On the Precision of a Length Measurement

0.16 MB·English

by  Xavier Calmet

EPJ manuscript No.
(will be inserted by the editor)
On the Precision of a Length Measurement
Xavier Calmet1
Universit´eLibre de Bruxelles, Service dePhysiqueTh´eorique, CP 225, Boulevard du Triomphe, B-1050 Bruxelles, Belgique
7
0 Received: date/ Revised version: date
0
2 Abstract. Weshowthatquantummechanicsandgeneralrelativityimplytheexistenceofaminimallength.

n To bemore precise, we show that no operational device subject to quantummechanics, general relativity
a and causality could exclude the discreteness of spacetime on lengths shorter than the Planck length. We
J then consider the fundamental limit coming from quantummechanics, general relativity and causality on

9 the precision of themeasurement of a length.

1 PACS. PACS-key 04.20.-q – PACS-key 03.65.-w
v
3
7 1 Introduction in particular the fact that such a measurement involves
0
actually two measurements.
1
0 Twentieth century Physics has been a quest for unifica- We then apply our framework to the old thought ex-
7 tion. The unification of quantum mechanics and special periment of Salecker and Wigner [2] and show that con-
0 relativity required the introduction of quantum field the- tractive states cannot beat the uncertainty due to quan-
/ ory. The unification of magnetism and electricity led to tum mechanics for the measurementof a length. We then
h
electrodynamics, which was unified with the weak inter- conclude.
t
- actions into the electroweak interactions. There are good
p
reasons to believe that the electroweak interactions and
e
h thestronginteractionsoriginatefromthesameunderlying 2 Minimal Length from Quantum Mechanics
: gauge theory: the grand unified theory. If general relativ-
v and General Relativity
ity is to be unified with a gauge theory, one first needs to
i
X understand how to unify general relativity and quantum
r mechanics, just like it was first necessary to understand In this section we review the results obtained in [3]. We
a how to unify quantum mechanics and special relativity show that quantum mechanics and classical general rela-
before three of the forces of nature could be unified. The tivity considered simultaneously imply the existence of a
aim of this paper is much more modest, we want to un- minimallength,i.e.nooperationalprocedureexistswhich
derstand some of the features of a quantum mechanical canmeasureadistancelessthanthisfundamentallength.
description of general relativity using some simple tools The key ingredients used to reach this conclusion are the
fromquantummechanicsandgeneralrelativity.Inpartic- uncertaintyprinciple fromquantummechanics,andgrav-
ularweshallshowthatifquantummechanicsandgeneral itational collapse from classical general relativity.
relativity are valid theories of nature up to the Planck Adynamicalconditionforgravitationalcollapseisgiven
scale, they imply the existence of a minimal length in na- by the hoop conjecture [4]: if an amount of energy E is
ture. confined at any instant to a ball of size R, where R<E,
We shall address two questions: Is there a minimal then that region will eventually evolve into a black hole1.
length in nature and is there a fundamental limit on the The hoop conjecture is, as its name says, a conjecture,
precision of a distance measurement? The first question however it is on a firm footing. The least favorable case,
will be addressed in the second section while the second i.e. as asymmetric as possible, is the one of two particles
will be considered in the third section. colliding head to head. It has been shown that even in
The usual approachto address the question of a mini- that case, when the hoop conjecture is fulfilled, a black
mallengthistodoascatteringthoughtexperiment[1],i.e. hole is formed [5].
one studies the high energy regime of the scattering and From the hoop conjecture and the uncertainty princi-
finds that one cannot measure a length shorter than the ple, we immediately deduce the existence of a minimum
Plancklength.Hereweshallarguethatthisisnotenough ball of size l . Consider a particle of energy E which is
P
to exclude a discreteness ofspacetime with a lattice spac-
ing shorter than the Planck length. The key new idea is 1 Weusenaturalunitswhere¯h,candNewton’sconstant(or
a precise definition of a measurement of a distance and lP) are unity.Wealso neglect numerical factors of order one.
2 XavierCalmet: On thePrecision of a Length Measurement
is kept fixed. This means that the displacement operator
a
t
xˆ(t)−xˆ(0)=pˆ(0) (2)
M
does not necessarily have discrete eigenvalues (the right
handsideof(2)assumesfreeevolution;weusetheHeisen-
berg picture throughout). Since the time evolution oper-
ator is unitary the eigenvalues of xˆ(t) are the same as
xˆ(0). Importantly though, the spectrum of xˆ(0) (or xˆ(t))
is completely unrelated to the spectrum of the pˆ(0), even
though they are related by (2). A measurement of arbi-
trarilysmalldisplacement(2)doesnotexclude ourmodel
ofminimumlength.Toexcludeit,onewouldhavetomea-
sure a position eigenvalue x and a nearby eigenvalue x′,
with |x−x′|<<l .
P
Many minimum length argumentsareobviatedby the
simple observation of the minimum ball. However, the
existence of a minimum ball does not by itself preclude
the localization of a macroscopic object to very high pre-
cision. Hence, one might attempt to measure the spec-
trumofxˆ(0) througha time offlightexperimentin which
wavepackets of primitive probes are bounced off of well-
Fig. 1. We choose a spacetime lattice of spacing a of the or- localised macroscopic objects. Disregarding gravitational
derofthePlanck lengthorsmaller. Thisformulation doesnot effects, the discrete spectrum of xˆ(0) is in principle ob-
depend on the details of quantumgravity.
tainable this way. But, detecting the discreteness of xˆ(0)
requires wavelengths comparable to the eigenvalue spac-
ing. For eigenvalue spacing comparable or smaller than
not already a black hole. Its size r must satisfy
l , gravitational effects cannot be ignored, because the
P
process produces minimal balls (black holes) of size l or
r >max[1/E, E] , (1) P
∼ larger.Thissuggestsadirectmeasurementoftheposition
spectrum to accuracy better than l is not possible. The
P
where λ ∼1/E is its Compton wavelength and E arises
C failure here is due to the use of probes with very short
from the hoop conjecture. Minimization with respect to
wavelength.
E results in r of order unity in Planck units or r ∼ l .
P A different class of instrument, the interferometer, is
If the particle is a black hole, then its radius grows with
capableofmeasuringdistancesmuchsmallerthanthesize
mass: r ∼ E ∼ 1/λ . This relationship suggests that an
C of any of its sub-components. Nevertheless, the uncer-
experimentdesigned(intheabsenceofgravity)tomeasure
taintyprincipleandgravitationalcollapsepreventanarbi-
a short distance l << l will (in the presence of gravity)
P trarilyaccuratemeasurementofeigenvaluespacing.First,
only be sensitive to distances 1/l.
the limit from quantum mechanics. Consider the Heisen-
Let us give a concrete model of minimum length. Let
berg operators for position xˆ(t) and momentum pˆ(t) and
the position operator xˆ have discrete eigenvalues {x },
i recall the standard inequality
with the separation between eigenvalues either of order
lP or smaller. (For regularly distributed eigenvalues with (∆A)2(∆B)2 ≥ − 1(h[Aˆ,Bˆ]i)2 . (3)
aconstantseparation,thiswouldbeequivalenttoaspatial 4
lattice, see Fig. 1.) We do not mean to imply that nature
Suppose that the position of a free test mass is measured
implements minimum length in this particular fashion -
at time t = 0 and again at a later time. The position
most likely, the physical mechanism is more complicated,
operator at a later time t is
and may involve, for example, spacetime foam or strings.
However,our concrete formulation lends itself to detailed t
xˆ(t)=xˆ(0) + pˆ(0) . (4)
analysis. We show below that this formulation cannot be M
excludedbyanygedankenexperiment,whichisstrongev-
WeassumeafreeparticleHamiltonianhereforsimplicity,
idence for the existence of a minimum length.
buttheargumentcanbegeneralized[3].Thecommutator
Quantizationofpositiondoesnotbyitselfimplyquan-
between the position operators at t=0 and t is
tizationofmomentum.Conversely,acontinuousspectrum
ofmomentumdoesnotimplyacontinuousspectrumofpo- t
[xˆ(0),xˆ(t)] = i , (5)
sition.Inaformulationofquantummechanicsonaregular M
spatial lattice, with spacing a and size L, the momentum
so using (3) we have
operator has eigenvalues which are spaced by 1/L. In the
infinite volume limit the momentum operator can have t
continuous eigenvalues even if the spatial lattice spacing |∆x(0)||∆x(t)| ≥ . (6)
2M
XavierCalmet: On thePrecision of a Length Measurement 3
We see that at least one of the uncertainties ∆x(0) or
∆x(t) must be larger than of order t/M. As a mea-
surement of the discreteness of xˆ(0) repquires two position
measurements, it is limited by the greater of ∆x(0) or
∆x(t), that is, by t/M,
p
t
∆x≡max[∆x(0),∆x(t)]≥ , (7)
r2M
where t is the time over which the measurement occurs
andM themassoftheobjectwhosepositionismeasured.
In order to push ∆x below l , we take M to be large.
P
In order to avoid gravitational collapse, the size R of our
Fig. 2. Salecker and Wigner thought experiment to measure
measuringdevice must also growsuchthat R>M. How-
a length. A clock emits a light beam at a time t=0 which is
ever, by causality R cannot exceed t. Any component of
reflected by a mirror and reabsorbed by the clock at a later
the device a distance greater than t away cannot affect time t =τ. Quantum mechanics implies a spread of the wave
the measurement, hence we should not consider it part function of theclock and of themirror.
of the device. These considerationscan be summarized in
the inequalities
t>R>M . (8) Euclidean path integral formulation), by checking to see
if they yield finite results even in the continuum limit.
Combined with (7), they require ∆x>1 in Planck units,
or
∆x>lP . (9) 3 Limits on the Measurement of Large
Notice that the considerations leading to (7), (8) and Distances from Fundamental Physics
(9)wereinnowayspecifictoaninterferometer,andhence
are device independent. We repeat: no device subject to Inthe section,we study whether quantummechanics and
quantummechanics,gravityandcausalitycanexcludethe generalrelativitycanlimittheprecisionofadistancemea-
quantizationofpositionondistances less thanthe Planck surement.Inordertoaddressthisquestionweshallrecon-
length. sider the thought experiment first proposed by Salecker
It is important to emphasize that we are deducing a and Wigner almost 50 years ago. In order to measure the
minimum length which is parametrically of order l , but distancelweshallconsideraclockwhichemitsalightray
P
maybelargerorsmallerbyanumericalfactor.Thispoint atatimet=0.Theclockwillsufferarecoilfromtheemis-
is relevant to the question of whether an experimenter sion of the light ray which induces a position uncertainty
might be able to transmit the result of the measurement in the position of the clock x(0). The mirror which is at
before the formation of a closed trapped surface, which a distance l of the clock will reflect the light ray which is
prevents the escape of any signal. If we decrease the min- reabsorbedat a time t at x(t) by the clock,again there is
imum length by a numerical factor, the inequality (7) re- arecoileffect andthe positionofthe clockwill havesome
quires M >> R, so we force the experimenter to work uncertainty (see Fig. 2).
fromdeepinside anapparatuswhichhasfar exceededthe Consider the Heisenberg operators for position xˆ(t)
criteria for gravitational collapse (i.e., it is much denser and momentum pˆ(t) and recall the standard inequality
than a black hole of the same size R as the apparatus).
For such an apparatus a horizon will already exist before (∆A)2(∆B)2 ≥ − 1(h[Aˆ,Bˆ]i)2 . (10)
themeasurementbegins.Theradiusofthehorizon,which 4
is of order M, is very large compared to R, so that no
Suppose that the position of a free test mass is measured
signal can escape.
at time t = 0 and again at a later time. The position
An implication of our result is that there may only be
operator at a later time t is
a finite number of degrees of freedom per unit volume in
our universe- no true continuumof spaceor time. Equiv-
t
alently, there is only a finite amount of information or xˆ(t)=xˆ(0) + pˆ(0) . (11)
M
entropy in any finite region our universe.
One of the main problems encountered in the quanti- The commutator between the position operators at t=0
zation of gravity is a proliferation of divergences coming and t is
from short distance fluctuations of the metric (or gravi- t
[xˆ(0),xˆ(t)] = i , (12)
ton). However, these divergences might only be artifacts M
of perturbation theory: minimum length, which is itself a
so using (10) we have
non-perturbative effect, might provide a cutoff which re-
moves the infinities. This conjecture could be verified by t
latticesimulationsofquantumgravity(forexample,inthe |∆x(0)||∆x(t)| ≥ . (13)
2M
4 XavierCalmet: On thePrecision of a Length Measurement
Since the total uncertainty for the measurement of the theyhaveorganizedandwherepartofthisworkwasdone.
distance l is given by the sum of the uncertainties of x(0) It is with great pleasure that I acknowledge that the re-
and x(t) we find: sultsofthethirdsectionwereworkedoutduringthissum-
t mer school with S. Hsu. I am very grateful to A. Zichichi
δl ∼ . (14) for the financial support that made my participation to
r2M
thisschoolpossibleaswellastoH.Fritzschwhohadnomi-
Note that we are not forced to take the mass of the clock
natedmeforthisschool.Finally,IwouldliketothankF.R.
tobe largelikeinthe previoussection.Thereareactually
Klinkhamer for enlightening discussions and for drawing
twooptions,oneistoallowthemassoftheclocktogrowat
myattentionto the workofSaleckerandWigner.Iwould
thesamerateast,thetimenecessaryforthemeasurement
like to thank M. Ozawa for a helpful communication and
in which case we have
for sending me a copy of his work on contractive states.
This work was supported in part by the IISN and the
δl∼1 , (15)
Belgian science policy office (IAP V/27).
or
δl∼l . (16)
p
Appendix: Contractive States
The other option is to consider a fixed, finite, mass. This
case applies to e.g. the measurement of a distance per-
Herewebrieflyreviewcontractivestates,followingOzawa’s
formedwithaninterferometersuchasLIGO[6].Themass
original work [9]. One introduces the operator aˆ defined
is at most the mass of the regionof spacetime which feels
by
one wavelength of the gravitational wave. In that case,
mω 1
the standard quantum limit [7] applies, and this is the aˆ= xˆ+ ipˆ. (17)
r 2h¯ r2h¯mω
well-knownstatementthatLIGOoperatesatthestandard
quantum limit. Note that contractive states [8,9] cannot The quantization of xˆ and pˆimplies [aˆ,aˆ†] = 1. The pa-
help to beat the standard quantum limit in a paramet- rameter ω is free. The twisted coherent state |µναωi is
ric manner. Again as in [3] the reason is that we need the eigenstate of µaˆ+νaˆ† with eigenvalue µαˆ+ναˆ⋆. The
twomeasurements.Contractivestatesallowtomakeδx(t) normalizationofthe wavefunctionimplies|µ|2−|ν|2 =1.
very smallat the price of losing all the informationabout The free Hamiltonian is given by
the uncertainty of x(0) (see appendix).
1 1 1
Hˆ =pˆ/2m=h¯ω/2(aˆ†aˆ+ − aˆ2− aˆ†2) (18)
2 2 2
4 Conclusions
and the wave function of this state is given by
In this work we have shownthat quantum mechanics and 1/4
mω
classical general relativity considered simultaneously im- hx|µναωi = × (19)
(cid:18)π¯h|µ−ν|2(cid:19)
ply the existence of a minimal length, i.e. no operational
procedure exists which can measure a distance less than mω 1+2ξi 2
thisfundamentallength.Thekeyingredientsusedtoreach exp(cid:18)−π¯h |µ−ν|2(x−x0) +ip0(x−x0)(cid:19)
this conclusion are the uncertainty principle from quan-
tum mechanics, and gravitational collapse from classical with ξ = Im(µ⋆ν), α = (mω/2h¯)1/2x0 +1/(2h¯mω)1/2ip0
general relativity. Furthermore we have shown that con- where x0 and p0 are real. The position fluctuation for a
tractive states cannot be used to beat the limit obtained free-mass is given by:
bySaleckerandWignerontheprecisionofameasurement
1 ¯hτ 2h¯ ω 2
of a length. Note that in that case we are not forced to ∆x(t)2 = + |µ+ν| (t−τ)2 (20)
4ξ m mω 2
consider very massive objects and thus the gravitational (cid:16) (cid:17)
collapse condition does not necessarily provide a bound.
with
If we are forced to consider very massive objects, then
the best precision for the measurement of a length which τ =ξ¯hm/∆p(0)2. (21)
can be archived is the minimal length itself. Our results
have deep consequences for the detectability of quantum Whenξ >0the x-depdendentphase leadsto anarrowing
foam using astrophysical sources [10,11,12,13,14,15,16, of ∆x(t) compared to ∆x(0). States with the property
17]. This however goes beyond the scope of this paper ξ >0arecalledcontractivestates.Theabsoluteminimum
and shall be considered elsewhere. is achieved for a time τ given by
2ξ ξ¯hm
τ = = (22)
ω|µ+ν|2) ∆p(0)2
Acknowledgments
and one obtains
I would like to thank G. ’t Hooft and A. Zichichi for the ¯h ∆x(0)
∆x(τ)= = . (23)
wonderful Erice Summer School on Subnuclear Physics 2∆p(0) 1+4ξ2
p
XavierCalmet: On thePrecision of a Length Measurement 5
The price to pay to make ∆x(τ) very small, i.e. smaller
than e.g. the Planck length, is to pick ξ very large,which
impliesthatτ isverylargeandthus∆x(0)2isverylargeas
well.Keepinginmindthatthemeasurementofadistance
impliestwomeasurementsweseethatitisnotpossibleto
parametricallymake the uncertainty on the measurement
of a distance arbitrarily small. This is equivalent to the
statement that LIGO and similar interferometers operate
at the quantum limit, one can beat by some small fac-
tor the standard quantum limit, but it cannot be beaten
parametrically.
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