R. Fürth's 1933 paper "On certain relations between classical Statistics and Quantum Mechanics" ["Über einige Beziehungen zwischen klassischer Statistik und Quantenmechanik", \textit{Zeitschrift für Physik,} \textbf{81} 143-162]
RR. Fürth’s 1933 paper “On certain relations betweenclassical Statistics and Quantum Mechanics”[“Über einige Beziehungen zwischen klassischerStatistik und Quantenmechanik”,
Zeitschrift fürPhysik, introduced by Luca PelitiSanta Marinella Research Institute00058 Santa Marinella, [email protected] translated and with a commentary by
Paolo Muratore-GinanneschiDepartment of Mathematics and Statistics, University of HelsinkiP.O. Box 68, 00014 Helsinki, Finlandpaolo.muratore-ginanneschi@helsinki.fiJune 9, 2020
Abstract
We present a translation of the 1933 paper by R. Fürth in which a pro-found analogy between quantum fluctuations and Brownian motion ispointed out. This paper opened in some sense the way to the stochas-tic methods of quantization developed almost 30 years later by EdwardNelson and others.
Introduction
Thermodynamic uncertainty relations are a remarkable set of inequalities in Stochas-tic Thermodynamics that bound the coefficient of variation of empirical currentsto their averages and to the entropy production rate (see, e.g., the research pa-pers [1, 2] and/or the monograph [3] for overview). In a nutshell, they intimatethat achieving very accurate currents, with a very small coefficient of variation,requires in general a minimal cost in terms of entropy production. Their name areevocative of the uncertainty relations of quantum physics, which set a bound onthe accuracy by which the position of a particle and its velocity can be evaluated.It is interesting to remark that a similar analogy between the inequality ex-pressing Heisenberg uncertainty relations and a similar one which applies to dif-fusion processes like Brownian motion was pointed out in a remarkable paper by1 a r X i v : . [ phy s i c s . h i s t - ph ] J un einhold Fürth in 1933. The paper, which points to a profound analogy betweenthe uncertainty arising from quantum fluctuations and that due to random forcesacting on a diffusing particle, opened the way in some sense to more recent devel-opments like Nelson’s stochastic mechanics approach [4] to quantum mechanics,and the stochastic quantization approach championed by Parisi and Wu [5, 6].We hope to be helpful to the community by providing a translation of thiscomparatively little known paper. The translation is preceded by a brief biograph-ical skecth of its author and is followed by some remarks on the translation andon some early developments of the approach initiated by the paper. Notice thatthe author’s references appear as footnotes as in the original paper. The referencesdue to the curator appear in brackets and are listed at the end. Reinhold Fürth
Reinhold Fürth was born in Prague (then Austria-Hungary) in 1893. He obtaineda doctorate from the German Charles-Ferdinand University in Prague in 1916,where he became a professor of experimental physics in 1931. After the Germantakeover in 1938-39 he moved to Scotland and became a research fellow at theUniversity of Edinburgh. After having been elected fellow of the Royal Society in1943, he moved in 1947 to Birkbeck College, London, to become there professor oftheoretical physics. He died in 1979.His name is well known as the editor of a collection of papers by Albert Ein-stein on the theory of Brownian movement [7], whose English translation [8] iswidely read. His lecture on the physics of social equilibrium [9], read before theBritish Association for the Advancement of Science in Edinburgh in 1951, can beconsidered as one of the earliest examples of the approach known as sociophysics.
Text
From the Physics Institute of the German University in Prague
On certain relations between classical Statistics and QuantumMechanics.
From Reinhold Fürth in Prague.With 4 figures. (Received on January 19, 1933.)
Abstract
It is highlighted the formal analogy between the differential equationsfor the probability distribution of the position of a mechanical systemaccording to classical Statistics and to Quantum Mechanics which canalso be interpreted as equations for the motion of a cluster of identi-cal particles, a diffusion. The physical origin of such diffusion will beascribed in the classical case to the collision with molecules of the sur-rounding matter, in the case of Quantum Mechanics to the uncertaintyrelations. In the last case, diffusion in the absence of forces is discussedand a simple derivation of the uncertainty relations is given on this ba-sis. The line of reasoning can be carried over to classical diffusion and2t is possible to derive an inequality for the variance of the positionand the velocity which is in strict analogy with H e i s e n b e r g’s un-certainty relations. The relation found can be also applied to a singleparticle and more generally to an arbitrary mechanical system, sinceit states that the simultaneous measurement of the position and of thecorresponding velocity is possible only up to a maximal accuracy inconsequence of the B r o w nian motion. It is discussed the relationof this finding with the problem of [determining] with which accuracyit is possible to measure a physical quantity with a mechanical mea-surement device and as a result it turns out that there exists also herein analogy [with Quantum Mechanics] an accuracy limit which cannotbe overcome. Finally, it is shed light from the point of view of waveMechanics on the question why the classical diffusion equation holdsfor a real density function with a real diffusion coefficient in contrastto the Schrödinger equation [which holds] for a complex function withan imaginary coefficient, and [this fact] is related to the problem ofthe observability of physical quantities and of the reversibility versusirreversibility of natural processes.In what follows there shall be a discussion of certain relations between classicalstatistics – the classical diffusion theory and the theory of B r o w nian motion –on the one hand and, Quantum Mechanics on the other, [discussion] which arisesfrom formal reasons and [which], although it might be already known to some, tothe best of my knowledge has yet not been addressed in this context. In particularit is possible to show that H e i s e n b e r g’s uncertainty relations carry over toprocesses which are governed by classical Statistics and that it is thus possibleto bring about new perspectives on the often addressed question of the limit ofmeasurability with an measurement device. It is furthermore attempted to makeprecise the physical meaning of the aforementioned similarities and differences. The classical theory of diffusion is governed by the generalised diffusion equa-tion ∂ u ∂ t = D ∆ u − div ( u v ) (1)where u ( x , y , z , t ) denotes the density as function of the position and time, D (as-sumed constant) the diffusion coefficient and v the velocity vector of the convec-tion current occasioned by external forces. The solution of this equation undergiven boundary conditions determines the distribution of the density at any fu-ture instant of time if the distribution is known in the present. See, e.g., Frank-Mises,
Differential-u. Integralgleichungen d. math. Physik
2, 248
3f one interpret the diffusion experiment as a collective experiment with a spa-tial ensemble of many identical particles then u d V is the relative frequency withwhich any element of the ensemble is found in the volume element d V at time t during the collective experiment if u satisfies the normalization condition (cid:90) (cid:90) (cid:90) u d V = t . The replacement of the spatial ensemble with a virtual ensemble turnsthe diffusion equation (1) into an equation for the “probability density” u of theposition of an individual particle which can be computed as a function of timewhen it is known at time zero: S m o l u c h o w s k i’s differential equation forB r o w nian motion of an individual particle under the action of external forces .It is possible to show that S m o l u c h o w s k i’s equation is a special case ofanother differential equation which can be derived under very general conditionsfor the B r o w nian motion of an arbitrary mechanical system and [which] is usu-ally referred to as the F o k k e r - P l a n c k equation . Following S c h r ö d i n g e r ,[the Fokker-Planck equation] can be written as ∂ u ∂ t = Fu (3)where F denotes a certain differential operator which, in agreement with (1), re-duces to F = D ∆ − div v in the case when the system is a particle under the actionof a force.The differential equation (3) is, as S c h r ö d i n g e r also already pointedout, formally identical to the time dependent S c h r ö d i n g e r differential equa-tion of wave mechanics for the wave function ψ which is usually written in theform ∂ψ∂ t = H ψ (4)where H denotes the H a m i l t o n operator for the mechanical problem of inter-est. According to the statistical formulation of wave mechanics also this equationis a “probability equation” inasmuch it allows one to compute this quantity at anyarbitrary later instant of time from the knowledge of ψ ( q ) at time zero and the“probability amplitude” ψ is linked to the probability density for the sojourn ofthe system in a certain volume element of the q -space by the relation w = ψψ ∗ (5)( ψ ∗ is the complex conjugate of ψ ) as far as ψ satisfies the normalization condition (cid:90) (cid:90) (cid:90) ψψ ∗ d V = M. v. Smoluchowski,
Ann. d. Phys.
43, 1105, 1915 See, among others, F. Zernike,
Handb. d. Phys. Bd. III,
S. 457 E. Schrödinger,
Ann. de l’Inst. H. Poincaré
S. Ber. Berl. Akad.
C. R. E. Schrödinger,
Ann. de l’Inst. H. Poincaré
S. Ber. Berl. Akad.
C. R.
4y reversing the line of reasoning, one can also construe the quantity w definedvia (5) as the phase point density of a large number of identical non interactingsystems in q -space. Equation (4) then determines the evolution of such distributiondensity and permits to compute the density at any further time if the densityfunction is assigned at time zero.In the case of special importance of an individual point system of mass m subject to the action of a force which can be derived from a potential U , equation(4) reads − h π i ∂ψ∂ t = − h π m ∆ ψ + U ψ (7)The discussion of this equation teaches, as E h r e n f e s t first showed that thecentre of mass of a cluster of particles obeying the conditions expounded aforemoves in the usual three dimensional space according to the prescription of classi-cal mechanics when the assigned forces act on the particle but also that the clusterof particles spreads around the centre of mass via a sort of diffusion. We thereforeencounter here a convection current with overlaid a diffusion in analogy with themotion of a cluster of particles according to the classical theory of diffusion.As we are interested only in the last phenomenon, we wish in what followsset to zero the external force. The equations (4) and (1) become then formallyidentical, namely ∂ u ∂ t = D ∆ u (8)and ∂ψ∂ t = (cid:101) ∆ ψ (9)where the shortcut (cid:101) = i h π m (10)is used. Subject to the same boundary and initial conditions the solutions of (8)and (9) read hence completely the same. Sure enough a substantial differencearises from the fact that in the case of Quantum Mechanics not the function (ingeneral complex) ψ but rather according to (5) its norm plays the role of densityfunction and that according to (10) the diffusion coefficient is here purely imagi-nary. We return to the physical meaning of this fact later below. The deeper reason for the analogy emerging in the comparative presentation of§ 1 between the motion of a cluster of particles according to the classical theoryof diffusion and Quantum Mechanics resides in the fact that in both cases thevelocities of individual particles in the cluster differ and obey a statistical law. P. Ehrenfest,
ZS. f. Phys. , 455, 1927
5n the first case, this (phenomenon) stems from the fact that particles incur inirregular collisions with molecules of the surrounding element whereby the par-ticles’ momentum is continuously varied in intensity and direction in such a waythat there is no relation between the change of momenta of distinct particles. This[fact] becomes manifest when considering an individual particle in its irregularB r o w nian motion, and, when considering a particle cluster, in the fact that foran assigned initial state of the cluster and initially vanishing “macroscopically”measured velocity, the particles actually possess velocity irregularly distributedacross the cluster and that in the course of time the initial distribution varies inthe characteristic way of a diffusion.In the case of Quantum Mechanics, the very assumption of an initial densitydistribution implies that the condition of vanishing initial velocity of all the par-ticles cannot be strictly satisfied. According to H e i s e n b e r g’s fundamentaluncertainty relations governing Quantum Mechanics a complete assignment ofthe initial velocity of the particles would be possible only in the presence of acomplete uncertainty about the initial positions.As a certain information about the initial position of the particles is conveyedby the assignement of the initial distribution, one must admit a certain blurring ofthe initial velocities, i.e., a certain statistical distribution of the initial velocities ofthe cluster particles. But a necessary consequence of this is that a variation of theinitial density distribution as well as a diffusion of the cluster must have occurredafter a certain time.That the uncertainty on the value of position of the particles of the diffus-ing cluster really satisfies H e i s e n b e r g’s uncertainty relations with the uncer-tainty about the value of the velocity (momentum), has been shown by H e i s e n -b e r g and K e n n a r d among others. A brief derivation may be given herefor the one-dimensional case, which, without resorting to the theory of transfor-mations, makes use only of equation (9) and its complex conjugate taking in onedimension the form ∂ψ∂ t = (cid:101) ∂ ψ∂ x ∂ψ ∗ ∂ t = − (cid:101) ∂ ψ ∗ ∂ x (11)Let x the initial position of one particle of the cluster, v its initial velocity and x its position after a time t , then x = x + v t (12)holds. If the centre of mass at time zero is located in the origin of the coordinatesand its velocity is zero, i.e. x = v = x = t . Evaluating the quadratic expectation value of (12) one getsinto x = x + x v t + v t (13) W. Heisenberg,
ZS. f. Phys. E. H. Kennard, loc. cit. I need to thank here Mr. K. L ö w n e r, Prague, for some hints.
6y definition x = (cid:90) + ∞ − ∞ x ψψ ∗ d x (14)holds true. Using equation (11) and under the assumption that ψ vanishes suffi-ciently fast at infinity, one gets into after a simple calculationdd t x = (cid:101) (cid:90) ∞ − ∞ x (cid:18) ψ ∂ψ ∗ ∂ x − ψ ∗ ∂ψ∂ x (cid:19) d x (15)d d t x = − (cid:101) (cid:90) ∞ − ∞ ∂ψ ∗ ∂ x ∂ψ∂ x d x (16)d d t x = x must be a quadratic function of time in agreementwith (13) ; it also follows from (16) that v as coefficient of t (13) satisfiesd d t v =
12 d d t x = − (cid:101) (cid:90) ∞ − ∞ (cid:12)(cid:12)(cid:12)(cid:12) ∂ψ∂ x (cid:12)(cid:12)(cid:12)(cid:12) d x (18)According to H e i s e n b e r g , it now follows from the self-evident inequality (cid:12)(cid:12)(cid:12)(cid:12) x x ψ + ∂ψ∂ x (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:90) ∞ − ∞ (cid:12)(cid:12)(cid:12)(cid:12) ∂ψ∂ x (cid:12)(cid:12)(cid:12)(cid:12) d x ≥ x whence from (17) x v ≥ − (cid:101) (20)If one introduces the uncertainties on the position and momentum of the particlecluster under consideration with the avail of the relations ∆ x = (cid:112) x ∆ p = m (cid:112) v (21)using (10) from (20) follows for their product H e i s e n b e r g’s relation ∆ x ∆ p ≥ h π (22) W. H e i s e n b e r g,
Die physikalischen Prinzipien der Quantentheorie.
Leipzig 1930. Page 13 andfollowing. ψψ = Const. e − x /4 ( ∆ x ) (23)and also for the density of the particle cluster (5) in consideration of (2) the Gaus-sian distribution w = √ π ∆ x e − x /2 ( ∆ x ) (24)If ψ takes at time t = dd t x = t disappears from (13). If there is a correspond-ing initial distribution of the position in the particle cluster under consideration,so that ( x v ) = In accordance with what was said at the beginning of section 2, it is natural toapply the above reasoning, which is based on the H e i s e n b e r g uncertaintyrelation in the quantum mechanical case, to the case of classic diffusion. Also inthis case we restrict ourselves to the one dimensional case with vanishing convec-tion current and so we start from equation (8) which in one dimension reads ∂ u ∂ t = D ∂ u ∂ x (25)where in agreement with (2) u satisfies the condition (cid:90) ∞ − ∞ u d x = x as x = (cid:90) ∞ − ∞ x u d x (27)At t = x =
0. To start with, we look for the derivation of the analog of equa-tion (13) which expresses how the uncertainty x initially present in the diffusingparticle cluster grows in the course of time.dd t x = dd t (cid:90) ∞ − ∞ x u d x = (cid:90) ∞ − ∞ x ∂ u ∂ t d x (28) = D (cid:90) ∞ − ∞ x ∂ u ∂ x d x = x = t x = (cid:90) ∞ − ∞ x ∂ u ∂ t d x = D (cid:90) ∞ − ∞ x ∂ u ∂ x d x = D (29)and therefore that x is a linear function of time of the form x = x + D t (30)The comparison of (30) with (13) shows that by all means in both cases the uncer-tainty over the position indefinitely grows over a sufficient long time and thus thata diffusion of the cluster occurs. Whereas here, however, the growth of x occursindependently of x and linearly in time, there the growth in time is quadratic andin consequence of (20) is itself dependent upon x (it takes place in a particularlysudden way if x = v becomes infinitely large); finally, if the linearterm in t is non-vanishing the so that dispersion of the positions and of the ve-locities are not statistically independent at time zero, it may be that the clusterundergoes first a contraction to a minimum and only afterwards a spreading.The formal causes for the aforementioned differences have been already dis-cussed at the end of § 1. The differences are physically explained by the factthat in the case of classical diffusion there is no “initial velocity” of the particlesand therefore no equation of the form (12) exist, and that furthermore the in-stantaneous speed of the particles is due to the collisions with the molecules, asalready mentioned . On the basis of the statistical independence of the disper-sion process and the initial distribution in the classic case, one can immediatelywrite equation (30), since it expresses that this is due to two causes: the “squareerror” of x resulting from initial spread and diffusion is the sum of these two justmentioned ingredients, the second being the well-known E i n s t e i n law for themean square of the B r o w nian motion.In order to find the analog of the uncertainty relations (20) to (22), we need infirst place to find a suitable definition of velocity for the classical diffusion. Fromthe above it is clear that by no means this role can be played by the microscopicvelocity produced by molecular collisions. Likewise, as we have already seen, themacroscopic velocity of the cluster regarded as a single entity, or strictly speakingthe velocity of its centre of mass, [is not a good candidate since it] vanishes. Asuitable quantity comes about from the consideration of the “diffusion current”,i.e., the quantity of diffusing matter crossing in the unit of time a fixed section inthe diffusion domain. As it is well known , the vector Q of the diffusion current If one, following S c h r ö d i n g e r (Ber. Ber. 1930, S. 296, Nr. 19), sets out to choose the valueof x such that ∆ x is as small as possible and the product ∆ x ∆ p = h π holds true so that the initialdistribution (23) is satisfied, then in (13) the second term of the right hand side vanishes whereasthe first and the third become equal upon setting v = − (cid:101) x . This implies for x x = h π m t (31)In this case, the formal analogy with (30) is strikingly evident when one thereby replaces D with theabsolute value of the “imaginary diffusion coefficient” (cid:101) according to (10). Compare ref. 1.
9s a local function in the diffusion domain, connected with the scalar u by therelation Q = − D grad u (32)Based on the fact that u is nothing else than density of matter of the diffusingelement, we find the corresponding velocity vector v according to v = u Q = − D u grad u (33)which in the one-dimensional case becomes v = − D u ∂ u ∂ x (34)If we now compute the particle cluster mean value of v at a certain time instant,we obtain by definition using (25) v = (cid:90) ∞ − ∞ v u d x = − D (cid:90) ∞ − ∞ ∂ u ∂ x d x = v is nothing else than the macroscopic velocity of the centre ofmass.For the mean value of v , one finds v = (cid:90) ∞ − ∞ v u d x = D (cid:90) ∞ − ∞ u (cid:18) ∂ u ∂ x (cid:19) d x (36)By a straightforward application of the reasoning of § 2 one can establish thederivation of an inequality for the product v x , by proceeding once again fromthe self-evident inequality (cid:18) u ∂ u ∂ x + xx (cid:19) ≥ u (cid:18) ∂ u ∂ x (cid:19) ≥ − xx ∂ u ∂ x − x u ( x ) Upon integrating, a simple calculation making use of (26) and (27) yields (cid:90) ∞ − ∞ u (cid:18) ∂ u ∂ x (cid:19) ≥ x whence finally according to (36) x v ≥ D (38)As one can see, the inequality (38) has the same form of the inequality (20), whichturns into (38) if one again replace the absolute value of (cid:101) with D . Introducing the10otation ∆ x and ∆ p in analogy with (21), we write our uncertainty relation in thesimpler form ∆ x ∆ p ≥ D (39)stating that in a classically diffusing particle cluster the position and the velocityof the particles at any instant of time cannot be simultaneously determined witharbitrary accuracy and that furthermore the product of the uncertainties must bealways larger than the diffusion coefficient D .The lower bound is attained, i.e., the inequality turns into an equality if andonly if (37) [also] holds as an equality. The solution of the differential equationobtained in this way immediately yields u = √ π ∆ x e − x /2 ( ∆ x ) (40)having taken (26) into account, and [is] therefore again the G a u s sian distribu-tion, as in the quantum mechanical case, in formal agreement with (24).Whereas in the present case, from the occurrence of the distribution (40) theequality ∆ x ∆ p = ∆ x ∆ p to attainits minimum. Furthermore, whereas in a cluster of particles left to itself and sat-isfying at time zero the minimum uncertainty condition this condition continuesto hold at any further time (because the distribution (40) is self-sustaining) in thequantum mechanical case the minimum condition is only instantaneously satis-fied, e.g., at time zero and later no more (since the form of the distribution (23)is not preserved by the motion of the particles). Finally, it should be emphasizedthat in the classical case one can always think a cluster of particles satisfying theminimum condition as one brought about by the diffusion of one which at a cer-tain instant of time was completely concentrated in the origin of the coordinates.In order to see this, one needs only to make use in (40) of (30) where one insertthe abbreviation x = D t ; one then obtains u = (cid:112) π D ( t + t ) e − x /4 D ( t + t ) (41)which entails that indeed, for t = − t , u vanishes in the full space with the excep-tion of x =
0. In the quantum mechanical case this reduction, as we have alreadyseen, is not possible. In the two preceding paragraphs we discussed the application of uncertainty re-lations to a spatial aggregate of identical particles in the quantum and in theclassical case. As it is well known, the fundamental significance of the uncertaintyrelation in Quantum Mechanics appears, however, when it is applied to an in-dividual system. It teaches that the simultaneous measurement of the positionand the momentum of a force-free particle can be performed with the maximum11ccuracy h /4 π predicted by formula (22) since the measurement process duringthe measurement of one of the two quantities disturbs the other to an amountthat the product of the uncertainties of both quantities cannot be lower than theaforementioned value. One can reformulate the statement for a general mechan-ical system by saying that the simultaneous measurement of a coordinate q andof the impulse canonically conjugated to it is only possible with an uncertainty ofthe order of magnitude of h .We can now also in a straightforward way apply the relation (39) obtained in§ 3 to the problem of the simultaneous measurement of the position and speed ofa particle, which is under the action of irregular impacts, and therefore performsa B r o w nian motion. Our relation teaches that the product of the uncertaintyof a simultaneous measurement of position and velocity cannot be lower than thevalue D , whereby velocity must be understood as the macroscopic speed of theparticle, i.e. the quantity δ x / δ t (assuming that δ t is large compared to the timebetween two successive molecular collisions of the particle). One sees that, as inthe quantum mechanical case, there is an actual impossibility of a simultaneous,precise measurement of position and velocity, which, however, is not, as in Quan-tum Mechanics, determined by the process of measurement itself and governedby a universal constant, but it is rather caused by the influence of the environmenton the observed system, and as a consequence it is clearly not of universal nature(for example, by lowering the temperature, which determines the value D , [theeffect] can be made arbitrarily small).The following argument evinces that formula (39) holds true also in the caseof the measurement of an individual particle: we consider a force-free particlewhich at time zero is located at the origin of the coordinates and has vanishingmacroscopic velocity. If we measure the position of the particle after a short time t then the expected value x satisfies Einstein’s formula x = D t (42)whence it follows dd t x = D If we now exchange the order between time differentiation ad expectation value,we get furthermore (cid:18) dd t x (cid:19) = (cid:18) x dd t x (cid:19) = D (43) x is evidently now the uncertainty over the position of the particle (we assumed x = x /d t is theuncertainty over the velocity (which we assumed vanishing at time zero) broughtabout by the very same causes. The product (cid:16) x dd t x (cid:17) thus specifies the value to beexpected by averaging over many measurements of the uncertainty product ∆ x ∆ v which according to equation (43) is equal to D . The fact that we obtained exactlythe minimum value here instead of equation (39) is due to the fact we evaluatedthe mean value over repeated measurements of a particle, which we assumed to12ave always the same starting position and starting velocity at time zero. It isimmediately obvious that without this assumption the uncertainty can in any caseonly increase, so that the product ∆ x ∆ v is actually larger than D , as required bythe relationship (39).Our relation states that an increase in the measurement accuracy of the positionof a B r o w nian particle reduces the accuracy of a simultaneous measurementof the velocity and vice versa. The physical meaning of this statement can bevisualized with the help of the following Figs 1-4 of which Fig 1 plots as thefunction x ( t ) the position as a function of time of a particle falling under the effectof gravity in a liquid, observed with a certain magnification, and Fig. 2 representsthe function v ( t ) = ˙ x ( t ) obtained from it. Fig 3 shows the beginning of Fig. 1,plotted with a stronger magnification, and Fig. 4 again the velocity curve obtainedtherefrom.Figure 1: Plot of the position of a B r o w nian particle as a function of time(stylized).Figure 2: Velocity v of the particle as a function of time computed from Fig.1(dashed line mean value v ).One can immediately see how increasing the accuracy in the determination ofthe position by increasing the magnification necessarily increases the uncertaintyin the simultaneous determination of the velocity. Our relation thus expresses inan exact way the fact known to everyone familiar with B r o w nian motion that thetrajectory of a B r o w nian particle exhibits more discontinuities with increasing13igure 3: 5-times magnification of the beginning of the plot in Fig. 1 (stylized).Figure 4: Velocity v computed from Fig. 3 (dashed line mean value v ).magnification.Exactly as in the case of Quantum Mechanics, we can extend the uncertaintyrelation (39) also to any mechanical system in contact with a surrounding tempera-ture bath. Then, to every degrees of freedom is associated the B r o w nian motionof the corresponding coordinate which we denote again by x . The F o k k e r -P l a n c k equation (3) takes the place of the differential equations (25) or (8). It isplausible that also in this general case an uncertainty relation on the form ∆ x ∆ v ≈ D (44)holds true where v is the velocity associated to the coordinate x , and D denotesthe coefficient of the term ∂ u ∂ x on the right hand side of (8) and expresses thecharacteristic constant of this B r o w nian motion. The relation states that thesimultaneous measurement of the coordinate x and of its associated speed v ispossible only with an uncertainty of order D .14 We can also extend the domain of validity of formula (44) to any non-mechanicalquantity since any physical quantity, even of non-mechanical nature, is measuredusing mechanical measurement instruments, for example a current [is measured]using a galvanometer itself consisting of mechanical components. We assumethat the “deflection” x of the mechanical instrument in use be proportional tothe quantity J to be measured (for example the deflection of a galvanometer [isproportional] to the intensity of the current). When this is not the case from thestart, one can always apply a compensation method in order to implement thedesired condition within strict accuracy. Let ˙ J be the speed of variation of J . Thenit holds true that J = a x ˙ J = a ˙ x = a v ∆ J = a ∆ x ∆ ˙ J = a ∆ ˙ x = a ∆ v (45)whence with the help of (44) ∆ J ∆ ˙ J ≈ a D (46)The relation (46) teaches that although one can arbitrarily increase the measure-ment accuracy by choosing an appropriate measurement device, specifically byreducing a , simply increasing the reading accuracy of the pointer cannot improveabove a certain value the precision of a simultaneous measurement of the quantity J and its speed-of-variation owing to the B r o w nian motion of the measuringinstrument. One can thus reduce a by reinforcing the magnetic field in a movingcoil galvanometer with given mechanical properties and as a consequence enhancethe accuracy of current measurement at least in principle arbitrarily; one cannot,however, achieve any reduction of the product ∆ J ∆ ˙ J by a simple increase of thereading accuracy for example by magnifying the deflection using a microscopicreading pointer ) or using a thermal relay or a light electric relay )The problem of the limits of measurement accuracy due to B r o w nian mo-tion of instruments, in particular galvanometers, has been recently repeatedly dis-cussed by several authors and it has been thoroughly discussed with whichprocedures one can perform the most accurate possible measurement of a quan-tity of interest with an instrument of a given type. In my opinion, these discussionhave always overlooked an important point. The task of the experimentalist iscertainly that of recording the quantity J of interest as a function of time, i.e. thefunction J ( t ) with the highest accuracy possible. If one restricts [the attention] toa short interval of time, this requirement is equivalent to the task of determininga quantity J and its variation speed ˙ J at a given instant of time with the highest possibleaccuracy . The relation (46) teaches that with a given instrument this is possibleonly with a certain uncertainty completely independent from any procedure toincrease the reading accuracy of the pointer. G. Ising,
Ann. d. Phys. N. Moll u. N. Burger,
Phil. Mag. L. Bergmann,
Phys,ZS. G. Ising,
Phil. Mag. Ann.d.Phys. ZS. f. Phys.
Schriften d. Königsberger Gel. Ges. Ann.d. Phys.
12, 993, 1932 J despite the B r o w nian motion by taking many readings and taking theiraverage which should be then more precise than an individual measurement orby using an integrating measuring instrument makes sense only when one knowsin advance that the quantity of interest is exactly constant. But how can one knowthis without having performed first a corresponding measurement to ascertainsuch stipulation? If one really tries this, then one would obtain by repeated obser-vation or by continuous recording a time dependence of the [pointer] deflection(because of the B r o w nian motion) whence it is certainly not possible to deter-mine whether the observed quantity remains constant or whether it varies in timewithin the limit of accuracy of the recorded oscillations. This circulus vitiosus isthe reason why the method proposed to increase measurement accuracy is notreally feasible.Actually we can even say with certainty that the requirement of constancy of J implied by the mentioned procedure is certainly not satisfied because any macro-scopically defined quantity, which can be measured by a macroscopic measure-ment instrument, undergoes oscillations. For instance, in reality there is certainlyno constant electromotive force even if the power source is protected from externalinterference with all possible refinement because of the occurrence of spontaneouspotential oscillations induced by the thermal motion of electrons how it has beenexperimentally shown by several researchers over the last years . Thus to mea-sure an electromotive force with the highest possible accuracy obviously meansto record as precisely as possible its time dependence or in a short time intervalto simultaneously measure as precisely as possible the electromotive force andits variation velocity. But, as we have shown above, this accuracy has becauseof the B r o w nian motion an upper limit which is independent of the way themeasurement is performed. The results reported in the previous paragraphs are, as it has been repeatedlymentioned, due to the formal analogy between the fundamental differential equa-tions of classical diffusion theory and quantum mechanics, a fact which becomesparticularly evident when contrasting the equations (8) and (9) of § 1. Alreadythere we have however pointed out essential formal differences between the twoequations. We now want to try to understand the physical origins of these differ-ences. The following considerations should at the same time contribute to clarifycertain ambiguities, which have recently been highlighted by E h r e n f e s t withthe invitation to the physicists to tackle these problems.Classical diffusion can be regarded as a current which, as we saw in § 1, isgoverned by a differential equation of the form (8), where F is a real differentialoperator and u is a real function of position and time, representing the density ofthe diffusing element. It follows that it must be possible from the assignement of u at any instant of time to compute the density distribution at any later (and of J. B. Johnson,
Phys. Rev.
97, 1928; N. H. Williams, ibidem
Proc. Roy. Soc. London (A) u and the coordinates aloneand does not depend on the history of the system. Thus if u ( x , y , z ) is known, thenit also specifies v ( x , y , z ) and therefore the evolution of the system in the followingtime step is completely determined in the sense of classic hydrodynamics.We also note that a time reversal operation, an exchange of t with − t in equa-tion (8) is not possible because D , the diffusion coefficient, owing to its moleculartheoretical meaning, is positive-definite. The diffusion process is therefore “irre-versible”. This is also evident from the fact that the velocity current is for given u a pure function of the position, so the initial velocities are not reversible and aredetermined solely by the collisions with the surrounding molecules.The situation is quite different in the quantum mechanical case. Since theparticle motion is not disturbed here by collisions with the molecules of the sur-rounding element, the motion of the particle cluster is essentially determined bythe initial positions and the initial velocities of the particles. It is therefore clearthat there cannot be a differential equation for the density function w in the sameway as it occurs for classic diffusion. That on the contrary an equation of the form(9) holds can be most easily seen from the point of view of wave mechanics. Fromthis point of view, the particle cluster forms a “wave packet”, i.e., a superpositionof harmonic partial waves of the form ψ k = ϕ k e π i E k t / h the number whereof has the cardinality of the continuum for the boundary con-ditions considered here. Here ϕ k stands for the “ amplitude function ” a complexfunction of the position of the form ϕ k = a k e i S k containing two real functions of the position, the amplitude A k and the phase S k . The assignment of all the A k ’s and S k ’s as functions of the position fullyspecifies the ϕ in the wave packet under consideration at a given instant time aswell as for every later (or earlier) instants of time in consequence of the differentialequation (9), which is physically obvious, since the fate of each partial wave isdetermined by the specification of amplitude and phase at time zero and thus alsothe fate of the wave packet created by interference from the partial waves. So it isimmediately comprehensible that for description of the state of the wave field twoscalars or one complex function, the S c h r ö d i n g e r function, are necessary.Since the density of the cluster under consideration (now considered from thecorpuscular point of view) is specified solely by | ψ | according to equation (5), theassignement of ψ as a function of the position entails more detailed informationthan the distribution of the particles’ positions at a certain instant of time. Accord-ing to what said above, as the fate of the cluster is determined by ψ , it is evidentthat the assignement of ψ contains information also about the distribution of the velocities at a certain instant of time. If, conversely, the initial velocities are not17nown, then it is not possible from the initial distribution alone to make predic-tions about the motion of the particles’ cluster. In fact there cannot be a differentialequation for | ψ | . Nevertheless only the density w = ψψ ∗ or, interpreted as a virtualentity, the probability density of the position, is observable and not ψ itself. Thisparadoxical state of the matter can be immediately explained as a consequence ofthe uncertainty relations. Were ψ indeed observable then according to our discus-sion the position and velocity distribution would be simultaneously assigned forour particle cluster which is not possible!The fact that the coefficient on the left side of equation (9) must be purelyimaginary or the diffusion coefficient (cid:101) in (10) must be purely imaginary can beseen as follows: if at an arbitrary instant of time the phases S k of all the partialwaves are reversed by 180 ◦ , then every φ k turns into φ ∗ k and therefore ψ into ψ ∗ . Atthe same time, however, the reversal of all phases means turning all wave processesin the opposite direction or the complete reversal of the motion of the wave packet.The exchange of ψ with its conjugate complex value ψ ∗ means nothing else than atime reversal, and the differential equation (9), which ψ satisfies , must thereforeremain unchanged under the simultaneous replacement of ψ with ψ ∗ and of t with − t . This is actually only possible, provided that the H a m i l t o n operator H is time independent, if the coefficient of ∂ψ∂ t is purely imaginary. The occurrenceof the imaginary diffusion coefficient means, as S c h r ö d i n g e r has alreadypointed out , simply the reversibility of the quantum mechanical “diffusion” incontrast to the classical one, a discrepancy that was already emphasized in § 2 and3 in the [discussion of the] differences between equations (13) and (30).Prague, January 1933. A Translation notes
A.1 Translation style
We tried to reproduce the style of the original prose by not splitting the long(!)sentences. We used square brackets [ . . . ] for text added either to maintain syn-tactic congruence or to emphasize the meaning implicit in the construction of theoriginal sentence. A.2 Sources
Some references are not exact. We could not retrieve in particular N. L. Williams’spaper. Furthermore we could not find any paper on Physical Review by J. B.Johnson in 1927. Notice that the references that Fürth displays in footnotes, ascostumary in the original journal, have been repeated at the end of the presentwork. Erwin S c h r ö d i n g e r, loc. cit. , ref 5. .3 Influence on stochastic mechanics: Fürth’s paper is discussed in Fényes [10]. This paper lays down the foundationof what will be Nelson’s “stochastic mechanics” program [4]. In the introductionof [10] Fe ´nyes states:
Although F ü r t h has demonstrated the existence in diffusion theory of arelation that is formally analogous to H e i s e n b e r g’s, in his opinion thetwo relations cannot have the same meaning, because the F o k k e r equationcannot be valid in quantum mechanics.
Fe ´nyes’s findings in [10] as summarized in the paper’s abstract are:
There are also certain uncertainty relations for M a r k o v processes. A cer-tain probability-amplitude function can also be assigned to a M a r k o v pro-cesses. The F o k k e r equation is also valid in quantum mechanics. TheH e i s e n b e r g relation is a special case of the uncertainty relation of theM a r k o v processes. The wave-mechanical wave function is a special caseof probability-amplitude functions governed by M a r k o v processes. Thewave-mechanical processes are special M a r k o v processes. The H e i s e n -b e r g relation is (in contrast to the previous interpretation) exclusively aconsequence of the statistical approach, and is independent of the disturbancesoccurring in the two measurements.
Finally Fürth and Fényes are known in the stochastic quantization communitywhere are somewhat considered as precursors of the Parisi-Wu method (see e.g,the discussion in the introduction of [6]).
A.4 General derivation of the uncertainty relation
Here we reproduce for convenience Fe ´nyes’ argument. To start with, we recall thediffusion pathwise probabilistic definition of current velocity and its relation withthe “coefficients” of the Fokker-Planck equation.Let us consider a stochastic process { ξ t } t ≥ with drift b ( x , t ) = lim s (cid:38) E (cid:18) ξ t + s − ξ t s (cid:12)(cid:12)(cid:12)(cid:12) ξ t = x (cid:19) and diffusion D ( x , t ) = lim s (cid:38) E (cid:18) ( ξ t + s − ξ t ) ⊗ ( ξ t + s − ξ t ) s (cid:12)(cid:12)(cid:12)(cid:12) ξ t = x (cid:19) We assume that drift and diffusion enjoy regularity properties in R d such that thetransition probability density (cid:84) satisfies Kolmogorov’s forward (Fokker-Planck) ∂ t (cid:84) ( x , t | y , s ) + ∂ x · b ( x , t ) (cid:84) ( x , t | y , s )=
12 Tr ∂ x ⊗ ∂ x D ( x , t ) (cid:84) ( x , t | y , s ) (47)19nd backward equations ∂ s (cid:84) ( x , t (cid:12)(cid:12) y , s ) + b ( y , s ) · ∂ y (cid:84) ( x , t (cid:12)(cid:12) y , s )+
12 Tr D ( y , s ) ∂ y ⊗ ∂ y (cid:84) ( x , t (cid:12)(cid:12) y , s ) = t − s (cid:38) (cid:84) ( x , t (cid:12)(cid:12) y , s ) = δ d ( x − y ) for any x , y ∈ R d , and t ≥ s ≥
0. In such a case, the probability density of theprocess { ξ t } t ≥ evolving from any reasonable initial data (cid:112) ι ( x ) also satisfies theFokker-Planck equation by means of the Markov property (cid:112) ( x , t ) = (cid:90) R d d d y (cid:84) ( x , t (cid:12)(cid:12) y , s ) (cid:112) ι ( y , s ) ∀ t ≥ s ≥ v ( x , t ) = lim s (cid:38) E (cid:18) ξ t + s − ξ t − s s (cid:12)(cid:12)(cid:12)(cid:12) ξ t = x (cid:19) The evaluation the conditional expectation yields v ( x , t ) = b ( x , t ) − (cid:112) ( x , t ) ∂ x D ( x , t ) (cid:112) ( x , t ) (49)where (cid:112) is the probability density of { ξ t } t ≥ . The derivation of (49) may useKolmogorov’s time reversal relation between the probability and the forward (cid:84) andbackward (cid:84) R transition probability densities (cid:84) R ( y , t (cid:12)(cid:12) x , t + s ) (cid:112) ( x , t + s ) = (cid:84) ( x , t + s (cid:12)(cid:12) y , t ) (cid:112) ( y , t ) eq. (8) of Kolmogorov’s 1937 paper [11]. Kolmogorov’s paper was known toFényes but, obviously, could not be to Fürth. Namely, under our working hy-potheses the identityE (cid:0) ξ t − s (cid:12)(cid:12) ξ t = x (cid:1) = (cid:90) R d d d y y (cid:84) R ( y , t − s (cid:12)(cid:12) x , t )= (cid:90) R d d d y y (cid:84) ( x , t (cid:12)(cid:12) y , t − s ) (cid:112) ( y , t − s ) (cid:112) ( x , t ) holds true. We then obtain (49) observing that as functions of y the densityobeys the Fokker-Planck equation whereas the transition probability satisfies Kol-mogorov’s backward equation (48).An important consequence of (49) for the line of reasoning of Fényes is that theFokker-Planck equation satisfied by a given density (cid:112) ( x , t ) once expressed in termsof the current velocity takes the form of a mass continuity equation ∂ t (cid:112) ( x , t ) + ∂ x · v ( x , t ) (cid:112) ( x , t ) = { ξ t } t ≥ : V ξ t = (cid:90) R d d d x (cid:112) ( x , t ) (cid:107) x − E ξ t (cid:107) and V v ( ξ t , t ) = (cid:90) R d d d x (cid:112) ( x , t ) (cid:107) v ( x , t ) − E v ( ξ t , t ) (cid:107) Cauchy-Schwarz immediately yields ( V ξ t ) V v ( ξ t , t ) ≥ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R d d d x (cid:112) ( x , t ) ( x − E ξ t ) · ( v ( x , t ) − E v ( ξ t , t )) | We then use (supposing that the density vanishes at infinity sufficiently fast)E v ( ξ t , t ) = E b ( ξ t , t ) to write (cid:90) R d d d x (cid:112) ( x , t ) ( x − E ξ t ) · ( v ( x , t ) − E v ( ξ t , t )) = (cid:90) R d d d x (cid:112) ( x , t ) ( x − E ξ t ) · (cid:18) b ( x , t ) − E b ( ξ t , t ) − (cid:112) ( x , t ) ∂ x D ( x , t ) (cid:112) ( x , t ) (cid:19) We thus obtain Fe ´nyes’s inequality ( V ξ t ) V v ( ξ t , t ) ≥ (cid:12)(cid:12)(cid:12)(cid:12) E (cid:0) ξ t · b ( ξ t , t ) (cid:1) − ( E ξ t ) · E b ( ξ t , t ) +
12 E Tr D ( ξ t , t ) (cid:12)(cid:12)(cid:12)(cid:12) (50)We observe that • the bound reduces to Fürth’s whenever the drift is negligible (setting e.g. b =
0) or whenever the process is positively correlated with the drift. • At equilibrium v = ( V ξ t ) V v ( ξ t , t ) ≥ b ( x ) = − ∂ x U ( x ) & D = D Then equilibrium means (assuming U positive definite and confining) (cid:112) ( x ) ∝ e − U ( x ) D
21o that (cid:90) R d d d x (cid:112) ( x ) x · b ( x ) = − (cid:90) R d d d x (cid:112) ( x ) x · ∂ x U ( x )= D (cid:90) R d d d x x · ∂ x (cid:112) ( x ) = − D d whereas (cid:90) R d d d x (cid:112) ( x ) b ( x ) = D (cid:90) R d d d x ∂ x (cid:112) ( x ) = D = D (cid:83) (cid:116) (cid:102) , (cid:116) (cid:105) = E (cid:90) (cid:116) (cid:102) (cid:116) (cid:105) d t (cid:107) v ( ξ t , t ) (cid:107) is proportional to the average entropy production of by the process { ξ t } t ∈ [ (cid:116) (cid:102) , (cid:116) (cid:105) ] afact which exhibits the interest of Fürth-Féynes uncertainty relations for contem-porary developments in stochastic thermodynamics. A.5 Continuity of Brownian motion
A qualitative understanding of the result based on the present-day theory of Brow-nian motion is as follows. Paths of a Brownian motion are with probability oneHölder continuous with exponent 1/2. This means that they are nowhere differ-entiable. As a consequence if one observes a Brownian particle with increasingresolution will magnify the resolution of the particle position but at the same timeregister an increase without bound of derivatives of the trajectory.
A.6 Acknowledgement to K. Löwner
It is intriguing to read that Fürth acknowledges K. Löwner for hints in the deriva-tion of the quantum uncertainty relation. Karel Löwner, also known as CharlesLoewner after emigration to the U.S., was a mathematician whose work on con-formal mappings led to the discovery of what is now commonly known as the“Loewner differential equation”. The stochastic extension of his work is the stochas-tic Loewner equation or Schramm-Loewner evolution (SLE) a family Markov pro-cesses describing interfaces in two dimensional critical systems. The study of SLEhas attracted a lot of attention both in the physics and mathematics communityover the last two decades see e.g. [13, 14].
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