Polymer quantization, stability and higher-order time derivative terms
Patricio Cumsille, Carlos M. Reyes, Sebastian Ossandon, Camilo Reyes
aa r X i v : . [ h e p - t h ] M a r Polymer quantization, stability and higher-order time derivative terms
Patricio Cumsille , , Carlos M. Reyes , ∗ Sebastian Ossandon , and Camilo Reyes Departamento de Ciencias B´asicas, Universidad del B´ıo B´ıo, Casilla 447, Chill´an, Chile Centro de Biotecnolog´ıa y Bioingenier´ıa (CeBiB),Universidad de Chile, Beaucheff 851, Santiago de Chile Instituto de Matem´aticas, Pontificia Universidad Cat´olicade Valpara´ıso, Casilla 4059, Valpara´ıso, Chile and Departamento de Ciencias Fisicas, Facultad de Ciencias Exactas,Universidad Andres Bello, Republica 220, Santiago, Chile
The possibility that fundamental discreteness implicit in a quantum gravity theory mayact as a natural regulator for ultraviolet singularities arising in quantum field theory hasbeen intensively studied. Here, along the same expectations, we investigate whether a non-standard representation, called polymer representation can smooth away the large amountof negative energy that afflicts the Hamiltonians of higher-order time derivative theories;rendering the theory unstable when interactions come into play. We focus on the fourth-order Pais-Uhlenbeck model which can be reexpressed as the sum of two decoupled harmonicoscillators one producing positive energy and the other negative energy. As expected, theSchr¨odinger quantization of such model leads to the stability problem or to negative normstates called ghosts. Within the framework of polymer quantization we show the existenceof new regions where the Hamiltonian can be defined well bounded from below.
PACS numbers: 03.65.-w, 04.60.Ds, 04.60.Nc, 04.60.PpKeywords: higher time derivatives; polymer quantization; stability
I. INTRODUCTION
The standard model of particles has foundations on local quantum field theories having opera-tors of mass dimension d ≤
4. These operators are justified in order to implement the requirementsof stability and unitarity without further elaborations [1]. However, when going to higher energiesit is commonly believed that higher-order operators will play a key role in describing fundamentalphysics. This may be particularly true when they involve higher time derivatives since the newmodes that arise allow to describe effects from a high scale. Usually these new modes are very high ∗ Electronic mail: [email protected] when the coupling of the higher-order time derivatives are suppressed by the high scale as takenin effective approaches. Higher-order operators have received increased attention over the years.They have been investigated in the context of loop quantum gravity [2–7], Lorentz symmetry viola-tion [8–13], causality and stability [14], fine-tuning [15–18], the hierarchy problem [19, 20], radiativecorrections [21–23] and nonminimal couplings [24–29]. The presence of higher-order derivatives inthe gravitational sector are a key ingredient in order to achieve a consistent renormalization in thesemiclassical approach, where matter fields are quantized over classical curved background [30].Higher time derivative theories were introduced long time ago by Ostrogradsky [31]. The ap-proach is based on a variational formalism and involves a Lagrangian L ( x, ˙ x, ¨ x . . . , x ( n ) = d n xdt n )and an extended Hamiltonian H ( x, π ˙ x , π ¨ x . . . , π x ( n ) ) of 2 n variables. Ostrogradsky showed that theformalism leads a higher-order Hamiltonian not bounded from below as can be seen in the non-quadratic momenta terms that appear. This is the classical Ostrogradsky instability of higher-ordertime derivative theories which can be avoided in a few cases, for instance when the models hasconstraints [32].The quantization of higher time derivative theories can be implemented introducing a changeof variables in order to transform the original Lagrangian into a sum of decoupled normal-orderLagrangians. In general one of these Lagrangians has large negative energy leading to the instabilityor alternatively, by changing the vacuum state, to an indefinite metric theory [33]. The instabilityof the Hamiltonian has received much attention and has been tackled from different perspectives,such as phase space reduction [34–36], complex canonical transformations [37], PT symmetry [38–40], and Euclidean-path reduced amplitudes [41], gravitational ghosts and tachyons [42, 43]. Inquantum field theory, Lee and Wick showed that by imposing the negative norm states to decay itis possible to preserve unitarity order by order in perturbation theory [44, 45]. Resurgence of suchideas have been used to solve the hierarchy problem [19, 20] and in higher-order effective modelswith Lorentz symmetry violation [46–48].In this work we study the stability of higher-order time derivative models within the frameworkof polymer quantization. In particular, we focus on the Pais-Uhlenbeck (P-U) model. The polymerrepresentation is a non-standard representation of quantum mechanics inspired by some resultsthat emerge from loop quantum gravity. The possibility of space discreteness that appears in loopquantum gravity [49] has served to improve the convergence of quantum field theories [6, 7] andcosmological singularities [50]. In this paper our main goal is to test whether the fundamentaldiscreteness implicit in the polymer representation allows to improve the stability of higher-ordertheories. The polymer quantization has been considered in several studies such as two-point func-tions [51–53], cosmology [50, 54], central forces [55, 56], higher space derivatives [57], thermody-namics [58], compact stars [59] and low energy limits [60, 61].The organization of this work is as follows. In section II we introduce the P-U model and weexplicitly show the origin of the instability in the Schr¨odinger quantization. In the third section wegive a basic review of the polymer formalism. In section IV we polymer quantize the P-U modeland solve the Hamiltonian eigenvalue equation. From the previous results we analyze stability inthe region of validity of the theory. In the last section we give the conclusions. II. THE PAIS-UHLENBECK MODEL
The P-U model consists of an harmonic oscillator coupled to a higher-order term described bythe Lagrangian L = 12 m ˙ x − kx − g x , (1)where g is a small parameter. The equation of motion is − kx − m d xdt − g d xdt = 0 . (2)Inserting the plane wave ansatz x ( t ) = x e − iyt produces the four solutions y = ± s m ± p m − kg g . (3)We define the two positive solutions according to ω = ω s − √ − ε ε , ω = ω s √ − ε ε , (4)where we introduce ω = p k/m as the usual frequency and g = mεω with ε a small dimensionlessparameter. Now, in the limit ε → ω → ω and the second one ω → ω √ ε blows up. One expects this behavior of the last solution in theorieswith higher-order time derivative theories indicating a possible window to physics at higher energyscales. In order to avoid imaginary solutions we impose ε ≤ ε c = 1 / ε c is the critical valueat which the two solutions collapse ω = ω .The conjugate momenta to x and ˙ x are defined by the expressions p = ∂L∂ ˙ x − ddt (cid:18) ∂L∂ ¨ x (cid:19) , (5) π = ∂L∂ ¨ x . (6)From (1), they are given by p = m ˙ x + gx (3) , (7) π = − g ¨ x . (8)The Hamiltonian is constructed via the extended Legendre transformation H = p ˙ x + π ¨ x − L whichafter substitution yields H = − π g − m ˙ x xp + kx . (9)Let us consider the new set of variables x = ω x − π/gω − ω , x = − ω x − π/gω − ω , (10)and p = p − ( m − gω ) ˙ x , p = p − ( m − gω ) ˙ x , (11)where m ′ = p m − kg . They allow to define the ladder variables a = r m ′ ω ~ (cid:18) x + i m ′ ω p (cid:19) ,a = r m ′ ω ~ (cid:18) x + i m ′ ω p (cid:19) . (12)Using these variables the Hamiltonian can be expressed as the sum of two decoupled harmonicoscillators H = ~ ω a a † + a † a ) − ~ ω a a † + a † a ) , (13)where we refer to the first and the second term as the positive and negative sectors of the theory.The quantization of such model follows by imposing the usual commutation relations in theextended phase space [ˆ x, ˆ p ] = i ~ and [ˆ˙ x, ˆ π ] = i ~ . With the use of the canonical variables [ ˆ x , ˆ p ] = i ~ and [ ˆ x , ˆ p ] = i ~ one can check that the creation and annihilation operators satisfy h ˆ a , ˆ a † i = 1 , h ˆ a , ˆ a † i = 1 . (14)To find the ground state wave function denoted by Ψ , consider the explicit action of the operatorsin the Hilbert space ˆ xψ = xψ , ˆ pψ = − i ~ ∂ψ∂x , (15)ˆ˙ xψ = ˙ xψ , ˆ πψ = − i ~ ∂ψ∂ ˙ x . (16)Inserting these expression into the previous creation and annihilation operators (12) we identifytwo realizations for the vacuum. The first one is to define the vacuum as the one annihilated byˆ a and ˆ a ˆ a Ψ = ˆ a Ψ = 0 , (17)leading to Ψ = N e g ( ω − ω ~ ( ˙ x + ω ω x )+ ig ~ ω ω x ˙ x , (18)where N is a normalization factor. The Hamiltonian turns out to beˆ H = ~ ω ( ˆ N + 12 ) − ~ ω ( ˆ N + 12 ) , (19)where ˆ N = ˆ a † ˆ a and ˆ N = ˆ a † ˆ a are the number operators of positive and negative particlesrespectively. We see that the energy is not bounded from below since one can always create morenegative particles.A different vacuum Ψ ′ amounts to change the annihilation operator in the negative sector andto maintain the previous in the positive sector, namelyˆ a Ψ ′ = ˆ a ′ Ψ ′ = 0 , (20)with ˆ a ′ = ˆ a † . The vacuum state Ψ ′ in this case isΨ ′ = N e g ( ω ω ~ ( ˙ x − ω ω x ) − ig ~ ω ω x ˙ x . (21)Note that Ψ ′ can be obtained in (18) performing the transformation ω → − ω . The Hamiltonianis found to be ˆ H ′ = ~ ω ( ˆ N + 12 ) + ~ ω ( ˆ N ′ + 12 ) , (22)where now [ˆ a ′ , ˆ a ′† ] = − N ′ = − ˆ a ′† ˆ a ′ is the new number operator. In this case the theory haspositive defined Hamiltonian but the price to pay is to end up with negative norm states or ghoststhat may threaten the conservation of unitarity and with non-normalizable wave functions. III. POLYMER REPRESENTATION
In quantum mechanics when dealing with the adjoint operators ˆ x and ˆ p usually one encounterssome technical problems due to their unboundedness. Therefore, it is convenient to switch to theexponentiated versions ˆ U ( α ) and ˆ V ( β )ˆ U ( α ) = e iα ˆ x , ˆ V ( β ) = e iβ ˆ p/ ~ , (23)whose action are defined byˆ U ( α ) ψ ( x ) = e iαx ψ ( x ) , ˆ V ( β ) ψ ( x ) = ψ ( x + β ) , (24)for all state ψ ( x ) in the Hilbert space L ( R ). Both operators ˆ U ( α ) and ˆ V ( β ) satisfy the Weyl-Heisenberg algebra ˆ U ( α ) ˆ U ( α ′ ) = ˆ U ( α + α ′ ) , ˆ V ( β ) ˆ V ( β ′ ) = ˆ V ( β + β ′ ) , ˆ U ( α ) ˆ V ( β ) = e − iαβ ˆ V ( β ) ˆ U ( α ) , (25)where the parameters α and β have (length) − and length dimensions respectively. From theabove algebra one can obtain the usual commutations relations [ˆ x, ˆ p ] = i ~ . Due to the Stone-von-Neumann theorem any representation of the commutation relations have the form of the operators(23), modulo unitarity transformation, since the two operators ˆ U ( α ) and ˆ V ( β ) are strongly con-tinuous in their parameters, see [49, 62] and references therein.In the polymeric construction one starts with a graph given by a countable set of points in thereal line, denoted by γ = { x j : j ∈ N } , with some requirements [49]. We define the functionsassociated to a graph γ as f ( x ) = f j x = x j x γ (26)and their Fourier transform functions f ( k ) given by f ( k ) = X j f j e − ix j k , (27)satisfying the relation X j | f j | x nj < ∞ for n = 0 , , , . . . (28)We denote by Cyl γ the space of all cylindrical functions f ( k ) and Cyl the union of all
Cyl γ overall graphs γ . We add to the space Cyl all the limits of Cauchy sequences, that is the Cauchycompletion which is called the polymeric Hilbert space denoted by H poly endowed with the scalarproduct h e − ikx i | e − ikx j i = δ x i ,x j . (29)Recall, this is an alternative form to view the Hilbert space within the construction of the riggedHilbert space Ω ⊂ H ⊂ Ω ∗ , see Ref. [63].The main differences between the polymeric representation and the usual of quantum mechanicsis that H poly is a non-separable space and has an intrinsic fundamental discreteness leading to anonequivalent representation, see Ref. [64]. To be precise, the action of the operators ˆ U ( α ) andˆ V ( β ) given in Eq. (24) is well-defined in H poly , however, in the polymer representation there is noself-adjoint operator ˆ p = − i ~ ∂∂x such that the second equality in (23) is satisfied, that is to say, themomentum operator ˆ p is not well defined on H poly . This is due to the fact that ˆ V ( β ) is not weaklycontinuous in the parameter β , as can be verified using the modified product with the Kroneckerdelta in Eq. (29). Nevertheless, one can approximate the operator ˆ p with the expressionˆ p = − i ~ µ (cid:16) ˆ V ( µ / − ˆ V ( − µ / (cid:17) , (30)where µ is a fundamental length scale associated with a possible discreteness of space, comingfrom a more fundamental theory. The above approximation is natural, at least in the distributionalsense, since if we take the limit as µ → L ( R ). IV. STABILITY AND HIGHER-ORDER TIME DERIVATIVES
In section II we have expressed the P-U Hamiltonian as two decoupled harmonic oscillators, forinstance H = H − H . Their quantum counterparts represent normal particles and nonstandardones producing negative energy sometimes called Lee-Wick particles [44, 45]. Using the new set ofvariables (10) and (11) we find H = 12 k x + 12 m ′ p ,H = 12 k x + 12 m ′ p , (31)where k j = m ′ ω j with j = 1 ,
2. In other words, the P-U model involves two oscillators withthe same mass m ′ and so taking advantage of this fact we polymer quantize the system as twoindividual harmonic oscillators.The polymer Hilbert space H poly = H poly ( x ) ⊗ H poly ( x ) comprises the polymeric spaces foreach oscillator. In the Hilbert space we have the action of the operatorsˆ U ( α ) ψ ( x ) = e iα x ψ ( x ) , ˆ V ( β ) ψ ( x ) = ψ ( x + β ) , ˆ U ( α ) ψ ( x ) = e iα x ψ ( x ) , ˆ V ( β ) ψ ( x ) = ψ ( x + β ) . (32)Considering the wave function ψ ( x , x ) ∈ H poly we arrive at the Schr¨odinger equation (cid:18) k ˆ x + 12 m ′ ˆ p − k ˆ x − m ′ ˆ p (cid:19) ψ ( x , x )= Eψ ( x , x ) , (33)where E is the total energy of the system and the momentum operators areˆ p = − i ~ µ (cid:16) ˆ V ( µ / − ˆ V ( − µ / (cid:17) , ˆ p = − i ~ µ (cid:16) ˆ V ( µ / − ˆ V ( − µ / (cid:17) , (34)with µ and µ the fundamental lengths associated to H poly ( x ) and H poly ( x ).With the ansatz ψ ( x , x ) = ψ ( x ) ψ ( x ) and considering the cylindrical function for eachoscillator j , namely ψ j ( k ) = X ℓ ψ j ( x j,ℓ ) e − ix j,ℓ k , (35)together with Eqs (34), we obtain the equations for the coefficients ψ j ( x j,ℓ + µ j ) + ψ j ( x j,ℓ − µ j )= − E j µ j ~ ω j d j + µ j x j,ℓ d j ! ψ j ( x j,ℓ ) , (36)where we have introduced E = E − E and d j = ~ m ′ ω j . The previous equations suggest that one canfind solutions supported at the uniformly spaced points x j,ℓ = x + ℓµ j for some x ∈ [0 , min { µ j } ).Indeed, given the parameters of the equations { ψ ( x j,ℓ ) } ∞ ℓ = −∞ , by using (36) one can construct asolution ψ j supported on the lattice γ jx ,µ j := { x j,ℓ ∈ R | x j,ℓ = x + ℓµ j , ℓ ∈ Z } . Using this in Eq.(36) yields ψ j ( x j,ℓ +1 ) + ψ j ( x j,ℓ − )= − E j µ j ~ ω j d j + µ j ( x + ℓµ j ) d j ! ψ j ( x j,ℓ ) . (37)Without loss of generality, let us consider both graphs based on the point x = 0. Hence, with x j,ℓ = ℓµ j , ψ j ( k j ) = P ∞ ℓ = −∞ ψ j ( x j,ℓ ) e − ik j ℓµ j and multiplying the equation (37) by e − ikℓµ j andsumming over ℓ , we arrive at2 cos( k j µ j ) ψ j ( k j ) = 2 − E j µ j ~ ω j d j ! ψ j ( k j ) − µ j d j ψ ′′ j ( k j ) , (38)where k j ∈ ( − πµ j , πµ j ). Normalizing and arranging the terms we get ψ ′′ j ( k j ) + 2 d j E j ~ ω j + d j µ j (cos( k j µ j ) − ! ψ j ( k j ) = 0 . (39)Finally, by making the change of variables z j = k j µ j + π ∈ (0 , π ) we write ψ ′′ j ( z j ) + [ a j − q j cos(2 z j )] ψ j ( z j ) = 0 , (40)where q j = 4 λ − j , a j = λ j (cid:18) λ j E j ~ ω j − (cid:19) and λ j = µ /d j is a dimensionless parameter. Equation(40) is the well-known Mathieu equation in its canonical form. We seek for periodic solutions ofthe Mathieu equations, since ψ j (0) = ψ j ( π ) by construction.In order to approximate perturbatively to the harmonic oscillator we take λ to be small, whichproduces the the following asymptotic expansion for the first oscillator a ,n ( λ ) = − λ − + 4(2 n + 1) λ − −
14 (2 n + 2 n + 1)+ O (cid:0) λ (cid:1) , (41)for n = 0 , , , . . . . Considering the asymptotic expansion for a we can write E ,n = ~ ω (cid:18) (2 n + 1) −
116 (2 n + 2 n + 1) λ (cid:19) + O ( λ ) . (42)For the second oscillator we have two alternatives: the first one is to consider µ ≪ d whichanalogously produces E ,m = ~ ω (cid:18) (2 m + 1) −
116 (2 m + 2 m + 1) λ (cid:19) + O ( λ ) . (43)The second alternative is to consider a large λ leading to the expansion E ,m = ~ ω λ + ~ ω λ m + O ( λ − ) . (44)With exception of the ground state we have that the negative energy of the system increaseswithout limit. This case may be interesting to analyze from a field point of view, since in this casethe propagators turn to be suppressed by the high scale [51]. For the rigid rotator case in which λ is large as well as λ , achieved for example taking large values of the original mass parameter m ,one can see that the energy spectrum goes as E n,m ≈ ~ ( λ n − λ m ) which can be more negativewith respect to the Schr¨odinger representation for the same occupation numbers ( n, m ). In thiscase there is no improvement for stability. Let us focus on the case λ ≪ λ ≪ −5 0 5 10 15 x 10 −8−6−4−2024 x 10 −4 m E ¯ n , m FIG. 1: Comparison of the P-U energies as a function of m evaluated at ¯ n in the Schr¨odinger and polymericrepresentation. FIG. 2: The polymeric P-U energy E n,m as a function of n and m . the total energy E n,m of the system can be written as E n,m = ~ ω (cid:18) (2 n + 1) −
116 (2 n + 2 n + 1) λ (cid:19) − ~ ω (cid:18) (2 m + 1) −
116 (2 m + 2 m + 1) λ (cid:19) . (45)Analogously to the Schr¨odinger quantization the high energy oscillator seems to lead to the in-stability, however we will show below that in certain regions the Hamiltonian can be defined wellbounded from below. It is important to emphasize that in the absence of operators connectingthe two Hilbert spaces the negative energy is not to serious and the problem of instability appearsupon introducing the interactions.Let us consider the constraints imposed on the number of normal particles n that follows fromthe absence of any polymeric effect in quantum mechanics. From the first term in (45), it can beseen that the positive-oscillator corrections become significant or O (1) at the value ¯ n = √ h . For1example considering the vibrational modes of a carbon monoxide molecule with mass m = 10 − kg and frequency ω = 10 s − , and estimating the polymer scale to be µ = 10 − m, we find¯ n ≈ , see [49]. In addition, in several approaches to polymer quantum mechanics a cutoff in theenergy eigenvalues has been justified for the harmonic oscillator in order to implement a consistentrenormalization [60]. This upper limit is key to define our effective region, since now, in contrastto what happens in the usual P-U model for higher values of the occupation number m and forfixed n = ¯ n the polymeric energy is bounded from below, see Fig. 1. In the general setting whereboth occupation numbers n and m vary freely the total energy of the system is well bounded frombelow, avoiding possible stability problems, as shown in Fig. 2. V. CONCLUSIONS
In this work we have analyzed the stability of theories containing higher-order time derivativeswithin the framework of polymeric quantization. For this we have focused on the well-knownPais-Uhlenbeck model with fourth-order time derivatives in the Lagrangian. Using a canonicaltransformation we have cast the theory into a sum of two decoupled harmonic oscillators, one withpositive energy and the other with negative energy. The negative-energy oscillator is responsiblefor the instabilities that arise in the presence of interactions.We have shown that the discrete nature of the polymer Hilbert space introduces correctionsin the energy spectrum of the P-U model which allows to define a region of positive definedHamiltonian. For this we have set a cutoff from observational constraints from quantum mechanicsdue to the absence of any polymeric effect and further motivated by a consistent renormalizationprogram. We have established an effective region defined by small values of the parameters λ and λ at which the theory has a well bounded Hamiltonian. However, for the case with large λ and λ , we have found that the instability shows up for very low occupation numbers with noimprovement with respect to the usual Schr¨odinger quantization. We leave for future investigationsthe inclusion of interactions and the case of large λ in the context of quantum field theory. Acknowledgments
We want to thank H. A. Morales-Tecotl and T. Pawlowski for valuable comments on this work.P.C acknowledges support from Centre for Biotechnology and Bioengineering under PIA-ConicytGrant No. FB0001. C.M.R. acknowledges support from Grant Fondecyt No. 1140781, DIUBB No.2141709 4/R and the group of
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