Generalized Uncertainty Principle: from the harmonic oscillator to a QFT toy model

Several models of quantum gravity predict the emergence of a minimal length at Planck scale. This is commonly taken into consideration by modifying the Heisenberg Uncertainty Principle into the Generalized Uncertainty Principle. In this work, we study the implications of a polynomial Generalized Uncertainty Principle on the harmonic oscillator. We revisit both the analytic and algebraic methods, deriving the exact form of the generalized Heisenberg algebra in terms of the new position and momentum operators. We show that the energy spectrum and eigenfunctions are affected in a non-trivial way. Furthermore, a new set of ladder operators is derived which factorize the Hamiltonian exactly. The above formalism is finally exploited to construct a quantum field theoretic toy model based on the Generalized Uncertainty Principle.


Introduction
Understanding how to merge quantum theory and general relativity is one of the most demanding challenges in modern theoretical physics. Despite the sustained efforts [1][2][3][4][5][6][7][8][9][10], a consistent framework is still missing, the major obstacle being the lack of experimental evidences at the scale where quantum gravity (QG) effects are expected to become manifest, i.e. the Planck scale. Among the most robust predictions of QG models, the emergence of a minimum length of the order of Planck length p " 10´3 5 m appears as a distinctive signature of a classicalto-quantum transition of gravity. This has found confirmation in a series of gedanken experiments involving the formation of gravitational instabilities in high-energy string scatterings [1][2][3]5] and the creation of micro black holes [11]. Hence, in the would-be theory of QG, Planck length marks a threshold beyond which the common understanding of spacetime as a smooth continuum breaks down, giving way to a granular foamy structure due to inherent quantum fluctuations.
Due to the relatively simple structure and the wide range of physical applicability, a paradigmatic system where to study GUP effects is the harmonic oscillator (HO). The analytic resolution of the linear HO in the context of the GUP has been first considered in [12], deducing both the modified energy spectrum and the corresponding eigenfunctions. On the other hand, in [52] the algebraic method has been revisited to construct the exact HO coherent states, the weight functions and the related probability distributions. This has been achieved by resorting to the modified Heisenberg algebra of Ref. [53] and exploiting only the formal action of the generalized lowering and raising operators on the HO Fock space states. In passing, we mention that a perturbative construction of the generalized HO ladder operators has been exhibited in [54] for the case of an arbitrary self-adjoint polynomial perturbation of the commutator. Moreover, various forms of deformed oscillator algebra have been investigated in [55,56] in the context of q-Hopf structure.
Starting from the above premises, in this work we study the HO within the framework of a polynomial GUP. First, we solve the stationary Schrödinger equation analytically and evaluate GUP corrections to the HO eigenvalues and eigenfunctions. This paves the way for an ab initio construction of the generalized HO Heisenberg algebra, which leads to the exact derivation of the ladder operators as functions of the GUP position and momentum. We wish to emphasize that the novelty of our approach compared to Refs. [52,53] is that it allows featuring HO eigenstates not only in terms of their number of quanta and energy content, but also in terms of the expectation value of the position and momentum in the presence of a minimal length. This is especially useful in the study of Planck scale effects on high-precision oscillator frequency measurements and, in particular, in the computation of the trajectory of the oscillator in a (generalized) coherent state, as recently discussed in Ref. [57]. Besides its own interest, we show that the above formalism can be exploited to construct a quantum-field-theoretic toy model based on the GUP. To avoid technicalities, we focus on the simplest case of a real free scalar field in 1`1 dimensions. The extension to higher dimensions, as well as to the case of the Dirac field, deserves careful attention and will be investigated elsewhere.
The layout of the paper is as follows: in Sec. 2, we set the mathematical framework of the GUP. In Sec. 3, we solve the GUP harmonic oscillator by using both the analytic and algebraic methods. Inspired by these results, in Sec. 4 we propose a field-theoretic toy model to quantize the Klein-Gordon field with the GUP. Conclusions and outlook are discussed in Sec. 5. Two Appendices contain computational details. Throughout the manuscript we set c " 1, while we keep explicit.

GUP: the mathematical framework
In this Section we introduce some basic mathematical tools that are necessary for our next analysis of the GUP framework. We start by assuming that the commutator between two given physical quantitiesq andp can be expressed in terms of a function of one of them as rq,ps " i f ppq. (1) With a view to identifyingq,p with the physical position and momentum operators of our system and Eq. (1) with the modified commutator modeling gravity corrections in the Planckian regime, we assume that f ppq Ñ 1I when the characteristic momentum is much smaller than the Planck momentum, so as to reproduce the standard Heisenberg relation at low-energy scale. Clearly,q andp are not conjugated variables. Let us then say that there exists a pair of variablesQ andP such that rQ,P s " i .
We want to findq andp in terms ofQ andP . Thus, by consideringq " qpQ,P q andp " ppQ,P q, we have (see Appendix A) rqpQ,P q, ppQ,P qs " 1 2 As a specific example, consider the casep "P . Then, we find This relation can be used to find the explicit expression ofq " qpQ,pq. Indeed, suppose for a while to deal with c-number quantities. We can easily see that where we have required qpQ " 0, pq " 0. Clearly, in coming back to operators, we need to assign an ordering prescription. For example, one could choose one of the followinĝ For reasons that will appear clear below, henceforth we shall work in the representation in which the operatorp is multiplicative. In this case, we findQ which leads to the following relationŝ The representationq 1 can be recognized as the one used in Ref. [12]. However, the other two are equally applicable to the case of the commutator in Eq. (1).

Maximally localized states with a generalized position operator
Let us now identifyq andp with the position and momentum operators of our physical system. Based on Eq. (8), we are going to introduce the most general position operator in momentum space aŝ Notice that the three cases in Eq. (8) correspond to A " 0 (q 1 ), A " 1 (q 2 ) and A " 1 2 (q 3 ). The purpose of introducing a deforming function f ppq as in Eq. (1) is to accommodate a minimal position uncertainty mimicking the granular structure of spacetime. As discussed in [12], in this case it makes no physical sense to work in the position eigenbasis, since the related matrix elements obviously cannot be featured with the usual direct physical interpretation about positions. Nevertheless, information on position is still accessible by introducing the so-called Maximally Localized States (MLS), which by definition minimize the position uncertainty consistently with Eq. (1).
To find the explicit form of the MLS within our framework, let us consider the equation they satisfy. In the momentum representation, this takes the form of a differential equation [12] d dp ψ pAq xqy ppq " where we have denoted by ψ pAq xqy the maximally localized state around the generic position xqy and corresponding to the factor ordering described by the parameter A. It is worth noticing that this relation generalizes the MLS equation of Refs. [12,13] to the A ‰ 0 case.
The solution of Eq. (10) can be written, up to a normalization constant, as ψ pAq xqy ppq " χ pAq ppq exp with p 0 ppq " Strictly speaking, the MLS are obtained by further implementing the condition of absolutely smallest uncertainty into Eq. (11). Clearly, this condition is subjected to the specific form of f ppq. For instance, for f ppq " 1` p 2 (which is the case considered in [12,13]), the MLS are obtained by demanding xpy " 0 and ∆p " 1{ ? , with ą 0 having dimensions of an inverse momentum squared. Now, to make the comparison with Refs. [12,13] easier, it is convenient to recast χ pAq ppq as where It is then straightforward to see that are exactly the MLS found in Refs. [12,13] for A " 0.

Measure
Using the same argument as in Ref. [12], we are now interested in looking for the correct measure factor for the momentum space so as to makeq symmetric. In fact, we notice that neitherq 1 norq 2 in Eq. (8) are symmetric with the standard measure of momentum space, whileq 3 is such. Thus, using a generic factor gppq P R for the new measure and letting the operators act on the dense domain S 8 of functions decaying faster than any power [12], we have where we have performed a partial integration. Then, we demand that i Agppq df ppq dp φ 1 ppq`i d dp rgppqf ppqφ 1 ppqs " gppqi " f ppq d dp`A df ppq dp With the further condition that for small p we need to obtain back standard QM, i.e. gp0q " 1, this implies gppq " rf ppqs 2A´1 .
Notice that for A " 0, we obtain the measure used in Refs. [12,13]. Furthermore, for A " 1 2 we have gppq " 1 @p. This result is consistent withq 3 being a symmetric operator with the standard measure of momentum space.
In terms of the momentum eigenstates |py, the identity operator can thus be expanded as 1 " which indeed recovers the completeness relation used in Ref. [12] for A " 0. For practical purposes, it is worth noticing that the wavefunctions associated with different ordering prescriptions, i.e. with different values of A, are related to each other. In fact, consider the stationary Schrödinger equation for an arbitrary potential U pqq p 2 2m φ pAq ppq`pU pqq´Eqφ pAq ppq " 0.
Since an arbitrary power of the position operatorq can be written as pqq n φ pAq ppq " pi q n rf ppqs Aˆf ppq when U pqq can be written as a power series inq, we can define the function φ p0q ppq " rf ppqs A φ pAq ppq with respect to which Eq. (20) becomes p 2 2m φ p0q ppq`pU pq 1 q´Eqφ p0q ppq " 0.
Given the solution of such equation, one can then reconstruct the wavefunction associated to any specific ordering. Thus, the wavefunctions associated to the various ordering are correlated. Furthermore, this allows us to identify the corresponding Hilbert space. In fact, how we saw above, since for A " 1{2 the measure of momentum space is unchanged, the corresponding Hilbert space is the usual space of L 2 pa, bq functions on a given (eventually infinite) domain pa, bq. However, by the definition of the function φ p0q ppq we also get that Thus, given a particular ordering prescription A, the Hilbert space of the corresponding physical states φ pAq ppq is H " tφ pAq ppq : rf ppqs A´1{2 φ pAq ppq P L 2 u.

Momentum and (quasi-)position representations
We now proceed with the definition of an integral transform which is consistent with the deformed commutator (1) and has the function (11) as kernel. By use of Eq. (19), one can show that Here ψ pAq pp, ξq " ψ pAq xqy ppq and the parameter xqy " ξ is now treated as a variable. Notice that the transformations (24a) and (24b) are the inverse of each other, as it should be. Furthermore, T pAq rφs is related to the corresponding transformation for A " 0 defined in Ref. [13] by T pAq rφs " rf ppqs´AT p0q rφs.
An useful relation for the subsequent discussion involves the product of two transformed function. Specifically, we find where pφ 1 ‹ φ 2 qpΞq stands for the cross-correlation of the two functions, i.e.
With these integral transforms, we can now find the quasi-position representation of the various operators. Specifically, it follows that the quasi-position representation of the momentum operator isp " ppp 0 q, with ppp 0 q being the inverse function of p 0 ppq andp On the other hand, by using the integral transform (24a), the representation of the position operator in the new space readsq that is clearly independent of the particular value of A. Moreover, we can find that the momentum eigenfunction in quasi-position space is φppξq " Using the integral transforms defined above, the scalar product of states in terms of the quasi-position wave functions can be written as xφ 1 |φ 2 y " ż 8 8 dp rf ppqs 2A´1 T pAq rφ 1 s ‹ ppqT pAq rφ 2 sppq " 1 ? 2π where we have used the completeness relation in Eq. (19) along with Eq. (26). Again, by employing Eqs. (13) and (25), it follows that the scalar product does not depend on the specific choice of A, as it was expected.
For later convenience, we specialize the above transforms to the case of the quadratic model f ppq " 1` p 2 considered in Refs. [12]. Furthermore, we set A " 0, the other orderings being recovered via the prescription (25). In this context, the transforms in Eqs. (24) become T rφsppq " 1 ? 2π An interesting feature of this GUP can be highlighted by rewriting the anti-transform T´1 in terms of the the auxiliary momentum p 0 introduced in Eq. (12). First, we notice that the relation between p and p 0 in the present case is Therefore, we have It is worth observing that the anti-transform as described by the last expression resembles the sum of two transforms producing functions with argument ξ ˘? . A better insight can be obtained considering momentum eigenfunctions, that is φpp 0 q " δpp 0´p0 q, withp 0 associated to the momentum eigenvalue related with such function. In this case, the anti-transform produces two plane waves, both with wavenumber k " p 0 { , with a phase difference ∆ϕ " 2 ? p 0 . This result is a direct consequence of the minimal length framework considered in this paper. We can interpret it by saying that the combination of the two functions is in fact the object under study (in the example, the momentum eigenstate) and that the two components cannot be resolved because of the existence of a minimal uncertainty in position. Clearly, the standard single-wave picture is recovered in the limit f ppq Ñ 1I, i.e. Ñ 0.

GUP-deformed harmonic oscillator
In this Section we solve the linear harmonic oscillator (HO) in the presence of a generalized commutator like that in Eq. (1). We revisit both the analytic and algebraic methods, deriving corrections to the eigenfunctions and energy spectrum. In particular, the algebraic approach allows us to understand how the Fock space structure is affected when deforming the fundamental algebra. This finds non-trivial applications in the study of minimal length effects on oscillator frequency measurements, as we shall discuss below.

Analytic method
Let us start from the stationary Schrödinger equation for the linear HO written for A=0 (any other ordering can be recovered from this as discussed above). This reads d dp " f ppq d dp ψppq ppq " 0.
Inspired by previous works in the literature, here we consider a polynomial deformation of the commutator in the form f ppq " 1´2δp`pδ 2` qp 2 .
In order for the r.h.s. of Eq. (1) to be non-vanishing, we require ą 0. Furthermore, for dimensional reasons we assume that δ goes as the inverse of the characteristic energy-scale where corrections to the HUP become relevant 1 .
It is a matter of trivial calculations to show that the GUP (1) with the above deforming function still admits a minimal position uncertainty, the MLS being now identified by the conditions xpy " δ{pδ 2` q and ∆p " ? {pδ 2` q. Clearly, the framework of Ref. [12] with a quadratic GUP model and the ordinary quantum mechanics are recovered for δ Ñ 0 and δ, Ñ 0, respectively.
Since the coefficients of the differential equation (36) are rational functions of p and since it has three regular singular points (two of which lying at finite values of p and the other at p Ñ 8), it is possible to recast the same equation in the form [58] ζp1´ζq whose solution can be written in general as with ρ 1 , σ 1 , a 1 , b 1 , c 1 linear functions of the parameters a, b, c and ζ 1 a linear fractional function of ζ to be determined.
In order to find the solution of Eq. (36) explicitly, it is useful to shift the two finite singularities toward the two reference values 0 and 1. This can be done by observing that f ppq has two roots p˘" δ˘i ?
We can then define a new variable such that In this way, the Schrödinger equation becomes whose finite singularities are indeed located at ξ " 0 and ξ " 1. This equation cannot yet be compared with Eq. (38), since the term multiplying ψpζq depends explicitly on ζ, while in (38) it does not. Let us then consider a test function in the form The Schrödinger equation reads As it stands, the last term still depends on ζ. However, we can now set the parameters ρ and σ so as to trivialize such dependence. Specifically, we can choose It is worth noticing that both ρ and σ are in general complex numbers, the one being the conjugate of the other if chosen with the same sign. However, they become real for vanishing δ, in which case they reduce to the parameters α 1 , α 2 , β 1 , and β 2 of Ref. [12], as it should be. Furthermore, in the limit Ñ 0 they are real only when δ´2 ă 2Em, which means that δ´1 should be (in module) smaller than the characteristic momentum ? 2Em of a quantum harmonic oscillator. Were this true, we should have already observed the implications of a minimal length (see Eq. (37)). Thus, we see that either δ " 0 or ρ and σ must necessarily be complex quantities. In what follows, we start by considering the δ ‰ 0 case. Going back to Eq. (45) and comparing it to Eq. (38), we can also identify recalling that this last expression is independent of ζ for the above choice of ρ and σ. We then find c " 1`2ρ.
With the above settings, the solution ψpζq is given in terms of Gauss hypergeometric series as [59] ψpζq " Aψ 1 pζq`Bψ 2 pζq, where and A and B are two constants. A remark is now in order. We know that the quantum mechanical HO eigenfunctions are given by the product of a Gaussian factor by Hermite polynomials. In order to recover this behavior for vanishing deformations parameters, we look for GUP solutions in the cases where the hypergeometric series reduces to a polynomial. This occurs whenever one of the first two entries of the hypergeometric series is a negative integer. Then, for ψ 1 to be acceptable, it must be verified that either a or b is equal to´n, where n P N will be later identified with the energy level of HO states. In turn, this requires that ρ and σ in Eq. (46) must be chosen with the same sign, as this is the only way for a and b to have vanishing imaginary parts (indeed, we recall that ρ and σ are complex conjugates if they are chosen with the same sign). On the other hand, from Eq. (54) the solution ψ 2 gives back the correct limit if either a`1´c or b`1´c is a negative integer. For the same reasons as above and since c P C, ρ and σ must now have opposite signs. It is thus clear that the consistency with QM forces us to select either A " 0 or B " 0 in Eq. (52). Without loss of generality, we focus on the latter case, though it is possible to show that the other choice leads to the same physical predictions.
Let us then assume ψpζq " Aψ 1 pζq, with the condition that ρ and σ have the same sign. In order to study the asymptotic behavior of ψ, it is worth analyzing the two terms multiplying the hypergeometric function in Eq. (53) separately. As for the first term, since ρ P C, it is characterized by an oscillating part, due to pρq, and a part evolving as a power of 1´2δp`pδ 2` qp 2 with exponent pρq. As for the second term, we can write it as δ´`δ 2` ˘p 2 ?
where we used the relations Therefore, this term consists of an oscillating part modulated in amplitude by a monotonic term which for p Ñ˘8 tends to the finite quantity exp r¯π pσqs. Altogether, the requirement that ψpζq does not diverge at infinity implies that pρq (and, thus, pσqq must be negative, which means that Eq. (46) should be considered with negative signs. 2 From Eq. (47), we then demand that where we have exploited the fact that a and b are both real in the present setting.
As discussed above, let us now impose the condition that either a or b is a negative integer. For instance, by choosing a "´n, we get b " a˘b m 2 ω 2 2 pδ 2` q 2`4 mω pδ 2` q "´n˘b Thus, we will consider the bottom sign so that the condition (57) is satisfied for every value of n. Alternatively, one can impose b "´n and choose the top sign for a. In fact, these two choices are equivalent since the solution is symmetric under exchange of a and b. Now, the conditions (58) allow us to find the energy spectrum. In fact, since a`b " 1`2pρ`σq, using Eq. (46) with the minus sign for both ρ and σ, we find that for the n-th level to be compared with the standard quantum mechanical result One can easily check that the harmonic oscillator spectrum of Ref. [12] is recovered for δ Ñ 0, while the limit δ, Ñ 0 gives back the standard result (60). Let us also remark that Eq. (59) agrees, up to second order in the inverse Planck momentum, with the results obtained in [19]. Now, for large values of n, it is possible to show that the dominant term in the spectrum (59) goes as We see that, as long as ą 0, the energy increases quadratically with n, which is the same behavior found in [12]. Thus, for n large enough, the linear GUP-correction in Eq. (37) does not affect the energy levels of the harmonic oscillator. Another remarkable feature can be derived from the study of the zero-point energy. Indeed, by imposing that the modified vacuum energy E pGU P q n"0 is equal to the standard value E pQM q n"0 " ω{2, we obtain the following condition between the GUP parameters Let us denote the r.h.s of this equation by¯ . The above relation allows us to distinguish among three different regimes (see Fig. 1): Figure 1: Energy spectrum as a function of the level number n. We have set for simplicity " m " ω " 1. The log-scale has been used on the vertical axis.
-for ă¯ , we have E pGU P q n"0 ă E pQM q n"0 . In this case, there exists only one intersection between the function (59) and the ordinary spectrum (60). This is due to the fact that E pGU P q n is a monotonically increasing function of the level number n and, for n large enough, it grows faster than the linear QM spectrum, according to Eq. (61).
-For "¯ , we have two intersection points, one of which at n " 0. This occurs because, for small n, E pQM q n grows faster than E pGU P q n , while this trend is reversed for increasing n, as explained above.
-A similar behavior occurs for Á¯ . However, in this case E pGU P q n"0 ą E pQM q n"0 , which entails that both intersections lie at positive values of n.
-By further increasing , the two curves have no intersection at all. Indeed, in this regime the growth rate of the QM spectrum is never high enough to fill the initial gap between E pGU P q n"0 and E pQM q n"0 . As a result, the modified spectrum lies above the QM prediction for any value of n.
Finally, let us focus on the characterization of the energy spacing between adjacent levels. In QM it is wellknown that the allowed energies are equally spaced and separated by a distance ω. Here the situation becomes far more complicated. The explicit expression of ∆E n " E pGU P q n`1´E pGU P q n is rather awkward to exhibit. However, an interesting property can still be derived in the regime of large level numbers n, where the spacing is found growing linearly with n. This is consistent with the behavior of the spectrum exhibited in Eq. (61) and implies that the higher n, the larger the amount of energy that must be supplied to the harmonic oscillator to push it to the next energy level. Qualitatively speaking, this result could be somehow expected, since the r.h.s. of the deformed commutator (1) is no longer a constant and increases for p large enough. More quantitative details on the ladder structure of the spectrum will be given below by studying the algebraic solution of the harmonic oscillator.

Algebraic method
Above we imposed a (or equivalently b) to be a negative integer number, which in turn corresponds to the energy quantum number. Thus, it is clear that increasing n by 1, which corresponds to climbing up the energy ladder, implies reducing a and b by 1. On the other hand, since under our assumptions c can be written as c "´n`1 2¨1´b increasing n by 1 does not correspond to changing c by an integer quantity, because of the n-dependence of pρq. Given the properties of Gauss hypergeometric functions, this poses an issue: it is not possible to find an operator composed by multiplicative and derivative terms which, applied to any energy eigenfunction of quantum number n, leads us to another eigenfunction of quantum number n˘1. However, in agreement with [12], imposing δ " 0 we have pρq " 0, allowing for a possible ladder operator. It is worth noticing that is the only possibility to implement efficiently the algebraic method. In other terms, the presence of a first degree term in p in the deforming function impedes the possibility of climbing the HO energy ladder. Therefore, from now on we will restrict our description to the case δ " 0. In particular, we have a n "´n, By using the properties of Gauss hypergeometric functions, in particular c ab we can define the operators A˘as 3 where N ψ n pζq " nψ n pζq, such that It is worth noticing that, up to a constant (dimensional) factor, such operators have the correct limit for Ñ 0, that is, where a and a : are the usual ladder operators of standard quantum mechanics. Furthermore, it is straightforward to see that Thus, we obtain rA´, A`s " ω 2 2 m 2 pN´2ρ 0 q, tA´, A`u " 3 Since there is no ambiguity, henceforth we drop the hats from operators to simplify the notation.
where in the last equality we have compared the expression for the anti-commutator tA´, A`u with Eq. (68). Such expressions resemble the canonical structure exhibited in quantum mechanics. However, unlike the standard case, the commutator rA´, A`s, when acting on an energy eigenstate, now produces terms that depend on the quantum number n. Therefore, such term can no longer be interpreted as a zero-point energy. Notice that Eq. (71) can be inverted to give where the position operator q is defined as in Eq. (9) with A " 0. One can then symbolically write q " ?
where we have retained the only solutions that recover the correct behavior in the Ñ 0 limit. Alternatively, using the properties of hyperbolic functions, the same expressions can be written in the following way Using Eqs. (77) and (78), it is in fact easy to see that such expressions correspond to the position and momentum operators q and p.
It is now possible to verify that the eigenfunctions ψ n with the norm |N n | 2 " xψ n |ψ n y " |ψ 0 | 2 n!p´4ρ 0 q n p1´2ρ 0 q n p´2ρ 0 q n , are normalized to unity with respect to the GUP-scalar product defined in Eq. (31). Here α n is the Pochhammer symbol pαq n " αpα`1q¨¨¨pα`n´1q.
Using the above result, it is convenient to redefine the normalized energy eigenfunctions with an abuse of notation as ψ n pζq " p´1q n N n ζ ρn p1´ζq ρn 2 F 1 pa n , b n ; c n ; ζq, where the factor p´1q n allows us to recover the standard expression for ψ n pζq in the ε Ñ 0 limit. Thus, by introducing the Dirac notation and denoting by |ny the normalized energy eigenstate corresponding to the n-th energy level, we have At this stage, it must be noted that A`and A´are not the adjoint of each other. In fact, it is straightforward to see that the following relations hold pA˘q : pN´2ρ 0 q " pN´2ρ 0 qA¯.
(89) Therefore, the matrix elements of pA˘q : and A¯in the basis of normalized states |ny are not equal, but differ by However, pA`q : still acts by lowering the HO state by one, while pA´q : acts by raising the state by one. These considerations suggest that A˘are not yet eligible candidates for the rôle of generalized ladder operators, although providing a good starting point for constructing them.
With the purpose of finding the proper generalized ladder operators and following Ref. [60], it is convenient to introduce new operators C˘such that It is fairly immediate to see C˘behave as full-fledged ladder operators for the GUP-harmonic oscillator. First, from Eq. (89) one can prove that they are each other's Hermitian conjugates, i.e. pC˘q : " C¯. For this reason, from now on we are allowed to drop the˘index and use the simpler notation C´" C and C`" C : . Furthermore, they inherit the raising and lowering character of the operators A˘. In fact, we have where the energy spectrum E n is defined as in Eq. (68). We also notice that C and C : retain the correct ε Ñ 0 limit, since they reduce to the standard annihilators and creators for the harmonic oscillator, up to the dimensional factor ? ω, The advantage of this new set of operators is that they allow to factorize the hamiltonian in the canonical form Additionally, they obey the following relations where the characteristic function g is defined recursively as E n`1 " gpE n q. In our case, it takes the form Finally, it is possible to express the position and momentum operators q and p in terms of the operators C, C : by first inverting Eqs. (91) and (92) and then using Eqs. (77) and (78). We then find A comment is in order here. Apart from their relevance in the theoretical study of the GUP, we emphasize that Eqs. (99) and (100) might also be of interest at phenomenological level. Indeed, in Ref. [57] tests of a minimal length have been proposed based on the computation of GUP effects on the period of the pendulum and the trajectory of the harmonic oscillator in the Gazeau-Klauder coherent state. In particular, the latter calculation has been carried out by resorting to the relation between the position operator and the deformed ladder operators in the approximation of small GUP parameter. Here, we have derived such relation in its exact form. We expect that this result can prove useful in view of future high-precision experiments aimed at searching for quantum gravity signatures in harmonic oscillator-like systems. This application, however, goes beyond the scope of the present analysis and will be investigated elsewhere.

QFT toy model
In this section we propose a quantum-field-theoretic toy model based on the results found above. To facilitate the description and highlight the main points, we propose here the quantization of a free real scalar field in p1`1qdimensional flat spacetime. A similar approach can in principle be followed in higher-dimensional GUP theories and/or for other spin fields. Following the guidelines of the standard QFT treatment, we consider the anti-transform of the annihilation operator C p , where the subscript p labels the generic field mode. We have where we have written the transform in terms of the auxiliary momentum p 0 according to the results of Sec. 2.3. Thus, the mode p is to be understood as a function of p 0 as in Eq. (34). The factor N p0 is the normalization coefficient to be fixed. As it stands, the object in Eq. (101) is not real, nor can it be made as such. We then introduce its Hermitian conjugate and build up the following self-adjoint combination φpξq " r φpξq`r φ : pξq " 1 ? 8π where we have exploited the fact that p is an odd function of p 0 in our GUP model, while N p0 is expected to depend on the modulus of p 0 as in the ordinary formalism. The above relation provides us with the fundamental object on which our next analysis is developed 4 . In passing, we mention that a similar application of the GUP to the QFT framework has been studied in [17]. Even in that case the approach involves applying a (3-dimensional) spatial Fourier transform to the k-space variables and then enforcing the deformed commutator on such variables. Corrections of the deformed commutator are then explored on first quantized-like equations of a field theory [17]. On the other hand, the rôle of the GUP in gauge field theories has been addressed in [61], showing that the implementation of the minimal length condition induces a generalized interaction between the matter field and the gauge field, as well as an additional self interaction of the gauge field. Now, inspired by Eq. (97), we impose the following commutation relations with all other commutators vanishing, where H p " C : p C p corresponds to the Hamiltonian density for the p-mode up to the zero-point energy. Accordingly, the total Hamiltonian describing the system is Here, α p is a c-number with dimensions of energy that tends to ω p for Ñ 0, with ω p " a m 2`p2 { being the standard frequency for the p-mode. Furthermore, here we will use Additionally, we have assumed that field modes with different p are still decoupled from each other. Consistently with the canonical field quantization, we can now provide the physical interpretation of the state φpξq|0y, where |0y denotes as usual the field vacuum such that C p |0y " 0 , @p. (106) From the expansion (102), it is straightforward to check that The above relation corresponds to a linear superposition of single-particle states of well-definite momentum p " ppp 0 q. Furthermore, it is worth noticing that We thus obtain a plane wave of wavenumber p 0 { , multiplied by a cosine factor encoding GUP corrections. Given the similarity with the quantum mechanical case [12,13], we can interpret this as the position-space representation of the single-particle wavefunction of the state |py in the GUP framework. The pure plane wave behavior is naturally recovered for Ñ 0.
It is now convenient to describe the system in the Heisenberg picture. We then introduce the time-dependent annihilation operators C p ptq " e iHt{ C p e´i Ht{ " e´i Ωpt C p , and similarly for C : p ptq " C : p e iΩpt , where we have used the commutators (B38) and introduced the shorthand notation Ω p " rgpH p q´H p sα p { . Notice that the above formula gives back the standard time-dependence of the annihilation operators in the limit of vanishing GUP, since in this case Ω p Ñ ω p I (see Eq. (105)). Thus, a remarkable property featuring the GUP framework is that the proportionality factor ruling the time-evolution is now a non-trivial operator.
In what follows, we will assume that the Hamiltonian density for our free field still depends on the absolute value of the physical momentum p only, i.e. H p " Hp|p|q. This is reasonable in the presence of a quadratic GUP. Combining with Eqs. (102) and (109), the time-dependent field can be written as We notice that the expression for the time-dependent field reduces to the usual expression of standard quantum field theory in the limit Ñ 0. From the above relation, we can also compute the conjugate momentum field π 0 pξ, tq defined as π 0 pξ, tq " i rH, φs " 9 φpξ, tq π 0 pξ, tq "´i ? 8π This allows us to derive the equation of motion 9 π 0 pξ, tq " : φpξ, tq "´1 ? 8π It is convenient considering the expressions above for each mode p. In this case, the field reads, as found above, φpp, tq " N p pe´i Ωpt C p`C : p e`i Ωpt q.
Furthermore, we find π 0 pp, tq "´iN p pΩ p e´i Ωpt C p´C and 9 π 0 pp, tq " : φpp, tq "´N p pΩ 2 p e´i Ωpt C p`C : p Ω 2 p e`i Ωpt q.
This last expression can be written in term of the field φpp, tq as : φpp, tq "´Ω p φpp, tqΩ p`1 2 rΩ p φpp, tq`φpp, tqΩ p s pα p ω p mq´1 2 φpp, tqpα p ω p mq 2 , where we used the results of Appendix B. It is worth noticing that the last two terms on the right hand side vanish in the limit Ñ 0, thus leaving the first term and the usual equation of motion in the low energy limit. Furthermore, since Ω p is related to the Hamiltonian density, the presence of an interaction in the form a minimal coupling intervenes in such object. We also observe that the equation of motion above has been found in momentum space. In fact, due to the modification introduced here related to a minimal length, the equation of motion in configuration space becomes non-trivial, involving an integro-differential equation. In fact, such equation of motion can be obtained by transforming Eq. (116) according to the prescription of Sec. 2.3.

Conclusions and Outlook
Several models of quantum gravity predict the emergence of a minimal length at Planck scale. A widespread way of dealing with this supposed quantum feature of spacetime is to modify the Heisenberg Uncertainty Principle so as to accommodate a minimal position uncertainty of order p . The ensuing uncertainty relation is referred to as Generalized Uncertainty Principle. In this paper, we studied the implications of a GUP with linear and quadratic momentum corrections on the harmonic oscillator. Following the standard treatment in low-energy quantum mechanics, we presented both the analytic and algebraic analyses. Remarkably, the latter method works efficiently only with the quadratic GUP, the linear correction impeding to define ladder operators. We found that the HO eigenfunctions and spectrum are non-trivially modified, the results being consistent with previous achievements in literature. The above formalism was finally exploited to develop a field theoretical toy model based on the GUP. While timely and largely investigated in literature [52,54,62], to the best of our knowledge this is the first time the derivation of the generalized HO algebra is carried out exactly in terms of the position and momentum operators. For comparison, we remark that the analysis of Ref. [52] is based on the generalized Heisenberg algebra given in [53], which exploits the formal action of the ladder operators on the Fock space without referring to their explicit dependence on p and q. On other hand, in [54] it was proposed a perturbative construction of the HO algebra for the case of an arbitrary self-adjoint polynomial perturbation of the commutator. Apart from its intrinsic theoretical interest, our study might be relevant for experimental purposes as well. For instance, in [57] phenomenological tests of the GUP were explored in connection with optomechanical system and oscillator frequency measurements. In particular, concerning the latter framework the strategy is to search for quantum gravity signatures by looking at deviations of the oscillatory trajectory from the expected result. In Ref. [57] this was done by computing the expectation value of the HO position in the Gazeau-Klauder state through the relation between q and a, a : approximated for small GUP parameter. Although quite satisfactory in describing QG effects at present, it is reasonable to expect that quantum tests of gravity might achieve an increasingly high level of precision in the next future, thus demanding theoretical models to be as accurate as possible. From this perspective, our work is to be intended as a further step towards the full theoretical understanding of GUP effects on harmonic oscillator-like systems.
Despite the non-trivial results, some issues are yet to be investigated. For instance, in our analysis we resorted to the most common polynomial form of GUP. Nevertheless, several deformations of the HUP with a minimal length have been proposed in the last years [63][64][65]. The question thus arises as to how our formalism and results are modified in these alternative frameworks. In particular, it would be interesting to investigate the impact of different commutation rules on the field quantization. Along this line, a more demanding task is to explore the consequences of a minimum length in a fully covariant Quantum Field Theory. Notice that a similar problem has been recently addressed in Refs. [23], where a covariant generalization of the GUP with a Lorentz invariant minimum length [66] was used to analyze quantum gravity effects on scattering amplitudes in scalar electrodynamics. Clearly, such a covariant formulation would also pave the way for a direct extension of the QFT with the GUP to some other background besides Minkowski spacetime. Of particular interest is the field quantization on black hole geometry.
Finally, given the new definition of ladder operators with respect to the previous literature, the formalism presented in the current work allows for a new study of coherent states in both quantum mechanics and field theory. For instance, inspired by Ref. [19], one might extend the above quantization to the electromagnetic field and compute Planck scale corrections to sources of noise in interefrometry experiments, with particular focus on gravity wave detectors such as LIGO. The strategy is to evaluate the differential momentum transferred to the end mirrors of the two arms of the interferometer. This can be done by computing the expectation value of the difference of photon number in each arm, considering a coherent state for one input channel of the interferometer and a squeezed state for the other [67]. In the presence of GUP effects, one then expects a shift in the interference fringes, due to a pressure gradient experienced by the two mirrors. Therefore, the analysis of noise data from advanced interferometry experiments may provide very stringent bounds on the existence of a minimal length, if not an indirect observation of it. Work along the above directions is under active investigation and will be presented elsewhere.
Using the above results, we can write rf pq, pq, Q A s "´A rf pq, pq, P B s " Here, differentiating the function f pQ, P q with respect to Q or P amounts to considering f pQ, P q as a c-valued function of the quantities Q and P and differentiating with respect to them. Only after having performed the differentiation, we can consider Q, P and the derivatives of the function f pQ, P q as operators again. To distinguish between the c-valued and q-valued quantities, we will mark the second by a p . In this way, the relations above become rf pQ,P q,Q A s "´A rf pQ,P q,P B s " Furthermore, we have the following relations rσpa, bq, σpA, Bqs " σpa, bqσpA, Bq minpa,Bq ÿ j"0 minpA,bq ÿ j 1 "0 p´i q j j! pi q j 1 j 1 ! pa`bq j`j 1 pA`Bq j`j 1 σpa´j, b´j 1 qσpA´j 1 , B´jq (A12) " minpa,Bq ÿ j"0 minpA,bq ÿ j 1 "0 pi q j j! p´i q j 1 j 1 ! pa`bq j`j 1 pA`Bq j`j 1 σpA´j 1 , B´jqσpa´j, b´j 1 q σpA, Bqσpa, bq, from which rσpa, bq, σpA, Bqs " minpa,Bq ÿ j"0 minpA,bq ÿ j 1 "0 pi q j`j 1 j!j 1 ! pa`bq j`j 1 pA`Bq j`j 1 " p´1q j 1 σpA´j 1 , B´jqσpa´j, b´j 1 q´p´1q j σpa´j, b´j 1 qσpA´j 1 , B´jq ı . (A14) The result above allows us to find the commutator between two functions f " f pQ,P q and g " gpQ,P q, i.e.
rf pQ,P q, gpQ,P qs " 1 2 It is worth noticing that in general rf ,ĝs ‰ i { tf, gu, with t¨,¨u representing the Poisson brackets of classical mechanics.
As a particular example, consider the case in which f and g are functions of only one variable, that is f " f pQq and g " gpP q. In this case, we have rf pQq, gpP qs " 1 2 The result above allows for finding the commutator of C and f pHq, with f pxq an infinitely differentiable function at x " 0. In fact, expanding in Taylor series the function f pHq in the following commutator, we get rC, f pHqs " Clearly, the same result can be applied to the case of the commutator of C and gpHq, for which we find rC, gpHqs " rgpgpHqq´gpHqsC.
Similar relations are valid for C : . Specifically, rC : , f pHqs "´C : rf pgpHqq´f pHqs "´rf pHq´f pg´1pHqqsC : , and C : f pHq " f pg´1pHqqC : . (B33) We now consider the commutation relations between the ladder operators and exponential functions of the Hamiltonian. Based on the results above, we find rC, e iHt{ s " re igpHqt{ ´e iHt{ sC " Cre iHt{ ´e ig´1pHqt{ s, rC : , e iHt{ s "´re iHt{ ´e ig´1pHqt{ sC : "´C : re igpHqt{ ´e iHt{ s. (B35)

B.1: QFT extension
In the case of QFT, we have the following commutators rC p , Hs " ż π{2 ? ´π{2 ? dp 1 0 rC p , H p 1 s " ż π{2 ? ´π{2 ? dp 1 0 Ω p C p δpp 0 , p 1 p q " Ω p C p , with Ω p " rgpH p q´H p s α p , and α p defined as in Sec. 4. It is easy to see that the relations found in QM are valid here as well when a function of the Hamiltonian density is concerned rather than the full Hamiltonian, namely f pH p qC p " C p f pg´1pH p qq, C : p f pH p q " f pg´1pH p qqC : p .
In the explicit case presented here, in which gpH p q´H p " a 1`2 m ω p H p` ω p 2 m, we find g´1pH p q´H p "´a1`2 m ω p H p` ω p 2 m "´rgpH p q´H p s` ω p m "´Ω p α p` ω p m.