We show that no device or gedanken experiment is capable of measuring a
distance less than the Planck length. By "measuring a distance less than the
Planck length" we mean, technically, resolve the eigenvalues of the position
operator to within that accuracy. The only assumptions in our argument are
causality, the uncertainty principle from quantum mechanics and a dynamical
criteria for gravitational collapse from classical general relativity called
the hoop conjecture. The inability of any gedanken experiment to measure a
sub-Planckian distance suggests the existence of a minimal length.Comment: 8 pages, Honorable Mention in the 2005 Gravity Research Foundation
  Essay Competitio

Calmet, Xavier

Graesser, Michael

Hsu, Stephen D. H.

English

arXiv

XAVIER CALMET

MICHAEL L. GRAESSER

STEPHEN D. H. HSU

Crossref

MINIMUM LENGTH FROM FIRST PRINCIPLES

We show that no device or gedanken experiment is capable of measuring a distance less than the Planck length. By ”measuring a distance less than the Planck length” we mean, technically, resolve the eigenvalues of the position operator to within that accuracy. The only assumptions in our argument are causality, the uncertainty principle from quantum mechanics and a dynamical criteria for gravitational collapse from classical general relativity called the hoop conjecture. The inability of any gedanken experiment to measure a sub-Planckian distance suggests the existence of a minimal length

CALMET, XAVIER

GRAESSER, MICHAEL

HSU, STEPHEN

Carolina Digital Repository

arXiv:hep-th/0505144v1  16 May 2005Minimum Length from First PrinciplesXavier Calmeta∗, Michael Graesserb†and Stephen D.H Hsuc‡aDepartment of Physics and Astronomy, UNC Chapel Hill, NC 27599-3255.bCalifornia Institute of Technology, Pasadena, CA 91125cInstitute of Theoretical Science, University of Oregon, Eugene OR 97403.March 12, 2018AbstractWe show that no device or gedanken experiment is capable of measuring a distanceless than the Planck length. By ”measuring a distance less than the Planck length”we mean, technically, resolve the eigenvalues of the position operator to within thataccuracy. The only assumptions in our argument are causality, the uncertainty princi-ple from quantum mechanics and a dynamical criteria for gravitational collapse fromclassical general relativity called the hoop conjecture. The inability of any gedanken∗calmet@physics.unc.edu†graesser@theory.caltech.edu‡hsu@duende.uoregon.edu1experiment to measure a sub-Planckian distance suggests the existence of a minimallength.2In this work we show that quantum mechanics and classical general relativity consideredsimultaneously imply the existence of a minimal length, i.e. no operational procedure ex-ists which can measure a distance less than this fundamental length. The key ingredientsused to reach this conclusion are the uncertainty principle from quantum mechanics, andgravitational collapse from classical general relativity.A dynamical condition for gravitational collapse is given by the hoop conjecture [1]: ifan amount of energy E is confined at any instant to a ball of size R, where R < E, thenthat region will eventually evolve into a black hole1.From the hoop conjecture and the uncertainty principle, we immediately deduce theexistence of a minimum ball of size lP . Consider a particle of energy E which is not alreadya black hole. Its size r must satisfyr ∼> max [ 1/E , E ] , (1)where λC ∼ 1/E is its Compton wavelength and E arises from the hoop conjecture. Min-imization with respect to E results in r of order unity in Planck units or r ∼ lP . If theparticle is a black hole, then its radius grows with mass: r ∼ E ∼ 1/λC . This relationshipsuggests that an experiment designed (in the absence of gravity) to measure a short distancel << lP will (in the presence of gravity) only be sensitive to distances 1/l.Let us give a concrete model of minimum length. Let the position operator x̂ have discreteeigenvalues {xi}, with the separation between eigenvalues either of order lP or smaller. (Forregularly distributed eigenvalues with a constant separation, this would be equivalent to1We use natural units where h̄, c and Newton’s constant (or lP ) are unity. We also neglect numericalfactors of order one.3a spatial lattice.) We do not mean to imply that nature implements minimum length inthis particular fashion - most likely, the physical mechanism is more complicated, and mayinvolve, for example, spacetime foam or strings. However, our concrete formulation lendsitself to detailed analysis. We show below that this formulation cannot be excluded by anygedanken experiment, which is strong evidence for the existence of a minimum length.Quantization of position does not by itself imply quantization of momentum. Conversely,a continuous spectrum of momentum does not imply a continuous spectrum of position. In aformulation of quantum mechanics on a regular spatial lattice, with spacing a and size L, themomentum operator has eigenvalues which are spaced by 1/L. In the infinite volume limitthe momentum operator can have continuous eigenvalues even if the spatial lattice spacingis kept fixed. This means that the displacement operatorx̂(t)− x̂(0) = p̂(0)tM(2)does not necessarily have discrete eigenvalues (the right hand side of (2) assumes free evolu-tion; we use the Heisenberg picture throughout). Since the time evolution operator is unitarythe eigenvalues of x̂(t) are the same as x̂(0). Importantly though, the spectrum of x̂(0) (orx̂(t)) is completely unrelated to the spectrum of the p̂(0), even though they are related by(2). A measurement of arbitrarily small displacement (2) does not exclude our model ofminimum length. To exclude it, one would have to measure a position eigenvalue x and anearby eigenvalue x′, with |x− x′| << lP .Many minimum length arguments are obviated by the simple observation of the minimumball. However, the existence of a minimum ball does not by itself preclude the localizationof a macroscopic object to very high precision. Hence, one might attempt to measure the4spectrum of x̂(0) through a time of flight experiment in which wavepackets of primitive probesare bounced off of well-localised macroscopic objects. Disregarding gravitational effects, thediscrete spectrum of x̂(0) is in principle obtainable this way. But, detecting the discretenessof x̂(0) requires wavelengths comparable to the eigenvalue spacing. For eigenvalue spacingcomparable or smaller than lP , gravitational effects cannot be ignored, because the processproduces minimal balls (black holes) of size lP or larger. This suggests a direct measurementof the position spectrum to accuracy better than lP is not possible. The failure here is dueto the use of probes with very short wavelength.A different class of instrument, the interferometer, is capable of measuring distances muchsmaller than the size of any of its sub-components. Nevertheless, the uncertainty principleand gravitational collapse prevent an arbitrarily accurate measurement of eigenvalue spacing.First, the limit from quantum mechanics. Consider the Heisenberg operators for positionx̂(t) and momentum p̂(t) and recall the standard inequality(∆A)2(∆B)2 ≥ −14(〈[Â, B̂]〉)2 . (3)Suppose that the position of a free test mass is measured at time t = 0 and again at a latertime. The position operator at a later time t isx̂(t) = x̂(0) + p̂(0)tM. (4)We assume a free particle Hamiltonian here for simplicity, but the argument can be gener-alized [4]. The commutator between the position operators at t = 0 and t is[x̂(0), x̂(t)] = itM, (5)5so using (3) we have|∆x(0)||∆x(t)| ≥t2M. (6)We see that at least one of the uncertainties ∆x(0) or ∆x(t) must be larger than of order√t/M . As a measurement of the discreteness of x̂(0) requires two position measurements, itis limited by the greater of ∆x(0) or ∆x(t), that is, by√t/M ,∆x ≡ max [∆x(0),∆x(t)] ≥√t2M, (7)where t is the time over which the measurement occurs and M the mass of the object whoseposition is measured. In order to push ∆x below lP , we take M to be large. Note that thisis not the standard quantum limit [2] which can be overcome using refined techniques [3].In order to avoid gravitational collapse, the size R of our measuring device must also growsuch that R > M . However, by causality R cannot exceed t. Any component of the devicea distance greater than t away cannot affect the measurement, hence we should not considerit part of the device. These considerations can be summarized in the inequalitiest > R > M . (8)Combined with (7), they require ∆x > 1 in Planck units, or∆x > lP . (9)Notice that the considerations leading to (7), (8) and (9) were in no way specific to aninterferometer, and hence are device independent. We repeat: no device subject to quantummechanics, gravity and causality can exclude the quantization of position on distances lessthan the Planck length.6It is important to emphasize that we are deducing a minimum length which is paramet-rically of order lP , but may be larger or smaller by a numerical factor. This point is relevantto the question of whether an experimenter might be able to transmit the result of the mea-surement before the formation of a closed trapped surface, which prevents the escape of anysignal. If we decrease the minimum length by a numerical factor, the inequality (7) requiresM >> R, so we force the experimenter to work from deep inside an apparatus which hasfar exceeded the criteria for gravitational collapse (i.e., it is much denser than a black holeof the same size R as the apparatus). For such an apparatus a horizon will already existbefore the measurement begins. The radius of the horizon, which is of order M , is very largecompared to R, so that no signal can escape.An implication of our result is that there may only be a finite number of degrees offreedom per unit volume in our universe - no true continuum of space or time. Equivalently,there is only a finite amount of information or entropy in any finite region our universe.One of the main problems encountered in the quantization of gravity is a proliferation ofdivergences coming from short distance fluctuations of the metric (or graviton). However,these divergences might only be artifacts of perturbation theory: minimum length, whichis itself a non-perturbative effect, might provide a cutoff which removes the infinities. Thisconjecture could be verified by lattice simulations of quantum gravity (for example, in theEuclidean path integral formulation), by checking to see if they yield finite results even inthe continuum limit.This research supported in part under DOE contracts DE-FG06-85ER40224, DE-FG03-92ER40701 and DE-FG02-97ER-41036.7References[1] K. S. Thorne, Nonspherical gravitational collapse: A short review, in J. R . Klauder,Magic Without Magic, San Francisco 1972, 231–258; D. M. Eardley and S. B. Giddings,Phys. Rev. D 66, 044011 (2002) [arXiv:gr-qc/0201034]; S. D. H. Hsu, Phys. Lett. B555, 92 (2003) [arXiv:hep-ph/0203154].[2] V. B. Braginsky and Yu. I. Vorontsov, Usp. Fiz. Nauk 114, 41 (1974) Sov. Phys. Usp.17, 644 (1975); C. M. Caves, Phys. Rev. Lett. 54, 2465 (1985).[3] M. Ozawa, ”Realization of measurement and the standard quantum limit, Squeezedand Nonclassical Light,” in squeezed and nonclassical light, edited by P. Tombesi andE. R. Pike, Plenum (1989) 263–286.[4] X. Calmet, M. Graesser and S. D. H. Hsu, Phys. Rev. Lett. 93, 211101 (2004)[arXiv:hep-th/0405033].8

We show that no device or gedanken experiment is capable of measuring a distance less than the Planck length. By "measuring a distance less than the Planck length" we mean, technically, resolve the eigenvalues of the position operator to within that accuracy. The only assumptions in our argument are causality, the uncertainty principle from quantum mechanics and a dynamical criteria for gravitational collapse from classical general relativity called the hoop conjecture. The inability of any gedanken experiment to measure a sub-Planckian distance suggests the existence of a minimal length. © World Scientific Publishing Company.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

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