A novel mechanism for probing the Planck scale
AA novel mechanism for probing the Planck scale
Saurya Das ∗ Theoretical Physics Group and Quantum Alberta,Department of Physics and Astronomy, University of Lethbridge,4401 University Drive, Lethbridge, Alberta T1K 3M4, Canada
Sujoy K. Modak † Facultad de Ciencias - CUICBAS, Universidad de Colima, Colima, C.P. 28045, M´exico
The Planck or the quantum gravity scale, being 16 orders of magnitude greater than the elec-troweak scale, is often considered inaccessible by current experimental techniques. However, it wasshown recently by one of the current authors that quantum gravity effects via the GeneralizedUncertainty Principle affects the time required for free wavepackets to double their size, and thisdifference in time is at or near current experimental accuracies [1, 2]. In this work, we make animportant improvement over the earlier study, by taking into account the leading order relativisticcorrection, which naturally appears in the sytems under consideration , due to the significant meanvelocity of the travelling wavepackets. Our analysis shows that although the relativistic correctionadds nontrivial modifications to the results of [1, 2], the earlier claims remain intact and are in factstrengthened. We explore the potential for these results being tested in the laboratory.
The Planck length, energy and time scales, first men-tioned by Planck himself in [3, 4], continues to play spe-cial roles in physics. While this are believed to be thescales where Quantum Gravity (QG) effects will mostcertainly appear, given the immense gap between theelectroweak scale ( (cid:39) (cid:39) TeV), it is conceivable that some of these effects mayshow up in this intermediate region, even if indirectly. Itis also believed that the Planck scale signifies an abso-lute minimum measurable length scale in Nature, beyondwhich the notion of a continuum spacetime seizes to exist.Arguments in favour of a Minimum Length scale (MLS)can also be found in early works of Heisenberg [5], Yang[6], Deser [7] and Mead [8, 9]. They have been refinedfurther in many recent works (see e.g. [10]).Although the Planck scale, MLS and the QG scale areoften assumed to be of the same order of magnitude, perse , there is no reason or evidence behind this assumption.We will therefore relax this, and assume that new physi-cal effects, including QG effects may potentially show upin the vast arena of 15 orders of magnitude interveningbetween the electroweak and the Planck scales. There-fore, in the absence of a direct probe beyond the LHCscale energy ( (cid:39)
10 TeV), it is imperative that one looksfor potential experimental signatures and new physicsthat may be present in the aforementioned energy range.In this letter, we examine this idea and expand onthe related work first proposed by one of us in [1, 2],in which a concrete proposal was made to examine thehypothesized fundamental minimal scale in Nature in anindirect manner. The way it works is as follows: weknow that wavepackets in quantum mechanics broaden intime as they evolve via a free Hamiltonian, and the rateof this broadening can be estimated accurately. In par-ticular, it is straightforward to compute the time takenfor wavepackets to double their size as they evolve via a free Hamiltonian. Width of wavepackets are often mea-sured in Atomic-Molecular-Optical (AMO) experimentsfor various purposes ( e.g. [11], [12]). In this work, we re-examine this effect, but in light of a Hamiltonian whichis still free, but modified from the canonical Hamiltoniandue to the Generalized Uncertainty Principle (GUP),which encapsulates a MLS and is implied by it. Such ageneric modification of the Heisenberg Uncertainty Prin-ciple (HUP) has been argued from many theories of QG,including String Theory, Loop Quantum Gravity, DoublySpecial Relativity, black hole physics etc, and its impli-cations were examined [13–23].Following earlier work by one of the current authors,in this paper, we examine promising experimental pathswhich might be able to detect GUP modifications with ahigh accuracy. In particular, the “doubling time differ-ence (DTD)” (difference in times taken for a wavepacketto double in size, with and without GUP) was com-puted in [1, 2]. It was also shown there however, thatthe DTD only becomes experimentally measurable, oncethe velocity of the travelling wave-packets are quite large( ≈ − m/s). This is because the GUP effectsare momentum (and hence velocity) dependent and getsenhanced with increasing velocity of the wavepackets.While this is encouraging, one encounters the followingissue: for these velocities, the relativistic corrections areof the order of ( v/c ) (cid:39) − − − , and it has to be de-termined whether these corrections will be comparable orexceed the GUP corrections for the energy and momentarange under consideration. It is precisely this importantpoint that we will examine in this paper and show thatthe GUP effects are still potentially measurable! In anattempt to systematically study both the relativistic andGUP effects, we in fact find that the two get mixed ina non-trivial way. However, it is still possible to appro-priately ‘filter out’ the relativistic effects and extract the a r X i v : . [ g r- q c ] F e b GUP corrections, which are again just within the realmof current and future experimental acuracies.We start by considering the Hamiltonian for a free par-ticle of mass m in (1 + 1)-dimensions, including the lead-ing order relativistic correction term H = p m − p m c (1)Now, as per GUP, the fundamental commutator betweenposition and momentum is modified to [21][ x, p ] = i (cid:126) [1 − αp + 4 α p ] . (2)The above defines a minimum measurable length and amaximum measurable momentum, in terms of the GUPparameter α [1](∆ x ) min = 3 α (cid:96) P l ; (∆ p ) max = M P l c α (3)where we have defined α = α /M P l c , α being dimen-sionless. M P l is the Planck mass, M P l c the Planckmomentum, M P l c ≈ TeV the Planck energy and (cid:96)
P l ≈ − m is the Planck length. We do not assumeany specific value of α , rather we hope that experimentswill shed light on the allowed values of α . Since no ev-idence of a MLS has not been found in experiments atthe LHC, one is forced to put an upper bound on α .Together with a lower bound on it corresponding to thePlanck scale, one arrives at the following allowed range:1 ≤ α ≤ .Next, for calculational convenience, we define an aux-iliary momentum variable p , which is ‘canonical’ in thesense that [ x, p ] = i (cid:126) , and therefore as an operator,one can write p = − i (cid:126) d/dx . This is related to thephysical (i.e. measurable) momentum p via the relation p = p (1 − αp + 2 α p ). Substituting in Eq.(1), one ob-tains the following effective Hamiltonian for a relativisticsystem, incorporating GUP H = H NR + H rel + H LGUP ++ H QGUP + H relLGUP (4)where, (i) H NR = p m , (ii) H rel = − p m c , (iii) H LGUP = − αm p , (iv) H QGUP = α m p , and (v) H relLGUP = α m c p .In the above, (i) is the standard non-relativistic Hamil-tonian, (ii) the leading order relativistic correction, (iii)the linear GUP correction (proportional to α ), (iv) thequadratic GUP correction (proportional to α ) and (v)the hybrid or mixed term, which includes both the rela-tivistic and linear GUP correction.Next, we move on to the study of evolution of freewavepackets under the above Hamiltonian. It is textbookknowledge that a free wave-packet tends to broaden itselfdue to the Heisenberg’s uncertainty principle. Use of the Ehrenfest theorem is one of the direct ways of estimatingthis broadening. Here our interest is to consider the mod-ified broadening rate of the free wave-packet with the fullHamiltonian (4). As is well-known, the Ehrenfest’s theo-rem gives the time derivative of the expectation values ofthe position ( x ) and its canonically conjugate momentum( p ) operators as follows: ddt (cid:104) x (cid:105) = i (cid:126) (cid:104) [ x, H ] (cid:105) = (cid:68) ∂H∂p (cid:69) and ddt (cid:104) p (cid:105) = i (cid:126) (cid:104) [ p , H ] (cid:105) = − (cid:68) ∂H∂x (cid:69) . These can be ex-tended to the expectation of any operator of course, andin particular to p n , which appear in (4) for various in-teger values of n . For the above, one obtains ddt (cid:104) p n (cid:105) = i (cid:126) (cid:104) [ p n , H ] (cid:105) = 0, implying that (cid:104) p n (cid:105) = constant in time.Next, to estimate the DTD, we first write the first andsecond time-derivatives of the square of the width (orvariance) of the quantum mechanical wave-packet, whichis defined as ξ = ∆ x = (cid:104) x (cid:105) − (cid:104) x (cid:105) :˙ ξ = dξdt = ddt (cid:104) x (cid:105) − (cid:104) x (cid:105) d (cid:104) x (cid:105) dt (5)¨ ξ = d ξdt = d dt (cid:104) x (cid:105) − (cid:18) d (cid:104) x (cid:105) dt (cid:19) − x d (cid:104) x (cid:105) dt . (6)The above can be simplified using the Ehrenfest theoremand the Hamiltonian given in (4).To calculate the contributions for all the terms in (4),we consider each term in addition to the free nonrelativis-tic term ( p / m ) separately, and write H = p m + Dp n with n > D a constant, and compute the cor-responding correction, using the Ehrenfest theorem and[ x, p ] = i (cid:126) . Finally, we plug-in the appropriate value of n and D for each correction term in (4) and add themtogether to find the total correction. A straightforwardcalculation of the (5) and (6) then yields,˙ ξ = 1 m (cid:0) (cid:104) xp + p x (cid:105) − (cid:104) p (cid:105)(cid:104) x (cid:105) (cid:1) + nD (cid:0) (cid:104) xp n − + p n − x (cid:105) − (cid:104) p n − (cid:105)(cid:104) x (cid:105) (cid:1) (7)¨ ξ = 2 m ∆ p + 4 nDm ( (cid:104) p n (cid:105) − (cid:104) p (cid:105)(cid:104) p n − (cid:105) )+2 n D ∆ p ( n − . (8)In the above, ∆ p = (cid:104) p (cid:105) − (cid:104) p (cid:105) is the variance ofthe canonical momentum and ∆ p ( n − = (cid:104) p n − (cid:105) −(cid:104) p n − (cid:105) , that of the ( n − n and D for all higherorder corrections to the NR Hamiltonian and put themin the above expression of ¨ ξ to obtain¨ ξ full = 2 m ∆ p + C rel + C LGUP + C QGUP + C relLGUP , (9)where C rel = − m c ( (cid:104) p (cid:105) − (cid:104) p (cid:105)(cid:104) p (cid:105) ) + ∆ p (3)0 2 m c (10) C LGUP = − αm ( (cid:104) p (cid:105) − (cid:104) p (cid:105)(cid:104) p (cid:105) ) + 18 α m ∆ p (2)0 2 (11) C QGUP = 40 α m ( (cid:104) p (cid:105) − (cid:104) p (cid:105)(cid:104) p (cid:105) ) + 200 α m ∆ p (3)0 2 (12) C relLGUP = 2 αm c ( (cid:104) p (cid:105) − (cid:104) p (cid:105)(cid:104) p (cid:105) ) + 25 α m c ∆ p (4)0 2 . (13) The master equation (9) has the following solution giv-ing the rate of broadening of the free wavepacket underthe combined influence of the relativistic and GUP cor-rections∆ x ( t ) = (cid:114) ξ in + ˙ ξ in t + (∆ p ) in m t + 12 (cid:0) C rel + C LGUP + C QGUP + C relLGUP (cid:1) t , (14)where, the subscript “in” corresponds to the initial valueof the various quantities, such as the initial width ( √ ξ in ),the initial rate of expansion ˙ ξ in and the initial varianceof the canonical momentum (∆ p ) in , and new correctionsdue to the relativistic and GUP effects appearing in (9).We now compute the expansion rates by considering anormalized Gaussian wave-packet of the form ψ ( x ) = 1(2 πξ ) / exp (cid:18) i (cid:126) p x − ( x − x ) ξ (cid:19) , which represents a minimum wave-packet with (cid:104) x (cid:105) = x , (cid:104) p (cid:105) = p , ∆ x = (cid:104) x (cid:105) − (cid:104) x (cid:105) = ξ , and ∆ p = (cid:126) √ ξ . ItsFourier transformation in momentum space is φ ( p ) = 1 √ π (cid:126) (cid:90) + ∞−∞ ψ ( x ) e − ip x/ (cid:126) = (cid:18) ξπ (cid:126) (cid:19) / exp (cid:18) − ix (cid:126) ( p − p ) − ( p − p ) ξ (cid:126) (cid:19) Since our results contain moments of p upto the eighthorder, and using the standard quantum mechanical defi-nition (cid:104) p n (cid:105) = (cid:82) + ∞−∞ φ ∗ ( p ) p n φ ( p ) we calculated following coefficients for the gaussian wavepacket, C rel = 3 (cid:126) c m ξ (cid:0) − c m ξ in (cid:0) p ξ in + (cid:126) (cid:1) +48 p ξ in (cid:126) + 48 p ξ + 5 (cid:126) (cid:1) (15) C LGUP = 3 α (cid:126) m ξ (cid:0) α (cid:126) + 24 αp ξ in − p ξ in (cid:1) (16) C QGUP = 15 α (cid:126) m ξ (cid:0) α (cid:126) + 16 p (cid:0) α ξ in (cid:126) + ξ (cid:1) +240 α p ξ + 4 ξ in (cid:126) (cid:1) (17) C relLGUP = α (cid:126) c m ξ (cid:2) c m p ξ (cid:0) p ξ in + 3 (cid:126) (cid:1) +25 α (cid:0) p ξ (cid:126) + 48 p ξ in (cid:126) +32 p ξ + 3 (cid:126) (cid:1)(cid:3) . (18)As can be seen from (10-13), the above modificationscontain moments up to the eighth order in momentumspace. It is indeed a nontrivial result, as it shows thatnot only the standard deviation, but also higher ordermoments such as the skewness (3 rd ), kurtosis (4 th ), hy-perkurtosis (5 th ), hypertailedness (6 th ) etc. all dictatethe broadening rate, albeit with decreasing importance.Equipped with the above, we ask our primary questionof interest - can the above GUP modifications be ob-served in an experiment, similar to earlier analyses whereit was shown that a large parameter space of GUP canbe probed by measuring the DTD for large molecularwavepackets such as C , C [1, 2]? The correspondingresults incorporating relativistic effects are given by (15)- (18). It can be easily checked the results of [1, 2] arerecovered in the c → ∞ limit. As in the above references,we address this question numerically.The DTD is defined as∆ t double = t GUPdouble − t HUPdouble , (19) Molecules Mass (kg) Width (m) Velocity (ms − ) C . × − × − C . × − . × − TPPF152 8 . × − × − TABLE I: Physical parameters of the wavepackets under con-sideration. While mass and width of the wavepackets areknown from experiments, mean velocity is assumed by uswhich provides measurable effects. where the first and second terms on the right hand sidesignify the times required for a free wavepacket to doubleits width following (14), and by the same equation in the α → C rel is alwayspresent in (14).To calculate DTD using (14), we first notice that, sincewe are working with a Gaussian wavepacket, the term˙ ξ in = 0. Next, one can replace initial value of ∆ p interms of the initial position uncertainty ∆ x in (14), usingthe following minimum uncertainty relation (again since[ x, p ] = i (cid:126) ) (∆ x ) in (∆ p ) in = (cid:126) , (20)and solve for the doubling time with GUP in which thewidth becomes 2∆ x . To find the doubling time with-out GUP, we simply set the GUP parameter to zero inthe above result. This enables us to calculate the dou-bling time difference (19) which becomes a function of theinitial width (∆ x ), mass ( m ), mean velocity ( v ) of thewavepacket, as well as, the Planck constant ( (cid:126) ), speed oflight ( c ), and the value of the GUP parameter ( α ).We calculate the DTD numerically for three differ-ent molecular wavepackets; these are - (i) Buckyball C , (ii) Buckyball C and (iii) Tetraphenylporphyrinor TPPF152 molecule ( C H F O N S ). Relevantphysical parameters for these molecular wavepackets aregiven in the accompanying table. It is important to notethat the aforementioned systems behave quantum me-chanically and are stable against decoherence, at leastfor their assumed widths, as shown for example by meansof double-slit experiments [24]-[26]. Therefore the GUPapplies to them and would affect the broadening rates ofthese wavepackets. Results of our numerical analysis are depicted in theleft panel of Figure 1, which is a log-log plot relatingthe GUP parameter α with the doubling time difference∆ t double . The plot covers the entire region between the Although it has been claimed that the GUP needs to be appliedcautiously for a composite system (such as one with many con-stituent atoms) [27], we adopt the point of view that GUP wouldapply to the quantum system as a whole [22, 28, 29]. In the end,it is for experiments to decide on its correctness. electroweak scale and the Planck scale. The shaded blueregion corresponds to the region of parameter space thatcan be probed by using these wavepackets and with anatomic clock with attosecond (10 − s) accuracy, whichhas already been achieved more than a decade ago (see,for instance [30]). With the C molecule, we are ex-pected to indirectly probe 8 orders of magnitude up fromthe electroweak energy scale (and an equal order of mag-nitude down in the corresponding length scale). With the C we get an improvement of a further order of mag-nitude, while with TPPF152 one may be able to probedown to 11 orders of magnitude away from electroweakscale. In other words, one should be able to probe theregion of the parameter space 10 ≤ α ≤ alreadywith these wavepacket expansion experiments with anaccuracy of attosecond timescale. Further improvementsare expected, with the advancement of atomic clock tech-niques and availability of larger wavepackets, with whichone may be be able to probe reasonably close to thePlanck scale itself. In fact, very recently the measure-ment of zeptosecond time delay (10 − s) was reported in[31]. As we can see from Fig. 1, zeptosecond accuracy to-gether with TPPF152 can probe all the way to the Planckscale, provided we can measure the DTD of this expand-ing wavepacket, moving with a velocity of 10 m/s. Thisamount to scanning the full parameter space of (linear)GUP 1 ≤ α ≤ . We know no other mechanism whichcan provide such an extraordinary and possibly completescanning of the GUP/QG parameter space. Note that,because of the relationship between the GUP parameter α with the minimal length, via eq. (3), this is equivalentof searching for the minimal length up to 10 l Planck withattosecond accuracy and up to the Planck length withzeptosecond accuracy.Finally, we provide a comparison of the results with orwithout relativistic correction. To do this, we include aright panel in figure 1, which is without the relativisticcorrections, first carried out by one of us in an earlierwork [1]. By comparison, we see that the relativistic cor-rections do change results for the doubling time differencefor all of the three wavepackets and for the entire limit1 ≤ α ≤ . Understandably, the difference is morepronounced for larger values of α . Also, as expected,the less massive wavepackets, such as C and C arestrongly affected as compared with the heavy TPPF152wavepacket.To conclude, the present study attempts to bridge theapparently formidable gap between QG theory and itspotential verification by experiments. In particular, wehave proposed to study wavepacket expansion experi-ments with the hope of either seeing some of the predictedeffects, or in their absence, imposing stringent bounds onQG parameters. In particular, we have considered thebroadening of molecular wave packets for a set of well-studied large molecular systems moving at relatively highspeeds, such that neither relativistic nor QG effects in ���� �� TPPF152 C C ���� �� TPPF152 C C FIG. 1: Doubling time difference versus GUP parameter plot in logarithmic scale. The left panel of the figure corresponds tothe results with the relativistic correction, as carried out in the present paper, and the right panel is without considering therelativistic correction, as carried out in the earlier work [1]. The blue shaded regions correspond to the GUP parameter spacethat could be scanned by measuring DTD with attosecond (10 − s) accuracy. The full GUP parameter space, up to the Planckscale could be scanned with zeptosecond accuracy (10 − s). For details see text. their evolution are insignificant. We computed the timetaken for the corresponding wavepackets to double in sizeand the showed that QG/GUP effects entail a measur-able difference in the doubling times, which may just bemeasurable with current precision of time-measurements,or those that are projected in the near future, as clearlydemonstrated in the accompanying figures! Taking therequired relativistic effects into account, we showed thatthe some of the earlier conclusions don’t just remain,they in fact get further solidified. Again, the unprece-dented accuracy of time measurements should aid in thismeasurement, which of course would get progressivelyeven better in the future. Note that the detection of a α (cid:29) α to date. Forexample, with an attosecond accuracy, we would provideup to 5 orders of magnitude tighter bound than previousbest bound by measuring Lamb shift [21]. Furthermore,with the latest implementation of time measurement inthe zeptosecond order, we would be able scan the wholeGUP parameter space, and thus verifying or rulling outthe linear GUP modification altogether! We hope to con-tinue our study of similar effects in other quantum sys-tems that can be prepared in the laboratory and reportelsewhere. Acknowledgement:
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