aa r X i v : . [ phy s i c s . g e n - ph ] S e p An Ahistorical Approach to Elementary Physics
B.C. Regan
1, 2, ∗ Department of Physics and Astronomy,University of California,Los Angeles, California 90095,U.S.A. California NanoSystems Institute,University of California,Los Angeles, California 90095,U.S.A. (Dated: October 21, 2020)
A goal of physics is to understand the greatest possible breadth of natural phenomenain terms of the most economical set of basic concepts. However, as the understanding ofphysics has developed historically, its pedagogy and language have not kept pace. Thisgap handicaps the student and the practitioner, making it harder to learn and applyideas that are ‘well understood’, and doubtless making it more difficult to see past thoseideas to new discoveries. Energy, momentum, and action are archaic concepts repre-senting an unnecessary level of abstraction. Viewed from a modern perspective, thesequantities correspond to wave parameters, namely temporal frequency, spatial frequency,and phase, respectively. The main results of classical mechanics can be concisely repro-duced by considering waves in the spacetime defined by special relativity. This approachunifies kinematics and dynamics, and introduces inertial mass not via a definition, butrather as the real-space effect of a reciprocal-space invariant.
I. INTRODUCTION
Physics is customarily taught following the samechronological order in which it was developed. Studentsfirst learn the classical mechanics of Galileo and Newton,then the classical electricity and magnetism of Faradayand Maxwell, and then more modern topics such as spe-cial relativity and quantum mechanics. This approachhas pedagogical advantages, for it features a steady pro-gression of decreasing intuitiveness and increasing math-ematical sophistication.However, the opportunity cost is high. The chronolog-ical approach misses the chance to present the subjectfrom the clearest, most logically consistent perspective,weighed down as it is with so much historical baggage.Repeatedly the student is asked to unlearn concepts pre-viously presented as foundational in favor of new ones,which in turn are overthrown in the next course.Not only is physics not taught in a sequence that em-phasizes our best, present understanding, this perspec-tive is never presented. The standard physics curriculumnever gets around to saying, “if we were starting fromscratch, this is how we would do it.” While classicalphysics is known to be only an approximation of moremodern theories, we persist in using its dated vocabu-lary, even in, for example, quantum mechanical situa-tions, where it is ill-suited to the purpose. Of course, weexperience the world as classical beings, and thus benefitfrom an understanding of the approximate theories. But ∗ [email protected] it is possible, and indeed easier, to access and admirethe full power of the approximate theories if they are notcloaked in classical language, which was developed wellbefore the underlying and unifying ideas were discovered.The goal here is to present a formulation of classicalmechanics that is concise and more closely based on ourcurrent, best understanding of physics as a whole. Un-like the chronological approach, this perspective does notrequire extensive revision as more advanced subjects areintroduced. With only two main ideas, both already partof the undergraduate physics curriculum, we can derivemost of classical mechanics (the exception being gravity,which even according to our best understanding requiresspecial treatment), and lay a firm foundation upon whichmore advanced topics (e.g. electricity and magnetism,quantum mechanics) can be based.No new predictions result. We cannot calculate any-thing that was previously incalculable. In fact, much ofour presentation reiterates what has already been writ-ten elsewhere. For instance, it has been noted previ-ously that, had quantum mechanics been discovered first,surely Planck’s constant would have been chosen to beunity (Abers, 2004; Sakurai and Napolitano, 2017), andthat quantum mechanics subsumes classical mechanics(Ogborn and Taylor, 2005). However, the shape and con-sequences of an ahistorical development of classical me-chanics seem to be relatively unexplored. As we willshow, the logical underpinnings of classical mechanicscan be recast by exploiting the developments of the earlytwentieth century. This route abbreviates the lexiconthat other physics theories subsequently adopt — by notstarting with Newton, we discover that much of his vo-cabulary is unnecessary. It also provides a satisfyingsimplification, one that removes unnecessary definitions,convoluted developments, and unmotivated postulates.The resulting edifice offers some insight, and the pleasurethat comes from seeing old ideas from a new perspective(Feynman, 1948). II. SPACETIME
The spacetime of special relativity serves as thearena. We follow the customary development ofthe subject (Goldstein et al. , 2002; Jackson, 1999;Taylor and Wheeler, 1992), postulating that the rules ofphysics are the same when viewed from any inertial refer-ence frame, and that, in particular, all inertial observersobserve the same value c for the speed of light in vac-uum. Applying this idea to simple physical systems (e.g.a pulsed light source with a mirror on a train) gives, viabasic geometry, the Lorentz contraction and time dilationeffects. From these we derive the Lorentz transforma-tions and the idea of an invariant, the spacetime interval s , which is seen to have the same value by all inertial ob-servers. In a standard 4-vector notation with x µ = ( ct, r )we have, x µ x µ = c t − r = s , (1)where x µ indicates a location, or ‘event’, in spacetime interms of its time t and its position r . We have chosento put the c with the time t , so that every term in thisexpression has the dimension of length-squared.Defining the speed of light c allows us to describeevents using just one dimensioned unit, and to more eas-ily treat spacetime as a unified whole. Physicists workingwith very large length scales, e.g. astronomers, tend tomeasure distances using a time unit: the light-year. Oth-ers are generally better off measuring times in distanceunits, simply because short distances have a greater num-ber of familiar landmarks. Outside of the ultrafast com-munity, an angstrom and a fermi are more recognizableas atomic and nuclear size scales, respectively, than theirnear counterparts the attosecond and the yoctosecond.To lay the groundwork for future developments, it isimportant to emphasize here that, given the strong geo-metric basis for Eq. 1, it would never occur to anyone tomeasure its right-hand side in terms of a new, dimension-ful unit. That the customary units designating time ( t )and place ( r ) differ by a factor of c is already regrettable.Introducing yet another unit specifically to describe theinterval s would only further muddle what ought to bea straightforward geometric (i.e. pseudo-Pythagorean)relationship. But, as we will see, just such a muddlingoccurs when the dynamic counterpart of Eq. 1 is pre-sented in its traditional form. III. SPACETIME CONTAINS WAVES
We postulate that spacetime contains objects that wewill, in a convenient shorthand, refer to as waves. (Forfurther discussion of this choice, see Appendix E.) In ad-vanced treatments these objects might be described morecarefully as, for example, fields or wave functions, but forour present purposes such specificity is unnecessary. Infact, we do not even need to specify a particular waveequation, only that these waves have a phase φ of theform − φ = f t − k · r , (2)as is found in the solutions u ( φ ) to the generic wave equa-tion − v ∂ u∂t + ∇ u = 0 , (3a)with the phase velocity v ≡ f / | k | . For our present pur-pose of reformulating classical mechanics, the solutions ofinterest are wave packets that are well-localized in realand reciprocal space, e.g. Gaussian functions viewed ata zoom level such that their widths are negligible. Thissecond assumption (waves) clearly has deep connectionsto the first (special relativity): Eqs. 3a and 1 are struc-turally similar, Eq. 3a is Lorentz invariant, and Eq. 3adoes not permit instantaneous action-at-a-distance (un-like, e.g. diffusion-type equations).The phase φ ( x µ ) is a number (defined here with anoverall minus sign for consistency with a historical con-vention to be introduced later) associated with a particu-lar event x µ = ( ct, r ) in spacetime. It can be determinedby a counting operation, and thus all inertial observersagree upon its value (Dirac, 1933; Jackson, 1999; Symon,1971; Wichmann, 1971). Therefore φ is a Lorentz invari-ant, and it can be written φ ( x µ ) = − k µ x µ , where wehave defined a new 4-vector k µ ≡ ( f /c, k ).Since both s = x µ x µ (Eq. 1) and φ = − k µ x µ (Eq. 2)are Lorentz invariant, x µ and k µ must have the same be-havior under a Lorentz transformation. These transfor-mation properties then ensure that the 4-vector product k µ k µ is also a Lorentz invariant. Thus k µ k µ = ( f /c ) − k = k C , (4)where the quantity k C , which at this point is just a num-ber seen to have the same value by all inertial observers,will shortly be identified as the Compton wavenumber.Equation 4 is the reciprocal-space, or Fourier-space,analog of Eq. 1. Although we have not yet introducedenergy, momentum, or mass, Eq. 4 contains the physicalcontent of Einstein’s famous energy-momentum relation.Multiplying through by ( hc ) , where h is Planck’s con-stant, on both sides of Eq. 4 puts this equation in itstraditional form: p µ p µ c = E − p c = m c , (5)where we have defined the four vector p µ = ( E/c, p ), aswell as three ‘new’ physical quantities: the energy E ≡ hf , the momentum p ≡ h k , and the invariant (or rest)mass m ≡ hk C /c .These three relationships (Table I) between waveparameters and their corresponding classical variableswere discovered and introduced separately through thePlanck-Einstein relation (1900–1905), the de Broglie hy-pothesis (1924), and the aforementioned Compton for-mula (1923), respectively. Typical pedagogy does notapply the Planck-Einstein relation to massive objects,and connects these three ideas only to the extent thatthey were all instrumental to the development of quan-tum mechanics. In particular, it does not present theunifying, geometric picture implied by Eq. 4.The path to Eq. 4 as presented here is fast and di-rect: find the invariant length in real space, introducethe Fourier-conjugate variables, and find the equiva-lent invariant length in reciprocal space. In compari-son, the traditional route to Eq. 5 is lengthy and cir-cuitous. One first elevates momentum conservation tothe status of a foundational principle. Then, work-ing backwards from Newton’s laws and employing thework-energy theorem, one reverse-engineers definitionsof momentum and energy that satisfy this principle(Eisberg and Resnick, 1985; Kleppner and Kolenkow,2014; Taylor and Wheeler, 1992; Thornton and Marion,2004). With a little more effort, one can obtain thesame result with momentum and energy conservationalone (Jackson, 1999; Symon, 1971). These argumentsare subtle (it is not immediately obvious that momentumis something other than merely that-quantity-which-is-conserved) and nearly circular (they justify the exact lawby appeal to the inexact). These traditional approachesalso rely on an unnecessarily large number of axioms. Aswe will see in Section IV, we can derive energy conser-vation, momentum conservation, and Newton’s laws allfrom Eq. 2.Furthermore, getting to the traditional Einsteinenergy-momentum relation (Eq. 5) via reciprocal spacereveals that the final step — multiplying through by ( hc ) — has no physical content, and in fact buries some valu-able ideas. First, the Fourier-transform relationship be-tween the conjugate pairs ( t, f ) and ( r , k ) is obscured inthe ‘new’ units. Second, mass has been defined such thatit agrees dimensionally with neither E nor p , thus hid-ing what would otherwise be a straightforward pseudo-Pythagorean relationship in reciprocal space. In the lan-guage of Taylor and Wheeler’s parable of the survey-ors (Taylor and Wheeler, 1992), we are measuring north-south distances in joules, east-west distances in kg · m/s,and the distance between two points in kilograms. ThusEq. 5, one of the most famous equations in all of physics,has not just three units (in the sense of meters and feet),but three different dimensions (in the sense of joules andkilograms) for its three terms, and could not more thor- oughly obfuscate its geometric content. It is small won-der that physics is considered an arcane discipline.Equation 5 might seem to have more content thanEq. 4, or at least to be more intuitive. In fact, energyand momentum are intuitive only to those who have beensuitably trained — one might say indoctrinated — intheir use. These quantities are actually relatively ab-stract, since they cannot be measured directly, even inprinciple. The lack of experimental access is a substan-tial failing. To appreciate the advantages of explicitlyconsidering the means of measurement, recall that a keyadvance in the development of special relativity was thesedeliberate definitions: time is what clocks measure, anddistance is what rulers measure. What then, are the cor-responding operational definitions (Hecht, 2017) of en-ergy and momentum? Such definitions do not exist; wedo not have meters for either energy or momentum. It isan interesting exercise to ponder how these quantities aredetermined experimentally in various situations. For me-chanics problems, we have only three basic measurementoperations: counting, measuring position (with rulers),and measuring time (with clocks). Some combination ofthese three basic measurements determine other physi-cal quantities : frequency (both spatial and temporal),angle, velocity, acceleration, energy, momentum, torqueetc. For example, to determine the momentum of a foot-ball, we would determine its mass and velocity separately,and then do a simple calculation. We might weigh thefootball with a scale, thereby determining its mass viaposition (and an auxiliary determination of the acceler-ation due to gravity), i.e. the deflection of the spring.And we might use clocks separated by a pre-measureddistance to determine the velocity.In the list of ‘physical quantities’ just given, the firsttwo items, spatial and temporal frequency, are in a dif-ferent category than the others. They are not less funda-mental than distance and time, in that distance and timecannot be defined without reference to frequency. To beuseful, clocks and rulers must contain some repeatingunit. A ‘clock’ must tick more than once, and ‘rulers’must either be subdivided into sub-units, or be applica-ble in multiple instances (e.g. laid end-to-end). Thuswe cannot measure time without access to a frequency,nor can we measure distance without access to a spatialfrequency. While the circularity of these base definitionsmight puzzle logicians and philosophers, this lexical issuecauses no difficulty for physicists in practice. It also em-phasizes the symmetry (in the sense of a lack of prece-dence) between the real-space (i.e. spacetime-domain)and the reciprocal-space (i.e. frequency-domain) vari-ables. Since number is determined by counting, from a physics stand-point we consider it to be well-defined. Mathematically its def- kinematic dynamicGalileo Lagrange NewtonHamiltonreal space Fourier reciprocal space × h traditionaltime ct ⇐⇒ temporal frequency f/c Planck-Einstein energy
E/c = hf/c position r ⇐⇒ spatial frequency k de Broglie momentum p = h k Pythagoras, Einstein, Minkowskiinterval s ⇐⇒ Compton wavenumber k C Compton mass mc = hk C event x µ wave k µ object p µ TABLE I Every event in real space has three distinct properties: a time, a position, and an interval. The constant ‘ c ’accommodates the pseudo-Pythagorean relationship between these properties by putting them in the same units. The constant‘ h ’, on the other hand, conceals the Fourier relationship between the kinematic and the dynamic quantities by taking themout of reciprocal units. As for the dynamic quantities, every object has an energy, a momentum, and a mass, and every wavehas a frequency, a spatial frequency (or wavevector), and a Compton wavenumber a . The ‘object’ and the ‘wave’ pictures givephysically equivalent descriptions of the dynamic variables. Because ‘mass’ is just another name for the Compton wavenumber,which is a derived property of waves in spacetime, we can say that objects have mass because they are waves. a Time, position, frequency, and spatial frequency can all have well-defined values (i.e. with negligible spread) simultaneously in theclassical limit. See Appendix E.
Compared to energy and momentum, frequency andspatial frequency are much less abstract, in that they canbe measured directly. A frequency is measured by count-ing cycles in a known time. A spatial frequency is mea-sured by counting cycles in a known distance. Of course,practical considerations make such direct measurementshopeless for macroscopic objects such as footballs (seenote at the end of Section V).Replacing the archaic concepts of energy and momen-tum (Eq. 5) with frequency and spatial frequency (Eq. 4),respectively, anticipates questions that can arise at an el-ementary level and yet are difficult to answer in the oldlanguage. The modern development avoids the question,“how can photons have energy and momentum, if theydo not have mass?” Since a photon has zero mass (kg, inSI units) and non-infinite velocity (m/s), it is legitimateto ask how it can have non-zero energy (kg · m / s ) andmomentum (kg · m / s). The answer, we see here, is thatenergy and momentum are better thought of as repre-senting (inverse) time and length scales (i.e. frequencies), inition is just as circular — counting determines number, andnumber is what counting determines — but we view mathematicsas being on a firmer logical footing than physics. We understandnumber better than we understand time.Thus number is preeminent among the measurable quan-tities, as can be seen from the current (circular) statementdefining the units of time and frequency in terms of num-ber. With the ground-state hyperfine splitting of the cesium-133 atom as the standard oscillator, by definition one second isthe time required to complete a number (9,192,631,770) of cy-cles, and the oscillator’s frequency in hertz is that same number(Stenger and Ullrich, 2016).While classical mechanics treats both time and distance asmeasurable, more advanced physics makes the preeminence ofnumber manifest. Quantum mechanics treats time as a parame-ter while maintaining position as an observable. Quantum fieldtheory treats both time and position as parameters, which leavesnumber as the only measurable quantity remaining. respectively, which the photon does have. Mass too cor-responds to a length scale, and for the photon this lengthhappens to be infinite. Mass was given its own, dimen-sioned unit long before the connection between mass andlength was understood, and applying the historical lan-guage to a modern concept such as the photon leads tothis inconsistency.Thus, assuming waves in the spacetime of special rel-ativity gives us mass. In other words, objects havemass because they are waves (Table I). This statementis exactly counter the orthodox presentation — usuallythe wave nature of a massive object is considered toarise despite its mass. The conflict in the concept of‘wave/particle duality’ that is so often emphasized ispartly an illusion.Having arrived at the concept of mass via this route(i.e. Fourier conjugate variables) further encourages usto re-examine the various classical conceptions of mass.Historically mass has been considered “a quantity of mat-ter”, a disinclination to accelerate (i.e. inertia), or ameasure of participation in the gravitational interaction(Hecht, 2010). These ideas date back to Newton andhave been continuously refined in the intervening cen-turies, but none of them is entirely satisfactory (Coelho,2012; Hecht, 2005, 2010). As we will show shortly, iner-tial mass does not need to be introduced as a postulate,since the inertial effect can be derived from other (wave)axioms. Moreover, instead of having inertia defined inrelation to the positions of the “fixed stars”, as Machwould have it, inertia is defined in relation to positionsin reciprocal space, a place much less remote. The in-ertia of a mass is not “an effect due to the presence ofall other masses” (Pais, 2005), but rather a real-space ef-fect arising from an invariance constraint (namely Eq. 4)in reciprocal space. Distant, massive objects are not re-quired; Eq. 4 and the equations that follow (Section IV)suffice to derive the inertial effect.Thus we can take Eq. 4 to define mass: mass is theinvariant reciprocal length associated with a wave of fre-quency f and wavevector k (times h/c in traditionalunits). This definition is exactly analogous to that of theinterval: the interval is the invariant length associatedwith an event at time t and position r . Both definitionsare closely tied to the structure of spacetime, and neitherrelies on the concept of ‘force’, which is both difficult todefine and almost ignored by advanced physics theories(e.g. quantum mechanics and quantum field theory).The modern, wave-based approach further suggestsa possible strategy for addressing a long-standing openproblem, namely implementing an operational definitionof mass (Hecht, 2005). Intervals are generally determinedwith rulers and clocks (Eq. 1), and, in the specific caseof a purely time-like separation, an interval can be mea-sured with just a clock. A mass is the reciprocal-spaceanalog of a purely time-like separation (Eq. 4), which sug-gests that a mass might be determined by measuring itsCompton frequency. In fact, the use of ‘rocks as clocks’has been proposed (Lan et al. , 2013), although the ideais not yet well understood (Pease, 2013; Schleich et al. ,2013; Wolf et al. , 2012). IV. STATIONARY PHASE
It is hardly an exaggeration to say that the rest ofphysics, excepting thermodynamics, will be built aroundthe phase φ . One more postulate — that φ is invari-ant under ‘gauge’ (or phase) transformations — willbe used to introduce interactions into the theory. In-variance under local U (1) gauge transformations givesthe electromagnetic interaction. Invariance under local SU (2) and SU (3) gauge transformations leads to theweak and strong interactions respectively (Cheng and Li,1991; Griffiths, 2010; Halzen and Martin, 1984).Postponing these more advanced topics, we can de-rive the results of classical mechanics by invoking anextremum principle: the principle of stationary phase.We postulate that classical trajectories or paths in space-time are determined by the condition that φ is station-ary along the trajectory (Cohen-Tannoudji et al. , 1977;Commins, 2014; Dirac, 1967; Peskin and Schroeder,1995; Sakurai and Napolitano, 2017), i.e. that δφ = 0.Trajectories not satisfying this condition are suppressedby destructive interference with neighboring trajecto-ries; minute changes in the trajectory give an ampli-tude of the opposite sign. This idea, which is moti-vated by the wave picture, ultimately leads to the pathintegral formulation of quantum mechanics (Feynman,1948; Feynman et al. , 2010, 1964; Itzykson and Zuber,1980; Styer et al. , 2002). Classically it is knownas the principle of stationary action (Ogborn et al. ,2006) (or, in some nomenclatures, Hamilton’s principle(Feynman et al. , 1964; Landau et al. , 1976)), and it is generally presented as a postulate that has no underly-ing physical motivation, other than that it gives the de-sired equations of motion (Dirac, 1967; Feynman et al. ,2010, 1964; Goldstein et al. , 2002; Itzykson and Zuber,1980; Landau et al. , 1976; Peskin and Schroeder, 1995;Sakurai, 1967; Schiff, 1968; Thornton and Marion, 2004).Unlike the Planck-Einstein relation, the de Broglie hy-pothesis, and the Compton formula (Table I), the relation S = hφ that connects the wave parameter ( φ , the phase)and the classical quantity ( S , the action) is not wellknown, with some authors declining to state it clearlywhen the opportunity arises (Peskin and Schroeder,1995; Sakurai and Napolitano, 2017).Having established that considerations of interferencemotivate the principle, we can derive the classical equa-tions of motion by considering the phase φ as it de-velops along a path, and then applying the calculus ofvariations. Then the derivation, based on the ‘modi-fied Hamilton’s principle’, is standard (Goldstein et al. ,2002; Landau et al. , 1976; Thornton and Marion, 2004),but for the fact that the phase φ , the frequency f , andwavevector k are taking the places of the action S = hφ ,the energy E = hf (or, better, the Hamiltonian H = hf ),and the momentum p = h k , respectively (Table II). The phase developed along a path from ( t a , a ) to ( t b , b )is given by φ = Z path k d r − f dt, (6)which can be converted into an integral over time aloneby defining the velocity ˙ r ≡ d r /dt and making the othervariables parametric functions of time, which gives φ = Z t b t a ˙ φ dt = Z t b t a ( k˙r − f ) dt. (7)(In traditional notation this expression reads S = R L dt = R ( p˙r − H ) dt , where L is the Lagrangian.) Theextremum condition is enforced by setting δφ = 0. Ap-plying the chain rule then gives δφ = Z t b t a ( δ k ˙r + k δ ˙r − δf ) dt. (8)Interchanging the order of operations in the second termgives k δ ˙r dt = k d ( δ r ), which can be integrated by parts: Z k d ( δ r ) = k δ r | ba − Z δ r d k . (9) When viewed from a historical perspective, this equivalence be-tween classical mechanics and an underpinning wave theory isunsurprising, given that Hamilton developed his formulation byanalogy with optics (Symon, 1971).
We have defined the path to begin at a and end at b .While the path varies by δ between a and b , at the end-points δ = 0, so the surface term vanishes by construc-tion. Thus we have δφ = Z t b t a ( δ k ˙r − δ r ˙k − ∂f∂ k δ k − ∂f∂ r δ r ) dt, (10)where we are taking f to be a function of the positioncoordinates r , the corresponding spatial frequencies k ,and the time t . Collecting terms gives δφ = Z t b t a (cid:18) δ k ( ˙r − ∂f∂ k ) − δ r ( ˙k + ∂f∂ r ) (cid:19) dt. (11)Because the variations δ r and δ k are independent, theirpre-factors must vanish to guarantee δφ = 0. Thus wehave two relations, ˙r = d r dt = ∂f∂ k and (12a) ˙k = d k dt = − ∂f∂ r , (12b)which are the canonical equations of motion, one for realspace and one for reciprocal space, respectively. Multi-plying the first by h/h and the second by h puts them inHamilton’s form: ˙r = d r dt = ∂H∂ p and (13a) ˙p = d p dt = − ∂H∂ r , (13b)which define the group velocity and relate the time rateof change of the momentum p to the gradient in theHamiltonian H , respectively. With the force F definedas the negative of that gradient, Eq. 13b is Newton’ssecond law. (By definition the potential energy U is theposition-dependent portion of H , so ˙p = − ∂H∂ r = − ∂U∂ r = −∇ U ≡ F .)In traditional language, one says that Newtonian me-chanics is equivalent to the statement that the classicalphysical path extremizes (usually minimizes) the action S (Abers, 2004). In modern language, one says that theclassical path extremizes the phase φ , a property intrinsicto waves. Far from being at odds with wave physics, as isusually implied, classical mechanics follows directly fromthe most basic and general elements of the wave picture.Some comment is required here, since, by writing φ asan integral (Eq. 6) instead of the simple product (Eq. 2),we have passed from a discussion of free waves (Eqs. 4–5)to one that can accommodate a potential (Eqs. 12–13).At the level of fundamental particles, interactions areintroduced by postulating the gauge symmetries men-tioned earlier and considering one-on-one interactions.For macroscopic objects the number of waves involvedmakes this approach impractical, and the sum interac-tion of many-on-one, or many-on-many is described phe-nomenologically by adding a potential term that gives k and f , or, equivalently, the Hamiltonian H , some depen-dence on (most commonly) position. Usually this poten-tial term is not obviously Lorentz invariant, as the coor-dinate dependence is written with respect to a preferredreference frame. And it may show no sign of its originas a Lorentz invariant gauge interaction. For instance,the forces that maintain the tension in a pendulum’sstring, or that push back when a spring is compressed, areat some level electromagnetic, but in these phenomeno-logical descriptions electric charges and electromagneticfields never appear. Nonetheless the dynamics that followfrom the canonical equations of motion (Eqs. 12–13) canbe understood as ultimately arising from the symmetriesof φ under gauge transformations.Also, for simplicity of exposition, to indicate the real-space position we have been writing r , which in a Carte-sian system decomposes to r = x ˆx + y ˆy + z ˆz . Thequantity ˙ φ , the equivalent of the Lagrangian in thismodern picture (i.e. L = h ˙ φ ), can be written alter-natively in terms of generalized coordinates q i , wherethe q i are any set of quantities that completely spec-ify the state of the system. Then the wave-equivalentof the generalized momentum canonically conjugate to q i is k i ≡ ∂ ˙ φ/∂ ˙ q i . In other words, instead of ˙ φ ( r , ˙ r , t )we can write ˙ φ ( q i , ˙ q i , t ) without invalidating any ofthe arguments presented above (Goldstein et al. , 2002;Landau et al. , 1976; Thornton and Marion, 2004). Thisflexibility, one of the main advantages of the Hamilto-nian and Lagrangian formulations relative to the New-tonian formulation of classical mechanics (Symon, 1971),is of course unimpaired by the switch to the wave-baseddescription. (If some q i depend explicitly on t , or theinteraction-derived component of f depends on ˙ q i , thenthe Hamiltonian is not equal to the total energy, whichleads to the preference expressed parenthetically aboveEq. 6 (Thornton and Marion, 2004).)The equivalents of energy and momentum conser-vation, and other consequences of Noether’s theorem(Goldstein et al. , 2002; Hanc et al. , 2004) (e.g. angu-lar momentum conservation), follow immediately fromEqs. 12. The total derivative of f ( r , k , t ) with respect totime is dfdt = ∂f∂ r ˙r + ∂f∂ k ˙k + ∂f∂t . (14)The first two terms cancel by Eqs. 12. Thus if the fre-quency f has no explicit time dependence, df /dt = 0and f is a conserved quantity. Similarly, by Eq. 12b,if f is independent of some component q i of r , then˙ k i = − ∂f /∂q i = 0 and k i , the component of the spatialfrequency (or wavevector) conjugate to q i , is conserved. V. HISTORICAL PERSPECTIVE
Let us compare the traditional and the wave-based de-velopments of classical mechanics. When Newton wrote traditional waveaction S = h × φ phaseLagrangian L = h × ˙ φ phase time derivativeHamiltonian, energy H , E = h × f frequencymomentum p = h × k spatial frequency,wavevectorangular momentum J = h × J angular momentumforce F − ∂H∂ r = h × − ∂f∂ r mass m = hc × k C Compton wavenumberTABLE II Traditional variables and their wave-based coun-terparts. In ‘wave’ units J is taken to be dimensionless. Thetheory can be completely specified in terms of wave parame-ters. To the extent that the concepts on the left are distinctfrom those on the right, none of them are necessary. F = m a , he was essentially defining the key concepts offorce F and mass m in a simultaneous, circular bootstrap.Operating in complete ignorance of both the space-timeconnection and the wave properties of matter, Newtonhad little to work with. He had to invent his dynamicalvariables out of nothing, and also to intuit useful rela-tionships between these variables. That his constructionsurvives, after more than three centuries of ceaseless sci-entific advances, as the practical description of everydayphysics is a testimony to the magnitude of his achieve-ment.The Lagrangian and Hamiltonian formulations of clas-sical mechanics, meanwhile, represent another leap in un-derstanding. These formulations make, via the action S ,the explicit connection between kinematic variables anddynamic variables that we now recognize as the phase φ = − k µ x µ , which sits between real space ( x µ ) and re-ciprocal space ( k µ ), taking equally from both (Table I).With the Planck-Einstein relation, the de Broglie hy-pothesis, and the Compton formula, the dynamical vari-ables are identified as wave parameters and the last con-nections are finally made.The wave-based formulation, on the other hand, doesnot require the incredible improvisations that produced F , m , and S . The analogous round of definition occurswith the introduction of f and k , but in this case thedefinitions follow directly from the wave postulate (oreven from the definitions of time and distance — seeAppendix E). Their meaning is already mathematicallyprecise: these quantities are conjugate to those variablesthat define the spacetime arena, t and r , and provide analternative but equivalent description of the arena’s con-tent via the Fourier transform (Appendix A). While thewave-based formulation does not remove circularity fromthe base definitions — see Section III and Appendix E —such an achievement is not to be expected (G¨odel, 1931;Tarski, 1936). And progress is evident, in that by deriv-ing inertial mass we have removed this important con-cept from the circle of definitions. The progression fromthe definitions is also more orderly, proceeding from real space ( x µ ) to the boundary ( φ = − k µ x µ ), and then toreciprocal space ( k µ ), instead of via a giant leap to thedynamical variables followed by a tortuous journey back.Returning to the discussion of measurement from Sec-tion III, we see that identifying the wave origins of theaction S moves this quantity from the realm of the ab-stract to that of the very concrete, for now we can give itan operational definition. Namely, the action is definedby S ≡ hφ , where φ is the number of cycles in a wave,relative to some arbitrary origin. This definition can becompared to that of time t , which is the number of ticksof a clock, relative to some arbitrary origin, and position r , which is the number of ruler units in three dimensions,relative to some arbitrary origin. Unlike time and dis-tance, however, the phase definition is self-contained, inthat the counting operation does not rely on the circular(e.g. time-frequency) bootstraps discussed in Section III.In this sense, the action S , far from being abstract anddifficult to understand, is, via its connection to the phase φ , preeminent in the list of well-defined and measurable quantities. (Here we are speaking from a perspective thatis both ‘in principle’ and strictly classical, in that we areignoring both problems having to do with the large sizeof φ in all but the smallest systems, and those havingto do with quantum measurement (Bassi and Ghirardi,2003; Commins, 2014; d’ Espagnat, 1999; Jammer, 1966;Lalo¨e, 2019; Lamb and Fearn, 1996).)Thus the wave-based development requires neither adhoc definitions nor unmotivated postulates (i.e. δS =0). Three centuries of effort have done more than builda theory that is more accurate and broadly applicable.They have also revealed underlying principles that makethe theory more coherent and easier to understand. VI. CONSTANTS AND CONCEPTS
Traditional pedagogy gives the impression of separateregimes: small systems have wave-like properties, whilelarger, classical systems do not. The development pre-sented here shows that these regimes are not so sep-arated. The subject of classical mechanics is waves:well-localized waves with unmeasurable phases. In otherwords, classical mechanics treats the limit where the wave‘packets’ have phases that are uncountably large andwidths that are negligible in comparison to the otherlength and frequency scales in the problem (see Appen-dices C and E).By invoking only three closely-related ideas — specialrelativity, waves, and an extremum principle — we havereproduced the exact (i.e. relativistically correct) dis-persion relations, equations of motion, and conservationlaws of classical mechanics. Mass has not been insertedas a distinct postulate of the theory, but has emerged asa geometric property of waves in Minkowski spacetime.We find that the traditional dynamical quantities (Ta-ble II) such as action, energy, and momentum are waveproperties cloaked in units that disguise these origins.According to this viewpoint, c and h are not ‘fun-damental’ constants, as they are often called, but ves-tiges of a more primitive (Newtonian) understanding ofthe underlying physics. Time, distance, energy, momen-tum, and mass were introduced as distinct aspects ofphysical theory before the interrelationships were under-stood. When, for instance, mass and time seemed un-related to distance, each was measured in its own dis-tinct units. Later the relationships were discovered, andconstants were introduced to describe the conversions(Abers, 2004).Mass is a particularly interesting case, for the follow-ing reason: we experience everyday life in real space.This fact makes it remarkable, and perhaps even counter-intuitive, that the real-space constraint described by theinterval (Eq. 1) is not noticeable without scientific equip-ment, while the reciprocal-space constraint correspond-ing to mass (Eqs. 4 and 5) has obvious real-space mani-festations. Thus the interval s was not measured beforeit was discovered theoretically — if it had been knownpreviously, a special unit would have been invented to de-scribe it! But mass has been known since Newton, and itsproxy, weight, has been measured since antiquity. Thusspecial units were invented to quantify mass before itsrelationship to other physical quantities was understood.Despite the misleading language, it is well (if not uni-versally (Eisberg and Resnick, 1985; Inglis et al. , 2019;Young et al. , 2020)) known that the ‘fundamental con-stants’, c and h , are merely conversion factors set by hu-man convention (Abers, 2004; Duff, 2015; Ralston, 2012,2013; Stenger and Ullrich, 2016; Taylor and Wheeler,1992). The revised International System of Units, orSI, of 2018 recognizes the distinction by purposefullyreferring to them as ‘defining’, as opposed to ‘funda-mental’, constants (Stenger and Ullrich, 2016). In 1983and 2018 mortal scientists defined the exact values c ≡ , ,
458 m/s and h ≡ . × − kg · m /s,respectively (Inglis et al. , 2019). The value of c sets therelationship between the marks on the face of a clock andthe marks on a ruler. The value of h sets the relationshipbetween those marks and the marks on a scale, a bal-ance, or a reference mass. Great pains have been takento define these constants so as to maintain compatibilitywith the historical values, and to take full advantage ofthe best available metrology (Bord´e, 2005; Inglis et al. ,2019). But the values human scientists choose to recog-nize as c and h would not be recognized by alien scien-tists. In that sense they are not fundamental, but ratherthe nearly random products of human custom and his-tory.Adjusting these constants would be like getting rid ofgallons, pounds, and feet in the United States: substan-tial economic and cultural — but no physical — con-sequences would result. It is sometimes implied that relativistic and quantum effects are not easily accessedbecause c is large and h is small, and that, with a god-like power to adjust these constants, we could makeour universe appear less classical to our un-aided senses(Gamow, 1964). Such statements depend upon non-obvious assumptions, seldom given explicitly, on whatconstraints are enforced as these constants are adjusted.But if the values of these constants are immaterial, whythen do we experience the world classically? The answeris that, as complex, sentient beings, we consist of manyfundamental particles. Any being sophisticated enoughto do science or metrology must necessarily have a hugenumber of degrees of freedom. The question, “why isPlanck’s constant so small?” would be better phrased,“why is the kilogram so big?” The answer is that thekilogram consists of a number of atoms that has been(somewhat arbitrarily) chosen to represent a quantity ofmatter that humans can easily access and manipulate.Planck’s constant is small and the kilogram is big be-cause we consist of very many — more than Avogadro’snumber N A ≡ . × (Inglis et al. , 2019) —fundamental particles. The Planck’s constant’s small nu-merical value is not “responsible for the fact that quan-tum phenomena are not usually observed in our everydaylife” (Gamow, 1964). Rather, quantum phenomena arenot usually observed because we consist of many fun-damental particles. The small value for Planck’s con-stant represents a choice that humans have made, andthis choice has no impact on any physical phenomenon.For objects consisting of few fundamental particles,large boosts and coherent phase control are achievablewith current technology, which allows access to obvi-ously relativistic and quantum-mechanical phenomena,respectively. As the number of potentially independentdegrees of freedom grows, however, it becomes progres-sively more difficult to reach the non-classical regimes.Many particles might be boosted to relativistic velocitiesvia a nuclear explosion, for instance, but such an event isgenerally hostile to the coherences that maintain a com-posite object as a unified whole. Quantum phenomenaare inaccessible to many-particle objects both becausethe objects are large in comparison to their wavelengths,and because objects with many degrees of freedom cannotbe sufficiently well-isolated from the decohering effects ofinteractions with the environment (Cronin et al. , 2009).Thus making an observationally less-classical universewould be substantially more involved than simply ad-justing c and h — somehow the observers (e.g. Gamow’sMr. Tompkins) must be constructed from far fewer con-stituent parts.While the numerical value of Planck’s constant h hasno physical significance, we can attribute a significance toits value as the ‘quantum of action’ (Messiah, 2014), thevery existence of which puzzled Planck himself and manyothers since (Commins, 2014; Jammer, 1966). Here wesee that one quantum of action, i.e. S = h , correspondsto a phase φ = 1 cycle (or 2 π radians), which could hardlybe more natural. After all, phase is introduced (Eq. 2) asa countable quantity, and the natural units for countingphase are cycles (see also Appendix A). From the waveperspective, the quantization that accompanies the quan-tum of action also becomes less mysterious. For instance,if energy and frequency are considered to be distinct con-cepts, one can ask why photons of frequency f necessar-ily carry energy quantized in increments of E = hf . Butdoes it make sense to ask why photons of frequency f carry frequency f ?‘Natural’ unit systems where ~ = c = 1, suchas are sometimes employed by particle physicists andcosmologists (Halzen and Martin, 1984; Jackson, 1999;Peskin and Schroeder, 1995; Sakurai, 1967; Tanabashi,2018), bear a resemblance to the wave-based approachpresented here. Unfortunately, however, such systemsare not employed at the elementary level, and commonimplementations fail to reap many of the possible bene-fits. For instance, the electron-volt ( e V), instead of thesecond or the meter, is often chosen as the base unit. Thischoice is confounding on several levels. As an energy unit,the e V base unit obscures the Fourier transform relation-ship between physical quantities, such as, for instance,a resonance energy width and the corresponding particlelifetime. The use of the e V is usually restricted to recipro-cal space, which further hides the real-space/reciprocal-space connection; adherents to this system rarely quotetimes or distances in e V − (Halzen and Martin, 1984;Sakurai, 1967; Tanabashi, 2018). But this particular unitalso references electromagnetism. Earlier we remarkedon the logical problems with using energy units to de-scribe the massless photon. Describing particles such asthe neutrino, the π , and the neutron, which have noelectronic charge, in terms of the electronic charge e isequally inelegant. Energy, momentum, and mass — orbetter, the corresponding frequencies — are general con-cepts that apply outside the more limited scope of elec-tricity and magnetism, and as such should not be tied toelectromagnetic units.Another deficiency with ~ = c = 1 unit systems, asusually implemented, is not the units per se , but ratherthe vocabulary. These systems maintain energy and mo-mentum as distinct concepts. Just as alien scientistswould not recognize our values for c and h , so too theymight not understand why we distinguish between energyand frequency. The distinction separating these conceptsis historical and cultural, not physical. In the wave-basedformulation of elementary physics, energy, momentum,and action are superseded. While not in the same cate-gory as phlogiston, caloric, and the luminiferous aether(in that they are not incorrect), these concepts are un-necessary physically and inefficient pedagogically. Theircontinued use should be discouraged. In the wave-basedformulation they never appear, and neither does the con-version factor they require. The value of Planck’s con- stant, unity or otherwise, is irrelevant.As a conversion factor, Planck’s constant h is moredifficult to defend than the speed of light c . The speedof light connects time and distance. These conceptsare physically and mathematically distinguishable: timehas an arrow, while space does not, and time-like co-ordinates appear with important minus signs relativeto their space-like counterparts in the invariants (Ap-pendix E). While alien scientists would not recognize ourvalue for c , they would at least recognize the distinctionswe make between time and space. Planck’s constant, onthe other hand, connects energy to frequency, momen-tum to wavevector, and action to phase. But no physicalor mathematical distinction can be made between thesequantities, other than the artificial one introduced byPlanck’s constant itself.As has been noted previously (Bord´e, 2005), the ex-istence of ‘fundamental’ constants with dimension indi-cates that we are calling the same thing by two differentnames. Some authors see no problem, emphasizing thatthe number of dimensional constants is arbitrary (Abers,2004; Jackson, 1999; Wichmann, 1971), while others re-gard dimensional constants as evil (Bridgman, 1931). Ata minimum having multiple names for one thing can beconsidered uneconomical, and therefore in violation of aprinciple that has a long tradition in physics. In his Prin-cipia
Newton himself declares (Newton et al. , 2016), asthe first rule of reasoning, thatNo more causes of natural things should beadmitted than are both true and sufficient toexplain their phenomena.As we have argued here, the Newtonian concepts of en-ergy, momentum, and action are not necessary to explainmechanical phenomena — the more elementary conceptsof frequency, spatial frequency, and phase are fully suf-ficient to construct a complete explanation. Thus New-ton’s principle of economy dictates that the unnecessaryconcepts should not be admitted as “causes”. Addingthese concepts to the theory is like adding unnecessarylines to a drawing or unnecessary parts to a machine, tore-purpose a famous analogy (Strunk and White, 2009).Jettisoning energy, momentum, action, and the multi-tude of conversion formulas involving Planck’s constantleaves behind a leaner and cleaner — but equally capable— physical theory.Replacing historical language with wave language givesanother perspective on familiar ideas. On first contactthis perspective might seem helpful, or might not. Forinstance, the statementThe energy describes the rate at which theaction of an object evolves as a function oftime.is quite abstract, and does not evoke a physical picture.In wave language this statement reads0The frequency describes the rate at which thephase of a wave evolves as a function of time.which is easy to visualize. On the other hand, making asimilar substitution inThe total energy is the sum of the potentialenergy and the kinetic energy.gives The total frequency is the sum of the poten-tial frequency and the kinetic frequency.which might initially seem bizarre. That wave languageclarifies in some instances, and appears strange in others,can be taken as prima facie evidence that the historicalvocabulary is hiding potentially valuable ideas.For instance, Feynman in his eponymous lectures intro-duces energy as an abstract-but-conserved quantity. Asrecorded with his own emphasis, he says (Feynman et al. ,1964) It is important to realize that in physics to-day, we have no knowledge of what energy is .From the traditional perspective this statement is per-fectly reasonable — Newton’s ideas offer no underlyingfoundation for the concept of energy. However, it is dif-ficult to imagine Feynman sayingIt is important to realize that in physics to-day, we have no knowledge of what frequency is .These statements are physically equivalent, and thus ifthe second one is wrong, the first one must be too.Twentieth-century developments not available to New-ton provide good insight into what energy is. To withina ‘trivial’ unit conversion, energy is frequency, whereenergy/frequency describes the rate at which the ac-tion/phase of an object/wave evolves in time.The traditional and the modern, wave-based develop-ments can be contrasted as follows. In the former, thetheory is constructed on the makeshift foundation formedwith Newton’s circular (or nearly so (Feynman et al. ,1964)) definitions. When the Planck-Einstein, de Broglie,and Compton relations are introduced as 20 th centurydevelopments, they appear rich but unconnected. With-out any particular organization or underlying, unifyingprinciple, it is not obvious, for instance, whether thede Broglie relation is better thought of as p = h k , or p = h/λ . Both make the wave connection, but onlythe former makes the important, additional link to theFourier conjugate variable. The wave-based develop-ment, on the other hand, has a sturdier foundation,with independent and mathematically precise definitions.Here the Planck-Einstein, de Broglie, and Compton rela-tions appear as discoveries that connect unit conversionson the three sides of a triangle. While historically impor-tant, they are physically trivial. VII. SCALES
As a practical matter, and to help internalize somesense of the scales involved, it is good to know a fewconversion factors. To an accuracy of better than 1%,energies and distances are inter-converted with hc =1240 e V · nm = 2 × − J · m. While e V · nm are usuallyconvenient in condensed matter and atomic physics, tobetter fit the typical energy and length scales one mightprefer the equivalent units m e V · µ m in biological physics,k e V · pm in electron microscopy, or M e V · fm in particlephysics.Masses are inter-converted with h/c = 2 . × − kg · m and c /h = 1 . × Hz/kg. The fantas-tic size of these numbers reflects the human scale and itscomplexity. It is curious to contemplate how such scalesare simultaneously both inaccessible (from a direct mea-surement standpoint) and commonplace (as features ofeveryday life). We rarely discuss, for example, the Comp-ton wavelengths of macroscopic objects, because they areso small so as to seem unphysical. The Planck length ℓ P ≡ p ~ G/c ∼ . × − m, which corresponds tothe Planck mass m P ≡ p ~ c/G ∼ µ g, is sometimesthought (Commins, 2014; Garay, 1995; Hossenfelder,2013; Misner et al. , 1973; Peskin and Schroeder, 1995)to represent a minimal length scale, for instance. Buta mere kilogram, never mind an astrophysical object,represents a length scale that is far smaller. Thus thesampling theorem (Bracewell, 2000) argues against anysimple interpretation of the Planck length in terms of adiscretization of spacetime. VIII. CONCLUSION
Physics is currently taught with a historical bent thatfails to embrace ideas that have been well-establishedfor nearly a century. Planck’s constant is taken to bethe signature of quantum mechanical modernity, when itshould be viewed as a vestige of ‘antediluvian’ language(the flood here being the development of relativity andquantum mechanics) and a more primitive understand-ing. The traditional development of classical mechanicsis ad hoc, convoluted, and conflates the exact with theapproximate. Newtonian ideas and language that are in-troduced during this development (e.g. energy and mo-mentum) permeate the field from top to bottom. Thesecultural and linguistic artifacts make physics more diffi-cult than necessary.An ahistorical, wave-based formulation of ‘classical’mechanics, on the other hand, is more systematic andeconomical. Geometry and symmetry play clear andcentral roles. A one-to-one correspondence exists be-tween the kinematic variables r and t and their dynami-cal counterparts, k and f . The equations of motion fol-low from an extremum principle that is physically moti-1vated. Mass is not inserted into the theory by hand, butemerges naturally from the geometry of spacetime. Moreadvanced theories, e.g. quantum mechanics, extend themore elementary material, instead of discrediting it. Inshort, the advances of the 20 th century streamline andunify physics from the 17 th century to the present, sothat it is no longer necessary or desirable to so conscien-tiously retrace the exploratory steps taken by the earliestdevelopers of the theory. IX. AFTERWORDA. Waves and the Classical Theory
By assuming waves in this development of ‘classical’mechanics, we have constructed a theory that is ‘quan-tized’ from the beginning. After all, ‘first quantization’associates the classical variables E and p with wave pa-rameters f and k , and waves in bound or bounded statescan only accommodate quantized values of f , k , and L ≡ r × k . Thus this reformulation of classical mechan-ics is hardly classical at all.The wave/quantum genesis of this formulation is astrength, not a defect. Our goal is to present physics froma more unified and coherent standpoint. To the extentthat this approach blurs bright lines separating classicalfrom quantum physics, so much the better. The wave-based formulation moves the boundary between these twoareas, historically speaking, from 1900 to the mid 1920’s,as it explicitly includes (or, more accurately, obviates)the Planck-Einstein, de Broglie, and Compton relations,which are usually considered non-classical. It also in-cludes the Heisenberg uncertainty relation (1927), whichis traditionally considered the archetypical quantum me-chanical result: by assuming waves, we have automati-cally introduced incompatible observables. Fourier con-jugate variables — x and k x , for instance — have a re-striction on the product of their widths given by∆ x ∆ k x ≥ / π, (15)which is a purely mathematical result (Bracewell, 2000;Cohen-Tannoudji et al. , 1977). Thus while the Heisen-berg uncertainty relation,∆ x ∆ p x ≥ ~ / , (16)might seem mysterious in traditional units, it has both asolid mathematical foundation and a simple, intuitive ex-planation in wave units. One does not need a physicist’smathematical training to understand that it is not pos-sible to simultaneously measure, for example, time andfrequency to arbitrary precision. B. Choice of Wave Equation
As the attentive reader might have noticed, the moti-vation accompanying the introduction of φ in Section IIIhas a shortcoming. The generic wave equation (Eq. 3a),which is invoked but not actually used, is not invariantunder a Lorentz transformation unless the phase velocity v = c , the speed of light. Thus it cannot be applied tomassive bodies and a more general wave equation mustbe sought.Unlike Eq. 3a, the Klein-Gordon equation − c ∂ u∂t + ∇ u = (2 πk C ) u, (3b)provides a natural fit with Eq. 4. We decline to take thisstep in the first pass because the choice of wave equationis unimportant for the development of classical mechan-ics, and the generic wave Eq. 3a, unlike the Klein-GordonEq. 3b, is familiar from a variety of classical problems(e.g. sound, vibrating strings, electromagnetic waves).However, in a second pass Eq. 3b is preferred for itsability to accommodate mass while preserving Lorentzinvariance. While seen less frequently, Eq. 3b can be em-ployed in classical contexts — for instance, to describethe braced string (Crawford, 1968; Gravel and Gauthier,2011).Because it fails to directly address (or even iden-tify) the underlying wave equation and associatedconcepts such as superposition and dispersion, thewave-based formulation of classical mechanics ismanifestly incomplete. Rectifying this shortcomingleads to developing the full quantum theory. Thisdevelopment dramatically expands the number ofpostulates (adding e.g. states are represented byvectors in a Hilbert space, etc.) (Bassi and Ghirardi,2003; Cohen-Tannoudji et al. , 1977; Commins, 2014;d’ Espagnat, 1999; Griffiths and Schroeter, 2018;von Neumann, 2018; Saxon, 1968), which is an indica-tion that this next level is not as well understood. Appendix A: Fourier conjugate variable choice
While employed without comment previously, thechoice of the Fourier conjugate variable f for frequencyover its radial counterpart ω ≡ πf is deliberate, and inkeeping with the goal of economy. Introducing radial fre-quencies does not obviously reduce the number of 2 π ’s,but it increases the number of variables in use (unlessone decides to forgo f entirely). The additional variable ω adds a new way to write any formula containing f , butwithout adding any new idea or perspective. Thus it in-creases algebraic clutter for an at-best marginal decreasein the number of 2 π ’s.The ordinary frequency f is preferred over the angularfrequency ω for two reasons. First, it puts the Fourier2transforms in their symmetric, unitary form¯ u ( f ) = Z ∞−∞ u ( t ) e πift dt (A1a) u ( t ) = Z ∞−∞ ¯ u ( f ) e − πift df. (A1b)This form simplifies the interpretation of many transformpairs. For instance, the Dirac δ -function and unity areFourier transforms of one another (i.e. for u ( t ) = δ ( t ),¯ u ( f ) = 1, and for ¯ u ( f ) = δ ( f ), u ( t ) = 1), without factorsof 2 π or √ π . (As Fourier conjugate variables E and p are worse than either choice of the frequency variables,for then normalization factors of Planck’s constant h orDirac’s reduced-Planck’s-constant ~ also appear.) Sec-ond and most importantly, f preserves the simplest andmost symmetrical reciprocal relationship between the pe-riod τ ≡ /f and the temporal frequency f .The wavevector, or spatial frequency, k is definedanalogously, in that the wavelength λ ≡ / | k | is likethe period τ ≡ /f . This wavevector convention dif-fers from the usual physicist’s convention by a factorof 2 π . However, it is a standard choice in mathemat-ics (Folland, 1989; Stein and Shakarchi, 2003), engineer-ing (Bracewell, 2000), crystallography (Schwarzenbach,1996), and in X-ray (Ewald, 1969; Jacobsen, 2019) andelectron microscopy (Cowley, 1995; Reimer and Kohl,2008), where the Fourier conversion between realspace (imaging) and reciprocal space (diffraction) is afrequently-performed, often push-button operation. Theconvenience advantage in being able to go back and forth,without having to wrestle with 2 π ’s, between wavevec-tors and real-space distances is difficult to overstate.Besides giving the simplest (reciprocal) relationship be-tween wavelength λ and wavevector k , the microscopist’sconvention also streamlines concepts such as the densityof states — compare, for example the formulas ( L/ π ) D and L D for the density of states in k -space for a D -dimensional volume L D .From the perspective of the phase φ (a ‘countable’quantity — see discussion after Eq. 2) as well, ordinaryfrequency is preferred over radial frequency. Countingcycles in a waveform can often be accomplished by in-spection. Counting radians is not so easy. Appendix B: Sign conventions
We choose a metric (+ − −− ) that gives x µ x µ > p µ p µ ∝ k µ k µ ∝ m > H = hf and p = h k (firsttwo lines of Table III) we choose u ( x µ ) = e πi ( k ′ r − f ′ t ) = e − πik ′ · x = e πiφ to describe a plane wave with k ′ > f ′ >
0. Because of the hyperbolic metric, we must have (compare the signs in Eqs. A1),¯ u ( k ) = Z Z Z ∞−∞ u ( r ) e − πi k · r d r (B1a) u ( r ) = Z Z Z ∞−∞ ¯ u ( k ) e πi k · r d k . (B1b)Equations A1–B1 can be summarized¯ u ( k ν ) = Z Z Z Z ∞−∞ u ( x ν ) e πik · x d x (B2a) u ( x ν ) = Z Z Z Z ∞−∞ ¯ u ( k ν ) e − πik · x d k, (B2b)where k · x ≡ k µ x µ . In Dirac’s notation h k | x i = e πik · x .We choose φ = − k µ x µ (Eq. 2) with the leading minussign so that the conversion between wave units and tra-ditional units for the action S = hφ will be like those forthe energy H = hf and the momentum p = h k . Thesign of the action/phase is purely conventional, since thetrajectories are independent of whether the extrema is aminimum or a maximum (Section IV). Historically thechoice has been to have dS/dt ≡ L ≡ p˙r − H and not dS/dt ≡ L ≡ H − p˙r , which would have been betterfor the uniformity of our sign conventions. The choice ofthe sign defining the Lagrangian L , and thus the action,in this Legendre transformation was likely made because p˙r > H for a free particle if its rest energy is ignored,and the concept of rest energy was unknown when thisconvention was established. Appendix C: Quantum mechanics and its classical limit
Quantum mechanics is purported to present a picturethat is wildly distinct from that of classical mechanics(Griffiths and Schroeter, 2018). Classical mechanics isthe intuitive, every-day world of massive bodies, whilequantum mechanics involves spooky, hard-to-understandwaves (Mermin, 1985). Our purpose in this article is tode-emphasize that distinction. Classical mechanics is alsoa wave theory. It happens that classical mechanics wasinvented before its wave underpinnings were understood,but we can understand both it and subsequent devel-opments better by exploiting those same developments.Emphasizing the differences between classical and quan-tum physics has few pedagogical advantages, and under-standing quantum mechanics is much easier if one has aclassical intuition that is already wave-based.Quantum mechanics has some real mysteries: en-tanglement, locality, and the measurement problem,for instance. However, other aspects of quantum me-chanics that can seem problematic are not, such asthe uncertainty principle and incompatible observables.It might seem deep and mysterious that one can-not simultaneously measure position and momentum3(Griffiths and Schroeter, 2018). But, as discussed in Sec-tion IX, it is comparatively obvious that one cannot si-multaneously measure position and spatial frequency.Using modern, wave-based units (i.e. dispensingwith Planck’s constant) will improve the clarity ofany presentation of quantum mechanics, which ismanifestly a wave theory. But given that manyexcellent references (Abers, 2004; Cheng and Li,1991; Cohen-Tannoudji et al. , 1977; Commins,2014; Dirac, 1967; Feynman et al. , 2010; Gamow,1964; Goldstein et al. , 2002; Griffiths and Schroeter,2018; Landau and Lifshitz, 1981; Messiah, 2014;Sakurai and Napolitano, 2017; Saxon, 1968) containconfusing statements about the role of Plank’s constant,particularly in regards to the classical limit, a fewadditional words are in order.Taking ~ → et al. , 2010;Goldstein et al. , 2002; Griffiths and Schroeter, 2018;Jackson, 1999; Jammer, 1966; Landau and Lifshitz, 1981;Messiah, 2014; Sakurai and Napolitano, 2017; Saxon,1968). As we have shown, Planck’s constant h = 2 π ~ is not a necessary element of physical theory: if wave-based units are used from the beginning, this conversionfactor need never enter the discussion. And even if we dodecide to use traditional units, then adjusting the size ofPlanck’s constant is equivalent to changing the definitionof the kilogram relative to the meter and the second (Sec-tion VI). Such re-definition of the agreed-upon standardunit of mass ought not to have physical effects. Thus,according to the perspective presented here, the valueof Planck’s constant — physically irrelevant, and set bycustom — should have no bearing on the classical limit.Another way to recognize this problem is to note thatsome authors set ~ = 1, others state that the classicallimit occurs as ~ →
0, and some do both (Abers, 2004;Cheng and Li, 1991; Jackson, 1999). Taking both state-ments at face value gives the nonsensical claim “ ~ = 1 → ~ → x, p x ] = i ~ (C1)and its corresponding uncertainty relation ∆ x ∆ p x ≥ ~ / ~ → et al. , 1964). Applying the canonical commu-tation relation C1 to the angular momentum operators L ≡ r × p gives the commutator[ L x , L y ] = i ~ L z . (C2)The angular momentum operators are the gen-erators of rotations (Cohen-Tannoudji et al. , 1977;Goldstein et al. , 2002). Their non-zero commutator saysthat rotations in three dimensions do not commute, as istrue both quantum-mechanically and classically.Not only does ~ → do commute)(Ralston, 2012), it also muddies an oppor-tunity to examine a general topic of wide applicability.Non-commuting rotations give rise to ‘geometric’ phases,which appear in the theory at the same level as the‘dynamical’ phase φ . (In other words, the ‘geometric’and the ‘dynamical’ phases are additive.) While elemen-tary enough to be demonstrated with just an arm and athumb (Chiao et al. , 1989), geometric phases have classi-cal, quantum mechanical, and gauge theoretic relevance.Examples of topics elucidated with geometric phasesinclude the Foucault pendulum and cyclotron motion,the AharonovBohm and AharonovCasher effects, Wess-Zumino terms, their anomalies, and fractional statistics(Cohen et al. , 2019; Shapere and Wilczek, 1989).In wave-based units the angular momentum operatorsare dimensionless, and the ~ → ~ → u, v ] → (1 /i ~ )( uv − vu ) gives the corre-spondence between classical functions on the left andquantum operators on the right (Goldstein et al. , 2002;Sakurai and Napolitano, 2017; Schiff, 1968), but the ap-pearance of ~ in the denominator of the patch just em-phasizes that ~ must drop out of the problem.Thus the blanket prescription ~ → ~ → x, k x ] = i/ π, (C3)and the associated uncertainty relation,∆ x ∆ k x ≥ / π, (15)provides a more constructive perspective. The classi-cal limit does not correspond to taking the right-handsides to zero, for i/ π → / π → et al. ,1977; Griffiths and Schroeter, 2018; Saxon, 1968; Schiff,1968), provide well-known examples of this general phe-nomenon.In the classical limit trajectories are well-defined, so x ≫ ∆ x and k x ≫ ∆ k x (Cohen-Tannoudji et al. , 1977;Messiah, 2014). If k x ≫ ∆ k x we can take ∆ k x /k x ≃ ∆ λ/λ , which gives λ ≫ ∆ λ . Thus, the uncertainty re-lation (Eq. 15) can be written ∆ x ∆ k x ≃ ∆ x ∆ λ/λ ≥ / π . Satisfying the requirements for classical trajec-tories and the uncertainty relation simultaneously thusrequires that x ≫ ∆ x ≫ λ π ≫ ∆ λ π , (C4)which holds even in the limiting case (equality) of theuncertainty relation. Note a subtlety here — we expectthe uncertainties to be small, not large, in the classicallimit. However, one can legitimately consider ∆ x → ∞ or ∆ x →
0, depending on the basis for the comparison(which points out the hazards of limits in dimensionedquantities). In other words, with two (or more) otherlength scales in the problem, the uncertainty ∆ x can beboth large and small. It is large in comparison to thewavelength λ . And it is small in comparison to x , aclassical dimension in the problem (not necessarily theposition coordinate, which can be set to zero by choiceof origin).Simple estimates of x and λ put macroscopic objectsproperly in the classical limit. The atomic size scale Exploring the quantum/classical boundary with nanoscopic ob-jects is an active area of research (Cronin et al. , 2009). Quantuminterference with ∆ x ≫ ℓ has been demonstrated using 2000-atom molecules (e.g. C H F N S Zn ) with masses of2 . × Da and size-to-wavelength ratios ℓ/λ of 10 (Fein et al. ,2019). (One dalton, or unified atomic mass unit, is equiva-lent to approximately 0 .
75 fm − .) Achieving interference withmasses exceeding 10 Da looks feasible with current technology(Kia lka et al. , 2019). is the Bohr radius a = ~ / ( αm e c ), where α is the fine-structure constant and m e is the electron mass, so anobject’s size ℓ ∼ x ∼ a N / , where N is the numberof atoms in the object. For λ , a conservative (i.e. large λ ) estimate corresponds to an object that has no appar-ent motion. Ascribing the object’s motion to its thermalenergy k B T alone gives p /M = k /k C ∼ k B T . Then λ ∼ h/ √ N Am n k B T , where the object’s mass M comesfrom its N atoms having A nucleons of mass m n . Com-paring N / to N − / , we see that, for a macroscopicquantity N (e.g. N A , Avogadro’s number), ℓ ≫ λ inde-pendent of any other details, and that the range where∆ x can simultaneously satisfy the inequalities C4 spansmany orders-of-magnitude.In passing we note that applying ~ → N → ∞ .Many authors draw an analogy between the routefrom physical (i.e. wave) optics to geometrical op-tics given by λ →
0, and the route from quan-tum mechanics to classical mechanics given by ~ → et al. , 1977; Landau and Lifshitz,1981; Messiah, 2014; Sakurai and Napolitano, 2017). Intraditional units this comparison is fairly called an anal-ogy, but in wave units the analogy is so apt that it is bet-ter viewed as an equivalence. (The wave picture tightensmany analogies between optics and mechanics, such asthat between a refractive index and a potential energy(Cronin et al. , 2009).) Just as one passes from physicaloptics to geometric optics by taking λ → f → ∞ ),so one passes from quantum mechanics to classical me-chanics.The ~ → E = hf or the deBroglie hypothesis p = h k , the classical variable ( E or p ) is held constant implicitly while Planck’s constant istaken to zero, which means that the frequencies are takento infinity. In this limit the phase φ is not being trackedat the single cycle level, which is equivalent to the action S not being tracked with a precision of h (Hanc et al. ,2003).The ~ → traditional wave H = i ~ ∂∂t f = i π ∂∂t p = ~ i ∇ k = πi ∇ [ x, p x ] = i ~ [ x, k x ] = i/ π ∆ x ∆ p x ≥ ~ / x ∆ k x ≥ / π L = r × p L = r × k [ L x , L y ] = i ~ L z [ L x , L y ] = i π L z U ( t ′ ) = e iHt ′ / ~ = e i πft ′ T ( r ′ ) = e − i p · r ′ / ~ = e − i π k · r ′ B ( p ′ ) = e i p ′ · r / ~ B ( k ′ ) = e i π k ′ · r R ( θ ′ ) = e − i L · θ ′ / ~ = e − i π L · θ ′ dAdt = i ~ [ H, A ] + ∂A∂t = 2 πi [ f, A ] + ∂A∂t TABLE III Standard quantum mechanical relationswritten in traditional (Cohen-Tannoudji et al. , 1977;Griffiths and Schroeter, 2018; Styer et al. , 2002) and wave-based units, i.e. with and without the reduced Planckconstant ~ . Some important quantum mechanical operators,such as the time evolution, translation, boost, and rotationoperators, are functions of 1 / ~ in traditional units, and thusbecome meaningless in the limit ~ → ciated energy E classically, and thus taking ~ → f → ∞ as just described. A massless (e.g. electromag-netic) wave, on the other hand, is classically identified byits frequency. Here ~ → E →
0, which indeed isthe classical limit. For instance, the quantum result for ablackbody’s energy per mode, hf / (exp[ hf /k B T ] −
1) = E/ [exp( E/k B T ) − k B T in the classicallimit.Turning these arguments around reminds us that the‘quantum vs. classical’ limit is not the same as the ‘wavevs. object/particle’ limit. Traditional language gives aconfusing, crossover behavior, namely that, in the quan-tum limit, classical particles adopt wave properties, whileclassical waves adopt particle properties (d’ Espagnat,1999). For example, consider a double-slit experimentperformed with electrons or low-intensity electromag-netic waves. The former show quantum behavior bygoing through both slits, while the latter show quan-tum behavior by hitting the screen in localized quanta,i.e. photons. In wave-based language the perspectiveis simpler: λ → f → ∞ ) produces particle be-havior and λ → ∞ (or f →
0) produces wave behavioralways. In traditional language the correspondence be-tween these limits and the quantum/classical limit de-pends on whether the objects under consideration aremassive or massless.Table III gives a list of standard quantum mechanicalformulas in traditional units side-by-side with their wave-unit counterparts. The equations with ~ in the denomi-nator become meaningless if one approaches the ‘classicallimit’ by taking ~ →
0. In this way the ~ → ~ in thedenominator also illustrate a general principle: if a phys- ical quantity is being tracked in classical units, it must beconverted out of that unit system before it can be usedto actually do anything. In this sense the wave-basedunit system is ‘natural’ and the traditional system, withits energies and momenta, is not. The time-evolution,translation, boost, and rotation operators all illustratethis principle. Appendix D: The quantum limit of classical mechanics
Traditional classical mechanics gives no prescriptionfor finding the quantum limit, but with the wave-basedformulation we can create one by reversing the argumentsof Appendix C. The situation is not exactly symmetric, inthat we cannot reproduce a prediction of the exact theoryby taking a limit in the approximate theory, but we canat least identify where the approximate theory breaksdown. Just as the classical limit of quantum mechanicsis found by taking λ to be small, so the quantum limit ofclassical mechanics is found by taking λ to be not small.When we interpret such limits involving a dimensionedquantity, the existence of another, comparison scale is al-ways implied. Quantum mechanics is widely understoodto be the theory of small systems. Here ‘small’ meansthat other length scales in the problem are comparableto the wavelength λ . At this point the crude, approx-imate description generated by considering the station-ary behavior of the phase (i.e. the action) alone is nolonger sufficiently accurate, and the specifics of the waveequation must be considered. Expressing these ideas intraditional language, one expects the classical theory tobreak down when the energy E and momentum p becomesmall (giving small frequencies and large wavelengths).However, while wave units provide a comparison scale —namely the size of the system — traditional units do notprovide comparison scales for the dimensioned quantities E and p . Viewed from a purely Newtonian perspective,the energy and momentum can be made arbitrarily smallwithout entering a new regime that might be governedby qualitatively different physical laws. Thus, withoutthe advantages of the wave-based perspective, traditionalclassical mechanics gives no hint that it is incomplete —hence the surprise accompanying the quantum revolutionof the early 20 th century. Appendix E: Glossary
To avoid slowing our development, in a few cases wehave adopted the practice borrowing ‘physics words’,which ideally have precise meanings, from common En-glish without pausing to define them. Because of thebootstrapping problem, it may be impossible to artic-ulate definitions that are wholly satisfactory (Symon,1971), so this practice is understandably common. Most6authors adopt it implicitly. Others warn that they arecapitulating with a few words such as “we can intuitivelysense the meaning of mass” (Feynman et al. , 1964), or“length, time, and mass are concepts normally alreadyunderstood” (Thornton and Marion, 2004), or “funda-mental physical concepts, such as space, time, simultane-ity, mass, and force...will not be analyzed critically here;rather, they will be assumed as undefined terms whosemeanings are familiar to the reader” (Goldstein et al. ,2002). Newton himself writes, “I do not define time,space, place, and motion, as being well known to all”(Hawking, 2002).Here we have carefully defined space and time in termsof their respective clock frequencies, and vice versa (Sec-tion III). We have argued that the dynamical quantitiesenergy and momentum are just frequencies multiplied bya dimensionful constant that is a historical legacy. Andwe have defined mass as the corresponding hypotenuse ofa hyperbolic right triangle that has temporal frequencyand spatial frequency as its other sides. In so doing wehave defined most of the concepts that appear in Tables Iand II, which together cover classical mechanics.However, the terms appearing in the bottom row ofTable I merit further discussion. As stated earlier, anevent is a position x µ in real space. From the structureof Table I then we deduce that a wave is a position k µ in reciprocal space, and an object is that same position,given instead in historical units. This usage of the term‘wave’ is in keeping with the common usage of the term: awave has a well-defined frequency and spatial frequency.However, as the Heisenberg uncertainty principle(Eq. 15) explains, a wave/object exactly located in recip-rocal space is completely delocalized in real space. Tem-poral delocalization is familiar — we see no problem withan object persisting for a long time — but spatial delo-calization is in contradiction with everyday experience.Thus, while waves corresponding to classical objects musthave well-defined frequencies/momenta, these cannot beexact: they must have some spread. The waves/objectsunder discussion (i.e. those in the classical limit) musttake the form of wave packets that are localized in bothreal- and reciprocal-space.Students familiar with Fourier transforms mightprotest that, since a small width in one space impliesa large width in the reciprocal space (∆ x ∝ / ∆ k ), itis impossible to be well-localized in both. The responseto this objection is that, in this context, we are not de-termining size by comparing ∆ x to 1 / ∆ k . Rather, hereby well-localized we mean x ≫ ∆ x and k ≫ ∆ k . Inother words, the magnitudes of the classical coordinatesin real- and reciprocal-space are large compared to thewidths of these wave packets (Cohen-Tannoudji et al. ,1977; Messiah, 2014).By a change of origin the coordinates of any one loca-tion, either in real or reciprocal space, can be made to bezero. However, this condition cannot be simultaneously achieved for all the constituents of an object consistingof many. In real space the classicality condition can besatisfied by taking the object’s classical length scale ‘ x ’to be its spatial dimension ℓ (i.e. its size), and that scaleto be much larger than the real-space packet’s width.But the reciprocal-space case requires a more nuancedargument. For instance, one might wonder why an object— say a billiard ball — at rest is not delocalized. Afterall, in the frame where the billiard ball is motionless, onemight think that its momentum is both small ( p ∼
0) andwell-specified (∆ p ∼ λ and∆ r are both large (via the de Broglie and uncertaintyrelations respectively). This conundrum is resolved bynoting that an “object’s rest frame” only exists in theclassical approximation. One can be in, or transformto, a frame where an object’s center of mass is at rest.However, for a multi-particle object this point is only amathematical construction relative to which the object’sconstituent parts are generally in motion.Consider, for example, the object as a collection of N coupled harmonic oscillators, which is a reasonablefirst approximation of a solid. Any collection of coupledharmonic oscillators of natural frequency f will producenormal modes with frequencies both above and below f (Thornton and Marion, 2004), and the frequencies of thelowest-frequency normal modes scale like f /N (Kittel,2005). At any finite temperature these low-frequencymodes are thermally excited, and thus the object con-sists of, say, halves in thermal motion moving aboutthe center of mass. The equipartition theorem then im-plies that these halves have thermal momenta such that λ ∼ h/ √ M k B T , a small length for a macroscopic mass M (Appendix C). Thus the halves are localizable, andso must be the object itself. This argument applies toall three spatial directions separately, in that one can-not transform, by either rotation or boost, to a framewhere a classical (i.e. many-particle, finite-temperature)object is delocalized in any spatial dimension. Thus the x ≫ ∆ x ≫ λ π ≫ ∆ λ π condition discussed in Appendix Cis achieved.We have now introduced enough ideas from both classi-cal and quantum physics that we can examine the defini-tion and usage of these three key words: particle, wave,and object. (Other terms, such as ‘corpuscle’, ‘thing’,‘entity’, and ‘body’, are sometimes used synonymouslyand interchangeably with one or more of the three termswe have chosen.) The first term in particular is usedwith a bewildering array of meanings, some quite contra-dictory — even compared to other physics concepts, thedefinition of ‘particle’ is particularly elastic. Standardclassical mechanics textbooks (Goldstein et al. , 2002;Kleppner and Kolenkow, 2014; Landau et al. , 1976;Symon, 1971; Thornton and Marion, 2004; Young et al. ,2020) use ‘particle’ to denote the concept of an ‘ob-ject’ (as used here) or a ‘body’ (in Newton’s language)(Newton et al. , 2016). Some emphasize a negligible spa-7tial extent, defining a particle to be a point-like ob-ject (Landau et al. , 1976; Omn`es, 1971; Symon, 1971).But in more advanced textbooks usage inconsistent withthese definitions is common. For instance, the wave-function ψ ∼ e i ( p · x − Et ) / ~ might be given to describe afree particle of energy E and momentum p (d’ Espagnat,1999; Halzen and Martin, 1984; Saxon, 1968). Thisstate is completely delocalized in real space, and thuscould not be less point-like. Unfortunately, terminologythat provides an explicit warning (e.g. use of ‘wavicle’(Eddington, 1928)) has not been widely adopted.Identifying, for instance, a particle such as anelectron or the muon as a “point particle” (Abers,2004; Eisberg and Resnick, 1985; Griffiths, 2010;Halzen and Martin, 1984; Harrison, 2000; Jackson,1999) is particularly misleading. An electron can benever be localized to a point, and is often delocalizedover macroscopic dimensions (as in an electron micro-scope, for instance). Under certain circumstances (e.g.only the center of mass motion is of interest, or the pointof observation is distant from an object with sphericalsymmetry) we can treat an extended object/wave as amathematical point, but under no circumstances is itproper to think of the “point particle” as anything otherthan a convenient approximation. A diffracted electrondoes not hit a phosphor screen at a point. Rather, ithits the screen in a locale that might be identified withoptical ( ≃ µ m) or even atomic ( ≃ . t is measured by counting clock cycles relativeto an arbitrary origin.2. A clock cycles with time t at a temporal frequency f c .3. Temporal frequency f is measured by counting cy-cles per interval in time t . 4. Position r is measured by counting ruler cycles rel-ative to an arbitrary origin.5. A ruler cycles with position r at the spatial fre-quency k c of a massless ( m = 0) wave cycling intime t at the clock frequency f c .6. Spatial frequency k is measured by counting cyclesper interval in position r .7. The interval s separating two events, one at time t and position r and the other defined to be theorigin, is given by s = t − r .8. At time t and position r , the phase φ of a wavewith constant temporal frequency f and spatial fre-quency k is given by φ = k · r − f t .9. For a wave with temporal frequency f and spatialfrequency k , the mass m is given by m = f − k .10. A wave is an object that can be equivalently de-scribed as a function of time t and position r oras a function of temporal frequency f and spatialfrequency k .11. A particle is an object that can be counted as aninteger unit.To keep these definitions as self-contained as possible,we use ahistorical units where h = c = 1 (which avoidshaving to define ‘light’) and avoid jargony shorthand(“A wave is an object with a Fourier transform.”). Thedefinitions are necessarily self-referential (i.e. circular),but not completely tautological (“A wave is an objectthat satisfies a wave equation.”). They are also incom-plete, but the undefined terms (e.g. ‘function’) are log-ical or mathematical. Measuring is defined implicitlyto be counting. Objects exhibit wave/particle duality:they can have wave properties and particle properties(d’ Espagnat, 1999; Harrison, 2000; Saxon, 1968).Definitions The frequency definitions defining characteristic is that it ordersevents (’t Hooft, 2018). While these definitions x ′ , its mean position in reciprocal space k ′ , and itswidth σ k in reciprocal space, σ k = 14 πσ x , (E1)where σ x is the wavepacket’s corresponding width in realspace. Normalizing the wavepacket establishes the corre-spondence with the integers required to make the particle‘countable’.This wavepacket can be constructed by shifting andthen boosting a wavepacket initially centered on the real-and reciprocal-space origins. It can be described equallywell in real space: ψ g ( x ; σ k , x ′ , k ′ ) = e πik ′ x q σ k √ πe − (2 πσ k ( x − x ′ )) , (E2)or reciprocal space:¯ ψ g ( k ; σ k , x ′ , k ′ ) = e − πi ( k − k ′ ) x ′ p σ k √ π e − ( k − k ′ σk ) . (E3)Now we wish to combine two particles to make a mini-mal, composite object. Normally we imagine assemblingobjects additively. For instance, we might add beans to abag to make a beanbag, or we might add an oxygen atomto a carbon atom to make a carbon monoxide molecule.According to the wave picture, composite objects are as-sembled not by addition, but by multiplication. Just asamplitudes for events occurring in succession multiply(Feynman et al. , 2010), so amplitudes for particles com-prising an object multiply. In other words, we do notadd the beans and the bag, we multiply the beans andthe bag.That multiplication, not addition, describes the combi-nation of two particles is a consequence of definition N increases. Infact, without definition N .Quantum mechanical superposition is fundamentallydifferent than classical superposition (Dirac, 1967), andthis difference is essential for realizing the classical limit.9Definition Superposition in the senseof addition occurs with the various states of one particle,but not with states of different particles. The ‘quantum’countability definition N , and, as we willnow show, the large- N limit gives the wave-objects thatwe recognize as ‘classical’ particles. Initially we consider the two particles to be non-interacting, in which case their Hilbert spaces are notcoupled. Combining the spaces in a tensor productcorresponds to simple multiplication in both real- andreciprocal-space, so a minimal object consisting of twoparticles ( ψ g ( x ; σ , x ′ , k ′ ) and ψ g ( x ; σ , x ′ , k ′ )) can bewritten ψ g ( x ; σ , x ′ , k ′ ) · ψ g ( x ; σ , x ′ , k ′ ) in a real-spacerepresentation. Some algebra shows that this compositestate has a second factorization (one of an infinite num-ber), ψ g ( x ; σ , x ′ , k ′ ) · ψ g ( x ; σ , x ′ , k ′ ) = (E4) ψ g ( x ; σ k , x ′ , k ′ ) · ψ g ( X ; σ K , X ′ , K ′ ) , where the coordinate transformation ( x , x ) → ( x, X ) isgiven by x = σ x + σ x σ + σ , and X = x − x , (E5)and the Gaussian parameters by x ′ = σ x ′ + σ x ′ σ + σ , X ′ = x ′ − x ′ , (E6) σ k = q σ + σ , σ K = σ σ p σ + σ ,k ′ = k ′ + k ′ , and K ′ = σ k ′ − σ k ′ σ + σ . We are making the usual classical assumption that the particlesare distinguishable (Schiff, 1968). If the particles are indistin-guishable then ‘different’ particles can superpose and interfere(Hanbury Brown, 1974; Purcell, 1956). Feynman argues (Feynman et al. , 1964) that the sentence “allthings are made of atoms” condenses much of our scientificknowledge into very few words. “All things are made of count-able waves” is written in the same spirit, and, for the price ofone additional word, communicates a good deal more.
The formulas for the inverse transformation are x , = x ± X ∓ s − σ K σ k ! , (E7) x ′ , = x ′ ± X ′ ∓ s − σ K σ k ! , (E8) σ , = σ k vuut ± s − σ K σ k ! , and (E9) k ′ , = k ′ ± s − σ K σ k ! ± K ′ , (E10)where the upper (lower) sign is chosen for particle 1 (2).If the particles do not have identical widths the axesscale, so the coordinate transformations E5 are gener-ally more than a simple rotation, but such stretching andcompressing is permissible in this abstract vector space(as opposed to actual physical space) (Symon, 1971).Using an obvious shorthand, if ψ g ( x ) and ψ g ( x ) de-scribe the first and second particles, respectively, then ψ g ( x ) and ψ g ( X ) can be taken to describe ‘external’ and‘internal’ degrees-of-freedom, respectively (i.e. ‘extrac-ules’ and ‘intracules’ (Coleman, 1967; Eddington, 1953;Gill et al. , 2003; Proud and Pearson, 2010)). We alsonote that, if the σ ’s are proportional to the square-rootsof their respective masses, then the ‘external’ and ‘in-ternal’ degrees-of-freedom may be associated with thecenter of mass and the reduced mass, respectively. Atthis point this connection is unexpected — the Gaussianwidths seem to be parameters that are adjustable inde-pendent of the dispersion relation, and the dimensionsof σ k and m differ by a length scale. We will return toexamine this connection shortly.In a standard quantum mechanical treatment, the in-ternal degrees of freedom are of interest and the externalcoordinates are ignored. Here, in order to understandhow classical objects arise from waves, we take the oppo-site approach. We integrate over all values of the inter-nal coordinate X . The (normalized) wavefunction ψ g ( X )then drops out of the problem. The object remaining nowhas a (Gaussian) particle description in terms of the ‘ex-ternal’ wavefunction ψ g ( x ) centered in reciprocal spaceon the sum wavevector k ′ (i.e. momentum) and in realspace between the constituent particles at x ′ .A complex, composite object can be built up by addingone particle after another, which corresponds to repeat-ing this multiplication process again and again. By in-duction (Bromiley, 2018) we find that the Gaussian pa-rameters describing the final, external degree of freedom0are σ k = N X i =1 σ ki , (E11) x ′ = 1 σ k N X i =1 σ ki x ′ i , and k ′ = N X i =1 k ′ i . The assembly process is similar in reciprocal-space andgives the same result as that obtained by Fourier trans-forming the final expression in its real-space representa-tion.This path already looks promising, in that it repro-duces properties expected for an object made of multipleparticles. For instance, in real space the composite ob-ject is centered around a weighted mean of the centers ofits constituents. And in reciprocal space the wavevector(i.e. momentum) of the composite object is equal to thesum of the wavevectors of its constituents.We now consider the case where the N constituent par-ticles are all similar. For instance, the beans of the com-posite beanbag are moving together on a ballistic trajec-tory. (Once the beanbag is launched, each bean’s path isapproximately unaltered by the presence of its neighbors,which illustrates how individual particles with negligibleinteractions can still be grouped as a composite object.)Then σ k ≃ √ N σ ki , k ′ ≃ N k ′ i , and x ′ is the mean of the x ′ i . Calculating the probability density for this compositeobject, we see that ψ g ∗ ( x ) · ψ g ∗ ( x ) = 2 σ k √ πe − πσ k ( x − x ′ )) (E12) → δ ( x − x ′ )in the limit that the number of constituent particles N → ∞ because σ k → ∞ . Thus the real-space prob-ability density for a many-particle object approaches a δ -function with a real-space width σ x ∝ / √ N .In reciprocal space the argument is again more subtle.The function¯ ψ g ∗ ( k ) · ¯ ψ g ( k ) = 1 σ k √ π e − k − k ′ σk ) (E13)does not approach a delta function because σ k ∝ √ N is becoming large. However, k ′ ∝ N is becoming largefaster, so that the function E13 looks like a delta functionfrom a perspective that encompasses both the reciprocalspace origin and the peak. From this perspective, thenatural perspective for the problem, the relative width σ k /k ′ is decreasing ∝ / √ N in reciprocal space, just as itis in real space. Thus the reference to the ‘zoom level’ inSection III is made precise: a wave can be well-localizedin both real- and reciprocal-space without violating theuncertainty principle. According to definition x ′ , k ′ , and awidth (either σ x or σ k ). In the classical limit, the phase ∼ k ′ x ′ ∝ N is unmeasurable , leaving Eqs. E12–E13 asthe information available. These equations are effectively δ ( x − x ′ ) and δ ( k − k ′ ) respectively, and knowledge of oneprovides no information about the other. In the classicallimit, the position and the spatial frequency (i.e. mo-mentum) descriptions are effectively decoupled.Here we can also get a more quantitative sense of howthe inequalities C4 develop as we move into the classicallimit. For a 1D, N -particle composite object, we canre-write x ≫ ∆ x ≫ λ π ≫ ∆ λ π (C4)as N σ xi ≫ σ x ≫ πk ′ ≫ σ k πk ′ , (E14)where the replacements are justified by arguments givenpreviously, except for x → N σ xi . This replacementbounds the size of the object by requiring that its con-stituent particles occupy distinct regions of space . Jus-tifying this bound requires ideas (e.g. Coulomb repulsionand the Pauli exclusion principle) not discussed here, but It is more accurate to say that the phase discussed here is un-measurable not because it is large, but because it changes rapidlywhen varied separately in either time or space. The phase dis-cussed here is only a (spatial) piece of the ‘full’ phase — Eq. 2or Eq. 6 — which is unchanging along the classical trajectory.(By choice of origin the phase might be set to zero, in whichcase it is not large at all.) Thus as we move into the classicallimit the ‘full’ phase controls the observable dynamics — Sec-tion IV — while the ‘partial’ phases, i.e. the leading terms inEqs. E2–E3, are becoming unobservable. Also note that, whileunobservable, these phases are not incoherent. The phase rela-tionships, i.e. coherences, are what maintain the object as a uni-fied whole. Fiddling with a particle’s phase changes its positionor its wavevector/momentum (Eqs. E2–E3). To the extent thatthese physical quantities are constrained in a composite object,the phases must be coherent. Coherent states of the ‘quantum’harmonic oscillator (Appendix C) provide a concrete example ofthe coherence of ‘classical’ objects. While the spatial uncertainty relation (Eq. 15) has a direct ana-log ∆ t ∆ f ≥ / π , and k ≫ ∆ k implies f ≫ ∆ f , this replace-ment and x ≫ ∆ x do not have such obvious temporal analogs.Objects do not overlap in space, but they do in time. Despitethe apparent similarity of Eqs. A1 and Eqs. B1, the uncertaintiesin the variables t and x seem to require very different interpreta-tions. Usually we assume that a measurement of x returning oneparticular value in the range ∆ x thereby excludes others, whilewe expect that a particle occupies all t in ∆ t (Mermin, 1985). N πσ ki ≫ π √ N σ ki ≫ πN k ′ i ≫ σ ki π √ N k ′ i . (E15)Multiplying through by 4 πσ ki puts this inequality chainin a dimensionless form, N ≫ √ N ≫ (cid:0) σ ki k ′ i (cid:1) N ≫ (cid:0) σ ki k ′ i (cid:1) N √ N . (E16)These four terms scale like N , N − / , N − , and N − / ,respectively. Thus as the number of particles N → ∞ ,the classical limit is inevitable, provided that σ ki ≫ k ′ i is not the case. If k ′ i is not obviously appreciable, thethermal arguments given earlier in Appendix C and herein this Appendix E can be applied to determine whetherthe classical limit obtains.In assembling the composite object we have neglectedinteractions between the constituent particles. However,reviewing this development we see that, because we in-tegrate over the internal degrees of freedom, the exactforms of the internal wavefunctions are unimportant. Forinstance, if the minimal, two-particle object is a hydro-gen atom consisting of a proton and an electron, theCoulomb interaction gives internal coordinates describedby hydrogenic wavefunctions. These wavefunctions arenot of a Gaussian form ψ g (Eqs. E2–E3). In this casethe states can only be factored in the external/internalcoordinate system ( r , R ), for in the electron/proton co-ordinate system ( r e , r p ) the states are entangled. How-ever, our goal here is see how a single object arises bycombining many particles, not to specify how the par-ticles arise by decomposing the object. The compositeobject’s state need not be factorizable in terms of the con-stituent particle coordinates. We only require the factor-ization on right-hand side of Eq. E4. Often the transfor-mation necessary to produce this separation of variablescan be found, with the transformation to center of massand reduced mass coordinates being the classic example(Feynman et al. , 2010; Ghirardi et al. , 1986). In fact, inthe many particle limit it is not even necessary that theinitial external wavefunctions be Gaussian. The centrallimit theorem (Bracewell, 2000) ensures that, with suit-ably well-behaved constituents , the external wavefunc-tion will eventually adopt a Gaussian form as the num-ber of constituent particles increases, regardless of thespecifics of the problem.Thus this argument showing how classical objects (lo-calized in real and reciprocal space) arise from waves (lo-calized in neither space) is more general than initially A double slit or a grating, for instance, can induce poor behavior. advertised. The passage to the classical limit is not re-stricted to non-interacting particles described by Gaus-sian wavepackets, but is, as it must be, a process thatgenerally occurs with large numbers of particles, regard-less of their interactions.To recapitulate while generalizing to three dimensions:we start with N wavepackets, each of which is describedin its own real and reciprocal spaces in terms of its meanposition r ′ i , mean wavevector k ′ i , and reciprocal-spacewidths ↔ σ ki . These N wavepackets represent particlesthat might be the constituent beans of a beanbag, theatoms that make up the beans, or the fundamental par-ticles that make up the atoms. We rearrange the coor-dinates of the 3 N -dimensional space and integrate overthe 3( N −
1) ‘internal’ coordinates. The ‘internal’ coordi-nates then drop out of the problem, one by one, becausethe normalization conditions from each of the constituentparticles have been transferred to the new degrees offreedom (i.e. pseudo-particles). The remaining ‘exter-nal’ 3D space contains a single normalized wavepacketrepresenting the composite (pseudo-)particle. In realspace this wavepacket is centered between the constituentwavepackets’ positions, and in reciprocal space it is cen-tered on the sum of the constituent wavepacket’s posi-tions. For large N this final wavepacket’s widths are neg-ligible in both real- and reciprocal space, where the basesfor comparison are the object’s size and its total wavevec-tor, respectively. The Fourier connection between thereal- and reciprocal-space descriptions is practically sev-ered, and the resulting object can be treated like a ‘par-ticle’ in the sense of the word as employed by standardtextbooks on classical mechanics.Returning to review the arc of the argument fromits beginning (Section III), eliminating the ‘uninterest-ing’ degrees of freedom from a complex, large- N collec-tion of particles produces a single ‘particle’ that is well-localized (though not small) and behaves classically. Inreal space its location is effectively described by a delta-function ∼ δ ( r − r ′ ), where r ′ develops in time accordingto Eq. 12a. In reciprocal space its location is effectivelydescribed by a delta-function ∼ δ ( k − k ′ ), where k ′ de-velops in time according to Eq. 12b. Thus the completemotion of the wavepacket is dictated by the conditionthat the phase φ = k · r − f t is stationary. In traditionallanguage, one says that the motion of the object is dic-tated by the condition that the action S is stationary.This ‘stationary action’ condition is abstract and unmo-tivated, while the wave formulation’s ‘stationary phase’ iscomparatively elementary and might even supply a men-tal picture — see, for example, the case of a masslessobject (Eq. 3a), where ‘stationary’ becomes ‘constant’.The relationship between the average position of aGaussian wavepacket in real-space and the center ofmass remains obscure. Proportionality between σ k andmass m is seen in some important and basic contexts.It appears in the propagator for a free particle, where2 σ k = − im/ (4 π ∆ t ) and ∆ t is the time interval of prop-agation (Feynman et al. , 2010). And it appears in theground state of a harmonic oscillator, where σ K = µf / µ is the reduced mass, and f is the oscillator’s fre-quency. In other words, in the position representation(with h = c = 1) a 1D harmonic oscillator’s ground state ψ ho can be written ψ ho ( X ) = (4 πµf ) / e − π µf X (E17)= ψ g ( X ; σ K , , . This Gaussian state gives σ K proportional to the cor-rect mass. It also has a parameter ( f ) that can be ad-justed to fit the problem: increasing the oscillator fre-quency localizes the state in real-space and delocalizesit in reciprocal-space, and vice-versa. However, for x ′ to be equivalent to the center of mass, we require that m , m , the total mass m = m + m , and the reducedmass µ = m m / ( m + m ) all be proportional to theirrespective σ k ’s with the same constant of proportional-ity (E6). This requirement seems to over-constrain theproblem. For example, a carbon monoxide molecule inits vibrational ground state might be trapped in a har-monic potential, with no specific relationship between themolecular binding and the trapping potential’s depth.The center of mass concept itself is not without prob-lems. It is most useful in situations with a great dealof symmetry (e.g. spherical), where the equivalencewith the Gaussian center will probably hold. With-out such symmetry a center of mass decomposition withas few as three particles is only approximate, and dis-tinctions arise between the center of mass and the cen-ter of gravity (Symon, 1971). Moreover, the center ofmass concept is seriously limited in that it does not havean obvious relativistic generalization (Alba et al. , 2007;Newton and Wigner, 1949; Pryce and Chapman, 1948).Thus the lack of an exact agreement between x ′ and thecenter of mass is intriguing, but not so disturbing that itshakes our confidence in the general soundness of thesearguments. REFERENCES
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