Minimum length uncertainty relations for a dark energy Universe

Minimum length uncertainty relations for a dark energy Universe

Matthew J. Lake matthewj@nu.ac.th School of Physics, Sun Yat-Sen University, Guangzhou 510275, China School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore The Institute for Fundamental Study, “The Tah Poe Academia Institute”, Naresuan University, Phitsanulok 65000, Thailand Thailand Center of Excellence in Physics (ThEP), Ministry of Education, Bangkok 10400, Thailand
October 1, 2019
Abstract

We introduce a dark energy-modified minimum length uncertainty relation (DE-MLUR) or dark energy uncertainty principle (DE-UP) for short. The new relation is structurally similar to the MLUR introduced by Károlyházy (1968), and reproduced by Ng and van Dam (1994) using alternative arguments, but with a number of important differences. These include a dependence on the de Sitter horizon, which may be expressed in terms of the cosmological constant as . Applying the DE-UP to both charged and neutral particles, we obtain estimates of two limiting mass scales, expressed in terms of the fundamental constants . Evaluated numerically, the charged particle limit corresponds to the order of magnitude value of the electron mass (), while the neutral particle limit is consistent with current experimental bounds on the mass of the electron neutrino (). Possible cosmological consequences of the DE-UP are briefly discussed and we note that these lead naturally to a holographic relation between the bulk and the boundary of the Universe, which strongly implies time-variation of the ratio . Low and high energy regimes in which dark energy effects may dominate canonical quantum behaviour at the present epoch are identified and the possibility of testing the model using table-top measurements of partial decoherence is briefly discussed.

Keywords: Minimum Length Uncertainty Relations, Generalized Uncertainty Principle, Dark Energy, Quantum Gravity, Neutrino Mass Problem, Holography, Variation of Fundamental Constants, Quantum Measurement Problem, Károlyházy model, Dirac Large Number Hypothesis

pacs:
01.30.Cc; 01.65.+g; 04.20.Cv; 04.50.Kd; 04.60.Bc; 05.10.Cc; 89.70.+c; 95.36.+x

I Introduction

The concept of superposition is the very essence of quantum theory. As the mathematical embodiment of wave-particle duality, it determines the state space structure of canonical non-relativistic quantum mechanics (QM) and its relativistic extension, quantum field theory (QFT). However, despite the unparalleled success of both QM and QFT in describing the micro-world, such duality does not manifest itself in our every day experience: the macro-world does not admit superpositions of states. This gives rise to the so-called measurement problem, recognised since the early days of quantum theory, whereby a classical ‘observer’ (an experimenter or apparatus not subject to the quantum formalism) is required to reduce the quantum superposition via the act of ‘measurement’.

This glaring ontological disparity, yet otherwise arbitrary distinction between observer and observed, has led many physicists to argue that canonical quantum theory is incomplete. Though proposals for the resolution of the measurement problem are varied (see Wheeler:1984dy ; Giulini:1996nw ; Schlosshauer:2003zy for reviews of contemporary approaches, plus Bell:1987hh for a discussion of foundational issues), many involve modifications of the quantum dynamics that lead to spontaneous reduction of the state vector in some mesoscopic regime, which interpolates between the microscopic (quantum) and macroscopic (classical) worlds Pearle:1988uh ; Adler:1999dv ; Bassi:2003gd ; Bassi:2012mx ; Bassi:2012bg ; Banerjee:2016udy . In modern terminology, this spontaneous reduction is known as decoherence, and is believed to be caused by the interaction of the system with its environment Zurek:1991vd . Thus, prior to the act of measurement, micro-systems are weakly coupled to their environment, whereas meso- or macro-systems are strongly coupled. The former behave quantum mechanically, whereas the latter behave classically.

With the measurement problem in mind, it is natural to consider the weakness of gravity, as compared to the three other known fundamental forces – electromagnetic, weak and strong. Indeed, classical gravitational interactions may typically be ignored in the micro-world and only become relevant on macroscopic, even astrophysical or cosmological, scales Davies:1979mj . Nonetheless, the exact nature of quantum gravitational interactions is unknown and their description remains the holy grail of theoretical physics research Kiefer:2007mf ; Ellis:1999rz . It is therefore natural to suppose that what is missing from canonical quantum theory is not an adequate description of the observer, vis-à-vis the observed, but gravity. Since the gravitational interaction is universal, affecting all forms of matter and energy, it may be hoped that gravity, or space-time itself, may play a fundamental role in the ‘spontaneous’ decoherence of quantum systems.

In fact, the idea that quantum gravitational effects may play an important role in the resolution of the measurement problem encountered in canonical non-gravitational QM has a long and distinguished history Karolyhazy:1966zz ; KFL ; Diosi:1988uy ; Ghirardi:1989bb ; Penrose:1996cv ; Penrose:1998dg ; Power:1998wh ; Diosi:2013toa ; Gao:2013fxt ; Bera:2014gwa ; Tilloy:2015zya ; Singh:2015sua ; Singh:2017msf . Originally published in 1966, Károlyházy’s model Karolyhazy:1966zz ; KFL was one of the first to consider the possibility of gravitationally-induced wave function collapse. The fundamental idea proposed in Karolyhazy:1966zz is that quantum fluctuations of the metric give rise to an intrinsic and irremovable ‘haziness’ in the space-time background, corresponding to a superposition of classical geometries. As a result, an initially pure state vector develops, over time, into a mixed state. Coherence is maintained only over a small region, known as a ‘coherence cell’, whose size depends on the space-time curvature induced by the body and, hence, on its mass. For micro-objects, the effect of curvature is small, giving rise to canonical quantum behaviour but, for macro-objects, the maximum size of a coherence cell lies within the classical radius of the body itself. Thus, the quantum nature of the macro-body remains ‘hidden’, as the wave function associated with its centre of mass (CoM) spontaneously decoheres on extremely small scales: the larger the body, the smaller the size of the cell.

From a theoretical perspective, a major advantage of the Károlyházy model is that it contains no free parameters. It is therefore able to make clear predictions regarding gravitational modifications of the canonical quantum dynamics, utilising only the known constants , and . Specifically, the existence of a minimum length uncertainty relation (MLUR), representing a modification of the canonical Heisenberg uncertainty principle (HUP), necessarily follows from the intrinsic ‘haziness’ of space-time assumed in the K-model. The resulting uncertainty, inherent in the measurement of a space-time interval , is

(1)

where is the Planck length Karolyhazy:1966zz ; KFL . For space-like intervals, this represents the minimum possible uncertainty in the position of a quantum mechanical particle, used to ‘probe’ the distance . When is identified with the Compton wavelength, , may be identified with Károlyházy’s estimate of the width of a coherence cell for a fundamental particle,

(2)

Though motivated by an attempt to resolve the measurement problem, the MLUR (1) represents an important theoretical prediction in its own right. Since its inception, the literature on quantum gravity phenomenology has expanded significantly and many modifications of the HUP, known as generalised uncertainty principles (GUPs), have been proposed Tawfik:2015rva ; Tawfik:2014zca ; Scardigli:2014qka . These share a common feature, giving rise to a minimum resolvable resolvable length in nature, which is usually assumed to be of the order of the Planck length Adler:1999bu ; Adler:2001vs . Hence, the existence of some form of MLUR is now regarded as a generic feature of candidate quantum gravity models Hossenfelder:2012jw ; Garay:1994en .

In the present paper, we will not concern ourselves with the measurement problem per se, though the possible implications of our model for this important open question are briefly discussed in Sec. V. Instead, we will focus on the second major prediction stemming from the introduction of a ‘hazy’ space-time, i.e., that of a fundamental MLUR in nature. In particular, we will focus on a major advance in fundamental physics, which should have radical implications for any model of gravitationally-induced wave function collapse, as well as for quantum gravity phenomenology in general, including MLURs AmelinoCamelia:2008qg ; Plato:2016azz ; Howl:2016ryt , namely, the discovery of dark energy Reiss1998 ; Perlmutter1999 . Though the precise microphysical origin of dark energy remains unknown, and is an active area of research within the cosmology/astrophysics community, the current best-fit to all available cosmological data favours a ‘cosmological concordance’ or CDM model Ostriker:1995rn , in which dark energy takes the form of a positive cosmological constant, . This accounts for approximately 69 of the total energy density of the Universe, whereas cold dark matter (CDM) accounts for around 26 and ordinary (visible) matter for around 5 Betoule:2014frx ; PlanckCollaboration .

For our purposes, it is important to note that, although dynamical dark energy models cannot be excluded on the basis of presently available data, any viable dark energy model must give rise to an effective cosmological constant at late times, comparable to the present epoch. (See Tsujikawa:2010zz ; AmendolaTsujikawa(2010) ; Li:2013hia ; Li:2012dt for reviews of current dark energy research.) Furthermore, though may, ultimately, turn out to have a particle physics origin (i.e., the dark energy field may correspond to a form of ‘matter’ in the usual sense, albeit of an exotic kind), its precise origin is unimportant for the derivation of dark energy-modified MLURs. What is important are its gravitational effects. Specifically, regarding the influence of dark energy on physical bodies, it makes no difference whether we write the -dependent term on the left-hand side or the right-hand side of Einstein’s field equations. On the right, it may be interpreted as a form of matter, on the left, as a geometrical effect.

As a geometrical effect, may be interpreted as a minimum space-time curvature, or minimum gravitational field strength. This clearly has implications for any model of gravitationally-induced wave function collapse, including the K-model, as well as for any MLUR purporting to include quantum gravity effects, irrespective of the measurement problem. Nonetheless, even if the true origin of dark energy is of a particle nature, the exotic form of matter to which it corresponds necessarily sources a minimum positive curvature, , in otherwise ‘empty’ space. As we will see, this has profound implications for Károlyházy’s model, which originally assumed quantum fluctuations of asymptotically flat (i.e. Minkowski) space Karolyhazy:1966zz ; KFL .

By contrast, we embed a K-type model in a realistic cosmological background, incorporating the effects of dark energy and (in principle) additional matter components in the observable Universe. A key consequence of the existence of a positive cosmological constant is the existence of a fundamental horizon for all observers (including quantum mechanical ‘particles’), the de Sitter horizon, . We argue that this necessarily implies a modification of the MLUR (1), including minimum curvature/finite-horizon effects. As with the original model presented in Karolyhazy:1966zz ; KFL , our model has the theoretical advantage of involving no free parameters. The main difference is that the MLUR obtained by considering a ‘hazy’ space-time, à la Károlyházy, in the presence of dark energy, necessarily involves , , and .

The structure of this paper is as follows. In Sec. II.1, we consider classical perturbations of the cosmological Friedmann-Lemâitre-Robertson-Walker (FLRW) metric, induced by the presence of point particles. In Sec. II.2, we show how the formula for the perturbed line element relates to Károlyházy’s scheme for measuring the minimum positional uncertainty of a gravitating, quantum mechanical, ‘point’ particle. Sections II.3-II.4 review the original derivation of the MLUR given in Karolyhazy:1966zz ; KFL and an alternative derivation formulated in Ng:1993jb ; Ng:1994zk , respectively, while Sec. II.5 outlines motivations for dark energy-induced modifications of the standard result. The physical basis of the resulting dark energy uncertainty principle (DE-UP) is laid out in Secs. III.1-III.2 and its basic properties, including applications to both neutral and electrically charged particles (Secs. III.3-III.4), as well as its implications for the holographic conjecture tHooft:1999rgb ; Bousso:2002ju (Sec. III.5), are explored. Possible cosmological consequences of the DE-UP are considered in Sec. IV and Sec. V contains a summary of our main conclusions together with a brief discussion of prospects for future work. Potential conceptual issues regarding the limits of applicability of the model, which arise at various points throughout the text, are discussed at greater length in the Appendix.

Ii Károlyházy’s MLUR – new perspectives

In Karolyhazy:1966zz ; KFL , Károlyházy et al consider ‘resolving’ a space-time interval , traversed by a quantum mechanical particle of mass , by projecting it into the lab frame using light signals emitted by the particle over the course of its path. They claim that classically, the observed interval is related to the original (‘true’) interval via

(3)

where is the Schwarzschild radius associated with the mass . By explicitly taking into account the quantum nature of the particle traversing , they then obtain an estimate of the minimum uncertainty in the measurement of , denoted . The derivation of the MLUR given in Karolyhazy:1966zz ; KFL is considered in detail in Sec. II.3 and Károlyházy’s measurement procedure is illustrated in Fig. 1.

In Sec. II.1, we show that a formally similar result, in which the quantities and in Eq. (3) have different physical meanings, may be obtained using gravitational perturbation theory. In this formulation, the quantities and do not a priori represent ‘true’ (CoM frame) and ‘measured’ (lab frame) values of the length of a space-time interval but, instead, the lengths of an interval in an unperturbed background space and in the perturbed space induced by the presence of the particle, respectively. Nonetheless, the new formulation may be reconciled with Károlyházy’s picture, since we are free to consider receiving light signals in a lab frame far away from the particle’s CoM, in which the gravitational perturbation induced by it is small. The formal equivalence of the two pictures is shown explicitly in Sec II.2.

ii.1 Classical intervals in perturbed and unperturbed backgrounds: and

We now consider the classical perturbation induced by the presence of a point particle in a realistic space-time background, requiring the perturbed metric to satisfy the linearised Einstein equations. In the presence of dark energy, represented by a positive cosmological constant , the gravitational action is

(4)

and the field equations take the form

(5)

where denotes the space-time metric, is the Einstein tensor, is the Ricci tensor, is the scalar curvature and is the matter energy-momentum tensor. For a perfect fluid, may be represented covariantly as

(6)

where denotes the rest-mass density, is the isotropic pressure and is the -velocity of an infitesimal fluid element.

The Friedmann-Lemaître-Roberston-Walker (FLRW) metric, describing a homogenous, isotropic, expanding Universe, may be written as

(7)

where is the cosmic time and is the cosmological scale factor which is normalized to one at the present epoch, . In spherical polar coordinates, takes the form

(8)

where is the line-element for the unit -sphere and is the Gaussian curvature, with dimensions . In appropriate units, for negative, zero, and positive curvature, respectively. Substituting Eqs. (6), (7) and (8) into (5) yields the well-known Friedmann equations

(9)
(10)

where a dot represents differentiation with respect to Islam:1992nt .

For future reference, we note that the Hubble parameter as is defined as

(11)

and that its present day value is , or (ignoring error bars) in cgs units PlanckCollaboration . The critical density is defined as

(12)

giving . This is the value of required to give zero curvature () in the absence of a cosmological constant ().

Dividing Eq. (9) by , it may be rewritten in terms of the density parameters , , and . These denote the present day contributions, as fractions of the critical density, to the total energy density of the Universe for radiation, matter, curvature and dark energy, respectively. To three significant figures, the values obtained from current observations are , , and , where the matter sector is composed of both non-relativistic baryons and non-relativistic (‘cold’) dark matter PlanckCollaboration . Thus,

(13)

where . In other words, the present day density is very close to the critical density () and the Universe is approximately flat on large scales, with the exception of the minimal curvature induced by .

In an arbitrary spatial coordinate system, Eq. (7) may be written in the general form

(14)

where is the spatial part of the metric, and an arbitrary metric perturbation may be written as

(15)

The gauge invariant tensor perturbations (‘gravitons’) satisfy the transverse-traceless conditions, , , where is the covariant derivative for the three-dimensional metric .

Let us now switch back to spherical polar coordinates and consider a spherically symmetric perturbation, induced by the ‘birth’ of a particle of mass , at some time . Our ansatz for the perturbative part of the energy-momentum tensor then takes the form

(16)

where is the Heaviside step function and all other components are zero. Strictly, Eq. (16) models the birth of a particle, at , which remains at rest with respect to a comoving coordinate system at all later times. It also holds approximately for particles that are not subjected to extreme accelerations. In this case, non-static tensor perturbations, which would otherwise lead to gravitational wave emission, may be neglected. In addition, we may set since, at linear order, vector perturbations are associated with vorticity in the cosmic fluid and do not arise in this scenario Uggla:2011jn ; Christopherson:2011ra . The full evolution of the scalar and tensor-type perturbations for the birth of a point-like mass may be determined by following a procedure analogous to that used in Lake:2011aa , though such a detailed treatment is unnecessary for our current purposes.

Instead, we note that the covariant metric (14) contains four extraneous degrees of freedom associated with coordinate invariance. In the Newtonian gauge, which holds approximately for situations in which and where wave-like tensor perturbations can be neglected, this ‘gauge’ freedom may be used to diagonalise the perturbed metric, giving

(17)

where and are Newtonian potentials obeying Poisson’s equation Uggla:2011jn ; Christopherson:2011ra . In our scenario, this is consistent with the fact that, since the source term is time-dependent only instantaneously, the time-dependence of the perturbations must be small on scales , where is the maximum extent of the particle’s light cone. In other words, we assume that the metric perturbation induced by the particle’s creation propagates radially outwards at the speed of light, but remains approximately ‘static’, with respect to comoving coordinates, within its horizon. Any additional time-dependence is confined to a thin spherical shell at .

In the absence of anisotropic stresses, Uggla:2011jn ; Christopherson:2011ra , and Poisson’s equation for a mass distribution immersed in a dark energy background in an expanding Universe is

(18)

where

(19)

is the dark energy density and is the Laplacian, defined with respect to comoving coordinates. For spherically symmetric systems, this reduces to , where

(20)

The current experimental value of , inferred from observations of high-redshift type 1A supernovae (SN1A), Large Scale Structure (LSS) data from the Sloan Digital Sky Survey (SDSS) and Cosmic Microwave Background (CMB) data from the Planck satellite, is Betoule:2014frx ; PlanckCollaboration . This is equivalent to the vacuum energy density .

Now let us consider the case in which is given by a -function density profile corresponding to a classical point-like mass , . In this scenario, Eq. (18) is simply Poisson’s equation with two source terms, a regular point mass () and an ‘irregular’ constant negative density, . (Recall that, when written on the right-hand side of the field equations, may be interpreted as a negative energy density belonging to the matter sector.) This is satisfied by the modified Newtonian potential

(21)

which gives rise to the gravitational field strength Hobson:2006se

(22)

Thus, the cosmological constant corresponds to an effective gravitational repulsion whose strength increases linearly with the comoving distance and we note that, for , where

(23)

and cm, the force between two particles is attractive (repulsive). is the asymptotic de Sitter horizon and is of the same order of magnitude as the present day radius of the Universe cm (13.8 billion light years). In Burikham:2015sro ; Burikham:2015nma , it was also referred to as the first Wesson length, after the pioneering work Wesson:2003qn , and denoted .

In the Newtonian picture, marks the separation distance beyond which the effective gravitational force between two spherically symmetric bodies becomes repulsive (i.e. beyond which the repulsive effect of dark energy overcomes the canonical gravitational attraction). Up to numerical factors of order unity, the same result may be obtained in general relativity by evaluating the Kretschmann invariant, , for a Schwarzschild-de Sitter metric embedded in a cosmological background, and noting the value of at which it changes sign.

Including contributions to from the background baryonic and dark matter densities – that is, embedding the perturbation in a full FLRW background – similar arguments yield

(24)

where is a solution to Eqs. (9)-(10), so that

(25)

This is known as the gravitational turn-around radius, and may also be derived rigorously in a fully general relativistic context Bhattacharya:2016vur . For , the matter density is diluted to such an extent that and the space-time becomes asymptotically de Sitter, yielding Eq. (23). The implications of de Sitter-type cosmological evolution are discussed further in Sec. III.2.

Hence, in general, the infitesimal line-elements of the perturbed and unperturbed metrics, and , are related via

(26)

Next, we rewrite the unperturbed line-element Eq. (14) as . Restricting ourselves to time-like intervals within the present day horizon then gives

(27)

This assumption is valid if represents the embedding of a non-relativistic particle. Substituting Eq. (27) into Eq. (26) then allows us to obtain a lower bound on , such that

(28)

where . For , which implies and , the final term is subdominant for , so that Eq. (28) reduces to Eq. (3) in this regime.

Note that, since explicit -dependence drops out of the expression for the unperturbed line element, the coordinate distance need not be equal to . Formally, Eq. (28) gives the difference between the perturbed and unperturbed line-elements, traversed by a light-like signal (e.g. a photon) over time , as seen by an observer at . The flight time of the photon(s) and the position of the observer relative to the mass are independent. Hence, so are and .

This corresponds to the following experimental procedure. Suppose we place a ‘detector’ at a coordinate distance from a specified origin. (We assume throughout that our detector represents an idealized observer whose gravitational field may be considered negligible, even compared to that of the perturbing particle: though unrealistic, this is a valid assumption in our idealized gedanken experiment.) If the massive particle is absent, a photon travelling for a time traverses a space-like interval . In flat space, this may simply be identified with the coordinate distance, so that .

However, if, instead, we assume the photon is emitted by a massive particle located at , and absorbed by a detector at after the same time , the traversed interval ‘seen’ by an observer at is (28). The simple relationship between the coordinate distance and space-like interval is destroyed by the gravitational field of the particle and, in general, the light signal will not reach the same value of at the same time (i.e., ).

Furthermore, need not correspond to the flight time of single photon. Instead, we may consider spitting the measurement of the interval into two (or more) parts. For simplicity, however, we consider only a two part measurement process. In the first part, a photon travels from the perturbing particle at to the detector at . In the second, an additional photon travels from a (generally different) point, to . If the total flight time of both photons is , the space-like interval that would have been traversed if the particle had not been present is , but the interval traversed in the perturbed space in .

Since can label any point in space, regardless of the value of , which we here identify with flight times of the photons used to perform the measurement, it follows that the measured interval depends on where we place our detector in relation to the perturbing particle. This fact also enables us to reinterpret Eq. (28) in terms of an experimental procedure to resolve time-like intervals traversed by massive, self-gravitating particles, à la Károlyházy. During the photon flight time , the CoM of a classical non-relativistic particle also traverses a time-like interval approximately equal to . Hence, and may be interpreted as the ‘observed’ (lab frame) and ‘true’ (CoM frame) values of the space-time interval traversed by a massive particle, as claimed in KFL . This procedure is discussed in greater detail in Sec. II.2 and is illustrated in Fig. 2.

For , the Hubble expansion term in Eq. (28) dominates, so that marks the limit of the validity of the perturbative Newtonian gauge picture. Though it is, unfortunately, not obvious from the explicit form of the modified potential Eq. (24), we must remember that, for , we require the scalar perturbation to vanish also, . (That this is indeed the case can be seen by taking the limit in the source term for the perturbed Einstein equations, Eq. (16), which leads directly to the desired result.) Physically, corresponds to a region in which the effect of the perturbation is negligible and the standard Hubble expansion takes over.

Thus, the Hubble expansion gives rise to small, additive correction term to Károlyházy’s formula (3), plus a modification of the original canonical gravitational term, corresponding to the substitution . Since the additive term is subdominant within the region of physical interest, , the latter modification is the most important.

Equation (28) holds both at the current epoch s (13.8 billion years) and at all earlier times for which the FLRW metric is valid, including epochs where the average curvature was far higher than today. In addition, it holds for regions of the present day Universe in which space-time curvature is well above the FLRW background level (). This may be seen by taking the static, spherically symmetric, weak-field limit of the full Einstein equations (5), which also reduce to Eq. (24) with and . This limit applies to all experiments carried out on (or near) the surface of the Earth at the present epoch.

Taking both these factors into account, it is reasonable to suppose that Eq. (28) holds (at least approximately), far more generally, remaining valid at any epoch under non-extreme conditions. We may expect it to break down close to the inflationary era MartinPeter2009 ; Guth:2004tw ; Liddle:2000cg , or for space-time intervals close to the event horizon of a black hole. However, we note that, using and substituting the Newtonian potential (21) into Eq. (17), we obtain a time-like metric component directly proportional to that of the Schwarzschild-de Sitter solution, which describes a black hole in the presence of a cosmological constant . It therefore seems probable that Eq. (28) is valid in all physically interesting scenarios.

ii.2 Classical equivalence of Károlyházy’s measurement procedure and the perturbative result: reinterpreting and

Based on the arguments presented in Sec. II.1, we see that, in addition to describing the difference between the ‘true’ and ‘measured’ values of a time-like interval traversed by a self-gravitating particle, Eq. (3) also describes the difference between the perturbed space-time interval, , induced by the presence of the particle, and the unperturbed space-time interval, , that would have existed if the particle had not been present (assuming and ). Physically, this makes sense, since we may consider projecting the particle’s world-line onto a detector in the lab frame, a distance from the CoM, at which the induced gravitational potential is . For relatively small , we project this interval onto a region of locally curved space (i.e., a region in which the curvature is above the background level of the FLRW metric), induced by the particle’s self gravity.

Similar arguments apply even when the background curvature is well above the FLRW average, for example, due to the presence of macroscopic lab equipment, or the lab’s proximity to the surface of the Earth. Practically, we may restrict our attention to projections within a very small region in the vicinity of the CoM, over which the particle’s (extremely small) self-gravity may be considered non-negligible compared to the background level, whatever this may be. Classically, such a region is well defined for any perturbed metric and traces out a ‘world-tube’ of width (c.f. Eq. (25)) surrounding the CoM world-line Karolyhazy:1966zz ; KFL . Projecting the world-line onto a ‘detector’ within this tube gives rise to significant deviations in the measured value of the interval, as compared to its ‘true’ value, due to the space-time curvature induced by the particle.

Clearly, once the ‘fuzziness’ of the CoM due to canonical quantum mechanics is taken into account things become even more complicated, as a second radius – the Compton radius – may be associated with the particle. Nonetheless, in our model, we will find that the counter-intuitive results implied by the considerations above remain the same: once the particle’s self-gravity is taken into account, physical measurements of space-time intervals – for example, the space-like position of a particle, relative to a predefined origin – yield more accurate results if the measurements are made from further away. Below a certain optimum length-scale, attempting to probe the position of the particle’s CoM with greater accuracy becomes self-defeating. The resulting ‘gravitational uncertainty’ caused by the fuzziness of the space-time close to the particle’s CoM outweighs the gain in localising the canonical quantum wave packet. By contrast, far away from the CoM, metric fluctuations reduce to the background level (assumed to be of order ), and canonical quantum behaviour is recovered. The measurement scheme considered above is shown, for particles with both classical gravitational (turn-around) and quantum mechanical (Compton) radii, in Fig. 1.

Figure 1: Measurement of the time-like interval traversed by a massive particle located at , by projecting light-like signals emitted over the course of its path onto a ‘detector’ at . The outer tube surrounding the centre of mass (CoM) represents the region , in which the particle’s gravitational field may be considered non-negligible compared to the background curvature. Placing the detector within leads to significant differences between the measured (lab frame) and ‘true’ (CoM frame) values, even in the classical regime (28). The inner tube represents the fuzziness of the particle’s CoM due to the nonzero width of the canonical quantum wave packet. Generally, the tubes defined by the gravitational and quantum mechanical radii have different thicknesses, but coincide for the minimum-mass particle predicted by the DE-UP. (See Sec. III.3.)

The explicit connection between this procedure and the perturbed space-time induced by the presence of the particle is illustrated in Fig. 2. For simplicity, let us begin by assuming that the gravitational effect of the particle mass can be neglected, so that in our notation. This scenario is represented by the flat blue line. Now let us consider measuring a space-like distance by means of a photon, emitted from the particle at and absorbed by a detector in the lab frame at some distance , where is the proper time measured by particle’s CoM. Note that, in general, this need not be identified with the cosmic time , so that we are free to consider . If the particle’s recoil velocity is non-relativistic, it may be considered negligible at the classical level, so that . Thus, if is small compared to the cosmic time (), we may set .

In this case, it is clear that the time-like interval traversed by the particle in time is identical to the space-like interval measured by the experimental apparatus (i.e the particle-photon-detector system). In Károlyházy’s notation, we have , where and denote the world-lines traversed by the particle and measured in the lab frame, respectively.

Now let us consider the more general case, in which the space-time curvature induced by the presence of the particle cannot be ignored. This scenario is represented by the curved red line in Fig. 2. In this case, if the photon travels from the particle at to the detector at in time , this corresponds to the measurement of a space-like interval . As shown in Sec. II.1, once the gravitational field of the particle is taken into account, the simple relation between the coordinate distance , traversed by the photon, and the space-like interval this corresponds to breaks down (). Likewise, the simple relationship between the coordinate distance and the time elapsed no longer holds ().

The time-like interval traversed by the particle is still , so that , as in Eq. (3). However, also represents the space-like interval that would have been measured, had the particle’s mass not perturbed the flat background. Hence, Károlyházy’s interpretation of the symbols and , as representing the ‘true’ (CoM) frame and measured (lab frame) values of the space-time interval traversed by the particle, is equivalent to ours, in which they represent intervals in the non-perturbed and perturbed backgrounds, respectively.

Figure 2: If the gravitational field of the particle is considered negligible, space-time is approximately flat. In this case, a photon emitted from the particle at travels to the point in time . This completes a measurement of the space-like interval . During this time (ignoring recoil), the particle traverses a time-like interval , so that . Taking the particle’s gravity into account, if the photon travels from to in time , this corresponds to a measurement of the space-like interval , . The time-like interval traversed by the particle is still , so that . This formula relates the perturbed line element to the unperturbed line element or, equivalently, the space-like interval measured at to the ‘true’ time-like interval traversed by the particle. Hence, the relation between and obtained from Károlyházy’s measurement procedure is equivalent to the perturbative result.

As stated in Sec. II.1, we now show explicitly that Eq. (3) holds even more generally. Suppose that, rather than measuring the space-like interval between the particle and the detector – which corresponds to the coordinate distance , even if the two are not equivalent – we instead choose to measure a much larger interval. For example, let us imagine that the particle is surrounded by a horizon, at a (classically) fixed distance from its CoM. Furthermore, let us imagine that, if the gravitational field of the particle were absent, the horizon would be located at a fixed distance rather than .

Our experimental procedure is then as follows. A photon is emitted from the particle at and absorbed by the detector in the lab frame (as before) after a time . This completes a measurement of the space-like interval , where . Simultaneously, or near simulataneously, a photon emitted from a point on the horizon at , where , also arrives at and is absorbed by the detector. This completes a measurement of the space-like interval , where . This result follows directly from the independence of the space-time coordinates and where, in our experimental procedure, is identified with the flight time of a photon and is identified with the position of the detector. Together, these interactions complete the measurement of a space-like interval given by

(29)

The time-like interval traversed by the particle during the flight time of both photons is , so that this procedure is equivalent to projecting the entire world-line of the particle, traced out over , onto the detector at .

Modifying this argument to include the effects of universal expansion, dark energy () and the background matter density on the Newtonian potential induced by the perturbation, gives

(30)

which is simply Eq. (28) with . In this case, we may identify , and the relevant horizon is the particle horizon, , given by

(31)

where is the conformal time Hobson:2006se

(32)

Hence, the measured value of the space-like distance between the particle and the horizon depends on where we place our detector in relation to each. This is a simple consequence of the fact that the perturbation breaks the global symmetry (i.e. homogeneity or, equivalently, isotropy about every point) of the FLRW background. If is very small, the detector sits within a (relatively) deep potential well, in which the difference between the curvature of the perturbed and the unperturbed backgrounds is large.

From Károlyházy’s viewpoint, the time-like interval traversed by the particle, over the time taken for a photon to reach the horizon, is projected onto a detector in the lab frame at . If , where is the coordinate distance corresponding to the position of the horizon, the distortion induced by the gravitational field of the particle renders the measured value significantly different from the true (CoM frame) value.

Implicitly, this argument assumes that the particle formed in the very early Universe (). However, even if this is not the case, still marks the furthest point in causal contact with the particle at the cosmic epoch . As such, it still represents the largest distance that can be measured by means of the particle-photon-detector system, at time . Strictly, for , Károlyházy’s interpretation is not applicable to Eq. (30), since the world-line of the particle is much shorter than . Nonetheless, this formula remains physically meaningful in relation to the gedanken experiment described above, in which the detector at receives signals from both the particle at and its horizon at .

The above argument demonstrates the classical equivalence of Károlyházy’s measurement scheme and the perturbative result, Eq. (28). In canonical QM, the picture of the classical point-particle is replaced by the wave function , representing a superposition of position or, equivalently, momentum states of the particle’s CoM. Thus, it is not difficult to imagine that, in the quantum regime, the classical region over which the particle’s self-gravity cannot be neglected gives rise to an irreducible ‘haziness’ of the underlying space-time metric, induced by the presence of the wave function. This is equivalent to an irreducible ‘smearing’ out of the particle mass or, equivalently, of the CoM associated with .

This observation, which formed the basis of Károlyházy’s predictions Karolyhazy:1966zz ; KFL , will also form the basis of our own analysis, though we will depart from his original prescription in a number of crucially important ways. In particular, we will attempt to incorporate the effects of a space-filling dark energy, which exists in the form of a cosmological constant , with effective energy density and pressure given by Eq. (19).

We note that in this model, as in Károlyházy’s original Karolyhazy:1966zz ; KFL , Dirac -function position states do not exist. Even if the position of a quantum particle is ideally localised, from the perspective of the gravitationally-modified quantum theory, its CoM remains ‘smeared’ over some minimum length-scale, which is a function of the size, mass and possibly charge of the body, and of fundamental physical constants. This point is discussed in detail in Sec. III, in which the dark energy-modified MLUR is derived.

ii.3 Derivation of the MLUR (Károlyházy, 1968)

To highlight both the similarities and the differences between the arguments presented in Karolyhazy:1966zz ; KFL and those presented in the present work, we briefly review the original derivation of Károlyházy’s minimum length uncertainty relation (MLUR). Special emphasis is placed on the physical assumptions that underly the model and on the chain of reasoning that gives rise to the final result. For clarity, where new or supplementary assumptions are introduced for the first time, they are explicitly stated.

Beginning with Eq. (3), Károlyházy effectively defines the uncertainty in in terms of an assumed uncertainty in , via

(33)

where is a positive numerical constant of order unity. In fact, following Eq. (3), is set exactly equal to one in Károlyházy’s original derivation Karolyhazy:1966zz ; KFL . We explicitly include it, from here on, for the sake of comparison with the results of Ng and van Dam Ng:1993jb ; Ng:1994zk , presented in Sec. II.3, and their modification in the presence of dark energy, given in Sec. III.

While this idea is reasonable from a gravitational perspective – where one may expect statistical fluctuations in space-time configurations to be equivalent to fluctuations in the mass that ‘sources’ the gravitational field (or at least correlated with them) – it is problematic from the quantum point of view, since ‘uncertainty’ refers to the statistical spread of measurement outcomes, where the physical quantity in question is represented by a Hermitian operator. However, in both canonical QM and QFT, mass is a parameter, not an operator.

In Karolyhazy:1966zz ; KFL , Károlyházy obtains the expression for from the ‘canonical’ uncertainty relation , though this too is potentially problematic, as time is not an operator in the canonical non-relativistic theory. Defining the uncertainty in the rest-energy of the particle as

(34)

and using to infer , yields

(35)

By substituting (35) into (33), then assuming that the self-gravity associated with the particle’s wave function is non-negligible only over the interval [i.e., replacing in (33)] and noting that the minimal value of is , then yields

(36)

where we define the Planck length and mass , for later convenience, as

(37)

In his original papers Karolyhazy:1966zz ; KFL , Károlyházy’s MLUR was related to the concept of a coherence cell via a special gravitationally-modified dispersion relation, which enabled estimates of the cell width, , and Eq. (36) to be satisfied simultaneously. However, in the present paper, we will not consider the implications of dark energy for models of gravitationally-induced wave function collapse. Their detailed examination is left to future work LakePaterek-DE-K-model .

ii.4 An alternative derivation (Ng and van Dam, 1994)

An alternative derivation of Eq. (36) is based on a gravitational ‘extension’ of the MLUR obtained in canonical QM, and was originally proposed by Ng and van Dam Ng:1993jb ; Ng:1994zk . That an MLUR exists, even in the canonical non-gravitational theory, can be seen by considering the dependence of the positional uncertainty on the time interval over which measurements are made. (Note that we again distinguish between this and the cosmic time .)

The approximate dependence of on the time interval may be determined from the non-relativistic quantum dispersion relations, , which give rise to the group velocity

(38)

The uncertainties in and at any time are related via , or, equivalently

(39)

Using the fact that for then gives

(40)

Next, we define the uncertainty over all measurements, made at both and , as the geometric mean of the canonical uncertainties at both times, i.e.

(41)

This yields

(42)

where is the Compton wavelength and where we have defined the distance , assuming that the wave function is spherically symmetric and spreads radially outwards. Interestingly, Eq. (42) may also be derived using the ‘canonical’ energy-time uncertainty relation, , by identifying .

More rigorously, it may be obtained as a direct solution to the Schrödinger equation in the Heisenberg picture Calmet:2004mp ; Calmet:2005mh . In the absence of an external potential (), the time evolution of the position operator is given by

(43)

which may be solved directly, yielding

(44)

The spectra of any two Hermitian operators, and , obey the general uncertainty relation Rae00 ; Ish95

(45)

so that setting , gives

(46)

and

(47)

Using the definition of , Eq. (41), together with , we recover Eq. (42).

Historically, this result was first obtained by Salecker and Wigner using a gedanken experiment in which a quantum ‘particle’ is used to measure a distance by means of the emission and reabsorption of a photon Salecker:1957be . In this description , given by Eq. (42), represents the minimum possible canonical quantum uncertainty in the measurement of .

The argument presented in Salecker:1957be proceeds as follows. Suppose we attempt to measure using a ‘clock’ consisting of a classical mirror and a quantum mechanical device (e.g. a charged particle such as an electron), initially located at , that both emits and absorbs photons. A photon is emitted at and reflected by the mirror, which is placed at some unknown distance . The photon is then reabsorbed by the particle after a time (not ).

Assuming that the velocity of the particle remains well below the speed of light, it may be modelled non-relativistically. By the standard Heisenberg uncertainty principle (HUP), the uncertainty in its velocity at any time obeys the inequality

(48)

where is the positional uncertainty obtained by evolving the initial wave function via the Schrödinger equation (i.e. neglecting recoil). However, if the initial positional uncertainty is then, in the time required for the photon to travel to the mirror and back, , the particle acquires an additional positional uncertainty

(49)

The total canonical positional uncertainty is now defined as

(50)

and obeys the inequality

(51)

Minimizing this expression with respect to , or equivalently , and using the fact that , gives

(52)

where we have again used .

We note that similar arguments apply if we consider a modified experimental set up, in which a photon is emitted by the particle at and absorbed by a device in the lab frame at , or vice versa. (In other words, we note that reflection by the mirror is not an essential part of the experimental procedure and, in addition, that it does not affect the order of magnitude estimates of the minimum quantum uncertainty inherent in the measurement.)

We also note that requiring (i.e., that photons cannot be emitted from within the Schwarzschild radius of our ‘probe’ particle), we obtain . Alternatively, requiring , the measurement process devised by Salecker and Wigner gives rise to a MLUR which is consistent with the standard Compton bound of the non-relativistic theory.

For fundamental particles, it is therefore interesting to ask, what happens if a photon is emitted from the particle and reabsorbed within the interval ? Strictly, the answer is that, for , the non-relativistic theory breaks down and we must switch to a field theoretic picture. In this, the ‘measurement’ of corresponds to a self-interaction, described by a one-loop process in the relevant Feynman diagram expansion, in which the photon remains virtual. However, it is important to remember that interactions corresponding to ‘measurements’ of in the non-relativistic theory are physical. It is therefore reasonable to apply the non-relativistic formulae, such as Eq. (II.4) and its gravitational ‘extensions’, in this regime, on the understanding that ‘measuring’ distances via photon emission/reabsorption corresponds to virtual photon exchange via a one-loop process.

A related point concerns the existence of superluminal velocities for , as implied by Eq. (II.4). However, though virtual particles can travel faster than the speed of light, this does not imply a violation of causality, as information is not transmitted outside the light cone of a given space-time point Peskin:1995ev . In fact, a similar effect occurs with respect to the standard Heisenberg term: for , the HUP implies , or equivalently . Hence, superluminal velocities and sub-Compton probe distances in the non-relativistic theory are associated with the regime in which field theoretic effects become important. Nonetheless, we may continue to apply the non-relativistic formulae in this region, subject to the caveats stated above. These issues are discussed in detail in the Appendix.

It is straightforward to extend the arguments presented in Calmet:2004mp ; Calmet:2005mh and Salecker:1957be to include an estimate of the uncertainty in the position of the particle due to gravitational effects, . By assuming that this is proportional to the Schwarzschild radius , Ng and van Dam defined the the total uncertainty due to canonical quantum effects, plus gravity, as

(53)

where , which is also assumed to be of order unity Ng:1993jb ; Ng:1994zk . (For , we recover exactly.) Minimizing Eq. (53) with respect to yields

(54)

and, substituting this back into Eq. (53), we obtain

(55)

Neglecting numerical factors of order unity, and relabelling in Eq. (36), in accordance with standard QM notation for distance measurements, we see that Eq. (55) is equivalent to Károlyházy’s result with .

Equivalently, may be written as a function of , using Eq. (54). By performing the minimization procedure with respect to , we have effectively asked the question “what mass must the probe particle have, in order to measure the distance with minimum quantum uncertainty?”. Physically, this is equivalent to asking, “if our particle has mass , what distance can be measured with minimum uncertainty?”. However, although Eq. (54) fixes the relation between and for an uncertainty-minimizing measurement, we note that there is no minimum of the function , given by Eq. (53), in the -direction of the plot. Intuitively, we may expect to be able to minimize with respect to either or , and to obtain the same result in either case, since this gives rise to a procedure which is self-consistent in the limit (i.e. when the ‘probe’ distance tends to the Compton wavelength of the particle, either from above or below). This point is discussed further in Sec. II.5.

Finally, before concluding the present subsection, we note that similar results hold, even for electrically neutral particles, whose interactions are mediated by massive, short-range bosons. For electrically charged particles, real photons may be emitted or absorbed, or virtual photon exchange may take place via a one-loop self-interaction. For uncharged particles, photons (either real or virtual) are replaced by the appropriate force-mediating boson(s). For example, in the case of the weak nuclear force, the and bosons are massive, and hence short-range, giving rise to short-range probe distances . To realize the measurement scheme outlined in Sec. II.2, in which a neutral particle communicates with – and effectively ‘measures’ the distance to – its own horizon, we must instead imagine a higher order self-interaction processes involving exchange, taking place on some scale , coupled with the exchange of virtual photons between the intermediate bosons and a charged particle located at .

ii.5 Motivations for the DE-UP

As shown in Sec. III.3, in Karolyhazy:1966zz ; KFL Eq. (36) was obtained by considering a gedanken experiment to measure the length of a space-time interval with minimum quantum uncertainty. This derivation relies on the fact that the mass of the measuring device (probe particle) distorts the background space-time. Equating the uncertainty in the particle’s rest energy with uncertainty in its mass then implies an irremovable uncertainty or ‘fuzziness’ in the space-time in the vicinity of the particle itself. This results in an absolute minimum uncertainty in the precision with which a gravitating system can be used to measure the length of any given world-line, . By contrast, the arguments presented in Ng:1993jb ; Ng:1994zk circumvent the need to assume quantum fluctuations in the rest mass, and hence the need to define a rest-energy Hamiltonian, .

Nonetheless, Károlyházy’s arguments Karolyhazy:1966zz ; KFL are similar to those of Ng and van Dam Ng:1993jb ; Ng:1994zk , in that arises as a direct result of the assumption that the Schwarzschild radius of a body, , represents the minimum ‘gravitational uncertainty’ in its position. In fact, for MLURs of the form (36)/(55), it is usually assumed that in most of the existing quantum gravity literature Hossenfelder:2012jw ; Garay:1994en . For all the scenarios leading to Eq. (55) considered above, this is directly equivalent to assuming a minimum gravitational uncertainty of order .

An important physical consequence is that, since Eq. (55) holds if and only if Eq. (54) also holds, it is straightforward to verify

(56)

Substituting the minimization condition for , Eq. (54), into Eq. (56) then gives

(57)

For , we require the ‘’ inequality in Eq. (57), since many arguments imply that represents the minimum resolvable length-scale due to quantum gravitational effects. (See, for example Padmanabhan:1985jq ; Padmanabhan:1986ny ; Padmanabhan:1987au , plus Hossenfelder:2012jw ; Garay:1994en for reviews of minimum length scenarios in phenomenological quantum gravity.) This implies that the ‘’ inequality also holds in Eq. (56) and, hence, that the minimum quantum gravitational uncertainty predicted by Károlyházy/Ng and van Dam is always greater than the Compton wavelength of the particle that minimizes it.

However, from a physical perspective, the assumption may be questioned on at least two grounds. First, we see that, for fundamental particles with masses , . Although the total uncertainty may remain super-Planckian, the assumption of simple additivity, , on which Eq. (55) is ultimately based, implies that canonical quantum uncertainty and the gravitational uncertainty arise independently, without influencing one other (i.e., that the gravitational uncertainty remains fixed, regardless of how dispersed the quantum wave packet becomes). It is therefore not clear whether a gravitational uncertainty given by is physically meaningful. Second, gravity is a long range force. Intuitively, we may expect that, however it is defined, the gravitational uncertainty induced by the presence of a point-like or near point-like particle at should fall with the gravtational field strength. Naïvely, we may assume that the gravitational uncertainty varies in proportion to the classical Newtonian potential, as .

If this is indeed the case, we see that, rather than being a simple constant, must take the form of a ratio, , where and is a phenomenologically significant length-scale which is well motivated by fundamental physical considerations. In the context of a dark energy Universe, it is clear that the de Sitter horizon, , fulfils this criterion. As we shall see, one consequence of this is that states for which and become possible, in contrast to the predictions obtained from Eqs. (53)-(55). We also note that replacing in Eq. (53) allows us to minimize with respect to either or . It is straightforward to demonstrate that this minimum is unique and is independent of both and . As a result, the minimization procedure remains consistent in the limit .

In Sec. III, we derive MLURs in which the minimum uncertainty in a physical quantity is given by the cube root of of three (possibly distinct) scales, , , , but which differ from relations derived from Eqs. (36)/(55) in two important ways. First, the new relations attempt to incorporate the effects of dark energy, in the form of a cosmological constant, on the ‘smearing’ of space-time and, thus, on the minimum quantum gravitational uncertainty inherent in a measurement of position and related physical observables. Second, they lead to substantially different but physically reasonable predictions in a number of scenarios. Specifically, they may be combined with other results obtained in general relativity and canonical quantum theory to give estimates of both the electron () and electron neutrino () masses, in terms of fundamental constants. These estimates yield the correct order of magnitude values obtained from experiment.

In deriving the new relations we follow a procedure analogous to that used by Ng and van Dam Ng:1993jb ; Ng:1994zk (outlined in Sec. III.2) but assume the existence of an asymptotically de Sitter/FLRW, rather than Minkowski, space-time. The results are obtained in two different ways. In the first, it is unnecessary to assume fluctuations in basic parameters, such as the mass . This avoids the need to promote parameters to observables, represented by Hermitian operators in the non-relativistic quantum-gravitational regime. (From a technical point of view, it removes the need to define the operator or, equivalently, the rest Hamiltonian .) In this case, it is, however, necessary to make certain assumptions about the properties of space-time superpositions in the Newtonian limit. In particular, we assume the existence of an upper bound on , given by the difference between line-elements in two classical space-times: one in which the particle is present and one in which it is absent. This is equivalent to assuming that the ‘spread’ of quantum states cannot exceed the difference between the two classical extremes.

In the second, we promote the classical Newtonian potential to an operator, , à la Károlyházy, and estimate the associated uncertainty, , by considering a superposition of CoM position states. We then relate to by considering the associated uncertainty induced in the measurement of space-time line-elements. From here on, we refer to all minimum quantity uncertainty relations of the form as ‘cubic’, due to the value of the exponent on the right-hand side.

Iii Dark energy-induced modifications of the MLUR – the DE-UP

iii.1 Space-time uncertainty and classical perturbations – a connection?

Like Károlyházy, for , we take Eq. (3) as our starting point for the quantum mechanical definition of a ‘hazy’ space-time. In this case, the Hubble flow correction term in Eq. (28) is subdominant within the turn-around radius, . However, rather than following the steps expressed in Eqs. (34)-(35), leading to Eq. (36), we instead make the following physical assumption.

We assume that the quantum mechanical uncertainty in the space-like interval between a particle of mass (located at ) and the coordinate distance , is of the order of the difference between the classical values and , where .

Classically, the presence of the particle induces a perturbation in the background space-time, whose magnitude at is given by

(58)

so that our assumption is equivalent to setting

(59)

where and represent the two (classical) extremes.

In the classical picture, the underlying space-time may be in one of two distinct states. In the first, in which the particle is absent, the underlying metric corresponds to the unperturbed line element . In the second, in which the particle is present, the metric corresponds, instead, to the perturbed line element . It is reasonable to suppose that, whatever the final theory of quantum gravity may be, a wave function of the form

(60)

describing a superposition of space-time background states, is possible in at least some limiting cases. Here, we use the notation to distinguish between wave functions representing space-time superpositions and , which represents a canonical quantum wave function that exists on a definite classical space-time background. Though the mathematical formalism of a theory that contains both is not developed in the present work, we have in mind a composite wave function, that reduces to when corresponds to a particular geometry.

More realistically, we may assume that the space-time background on which the canonical quantum wave function propagates is, in fact, in a superposition of an infinite number of states, each corresponding to a unique classical line element , i.e.

(61)

An expansion of this form will yield Eq. (59) if either the limits of integration are such that , the unperturbed line element, and , the perturbed line element, or, more generally, if and take arbitrary values but the wave packet maintains a standard deviation of order . This holds true even for , , as