
The operator (resp., ) is the bosonic annihilation (resp., bosonic creation) operator for the boson when and for the boson when (see, e.g., [20, 26], or [27]). Note that commutes with and .
The following canonical anticommutation and commutation relations hold:
where we used the notation .
We recall that the following operators, with ,
are bounded operators in such that
where (resp., ) is (resp., ) or (resp., ).
The operators and satisfy similar anticommutaion relations (see, e.g., [28]).
The free Hamiltonian is given by
where
where is the mass of the bosons and such that .
The spectrum of is and is a simple eigenvalue with as eigenvector. The set of thresholds of , denoted by , is given by
and each set , , is a branch of absolutely continuous spectrum for .
The interaction, denoted by , is given by
where
The kernels , , are supposed to be functions.
The total Hamiltonian is then
where is a coupling constant.
The operator describes the decay of the bosons and into leptons. Because of the bare vacuum will not be an eigenvector of the total Hamiltonian for every as we expect from the physics.
Every kernel , computed in theoretical physics, contains a -distribution because of the conservation of the momentum (see [1] and [24, Section 4.4]). In what follows, we approximate the singular kernels by square integrable functions.
Thus, from now on, the kernels are supposed to satisfy the following hypothesis.
Hypothesis 2.1. For , , , we assume
Remark 2.2. A similar model can be written down for the weak decay of pions and (see [1, Section 6.2]).
Remark 2.3. The total Hamiltonian is more general than the one involved in the theory of weak interactions because, in the Standard Model, neutrinos have helicity and antineutrinos have helicity .
In the physical case, the Fock space, denoted by , is isomorphic to , with
The free Hamiltonian, now denoted by , is then given by
and the interaction, now denoted by , is the one obtained from by supposing that if . The total Hamiltonian, denoted by , is then given by . The results obtained in this paper for hold true for with obvious modifications.
Under Hypothesis 2.1 a well-defined operator on corresponds to the formal interaction as it follows.
The formal operator
is defined as a quadratic form on as
where , .
By mimicking the proof of [29, Theorem X.44], we get a closed operator, denoted by , associated with the quadratic form such that it is the unique operator in such that is a core for and
as quadratic forms on .
Similarly for the operator , we have the equality as quadratic forms
Again, there exists two closed operators and such that , , and is a core for and and such that
as quadratic forms on .
We will still denote by and () their extensions to . The set is then a core for and .
Thus
is a symmetric operator defined on .
We now want to prove that is essentially self-adjoint on by showing that and are relatively -bounded.
Once again, as above, for almost every , there exists closed operators in , denoted by and such that
as quadratic forms on .
We have that (resp., is a core for (resp., for ). We still denote by and their extensions to .
It then follows that the operator with domain is symmetric and can be written in the following form:
Let denote the operator number of massive leptons in , that is,
The operator is a positive self-adjoint operator in . We still denote by its extension to . The set is a core for .
We then have the following.
Proposition 2.4. For almost every , , , and for one has
Proof. The estimates (2.38) are examples of estimates (see [25]). The proof is quite similar to the proof of [20, Proposition 3.7]. Details can be found in [27] but are omitted here.
Let
Then is a self-adjoint operator in , and is a core for .
We get the following.
Proposition 2.5. One has
for every and every .
Proof. Suppose that . Let
We have
Therefore, for , (2.40) follows from Proposition 2.4.
We now have
Furthermore
for every .
By (2.40), (2.45), and (2.46), we finally get (2.41) for every . It then follows that (2.40) and (2.41) are verified for every .
We now prove that is a self-adjoint operator in for sufficiently small.
Theorem 2.6. Let be such that
Then for every satisfying , is a self-adjoint operator in with domain , and is a core for .
Proof. Let be in . We have
Note that
where
We further note that
for , and
Combining (2.48) with (2.40), (2.41), (2.51), and (2.52) we get for ,
by noting
By (2.53) the theorem follows from the Kato-Rellich theorem.
3. Main Results
In the sequel, we will make the following additional assumptions on the kernels .
Hypothesis 3.1. (i) For , , ,
(ii) There exists such that for , , ,
(iii) For , , , and
(iv) There exists such that for , , ,
Remark 3.2. Hypothesis 3.1(ii) is nothing but an infrared regularization of the kernels . In order to satisfy this hypothesis it is, for example, sufficient to suppose that
where is a smooth function of in the Schwartz space.
Hypothesis 3.1(iv), which is a sharp ultraviolet cutoff, is actually not necessary, and can be removed at the expense of some additional technicalities. However, in order to simplify the proof of Proposition 3.5, we will leave it.
Our first result is devoted to the existence of a ground state for together with the location of the spectrum of and of its absolutely continuous spectrum when is sufficiently small.
Theorem 3.3. Suppose that the kernels satisfy Hypotheses 2.1 and 3.1(i). Then there exists such that has a unique ground state for . Moreover
with .
According to Theorem 3.3 the ground state energy is a simple eigenvalue of , and our main results are concerned with a careful study of the spectrum of above the ground state energy. The spectral theory developed in this work is based on the conjugated operator method as described in [5, 6, 30]. Our choice of the conjugate operator denoted by is the second quantized dilation generator for the neutrinos.
Let denote the following operator in :
The operator is essentially self-adjoint on . Its second quantized version is a self-adjoint operator in . From the definition (2.8) of the space , the following operator in
is essentially self-adjoint on .
Let now be the following operator in :
Then is essentially self-adjoint on .
We will denote again by its extension to . Thus is essentially self-adjoint on and we still denote by its closure.
We also set
We then have the following.
Theorem 3.4. Suppose that the kernels satisfy Hypotheses 2.1 and 3.1. For any satisfying there exists such that, for , the following points are satisfied. (i)The spectrum of in is purely absolutely continuous.(ii)Limiting absorption principle.
For every and , in , the limits
exist uniformly for in any compact subset of . (iii) Pointwise decay in time.
Suppose and with . Then
as .
The proof of Theorem 3.4 is based on a positive commutator estimate, called the Mourre estimate, and on a regularity property of with respect to (see [5, 6, 30]). According to [4], the main ingredient of the proof is auxiliary operators associated with infrared cutoff Hamiltonians with respect to the momenta of the neutrinos that we now introduce.
Let , with on , on and .
For we set
where .
The operator is the interaction given by (2.23) and (2.24) and associated with the kernels . We then set
Let
The space is the Fock space for the massive leptons , and is the Fock space for the neutrinos and antineutrinos .
Set
We have
Set
We have
Set
We have
Set
We have on
Here, (resp., ) is the identity operator on (resp., ).
Define
We get
In order to implement the conjugate operator theory, we have to show that has a gap in its spectrum above its ground state.
We now set, for and ,
Let
and set
Let
Let
Let be such that
We set
where has been introduced in Hypothesis 3.1(iv).
Let us define the sequence by
where .
Let be such that
For we have
Set
We then get the following.
Proposition 3.5. Suppose that the kernels satisfy Hypotheses 2.1, 3.1(i), and 3.1(iv). Then there exists such that, for and , is a simple eigenvalue of and does not have spectrum in .
The proof of Proposition 3.5 is given in the appendix.
We now introduce the positive commutator estimates and the regularity property of with respect to in order to prove Theorem 3.4.
The operator has to be split into two pieces depending on .
Let
Since , and , we obtain (see [4])
Note that we also have
The operators , and are essentially self-adjoint on (see [5, Proposition 4.2.3]). We still denote by , and their closures. If denotes any of the operator , and , we have
The operators , , and are self-adjoint operators in , and we have
By (2.8), the following operators in , denoted by and , respectively,
are essentially self-adjoint on .
Let and be the following two operators in :
The operators and are essentially self-adjoint on . Still denoting by and their extensions to , and are essentially self-adjoint on and we still denote by and their closures.
We have
The operators , and are associated to the following -vector fields in respectively:
Let be any of these vector fields. We have
for some and we also have
where the 's are defined by (3.46) and (3.48) and fulfil bounded for .
Let be the corresponding flow generated by :
is a -flow and we have
induces a one-parameter group of unitary operators in defined by
Let , and be the flows associated with the vector fields , , and respectively.
Let , and be the corresponding one-parameter groups of unitary operators in . The operators , , and are the generators of , and respectively, that is,
Let
Let be any of the one-parameter groups , and . We set
and we have
Here is the flow associated to .
This yields, for any , (see [11, Lemma 2.8])
Proposition 3.6. Suppose that the kernels satisfy Hypothesis 2.1.
For every one has, for ,
Proof. We only prove (i), since (ii) and (iii) can be proved similarly. By (3.56) we have, for ,
It follows from (3.50) and (3.60) that
This yields (i) because is a core for . Moreover we get
In view of , the operators and are bounded, and there exists a constant such that
Similarly, we also get
Let be the interaction associated with the kernels , where the kernels satisfy Hypothesis 2.1.
We set
We have for (see [11, Lemma 2.7]),
According to [5, 6], in order to prove Theorem 3.4 we must prove that is locally of class , and in and that and are locally strictly conjugate to in .
Recall that is locally of class in if, for any , is of class ; that is, is twice continuously differentiable for all and all .
Thus, one of our main results is the following one.
Theorem 3.7. Suppose that the kernels satisfy Hypotheses 2.1, 3.1(i) and 3.1(iii). (a) is locally of class , , and in . (b) is locally of class in .
It follows from Theorem 3.7 that , , , and are defined as sesquilinear forms on , where the union is taken over all the compact subsets of .
Furthermore, by Proposition 3.6, Theorem 3.7 and [4, Lemma 29], we get for all and all ,
The following proposition allows us to compute , , and as sesquilinear forms. By Hypotheses 2.1 and 3.1(iii.a), the kernels belong to the domains of , , and .
Proposition 3.8. Suppose that the kernels satisfy Hypotheses 2.1 and 3.1(iii.a). Then (a)for all one has (i), (ii), (iii), (iv); (b)and(i), (ii), (iii), (iv).
Proof. Part follows from part by the uniform boundedness principle. For part , we only prove (i), since other statements can be proved similarly.
By (3.50), we obtain
for .
By (3.56)–(3.58) and Lebesgue's Theorem we then get for all
By (3.66), we obtain, for all ,
This concludes the proof of Proposition 3.8.
Combining (3.67) with Proposition 3.8, we finally get for every and every
We now introduce the Mourre inequality.
Let be the smallest integer such that
We have, for ,
Let
We choose such that and
Note that for and .
We set, for ,
Let
Let denote the ground state projection of . It follows from Proposition 3.5 that, for and ,
Note that
Set
We have
We further note that
By (3.72), (3.74), and (3.85), we obtain
as sesquilinear forms with respect to .
Furthermore, it follows from the virial theorem (see [6, Proposition 3.2] and Proposition 6.1) that
By (3.81) and (3.87) we then get, for ,
We then have the following.
Proposition 3.9. Suppose that the kernels satisfy Hypotheses 2.1 and 3.1. Then there exists and such that and
for and .
Let be the spectral projection for the operator associated with the interval , and let
Note that
Theorem 3.10. Suppose that the kernels satisfy Hypotheses 2.1 and 3.1. Then there exists and such that and
for and .
4. Existence of a Ground State and Location of the Absolutely Continuous Spectrum
We now prove Theorem 3.3. The scheme of the proof is quite well known (see [9, 31]). It follows from Proposition 3.5 that has a unique ground state, denoted by , in :
Therefore has an unique normalized ground state in , given by , where is the vacuum state in :
Since , there exists a subsequence , converging to such that converges weakly to a state . We have to prove that . By adapting the proof of Theorem in [22] (see also [20]), the key point is to estimate in order to show that
uniformly with respect to .
The estimate (4.3) is a consequence of the so-called “pull-through’’ formula as it follows.
Let denote the interaction associated with the kernels . We thus have
with
This yields
By adapting the proof of Propositions 2.4 and 2.5 we easily get
where .
Let us estimate . By (2.53), (2.54), (3.26), and (3.28) we have
Therefore
By (3.82), (A.11), and (4.9), there exists such that
uniformly in and .
By (4.6), (4.7), and (4.10) we get
By Hypothesis 3.1(i), there exists a constant depending on the kernels and such that
The existence of a ground state for follows by choosing sufficiently small, that is, , as in [20, 22]. By adapting the method developed in [32] (see [32, Corollary 3.4]), one proves that the ground state of is unique. We omit here the details.
Statements about are consequences of the existence of a ground state and follows from the existence of asymptotic Fock representations for the CAR associated with the 's. For , we define on the operators
By mimicking the proof given in [21, 31] one proves, under the hypothesis of Theorem 3.3 and for , that the strong limits of when exist for :
The operators satisfy the CAR and we have
where is the ground state of .
It then follows from (4.14) and (4.15) that the absolutely continuous spectrum of equals to . We omit the details (see [21, 31]).
5. Proof of the Mourre Inequality
We first prove Proposition 3.9. In view of Proposition 3.8(a)(iii) and (3.73), we have, as sesquilinear forms,
Let (resp., ) be the Fock space for the massive leptons (resp., the neutrinos and antineutrinos ).
We have
Let
We have
is the Fock space for the massive leptons and the bosons , and is the Fock space for the neutrinos and antineutrinos.
We have, as sesquilinear forms and with respect to (5.4),
where
and where is the identity operator in .
By mimicking the proofs of Propositions 2.4 and 2.5, we get, for every ,
Noting that uniformly with respect to , it follows from Hypotheses 2.1 and 3.1 that there exists a constant such that
This yields
Combining (5.1), (5.5) with (5.9), we obtain
We have
By (3.76), (3.81), and (5.11) we get
for .
This, together with (5.10), yields for
Setting
we get
for .
Proposition 3.9 is proved by setting and .
The proof of Theorem 3.10 is the consequence of the following two lemmas.
Lemma 5.1. Assume that the kernels satisfy Hypotheses 2.1 and 3.1(ii). Then there exists a constant such that
for and .
Proof. Let (resp., ) be the unique normalized ground state of (resp., ). We have
with
Combining (2.53) and (2.54) with (3.26)–(3.28) and (5.18), we get
It follows from Hypothesis 3.1(ii), (4.10), and (5.19) that there exists a constant such that
for and .
By (5.17), this proves Lemma 5.1.
Lemma 5.2. Suppose that the kernels satisfy Hypotheses 2.1 and 3.1(ii). Then there exists a constant such that
for and .
Proof. Let be an almost analytic extension of given by (3.78) satisfying
Note that . We thus have
Using the functional calculus based on this representation of , we get
Combining (2.53) and (2.54) with (3.26)–(3.28) and Hypothesis 3.1(ii), we get, for every and for ,
This yields
for some constant and for .
By mimicking the proof of (A.21) we show that there exists a constant such that
or .
Combining Lemma 5.1 and (5.24) with (5.25)–(5.27) we obtain
for some constant and for .
Using (5.22) and one concludes the proof of Lemma 5.2.
We now prove Theorem 3.10.
Proof. It follows from Proposition 3.9 that
for and .
This yields