Quantization of scalar fields coupled to point-masses
J. Fernando Barbero, Benito A. Juárez-Aubry, Juan Margalef-Bentabol, Eduardo J. S. Villaseñor
aa r X i v : . [ m a t h - ph ] N ov Quantization of scalar fields coupled to point-masses
J. Fernando Barbero G.,
1, 2, ∗ Benito A. Ju´arez-Aubry, † JuanMargalef-Bentabol,
1, 4, ‡ and Eduardo J. S. Villase˜nor
4, 2, § Instituto de Estructura de la Materia,CSIC, Serrano 123, 28006 Madrid, Spain Grupo de Teor´ıas de Campos y F´ısica Estad´ıstica,Instituto Universitario Gregorio Mill´an Barbany,Universidad Carlos III de Madrid, Unidad Asociada al IEM-CSIC. School of Mathematical Sciences, University of Nottingham,Nottingham NG7 2RD, United Kingdom Instituto Gregorio Mill´an, Grupo de Modelizaci´on y Simulaci´on Num´erica,Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Legan´es, Spain (Dated: June 25, 2015)We study the Fock quantization of a compound classical system consisting of pointmasses and a scalar field. We consider the Hamiltonian formulation of the model byusing the geometric constraint algorithm of Gotay, Nester and Hinds. By relying onthis Hamiltonian description, we characterize in a precise way the real Hilbert spaceof classical solutions to the equations of motion and use it to rigorously construct theFock space of the system. We finally discuss the structure of this space, in particularthe impossibility of writing it in a natural way as a tensor product of Hilbert spacesassociated with the point masses and the field, respectively.
Keywords:
QFT in curved space-times; geometric Hamiltonian formulation; Fockquantization. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] I. INTRODUCTION
The main motivation of this work is to understand, in rigorous terms, the Hamiltonianformulation and subsequent Fock quantization of linear systems consisting of fields coupledto point masses (and, eventually, other low-dimensional objects). An important questionthat we will answer in the paper regards the possibility of writing the Hilbert space of sucha compound system as a tensor product of Hilbert spaces associated with the point massesand the field. The point masses are introduced to model external devices that can be usedboth to excite the system and to act as detectors sensitive to the “field quanta”. In this lastsense they can be thought of as generalizations of the Unruh-DeWitt particle detectors andsimilar devices, used in the discussion of quantum field theories in curved space-times andaccelerated frames [1–3]. We wish to emphasize from the start that the point masses –thatwe introduce already at the classical level– have nothing to do, in principle, with the fieldquanta despite the fact that the latter are usually interpreted as particles or quasiparticles.The types of systems that we consider here are related –but not equal– to field theoriesdefined in bounded spatial regions and share some features with them, as will be explainedin the following. There are several important field theoretic models that display interestingand non-trivial behaviors when defined in such regions (or more generally in the presence ofspatial boundaries). Among them we would like to mention Chern-Simons [4] and Maxwell-Chern-Simons models in 2+1 dimensions [5–7], Yang-Mills theories in 3+1 dimensions [8],and general relativistic models such as the isolated horizons used to study black hole entropyin loop quantum gravity [9, 10]. Part of our work is motivated by the comments appearingin [11] regarding the use of boundaries with the same purpose as the classical point-particlesthat we introduce here.The standard approach to derive the Hamiltonian formulation for field theories, especiallywhen gauge symmetries are present, relies on the methods developed by Dirac [12]. Theseare straightforward to use in the case of mechanical systems with a finite number of degreesof freedom and –in simple circumstances– can be adapted to field theories if one is willingto accept a certain lack of mathematical precision (regarding, for instance, the functionalspaces describing the field degrees of freedom). In the presence of boundaries, however, thenaive implementation of the Dirac algorithm is awkward and often leads to incomplete orplainly wrong results (see [13] and references therein). This is even more so for the modelthat we consider in the paper where we have both boundaries and particle-like objectsassociated with them. Some technical details regarding the difficulties of implementing theDirac algorithm for field theories can be found in [14].These difficulties –and other important ones– can be avoided by the use of the geometricconstraint algorithm developed by Gotay, Nester and Hinds [14–16]. This method providesa rigorous, geometric and global way to obtain the Hamiltonian description of field theoriesand pays due attention to functional analytic issues. In particular it provides a completelydetailed description of the spaces where the Hamiltonian dynamics takes place and can thusbe used as the starting point for quantization. This is especially useful for linear theories forwhich Fock quantization can be rigorously defined starting from the complexification of anappropriately defined real Hilbert space of classical solutions (along the lines described, forexample, in [17] in the context of quantum field theory in curved spacetimes). The preciseconstruction of the Fock space is important in order to discuss the eventual factorization ofthe Hilbert space H of the system in the form H = H masses ⊗ H field that would account fora clean separation between quantum point particle and field degrees of freedom.In the present paper we will study a model consisting of a finite length elastic stringattached, at the ends, to point particles subject to harmonic restoring forces (in additionto the ones exerted by the string). Although similar systems have been considered in theliterature [18, 19] our approach will concentrate on several mathematical issues —relevantfrom the physical point of view— that have not been discussed elsewhere. Specifically wewill introduce a natural way to deal with this model by using a particular class of measuresand the Radon-Nikodym (RN) derivatives defined with their help. This measure theoreticapproach is suggested by Mardsen and Hughes in [20, page 85].The methods and ideas put forward here can be exported to more complicated situations.For instance, it should be straightforward to generalize them to deal with higher dimensionalsystems where, in addition to point particles, other low-dimensional objects could be coupled.From the perspective of the standard quantum field theory in curved spacetimes, we areextending the usual approach by replacing the all-important Laplace-Beltrami operator byan elliptic operator defined with the help of a measure that combines physical features ofboth the field and point masses of the system. We would like to remark here that, as freetheories are essential building blocks for the perturbative quantization of non-linear models(for instance, they play a central role to define the Fock spaces used in their description),it is important to understand them well as a first step to consider their quantization in thepresence of boundaries and/or lower-dimensional objects.The layout of the paper is the following: After this introduction, section II will be devotedto the classical description of the system under consideration. In particular, in II.2 we presenta short discussion of the resolution of the evolution equations by separation of variables.This motivates the introduction of non trivial measures that play a relevant role in sectionII.3, where we provide an alternative Lagrangian formulation. In section II.4 we obtainthe Hamiltonian description by implementing the GNH algorithm [14–16]. We provide theprecise description of the functional spaces relevant for the model. As it will be shownthese are generalizations of Sobolev spaces that can be understood in a neat way with thehelp of a scalar product defined in terms of appropriate measures. In section III we usethe Hamiltonian description of the system to build a Fock space and quantize. We will payparticular attention to the characterization of this Fock space as a tensor product of Hilbertspaces associated with the field and the particles. We end the paper with our conclusionsin section IV and one appendix where we give a number of useful mathematical results.The reader interested only in the physical results of the paper should skip sections II.3and II.4, where a number of mathematical details are provided, and go directly from sectionII.2 to the quantum description in terms of physical modes discussed in section III. II. CLASSICAL DESCRIPTIONII.1. Lagrangian and field equations
Let us consider a model consisting of an elastic string of finite length (in 1+1 dimensions)coupled to two point masses located at the ends and attached to springs of zero rest length.Both the string and the masses are subject to restoring forces proportional to the deviationsfrom their equilibrium configurations. For definiteness we will consider that the motion ofthe system is longitudinal, although this is not essential. From a logical perspective theequations of motion for such a system should be obtained by analyzing with due care theforces acting on the string and the masses, and using Newton’s laws. In practice, however,it is more convenient to use an action written in terms of an easily interpretable Lagrangianand formally derive them by computing its first variations. Let us, then, start from theaction S ( u ) = Z t t L ( u ( t ) , ˙ u ( t )) d t (II.1)where the Lagrangian, for smooth Q and V , has the following form L ( Q, V ) = λ h V, V i − ǫ h Q ′ , Q ′ i − m h Q, Q i + X x ∈{ ,ℓ } (cid:18) M x V ( x ) − k x Q ( x ) (cid:19) . (II.2)In the preceding expression h· , ·i denotes the usual scalar product in the Hilbert space L (0 , ℓ )defined with the help of the Lebesgue measure µ L , ℓ is the length of the unstretched string, λ its longitudinal mass density, ǫ its Young modulus, m > , M and M ℓ are themasses of the point particles and k , k ℓ the elastic constants of the springs attached to them.The field Q ( x ) represents the deviation of the string point labelled by x from its equilibriumposition. Spatial derivatives are denoted by primes and time derivatives by dots.As it can be seen the Lagrangian has terms of “field” and “particle” types involvingspatial derivatives of first order, at most. It is convenient at this point to choose unitsof length, time and mass such that ℓ = ǫ = λ = 1 (which in particular implies that thespeed of sound is c := ǫ/λ = 1). Notice that by doing this we exhaust all the freedomin the choice of units so we will have to keep ~ explicit when quantizing the model. Toremind the reader of this choice we will rename the remaining constants in the Lagrangianas e ω := m /ǫ , e ω j := k j /M j , µ j := M j /ǫ so that (II.2) becomes L ( Q, V ) = 12 h V, V i − h Q ′ , Q ′ i − e ω h Q, Q i + X j ∈{ , } µ j (cid:16) V ( j ) − e ω j Q ( j ) (cid:17) . (II.3)It is interesting to mention at this point that the positions of the point particles are notclassical independent degrees of freedom as they are given by continuity by the position ofthe ends of the string. This fact will have an analogue in the quantum description of themodel (non-factorization of the Fock space, see section III).The equations of motion derived from (II.3) are:¨ u ( x, t ) − u ′′ ( x, t ) + e ω u ( x, t ) = 0 , x ∈ (0 , , (II.4) µ ¨ u (0 , t ) − u ′ (0 , t ) + µ e ω u (0 , t ) = 0 , (II.5) µ ¨ u (1 , t ) + u ′ (1 , t ) + µ e ω u (1 , t ) = 0 . (II.6)As can be readily seen the time evolution of the deformation of the string is governed by the1+1 Klein-Gordon equation whereas the point masses at the boundary points move underthe combined force exerted by the springs and the string (given by the spatial derivatives atthe boundary). It is important to notice at this point that the preceding equations are notconventional in the sense that (II.5) and (II.6) are not standard boundary conditions becausethey involve second order time derivatives. We will see in section II.2 that this feature We follow the custom of calling this constant m because the squared mass of the quantum excitations ofthe usual Klein-Gordon field is proportional to it. The case m = 0 differs slightly from the one that westudy but can be approached with the same methods. qualitatively changes the type of eigenvalue problem that has to be solved to identify thenormal modes and characteristic frequencies. Actually it renders the present problem quitenon-trivial because the relevant eigenvalue equations are not of the standard Sturm-Liouvilletype and hence the classical theorems that ensure that the normal modes form a completeset cannot be applied. There is however a workaround to prove such result that consists inintroducing a different measure space. We anticipate now the result of this approach: Proposition 1.
The equations (II.4) - (II.6) are contained in the equations: ¨ u − (∆ µ − e ω ) u = 0 x ∈ [0 , , µ -a.e. (II.7)( − j d u d µ ( j ) − A ( j ) u ( j ) = 0 j ∈ { , } , (II.8) that consist of a 1+1 dimensional Klein-Gordon equation on the interval [0 , subject toRobin boundary conditions written in terms of the Radon-Nikodym derivative ddµ with respectto the measure µ = α δ + µ L + α δ . The parameters α j are related to the physical parametersof the problem by α j (1 − α j µ j ( e ω j − e ω )) = µ j , A ( j ) := ( e ω j − e ω ) √ µ j α j , and the Laplace-likeoperator ∆ µ := (1 + C ) d d µ is defined in terms of the function C ( j ) := A ( j ) α j , C ( x ) := 0 if x = 0 , . In the previous proposition µ -a.e. stands for “ µ -almost everywhere”, (i.e. the equalitycan fail to be true in a set of zero µ -measure, at most). The measure µ is defined in theAppendix (see Equation A.2) and the domain of ∆ µ , which specifies the regularity conditionson the solutions to the equations of motion, is given in Appendix A.III. Instead of givinga direct proof of this result, we will apply separation of variables to (II.4)-(II.6). In thisprocess some issues will arise, the most important one being that the standard Laplacian isnot self adjoint. This will lead to the introduction of a new self-adjoint Laplace-like operator(in an appropriate functional space) in terms of which we obtain equations (II.7) and (II.8). II.2. Classical description of the model: solving the field equations
In this section we consider the resolution of the equations of motion (II.4)-(II.6) by usingthe method of separation of variables. By writing u ( x, t ) = X ( x ) T ( t ) we get¨ T = ( λ − e ω ) T (II.9) X ′′ = λ X (II.10) X ′ (0) = µ ( λ + e ω − e ω ) X (0) (II.11) X ′ (1) = − µ ( λ + e ω − e ω ) X (1) (II.12)where λ ∈ R . In this form these equations do define an eigenvalue problem for X with onekey (and relatively unusual) feature: the eigenvalue appears also in the boundary equations(II.11), (II.12). This means that we are not directly dealing with a Sturm-Liouville problemand, hence, we cannot directly import the usual results that characterize the eigenvalues λ (do they exist? are they isolated? are they bounded?) and the corresponding eigenfunctions(are they a complete set? are they orthogonal?). The answer to these questions is importantin order to expand the general solution to the equations of motion as a functional series ofeigenfunctions and also to quantize the system.In any case, a lot of information can be gathered in practice by solving the concreteeigenvalue problem that we have at hand so we sketch now the computation of the eigenvaluesand the eigenfunctions. i) Negative eigenvalues λ = − ω < X λ = a cos( ωx ) + b sin( ωx ) with a, b ∈ R . Theconditions (II.11),(II.12) imply that µ ( ω − ∆ e ω ) a + ωb = 0 (cid:16) µ ( ω − ∆ e ω ) cos ω + ω sin ω (cid:17) a + (cid:16) µ ( ω − ∆ e ω ) sin ω − ω cos ω (cid:17) b = 0Where we have introduced the shorthand ∆ e ω j := e ω j − e ω . These have non-trivial solutionsfor a , b if and only if (cid:16) ω − µ µ ( ω − ∆ e ω )( ω − ∆ e ω ) (cid:17) sin ω + (cid:16) µ ( ω − ∆ e ω ) + µ ( ω − ∆ e ω ) (cid:17) ω cos ω = 0 . (II.13)It is straightforward to see that (II.13) has an infinite number of solutions for ω for every(physical) choice of parameters µ , µ , e ω , e ω , e ω ; in fact, there exists n ∈ N , such thatevery interval of the form ( kπ, ( k + 1) π ) contains one and only one solution of (II.13) forevery k ∈ N , k > n . In the asymptotic limit k → ∞ we have ω k = kπ + (cid:18) µ + 1 µ (cid:19) kπ + O (cid:0) k − (cid:1) , (II.14)so we see that we actually have an infinite set of negative eigenvalues and also that theygrow without bound. Finally the eigenfunctions have the form (labelling them with k andwith a minus superscript to indicate that the eigenvalue is negative) X − k ( x ) = ω k cos( ω k x ) + µ (∆ e ω − ω k ) sin( ω k x ) . ii) Positive eigenvalues λ = ω > X λ ( x ) have the form X λ ( x ) = ae ωx + be − ωx where the realcoefficients a , b must satisfy now the conditions (cid:16) ω − µ ( ω + ∆ e ω ) (cid:17) a − (cid:16) ω + µ ( ω + ∆ e ω ) (cid:17) b = 0 e ω (cid:16) ω + µ ( ω + ∆ e ω ) (cid:17) a − e − ω (cid:16) ω − µ ( ω + ∆ e ω ) (cid:17) b = 0that have non-trivial solutions if and only if e − ω (cid:16) ω − µ ( ω + ∆ e ω ) (cid:17) (cid:16) ω − µ ( ω + ∆ e ω ) (cid:17) − e ω (cid:16) ω + µ ( ω + ∆ e ω ) (cid:17) (cid:16) ω + µ ( ω + ∆ e ω ) (cid:17) = 0 . It can be seen that this equation has a finite number of solutions N (maybe none) dependingon the particular choices of the physical parameters defining the problem (e.g. if both∆ e ω j ≥
0, there is no positive eigenvalue). Notice that, as the energy is constant, thefunction T cannot have an exponential growth so a ≤ e ω and, in fact, the limit case a = e ω (where T has a linear behavior) happens only if e ω j = 0 = e ω (in the paper we are assuming e ω > k ∈ {− , . . . , − N } as a negative and finite counter, the correspondingeigenfunctions are X + k ( x ) = (cid:0) ω k + µ ( ω k + ∆ e ω ) (cid:1) e ω k x + (cid:0) ω k − µ ( ω k + ∆ e ω ) (cid:1) e − ω k x . iii) Zero mode λ = 0It is easy to check that λ = 0 is an eigenvalue if and only if (1 + µ ∆ e ω )(1 + µ ∆ e ω ) = 1,in which case the solution is simply X ( x ) = X (0)(1 + µ ∆ e ω x ). This can only happen if∆ e ω = 0 = ∆ e ω or ∆ e ω < < ∆ e ω or ∆ e ω < < ∆ e ω .From now on we will collectively denote the eigenvectors as X n with X n = X + n when n ∈ {− , . . . , − N } , X = X (“when it exists”), and X n = X − n when n ∈ N . Theirassociated eigenvalues λ n are, respectively λ n = ω n when n ∈ {− , . . . , − N } , λ = 0, and λ n = − ω n when n ∈ N .Notice that, at this point, we have found the solutions of the form u n ( x, t ) = X n ( x ) T n ( t )to equations (II.9)-(II.12). In order to find all the solutions to (II.4)-(II.6) we have to provethat the eigenfunctions { X k } form a complete set in an appropriate functional space. This isnot straightforward because, as mentioned before, we do not have a standard Sturm-Liouvilleproblem.The generalized Sturm-Liouville problems of the form defined by (II.10)-(II.12) have along history both in mechanics and mathematics (see, for instance [18, 21–23] and referencestherein). A direct but important observation is the fact that eigenfunctions X k , correspond-ing to different eigenvalues λ k , are not orthogonal with respect to the standard scalar productin L (0 ,
1) but are orthogonal with respect to the following modified scalar product [23] hh u, v ii := µ u (0) v (0) + µ u (1) v (1) + Z u · v µ L . (II.15)This can be readily proved by taking two such eigenfunctions X m , X n (associated withdifferent eigenvalues λ m , λ n ), integrating the following identity over the interval [0 , X n X ′ m − X m X ′ n ) ′ = ( λ m − λ n ) X m X n and using the boundary conditions. The appearance of the scalar product (II.15) suggeststo look for Hilbert spaces adapted to it. We consider this issue in the next section.We want to point out here that, at this stage, the position of the point particles are thelimits X ( j ) = lim x → j X ( x ). However, as will be shown in the following, there are technicalreasons to introduce function spaces where generically this equality does not hold (instead,boundary conditions of the type (A.8) are satisfied). II.3. Alternative Lagrangian for the system
In order to get a precise Hamiltonian formulation for the dynamics of the system at hand,properly identify and characterize its degrees of freedom and deal with the delicate analyticand geometric issues posed by the presence of boundaries, it is most appropriate to use theGNH geometric algorithm developed in [14–16]. A convenient starting point is to introducea new Lagrangian defined on a manifold domain [24] of the tangent bundle of a configurationspace, that we will take to be a real Hilbert space. In practice this means that we will haveto extend the system somehow and also consider field configurations which are less smooththan the ones used in (II.3). We will require, nonetheless, that the solutions to (II.4)-(II.6)are appropriately contained in those corresponding to the equations of motion of the newLagrangian.In view of the results of previous section, instead of working with (II.3), it is naturalto look for a generalized Lagrangian written in terms of the scalar product defined by acertain measure µ and the associated RN derivatives. The hope –that will be realized– isthat the equations of motion give rise to a standard Sturm-Liouville problem in terms of thisderivative and also that the boundary conditions defining the elliptic operator that will playa central role in its solution are such that its self-adjointness (and other related propertiessuch as the completeness of the set of eigenfunctions) can be readily asserted and proved.Let us consider then D := (cid:26) u ∈ L µ [0 ,
1] : ∃ d u d µ ( x ) ∀ x ∈ [0 , , d u d µ is µ -a.c. , d u d µ ∈ L µ [0 , (cid:27) and the Lagrangian L : D × L µ [0 , → R of the following form L ( Q, V ) = 12 h V, V i µ − (cid:28) d Q d µ , d Q d µ (cid:29) µ − e ω h Q, Q i µ (II.16) − X j ∈{ , } α j (cid:18) ( − j d Q d µ ( j ) − A ( j ) Q ( j ) (cid:19) d Q d µ ( j ) − X j ∈{ , } A ( j ) Q ( j ) , where h· , ·i µ is the scalar product with respect to the measure µ = α δ + µ L | (0 , + α δ , dd µ denotes the associated RN derivative and µ -a.c. stands for “ µ -absolutely continuous”.In view of the scalar product (II.15), one might naively expect that α j = µ j , however α j ,as well as A ( j ), have to be taken as non-trivial functions of the physical parameters of themodel (see [21] and appendix A for more details, in particular equations (A.8) and (A.11)).The equations of motion can be obtained by computing the first variation of the actionderived from the Lagrangian. A straightforward computation gives δS ( u ) = Z t t d t − h ¨ u, δu i µ + h ∆ µ u, δu i µ − e ω h u, δu i µ − X j ∈{ , } (cid:16) ( − j α j γ j ( δu ′ ) − γ j ( δu ) (cid:17)(cid:16) ( − j d u d µ ( j ) − A ( j ) u ( j ) (cid:17)! , where δu ∈ D so that the traces γ j ( δu ′ ) and γ j ( δu ) are well defined (see Appendix A.II).From this last expression we get the equations of motion¨ u − (∆ µ − e ω ) u = 0 x ∈ [0 , , µ -a.e. (II.17)( − j d u d µ ( j ) − A ( j ) u ( j ) = 0 j ∈ { , } . (II.18)They have the form of the 1+1 dimensional Klein-Gordon equation on the interval [0 , µ is not self-adjoint in D . However, the solutions to (II.18) belong to b D := (cid:26) u ∈ D : ( − j d u d µ ( j ) − A ( j ) u ( j ) = 0 (cid:27) and in this domain ∆ µ is indeed self adjoint (see section A.III).In order to see that these equations describe the same dynamics as (II.4)-(II.6) we firstnotice that in the open interval (0 ,
1) equation (II.17) is simply the Klein-Gordon equation¨ u − u ′′ + e ω u = 0. If we write now the boundary conditions (II.18) in the form given by(A.8), plug the resulting expression into ¨ u ( j ) − ∆ µ u ( j ) + e ω u ( j ) = 0, and use the relationsthat fix α j and A ( j ) in terms of the physical parameters for the problem we immediatelyobtain (II.5)-(II.6). This completes the proof of Proposition 1. II.4. Hamiltonian formulation
The equations (II.17) and (II.18) derived from the Lagrangian (II.16) can be understoodas a particular case of the abstract wave equation (see discussion in [13]). By using theresults described there, it is possible to directly get both the expression of the Hamiltonianvector field and the manifold domains where the dynamics takes place.
Proposition 2.
The Hamiltonian dynamics takes place in the second class (generalized)submanifold M = b D × D of the weakly symplectic manifold ( M = D × L µ , ω ) , where ω isthe pullback to M of the strong, canonical, symplectic form on L µ × L µ . The dynamics isgoverned by the Hamiltonian vector field X = ( X Q , X P ) : M → M whose components are X Q ( Q, P ) = P (II.19) X P ( Q, P ) = − ( e ω − ∆ µ ) Q , (II.20) and the Hamiltonian is given by H ( Q, P ) = 12 h P, P i µ + 12 (cid:28) d Q d µ , d Q d µ (cid:29) µ + 12 e ω h Q, Q i µ (II.21)+ X j ∈{ , } α j (cid:16) ( − j d Q d µ ( j ) − A ( j ) Q ( j ) (cid:17) d Q d µ + 12 X j ∈{ , } A ( j ) Q ( j ) . A submanifold N → M of a presymplectic manifold ( M, ω ) is said to be second class if T N ⊥ ∩ T N = { } where T N ⊥ := { Z ∈ T M | N : ω | N ( Z, X ) = 0 ∀ X ∈ T N } , and T N := j ∗ T N . F L : D × L µ → L µ × L ∗ µ : ( Q, V ) ( Q, h V, ·i µ ) . (II.22)In order to conform with the standard notation we will write it as ( Q, h P, ·i µ ) with P := V .By using now the Riesz representation theorem we can simply consider, as in the standardcase of the scalar field with Dirichlet or Robin boundary conditions [13], that the fiberderivative is D × L µ → L µ × L µ with F L ( Q, V ) = (
Q, V ) and the primary constraintmanifold is M := D × L µ taken as a generalized submanifold of L µ × L µ .The space L µ × L µ carries a canonical strongly nondegenerate symplectic form (inheritedfrom the cotangent bundle L µ × L ∗ µ ) given byΩ ( Q,P ) (( q , p ) , ( q , p )) = h q , p i µ − h q , p i µ (II.23)where we have Q, P, q i , p i ∈ L µ . The pull-back of Ω to M is the weakly symplectic form ω := F L ∗ Ω given by ω ( Q,P ) (( q , p ) , ( q , p )) = h q , p i µ − h q , p i µ , (II.24)where we have now Q, q i ∈ D and P, p i ∈ L µ . It is interesting to mention that the “boundaryterms” of the scalar product give rise here to “boundary terms” in the symplectic form.However we will see that this does not imply the existence of boundary degrees of freedom.In order to obtain the Hamiltonian on M we compute the energy H ◦ F L ( Q, V ) = h V, V i µ − L ( Q, V ) . (II.25)This expression fixes the values of the Hamiltonian H only on the primary constraint sub-manifold M , however, this is the only information that we need to proceed with the GNHalgorithm. From (II.25) we find that H : M → R is given by Equation II.21.On the primary constraint submanifold M , vector fields are maps X : M → M ×M :( Q, P ) (( Q, P ) , ( X Q ( Q, P ) , X P ( Q, P )), such that X Q ( Q, P ) ∈ D and X P ( Q, P ) ∈ L µ . Wehave then ( i X ω ) ( Q,P ) ( q, p ) = h X Q , p i µ − h q, X P i µ . We must find now a submanifold M and an injective immersion M → M that allowsus to solve the equation ( i X ω − d H ) | ( M ) = 0 . (II.26)Notice that this will require us to identify M as a subspace of M and also to specify itstopology (by giving, for instance, a scalar product on it).The resolution of Equation (II.26) is relatively long but straightforward (see [13] forseveral similar computations). The final result is that M = b D × D and the Hamiltonianvector field is given by X Q ( Q, P ) = P (II.27) X P ( Q, P ) = − ( e ω − ∆ µ ) Q . (II.28)1The injection : M → M is just given by the inclusion map and is continuous if weuse the natural topologies defined by the scalar products in M and M . The Hamiltonianvector field ( X Q , X P ) is also continuous in the same topology.By using the same kind of argument employed in [13], it is straightforward to show thatthe closure of b D × D in b D × L µ satisfies cl D× L µ ( b D × D ) = b D × L µ . We then conclude that theHamiltonian vector field X is tangent to M so the GNH algorithm stops. We finish thissection with a comment. Although it is not obvious at first sight, the Hamiltonian (II.21)is positive definite when restricted to M . III. FOCK QUANTIZATION
The fact that the positions of the point particles attached at the ends of the string arenot independent physical degrees of freedom actually suggests that the Fock space for thissystem will not have the form of a tensor product of different Hilbert spaces associated withthe masses and the string. Although this is quite natural from this perspective there are,however, physical questions that come to mind. For instance, if we think about this modelas two masses connected by a physical spring (with “internal degrees of freedom”), how doesone recover the situation where the string just models an ideal spring? What is the originof the L ( R ) ⊗ L ( R ) Hilbert space that one would use to describe this system? As a firststep towards addressing these questions it is important to understand in detail why the Fockspace does not factorize. We do this in the following. III.1. Construction of the Fock space
An accepted way to quantize linear systems relies on the construction of a Fock space.In the specific case of quantum field theory in curved spacetimes the relevant details can befound in [17]. The starting point is the real vector space S of the classical (smooth) solutionsto the field equations. By introducing a complex structure (among the many available) acomplexified version of S is defined and endowed with a sesquilinear form obtained with thehelp of the symplectic structure. Taking a subspace of “positive frequency” solutions, thesesquilinear form defines a proper scalar product h· , ·i + . By completing the complex vectorspace of positive frequency solutions w.r.t. this scalar product we obtain the one particleHilbert space h . Finally, the Hilbert space of states of the quantum field theory is the Fockspace H := F ( h ).In our case, the Hamiltonian description of the preceding section has produced a linearmanifold domain of L µ × L µ given by M = b D×D , where the classical Hamiltonian dynamicstakes place, and a Hamiltonian vector field X tangent to the closure of M . Notice that wecan pullback the canonical symplectic form in phase spaceΩ ( Q,P ) (( q , p ) , ( q , p )) = h q , p i µ − h q , p i µ (III.1)to M (where ( Q, P ) , ( q i , p i ) ∈ M ). We introduce now a complexification of this vectorspace M C and use the symplectic form to define a scalar product. Vector addition is definedcomponentwise as the standard sum of real functions and multiplication by complex scalarsis defined by introducing the complex structure: J : M × M → M × M : (cid:0) ( q , p ) , ( q , p ) (cid:1) (cid:0) ( − q , − p ) , ( q , p ) (cid:1) (III.2)2and requiring that ( a + bi ) · V := ( a I + b J )( V ) (III.3)for a, b ∈ R and V ∈ M ×M . In this case one can think of the elements in the complexifiedvector space as complex functions in M with the standard sum and multiplication bycomplex scalars. The complexified symplectic form is the straightforward extension bycomplex linearity of (III.1)Ω C ( Q,P ) (( q , p ) , ( q , p )) := h q , p i µ − h q , p i µ . The integral curves of the Hamiltonian vector field X can be identified with the space ofsolutions to the equations of motion. As X is defined in terms of the elliptic operator ∆ µ ,its eigenvalues λ n and normalized eigenfunctions Y n satisfying ∆ µ Y n = λ n Y n (see appendixA.IV), play a relevant role in the following. It is important to point out here that these arenot necessarily classical, regular solutions (for instance, they are not smooth with compactsupport). Furthermore, as the evolution is a symplectic transformation, the symplectic formcan be pulled back to this space in the obvious way. In particular, the complexified solutiondefined by the (complex) Cauchy data ( Q, P ) := ( u ( · , , ˙ u ( · , ∈ M C at t = 0 can bewritten in the form u ( x, t ) = 12 X n (cid:18) e it √ e ω − λ n (cid:16) Q n − i P n √ e ω − λ n (cid:17) + e − it √ e ω − λ n (cid:16) Q n + i P n √ e ω − λ n (cid:17)(cid:19) Y n ( x )with Q n = h Y n , Q i µ ∈ C and P n = h Y n , P i µ ∈ C . Notice that, as mentioned in section II.2,our assumption that e ω > λ n < e ω . In order to select a subspace of positivefrequency solutions we require that iP + pe ω − ∆ µ Q = 0 . (III.4)This condition is equivalent to Q n + i P n √ e ω − λ n = 0 , ∀ n . It is now straightforward to see that, when equation (III.4) holds, we can indeed define ascalar product in this “positive frequency part” of the solution space as h Q (1) , Q (2) i + := − i Ω C (( Q (1) , P (1) ) , ( Q (2) , P (2) )) = 2 h Q (1) , pe ω − ∆ µ Q (2) i µ . (III.5)At this point, the only remaining step to finish the construction of h is to Cauchy completein the norm defined by the scalar product. In terms of the Fourier coefficients of Q (1) and Q (2) in the orthonormal basis that we are using the preceding scalar product becomes h Q (1) , Q (2) i + = 2 X n pe ω − λ n h Y n , Q (1) i µ h Y n , Q (2) i µ . Notice that the L µ -orthonormal basis { Y n } leads to an orthonormal basis of h consisting of Z n := 1 h Y n , Y n i / Y n = 1 √ e ω − λ n ) / Y n , h = ( ψ = X n ψ n Z n : ψ n ∈ C , X n | ψ n | < ∞ ) . Finally the Hilbert space H of our quantum field theory is given by the symmetric Fockspace over h , H = F ( h ).The standard procedure described in [17] can be used to introduce creation and annihila-tion operators, quantum fields and the quantum Hamiltonian ˆ H that generates the quantumdynamics of the system. In particular ˆ H is given by the lift to the Fock space of pe ω − ∆ µ .Notice that the unitarity of the quantum evolution is guaranteed by the self-adjointness ofthe operator ∆ µ in b D . III.2. Factorization of the Fock space
In this section we will discuss the impossibility of factorizing H = H masses ⊗ H string in a natural way, i.e. with “factors” associated, respectively, with the point masses and the string.For the sake of the argument let us see what would happen if the Hilbert space for our modelwere the Fock space over L µ . As discussed in appendix A.I, the space L µ is isomorphic to R ⊕ L (0 , ⊕ R where each R is associated with a point mass. Using the properties of theFock construction it is straightforward to see that F ( L µ ) = F ( R ) ⊗ F ( L (0 , ⊗ F ( R ) and,hence, if this were the case, we would have a Hilbert space corresponding to each of thepoint masses contributing a factor to the Fock space.However we have to consider h , instead of L µ , and we are prescribed to use the scalarproduct h· , ·i + involving the square root factor pe ω − ∆ µ . This changes crucially the out-come and prevents us, in particular, to write h = C ⊕ h string ⊕ C with the C subspacesassociated with the boundaries as in the previous example. Indeed, if such a decompositionwere available, the function F : [0 , → R : x F ( x ) = (cid:26) x = 00 x ∈ (0 ,
1] (III.6)considered in the appendix (see equation (A.12) and the discussion after it) would have tobe in h . However, it is straightforward to see that, in the limit n → ∞ , the coefficients h F, Z n i + satisfy h F, Z n i + = 2 √ α √ µ πn + O (cid:0) n − / (cid:1) , so they are not square summable and, hence, F h .The impossibility of achieving the previous decomposition immediately implies the non-factorizability of the Fock space built from h . Of course, this does not exclude the possibilityof finding such a factorization in other ways, for example for every decomposition of the type h = h ⊕ h we would have H = F ( h ) = F ( h ⊕ h ) = F ( h ) ⊗ F ( h ) , however, they are definitely not obvious from the present perspective and, in particular,there is no reason a priori to associate the factors to the point masses.4 IV. CONCLUSIONS
We have studied the Hamiltonian formulation and Fock quantization for a 1+1 dimen-sional model containing both fields and point masses. The combined description of differenttypes of physical objects poses some interesting and non-trivial questions related to theproper characterization of the physical degrees of freedom both at the classical and quan-tum levels. Some natural questions for these systems are: Is it possible to talk about independent degrees of freedom associated with the masses and the fields? or, is it possibleto split the quantum Hilbert space of the system in the form H = H masses ⊗ H field ? Oneof the main results of the paper is the proof that the Fock quantization of such compoundmodel leads to a Fock space that cannot be written in a natural way as the tensor productof Hilbert spaces associated with the masses and the field respectively. We want to em-phasize that this result comes about after a careful discussion of the spaces of solutions tothe equations of motion and the details of the construction of the Fock space, in particular,the scalar product appearing in the complexified solution space. This lack of factorizationmust be understood because one of the assumptions used in the description of compoundquantum systems is that their state spaces are tensor products of Hilbert spaces associatedwith each subsystem. This statement is singled out by some authors as the zeroth postulateof Quantum Mechanics [25, 26]. Of course, in the present case our construction does notviolate this postulate but, rather, shows that identifying subsystems in a proper way is notstraightforward, even at the classical level.From a technical point of view the method that we have employed relies on the intro-duction of Hilbert spaces endowed with scalar products defined with the help of modifiedmeasures describing the coupling of the fields with lower dimensional objects (point massesin the present case). This is useful both at the classical and quantum levels because theidentification of the relevant functional spaces is simplified in a significant way. One ofthe key elements of our approach relies on the ideas developed by Evans [21] to deal withmodified Sturm-Liouville problems of the type considered here. It is especially importantto work with elliptic, self-adjoint operators (generalized Laplacians) with spectrum and as-sociated eigenfunctions that can be mapped to the ones appearing in the resolution of thenon-standard eigenvalue problem that determines the normal modes of the system. Thesegeneralized Laplacians play a fundamental role in the Hamiltonian formulation of the modeland its subsequent Fock quantization.The Hamiltonian formulation that we have developed starts from a Lagrangian writtenin terms of the natural objects: measures, the scalar product defined with their help and theassociated Radon-Nikodym derivatives. A striking feature of the approach that we followis that although the constants appearing in the scalar product are functionally dependenton the physical parameters they are, generically, non-trivial functions of them. It is alsoimportant to emphasize the fact that the appropriate Lagrangians are non-trivial whenwritten in terms of these objects because of the necessity to have appropriate self-adjointoperators (in particular it is very important that the normal modes of the system can beinterpreted as eigenfunctions of the self-adjoint operator ∆ µ with eigenvalues given by thenormal frequencies).We have described the construction of the Fock space by relying of the methods custom-arily used in the context of quantum field theory in curved spacetimes (in particular thosedescribed by Wald in [17]). An advantage of combining those methods with the Hamiltonianformulation that we are using here is the possibility of having a precise and explicit char-5acterization of the solution space for the equations describing the dynamics. Basically theonly difference with the present case is due to the fact that the relevant elliptic self-adjointoperator is not a Laplace-Beltrami operator but one defined with the help of a measure withsingular contributions at the boundaries of the region where the fields are defined.We have illustrated the ideas developed in the paper with a simple 1+1 dimensionalmodel but they can be extended without difficulty to other more complicated systems, inparticular, our approach can be used to deal with higher-dimensional models and is flexibleenough to allow the coupling of different types of low-dimensional objects not restricted topoint masses. The model that we have studied provides an interesting way to interpolatebetween different types of boundary conditions (by, for instance, considering the limits ofsmall or large masses at the boundary). It is important to highlight here that, the presentmethods can be of use not only for linear systems but also for other non-linear ones thatare obtained by consistently adding interactions to them (for instance, gauge theories). Byworking with fields defined in unbounded space-time regions these techniques can also beadapted to the study of quantum dissipative systems (in the spirit of the Caldeira-Leggetor Rubin models) and decoherence.A final comment concerns the interpretation of the current system in terms of so-calledparticle detectors, which provide a clear-cut definition of a particle in general situations,such as curved spacetimes and non-inertial frames. In this approach, the particle detectoris a system with additional degrees of freedom that interacts with the field according tosome specific interaction Hamiltonian, as in e.g. [1–3]. The effect of the interaction is readin the evolution of the states of the Hilbert space, which is a tensor product, and in whichexcitations and de-excitations in the sector of the detector according to its interaction withthe field are interpreted as particles. For an in-depth review of the state of the art see [27].As we have shown this type of factorization does not take place for such natural models asthe one considered here so it is very important to understand and characterize the physicalsystems for which this is possible and the ensuing implications for the measurement problemin quantum mechanics. Appendix A: Hilbert spaces, modified measures and Radon-Nikodym derivativesA.I. The Hilbert space L µ The implementation of the GNH algorithm requires the introduction of appropriateHilbert spaces. After realizing that a scalar product of the type (II.15) plays a naturalrole in the eigenvalue problems that crop up in the study of the system that we are consider-ing here, it is natural to construct Hilbert spaces endowed with it. To see that this is not acompletely trivial task, it suffices to realize that (II.15) does not make sense for elements of L (0 ,
1) as they do not have well defined boundary values at x = 0 , H (0 , trace operator γ . The elements of the H ( I ) Sobolev space defined onan interval I ⊂ R have representatives given by continuous functions on its closure I [28].Their values at the boundary (or their limits for that matter) are the traces that we denoteas γ , γ because the boundary of an interval is disconnected.In the present context it is better to avoid demanding so much regularity so we consider,6to begin with, spaces with elements satisfying minimal regularity requirements. The simplestexample of such a space would be L [0 ,
1] := R × L (0 , × R consisting of elements thatwe will denote as ~v := ( v , v, v ). This is a Hilbert space with the scalar product h ~v, ~w i L := α v w + α v w + Z [0 , v · w d µ L (A.1)provided that α j >
0. Notice that the completeness of L [0 ,
1] is a direct consequence ofthat of L (0 ,
1) and R .A useful alternative way to describe this space makes use of a suitable measure µ . Letus consider the Borel σ -algebra B ([0 , µ = α δ + µ L | (0 , + α δ , wherethe measure δ j is defined for any A ∈ B ([0 , δ j ( A ) = (cid:26) j ∈ A j / ∈ A (A.2)Let us consider now the real Hilbert space of square integrable functions L µ [0 ,
1] := (cid:26) f : [0 , → R : f is µ -measurable , Z [0 , f d µ < + ∞ (cid:27) , (as usual, functions equal µ -a.e. are identified) endowed with the scalar product h u, v i µ = Z [0 , u · v d µ . (A.3)It is straightforward to see that the mapΦ : L µ [0 , −→ L [0 , f ( f (0) , f | (0 , , f (1))is a Hilbert space isomorphism, hence, in the following we will view elements of these Hilbertspaces as square integrable functions (w.r.t. the measure µ ) defined on the closed interval[0 ,
1] or as elements of R × L (0 , × R with the scalar products defined above. Wheneverno confusion may arise we will use the shorthand L µ to denote L µ [0 , A.II. Absolutely continuous functions and Radon-Nikodym derivatives
A useful way to write the Lagrangians that we use in the paper makes use of the Radon-Nikodym (RN) derivatives of appropriately defined functions. They also play a central role inthe description of the relevant elliptic operators that appear in the Hamiltonian formulationand the construction of the Fock space. Here we briefly review the central concepts and givea list of properties that are used throughout the paper.Given two measures µ and ν over B ([0 , ν is µ -absolutely continuous ( µ -a.c. usually denoted as ν ≪ µ ) if whenever µ ( A ) = 0 for some A ∈ B ([0 , ν ( A ) = 0.The RN theorem states that over finite measure spaces, such definition is equivalent to “weak µ -differenciability”, which means that there exists a µ -measurable function f ∈ L ( µ ) suchthat ν f ( A ) = Z A f d µ . f (usually denoted by f = d ν d µ ) is known as the RN derivative and it is uniquein the sense that any other function that satisfies the preceding properties is equal to fµ -a.e.If we want to µ -differentiate a function F : [0 , → R w.r.t. µ = α δ + µ L | (0 , + α δ wehave to associate to F a µ -a.c. Lebesgue-Stieltjes measure ν F . To this end we define ν F = (cid:16) F (0+) − F (0) (cid:17) δ + F ′ µ L | (0 , + (cid:16) F (1) − F (1 − ) (cid:17) δ where F | (0 , is a.c. in the usual calculus sense (so that F ′ µ L | (0 , is µ L -a.c.). Under theseconditions F is differentiable µ L -a.e. and has well defined limits F (0+) and F (1 − ), notnecessarily equal to its values at the boundary F (0) and F (1). In fact, considering that thea.c. functions over (0 ,
1) can be seen as elements of the Sobolev space H (0 ,
1) (functionsof L (0 ,
1) with distributional derivative in L (0 , { , } is placed by the condition of being µ -a.c. continuous, we conclude that thevector space of µ -a.c. functions H µ [0 ,
1] is isomorphic to R × H (0 , × R (compare with L µ [0 , ∼ = R × L (0 , × R ). An element F ∈ H µ [0 ,
1] has boundary values F (0), F (1) andtraces γ ( F ) = F (0+), γ ( F ) = F (1 − ) which are generically different from F (0) and F (1).This is related to the fact that, in general, α j = µ j . In our model the boundary valuesof functions (such as F (0), F (1)) play a secondary role from the physical point of view asthe positions of the point masses will be defined by the traces, but they play a central role from the mathematical point of view because they are crucial to prove that the fundamentalLaplace operator of the model is self-adjoint.Finally we define the RN derivative of F ∈ H µ [0 ,
1] as d F d µ := d ν F d µ . Notice that if d F d µ is in H µ [0 , d F d µ := dd µ (cid:16) d F d µ (cid:17) . The explicit form of the RN derivative of F ∈ H µ [0 ,
1] can be immediatelyobtained from the previous result by comparing the expressions of ν F and µ • d F d µ ( j ) = ( − j α j (cid:0) γ j ( F ) − F ( j ) (cid:1) . • d F d µ ( x ) = F ′ ( x ), µ L -a.e. in (0 ,
1) which is just the RN derivative of F w.r.t. µ L .It is important to notice that the standard Leibniz rule does not hold, in general, for RNderivatives though a modified rule exists. Indeed, by defining K : [0 , → R with K | (0 , = 0and K ( j ) = ( − j α j for j ∈ { , } we have:d( F G )d µ ( x ) = d F d µ ( x ) G ( x ) + F ( x ) d G d µ ( x ) + K ( x ) d F d µ ( x ) d G d µ ( x ) , µ -a.e. in [0 , . (A.4)This expression will be very useful whenever we have to perform integrations by parts ofRN derivatives.8 A.III. The fundamental Laplace operator
Next we will introduce a generalized Laplace operator, defined in terms of the RN deriva-tive, that plays a central role in the paper. Let us write D := (cid:26) u ∈ L µ [0 ,
1] : ∃ d u d µ ( x ) ∀ x ∈ [0 , , d u d µ is µ -a.c. , d u d µ ∈ L µ [0 , (cid:27) , (A.5) b D := (cid:26) u ∈ D : ( − j d u d µ ( j ) − A ( j ) u ( j ) = 0 (cid:27) , (A.6)∆ µ : D ⊂ L µ [0 , → L µ [0 ,
1] : u (1 + C ) d u d µ , (A.7)where C : [0 , → R and A ( j ) ∈ R for j ∈ { , } are to be determined together with α j .One condition that we have to impose is that we should recover the equations of motion,in particular for the lateral limits at the boundaries. Fulfilling this requirement does notcompletely fix these constants, but puts the eigenfunctions of ∆ µ in correspondence with theones of the original problem. Hence, if we make ∆ µ self-adjoint, then the set of eigenfunctionswill form a complete set of orthogonal functions w.r.t. the scalar product h· , ·i µ .So let us compare equations (II.10)-(II.12) with the eigenvalue problem ∆ µ Y = λY (defined in a subspace of L µ ). First, (II.10) requires that C | (0 , = 0 because then ∆ µ is the usual Laplacian (second derivative) when restricted to the interval (0 , C ( j ) = A ( j ) α j makes ∆ µ symmetric (i.e. h ∆ µ u, w i µ − h u, ∆ µ w i µ = 0) provided we restrictits domain to b D . In fact using (A.4) we have: h ∆ µ u, w i µ = − (cid:28) d u d µ , d w d µ (cid:29) µ − X j ∈{ , } ( − j d u d µ w − X j ∈{ , } α j d u d µ (cid:18) ( − j α j d w d µ − C ( j ) w (cid:19) = − (cid:28) d u d µ , d w d µ (cid:29) µ + X j ∈{ , } A ( j ) u ( j ) w ( j ) . where we have used that the boundary conditions ( − j d u d µ ( j ) − A ( j ) u ( j ) = 0 and ourparticular choice of C ( j ), leading to a symmetric expression in u and w .It is important to notice that ussing the expression given for the RN derivative at theboundary, we can show that the Robin-like boundary conditions ( − j d u d µ ( j ) − A ( j ) u ( j ) = 0are equivalent to γ j ( u ) = (1 + α j A ( j )) u ( j ) . (A.8)Now taking A ( j ) = µ j ∆ e ω j − α j µ j ∆ e ω j and using the previous equation for d F d µ and for γ j ( F ), weobtain that ∆ µ Y = λY on (0 ,
1) is equivalent to: Y ′′ = λY , (A.9) γ j ( Y ′ ) = ( − j (cid:16) α j (1 − α j µ j ∆ e ω j ) λ + µ j ∆ e ω j (cid:17) γ j ( Y ) . (A.10)If we finally require α j (1 − α j µ j ∆ e ω j ) = µ j , equations (A.10) become equations (II.11)-(II.12), recovering thus the original problem. Clearly from this condition and the fact that µ j >
0, we see that all the possible solutions for α j are positive, so they can be used todefine scalar products of the form (A.1),(A.3). A very simple case corresponds to ∆ e ω j = 09(some sort of resonance that cancels out the effect of the springs and the string) for which α j = µ j and A ( j ) = 0. It is important to realize that, as we have already mentioned before,the positions of the point masses are given by the traces γ j ( Y ) (which can be identified withthe X ( j ) of section II.2), however, the values Y ( j ) that appear in the Robin-like boundaryconditions (A.8) do not have a direct physical interpretation as far as we know.At this point, we have only managed to obtain a symmetric operator ∆ µ whose eigen-functions are in correspondence with the ones of the original problem, but in fact, it can beshown that the operator ∆ µ is self-adjoint in b D by using the method described in [21]. Asmentioned before, this implies that the set of its eigenfunctions can be turned into a Hilbertbasis of the Hilbert space b D showing in particular, that it is complete. A remark is in orderhere, if u, v ∈ b D we have, using equation (A.8), that: h u, v i µ = X j ∈{ , } α j u ( j ) v ( j ) + h u, v i = X j ∈{ , } µ j γ j ( u ) γ j ( v ) + h u, v i = hh u, v ii . (A.11)This establishes a connection between the results in the present section with the argumentleading to (II.15) (notice that the boundary values of the eigenfunctions appearing there can–and should– be interpreted as right or left limits or, in our notation as γ j ( u ) , γ j ( v )). A.IV. Normal modes and Fourier coefficients
An orthonormal (Hilbert) basis of L µ [0 ,
1] can be constructed by using the eigenfunctionsof the self-adjoint operator ∆ µ defined in (A.7) in the domain b D given in (A.6). Theseeigenfunctions, in turn, can be computed from the solutions to the eigenvalue problemconsidered in (II.10)-(II.12) by defining their values at x ∈ { , } in such a way that theRobin-like boundary conditions, or equivalently equation (A.8), are satisfied. The elementsof the Hilbert basis will, hence, be of the form { Y n } (in one to one correspondence withthe eigenfunctions { X n } introduced on section II.2) with Y n | (0 , = X n /g n (in particular γ j ( Y n ) = X n ( j ) /g n ) and Y n ( j ) = (1 − α j µ j ∆ e ω j ) X n ( j ) /g n . The factor g n guarantees that Y n is normalized w.r.t. the scalar product h· , ·i µ . It can be explicitly computed to be g n = 12 (cid:18) µ ∆ e ω + (1 + µ ) ω n + µ ( ω n − ∆ e ω ) + µ ( ω n + ∆ e ω ) ω n + µ ( ω n − ∆ e ω ) ω n + µ ( ω n − ∆ e ω ) (cid:19) . The asymptotic behavior of 1 /g n when n → ∞ can be obtained by using equation (II.14): g − n = √ µ π n + O (cid:0) n − (cid:1) . An element F ∈ L µ [0 ,
1] can be expanded as usual as F = P h Y n , F i µ Y n . It is interestingto notice at this point that functions f : [0 , → R supported at the boundary { , } arenon trivial elements of L µ [0 ,
1] and hence can be expanded in the basis introduced above.In the particular case of the function F ∈ L µ [0 ,
1] given by F : [0 , → R : x F ( x ) = (cid:26) x = 00 x ∈ (0 ,
1] (A.12)0we have h Y n , F i µ = α Y n (0) = α (1 − α µ ∆ e ω ) X n (0) /g n = √ µ α X n (0) /g n . In the dis-cussion of the Fock quantization of the model we need the asymptotic behavior of thesecoefficients for large values of n . This can be easily obtained by using the fact that in thislimit only the eigenvectors associated with negative eigenvalues matter. The asymptoticbehavior of g − n and the one of ω n given in equation (II.14) lead to h Y n , F i µ = √ µ α X n (0) g n = √ µ α ω n g n = p µ α πn + O (cid:0) n − (cid:1) . ACKNOWLEDGMENTS
We would like to thank J. Louko for some enlightening discussions. This work has beensupported by the Spanish MINECO research grants FIS2012-34379, FIS2014-57387-C3-3-P and the Consolider-Ingenio 2010 Program CPAN (CSD2007-00042). B. Ju´arez-Aubryis supported by CONACYT, M´exico REF 216072/311506 with additional support fromSistema Estatal de Becas, Veracruz, M´exico. Juan Margalef-Bentabol is supported by a “laCaixa” fellowship. [1] W. G. Unruh, Phys. Rev. D (1976) 870.[2] B. S. DeWitt “Quantum Gravity: the new synthesis”, in General Relativity: an Einsteincentenary survey , eds. S. W. Hawking and W. Israel, Cambridge University Press (1979).[3] W. G. Unruh and R. M. Wald, Phys. Rev. D (1984) 1047.[4] E. Witten, Commun. Math. Phys. 121 (1989) 351.[5] J. F. Schonfeld, Nucl. Phys. B185 (1981) 157[6] S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48 (1982) 975.[7] S. Deser, R. Jackiw and S. Templeton, Ann. Phys. 140 (1982) 372.[8] J. ´Sniatycki and G. Schwarz, Commun. Math. Phys. 168 (1995) 441.[9] A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. (1998) 904.[10] A. Ashtekar, A. Corichi and K. Krasnov, Adv. Theor. Math. Phys. (1999) 419.[11] J. ´Sniatycki, Rep. Math. Phys. (1999), 205.[12] P. A. M. Dirac, Lectures on Quantum Mechanics , Dover Publications Inc., (2001).[13] J. F. Barbero G., J. Prieto and E. J. S. Villase˜nor, Class. Quant. Grav. (2014) 045021.[14] M. J. Gotay, Presymplectic Manifolds, Geometric Constraint Theory and the Dirac-BergmannTheory of Constraints , Thesis, Center for Theoretical Physics of the University of Maryland(1979).[15] M. Gotay, J. Nester and G. Hinds, J. Math. Phys. (1978) 2388.[16] M. Gotay and J. Nester, Generalized constraint algorithm and special presymplectic manifolds (Lecture Notes in Mathematics 775, Geometric Methods in Mathematical Physics), Springer(1980).[17] R. M. Wald,
Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics ,University of Chicago Press (1994).[18] B. Yurke, Am. J. Phys. (1984) 1099.[19] A. Lewis Licht, Quantization of a string with attached mass , arXiv:1109.4849 [physics].[20] J. E. Marsden and T. J. Hughes,