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California Angeles, Los California, of University 2 90095, U.S.A. California Angeles, Los California, of University 1 aionaNnSsesInstitute, NanoSystems California Astronomy, and Physics of Department ,2, 1, ∗ ant. g aentkp ae This pace. kept not have age nc a ecnieyrepro- concisely be can anics rqec,sailfrequency, spatial frequency, edffiutt e atthose past see to difficult re sntvaadfiiin but definition, a via not ss t fntrlphenomena natural of dth eaiiy hsapproach This relativity. l trigwt etn edsoe htmc fhsvo- his of much that discover we not by lexicon Newton, — with the adopt starting subsequently abbreviates theories physics route other early This that the will of mechanics developments century. we the classical twentieth exploiting As by of recast underpinnings be unexplored. can logical relatively the be me- show, classical to of seem development chanics ahistorical con- an and shape of the However, sequences mechanics 2005). Taylor, classical and subsumes and (Ogborn 2017), mechanics be Napolitano, quantum to and that chosen Sakurai previ- been 2004; have (Abers, noted unity would been constant first, has Planck’s discovered been surely it mechanics quantum instance, had writ- that, For ously been already of elsewhere. has much ten what fact, In reiterates presentation incalculable. our previously was that thing magnetism, and based. be electricity can mechanics) (e.g. quantum which topics upon advanced foundation firm more a requires lay understanding and best treatment), special our gravity, to being according exception even derive (the which can mechanics we classical of curriculum, most physics part already undergraduate both the ideas, of main are two only subjects With advanced more introduced. Un- as not revision does whole. extensive perspective require a this as approach, our chronological physics on the of like based understanding closely more best and current, concise is that mechanics discovered. were well ideas developed unifying and was admire underlying not which the and are before language, they access if classical to theories in approximate cloaked easier, the of indeed power full and the possible, is it ,a h nesadn of understanding the as r, dr esetv,these perspective, odern drt er n apply and learn to rder onwpeitosrsl.W antcluaeany- calculate cannot We result. predictions new No classical of formulation a present to is here goal The caccnet repre- concepts rchaic 2 cabulary is unnecessary. It also provides a satisfying III. SPACETIME CONTAINS WAVES simplification, one that removes unnecessary definitions, convoluted developments, and unmotivated postulates. We postulate that spacetime contains objects that we The resulting edifice offers some insight, and the pleasure will, in a convenient shorthand, refer to as waves. (For that comes from seeing old ideas from a new perspective further discussion of this choice, see Appendix E.) In ad- (Feynman, 1948). vanced treatments these objects might be described more carefully as, for example, fields or wave functions, but for our present purposes such specificity is unnecessary. In fact, we do not even need to specify a particular wave II. SPACETIME equation, only that these waves have a phase φ of the form The spacetime of special relativity serves as the arena. We follow the customary development of φ = ft k r, (2) − − · the subject (Goldstein et al., 2002; Jackson, 1999; Taylor and Wheeler, 1992), postulating that the rules of as is found in the solutions u(φ) to the generic wave equa- physics are the same when viewed from any inertial refer- tion ence frame, and that, in particular, all inertial observers 1 ∂2u + 2u =0, (3a) observe the same value c for the speed of light in vac- − v2 ∂t2 ∇ uum. Applying this idea to simple physical systems (e.g. with the phase velocity v f/ k . For our present pur- a pulsed light source with a mirror on a train) gives, via pose of reformulating classical≡ mechanics,| | the solutions of basic geometry, the Lorentz contraction and time dilation interest are wave packets that are well-localized in real effects. From these we derive the Lorentz transforma- and reciprocal space, e.g. Gaussian functions viewed at tions and the idea of an invariant, the spacetime interval a zoom level such that their widths are negligible. This s, which is seen to have the same value by all inertial ob- second assumption (waves) clearly has deep connections servers. In a standard 4-vector notation with xµ = (ct, r) to the first (special relativity): Eqs. 3a and 1 are struc- we have, turally similar, Eq. 3a is Lorentz invariant, and Eq. 3a does not permit instantaneous action-at-a-distance (un- x xµ = c2t2 r2 = s2, (1) µ − like, e.g. diffusion-type equations). The phase φ(xµ) is a number (defined here with an µ where x indicates a location, or ‘event’, in spacetime in overall minus sign for consistency with a historical con- terms of its time t and its position r. We have chosen vention to be introduced later) associated with a particu- to put the c with the time t, so that every term in this lar event xµ = (ct, r) in spacetime. It can be determined expression has the dimension of length-squared. by a counting operation, and thus all inertial observers Defining the speed of light c allows us to describe agree upon its value (Dirac, 1933; Jackson, 1999; Symon, events using just one dimensioned unit, and to more eas- 1971; Wichmann, 1971). Therefore φ is a Lorentz invari- µ µ ily treat spacetime as a unified whole. Physicists working ant, and it can be written φ(x ) = kµx , where we with very large length scales, e.g. astronomers, tend to have defined a new 4-vector kµ (f/c,−k). 2 µ ≡ µ measure distances using a time unit: the light-year. Oth- Since both s = xµx (Eq. 1) and φ = kµx (Eq. 2) ers are generally better off measuring times in distance are Lorentz invariant, xµ and kµ must have− the same be- units, simply because short distances have a greater num- havior under a Lorentz transformation. These transfor- ber of familiar landmarks. Outside of the ultrafast com- mation properties then ensure that the 4-vector product µ munity, an angstrom and a fermi are more recognizable kµk is also a Lorentz invariant. Thus as atomic and nuclear size scales, respectively, than their µ 2 2 2 near counterparts the attosecond and the yoctosecond. kµk = (f/c) k = kC , (4) − To lay the groundwork for future developments, it is where the quantity kC , which at this point is just a num- important to emphasize here that, given the strong geo- ber seen to have the same value by all inertial observers, metric basis for Eq. 1, it would never occur to anyone to will shortly be identified as the Compton wavenumber. measure its right-hand side in terms of a new, dimension- Equation 4 is the reciprocal-space, or Fourier-space, ful unit. That the customary units designating time (t) analog of Eq. 1. Although we have not yet introduced and place (r) differ by a factor of c is already regrettable. energy, momentum, or mass, Eq. 4 contains the physical Introducing yet another unit specifically to describe the content of Einstein’s famous energy-momentum relation. interval s would only further muddle what ought to be Multiplying through by (hc)2, where h is Planck’s con- a straightforward geometric (i.e. pseudo-Pythagorean) stant, on both sides of Eq. 4 puts this equation in its relationship. But, as we will see, just such a muddling traditional form: occurs when the dynamic counterpart of Eq. 1 is pre- sented in its traditional form. p pµ c2 = E2 p2c2 = m2c4, (5) µ − 3 where we have defined the four vector pµ = (E/c, p), as oughly obfuscate its geometric content. It is small won- well as three ‘new’ physical quantities: the energy E der that physics is considered an arcane discipline. ≡ hf, the momentum p hk, and the invariant (or rest) Equation 5 might seem to have more content than ≡ mass m hkC /c. Eq. 4, or at least to be more intuitive. In fact, energy ≡ These three relationships (Table I) between wave and momentum are intuitive only to those who have been parameters and their corresponding classical variables suitably trained — one might say indoctrinated — in were discovered and introduced separately through the their use. These quantities are actually relatively ab- Planck-Einstein relation (1900–1905), the de Broglie hy- stract, since they cannot be measured directly, even in pothesis (1924), and the aforementioned Compton for- principle. The lack of experimental access is a substan- mula (1923), respectively. Typical pedagogy does not tial failing. To appreciate the advantages of explicitly apply the Planck-Einstein relation to massive objects, considering the means of measurement, recall that a key and connects these three ideas only to the extent that advance in the development of special relativity was these they were all instrumental to the development of quan- deliberate definitions: time is what clocks measure, and tum mechanics. In particular, it does not present the distance is what rulers measure. What then, are the cor- unifying, geometric picture implied by Eq. 4. responding operational definitions (Hecht, 2017) of en- The path to Eq. 4 as presented here is fast and di- ergy and momentum? Such definitions do not exist; we rect: find the invariant length in real space, introduce do not have meters for either energy or momentum. It is the Fourier-conjugate variables, and find the equiva- an interesting exercise to ponder how these quantities are lent invariant length in reciprocal space. In compari- determined experimentally in various situations. For me- son, the traditional route to Eq. 5 is lengthy and cir- chanics problems, we have only three basic measurement cuitous. One first elevates momentum conservation to operations: counting, measuring position (with rulers), the status of a foundational principle. Then, work- and measuring time (with clocks). Some combination of ing backwards from Newton’s laws and employing the these three basic measurements determine other physi- work-energy theorem, one reverse-engineers definitions cal quantities : frequency (both spatial and temporal), of momentum and energy that satisfy this principle angle, velocity, acceleration, energy, momentum, torque (Eisberg and Resnick, 1985; Kleppner and Kolenkow, etc. For example, to determine the momentum of a foot- 2014; Taylor and Wheeler, 1992; Thornton and Marion, ball, we would determine its mass and velocity separately, 2004). With a little more effort, one can obtain the and then do a simple calculation. We might weigh the same result with momentum and energy conservation football with a scale, thereby determining its mass via alone (Jackson, 1999; Symon, 1971). These arguments position (and an auxiliary determination of the acceler- are subtle (it is not immediately obvious that momentum ation due to gravity), i.e. the deflection of the spring. is something other than merely that-quantity-which-is- And we might use clocks separated by a pre-measured conserved) and nearly circular (they justify the exact law distance to determine the velocity. by appeal to the inexact). These traditional approaches In the list of ‘physical quantities’ just given, the first also rely on an unnecessarily large number of axioms. As two items, spatial and temporal frequency, are in a dif- we will see in Section IV, we can derive energy conser- ferent category than the others. They are not less funda- vation, momentum conservation, and Newton’s laws all mental than distance and time, in that distance and time from Eq. 2. cannot be defined without reference to frequency. To be Furthermore, getting to the traditional Einstein useful, clocks and rulers must contain some repeating energy-momentum relation (Eq. 5) via reciprocal space unit. A ‘clock’ must tick more than once, and ‘rulers’ reveals that the final step — multiplying through by (hc)2 must either be subdivided into sub-units, or be applica- — has no physical content, and in fact buries some valu- ble in multiple instances (e.g. laid end-to-end). Thus able ideas. First, the Fourier-transform relationship be- we cannot measure time without access to a frequency, tween the conjugate pairs (t,f) and (r, k) is obscured in nor can we measure distance without access to a spatial the ‘new’ units. Second, mass has been defined such that frequency. While the circularity of these base definitions it agrees dimensionally with neither E nor p, thus hid- might puzzle logicians and philosophers, this lexical issue ing what would otherwise be a straightforward pseudo- causes no difficulty for physicists in practice. It also em- Pythagorean relationship in reciprocal space. In the lan- phasizes the symmetry (in the sense of a lack of prece- guage of Taylor and Wheeler’s parable of the survey- dence) between the real-space (i.e. spacetime-domain) ors (Taylor and Wheeler, 1992), we are measuring north- and the reciprocal-space (i.e. frequency-domain) vari- south distances in joules, east-west distances in kg m/s, ables.1 and the distance between two points in kilograms.· Thus Eq. 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 and 1 Since number is determined by counting, from a physics stand- kilograms) for its three terms, and could not more thor- point we consider it to be well-defined. Mathematically its def- 4

kinematic dynamic Lagrange Galileo Newton Hamilton real space Fourier reciprocal space ×h traditional time ct ⇐⇒ temporal frequency f/c Planck-Einstein energy E/c = hf/c position r ⇐⇒ spatial frequency k de Broglie momentum p = hk Pythagoras, Einstein, Minkowski interval s ⇐⇒ Compton wavenumber kC Compton mass mc = hkC event xµ wave kµ object pµ

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 them out of reciprocal units. As for the dynamic quantities, every object has an energy, a momentum, and a mass, and every wave has a frequency, a spatial frequency (or wavevector), and a Compton wavenumbera. The ‘object’ and the ‘wave’ pictures give physically 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 the classical limit. See Appendix E.

Compared to energy and momentum, frequency and respectively, which the photon does have. Mass too cor- spatial frequency are much less abstract, in that they can responds to a length scale, and for the photon this length be measured directly. A frequency is measured by count- happens to be infinite. Mass was given its own, dimen- ing cycles in a known time. A spatial frequency is mea- sioned unit long before the connection between mass and sured by counting cycles in a known distance. Of course, length was understood, and applying the historical lan- practical considerations make such direct measurements guage to a modern concept such as the photon leads to hopeless for macroscopic objects such as footballs (see this inconsistency. note at the end of Section V). Thus, assuming waves in the spacetime of special rel- Replacing the archaic concepts of energy and momen- ativity gives us mass. In other words, objects have tum (Eq. 5) with frequency and spatial frequency (Eq. 4), mass because they are waves (Table I). This statement respectively, anticipates questions that can arise at an el- is exactly counter the orthodox presentation — usually ementary level and yet are difficult to answer in the old the wave nature of a massive object is considered to language. The modern development avoids the question, arise despite its mass. The conflict in the concept of “how can photons have energy and momentum, if they ‘wave/particle duality’ that is so often emphasized is do not have mass?” Since a photon has zero mass (kg, in partly an illusion. SI units) and non-infinite velocity (m/s), it is legitimate Having arrived at the concept of mass via this route to ask how it can have non-zero energy (kg m2/s2) and (i.e. Fourier conjugate variables) further encourages us · momentum (kg m/s). The answer, we see here, is that to re-examine the various classical conceptions of mass. · energy and momentum are better thought of as repre- Historically mass has been considered “a quantity of mat- senting (inverse) time and length scales (i.e. frequencies), ter”, a disinclination to accelerate (i.e. inertia), or a measure of participation in the gravitational interaction (Hecht, 2010). These ideas date back to Newton and inition is just as circular — counting determines number, and have been continuously refined in the intervening cen- number is what counting determines — but we view mathematics turies, but none of them is entirely satisfactory (Coelho, as being on a firmer logical footing than physics. We understand 2012; Hecht, 2005, 2010). As we will show shortly, iner- number better than we understand time. tial mass does not need to be introduced as a postulate, Thus number is preeminent among the measurable quan- tities, as can be seen from the current (circular) statement since the inertial effect can be derived from other (wave) defining the units of time and frequency in terms of num- axioms. Moreover, instead of having inertia defined in ber. With the ground-state hyperfine splitting of the cesium- relation to the positions of the “fixed stars”, as Mach 133 atom as the standard oscillator, by definition one second is would have it, inertia is defined in relation to positions the time required to complete a number (9,192,631,770) of cy- cles, and the oscillator’s frequency in hertz is that same number in reciprocal space, a place much less remote. The in- (Stenger and Ullrich, 2016). ertia of a mass is not “an effect due to the presence of While classical mechanics treats both time and distance as all other masses” (Pais, 2005), but rather a real-space ef- measurable, more advanced physics makes the preeminence of fect arising from an invariance constraint (namely Eq. 4) number manifest. Quantum mechanics treats time as a parame- ter while maintaining position as an observable. Quantum field in reciprocal space. Distant, massive objects are not re- theory treats both time and position as parameters, which leaves quired; Eq. 4 and the equations that follow (Section IV) number as the only measurable quantity remaining. suffice to derive the inertial effect. 5

Thus we can take Eq. 4 to define mass: mass is the generally presented as a postulate that has no underly- invariant reciprocal length associated with a wave of fre- ing physical motivation, other than that it gives the de- quency f and wavevector k (times h/c in traditional sired equations of motion (Dirac, 1967; Feynman et al., units). This definition is exactly analogous to that of the 2010, 1964; Goldstein et al., 2002; Itzykson and Zuber, interval: the interval is the invariant length associated 1980; Landau et al., 1976; Peskin and Schroeder, 1995; with an event at time t and position r. Both definitions Sakurai, 1967; Schiff, 1968; Thornton and Marion, 2004). are closely tied to the structure of spacetime, and neither Unlike the Planck-Einstein relation, the de Broglie hy- relies on the concept of ‘force’, which is both difficult to pothesis, and the Compton formula (Table I), the relation define and almost ignored by advanced physics theories S = hφ that connects the wave parameter (φ, the phase) (e.g. quantum mechanics and quantum field theory). and the classical quantity (S, the action) is not well The modern, wave-based approach further suggests known, with some authors declining to state it clearly a possible strategy for addressing a long-standing open when the opportunity arises (Peskin and Schroeder, problem, namely implementing an operational definition 1995; Sakurai and Napolitano, 2017). of mass (Hecht, 2005). Intervals are generally determined Having established that considerations of interference with rulers and clocks (Eq. 1), and, in the specific case motivate the principle, we can derive the classical equa- of a purely time-like separation, an interval can be mea- tions of motion by considering the phase φ as it de- sured with just a clock. A mass is the reciprocal-space velops along a path, and then applying the calculus of analog of a purely time-like separation (Eq. 4), which sug- variations. Then the derivation, based on the ‘modi- gests that a mass might be determined by measuring its fied Hamilton’s principle’, is standard (Goldstein et al., Compton frequency. In fact, the use of ‘rocks as clocks’ 2002; Landau et al., 1976; Thornton and Marion, 2004), has been proposed (Lan et al., 2013), although the idea but for the fact that the phase φ, the frequency f, and is not yet well understood (Pease, 2013; Schleich et al., wavevector k are taking the places of the action S = hφ, 2013; Wolf et al., 2012). the energy E = hf (or, better, the Hamiltonian H = hf), and the momentum p = hk, respectively (Table II).2 The phase developed along a path from (ta, a) to (tb, b) IV. STATIONARY PHASE is given by

It is hardly an exaggeration to say that the rest of φ = kdr fdt, (6) physics, excepting thermodynamics, will be built around − Zpath the phase φ. One more postulate — that φ is invari- ant under ‘gauge’ (or phase) transformations — will which can be converted into an integral over time alone by defining the velocity r˙ dr/dt and making the other be used to introduce interactions into the theory. In- ≡ variance under local U(1) gauge transformations gives variables parametric functions of time, which gives the electromagnetic interaction. Invariance under local tb tb SU(2) and SU(3) gauge transformations leads to the φ = φ˙ dt = (k˙r f)dt. (7) − weak and strong interactions respectively (Cheng and Li, Zta Zta 1991; Griffiths, 2010; Halzen and Martin, 1984). Postponing these more advanced topics, we can de- (In traditional notation this expression reads S = rive the results of classical mechanics by invoking an L dt = (p˙r H)dt, where L is the Lagrangian.) The extremum condition− is enforced by setting δφ = 0. Ap- extremum principle: the principle of stationary phase. R R We postulate that classical trajectories or paths in space- plying the chain rule then gives time are determined by the condition that φ is station- tb ary along the trajectory (Cohen-Tannoudji et al., 1977; δφ = (δk ˙r + k δ˙r δf) dt. (8) − Commins, 2014; Dirac, 1967; Peskin and Schroeder, Zta 1995; Sakurai and Napolitano, 2017), i.e. that δφ = 0. Trajectories not satisfying this condition are suppressed Interchanging the order of operations in the second term k ˙r k r by destructive interference with neighboring trajecto- gives δ dt = d(δ ), which can be integrated by parts: ries; minute changes in the trajectory give an ampli- tude of the opposite sign. This idea, which is moti- k d(δr)= k δr b δrdk. (9) |a − vated by the wave picture, ultimately leads to the path Z Z integral formulation of quantum mechanics (Feynman, 1948; Feynman et al., 2010, 1964; Itzykson and Zuber, 1980; Styer et al., 2002). Classically it is known 2 When viewed from a historical perspective, this equivalence be- as the principle of stationary action (Ogborn et al., tween classical mechanics and an underpinning wave theory is 2006) (or, in some nomenclatures, Hamilton’s principle unsurprising, given that Hamilton developed his formulation by (Feynman et al., 1964; Landau et al., 1976)), and it is analogy with optics (Symon, 1971). 6

We have defined the path to begin at a and end at b. and f, or, equivalently, the Hamiltonian H, some depen- While the path varies by δ between a and b, at the end- dence on (most commonly) position. Usually this poten- points δ = 0, so the surface term vanishes by construc- tial term is not obviously Lorentz invariant, as the coor- tion. Thus we have dinate dependence is written with respect to a preferred

tb reference frame. And it may show no sign of its origin ∂f ∂f δφ = (δk ˙r δrk˙ δk δr) dt, (10) as a Lorentz invariant gauge interaction. For instance, − − ∂k − ∂r Zta the forces that maintain the tension in a pendulum’s where we are taking f to be a function of the position string, or that push back when a spring is compressed, are coordinates r, the corresponding spatial frequencies k, at some level electromagnetic, but in these phenomeno- and the time t. Collecting terms gives logical descriptions electric charges and electromagnetic fields never appear. Nonetheless the dynamics that follow tb ∂f ∂f δφ = δk (˙r ) δr (k˙ + ) dt. (11) from the canonical equations of motion (Eqs. 12–13) can − ∂k − ∂r Zta   be understood as ultimately arising from the symmetries Because the variations δr and δk are independent, their of φ under gauge transformations. pre-factors must vanish to guarantee δφ = 0. Thus we Also, for simplicity of exposition, to indicate the real- have two relations, space position we have been writing r, which in a Carte- sian system decomposes to r = xˆx + yˆy + zˆz. The dr ∂f ˙ ˙r = = and (12a) quantity φ, the equivalent of the Lagrangian in this dt ∂k modern picture (i.e. L = hφ˙), can be written alter- dk ∂f natively in terms of generalized coordinates q , where k˙ = = , (12b) i dt − ∂r the qi are any set of quantities that completely spec- which are the canonical equations of motion, one for real ify the state of the system. Then the wave-equivalent space and one for reciprocal space, respectively. Multi- of the generalized momentum canonically conjugate to ˙ ˙ plying the first by h/h and the second by h puts them in qi is ki ∂φ/∂q˙i. In other words, instead of φ(r, r˙,t) ≡ ˙ Hamilton’s form: we can write φ(qi, q˙i,t) without invalidating any of the arguments presented above (Goldstein et al., 2002; dr ∂H ˙r = = and (13a) Landau et al., 1976; Thornton and Marion, 2004). This dt ∂p flexibility, one of the main advantages of the Hamilto- dp ∂H ˙p = = , (13b) nian and Lagrangian formulations relative to the New- dt − ∂r tonian formulation of classical mechanics (Symon, 1971), which define the group velocity and relate the time rate is of course unimpaired by the switch to the wave-based of change of the momentum p to the gradient in the description. (If some qi depend explicitly on t, or the Hamiltonian H, respectively. With the force F defined interaction-derived component of f depends onq ˙i, then as the negative of that gradient, Eq. 13b is Newton’s the Hamiltonian is not equal to the total energy, which second law. (By definition the potential energy U is the leads to the preference expressed parenthetically above ∂H ∂U Eq. 6 (Thornton and Marion, 2004).) position-dependent portion of H, so ˙p = ∂r = ∂r = U F.) − − The equivalents of energy and momentum conser- −∇In traditional≡ language, one says that Newtonian me- vation, and other consequences of Noether’s theorem chanics is equivalent to the statement that the classical (Goldstein et al., 2002; Hanc et al., 2004) (e.g. angu- physical path extremizes (usually minimizes) the action lar momentum conservation), follow immediately from S (Abers, 2004). In modern language, one says that the Eqs. 12. The total derivative of f(r, k,t) with respect to classical path extremizes the phase φ, a property intrinsic time is to waves. Far from being at odds with wave physics, as is df ∂f ∂f ∂f = ˙r + k˙ + . (14) usually implied, classical mechanics follows directly from dt ∂r ∂k ∂t the most basic and general elements of the wave picture. The first two terms cancel by Eqs. 12. Thus if the fre- Some comment is required here, since, by writing φ as quency f has no explicit time dependence, df/dt = 0 an integral (Eq. 6) instead of the simple product (Eq. 2), and f is a conserved quantity. Similarly, by Eq. 12b, we have passed from a discussion of free waves (Eqs. 4–5) if f is independent of some component qi of r, then to one that can accommodate a potential (Eqs. 12–13). k˙i = ∂f/∂qi = 0 and ki, the component of the spatial At the level of fundamental particles, interactions are − frequency (or wavevector) conjugate to qi, is conserved. introduced by postulating the gauge symmetries men- tioned earlier and considering one-on-one interactions. For macroscopic objects the number of waves involved V. HISTORICAL PERSPECTIVE makes this approach impractical, and the sum interac- tion of many-on-one, or many-on-many is described phe- Let us compare the traditional and the wave-based de- nomenologically by adding a potential term that gives k velopments of classical mechanics. When Newton wrote 7

traditional wave µ µ space (x ) to the boundary (φ = kµx ), and then to action S = h× φ phase reciprocal space (kµ), instead of via− a giant leap to the Lagrangian L = h× φ˙ phase time derivative dynamical variables followed by a tortuous journey back. Hamiltonian, energy H, E = h× f frequency Returning to the discussion of measurement from Sec- momentum p = h× k spatial frequency, tion III, we see that identifying the wave origins of the wavevector angular momentum J = h× J angular momentum action S moves this quantity from the realm of the ab- − ∂H × − ∂f stract to that of the very concrete, for now we can give it force F ∂r = h ∂r h × an operational definition. Namely, the action is defined mass m = c kC Compton wavenumber by S hφ, where φ is the number of cycles in a wave, TABLE II Traditional variables and their wave-based coun- relative≡ to some arbitrary origin. This definition can be terparts. In ‘wave’ units J is taken to be dimensionless. The compared to that of time t, which is the number of ticks theory can be completely specified in terms of wave parame- of a clock, relative to some arbitrary origin, and position ters. To the extent that the concepts on the left are distinct r from those on the right, none of them are necessary. , 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, in that the counting operation does not rely on the circular F = ma, he was essentially defining the key concepts of (e.g. time-frequency) bootstraps discussed in Section III. force F and mass m in a simultaneous, circular bootstrap. In this sense, the action S, far from being abstract and Operating in complete ignorance of both the space-time difficult to understand, is, via its connection to the phase connection and the wave properties of matter, Newton φ, preeminent in the list of well-defined and measurable had little to work with. He had to invent his dynamical quantities. (Here we are speaking from a perspective that variables out of nothing, and also to intuit useful rela- is both ‘in principle’ and strictly classical, in that we are tionships between these variables. That his construction ignoring both problems having to do with the large size survives, after more than three centuries of ceaseless sci- of φ in all but the smallest systems, and those having entific advances, as the practical description of everyday to do with quantum measurement (Bassi and Ghirardi, physics is a testimony to the magnitude of his achieve- 2003; Commins, 2014; d’ Espagnat, 1999; Jammer, 1966; ment. Lalo¨e, 2019; Lamb and Fearn, 1996).) The Lagrangian and Hamiltonian formulations of clas- Thus the wave-based development requires neither ad sical mechanics, meanwhile, represent another leap in un- hoc definitions nor unmotivated postulates (i.e. δS = derstanding. These formulations make, via the action S, 0). Three centuries of effort have done more than build the explicit connection between kinematic variables and a theory that is more accurate and broadly applicable. dynamic variables that we now recognize as the phase They have also revealed underlying principles that make µ µ φ = kµx , which sits between real space (x ) and re- the theory more coherent and easier to understand. ciprocal− space (kµ), taking equally from both (Table I). With the Planck-Einstein relation, the de Broglie hy- pothesis, and the Compton formula, the dynamical vari- VI. CONSTANTS AND CONCEPTS ables are identified as wave parameters and the last con- nections are finally made. Traditional pedagogy gives the impression of separate The wave-based formulation, on the other hand, does regimes: small systems have wave-like properties, while not require the incredible improvisations that produced larger, classical systems do not. The development pre- F, m, and S. The analogous round of definition occurs sented here shows that these regimes are not so sep- with the introduction of f and k, but in this case the arated. The subject of classical mechanics is waves: definitions follow directly from the wave postulate (or well-localized waves with unmeasurable phases. In other even from the definitions of time and distance — see words, classical mechanics treats the limit where the wave Appendix E). Their meaning is already mathematically ‘packets’ have phases that are uncountably large and precise: these quantities are conjugate to those variables widths that are negligible in comparison to the other that define the spacetime arena, t and r, and provide an length and frequency scales in the problem (see Appen- alternative but equivalent description of the arena’s con- dices C and E). tent via the Fourier transform (Appendix A). While the By invoking only three closely-related ideas — special wave-based formulation does not remove circularity from relativity, waves, and an extremum principle — we have the base definitions — see Section III and Appendix E — reproduced the exact (i.e. relativistically correct) dis- such an achievement is not to be expected (G¨odel, 1931; persion relations, equations of motion, and conservation Tarski, 1936). And progress is evident, in that by deriv- laws of classical mechanics. Mass has not been inserted ing inertial mass we have removed this important con- as a distinct postulate of the theory, but has emerged as cept from the circle of definitions. The progression from a geometric property of waves in Minkowski spacetime. the definitions is also more orderly, proceeding from real We find that the traditional dynamical quantities (Ta- 8 ble II) such as action, energy, and momentum are wave relativistic and quantum effects are not easily accessed properties cloaked in units that disguise these origins. because c is large and h is small, and that, with a god- According to this viewpoint, c and h are not ‘fun- like power to adjust these constants, we could make damental’ constants, as they are often called, but ves- our universe appear less classical to our un-aided senses tiges of a more primitive (Newtonian) understanding of (Gamow, 1964). Such statements depend upon non- the underlying physics. Time, distance, energy, momen- obvious assumptions, seldom given explicitly, on what tum, and mass were introduced as distinct aspects of constraints are enforced as these constants are adjusted. physical theory before the interrelationships were under- But if the values of these constants are immaterial, why stood. When, for instance, mass and time seemed un- then do we experience the world classically? The answer related to distance, each was measured in its own dis- is that, as complex, sentient beings, we consist of many tinct units. Later the relationships were discovered, and fundamental particles. Any being sophisticated enough constants were introduced to describe the conversions to do science or metrology must necessarily have a huge (Abers, 2004). number of degrees of freedom. The question, “why is Mass is a particularly interesting case, for the follow- Planck’s constant so small?” would be better phrased, ing reason: we experience everyday life in real space. “why is the kilogram so big?” The answer is that the This fact makes it remarkable, and perhaps even counter- kilogram consists of a number of atoms that has been intuitive, that the real-space constraint described by the (somewhat arbitrarily) chosen to represent a quantity of interval (Eq. 1) is not noticeable without scientific equip- matter that humans can easily access and manipulate. ment, while the reciprocal-space constraint correspond- Planck’s constant is small and the kilogram is big be- ing to mass (Eqs. 4 and 5) has obvious real-space mani- cause we consist of very many — more than Avogadro’s festations. Thus the interval s was not measured before number N 6.02214076 1023 (Inglis et al., 2019) — A ≡ × it was discovered theoretically — if it had been known fundamental particles. The Planck’s constant’s small nu- previously, a special unit would have been invented to de- merical value is not “responsible for the fact that quan- scribe it! But mass has been known since Newton, and its tum phenomena are not usually observed in our everyday proxy, weight, has been measured since antiquity. Thus life” (Gamow, 1964). Rather, quantum phenomena are special units were invented to quantify mass before its not usually observed because we consist of many fun- relationship to other physical quantities was understood. damental particles. The small value for Planck’s con- Despite the misleading language, it is well (if not uni- stant represents a choice that humans have made, and versally (Eisberg and Resnick, 1985; Inglis et al., 2019; this choice has no impact on any physical phenomenon. Young et al., 2020)) known that the ‘fundamental con- For objects consisting of few fundamental particles, stants’, c and h, are merely conversion factors set by hu- large boosts and coherent phase control are achievable man convention (Abers, 2004; Duff, 2015; Ralston, 2012, with current technology, which allows access to obvi- 2013; Stenger and Ullrich, 2016; Taylor and Wheeler, ously relativistic and quantum-mechanical phenomena, 1992). The revised International System of Units, or respectively. As the number of potentially independent SI, of 2018 recognizes the distinction by purposefully degrees of freedom grows, however, it becomes progres- referring to them as ‘defining’, as opposed to ‘funda- sively more difficult to reach the non-classical regimes. mental’, constants (Stenger and Ullrich, 2016). In 1983 Many particles might be boosted to relativistic velocities and 2018 mortal scientists defined the exact values c via a nuclear explosion, for instance, but such an event is 299, 792, 458 m/s and h 6.62607015 10−34 kg m2/s,≡ generally hostile to the coherences that maintain a com- respectively (Inglis et al.≡, 2019). The value× of c sets· the posite object as a unified whole. Quantum phenomena relationship between the marks on the face of a clock and are inaccessible to many-particle objects both because the marks on a ruler. The value of h sets the relationship the objects are large in comparison to their wavelengths, between those marks and the marks on a scale, a bal- and because objects with many degrees of freedom cannot ance, or a reference mass. Great pains have been taken be sufficiently well-isolated from the decohering effects of to define these constants so as to maintain compatibility interactions with the environment (Cronin et al., 2009). with the historical values, and to take full advantage of Thus making an observationally less-classical universe the best available metrology (Bord´e, 2005; Inglis et al., would be substantially more involved than simply ad- 2019). But the values human scientists choose to recog- justing c and h — somehow the observers (e.g. Gamow’s nize as c and h would not be recognized by alien scien- Mr. Tompkins) must be constructed from far fewer con- tists. In that sense they are not fundamental, but rather stituent parts. the nearly random products of human custom and his- While the numerical value of Planck’s constant h has tory. no physical significance, we can attribute a significance to Adjusting these constants would be like getting rid of its value as the ‘quantum of action’ (Messiah, 2014), the gallons, pounds, and feet in the United States: substan- very existence of which puzzled Planck himself and many tial economic and cultural — but no physical — con- others since (Commins, 2014; Jammer, 1966). Here we sequences would result. It is sometimes implied that see that one quantum of action, i.e. S = h, corresponds 9 to a phase φ = 1cycle(or2π radians), which could hardly stant, unity or otherwise, is irrelevant. be more natural. After all, phase is introduced (Eq. 2) as As a conversion factor, Planck’s constant h is more a countable quantity, and the natural units for counting difficult to defend than the speed of light c. The speed phase are cycles (see also Appendix A). From the wave of light connects time and distance. These concepts perspective, the quantization that accompanies the quan- are physically and mathematically distinguishable: time tum of action also becomes less mysterious. For instance, has an arrow, while space does not, and time-like co- if energy and frequency are considered to be distinct con- ordinates appear with important minus signs relative cepts, one can ask why photons of frequency f necessar- to their space-like counterparts in the invariants (Ap- ily carry energy quantized in increments of E = hf. But pendix E). While alien scientists would not recognize our does it make sense to ask why photons of frequency f value for c, they would at least recognize the distinctions carry frequency f? we make between time and space. Planck’s constant, on ‘Natural’ unit systems where ~ = c = 1, such the other hand, connects energy to frequency, momen- as are sometimes employed by particle physicists and tum to wavevector, and action to phase. But no physical cosmologists (Halzen and Martin, 1984; Jackson, 1999; or mathematical distinction can be made between these Peskin and Schroeder, 1995; Sakurai, 1967; Tanabashi, quantities, other than the artificial one introduced by 2018), bear a resemblance to the wave-based approach Planck’s constant itself. presented here. Unfortunately, however, such systems As has been noted previously (Bord´e, 2005), the ex- are not employed at the elementary level, and common istence of ‘fundamental’ constants with dimension indi- implementations fail to reap many of the possible bene- cates that we are calling the same thing by two different fits. For instance, the electron-volt (eV), instead of the names. Some authors see no problem, emphasizing that second or the meter, is often chosen as the base unit. This the number of dimensional constants is arbitrary (Abers, choice is confounding on several levels. As an energy unit, 2004; Jackson, 1999; Wichmann, 1971), while others re- the eV base unit obscures the Fourier transform relation- gard dimensional constants as evil (Bridgman, 1931). At ship between physical quantities, such as, for instance, a minimum having multiple names for one thing can be a resonance energy width and the corresponding particle considered uneconomical, and therefore in violation of a lifetime. The use of the eV is usually restricted to recipro- principle that has a long tradition in physics. In his Prin- cal space, which further hides the real-space/reciprocal- cipia Newton himself declares (Newton et al., 2016), as space connection; adherents to this system rarely quote the first rule of reasoning, that times or distances in eV−1 (Halzen and Martin, 1984; No more causes of natural things should be Sakurai, 1967; Tanabashi, 2018). But this particular unit admitted than are both true and sufficient to also references electromagnetism. Earlier we remarked explain their phenomena. on the logical problems with using energy units to de- scribe the massless photon. Describing particles such as As we have argued here, the Newtonian concepts of en- the neutrino, the π0, and the neutron, which have no ergy, momentum, and action are not necessary to explain electronic charge, in terms of the electronic charge e is mechanical phenomena — the more elementary concepts equally inelegant. Energy, momentum, and mass — or of frequency, spatial frequency, and phase are fully suf- better, the corresponding frequencies — are general con- ficient to construct a complete explanation. Thus New- cepts that apply outside the more limited scope of elec- ton’s principle of economy dictates that the unnecessary tricity and magnetism, and as such should not be tied to concepts should not be admitted as “causes”. Adding electromagnetic units. these concepts to the theory is like adding unnecessary Another deficiency with ~ = c = 1 unit systems, as lines to a drawing or unnecessary parts to a machine, to usually implemented, is not the units per se, but rather re-purpose a famous analogy (Strunk and White, 2009). the vocabulary. These systems maintain energy and mo- Jettisoning energy, momentum, action, and the multi- mentum as distinct concepts. Just as alien scientists tude of conversion formulas involving Planck’s constant would not recognize our values for c and h, so too they leaves behind a leaner and cleaner — but equally capable might not understand why we distinguish between energy — physical theory. and frequency. The distinction separating these concepts Replacing historical language with wave language gives is historical and cultural, not physical. In the wave-based another perspective on familiar ideas. On first contact formulation of elementary physics, energy, momentum, this perspective might seem helpful, or might not. For and action are superseded. While not in the same cate- instance, the statement gory as phlogiston, caloric, and the luminiferous aether The energy describes the rate at which the (in that they are not incorrect), these concepts are un- action of an object evolves as a function of necessary physically and inefficient pedagogically. Their time. continued use should be discouraged. In the wave-based formulation they never appear, and neither does the con- is quite abstract, and does not evoke a physical picture. version factor they require. The value of Planck’s con- In wave language this statement reads 10

The frequency describes the rate at which the VII. SCALES phase of a wave evolves as a function of time. which is easy to visualize. On the other hand, making a As a practical matter, and to help internalize some similar substitution in sense of the scales involved, it is good to know a few conversion factors. To an accuracy of better than 1%, The total energy is the sum of the potential energies and distances are inter-converted with hc = energy and the kinetic energy. 1240 eV nm = 2 10−25 J m. While eV nm are usually · × · · gives convenient in condensed matter and atomic physics, to better fit the typical energy and length scales one might The total frequency is the sum of the poten- tial frequency and the kinetic frequency. prefer the equivalent units meV µm in biological physics, keV pm in electron microscopy,· or MeV fm in particle which might initially seem bizarre. That wave language physics.· · clarifies in some instances, and appears strange in others, Masses are inter-converted with h/c = 2.21 can be taken as prima facie evidence that the historical 10−42 kg m and c2/h = 1.36 1050 Hz/kg. The fantas-× vocabulary is hiding potentially valuable ideas. tic size of· these numbers reflects× the human scale and its For instance, Feynman in his eponymous lectures intro- complexity. It is curious to contemplate how such scales duces energy as an abstract-but-conserved quantity. As are simultaneously both inaccessible (from a direct mea- recorded with his own emphasis, he says (Feynman et al., surement standpoint) and commonplace (as features of 1964) everyday life). We rarely discuss, for example, the Comp- It is important to realize that in physics to- ton wavelengths of macroscopic objects, because they are day, we have no knowledge of what energy so small so as to seem unphysical. The Planck length 3 −35 is. ℓP ~G/c 1.6 10 m, which corresponds to ≡ ∼ × the Planck mass m ~c/G 22 µg, is sometimes From the traditional perspective this statement is per- p P thought (Commins, 2014;≡ Garay,∼ 1995; Hossenfelder, fectly reasonable — Newton’s ideas offer no underlying p foundation for the concept of energy. However, it is dif- 2013; Misner et al., 1973; Peskin and Schroeder, 1995) ficult to imagine Feynman saying to represent a minimal length scale, for instance. But a mere kilogram, never mind an astrophysical object, It is important to realize that in physics to- represents a length scale that is far smaller. Thus the day, we have no knowledge of what frequency sampling theorem (Bracewell, 2000) argues against any is. simple interpretation of the Planck length in terms of a These statements are physically equivalent, and thus if discretization of spacetime. the 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 within VIII. CONCLUSION a ‘trivial’ unit conversion, energy is frequency, where energy/frequency describes the rate at which the ac- Physics is currently taught with a historical bent that tion/phase of an object/wave evolves in time. fails to embrace ideas that have been well-established The traditional and the modern, wave-based develop- for nearly a century. Planck’s constant is taken to be ments can be contrasted as follows. In the former, the the signature of quantum mechanical modernity, when it theory is constructed on the makeshift foundation formed should be viewed as a vestige of ‘antediluvian’ language with Newton’s circular (or nearly so (Feynman et al., (the flood here being the development of relativity and 1964)) definitions. When the Planck-Einstein, de Broglie, quantum mechanics) and a more primitive understand- and Compton relations are introduced as 20th century ing. The traditional development of classical mechanics developments, they appear rich but unconnected. With- is ad hoc, convoluted, and conflates the exact with the out any particular organization or underlying, unifying approximate. Newtonian ideas and language that are in- principle, it is not obvious, for instance, whether the troduced during this development (e.g. energy and mo- de Broglie relation is better thought of as p = hk, or mentum) permeate the field from top to bottom. These p = h/λ. Both make the wave connection, but only cultural and linguistic artifacts make physics more diffi- the former makes the important, additional link to the cult than necessary. Fourier conjugate variable. The wave-based develop- An ahistorical, wave-based formulation of ‘classical’ ment, on the other hand, has a sturdier foundation, mechanics, on the other hand, is more systematic and with independent and mathematically precise definitions. economical. Geometry and symmetry play clear and Here the Planck-Einstein, de Broglie, and Compton rela- central roles. A one-to-one correspondence exists be- tions appear as discoveries that connect unit conversions tween the kinematic variables r and t and their dynami- on the three sides of a triangle. While historically impor- cal counterparts, k and f. The equations of motion fol- tant, they are physically trivial. low from an extremum principle that is physically moti- 11 vated. Mass is not inserted into the theory by hand, but B. Choice of Wave Equation emerges naturally from the geometry of spacetime. More advanced theories, e.g. quantum mechanics, extend the As the attentive reader might have noticed, the moti- more elementary material, instead of discrediting it. In vation accompanying the introduction of φ in Section III short, the advances of the 20th century streamline and has a shortcoming. The generic wave equation (Eq. 3a), unify physics from the 17th century to the present, so which is invoked but not actually used, is not invariant that it is no longer necessary or desirable to so conscien- under a Lorentz transformation unless the phase velocity tiously retrace the exploratory steps taken by the earliest v = c, the speed of light. Thus it cannot be applied to developers of the theory. massive bodies and a more general wave equation must be sought. Unlike Eq. 3a, the Klein-Gordon equation

IX. AFTERWORD 1 ∂2u + 2u = (2πk )2u, (3b) − c2 ∂t2 ∇ C A. Waves and the Classical Theory provides a natural fit with Eq. 4. We decline to take this step in the first pass because the choice of wave equation By assuming waves in this development of ‘classical’ is unimportant for the development of classical mechan- mechanics, we have constructed a theory that is ‘quan- ics, and the generic wave Eq. 3a, unlike the Klein-Gordon tized’ from the beginning. After all, ‘first quantization’ Eq. 3b, is familiar from a variety of classical problems associates the classical variables E and p with wave pa- (e.g. sound, vibrating strings, electromagnetic waves). rameters f and k, and waves in bound or bounded states However, in a second pass Eq. 3b is preferred for its can only accommodate quantized values of f, k, and ability to accommodate mass while preserving Lorentz L r k. Thus this reformulation of classical mechan- ≡ × invariance. While seen less frequently, Eq. 3b can be em- ics is hardly classical at all. ployed in classical contexts — for instance, to describe The wave/quantum genesis of this formulation is a the braced string (Crawford, 1968; Gravel and Gauthier, strength, not a defect. Our goal is to present physics from 2011). a more unified and coherent standpoint. To the extent Because it fails to directly address (or even iden- that this approach blurs bright lines separating classical tify) the underlying wave equation and associated from quantum physics, so much the better. The wave- concepts such as superposition and dispersion, the based formulation moves the boundary between these two wave-based formulation of classical mechanics is areas, historically speaking, from 1900 to the mid 1920’s, manifestly incomplete. Rectifying this shortcoming as it explicitly includes (or, more accurately, obviates) leads to developing the full quantum theory. This the Planck-Einstein, de Broglie, and Compton relations, development dramatically expands the number of which are usually considered non-classical. It also in- postulates (adding e.g. states are represented by cludes the Heisenberg uncertainty relation (1927), which vectors in a Hilbert space, etc.) (Bassi and Ghirardi, is traditionally considered the archetypical quantum me- 2003; Cohen-Tannoudji et al., 1977; Commins, 2014; chanical result: by assuming waves, we have automati- d’ Espagnat, 1999; Griffiths and Schroeter, 2018; cally introduced incompatible observables. Fourier con- von Neumann, 2018; Saxon, 1968), which is an indica- jugate variables — x and kx, for instance — have a re- tion that this next level is not as well understood. striction on the product of their widths given by

∆x∆k 1/4π, (15) Appendix A: Fourier conjugate variable choice x ≥ While employed without comment previously, the which is a purely mathematical result (Bracewell, 2000; choice of the Fourier conjugate variable f for frequency Cohen-Tannoudji et al., 1977). Thus while the Heisen- over its radial counterpart ω 2πf is deliberate, and in berg uncertainty relation, keeping with the goal of economy.≡ Introducing radial fre- quencies does not obviously reduce the number of 2π’s, ∆x ∆p ~/2, (16) x ≥ but it increases the number of variables in use (unless one decides to forgo f entirely). The additional variable might seem mysterious in traditional units, it has both a ω adds a new way to write any formula containing f, but solid mathematical foundation and a simple, intuitive ex- without adding any new idea or perspective. Thus it in- planation in wave units. One does not need a physicist’s creases algebraic clutter for an at-best marginal decrease mathematical training to understand that it is not pos- in the number of 2π’s. sible to simultaneously measure, for example, time and The ordinary frequency f is preferred over the angular frequency to arbitrary precision. frequency ω for two reasons. First, it puts the Fourier 12 transforms in their symmetric, unitary form have (compare the signs in Eqs. A1),

∞ ∞ −2πik·r 3 u¯(f)= u(t)e2πiftdt (A1a) u¯(k)= u(r)e d r (B1a) −∞ −∞ Z ∞ ZZZ ∞ − 2πik·r 3 u(t)= u¯(f)e 2πiftdf. (A1b) u(r)= u¯(k)e d k. (B1b) −∞ −∞ Z ZZZ This form simplifies the interpretation of many transform Equations A1–B1 can be summarized pairs. For instance, the Dirac δ-function and unity are ∞ · Fourier transforms of one another (i.e. for u(t) = δ(t), u¯(kν )= u(xν )e2πik xd4x (B2a) −∞ u¯(f)=1, and foru ¯(f)= δ(f), u(t) = 1), without factors ZZZZ ∞ − · of 2π or √2π. (As Fourier conjugate variables E and p u(xν )= u¯(kν )e 2πik xd4k, (B2b) are worse than either choice of the frequency variables, −∞ ZZZZ for then normalization factors of Planck’s constant h or µ 2πik·x Dirac’s reduced-Planck’s-constant ~ also appear.) Sec- where k x kµx . In Dirac’s notation k x = e . · ≡ µ h | i ond and most importantly, f preserves the simplest and We choose φ = kµx (Eq. 2) with the leading minus − most symmetrical reciprocal relationship between the pe- sign so that the conversion between wave units and tra- riod τ 1/f and the temporal frequency f. ditional units for the action S = hφ will be like those for ≡ The wavevector, or spatial frequency, k is defined the energy H = hf and the momentum p = hk. The analogously, in that the wavelength λ 1/ k is like sign of the action/phase is purely conventional, since the the period τ 1/f. This wavevector≡ convention| | dif- trajectories are independent of whether the extrema is a fers from the≡ usual physicist’s convention by a factor minimum or a maximum (Section IV). Historically the of 2π. However, it is a standard choice in mathemat- choice has been to have dS/dt L p˙r H and not dS/dt L H p˙r, which≡ would≡ have− been better ics (Folland, 1989; Stein and Shakarchi, 2003), engineer- ≡ ≡ − ing (Bracewell, 2000), crystallography (Schwarzenbach, for the uniformity of our sign conventions. The choice of 1996), and in X-ray (Ewald, 1969; Jacobsen, 2019) and the sign defining the Lagrangian L, and thus the action, electron microscopy (Cowley, 1995; Reimer and Kohl, in this Legendre transformation was likely made because 2008), where the Fourier conversion between real p˙r > H for a free particle if its rest energy is ignored, space (imaging) and reciprocal space (diffraction) is a and the concept of rest energy was unknown when this frequently-performed, often push-button operation. The convention was established. convenience advantage in being able to go back and forth, without having to wrestle with 2π’s, between wavevec- tors and real-space distances is difficult to overstate. Appendix C: Quantum mechanics and its classical limit Besides giving the simplest (reciprocal) relationship be- tween wavelength λ and wavevector k, the microscopist’s Quantum mechanics is purported to present a picture convention also streamlines concepts such as the density that is wildly distinct from that of classical mechanics of states — compare, for example the formulas (L/2π)D (Griffiths and Schroeter, 2018). Classical mechanics is and LD for the density of states in k-space for a D- the intuitive, every-day world of massive bodies, while dimensional volume LD. quantum mechanics involves spooky, hard-to-understand waves (Mermin, 1985). Our purpose in this article is to From the perspective of the phase φ (a ‘countable’ de-emphasize that distinction. Classical mechanics is also quantity — see discussion after Eq. 2) as well, ordinary a wave theory. It happens that classical mechanics was frequency is preferred over radial frequency. Counting invented before its wave underpinnings were understood, cycles in a waveform can often be accomplished by in- but we can understand both it and subsequent devel- spection. Counting radians is not so easy. opments better by exploiting those same developments. Emphasizing the differences between classical and quan- tum physics has few pedagogical advantages, and under- Appendix B: Sign conventions standing quantum mechanics is much easier if one has a classical intuition that is already wave-based. µ We choose a metric (+ ) that gives x xµ > 0 for Quantum mechanics has some real mysteries: en- −−−µ µ 2 timelike separations and p pµ k kµ m > 0 for nor- tanglement, locality, and the measurement problem, mal objects. For consistency with∝ the standard∝ quantum for instance. However, other aspects of quantum me- mechanics conventions for H = hf and p = hk (first chanics that can seem problematic are not, such as ′ ′ two lines of Table III) we choose u(xµ)= e2πi(k r−f t) = the uncertainty principle and incompatible observables. ′ e−2πik ·x = e2πiφ to describe a plane wave with k′ > 0 It might seem deep and mysterious that one can- and f ′ > 0. Because of the hyperbolic metric, we must not simultaneously measure position and momentum 13

(Griffiths and Schroeter, 2018). But, as discussed in Sec- (Feynman et al., 1964). Applying the canonical commu- tion IX, it is comparatively obvious that one cannot si- tation relation C1 to the angular momentum operators multaneously measure position and spatial frequency. L r p gives the commutator ≡ × Using modern, wave-based units (i.e. dispensing with Planck’s constant) will improve the clarity of [Lx,Ly]= i~Lz. (C2) any presentation of quantum mechanics, which is manifestly a wave theory. But given that many The angular momentum operators are the gen- excellent references (Abers, 2004; Cheng and Li, erators of rotations (Cohen-Tannoudji et al., 1977; 1991; Cohen-Tannoudji et al., 1977; Commins, Goldstein et al., 2002). Their non-zero commutator says 2014; Dirac, 1967; Feynman et al., 2010; Gamow, that rotations in three dimensions do not commute, as is 1964; Goldstein et al., 2002; Griffiths and Schroeter, true both quantum-mechanically and classically. 2018; Landau and Lifshitz, 1981; Messiah, 2014; Not only does ~ 0 give a geometrically untenable → Sakurai and Napolitano, 2017; Saxon, 1968) contain result here (namely that rotations in three dimensions confusing statements about the role of Plank’s constant, do commute)(Ralston, 2012), it also muddies an oppor- particularly in regards to the classical limit, a few tunity to examine a general topic of wide applicability. additional words are in order. Non-commuting rotations give rise to ‘geometric’ phases, Taking ~ 0 is often cited as a means of reduc- which appear in the theory at the same level as the ing a quantum→ result to the classical limit (Abers, 2004; ‘dynamical’ phase φ. (In other words, the ‘geometric’ Cheng and Li, 1991; Dirac, 1967; Feynman et al., 2010; and the ‘dynamical’ phases are additive.) While elemen- Goldstein et al., 2002; Griffiths and Schroeter, 2018; tary enough to be demonstrated with just an arm and a Jackson, 1999; Jammer, 1966; Landau and Lifshitz, 1981; thumb (Chiao et al., 1989), geometric phases have classi- Messiah, 2014; Sakurai and Napolitano, 2017; Saxon, cal, quantum mechanical, and gauge theoretic relevance. 1968). As we have shown, Planck’s constant h = 2π~ Examples of topics elucidated with geometric phases is not a necessary element of physical theory: if wave- include the Foucault pendulum and cyclotron motion, based units are used from the beginning, this conversion the AharonovBohm and AharonovCasher effects, Wess- factor need never enter the discussion. And even if we do Zumino terms, their anomalies, and fractional statistics decide to use traditional units, then adjusting the size of (Cohen et al., 2019; Shapere and Wilczek, 1989). Planck’s constant is equivalent to changing the definition In wave-based units the angular momentum operators of the kilogram relative to the meter and the second (Sec- are dimensionless, and the ~ 0 problem with their com- → tion VI). Such re-definition of the agreed-upon standard mutator does not arise. There is never any implication unit of mass ought not to have physical effects. Thus, that rotations in three dimensions commute. A classical according to the perspective presented here, the value limit exists in the sense that, with a sufficiently large an- of Planck’s constant — physically irrelevant, and set by gular momentum, its three orthogonal components can custom — should have no bearing on the classical limit. be simultaneously determined. This limit corresponds to Another way to recognize this problem is to note that a failure to track the phase at the single cycle level (see some authors set ~ = 1, others state that the classical below). limit occurs as ~ 0, and some do both (Abers, 2004; The case of rotations, with its geometric roots and ap- Cheng and Li, 1991;→ Jackson, 1999). Taking both state- plications throughout classical and quantum physics, is ments at face value gives the nonsensical claim “~ =1 thus clearly important, and here the ~ 0 prescrip- → 0 in the classical limit”, indicating that one of these as-→ tion fails. The prescription can be patched by adding signments is of questionable generality. a special translation for relating the classical Poisson Some equations support the ~ 0 sophistry, but it bracket to the quantum mechanical commutator. The is not true generally and indiscriminate→ application can translation [u, v] (1/i~)(uv vu) gives the corre- → − lead to nonsense. For instance, applying this ‘rule’ to the spondence between classical functions on the left and canonical commutation relation quantum operators on the right (Goldstein et al., 2002; Sakurai and Napolitano, 2017; Schiff, 1968), but the ap- ~ [x, px]= i~ (C1) pearance of in the denominator of the patch just em- phasizes that ~ must drop out of the problem. and its corresponding uncertainty relation ∆x∆px ~/2 Thus the blanket prescription ~ 0 is not a reliable (Eq. 16) gives results that are sensible, if not reveal-≥ route to the classical limit. But we→ recognize real differ- ing. These relations imply that position and momentum ences between the quantum and the classical regimes, and are classically compatible, as indeed they are: both vari- it is helpful to have a prescription for passing from the ables can be simultaneously known to arbitrary precision former to the latter. How then is the classical limit found within a classical framework. without Planck’s constant, and to what extent does the But the ~ 0 rule gives results that are diffi- ~ 0 route to the classical limit make sense? → cult to defend→ in what might be the very next topic Examining the wave-based form of the canonical com- 14 mutation relation, is the Bohr radius a0 = ~/(αmec), where α is the fine- structure constant and me is the electron mass, so an [x, k ]= i/2π, (C3) 1/3 x object’s size ℓ x a0N , where N is the number of atoms in the∼ object.∼ For λ, a conservative (i.e. large and the associated uncertainty relation, λ) estimate corresponds to an object that has no appar-

∆x∆kx 1/4π, (15) ent motion. Ascribing the object’s motion to its thermal 2 2 ≥ energy kB T alone gives p /M = k /kC kBT . Then provides a more constructive perspective. The classi- λ h/√NAm k T , where the object’s mass∼ M comes ∼ n B cal limit does not correspond to taking the right-hand from its N atoms having A nucleons of mass mn. Com- sides to zero, for i/2π 0 and 1/4π 0 do not paring N 1/3 to N −1/2, we see that, for a macroscopic → → make sense. In fact, in the classical limit the product quantity N (e.g. NA, Avogadro’s number), ℓ λ inde- on the left-hand side of the uncertainty relation Eq. 15 pendent of any other details, and that the range≫ where does not even need to be large, and often it is not ∆x can simultaneously satisfy the inequalities C4 spans (see Appendix E). Coherent and displaced states of the many orders-of-magnitude. ‘quantum’ harmonic oscillator, which give the equality In passing we note that applying ~ 0 to the first in Eq. 15 and exhibit classical, oscillatory position and argument of the previous paragraph gives→ the conclusion momentum expectation values (Cohen-Tannoudji et al., that solids do not exist in the classical limit. This state- 1977; Griffiths and Schroeter, 2018; Saxon, 1968; Schiff, ment is true in the sense that ‘classical’ atoms are unsta- 1968), provide well-known examples of this general phe- ble to Coulombic collapse, and it leaves classical mechan- nomenon. ics bereft of practically all of its usual and fruitful fields In the classical limit trajectories are well-defined, so of application. From a wave perspective the stability of x ∆x and kx ∆kx (Cohen-Tannoudji et al., 1977; atoms is unsurprising, and the classical limit corresponds ≫ ≫ Messiah, 2014). If kx ∆kx we can take ∆kx/kx to taking the dimensionless N . ≫ ≃ → ∞ ∆λ/λ, which gives λ ∆λ. Thus, the uncertainty re- Many authors draw an analogy between the route ≫ 2 lation (Eq. 15) can be written ∆x∆kx ∆x∆λ/λ from physical (i.e. wave) optics to geometrical op- ≃ ≥ 1/4π. Satisfying the requirements for classical trajec- tics given by λ 0, and the route from quan- tories and the uncertainty relation simultaneously thus tum mechanics to→ classical mechanics given by ~ requires that 0 (Cohen-Tannoudji et al., 1977; Landau and Lifshitz,→ λ ∆λ 1981; Messiah, 2014; Sakurai and Napolitano, 2017). In x ∆x , (C4) traditional units this comparison is fairly called an anal- ≫ ≫ 4π ≫ 4π ogy, but in wave units the analogy is so apt that it is bet- which holds even in the limiting case (equality) of the ter viewed as an equivalence. (The wave picture tightens uncertainty relation. Note a subtlety here — we expect many analogies between optics and mechanics, such as the uncertainties to be small, not large, in the classical that between a refractive index and a potential energy limit. However, one can legitimately consider ∆x (Cronin et al., 2009).) Just as one passes from physical → ∞ or ∆x 0, depending on the basis for the comparison optics to geometric optics by taking λ 0 (or f ), → (which points out the hazards of limits in dimensioned so one passes from quantum mechanics→ to classical→ ∞me- quantities). In other words, with two (or more) other chanics. length scales in the problem, the uncertainty ∆x can be The ~ 0 approach to the classical regime for massive both large and small. It is large in comparison to the particles→ (this caveat is important — see next paragraph) wavelength λ. And it is small in comparison to x, a is thus better understood as concerning the limit where classical dimension in the problem (not necessarily the the frequencies are large in comparison to the other (in- position coordinate, which can be set to zero by choice verse) time or length scales in the problem. In other of origin). words, in the Planck-Einstein relation E = hf or the de Simple estimates of x and λ put macroscopic objects Broglie hypothesis p = hk, the classical variable (E or properly in the classical limit.3 The atomic size scale p) is held constant implicitly while Planck’s constant is taken to zero, which means that the frequencies are taken to infinity. In this limit the phase φ is not being tracked at the single cycle level, which is equivalent to the action 3 Exploring the quantum/classical boundary with nanoscopic ob- jects is an active area of research (Cronin et al., 2009). Quantum S not being tracked with a precision of h (Hanc et al., interference with ∆x ≫ ℓ has been demonstrated using 2000- 2003). atom molecules (e.g. C707H260F908N16S53Zn4) with masses of The ~ 0 prescription does hold one remarkable ad- 4 5 et al. 2.5×10 Da and size-to-wavelength ratios ℓ/λ of 10 (Fein , vantage for→ the expert practitioner. By judiciously choos- 2019). (One dalton, or unified atomic mass unit, is equiva- lent to approximately 0.75 fm−1.) Achieving interference with ing the ‘classical’ variable to hold fixed, one can suc- masses exceeding 105 Da looks feasible with current technology cessfully identify the classical limit for both the massive (Kia lka et al., 2019). and the massless cases. Massive particles have an asso- 15

traditional wave ical quantity is being tracked in classical units, it must be ~ ∂ i ∂ H = i ∂t f = 2π ∂t converted out of that unit system before it can be used p = ~ ∇ k = 1 ∇ i 2πi to actually do anything. In this sense the wave-based [x,p ] = i~ [x,k ] = i/2π x x unit system is ‘natural’ and the traditional system, with ∆x∆px ≥ ~/2 ∆x∆kx ≥ 1/4π L = r × p L = r × k its energies and momenta, is not. The time-evolution, ~ i translation, boost, and rotation operators all illustrate [Lx, Ly] = i Lz [Lx, Ly] = 2π Lz ′ ′ ~ ′ U(t ) = eiHt / = ei2πft this principle. ′ − · ′ ~ − · ′ T (r ) = e ip r / = e i2πk r ′ ′· ~ ′ ′· B(p ) = eip r/ B(k ) = ei2πk r ′ ′ R(θ′) = e−iL·θ /~ = e−i2πL·θ Appendix D: The quantum limit of classical mechanics dA i ∂A ∂A dt = ~ [H,A]+ ∂t = 2πi[f,A]+ ∂t Traditional classical mechanics gives no prescription TABLE III Standard quantum mechanical relations for finding the quantum limit, but with the wave-based written in traditional (Cohen-Tannoudji et al., 1977; formulation we can create one by reversing the arguments Griffiths and Schroeter, 2018; Styer et al., 2002) and wave- of Appendix C. The situation is not exactly symmetric, in based units, i.e. with and without the reduced Planck constant ~. Some important quantum mechanical operators, that we cannot reproduce a prediction of the exact theory such as the time evolution, translation, boost, and rotation by taking a limit in the approximate theory, but we can operators, are functions of 1/~ in traditional units, and thus at least identify where the approximate theory breaks become meaningless in the limit ~ → 0. down. Just as the classical limit of quantum mechanics is found by taking λ to be small, so the quantum limit of classical mechanics is found by taking λ to be not small. ciated energy E classically, and thus taking ~ 0 gives When we interpret such limits involving a dimensioned → f as just described. A massless (e.g. electromag- quantity, the existence of another, comparison scale is al- → ∞ netic) wave, on the other hand, is classically identified by ways implied. Quantum mechanics is widely understood its frequency. Here ~ 0 gives E 0, which indeed is to be the theory of small systems. Here ‘small’ means → → the classical limit. For instance, the quantum result for a that other length scales in the problem are comparable blackbody’s energy per mode, hf/(exp[hf/kBT ] 1) = to the wavelength λ. At this point the crude, approx- − E/[exp(E/kB T ) 1], reduces to kBT in the classical imate description generated by considering the station- − limit. ary behavior of the phase (i.e. the action) alone is no Turning these arguments around reminds us that the longer sufficiently accurate, and the specifics of the wave ‘quantum vs. classical’ limit is not the same as the ‘wave equation must be considered. Expressing these ideas in vs. object/particle’ limit. Traditional language gives a traditional language, one expects the classical theory to confusing, crossover behavior, namely that, in the quan- break down when the energy E and momentum p become tum limit, classical particles adopt wave properties, while small (giving small frequencies and large wavelengths). classical waves adopt particle properties (d’ Espagnat, However, while wave units provide a comparison scale — 1999). For example, consider a double-slit experiment namely the size of the system — traditional units do not performed with electrons or low-intensity electromag- provide comparison scales for the dimensioned quantities netic waves. The former show quantum behavior by E and p. Viewed from a purely Newtonian perspective, going through both slits, while the latter show quan- the energy and momentum can be made arbitrarily small tum behavior by hitting the screen in localized quanta, without entering a new regime that might be governed i.e. photons. In wave-based language the perspective by qualitatively different physical laws. Thus, without is simpler: λ 0 (or f ) produces particle be- the advantages of the wave-based perspective, traditional → → ∞ havior and λ (or f 0) produces wave behavior classical mechanics gives no hint that it is incomplete — → ∞ → always. In traditional language the correspondence be- hence the surprise accompanying the quantum revolution tween these limits and the quantum/classical limit de- of the early 20th century. pends on whether the objects under consideration are massive or massless. Table III gives a list of standard quantum mechanical Appendix E: Glossary formulas in traditional units side-by-side with their wave- unit counterparts. The equations with ~ in the denomi- To avoid slowing our development, in a few cases we nator become meaningless if one approaches the ‘classical have adopted the practice borrowing ‘physics words’, limit’ by taking ~ 0. In this way the ~ 0 prescription which ideally have precise meanings, from common En- fails for an operation→ such as, for instance,→ translation, glish without pausing to define them. Because of the which seems sufficiently mundane that it ought to have a bootstrapping problem, it may be impossible to artic- well-defined classical limit. The equations with ~ in the ulate definitions that are wholly satisfactory (Symon, denominator also illustrate a general principle: if a phys- 1971), so this practice is understandably common. Most 16 authors adopt it implicitly. Others warn that they are achieved for all the constituents of an object consisting capitulating with a few words such as “we can intuitively of many. In real space the classicality condition can be sense the meaning of mass” (Feynman et al., 1964), or satisfied by taking the object’s classical length scale ‘x’ “length, time, and mass are concepts normally already to be its spatial dimension ℓ (i.e. its size), and that scale understood” (Thornton and Marion, 2004), or “funda- to be much larger than the real-space packet’s width. mental physical concepts, such as space, time, simultane- But the reciprocal-space case requires a more nuanced ity, mass, and force...will not be analyzed critically here; argument. For instance, one might wonder why an object rather, they will be assumed as undefined terms whose — say a billiard ball — at rest is not delocalized. After meanings are familiar to the reader” (Goldstein et al., all, in the frame where the billiard ball is motionless, one 2002). Newton himself writes, “I do not define time, might think that its momentum is both small (p 0) and space, place, and motion, as being well known to all” well-specified (∆p 0), which would imply that∼ λ and (Hawking, 2002). ∆r are both large∼ (via the de Broglie and uncertainty Here we have carefully defined space and time in terms relations respectively). This conundrum is resolved by of their respective clock frequencies, and vice versa (Sec- noting that an “object’s rest frame” only exists in the tion III). We have argued that the dynamical quantities classical approximation. One can be in, or transform energy and momentum are just frequencies multiplied by to, a frame where an object’s center of mass is at rest. a dimensionful constant that is a historical legacy. And However, for a multi-particle object this point is only a we have defined mass as the corresponding hypotenuse of mathematical construction relative to which the object’s a hyperbolic right triangle that has temporal frequency constituent parts are generally in motion. and spatial frequency as its other sides. In so doing we Consider, for example, the object as a collection of have defined most of the concepts that appear in Tables I N coupled harmonic oscillators, which is a reasonable and II, which together cover classical mechanics. first approximation of a solid. Any collection of coupled However, the terms appearing in the bottom row of harmonic oscillators of natural frequency f0 will produce Table I merit further discussion. As stated earlier, an normal modes with frequencies both above and below f0 event is a position xµ in real space. From the structure (Thornton and Marion, 2004), and the frequencies of the of Table I then we deduce that a wave is a position kµ lowest-frequency normal modes scale like f0/N (Kittel, in reciprocal space, and an object is that same position, 2005). At any finite temperature these low-frequency given instead in historical units. This usage of the term modes are thermally excited, and thus the object con- ‘wave’ is in keeping with the common usage of the term: a sists of, say, halves in thermal motion moving about wave has a well-defined frequency and spatial frequency. the center of mass. The equipartition theorem then im- However, as the Heisenberg uncertainty principle plies that these halves have thermal momenta such that (Eq. 15) explains, a wave/object exactly located in recip- λ h/√MkBT , a small length for a macroscopic mass rocal space is completely delocalized in real space. Tem- M∼(Appendix C). Thus the halves are localizable, and poral delocalization is familiar — we see no problem with so must be the object itself. This argument applies to an object persisting for a long time — but spatial delo- all three spatial directions separately, in that one can- calization is in contradiction with everyday experience. not transform, by either rotation or boost, to a frame Thus, while waves corresponding to classical objects must where a classical (i.e. many-particle, finite-temperature) have well-defined frequencies/momenta, these cannot be object is delocalized in any spatial dimension. Thus the λ ∆λ exact: they must have some spread. The waves/objects x ∆x 4π 4π condition discussed in Appendix C under discussion (i.e. those in the classical limit) must is≫ achieved.≫ ≫ take the form of wave packets that are localized in both We have now introduced enough ideas from both classi- real- and reciprocal-space. cal and quantum physics that we can examine the defini- Students familiar with Fourier transforms might tion and usage of these three key words: particle, wave, protest that, since a small width in one space implies and object. (Other terms, such as ‘corpuscle’, ‘thing’, a large width in the reciprocal space (∆x 1/∆k), it ‘entity’, and ‘body’, are sometimes used synonymously ∝ is impossible to be well-localized in both. The response and interchangeably with one or more of the three terms to this objection is that, in this context, we are not de- we have chosen.) The first term in particular is used termining size by comparing ∆x to 1/∆k. Rather, here with a bewildering array of meanings, some quite contra- by well-localized we mean x ∆x and k ∆k. In dictory — even compared to other physics concepts, the ≫ ≫ other words, the magnitudes of the classical coordinates definition of ‘particle’ is particularly elastic. Standard in real- and reciprocal-space are large compared to the classical mechanics textbooks (Goldstein et al., 2002; widths of these wave packets (Cohen-Tannoudji et al., Kleppner and Kolenkow, 2014; Landau et al., 1976; 1977; Messiah, 2014). Symon, 1971; Thornton and Marion, 2004; Young et al., By a change of origin the coordinates of any one loca- 2020) use ‘particle’ to denote the concept of an ‘ob- tion, either in real or reciprocal space, can be made to be ject’ (as used here) or a ‘body’ (in Newton’s language) zero. However, this condition cannot be simultaneously (Newton et al., 2016). Some emphasize a negligible spa- 17 tial extent, defining a particle to be a point-like ob- 4. Position r is measured by counting ruler cycles rel- ject (Landau et al., 1976; Omn`es, 1971; Symon, 1971). ative to an arbitrary origin. But in more advanced textbooks usage inconsistent with these definitions is common. For instance, the wave- 5. A ruler cycles with position r at the spatial fre- function ψ ei(p·x−Et)/~ might be given to describe a quency kc of a massless (m = 0) wave cycling in free particle∼ of energy E and momentum p (d’ Espagnat, time t at the clock frequency fc. 1999; Halzen and Martin, 1984; Saxon, 1968). This 6. Spatial frequency k is measured by counting cycles state is completely delocalized in real space, and thus per interval in position r. could not be less point-like. Unfortunately, terminology that provides an explicit warning (e.g. use of ‘wavicle’ 7. The interval s separating two events, one at time (Eddington, 1928)) has not been widely adopted. t and position r and the other defined to be the Identifying, for instance, a particle such as an origin, is given by s2 = t2 r2. electron or the muon as a “point particle” (Abers, − 2004; Eisberg and Resnick, 1985; Griffiths, 2010; 8. At time t and position r, the phase φ of a wave Halzen and Martin, 1984; Harrison, 2000; Jackson, with constant temporal frequency f and spatial fre- quency k is given by φ = k r ft. 1999) is particularly misleading. An electron can be · − never be localized to a point, and is often delocalized 9. For a wave with temporal frequency f and spatial over macroscopic dimensions (as in an electron micro- frequency k, the mass m is given by m2 = f 2 k2. scope, for instance). Under certain circumstances (e.g. − only the center of mass motion is of interest, or the point 10. A wave is an object that can be equivalently de- of observation is distant from an object with spherical scribed as a function of time t and position r or symmetry) we can treat an extended object/wave as a as a function of temporal frequency f and spatial mathematical point, but under no circumstances is it frequency k. proper to think of the “point particle” as anything other than a convenient approximation. A diffracted electron 11. A particle is an object that can be counted as an does not hit a phosphor screen at a point. Rather, it integer unit. hits the screen in a locale that might be identified with To keep these definitions as self-contained as possible, optical ( 1 µm) or even atomic ( 0.1 nm) spatial we use ahistorical units where h = c = 1 (which avoids resolution,≃ but not better. The collision≃ with the screen having to define ‘light’) and avoid jargony shorthand produces a measurement of the electron’s position, (“A wave is an object with a Fourier transform.”). The but this measurement does not ‘convert’ the wave-like definitions are necessarily self-referential (i.e. circular), electron into a point-like particle, even for an instant but not completely tautological (“A wave is an object (Abers, 2004). The electron is still wave-like, and after that satisfies a wave equation.”). They are also incom- it collides with the screen it occupies wave-like states plete, but the undefined terms (e.g. ‘function’) are log- in the screen. Size is not an intrinsic property of the ical or mathematical. Measuring is defined implicitly electron, and thus one cannot ascribe it any permanent to be counting.4 Objects exhibit wave/particle duality: value, zero or otherwise. The size an electron manifests they can have wave properties and particle properties depends on how aggressively it is localized or probed. (d’ Espagnat, 1999; Harrison, 2000; Saxon, 1968). In a deep potential well, or subject to a collision with Definitions #1–11 define the real-space arena (Sec- a high energy particle, an electron can be well-localized tion II), the reciprocal-space arena dual to the real-space (down to the length scale corresponding to its Compton arena, and the arena content. Putting waves (defini- wavelength, at which point pair production prohibits tion #10) in the arena via the wave postulate (Sec- further progress). A free electron, or an electron in a tion III) generates all of the ‘classical’ dynamical laws clean crystal, can be delocalized over macroscopic length (Sections III–IV). scales. These definitions fail to explain two asymmetries that Having identified a few hazards, we here collect some of are obvious to our unaided senses. First, we experience our previous definitions and attempt explicit definitions of some of the terms remaining:

1. Time t is measured by counting clock cycles relative 4 to an arbitrary origin. The frequency definitions #3 and #6 recall a point raised earlier: classical physics combines the commonplace with the inaccessi- ble. These definitions are common and uncontroversial (which 2. A clock cycles with time t at a temporal frequency is to say mathematical) enough that most physics textbooks do fc. not bother to discuss them. In a classical context they are both theoretically essential (from the standpoint presented here) and, 3. Temporal frequency f is measured by counting cy- often, experimentally impracticable. See Section VII and the cles per interval in time t. discussion of footballs in Section III. 18 the real-space arena, not the reciprocal-space arena. Per- quired that the thermal radiation be localized. In fact, haps this asymmetry is ‘by definition’, but these defini- often the thermal radiation in a blackbody cavity is ana- tions do not explain its origin. Second, we experience lyzed in terms of standing waves, which are delocalized. the real-space arena asymmetrically: forward and back- Thus the blackbody problem is solved by postulating that ward directions in space are similar, while forward and the thermal electromagnetic field consists of countable backward directions in time are not. In fact, one can particles (i.e. photons). argue that time’s defining characteristic is that it orders The particle definition (#11) leads to one more per- events (’t Hooft, 2018). While these definitions #1–11 spective on the classical limit and allows us to clarify the define time separately from space, and include distin- footnote in Table I. We imagine a particle described by a guishing minus signs in the invariants, they and the laws Gaussian wavepacket. In one dimension the wavepacket of physics that follow (Sections II–IV) do not explain why is specified by three quantities: its mean position in real one spacetime coordinate so distinguishes before, now, space x′, its mean position in reciprocal space k′, and its and after (or past, present, and future), and the other width σk in reciprocal space, three do not. The so-called arrow of time is not under- 1 stood at any level (Savitt, 1997; Schulman, 1997). σk = , (E1) Here the defining characteristic of a particle is count- 4πσx ability, as opposed to localizability (d’ Espagnat, 1999). where σx is the wavepacket’s corresponding width in real In a double-slit experiment, the diffracting object mani- space. Normalizing the wavepacket establishes the corre- fests its particle nature when it hits the screen because at spondence with the integers required to make the particle that instant it can be counted, not because it is localized ‘countable’. to a point. Attempting to count the diffracting objects This wavepacket can be constructed by shifting and beforehand, for instance at the slit, destroys their (com- then boosting a wavepacket initially centered on the real- plementary) wave properties and the diffraction pattern and reciprocal-space origins. It can be described equally (Saxon, 1968). Capturing the particle on the screen does well in real space: constitute a measurement of the particle’s position, but ′ ′ 2 such measurements are better considered as counting the g ′ ′ 2πik x −(2πσk(x−x )) ψ (x; σk, x , k )= e 2σk√2πe , particle’s presence or absence relative to a pre-measured q (E2) coordinate system than as counting ruler cycles from the or reciprocal space: origin to where the particle is counted. When we capture k−k′ 2 a diffracting photon with a digital camera sensor, for in- ′ ′ − − ′ ′ 1 −( ) ψ¯g(k; σ , x , k )= e 2πi(k k )x e 2σk . stance, the position measurement answers the question, k σk√2π “which pixel registered the count?”, not “where is the (E3) p pixel that registered the count?”, which presumably is Now we wish to combine two particles to make a mini- known from the manufacturer’s datasheet. mal, composite object. Normally we imagine assembling Localizing is, of course, a great aid to counting. The objects additively. For instance, we might add beans to a strong association of these concepts leads to their con- bag to make a beanbag, or we might add an oxygen atom flation. However, the definition given here is preferred to a carbon atom to make a carbon monoxide molecule. because it can accommodate, for instance, a description According to the wave picture, composite objects are as- of a particle in a delocalized state, which the usual defi- sembled not by addition, but by multiplication. Just as nition cannot. Counting is thus associated with normal- amplitudes for events occurring in succession multiply izability. In the double slit experiment, a particle can be (Feynman et al., 2010), so amplitudes for particles com- launched at the slits and can hit the screen, for the ini- prising an object multiply. In other words, we do not tial and final states can be integrated to an integer value add the beans and the bag, we multiply the beans and (unity). However, a particle does not go through one slit the bag. or the other, as it cannot be counted (i.e. normalized) at That multiplication, not addition, describes the combi- either one. nation of two particles is a consequence of definition #11, The blackbody problem (Appendix C) gives another which is the definition that introduces the ‘quantum’. example — one of great historical importance (Jammer, Classical waves, such as those on a guitar string, a drum- 1966) — that illustrates how particles are more properly head, or a wind-swept sea, add with no particular restric- defined as countable, as opposed to localizable. Consider- tions on their amplitudes. Definitions #1–10 can accom- ing thermal radiation as classical electromagnetic waves, modate such waves, which show no tendency to become which have no normalization constraint on their ampli- classical objects as the number of waves N increases. In tudes, leads to the ultraviolet catastrophe. Assigning a fact, without definition #11 there is no number N. countable nature to these waves, such that electromag- Quantum mechanical superposition is fundamentally netic modes are occupied in integer units, gives Planck’s different than classical superposition (Dirac, 1967), and spectrum and agreement with experiment. It is not re- this difference is essential for realizing the classical limit. 19

Definition #11 requires that each particle have its own The formulas for the inverse transformation are space in the ledger, so that it is normalizable, which is to say that it can be counted.5 Superposition in the sense of addition occurs with the various states of one particle, 2 X 4σK but not with states of different particles. The ‘quantum’ x1,2 = x 1 1 , (E7) ± 2 ∓ − σ2 countability definition #11 creates N, and, as we will s k ! ′ 2 now show, the large-N limit gives the wave-objects that ′ ′ X 4σ 6 K we recognize as ‘classical’ particles. x1,2 = x 1 1 2 , (E8) ± 2 ∓ s − σk ! Initially we consider the two particles to be non- interacting, in which case their Hilbert spaces are not 1 4σ2 σ = σ 1 1 K , and (E9) 1,2 kv 2 coupled. Combining the spaces in a tensor product u2 ± s − σk ! corresponds to simple multiplication in both real- and u ′ t 2 reciprocal-space, so a minimal object consisting of two ′ k 4σK ′ g ′ ′ g ′ ′ k1,2 = 1 1 2 K , (E10) particles (ψ1 (x1; σ1, x1, k1) and ψ1 (x2; σ2, x2, k2)) can be 2 ± − σ ± g ′ ′ g ′ ′ s k ! written ψ1 (x1; σ1, x1, k1) ψ1 (x2; σ2, x2, k2) in a real-space representation. Some algebra· shows that this composite state has a second factorization (one of an infinite num- where the upper (lower) sign is chosen for particle 1 (2). ber), If the particles do not have identical widths the axes scale, so the coordinate transformations E5 are gener-

g ′ ′ g ′ ′ ally more than a simple rotation, but such stretching and ψ (x1; σ1, x1, k1) ψ (x2; σ2, x2, k2) = (E4) compressing is permissible in this abstract vector space · g ′ ′ g ′ ′ ψ (x; σk, x , k ) ψ (X; σK ,X ,K ), (as opposed to actual physical space) (Symon, 1971). · g g Using an obvious shorthand, if ψ (x1) and ψ (x2) de- scribe the first and second particles, respectively, then where the coordinate transformation (x1, x2) (x, X) is g g given by → ψ (x) and ψ (X) can be taken to describe ‘external’ and ‘internal’ degrees-of-freedom, respectively (i.e. ‘extrac- ules’ and ‘intracules’ (Coleman, 1967; Eddington, 1953; 2 2 Gill et al., 2003; Proud and Pearson, 2010)). We also σ1 x1 + σ2 x2 x = , and X = x1 x2, (E5) note that, if the σ’s are proportional to the square-roots σ2 + σ2 − 1 2 of their respective masses, then the ‘external’ and ‘in- ternal’ degrees-of-freedom may be associated with the and the Gaussian parameters by center of mass and the reduced mass, respectively. At this point this connection is unexpected — the Gaussian widths seem to be parameters that are adjustable inde- 2 ′ 2 ′ pendent of the dispersion relation, and the dimensions ′ σ x + σ x ′ ′ ′ x = 1 1 2 2 , X = x x , (E6) of σ2 and m differ by a length scale. We will return to σ2 + σ2 1 − 2 k 1 2 examine this connection shortly. 2 2 σ1σ2 σk = σ + σ , σK = , 1 2 σ2 + σ2 In a standard quantum mechanical treatment, the in- q 1 2 2 ′ 2 ′ ternal degrees of freedom are of interest and the external ′ ′ ′ ′ σ k σ k k = k + k , and K = p2 1 − 1 2 . coordinates are ignored. Here, in order to understand 1 2 σ2 + σ2 1 2 how 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 ψg(X) then drops out of the problem. The object remaining now has a (Gaussian) particle description in terms of the ‘ex- ternal’ wavefunction ψg(x) centered in reciprocal space ′ 5 We are making the usual classical assumption that the particles on the sum wavevector k (i.e. momentum) and in real are distinguishable (Schiff, 1968). If the particles are indistin- space between the constituent particles at x′. guishable then ‘different’ particles can superpose and interfere (Hanbury Brown, 1974; Purcell, 1956). A complex, composite object can be built up by adding 6 et al. Feynman argues (Feynman , 1964) that the sentence “all one particle after another, which corresponds to repeat- things are made of atoms” condenses much of our scientific knowledge into very few words. “All things are made of count- ing this multiplication process again and again. By in- able waves” is written in the same spirit, and, for the price of duction (Bromiley, 2018) we find that the Gaussian pa- one additional word, communicates a good deal more. rameters describing the final, external degree of freedom 20 are According to definition #10, a description of the wave in either real- or reciprocal-space can be converted into N σ2 = σ2 , (E11) an equivalent description in the conjugate space. In the k ki classical limit, the phase becomes uncountably large and i=1 X N this connection is lost from a practical (i.e. experimen- ′ 1 ′ x = σ2 x , and tal) standpoint. In our example, either Eq. E2 or Eq. E3 σ2 ki i ′ ′ k i=1 provides a complete description in terms of x , k , and a X N width (either σx or σk). In the classical limit, the phase ′ ′ k′x′ N is unmeasurable7, leaving Eqs. E12–E13 as k = ki. ∼ ∝ i=1 the information available. These equations are effectively X δ(x x′) and δ(k k′) respectively, and knowledge of one The assembly process is similar in reciprocal-space and provides− no information− about the other. In the classical gives the same result as that obtained by Fourier trans- limit, the position and the spatial frequency (i.e. mo- forming the final expression in its real-space representa- mentum) descriptions are effectively decoupled. tion. Here we can also get a more quantitative sense of how This path already looks promising, in that it repro- the inequalities C4 develop as we move into the classical duces properties expected for an object made of multiple limit. For a 1D, N-particle composite object, we can particles. For instance, in real space the composite ob- re-write ject is centered around a weighted mean of the centers of λ ∆λ its constituents. And in reciprocal space the wavevector x ∆x (C4) (i.e. momentum) of the composite object is equal to the ≫ ≫ 4π ≫ 4π sum of the wavevectors of its constituents. as We now consider the case where the N constituent par- ticles are all similar. For instance, the beans of the com- 1 σk Nσ σ , (E14) posite beanbag are moving together on a ballistic trajec- xi ≫ x ≫ 4πk′ ≫ 4πk′2 tory. (Once the beanbag is launched, each bean’s path is approximately unaltered by the presence of its neighbors, where the replacements are justified by arguments given previously, except for x Nσ . This replacement which illustrates how individual particles with negligible xi bounds the size of the object→ by requiring that its con- interactions can still be grouped as a composite object.) ′ ′ ′ stituent particles occupy distinct regions of space8. Jus- Then σk √Nσki, k Nki, and x is the mean of the ′ tifying this bound requires ideas (e.g. Coulomb repulsion x . Calculating≃ the probability≃ density for this composite i and the Pauli exclusion principle) not discussed here, but object, we see that

′ 2 g∗ g∗ −2(2πσk (x−x )) ψ (x) ψ (x)=2σk√2πe (E12) · ′ δ(x x ) 7 → − It is more accurate to say that the phase discussed here is un- measurable not because it is large, but because it changes rapidly in the limit that the number of constituent particles when varied separately in either time or space. The phase dis- N because σk . Thus the real-space prob- cussed here is only a (spatial) piece of the ‘full’ phase — Eq. 2 ability→ ∞ density for a many-particle→ ∞ object approaches a or Eq. 6 — which is unchanging along the classical trajectory. δ-function with a real-space width σ 1/√N. (By choice of origin the phase might be set to zero, in which x ∝ case it is not large at all.) Thus as we move into the classical In reciprocal space the argument is again more subtle. limit the ‘full’ phase controls the observable dynamics — Sec- The function tion IV — while the ‘partial’ phases, i.e. the leading terms in Eqs. E2–E3, are becoming unobservable. Also note that, while k−k′ 2 ∗ 1 −2( ) unobservable, these phases are not incoherent. The phase rela- ψ¯g (k) ψ¯g(k)= e 2σk (E13) · σ √2π tionships, i.e. coherences, are what maintain the object as a uni- k fied whole. Fiddling with a particle’s phase changes its position √ or its wavevector/momentum (Eqs. E2–E3). To the extent that does not approach a delta function because σk N these physical quantities are constrained in a composite object, ′ ∝ is becoming large. However, k N is becoming large the phases must be coherent. Coherent states of the ‘quantum’ faster, so that the function E13 looks∝ like a delta function harmonic oscillator (Appendix C) provide a concrete example of the coherence of ‘classical’ objects. from a perspective that encompasses both the reciprocal 8 While the spatial uncertainty relation (Eq. 15) has a direct ana- space origin and the peak. From this perspective, the log ∆t∆f ≥ 1/4π, and k ≫ ∆k implies f ≫ ∆f, this replace- natural perspective for the problem, the relative width ment and x ≫ ∆x do not have such obvious temporal analogs. ′ Objects do not overlap in space, but they do in time. Despite σk/k is decreasing 1/√N in reciprocal space, just as it is in real space. Thus∝ the reference to the ‘zoom level’ in the apparent similarity of Eqs. A1 and Eqs. B1, the uncertainties in the variables t and x seem to require very different interpreta- Section III is made precise: a wave can be well-localized tions. Usually we assume that a measurement of x returning one in both real- and reciprocal-space without violating the particular value in the range ∆x thereby excludes others, while uncertainty principle. we expect that a particle occupies all t in ∆t (Mermin, 1985). 21 that are probably familiar. Using the uncertainty rela- advertised. The passage to the classical limit is not re- tion for Gaussians (Eq. E1) and the relations that follow stricted to non-interacting particles described by Gaus- from E11 gives sian wavepackets, but is, as it must be, a process that generally occurs with large numbers of particles, regard- N 1 1 σki ′ . (E15) less of their interactions. 4πσ √ 4πNk √ 3 ′2 ki ≫ 4π Nσki ≫ i ≫ 4π N ki To recapitulate while generalizing to three dimensions: we start with N wavepackets, each of which is described Multiplying through by 4πσ puts this inequality chain ki in its own real and reciprocal spaces in terms of its mean in a dimensionless form, ′ ′ position r i, mean wavevector k i, and reciprocal-space σ↔ 1 σki 1 σki 2 1 widths ki. These N wavepackets represent particles N ′ ′ . (E16) that might be the constituent beans of a beanbag, the ≫ √N ≫ ki N ≫ ki N√N

atoms that make up the beans, or the fundamental par- These four terms scale like N 1, N −1/2, N −1, and N −3/2, ticles that make up the atoms. We rearrange the coor- respectively. Thus as the number of particles N , dinates of the 3N-dimensional space and integrate over → ∞′ the classical limit is inevitable, provided that σki ki the 3(N 1) ‘internal’ coordinates. The ‘internal’ coordi- ′ ≫ − is not the case. If ki is not obviously appreciable, the nates then drop out of the problem, one by one, because thermal arguments given earlier in Appendix C and here the normalization conditions from each of the constituent in this Appendix E can be applied to determine whether particles have been transferred to the new degrees of the classical limit obtains. freedom (i.e. pseudo-particles). The remaining ‘exter- In assembling the composite object we have neglected nal’ 3D space contains a single normalized wavepacket interactions between the constituent particles. However, representing the composite (pseudo-)particle. In real reviewing this development we see that, because we in- space this wavepacket is centered between the constituent tegrate over the internal degrees of freedom, the exact wavepackets’ positions, and in reciprocal space it is cen- forms of the internal wavefunctions are unimportant. For tered on the sum of the constituent wavepacket’s posi- instance, if the minimal, two-particle object is a hydro- tions. For large N this final wavepacket’s widths are neg- gen atom consisting of a proton and an electron, the ligible in both real- and reciprocal space, where the bases Coulomb interaction gives internal coordinates described for comparison are the object’s size and its total wavevec- by hydrogenic wavefunctions. These wavefunctions are tor, respectively. The Fourier connection between the not of a Gaussian form ψg (Eqs. E2–E3). In this case real- and reciprocal-space descriptions is practically sev- the states can only be factored in the external/internal ered, and the resulting object can be treated like a ‘par- coordinate system (r, R), for in the electron/proton co- ticle’ in the sense of the word as employed by standard ordinate system (re, rp) the states are entangled. How- textbooks on classical mechanics. ever, our goal here is see how a single object arises by Returning to review the arc of the argument from combining many particles, not to specify how the par- its beginning (Section III), eliminating the ‘uninterest- ticles arise by decomposing the object. The composite ing’ degrees of freedom from a complex, large-N collec- object’s state need not be factorizable in terms of the con- tion of particles produces a single ‘particle’ that is well- stituent particle coordinates. We only require the factor- localized (though not small) and behaves classically. In ization on right-hand side of Eq. E4. Often the transfor- real space its location is effectively described by a delta- ′ ′ mation necessary to produce this separation of variables function δ(r r ), where r develops in time according ∼ − can be found, with the transformation to center of mass to Eq. 12a. In reciprocal space its location is effectively ′ ′ and reduced mass coordinates being the classic example described by a delta-function δ(k k ), where k de- ∼ − (Feynman et al., 2010; Ghirardi et al., 1986). In fact, in velops in time according to Eq. 12b. Thus the complete the many particle limit it is not even necessary that the motion of the wavepacket is dictated by the condition initial external wavefunctions be Gaussian. The central that the phase φ = k r ft is stationary. In traditional · − limit theorem (Bracewell, 2000) ensures that, with suit- language, one says that the motion of the object is dic- ably well-behaved constituents9, the external wavefunc- tated by the condition that the action S is stationary. tion will eventually adopt a Gaussian form as the num- This ‘stationary action’ condition is abstract and unmo- ber of constituent particles increases, regardless of the tivated, while the wave formulation’s ‘stationary phase’ is specifics of the problem. comparatively elementary and might even supply a men- Thus this argument showing how classical objects (lo- tal picture — see, for example, the case of a massless calized in real and reciprocal space) arise from waves (lo- object (Eq. 3a), where ‘stationary’ becomes ‘constant’. calized in neither space) is more general than initially The relationship between the average position of a Gaussian wavepacket in real-space and the center of 2 mass remains obscure. Proportionality between σk and mass m is seen in some important and basic contexts. 9 A double slit or a grating, for instance, can induce poor behavior. It appears in the propagator for a free particle, where 22

2 σk = im/(4π∆t) and ∆t is the time interval of prop- Computer Engineering. Circuits and Systems (McGraw agation− (Feynman et al., 2010). And it appears in the Hill, Boston). 2 Bridgman, P W (1931), Dimensional Analysis (Yale Univer- ground state of a harmonic oscillator, where σK = µf0/2, µ is the reduced mass, and f is the oscillator’s fre- sity Press, New Haven). 0 Bromiley, Paul A (2018), Products and Convolutions of Gaus- quency. In other words, in the position representation sian Probability Density Functions, Internal Report 2003- (with h = c = 1) a 1D harmonic oscillator’s ground state 003 (School of Health Sciences, University of Manchester, ho ψ can be written Division of Informatics, Imaging and Data Sciences).

2 2 Cheng, Ta-Pei, and Ling-Fong. Li (1991), Gauge Theory of ho 1/4 −2π µf0 X ψ (X)=(4πµf0) e (E17) Elementary Particle Physics, Oxford Science Publications (Clarendon Press, Oxford). = ψg(X; σ , 0, 0). K Chiao, R Y, C. K. Hong, P. G. Kwiat, H. Nathel, and W. A. 2 Vareka (1989), “Optical Manifestations of Berry’s Topo- This Gaussian state gives σK proportional to the cor- logical Phase: Classical and Quantum Aspects,” in Coher- rect mass. It also has a parameter (f0) that can be ad- ence and Quantum Optics VI, edited by Joseph H. Eberly, justed to fit the problem: increasing the oscillator fre- Leonard Mandel, and Emil Wolf (Springer US) pp. 155– quency localizes the state in real-space and delocalizes 160. it in reciprocal-space, and vice-versa. However, for x′ Coelho, Ricardo Lopes (2012), “On the Definition of Mass in to be equivalent to the center of mass, we require that Mechanics: Why Is It So Difficult?” The Physics Teacher 50 (5), 304–306. m , m , the total mass m = m + m , and the reduced 1 2 1 2 Cohen, Eliahu, Hugo Larocque, Fr´ed´eric Bouchard, Farshad mass µ = m1m2/(m1 + m2) all be proportional to their Nejadsattari, Yuval Gefen, and Ebrahim Karimi (2019), 2 respective σk’s with the same constant of proportional- “Geometric phase from Aharonov–Bohm to Pancharat- ity (E6). This requirement seems to over-constrain the nam–Berry and beyond,” Nature Reviews Physics 1 (7), problem. For example, a carbon monoxide molecule in 437–449. its vibrational ground state might be trapped in a har- Cohen-Tannoudji, Claude, Bernard Diu, and Franck Lalo¨e monic potential, with no specific relationship between the (1977), Quantum Mechanics (Wiley, New York). 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