Advances in Mathematical Physics
VolumeΒ 2009Β (2009), Article IDΒ 978903, 52 pages
doi:10.1155/2009/978903
Research Article

Spectral Theory for a Mathematical Model of the Weak Interaction—Part I: The Decay of the Intermediate Vector Bosons π‘Š Β±

J.-M. Barbaroux1,2Β and J.-C. Guillot3

1Centre de Physique Théorique, Centre National de la Recherche Scientique (CNRS), Luminy Case 907, 13288 Marseille Cedex 9, France
2Département de Mathématiques, Université du Sud Toulon-Var, 83957 La Garde Cedex, France
3Centre de Mathématiques Appliquées, École Polytechnique, UMR-CNRS 7641, 91128 Palaiseau Cedex, France

Received 28 April 2009; Accepted 13 August 2009

Academic Editor: ValentinΒ Zagrebnov

Copyright Β© 2009 J.-M. Barbaroux and J.-C. Guillot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 π‘Š Β± 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 π‘Š + and π‘Š βˆ’ 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 𝑒 βˆ’ and the positron 𝑒 + , together with the associated neutrino 𝜈 𝑒 and antineutrino 𝜈 𝑒 , the muons πœ‡ βˆ’ and πœ‡ + together with the associated neutrino 𝜈 πœ‡ and antineutrino 𝜈 πœ‡ , and the tau leptons 𝜏 βˆ’ and 𝜏 + together with the associated neutrino 𝜈 𝜏 and antineutrino 𝜈 𝜏 .

It follows from the Standard Model that neutrinos and antineutrinos are massless particles. Neutrinos are left handed, that is, neutrinos have helicity βˆ’ 1 / 2 and antineutrinos are right handed, that is, antineutrinos have helicity + 1 / 2 .

In what follows, the mathematical model for the weak decay of the vector bosons π‘Š + and π‘Š βˆ’ 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 Β± 1 / 2 . 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 π‘Š Β± 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)]) ξ€œ 𝐼 = d 3 π‘₯ Ξ¨ 𝑒 ( π‘₯ ) 𝛾 𝛼 ξ€· 1 βˆ’ 𝛾 5 ξ€Έ Ξ¨ 𝜈 𝑒 ( π‘₯ ) π‘Š 𝛼 ( ξ€œ π‘₯ ) + d 3 π‘₯ Ξ¨ 𝜈 𝑒 ( π‘₯ ) 𝛾 𝛼 ξ€· 1 βˆ’ 𝛾 5 ξ€Έ Ξ¨ 𝑒 ( π‘₯ ) π‘Š 𝛼 ( π‘₯ ) βˆ— , ( 2 . 1 ) where 𝛾 𝛼 , 𝛼 = 0 , 1 , 2 , 3 and 𝛾 5 are the Dirac matrices and Ξ¨ β‹… ( π‘₯ ) and Ξ¨ β‹… ( π‘₯ ) are the Dirac fields for 𝑒 βˆ’ , 𝑒 + , 𝜈 𝑒 , and 𝜈 𝑒 .

We have Ξ¨ 𝑒 ξ‚€ 1 ( π‘₯ ) =  2 πœ‹ 3 / 2  𝑠 = Β± 1 / 2 ξ€œ d 3 𝑝  𝑏 𝑒 , + ( 𝑝 , 𝑠 ) 𝑒 ( 𝑝 , 𝑠 ) √ 𝑝 0 e 𝑖 𝑝 β‹… π‘₯ + 𝑏 βˆ— 𝑒 , βˆ’ ( 𝑝 , 𝑠 ) 𝑣 ( 𝑝 , 𝑠 ) √ 𝑝 0 e βˆ’ 𝑖 𝑝 β‹… π‘₯ ξƒͺ , Ξ¨ 𝑒 ( π‘₯ ) = Ξ¨ 𝑒 ( π‘₯ ) † 𝛾 0 . ( 2 . 2 ) Here 𝑝 0 = ( | 𝑝 | 2 + π‘š 2 𝑒 ) 1 / 2 where π‘š 𝑒 > 0 is the mass of the electron, and 𝑒 ( 𝑝 , 𝑠 ) and 𝑣 ( 𝑝 , 𝑠 ) are the normalized solutions to the Dirac equation (see [1, Appendix]).

The operators 𝑏 𝑒 , + ( 𝑝 , 𝑠 ) and 𝑏 βˆ— 𝑒 , + ( 𝑝 , 𝑠 ) (resp., 𝑏 𝑒 , βˆ’ ( 𝑝 , 𝑠 ) and 𝑏 βˆ— 𝑒 , βˆ’ ( 𝑝 , 𝑠 ) ) are the annihilation and creation operators for the electrons (resp., the positrons) satisfying the anticommutation relations (see below).

Similarly we define Ξ¨ 𝜈 𝑒 ( π‘₯ ) and Ξ¨ 𝜈 𝑒 ( π‘₯ ) by substituting the operators 𝑐 𝜈 𝑒 , Β± ( 𝑝 , 𝑠 ) and 𝑐 βˆ— 𝜈 𝑒 , Β± ( 𝑝 , 𝑠 ) for 𝑏 𝑒 , Β± ( 𝑝 , 𝑠 ) and 𝑏 βˆ— 𝑒 , Β± ( 𝑝 , 𝑠 ) with 𝑝 0 = | 𝑝 | . The operators 𝑐 𝜈 𝑒 , + ( 𝑝 , 𝑠 ) and 𝑐 βˆ— 𝜈 𝑒 , + ( 𝑝 , 𝑠 ) (resp., 𝑐 𝜈 𝑒 , βˆ’ ( 𝑝 , 𝑠 ) and 𝑐 βˆ— 𝜈 𝑒 , βˆ’ ( 𝑝 , 𝑠 ) ) are the annihilation and creation operators for the neutrinos associated with the electrons (resp., the antineutrinos).

For the π‘Š 𝛼 fields we have (see [24, Section  5.3]) π‘Š 𝛼 ξ‚€ 1 ( π‘₯ ) =  2 πœ‹ 3 / 2  πœ† = βˆ’ 1 , 0 , 1 ξ€œ d 3 π‘˜ √ 2 π‘˜ 0 ξ€· πœ– 𝛼 ( π‘˜ , πœ† ) π‘Ž + ( π‘˜ , πœ† ) e 𝑖 π‘˜ β‹… π‘₯ + πœ– βˆ— 𝛼 ( π‘˜ , πœ† ) π‘Ž βˆ— βˆ’ ( π‘˜ , πœ† ) e βˆ’ 𝑖 π‘˜ β‹… π‘₯ ξ€Έ . ( 2 . 3 ) Here π‘˜ 0 = ( | π‘˜ | 2 + π‘š 2 π‘Š ) 1 / 2 where π‘š π‘Š > 0 is the mass of the bosons π‘Š Β± . π‘Š + is the antiparticule of π‘Š βˆ’ . The operators π‘Ž + ( π‘˜ , πœ† ) and π‘Ž βˆ— + ( π‘˜ , πœ† ) (resp., π‘Ž βˆ’ ( π‘˜ , πœ† ) and π‘Ž βˆ— βˆ’ ( π‘˜ , πœ† ) ) are the annihilation and creation operators for the bosons π‘Š βˆ’ (resp., π‘Š + ) satisfying the canonical commutation relations. The vectors πœ– 𝛼 ( π‘˜ , πœ† ) are the polarizations of the massive spin 1 bosons π‘Š Β± (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 𝑔 ( π‘₯ ) such that 𝑔 ∈ 𝐿 1 ( ℝ 3 ) , together with momentum cutoffs πœ’ ( 𝑝 ) and 𝜌 ( π‘˜ ) for the Dirac fields and the π‘Š πœ‡ fields, respectively.

Thus when one develops the interaction 𝐼 with respect to products of creation and annihilation operators, one gets a finite sum of terms associated with kernels of the form πœ’ ξ€· 𝑝 1 ξ€Έ πœ’ ξ€· 𝑝 2 ξ€Έ 𝜌 ξ€· 𝑝 ( π‘˜ ) Μ‚ 𝑔 1 + 𝑝 2 ξ€Έ , βˆ’ π‘˜ ( 2 . 4 ) where Μ‚ 𝑔 is the Fourier transform of 𝑔 . 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, β„“ ∈ { 1 , 2 , 3 } denotes each species of leptons. β„“ = 1 denotes the electron 𝑒 βˆ’ the positron 𝑒 + and the neutrinos 𝜈 𝑒 , 𝜈 𝑒 . β„“ = 2 denotes the muons πœ‡ βˆ’ , πœ‡ + and the neutrinos 𝜈 πœ‡ and 𝜈 πœ‡ , and β„“ = 3 denotes the tau-leptons and the neutrinos 𝜈 𝜏 and 𝜈 𝜏 .

Let πœ‰ 1 = ( 𝑝 1 , 𝑠 1 ) be the quantum variables of a massive lepton, where 𝑝 1 ∈ ℝ 3 and 𝑠 1 ∈ { βˆ’ 1 / 2 , 1 / 2 } is the spin polarization of particles and antiparticles. Let πœ‰ 2 = ( 𝑝 2 , 𝑠 2 ) be the quantum variables of a massless lepton where 𝑝 2 ∈ ℝ 3 and 𝑠 2 ∈ { βˆ’ 1 / 2 , 1 / 2 } is the helicity of particles and antiparticles, and, finally, let πœ‰ 3 = ( π‘˜ , πœ† ) be the quantum variables of the spin 1 bosons π‘Š + and π‘Š βˆ’ where π‘˜ ∈ ℝ 3 and πœ† ∈ { βˆ’ 1 , 0 , 1 } is the polarization of the vector bosons (see [24, Section  5]). We set Ξ£ 1 = ℝ 3 Γ— { βˆ’ 1 / 2 , 1 / 2 } for the leptons and Ξ£ 2 = ℝ 3 Γ— { βˆ’ 1 , 0 , 1 } for the bosons. Thus 𝐿 2 ( Ξ£ 1 ) is the Hilbert space of each lepton and 𝐿 2 ( Ξ£ 2 ) is the Hilbert space of each boson. The scalar product in 𝐿 2 ( Ξ£ 𝑗 ) , 𝑗 = 1 , 2 is defined by ( ξ€œ 𝑓 , 𝑔 ) = Ξ£ 𝑗 𝑓 ( πœ‰ ) 𝑔 ( πœ‰ ) d πœ‰ , 𝑗 = 1 , 2 . ( 2 . 5 ) Here ξ€œ Ξ£ 1 d  πœ‰ = 𝑠 = + 1 / 2 , βˆ’ 1 / 2 ξ€œ d ξ€œ 𝑝 , Ξ£ 2 d  πœ‰ = πœ† = 0 , 1 , βˆ’ 1 ξ€œ d ξ€· π‘˜ , 𝑝 , π‘˜ ∈ ℝ 3 ξ€Έ . ( 2 . 6 )

The Hilbert space for the weak decay of the vector bosons π‘Š + and π‘Š βˆ’ is the Fock space for leptons and bosons that we now describe.

Let 𝔖 be any separable Hilbert space. Let ⨂ 𝑛 π‘Ž 𝔖 (resp., ⨂ 𝑛 𝑠 𝔖 ) denote the antisymmetric (resp., symmetric) 𝑛 th tensor power of 𝔖 . The fermionic (resp., bosonic) Fock space over 𝔖 , denoted by 𝔉 π‘Ž ( 𝔖 ) (resp., 𝔉 𝑠 ( 𝔖 ) ) , is the direct sum 𝔉 π‘Ž ( 𝔖 ) = ∞  𝑛 𝑛 = 0  π‘Ž 𝔖  r e s p . , 𝔉 𝑠 ( 𝔖 ) = ∞  𝑛 𝑛 = 0  𝑠 𝔖 ξƒͺ , ( 2 . 7 ) where ⨂ 0 π‘Ž ⨂ 𝔖 = 0 𝑠 𝔖 ≑ β„‚ . The state Ξ© = ( 1 , 0 , 0 , … , 0 , … ) denotes the vacuum state in 𝔉 π‘Ž ( 𝔖 ) and in 𝔉 𝑠 ( 𝔖 ) .

For every β„“ , 𝔉 β„“ 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, 𝔉 β„“ = 4  𝔉 π‘Ž ξ€· 𝐿 2 ξ€· Ξ£ 1 ξ€Έ ξ€Έ β„“ = 1 , 2 , 3 . ( 2 . 8 ) We have 𝔉 β„“ =  π‘ž β„“ β‰₯ 0 , π‘ž β„“ β‰₯ 0 , π‘Ÿ β„“ β‰₯ 0 , π‘Ÿ β„“ β‰₯ 0 𝔉 ( π‘ž β„“ , π‘ž β„“ , π‘Ÿ β„“ , π‘Ÿ β„“ ) β„“ ( 2 . 9 ) with 𝔉 ( π‘ž β„“ , π‘ž β„“ , π‘Ÿ β„“ , π‘Ÿ β„“ ) β„“ =  π‘ž β„“  π‘Ž 𝐿 2 ξ€· Ξ£ 1 ξ€Έ ξƒͺ βŠ— βŽ› ⎜ ⎜ ⎝ π‘ž β„“  π‘Ž 𝐿 2 ξ€· Ξ£ 1 ξ€Έ ⎞ ⎟ ⎟ ⎠ βŠ—  π‘Ÿ β„“  π‘Ž 𝐿 2 ξ€· Ξ£ 1 ξ€Έ ξƒͺ βŠ— βŽ› ⎜ ⎜ ⎝ π‘Ÿ β„“  π‘Ž 𝐿 2 ξ€· Ξ£ 1 ξ€Έ ⎞ ⎟ ⎟ ⎠ . ( 2 . 1 0 ) Here π‘ž β„“ (resp., π‘ž β„“ ) is the number of massive fermionic particle (resp., antiparticles) and π‘Ÿ β„“ (resp., π‘Ÿ β„“ ) is the number of neutrinos (resp., antineutrinos). The vector Ξ© β„“ is the associated vacuum state. The fermionic Fock space denoted by 𝔉 𝐿 for the leptons is then 𝔉 𝐿 = 3  β„“ = 1 𝔉 β„“ , ( 2 . 1 1 ) and Ξ© 𝐿 = ⨂ 3 β„“ = 1 Ξ© β„“ is the vacuum state.

The bosonic Fock space for the vector bosons π‘Š + and π‘Š βˆ’ , denoted by 𝔉 π‘Š , is then 𝔉 π‘Š = 𝔉 𝑠 ξ€· 𝐿 2 ξ€· Ξ£ 2 ξ€Έ ξ€Έ βŠ— 𝔉 𝑠 ξ€· 𝐿 2 ξ€· Ξ£ 2 ξ€Έ ξ€Έ ≃ 𝔉 𝑠 ξ€· 𝐿 2 ξ€· Ξ£ 2 ξ€Έ βŠ• 𝐿 2 ξ€· Ξ£ 2 . ξ€Έ ξ€Έ ( 2 . 1 2 ) We have 𝔉 π‘Š =  𝑑 β‰₯ 0 , 𝑑 β‰₯ 0 𝔉 ( 𝑑 , π‘Š 𝑑 ) , ( 2 . 1 3 ) where 𝔉 ( 𝑑 , π‘Š 𝑑 ) ⨂ = ( 𝑑 𝑠 𝐿 2 ( Ξ£ 2 ⨂ ) ) βŠ— ( 𝑑 𝑠 𝐿 2 ( Ξ£ 2 ) ) . Here 𝑑 (resp., 𝑑 ) is the number of bosons π‘Š βˆ’ (resp., π‘Š + ). The vector Ξ© π‘Š is the corresponding vacuum.

The Fock space for the weak decay of the vector bosons π‘Š + and π‘Š βˆ’ , denoted by 𝔉 , is thus 𝔉 = 𝔉 𝐿 βŠ— 𝔉 π‘Š , ( 2 . 1 4 ) and Ξ© = Ξ© 𝐿 βŠ— Ξ© π‘Š is the vacuum state.

For every β„“ ∈ { 1 , 2 , 3 } let 𝔇 β„“ denote the set of smooth vectors πœ“ β„“ ∈ 𝔉 β„“ for which πœ“ ( π‘ž β„“ , π‘ž β„“ , π‘Ÿ β„“ , π‘Ÿ β„“ ) β„“ has a compact support and πœ“ ( π‘ž β„“ , π‘ž β„“ , π‘Ÿ β„“ , π‘Ÿ β„“ ) β„“ = 0 for all but finitely many ( π‘ž β„“ , π‘ž β„“ , π‘Ÿ β„“ , π‘Ÿ β„“ ) . Let 𝔇 𝐿 = ξ‚Š ⨂ 3 β„“ = 1 𝔇 β„“ . ( 2 . 1 5 ) Here ξ‚Š ⨂ is the algebraic tensor product.

Let 𝔇 π‘Š denote the set of smooth vectors πœ™ ∈ 𝔉 π‘Š for which πœ™ ( 𝑑 , 𝑑 ) has a compact support and πœ™ ( 𝑑 , 𝑑 ) = 0 for all but finitely many ( 𝑑 , 𝑑 ) .

Let 𝔇 = 𝔇 𝐿  βŠ— 𝔇 π‘Š . ( 2 . 1 6 ) The set 𝔇 is dense in 𝔉 .

Let 𝐴 β„“ be a self-adjoint operator in 𝔉 β„“ such that 𝔇 β„“ is a core for 𝐴 β„“ . Its extension to 𝔉 𝐿 is, by definition, the closure in 𝔉 𝐿 of the operator 𝐴 1 βŠ— 1 2 βŠ— 1 3 with domain 𝔇 𝐿 when β„“ = 1 , of the operator 1 1 βŠ— 𝐴 2 βŠ— 1 3 with domain 𝔇 𝐿 when β„“ = 2 , and of the operator 1 1 βŠ— 1 2 βŠ— 𝐴 3 with domain 𝔇 𝐿 when β„“ = 3 . Here 1 β„“ is the operator identity on 𝔉 β„“ .

The extension of 𝐴 β„“ to 𝔉 𝐿 is a self-adjoint operator for which 𝔇 𝐿 is a core and it can be extended to 𝔉 . The extension of 𝐴 β„“ to 𝔉 is, by definition, the closure in 𝔉 of the operator  𝐴 β„“ βŠ— 1 π‘Š with domain 𝔇 , where  𝐴 β„“ is the extension of 𝐴 β„“ to 𝔉 𝐿 . The extension of 𝐴 β„“ to 𝔉 is a self-adjoint operator for which 𝔇 is a core.

Let 𝐡 be a self-adjoint operator in 𝔉 π‘Š for which 𝔇 π‘Š is a core. The extension of the self-adjoint operator 𝐴 β„“ βŠ— 𝐡 is, by definition, the closure in 𝔉 of the operator 𝐴 1 βŠ— 1 2 βŠ— 1 3 βŠ— 𝐡 with domain 𝔇 when β„“ = 1 , of the operator 1 1 βŠ— 𝐴 2 βŠ— 1 3 βŠ— 𝐡 with domain 𝔇 when β„“ = 2 , and of the operator 1 1 βŠ— 1 2 βŠ— 𝐴 3 βŠ— 𝐡 with domain 𝔇 when β„“ = 3 . The extension of 𝐴 β„“ βŠ— 𝐡 to 𝔉 is a self-adjoint operator for which 𝔇 is a core.

We now define the creation and annihilation operators.

For each β„“ = 1 , 2 , 3 , 𝑏 β„“ , πœ– ( πœ‰ 1 ) (resp., 𝑏 βˆ— β„“ , πœ– ( πœ‰ 1 ) ) is the fermionic annihilation (resp., fermionic creation) operator for the corresponding species of massive particle when πœ– = + and for the corresponding species of massive antiparticle when πœ– = βˆ’ . The operators 𝑏 β„“ , πœ– ( πœ‰ 1 ) and 𝑏 βˆ— β„“ , πœ– ( πœ‰ 1 ) are defined as usually (see, e.g., [20, 26]; see also the detailed definitions in [27]).

Similarly, for each β„“ = 1 , 2 , 3 , 𝑐 β„“ , πœ– ( πœ‰ 2 ) (resp., 𝑐 βˆ— β„“ , πœ– ( πœ‰ 2 ) ) is the fermionic annihilation (resp., fermionic creation) operator for the corresponding species of neutrino when πœ– = + and for the corresponding species of antineutrino when πœ– = βˆ’ . The operators 𝑐 β„“ , πœ– ( πœ‰ 2 ) and 𝑐 βˆ— β„“ , πœ– ( πœ‰ 2 ) are defined in a standard way, but with the additional requirements that for any β„“ , β„“ ξ…ž , πœ– and πœ– ξ…ž , the operators 𝑏 β™― β„“ , πœ– ( πœ‰ 1 ) and 𝑐 β™― β„“ β€² , πœ– β€² ( πœ‰ 2 ) anticommutes, where β™― stands either for a βˆ— or for no symbol (see the detailed definitions in [27]).

The operator π‘Ž πœ– ( πœ‰ 3 ) (resp., π‘Ž βˆ— πœ– ( πœ‰ 3 ) ) is the bosonic annihilation (resp., bosonic creation) operator for the boson π‘Š βˆ’ when πœ– = + and for the boson π‘Š + when πœ– = βˆ’ (see, e.g., [20, 26], or [27]). Note that π‘Ž β™― ( πœ‰ 3 ) commutes with 𝑏 β™― β„“ , πœ– ( πœ‰ 1 ) and 𝑐 β™― β„“ β€² , πœ– β€² ( πœ‰ 2 ) .

The following canonical anticommutation and commutation relations hold:  𝑏 β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ , 𝑏 βˆ— β„“ β€² , πœ– β€² ξ€· πœ‰ ξ…ž 1 ξ€Έ  = 𝛿 β„“ β„“ β€² 𝛿 πœ– πœ– β€² 𝛿 ξ€· πœ‰ 1 βˆ’ πœ‰ ξ…ž 1 ξ€Έ ,  𝑐 β„“ , πœ– ξ€· πœ‰ 2 ξ€Έ , 𝑐 βˆ— β„“ β€² , πœ– β€² ξ€· πœ‰ ξ…ž 2 ξ€Έ  = 𝛿 β„“ β„“ β€² 𝛿 πœ– πœ– β€² 𝛿 ξ€· πœ‰ 2 βˆ’ πœ‰ ξ…ž 2 ξ€Έ , ξ€Ί π‘Ž πœ– ξ€· πœ‰ 3 ξ€Έ , π‘Ž βˆ— πœ– β€² ξ€· πœ‰ ξ…ž 3 ξ€Έ ξ€» = 𝛿 πœ– πœ– β€² 𝛿 ξ€· πœ‰ 3 βˆ’ πœ‰ ξ…ž 3 ξ€Έ , ξ€½ 𝑏 β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ , 𝑏 β„“ β€² , πœ– β€² ξ€· πœ‰ ξ…ž 1 = ξ€½ 𝑐 ξ€Έ ξ€Ύ β„“ , πœ– ξ€· πœ‰ 2 ξ€Έ , 𝑐 β„“ β€² , πœ– β€² ξ€· πœ‰ ξ…ž 2 ξ€Ί π‘Ž ξ€Έ ξ€Ύ = 0 , πœ– ξ€· πœ‰ 3 ξ€Έ , π‘Ž πœ– β€² ξ€· πœ‰ ξ…ž 3 ξ€½ 𝑏 ξ€Έ ξ€» = 0 , β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ , 𝑐 β„“ β€² , πœ– β€² ξ€· πœ‰ 2 =  𝑏 ξ€Έ ξ€Ύ β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ , 𝑐 βˆ— β„“ β€² , πœ– β€² ξ€· πœ‰ 2 ξ€Έ  ξ€Ί 𝑏 = 0 , β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ , π‘Ž πœ– β€² ξ€· πœ‰ 3 = ξ€Ί 𝑏 ξ€Έ ξ€» β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ , π‘Ž βˆ— πœ– β€² ξ€· πœ‰ 3 = ξ€Ί 𝑐 ξ€Έ ξ€» β„“ , πœ– ξ€· πœ‰ 2 ξ€Έ , π‘Ž πœ– β€² ξ€· πœ‰ 3 = ξ€Ί 𝑐 ξ€Έ ξ€» β„“ , πœ– ξ€· πœ‰ 2 ξ€Έ , π‘Ž βˆ— πœ– β€² ξ€· πœ‰ 3 ξ€Έ ξ€» = 0 , ( 2 . 1 7 ) where we used the notation 𝛿 ( πœ‰ 𝑗 βˆ’ πœ‰ ξ…ž 𝑗 ) = 𝛿 πœ† πœ† β€² 𝛿 ( π‘˜ βˆ’ π‘˜ ξ…ž ) .

We recall that the following operators, with πœ‘ ∈ 𝐿 2 ( Ξ£ 1 ) , 𝑏 β„“ , πœ– ( ξ€œ πœ‘ ) = Ξ£ 1 𝑏 β„“ , πœ– ( πœ‰ ) πœ‘ ( πœ‰ ) d πœ‰ , 𝑐 β„“ , πœ– ( ξ€œ πœ‘ ) = Ξ£ 1 𝑐 β„“ , πœ– ( πœ‰ ) πœ‘ ( πœ‰ ) d 𝑏 πœ‰ , βˆ— β„“ , πœ– ξ€œ ( πœ‘ ) = Ξ£ 1 𝑏 βˆ— β„“ , πœ– ( πœ‰ ) πœ‘ ( πœ‰ ) d πœ‰ , 𝑐 βˆ— β„“ , πœ– ξ€œ ( πœ‘ ) = Ξ£ 1 𝑐 βˆ— β„“ , πœ– ( πœ‰ ) πœ‘ ( πœ‰ ) d πœ‰ ( 2 . 1 8 ) are bounded operators in 𝔉 such that β€– β€– 𝑏 β™― β„“ , πœ– β€– β€– = β€– β€– 𝑐 ( πœ‘ ) β™― β„“ , πœ– β€– β€– ( πœ‘ ) = β€– πœ‘ β€– 𝐿 2 , ( 2 . 1 9 ) where 𝑏 β™― (resp., 𝑐 β™― ) is 𝑏 (resp., 𝑐 ) or 𝑏 βˆ— (resp., 𝑐 βˆ— ).

The operators 𝑏 β™― β„“ , πœ– ( πœ‘ ) and 𝑐 β™― β„“ , πœ– ( πœ‘ ) satisfy similar anticommutaion relations (see, e.g., [28]).

The free Hamiltonian 𝐻 0 is given by 𝐻 0 = 𝐻 0 ( 1 ) + 𝐻 0 ( 2 ) + 𝐻 0 ( 3 ) = 3  β„“ = 1  0 π‘₯ 0 2 0 0 𝑑 πœ– = Β± ξ€œ 𝑀 β„“ ( 1 ) ξ€· πœ‰ 1 ξ€Έ 𝑏 βˆ— β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ 𝑏 β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ d πœ‰ 1 + 3  β„“ = 1  0 π‘₯ 0 2 0 0 𝑑 πœ– = Β± ξ€œ 𝑀 β„“ ( 2 ) ξ€· πœ‰ 2 ξ€Έ 𝑐 βˆ— β„“ , πœ– ξ€· πœ‰ 2 ξ€Έ 𝑐 β„“ , πœ– ξ€· πœ‰ 2 ξ€Έ d πœ‰ 2 +  πœ– = Β± ξ€œ 𝑀 ( 3 ) ξ€· πœ‰ 3 ξ€Έ π‘Ž βˆ— πœ– ξ€· πœ‰ 3 ξ€Έ π‘Ž πœ– ξ€· πœ‰ 3 ξ€Έ d πœ‰ 3 , ( 2 . 2 0 ) where 𝑀 β„“ ( 1 ) ξ€· πœ‰ 1 ξ€Έ = ξ‚€ | | 𝑝 1 | | 2 + π‘š 2 β„“  1 / 2 , w i t h 0 < π‘š 1 < π‘š 2 < π‘š 3 , 𝑀 β„“ ( 2 ) ξ€· πœ‰ 2 ξ€Έ = | | 𝑝 2 | | , 𝑀 ( 3 ) ξ€· πœ‰ 3 ξ€Έ = ξ‚€ | | π‘˜ | | 2 + π‘š 2 π‘Š  1 / 2 , ( 2 . 2 1 ) where π‘š π‘Š is the mass of the bosons π‘Š + and π‘Š βˆ’ such that π‘š π‘Š > π‘š 3 .

The spectrum of 𝐻 0 is [ 0 , ∞ ) and 0 is a simple eigenvalue with Ξ© as eigenvector. The set of thresholds of 𝐻 0 , denoted by 𝑇 , is given by ξ€½ 𝑇 = 𝑝 π‘š 1 + π‘ž π‘š 2 + π‘Ÿ π‘š 3 + 𝑠 π‘š π‘Š ; ( 𝑝 , π‘ž , π‘Ÿ , 𝑠 ) ∈ β„• 4 ξ€Ύ , , 𝑝 + π‘ž + π‘Ÿ + 𝑠 β‰₯ 1 ( 2 . 2 2 ) and each set [ 𝑑 , ∞ ) , 𝑑 ∈ 𝑇 , is a branch of absolutely continuous spectrum for 𝐻 0 .

The interaction, denoted by 𝐻 𝐼 , is given by 𝐻 𝐼 = 2  𝛼 = 1 𝐻 𝐼 ( 𝛼 ) , ( 2 . 2 3 ) where 𝐻 𝐼 ( 1 ) = 3  β„“ = 1  0 π‘₯ 0 2 0 0 𝑑 πœ– β‰  πœ– β€² ξ€œ 𝐺 ( 1 ) β„“ , πœ– , πœ– β€² ξ€· πœ‰ 1 , πœ‰ 2 , πœ‰ 3 ξ€Έ 𝑏 βˆ— β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ 𝑐 βˆ— β„“ , πœ– β€² ξ€· πœ‰ 2 ξ€Έ π‘Ž πœ– ξ€· πœ‰ 3 ξ€Έ d πœ‰ 1 d πœ‰ 2 d πœ‰ 3 + 3  β„“ = 1  0 π‘₯ 0 2 0 0 𝑑 πœ– β‰  πœ– β€² ξ€œ 𝐺 ( 1 ) β„“ , πœ– , πœ– β€² ( πœ‰ 1 , πœ‰ 2 , πœ‰ 3 ) π‘Ž βˆ— πœ– ξ€· πœ‰ 3 ξ€Έ 𝑐 β„“ , πœ– β€² ξ€· πœ‰ 2 ξ€Έ 𝑏 β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ d πœ‰ 1 d πœ‰ 2 d πœ‰ 3 , 𝐻 𝐼 ( 2 ) = 3  β„“ = 1  πœ– β‰  πœ– β€² ξ€œ 𝐺 ( 2 ) β„“ , πœ– , πœ– β€² ξ€· πœ‰ 1 , πœ‰ 2 , πœ‰ 3 ξ€Έ 𝑏 βˆ— β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ 𝑐 βˆ— β„“ , πœ– β€² ξ€· πœ‰ 2 ξ€Έ π‘Ž βˆ— πœ– ξ€· πœ‰ 3 ξ€Έ d πœ‰ 1 d πœ‰ 2 d πœ‰ 3 + 3  β„“ = 1  πœ– β‰  πœ– β€² ξ€œ 𝐺 ( 2 ) β„“ , πœ– , πœ– β€² ξ€· πœ‰ 1 , πœ‰ 2 , πœ‰ 3 ξ€Έ π‘Ž πœ– ξ€· πœ‰ 3 ξ€Έ 𝑐 β„“ , πœ– β€² ξ€· πœ‰ 2 ξ€Έ 𝑏 β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ d πœ‰ 1 d πœ‰ 2 d πœ‰ 3 . ( 2 . 2 4 ) The kernels 𝐺 ( 2 ) β„“ , πœ– , πœ– β€² ( β‹… , β‹… , β‹… ) , 𝛼 = 1 , 2 , are supposed to be functions.

The total Hamiltonian is then 𝐻 = 𝐻 0 + 𝑔 𝐻 𝐼 , 𝑔 > 0 , ( 2 . 2 5 ) where 𝑔 is a coupling constant.

The operator 𝐻 𝐼 ( 1 ) describes the decay of the bosons π‘Š + and π‘Š βˆ’ into leptons. Because of 𝐻 𝐼 ( 2 ) the bare vacuum will not be an eigenvector of the total Hamiltonian for every 𝑔 > 0 as we expect from the physics.

Every kernel 𝐺 β„“ , πœ– , πœ– β€² ( πœ‰ 1 , πœ‰ 2 , πœ‰ 3 ) , computed in theoretical physics, contains a 𝛿 -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 𝐺 ( 𝛼 ) β„“ , πœ– , πœ– β€² are supposed to satisfy the following hypothesis.

Hypothesis 2.1. For 𝛼 = 1 , 2 , β„“ = 1 , 2 , 3 , πœ– , πœ– ξ…ž = Β± , we assume 𝐺 ( 𝛼 ) β„“ , πœ– , πœ– β€² ξ€· πœ‰ 1 , πœ‰ 2 , πœ‰ 3 ξ€Έ ∈ 𝐿 2 ξ€· Ξ£ 1 Γ— Ξ£ 1 Γ— Ξ£ 2 ξ€Έ . ( 2 . 2 6 )

Remark 2.2. A similar model can be written down for the weak decay of pions πœ‹ βˆ’ and πœ‹ + (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 βˆ’ 1 / 2 and antineutrinos have helicity 1 / 2 .
In the physical case, the Fock space, denoted by 𝔉 ξ…ž , is isomorphic to 𝔉 ξ…ž 𝐿 βŠ— 𝔉 π‘Š , with 𝔉 ξ…ž 𝐿 = 3  β„“ = 1 𝔉 ξ…ž β„“ , 𝔉 ξ…ž β„“ =  2  π‘Ž 𝐿 2 ξ€· Ξ£ 1 ξ€Έ ξƒͺ βŠ—  2  π‘Ž 𝐿 2 ξ€· ℝ 3 ξ€Έ ξƒͺ . ( 2 . 2 7 ) The free Hamiltonian, now denoted by 𝐻 ξ…ž 0 , is then given by 𝐻 ξ…ž 0 = 3  β„“ = 1  πœ– = Β± ξ€œ 𝑀 β„“ ( 1 ) ξ€· πœ‰ 1 ξ€Έ 𝑏 βˆ— β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ 𝑏 β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ d πœ‰ 1 + 3  β„“ = 1  πœ– = Β± ξ€œ ℝ 3 | | 𝑝 2 | | 𝑐 βˆ— β„“ , πœ– ξ€· 𝑝 2 ξ€Έ 𝑐 β„“ , πœ– ξ€· 𝑝 2 ξ€Έ d 𝑝 2 +  πœ– = Β± ξ€œ 𝑀 ( 3 ) ξ€· πœ‰ 3 ξ€Έ π‘Ž βˆ— πœ– ξ€· πœ‰ 3 ξ€Έ π‘Ž πœ– ξ€· πœ‰ 3 ξ€Έ d πœ‰ 3 , ( 2 . 2 8 ) and the interaction, now denoted by 𝐻 ξ…ž 𝐼 , is the one obtained from 𝐻 𝐼 by supposing that 𝐺 ( 𝛼 ) ( πœ‰ 1 , ( 𝑝 2 , 𝑠 2 ) , πœ‰ 3 ) = 0 if 𝑠 2 = πœ– ( 1 / 2 ) . The total Hamiltonian, denoted by 𝐻 ξ…ž , is then given by 𝐻 ξ…ž = 𝐻 ξ…ž 0 + 𝑔 𝐻 ξ…ž 𝐼 . The results obtained in this paper for 𝐻 hold true for 𝐻 ξ…ž with obvious modifications.

Under Hypothesis 2.1 a well-defined operator on 𝔇 corresponds to the formal interaction 𝐻 𝐼 as it follows.

The formal operator ξ€œ 𝐺 ( 1 ) β„“ , πœ– , πœ– β€² ξ€· πœ‰ 1 , πœ‰ 2 , πœ‰ 3 ξ€Έ 𝑏 βˆ— β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ 𝑐 βˆ— β„“ , πœ– β€² ξ€· πœ‰ 2 ξ€Έ π‘Ž πœ– ξ€· πœ‰ 3 ξ€Έ d πœ‰ 1 d πœ‰ 2 d πœ‰ 3 ( 2 . 2 9 ) is defined as a quadratic form on ( 𝔇 β„“ βŠ— 𝔇 π‘Š ) Γ— ( 𝔇 β„“ βŠ— 𝔇 π‘Š ) as ξ€œ ξ‚€ 𝑐 β„“ , πœ– β€² ξ€· πœ‰ 2 ξ€Έ 𝑏 β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ πœ“ , 𝐺 ( 1 ) β„“ , πœ– , πœ– β€² π‘Ž πœ– ξ€· πœ‰ 3 ξ€Έ πœ™  d πœ‰ 1 d πœ‰ 2 d πœ‰ 3 , ( 2 . 3 0 ) where πœ“ , πœ™ ∈ 𝔇 β„“ βŠ— 𝔇 π‘Š .

By mimicking the proof of [29, Theorem X.44], we get a closed operator, denoted by 𝐻 ( 1 ) 𝐼 , β„“ , πœ– , πœ– β€² , associated with the quadratic form such that it is the unique operator in 𝔉 β„“ βŠ— 𝔉 π‘Š such that 𝔇 β„“ βŠ— 𝔇 π‘Š βŠ‚ π’Ÿ ( 𝐻 ( 1 ) 𝐼 , β„“ , πœ– , πœ– β€² ) is a core for 𝐻 ( 1 ) 𝐼 , β„“ , πœ– , πœ– β€² and 𝐻 ( 1 ) 𝐼 , β„“ , πœ– , πœ– β€² = ξ€œ 𝐺 ( 1 ) β„“ , πœ– , πœ– β€² ξ€· πœ‰ 1 , πœ‰ 2 , πœ‰ 3 ξ€Έ 𝑏 βˆ— β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ 𝑐 βˆ— β„“ , πœ– β€² ξ€· πœ‰ 2 ξ€Έ π‘Ž πœ– ξ€· πœ‰ 3 ξ€Έ d πœ‰ 1 d πœ‰ 2 d πœ‰ 3 ( 2 . 3 1 ) as quadratic forms on ( 𝔇 β„“ βŠ— 𝔇 π‘Š ) Γ— ( 𝔇 β„“ βŠ— 𝔇 π‘Š ) .

Similarly for the operator ( 𝐻 ( 1 ) 𝐼 , β„“ , πœ– , πœ– β€² ) βˆ— , we have the equality as quadratic forms ξ‚€ 𝐻 ( 1 ) 𝐼 , β„“ , πœ– , πœ– β€²  βˆ— = ξ€œ 𝐺 ( 1 ) β„“ , πœ– , πœ– β€² ( πœ‰ 1 , πœ‰ 2 , πœ‰ 3 ) π‘Ž βˆ— πœ– ξ€· πœ‰ 3 ξ€Έ 𝑐 β„“ , πœ– β€² ξ€· πœ‰ 2 ξ€Έ 𝑏 β„“ , πœ– ξ€· πœ‰ 1 ξ€Έ d πœ‰ 1 d πœ‰ 2 d πœ‰ 3 . ( 2 . 3 2 )

Again, there exists two closed operators 𝐻 ( 2 ) 𝐼 , β„“ , πœ– , πœ– β€² and ( 𝐻