Inspired by the work of Wheeler among others, we have studied the problem of quantum measurements of space-time distances by applying the general principles of quantum mechanics as well as those of general relativity. Contrary to folklore, the mimimum error in the measurement of a length is shown to be proportional to the one-third power of the length itself. This uncertainty in space-time measurements implies an uncertainty of the space-time metric and yields quantum decoherence for particles heavier than the Planck mass. There is also a corresponding minimum error in energy-momentum measurements

Dam, H

Ng, Y

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IASSNS-HEP-94/41Y. Jack Ng1;2;3 and H. van Dam31. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 085402. Center for Theoretical Physics, M.I.T., Cambridge, MA 021393. Institute of Field Physics, Department of Physics and Astronomy,University of North Carolina, Chapel Hill, NC 27599-3255Anything I have written on the topic is primarily testimony as to how important Iconsider it. | J.A. Wheeler, private correspondence.ABSTRACTInspired by the work of Wheeler among others, we have studied theproblem of quantum measurements of space-time distances by applying thegeneral principles of quantum mechanics as well as those of general relativ-ity. Contrary to the folklore, the minimum error in the measurement of alength is shown to be proportional to the one-third power of the length it-self. This uncertainty in space-time measurements implies an uncertaintyof the space-time metric and yields quantum decoherence for particles heav-ier than the Planck mass. There is also a corresponding minimum error inenergy-momentum measurements.1We start by recalling the fundamental nature of space-time distance measurements.1 4In quantum mechanics one species a space-time point simply by its coordinates, withoutbothering to give a prescription as to how these coordinates are to be measured. However,general relativity ordains that coordinates do not have any meaning independent of observa-tions; in other words, according to relativity, a coordinate system is dened only by explicitlycarrying out space-time distance measurements. We will pay heed to this dictum of generalrelativity, and will follow Salecker and Wigner1 to use clocks and light signals to measuredistances.5[f1]Suppose we want to measure the length between two spatially separated points Aand B. The measurement can be carried out in the following way. A clock is put at pointA. Set the clock to read zero when a light signal is sent from A towards B where a mirror isstationed to reect the light signal back to A. From the reading of the clock, to be denotedby t, when the light signal arrives at A, one deduces that the length AB is given by ` = ct=2where c denotes the speed of light. There are two major sources of errors in the lengthmeasurement: one comes from the uncertainty principle of quantum mechanics, and theother is due to spacetime curvature eects.First we note that the clock is not absolutely stationary, its spread in speed beinggiven by the uncertainty principle of quantum mechanics,v =pm>12hm`QM; (1)where m is the mass of the clock. Since the clock is the agent in measuring the length, that[f1] Our work has some overlap with that of Ref. 6.2it is not stationary implies an uncertainty in the length measurement given by, at time t,`QM (t) = tv >12hm`QM (0)t =1mh`QM (0)`c; (2)where in the last two steps we have used Eq.(1) and t = 2`=c respectively. Next we minimize`QM (0) + `QM (t) so that the uncertainty in the length measurement due to quantummechanical eects is given by(`QM )2 >h`mc: (3)But the uncertainty in the position of the clock cannot be made arbitrarily small bymaking the clock very massive as that would disturb the spacetime curvature.7 If one assumesthe clock to be spherically symmetric, and to have a radius (to be denoted a) larger thanthe Schwarzschild radius r = ( 2Gmc2) where G is the gravitational constant, `c, the error inthe length measurement caused by the curvature, may be calculated from the Schwarzschildsolution. The result is r multiplied by a logarithm involving raand ra+`. For a  r onends `c =12r `n a+`a, henceGmc2< `c : (4)The combined error in the length measurement, ` = `QM + `c, , due to quantummechanical and curvature eects, can be minimized by adjusting m. Using Eqs. (3) and (4)we nd` > (` `2P )1=3 ; (5)where `P = hGc31=2is the Planck length.[f2] We expect the presence of the mirror at pointB to contribute an error of comparable magnitude. We can also deduce the minimum error[f2] Carrying out the measurement at non-zero temperature results in an additional error.3in time interval measurements by using the same experimental set-up:t > (t t2P )1=3 ; (6)where tP = `P =c is the Planck time. These errors in spacetime measurements induce anuncertainty in the spacetime metric. Noting that `2 = `2g and using Eqs. (5) and (6) wereadily getgv >`p`2=3tpt2=3: (7)Our results (Eqs. (5)-(7)) should be contrasted with those according to the canonical8viewpoint. The derivation of the latter goes as follows. The vacuum functional for the theoryof pure gravity has roughly the formZ ZD[g] exp(ic3hGZ @g@x2d4x): (8)Thus if one is making measurements in a spacetime region of volume `4, contributions tothe Feynman integral are more or less in phase until variations in the gravitational eldamplitudes from their classical values become as large asg > (hG=c3)1=2=` = lP =` : (9)These represent the quantum uctuations of the gravitational elds. They give rise to errorsin spacetime measurements which are constants:` > `P ; t > tP : (10)Let us see how reasonable the canonical results (Eqs.(9) and (10)) are in the lightof the analysis given above for our actual experimental set-up involving a clock, a mirror,and light signals. There, to obtain the canonical results instead of Eqs. (5) - (7), all one4has to do is to replace Eq. (4) by the requirement Gmc2< `. But this requirement is triviallytrue, for otherwise the mirror would be required to be inside the Schwarzschild radius of theclock, a nonsensical situation. One would think one could impose a more restrictive (butstill physically sensible) requirement to arrive at a more useful and better bound for theminimum errors in spacetime measurements (such as those given by Eq. (5) - (7)).Two more comments on the results Eq. (5) - (7):(1) Mathematically it is not surprising that the uncertainty in the length ` involves thetwo lengths in the problem: ` and `P . There is an analogous result which is actuallyrelevant for long thin rulers. A quantum mechanical calculation for a 1-dimensionalchain of N ions with a spring of constant k between successive ions gives in the hightemperature limit, for the uncertainty in the length, [f3]` =1qNx2i =1sNbx2ib=1s`x2ib;where b is the distance between two successive ions when the spring is relaxed, andwhere x2i =12kBTk, i.e., the mean square displacement of a mass on a spring offorce constant k. Thus, for a long thin ruler, the uncertainty in the length dependson both the length ` itself and the lattice constant b. Note that ` is proportionalto `1=2. [In the zero temperature limit, one nds ` to be proportional toqlog `b.]This is one of the reasons that long thin rulers are not the best tools to measuredistances. In addition, being macroscopic objects, rulers will inuence other objectsin the measurement process through their gravitational attraction. The Lorentzcontraction of rulers will also complicate matters.[f3] This result is originally due to E.P. Wigner (in response to a question raised by oneof us (HvD),) private communication. Details of the calculation will appear elsewhere.5(2) Because the Planck length is so small, even for the universe [f4] for which ` =1010light-years, ` is only 10 13 cm long, the size of a nucleus, which is quite small,but in principle is there.As a simple application of Eqs. (5)-(7), we deduce the minimum error in momentum-energy measurements. Imagine sending a particle of momentum p to probe a certain structureof spatial extent ` so thatp  h`: (11)Consider the coupling of the metric to the energy-momentum tensor of the particle,(g + g)t = g(t + t) ; (12)where we have noted that the uncertainty in g can be translated into an uncertainty int . Eqs. (7) and (12) can now be employed to givep > p`P`2=3; (13)which, with the aid of Eq. (11), yieldsp > ppmP c2=3; (14)where mP = (hc=G)1=2 is the Planck mass.[f4] Since this conference is held in honor of Prof. J.A. Wheeler, it may not be out ofplace here at the mentioning of the word \universe" to quote him (private correspondence):`[I recall] the well-known statement of Rutherford, \When a student of mine uses the word\universe", I tell him it is time for him to leave." But maybe that's why so many of us livein America!'6[An alternative derivation of Eq. (14) is provided by considering p, the "uncer-tainty" of the momentum operator p = hi@@x, associated with x = (x`2P )1=3 (Eq. (5)). Forany function f(x), (p)f is given by(p)f =hix@2f@x2+@f@x@x@x:Taking the function f(x) to be the linear momentum eigenstate f = exp(ipx=h), one gets(p)e ipxh  = hi p2h2`2=3P x1=3 +13ph`2=3P x 2=3eipxh :The minimum value of jpj is attained at x  23hpyielding Eq. (14).]The analogous statement for the minimum error in energy measurements isE > EEmP c22=3: (15)For energy-momentum near the Planck scale, the error is not negligible; for example, at theGrand unication scale  1016 GeV, the error is of order 1%. In analyzing a high-energyexperiment an experimentalist should keep in mind that energy-momentum conservationholds only up to the errors given by Eqs. (14) and (15). In passing, we mention that theminimum errors in measurements given by Eq.(6), (7), (14) and (15) are xed by dimensionalanalyses once the minimum error in spatial distance measurements is found given by Eq.(5).As another application of the above results Eqs. (5)-(7), let us consider the quantum(de)coherence phenomenon for a scalar particle of mass m moving slowly. Let us assumethat the particle satises the Schrodinger-Klein-Gordon type equation,ih@@t =  h2m@2@x2+ V (x; t) + g00mc2 ; (16)7where we have kept only the most important term involving g . We are interested here inthe eects caused by the uncertainty in the metric given by Eq. (7). Obviously g inducesa multiplicative phase factor in the wave-function ! ei' ; (17)where' =1hZ tmc2g00dt0 : (18)For consistency the integral should be fairly insensitive to the lower integration limit as longas t tP . If one is making measurements in a short time interval, contributions to the phaseof the wave-function from the dierent time elements in this time interval will be more orless in phase (i.e. ' 1) until the time interval reaches the decoherence time tD when 'becomes sizable, i.e.,1  1hZ tD0mc2tPt02=3dt0  mc2ht2=3P t1=3D = `2=3P `1=3D =c ; (19)where c =hmcis the Compton wave-length of the particle of mass m, and `D = ctD is whatwe will call the decoherence length. The system can be treated classically if the decoherencelength is less than the Compton wave-length, in other words, if`D  c ;or, via Eq. (19),m > mP : (20)Therefore, due to the uncertainty of the space-time metric, it suces to give a particleheavier than the Planck mass a classical treatment.8We thank Prof. J.A. Wheeler for an enjoyable correspondence, and G. Domokos forcalling our attention to Ref. 6. The rst author thanks the faculty (especially S. Adler)at the School of Natural Sciences, Institute for Advanced Study and the faculty (especiallyA. Guth) at the Center for Theoretical Physics, M.I.T. for kind hospitality. This work wassupported in part by the U.S. Department of Energy under Grant No.DE-FG05-85ER40219and by the Z. Smith Reynolds Fund of the University of North Carolina.References1. Salecker, H. & E.P. Wigner. 1958. Phys. Rev. 109: 571-577.2. DeWitt, B.S. 1962. In Gravitation: An introduction to Current Research, L. Witten, Ed.:266-381. Wiley, New York.3. Marzke, R.F. & J.A. Wheeler. 1964. In Gravitation and Relativity. H.Y. Chiu & W.F.Homann, Eds.: 40-64. Benjamin, New York.4. Wheeler, J.A. 1984. In Quantum Theory of Gravity: Essays in Honor of the 60th birthdayof Bryce S. DeWitt, S.M. Christensen, Ed.: 223-233. Adam Hilger Ltd, Bristol,England.5. Ng, Y.J. & H. van Dam. 1994. Mod. Phys. Lett. A9: 335-340.6. Karolyhazy, F. 1966. Il Nuovo Cimento A42: 390-402.7. Wigner, E.P. 1957. Rev. Mod. Phys. 29: 255-268.8. Misner, C.W., K.S. Thorne & J.A. Wheeler. 1973. Gravitation: 1190-1194. W.H.Freeman & Co., San Francisco, CA.9

Limitation to Quantum Measurements of Space‐Time Distances

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Limitation to Quantum Measurements of Space‐Time Distances

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