On conservation laws in

Yakir Aharonova,b, Sandu Popescuc, and Daniel Rohrlichd aSchool of and Astronomy, Tel Aviv University, Tel Aviv 69978 , Israel. bDepartment of Physics, Chapman University, Orange CA, USA. cH. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL and dPhysics Department, Ben-Gurion University of the Negev, Beersheba 8410501, Israel.

We raise fundamental questions about the very meaning of conservation laws in quantum me- chanics and we argue that the standard way of defining conservation laws, while perfectly valid as far as it goes, misses essential features of nature and has to be revisited and extended.

I. INTRODUCTION II. SUPEROSCILLATIONS

Essential to this paper is a mathematical structure we Conservation laws, such as those for , momen- call “superoscillation”. Common wisdom assumes that tum, and angular , are among the most fun- no function can oscillate faster than its fastest Fourier damental laws of nature. As such they have been inten- component. Yet as we show here, there is a large class sively studied and extensively applied. First discovered of functions for which this assumption fails. Indeed we in classical Newtonian mechanics, they are at the core of have found functions that oscillate, on a given interval, all subsequent physical theories, non-relativistic and rel- arbitrarily faster than the fastest Fourier component. An ativistic, classical and quantum. Here we present a para- example of such a function is the following: doxical situation in which such quantities are seemingly not conserved. Our results raise fundamental questions 1 + α 1 − α N about the very meaning of conservation laws in quantum f() = eix/N + e−ix/N , (1) mechanics and we argue that the standard way of defin- 2 2 ing conservation laws, while perfectly valid as far as it goes, misses essential features of nature and has to be where α is a positive real number, |x| ≤ πN, and N is a revisited and extended. large integer. An extensive discussion of the properties of this function, first introduced in [4, 5], appears in [6–8]. To display its basic properties, we first write it, via the That paradoxical processes must arise in quantum me- binomial formula, as chanics in connection with conservation laws is to be ex- pected. Indeed, on the one hand, physics is local: causes and observable effects must be locally related, in the N X sense that no observations in a given space-time region f(x) = c(n; N, α)ei(2n/N−1)x , (2) can yield any information about events that take place n=0 outside its past light cone [1]. On the other hand, mea- surable dynamical quantities are identified with eigenval- where the c(n; N, α) are constants; ues of operators and their corresponding eigenfunctions 1 N are not, in general, localized. Energy, for example, is a c(n; N, α) = (1 + α)n(1 − α)N−n. (3) property of an entire wave function. However, the law 2N n of is often applied to processes in From (2) one can see that f(x) is a sum over wave num- which a system with an extended wave function interacts bers k = 2n/N − 1, ranging from -1 to 1. with a local probe. How can the local probe “see” an ex- n √ tended wave function? What determines the change in Now consider this function in the region |x| . N. arXiv:1609.05041v1 [quant-ph] 16 Sep 2016 energy of the local probe? These questions lead us to un- Here we can approximate the exponentials by their first cover quantum processes that seem, paradoxically, not to order Taylor expansion and obtain conserve energy. 1 + α x 1 − α x N The present paper (which is based on a series of unpub- f(x) ≈ (1 + i ) + (1 − i ) 2 N 2 N lished results, first described in [4] and [5]), presents the paradox and discusses various ways to think of conserva-  iαxN = 1 + ≈ eiαx. (4) tion laws but does not offer a resolution of the paradox. N A subsequent paper will present our resolution. The rea- son for publishing the paradox and resolution separately is that the paradoxical effect stands alone - its existence Hence, in the restricted region, f(x) behaves as an os- is independent of attempts to explain it - while readers cillation of wave number α. But, crucially, α need not may disagree with our proposed resolution. be smaller than 1. By taking α  1, we ensure that in 2 √ the region of validity of the approximation, |x| . N, the function f(x) oscillates with wave number α  1 although all its Fourier components have wave number smaller than 1. In other words, a superposition of long wavelengths, the longest being 2πN√ and the shortest be- ing 2π, can, in the region |x| . N, oscillate with the much shorter wavelength 2π/α. Furthermore, the region of these “superoscilations” can be made arbitrarily large and include arbitrarily many wavelengths by taking N sufficiently large. Note that there is no contradiction with the Fourier theorem since the region where this function is (almost) identical to an oscillation of a frequency not contained by its Fourier decomposition does not extend over the entire region where the function is defined. a) Although it is not essential for our present paper, it 1/α is interesting to note that outside the superoscillatory region f(x) increases exponentially. This is a generic property of functions with superoscillatory regions.

III. THE EXPERIMENT b) 1

The type of effect we describe here is common for all con- served quantities that depend on the shape of the wave function all over the space including energy, momentum and . Here we will focus on energy, FIG. 1: (a) A in the box in the specially prepared for which the proof is more intuitive. low-energy quantum state which, in the central region, oscil- lates with a spatial frequency greater than that of any Fourier Consider a box of length 2πNa (where a is some unit components (see (b)). As the opener is passing by, it opens the box and extracts the photon, if the photon is there. The length) that contains a single photon in the state ψ(x) shape of the wave function is not accurate but simply illus- i trative; the true wave function is exponentially larger away ψ(x) = f(x/a) − f ∗(x/a) (5) from the center and has a more complicated shape. N with f(x) defined in Eq. (4) and α  1. Here N is a normalisation factor. ψ(x) has properties similar to f but it obeys the boundary conditions ψ(−πNa) = ψ(πNa) = 0 at the walls of the box. (See Fig. 1.) E = α >> Emax = 1. (8) From now on however, for simplicity, we take a = 1 In other words, in the box we have a low-energy photon, and we work in the usual units ~ = c = 1. Given the relation between wavelength, frequency and which in the center of the box looks like a high-energy energy for the photon, the decomposition (2) shows that photon. the photon is in a superposition of different energy eigen- states with wave numbers Suppose now that a mechanism that we will call√ the “opener” opens the box in the center for a time T . N kn = (2n/N − 1) (6) and inserts a mirror, such that if the photon hits the mir- ror, it comes out of the box (as in Fig. 1). This happens all smaller than or equal to 1 (in absolute value), corre- if the photon is situated at a distance not larger than sponding to energy eigenvalues T from the mirror; otherwise it cannot get there while the box is open. The probability for this to happen is at most R T |ψ(x)|2dx. En = |kn| (7) −T with the maximal energy Emax = 1.√ On the other hand, Now suppose we find the photon out of the box. What we also know that in the region |x| . N around the cen- is its energy? ter of the box, the wave function of the photon resembles that of a monochromatic photon with wave number α Naively we would think that the emerging photon must hence of energy have one of the En = |2n/N − 1| ≤ 1 that it had 3 originally in the box - after all, reflection from a mirror doesn’t change the spectrum of light. On second thought, however, we realize that this cannot be so. Indeed, the box was open√ only for a time T . In the entire region |x| ≤ T ≤ N around the opening, the wave function of the photon was essentially indistinguishable from a plane wave of high energy E. The information revealing that the photon was not a true monochromatic photon of this high energy is contained in the shape of the wave func- tion in regions situated farther from the opening than T ; relativistic causality dictates that this information could not arrive at the opening during the time it was open. Hence the photon that emerges from the box must be in FIG. 3: The initial and final distribution of the energy of a state which is identical to that in which a genuine pho- the opener for the cases when the photon emerges from the ton of energy E would emerge from the box. But for this box. One would expect the final energy of the opener to second case it is trivial to see what happens: the mir- be lower than the initial, to compensate for the increase of ror just reflects the photon out of the box but it doesn’t the energy of the photon, but this does not happen; instead the final energy distribution is the same as the initial (up to change it frequency and energy. small perturbations due to the truncation of the photon wave In addition, the wave function is chopped into a wave- packet). train of length T by our closing the box after time T . Hence if a genuine photon of energy E = α emerged from the box its energy spectrum would have a peak at energy it emerges with the much higher energy α. Where does α and a spread in energy of 1/T . By increasing N we the extra energy come from? A first guess is that the can increase T and hence reduce by as much as we want energy comes from the mechanism used for extracting the disturbance produced by the finite opening time of the photon. Indeed, we need to open the box, insert the box; the photon thus emerges as close as we want to its mirror, then take out the mirror and close the box. This initial energy E. We thus conclude that when our “fake mechanism, which we call “opener”, effectively subjects high-energy” photon emerges from the box it must have the photon to a time-dependent Hamiltonian, and such energy E exactly as the genuine high-energy photon and a Hamiltonian need not conserve energy. not the low energies it originally had; see Figs. 2 and 3. To put it differently, we can look at the total Hamil- tonian, describing the photon and the opener. The total Hamiltonian is time independent since it describes the total system, with no parts left outside; the time depen- dence seen by the photon comes from the time evolution of the opener and its interaction with the photon. The total energy is conserved for time-independent Hamilto- nians. Thus, we are tempted to say, all that happens is that the opener and the photon exchange energy: when the photon emerges from the box, and, as we proved, has higher energy than it had inside the box, the opener must have lost the same amount - a trivial case of energy exchange. We now arrive at the crux of the problem. Although FIG. 2: The initial and final distributions of the energy of the this explanation is the most natural, it is wrong: the photon. Initially it was a superposition of low energies and strictly no energy higher than 1. Finally a peak at energy α photon could not have gotten its energy from the opener. appears, which corresponds to the extracted . The reason is again causality, as we shall now see. Consider the case of a monochromatic high-energy photon of energy α. When this photon emerges from To summarize, when the photon emerges from the box, the box its energy is unchanged (up to fluctuations of it has much higher energy than it initially had. Where order 1/T ). Hence in this case the opener does not give did the extra energy come from? This is the question it any energy. But then the opener cannot give energy that concerns us in this paper. to the fake photon either. Indeed, recall that the entire difference between the high-energy photon of energy α and our specially pre- IV. THE PARADOX pared low-energy photon√ lies in regions situated further from the center than N; this information cannot ar- Inside the box the photon was in a superposition of rive at√ the opener during the time of the experiment, different energies eigenstates, all smaller than 1, and out T . N. Immediately after the experiment is over, we 4 can measure the energy of the opener. Since the opener 1 eV). Yet, when we open the box the photon emerges is localized in a small region (around the center of the with energy of order of 1 GeV. We should definitely be box) we have immediate access to it and can measure entitled to ask where the energy came from. its energy in a short time; the measurement√ can be fin- What we showed in our example is that this energy ished long before information from |x| ≥ N can arrive. cannot come from the mechanism that extracts the pho- If by measuring the opener we could determine whether ton from the box. Since this is the single other system in the box contained the original low energy photon or the the problem, we are faced with energy non-conservation monochromatic high-energy photon, we would violate rel- in this individual case. So we do have a problem that ativistic causality. Thus, since the opener didn’t lose en- needs to be explained. ergy when the box contained a high-energy photon, it Furthermore, we also argue that the standard formu- cannot lose energy in the case of the low-energy photon lation is not only limited in that it cannot address indi- either! We must therefore conclude that the extra energy vidual cases, it is also unsatisfactory in the meaning of did not come from the opener. the story it tells. This is the paradox. The photon emerged from the box Suppose that we repeat our experiment a large number with energy much higher than it had inside, but the en- of times. Consider a large number of boxes, each box con- ergy did not come from the opener, the only other system taining just one single photon, prepared in the special low in the problem. Energy seems not to be conserved. energy state ψ(x) of Eq. (5), each box having its associ- ated opener. In some experiments the photon comes out of the box; in others the photon remains inside. In each V. ENERGY CONSERVATION case the energy of the opener remains unchanged (apart from fluctuations of the order 1/T << E − Emax). Since Faced with this paradox one can respond in various the photons that emerge from their boxes have increased ways. The conventional response is that there is problem energy, for the average energy to remain constant the whatsoever, and there cannot ever be. In quantum me- photons that stay in their boxes must be left with lower chanics, the standard formulation of a conservation law energy. But what this effectively means is that the pho- is that the probability distribution of the conserved vari- tons that emerged from their boxes got their energy from able over the entire ensemble should not change. This the photons in the other boxes. But the experiments are law applies to any time-independent Hamiltonian. On completely independent; they may even happen at dif- this basis, there should be absolutely no energy non- ferent times and in different places. Nevertheless, the conservation in our example either. And, of course, from photons which stay inside their boxes supply energy to this point of view, there is none. Indeed, in the preceding the ones that emerge from the other boxes! Clearly the sections we focused on what happens when the photon idea is absurd and it cannot be an acceptable resolution emerges from the box. But it is also possible (and actu- of our paradox. ally far more probable) that the photon does not emerge Another possible response to the paradox is to argue from the box. It happens, because the wave function of that it makes no sense to talk about the energy of a the photon extends all over the box, so the photon has photon as long as it is in a superposition of different a non-zero (and in fact quite large) probability to be far energy eigenstates. However we note that the photon had from the central region. If so, it cannot reach the opening zero probability to have any energy larger than Emax = 1, while the box is open therefore it cannot leave the box. yet it emerges with the high energy E >> Emax. So the To see standard energy conservation at work, we must paradox of the photon’s extra energy remains. consider these cases as well. What we find in these cases As noted in the introduction, we do not offer any res- is that again the opener didn’t lose any energy (since it olution here; we leave the paradox open. But below we did not collide with the photon), but the photon remains provide more details. First we present an explicit model; in the box with lower energy (as the wave function loses then we analyze in more detail the standard energy con- its superoscillatory piece) - again a paradox. Consider- servation as applied to our situation. Although there are ing these cases as well we find that, as expected, the no contradictions here, the specific way in which the en- probability distribution of the total energy (photon plus ergy is conserved in the statistical ensemble is extremely opener) did not change. unusual and instructive. But - and this is the main point of our paper - we would like to argue that the standard formulation of conserva- tion laws, though absolutely correct as far as it goes, is VI. EXPLICIT MODEL simply not enough. The standard conservation law is statistical and says nothing about individual cases. We We now give an explicit model. Since relativistic quan- would like to argue, however, that it is legitimate to ask tum field theory has well-known technical difficulties, we what happened in a particular individual case. In our ex- will formulate a non-relativistic model. The experiment ample, suppose we have in the box a photon of energy of is the same, the only difference being that instead of a order 1 eV (more precisely a photon in a superposition of photon, the box contains a non-relativistic particle. Of various energies but absolutely none of them larger than course, we are now no longer allowed to use relativistic 5 causality arguments, and we will prove our statements information that this is not a true high-energy particle is by explicit calculations. Nevertheless, the intuition for contained only in faraway regions of space and it takes a the non-relativistic model is exactly the same as in the finite time to arrive in the center of the box. But, as we relativistic case since also non-relativistic quantum me- mentioned above, the same is (approximately) true also chanics allows finite time intervals in which a part of a in non-relativistic quantum mechanics (see√ Supplemen- wave function can act, to a good approximation, inde- tary Information 1). Therefore for |x| . N the time pendently of the rest [9]. evolution of the particle (in the absence of the interac- 2 Consider the following Hamiltonian to model our ex- tion with the opener) is Ψ(x, t) = e−iα t/2mΨ(x, 0), i.e. periment: it just accumulates the same time dependent phase as the 2 bona-fide energy eigenstate sin(αx). This approximation p 1 + σz π √ H = + V (x) + p + g(x)δ(q)σ . (9) is valid for any time T N. It is during this time that 2 2 q 2 x . we extract the particle from the box. The low-energy particle in the box has coordinate x, mo- To ensure that the opener opens the box during this mentum p and m = 1, while the “opener” is mod- time window we choose its initial state φ(q) to have sup- eled by a particle with coordinate q and momentum pq. port only for q within −T ≤ q ≤ 0. V (x) is an infinite square potential well which represents We have now to calculate the time evolution of the the box: it is zero inside the box ( i.e. for |x| ≤ πN) and particle-opener system it is infinite outside. −iHT We let our particle have an internal degree of freedom, e φ(q)ψ(x)| ↑i = Θ↑(q, x)| ↑i + Θ↓(q, x)| ↓i. (10) a “spin”, which determines whether the particle is in the box or free. The states | ↑i and | ↓i are eigenstates of As noted above, if the particle is found in the central part of the box, after the interaction with the opener it gets σz. When the spin is | ↑i the potential V (x) confines the particle inside the box; when the spin is | ↓i the particle released; otherwise it remains inside the box. This cor- responds to the final state (10) being a superposition of is free since the term in 1+σz multiplying V (x) vanishes. two terms, one with the “spin” ↓ and one ↑ respectively. The opener’s free hamiltonian is pq while the last term in H describes the particle-opener interaction. The in- (Note that Θ↑ and Θ↓ are not normalised; the norm of teraction takes place when the opener is at q = 0; at Θ↓ is much smaller than the one of Θ↑ since the prob- all other times the opener is free. The opener moves at ability of the particle to emerge from the box is much constant speedq ˙ = 1 without spreading, both when it smaller than the probability to remain inside.) Here we is free and while the interaction takes place. The opener are interested in the case when the particle is released. moves from q < 0 where it is free, through the interaction The released particle and opener state is approximatively region, q = 0, to q > 0 where it is free again. (One may (see Suplementary Information 2) recognize the opener as the model of an ideal clock that turns on and off an interaction when the “clock time” Z 2 2 2 −i α T −i( k − α )q ikx indicated by the pointer q is q = 0.) Θ↑(q, x) = e 2 φ(q − T )e 2 2 h(k)e dk, The interaction term is designed to release the particle if it is situated in a window around the center of the box. (11) This works as follows. The operator σ can release the x with trapped particle by flipping | ↑i to | ↓i. But the particle is released only if it is situated in a window around the center of the box. The window is determined by g(x), Z L √ 1 0 i(α−k)x0 i(−α−k)x0 which is zero outside a window |x| ≤ L N and g(x) = h(k) = dx (e − e ). (12) . 2i −L 1 inside. The interaction term is thus non-zero only when the particle is in this window, hence only if it is here An essential thing to note about Eq. (11) is that it is can the particle be released. (Note that this toy model identical (up to normalisation) to what we would have differs slightly from the example in the previous sections: obtained had we started with the particle in the energy There the window was taken to be small, but open for a eigenstate sin(αx) instead of the “fake” state ψ(x) of eq. time long enough for a wave-train of length L to emerge (5). through it. Here the window is open for an infinitesimal To see the meaning of Eq. (11) we first note that time, but is large enough to let a wave-train of length L since the particle is now free the energy eigenstates are emerge from it.) the plane waves eikx. Hence, as far as the particle is Consider now that at time t = 0, we prepare the concerned, Eq. (11) is actually the decomposition of trapped particle in a state Ψ(x, 0)| ↑i with Ψ(x, 0) equal the state into energy eigenstates. Everything else be- to our√ special state ψ(x) given by Eq. (5). In the region ing phase factors, the probability of the released particle 2 |x| . N the wave function Ψ(x, 0) looks like a high- to have momentum k corresponding to energy k /2 is energy state. In relativistic quantum mechanics, when |h(k)|2. But all we have in (12) are two truncated wave- the particle is situated in this region its time evolution trains, corresponding to the (untruncated) plane waves is identical to that of a high-energy particle, since the e±iαx. Thus the most probable final plane-wave state is a 6 free particle in a superposition of eigenstates of momen- The total energy long before the interaction and long α2 after it is simply the sum of the free energies, E = tum ±α and energy 2 . The particle could also emerge k2 α2 Eγ + EΩ. With this definition, the standard energy con- with other energies 2 6= 2 but with smaller probability. The possibility of coming out with these energies is due servation relation is to the truncation of the wave-train to |x| ≤ L. Taking P tot(E) = P tot(E) (14) N larger, we can make L larger and thus decrease these i f probabilities, relative to the probability of having energy tot 2 where P denotes the probability distribution of the α /2, as much as we want. As noted before, this is ex- total energy and the indices i and f stand for “initial” actly what would have happened had we started in the and “final”. Note that all the probability distributions high-energy eigenstate sin(αx). discussed here are over the entire ensemble, including One’s natural suspicion is that when the particle 2 both the cases in which the photon emerged out of the emerges from the box with energy α /2, which is much box and the cases when it didn’t. larger than the maximal energy it had inside the box, the Before the interaction, there is no correlation between additional energy of the emitted particle comes from the the energies of the photon and opener, i.e. the initial joint opener. But the energy distribution of the opener before probability P γ,Ω(E ,E ) of their energies is the product and after the interaction is the same; φ(q) is merely dis- i p Ω of their respective individual probability distributions placed, as if there had been no interaction. This is the paradox. Note also that when the particle emerges with an en- P γ,Ω(E E ) = P γ (E )P Ω(E ). (15) ergy slightly different from α2/2, because of the trunca- i p Ω i γ i Ω tion of the wave-packet, the opener supplies this small Correspondingly, the initial total energy distribution is difference (mathematically expressed by the q-dependent given by phase accumulated by the opener) but not the difference between the true low energies that the particle originally Z P tot(E) = P γ (E0)P Ω(E − E0)dE0. (16) had and the high energy with which it emerges. i i i

After the interaction it is again the case that the prob- VII. THE STANDARD ENERGY ability distributions of the photon and opener are uncor- CONSERVATION related. More precisely, as discussed in the previous sec- tions, the truncation of the emerging wave packet leads As we discussed before, for any time independent to some correlations between the energy of the photon Hamiltonian, energy is always conserved in the standard and the opener (both when the photon emerges out of sense, that is, the probability distribution of the total the box as well as when it remains in the box), but these energy is time independent. Since this is a theorem, it correlations can be made as small as we want by taking holds in our case as well; our paradox appears only at the the length 2πN of the box long enough. Hence level of individual cases. Yet it is worth looking in more detail at the standard account of energy conservation as γ,Ω γ Ω it applies in our case. As our case has interesting char- Pf (Eγ EΩ) = Pf (Eγ )Pf (EΩ) (17) acteristics, the standard account of energy conservation and the final total energy distribution is turns out to be interesting as well. The total Hamiltonian is Z tot γ 0 Ω 0 0 Pf (E) = Pf (E )Pf (E − E )dE . (18) H = Hγ + HΩ + Hint (13) where Hγ and HΩ represent the free Hamiltonians of the The main feature of our experiment is that the energy photon and opener respectively, and Hint is the interac- distribution of the opener doesn’t change. Indeed, this tion Hamiltonian. Energy conservation strictly speaking feature remains also when we look separately at the cases refers to the distribution of the eigenvalues E of H, which in which the photon remains in the box and the ones in includes the interaction term. However, although the in- which the photon is emitted. The photon remains in the teraction Hamiltonian is present at all times, the system box if originally it was far from the central region; in is prepared such that the photon and the opener interact this case it doesn’t hit the mirror, so the mirror (and only for a finite time. Indeed, they evolve essentially free, its entire moving mechanism) doesn’t change energy. On then interact for a finite time and then continue the free the other hand, when the photon is in the central region evolution. Hence, long before the interaction and long it collides with the mirror. However, as we emphasized afterwards, we can ignore Hint and say that the proba- when we analyzed the paradox, the mirror energy distri- bility distribution of the free Hamiltonian, Hγ + HΩ, is bution can’t change its energy distribution because, by conserved. That is, the distributions of the total energies causality, it must act identically to the case in which a Hγ +HΩ long before the interaction, and long after, must true high-energy photon was in the box; just as the high- be the same. energy photon just emerges without changing its energy, 7 and thus leaving the mirror’s energy unchanged, so must reach the opening. In our explicit model, the opener must the fake high-energy photon. Hence thus move from far away, going from an initial state in which there is no interaction to an orthogonal state in which there is interaction and then again to a state with Ω Ω Pi (E) = Pf (E). (19) no interaction. This is accomplished by moving through a long sequence of orthogonal states both before and after On the other hand, the final probability distribution the interaction (as one can explicitly see in the model). for the energy of the particle is different from the initial γ γ Hence, since the interaction should only take a finite time distribution, Pf 6= Pi . Indeed, initially the particle was T , it must be the case that the wave function φ of the in a superposition of energy eigenstates all smaller than opener, as it evolves, must obey 1. (See Fig. 2.) After the interaction, however, there are cases when the photon emerges from the box and has hφ(t)|φ(t + τ)i = 0 (23) high energy, much higher than 1, while in other cases it remains inside the box and has low energies, and the for any time τ > T . Since (23) is independent of t, we low energies average to something less than the original can take t to be far in the past, when the opener evolved average in order to conserve the total energy average. under its free hamiltonian HΩ. Hence, the fact that the All this leads to a surprising situation: The distribu- interaction takes only a finite amount of time implies tion of the energy of the photon changes without being that P˜Ω(τ) = 0 for τ > T and this enables the strange accompanied by a corresponding change in the distribu- behavior of the energy distributions that characterizes tion of the energy of the opener, yet the distribution of our problem. the total energy is conserved. Note that the opener has the role of a catalyst: its Although the above situation is surprising, it is math- energy distribution doesn’t change, yet, without it, the ematically consistent, and it has remarkable implica- photon’s energy distribution could not change, because tot tions. Denoting the conserved distributions Pi (E) = the energy of the particle would then be the total energy tot tot Ω Ω Ω Pf (E) = P (E) and Pi (E) = Pf (E) = P (E) we and changing its distribution would violate the standard obtain energy conservation law.

tot R γ 0 Ω 0 0 P (E) = Pi (E )P (E − E )dE VIII. MODULAR ENERGY EXCHANGE R γ 0 Ω 0 0 = Pf (E )P (E − E )dE . (20) It is interesting to examine further the changes in the By making the Fourier transform of the convolutions energy distributions. Since neither the total energy dis- (16) and (18) we obtain tribution nor the opener energy distributions change, it is clear that the average energy of the photon cannot change: Indeed, both before the interaction and after ˜γ ˜Ω ˜γ ˜Ω Pi (τ)P (τ) = Pf (τ)P (τ) (21)

R iEτ where, for each index, P˜(τ) = e P (E)dE is the hHi = hHγ i + hHΩi; (24) Fourier transform of P (E). ˜γ ˜γ since hHi and hHΩi are constant, so is hHγ i. Equation (21) can have a solution with Pi (τ) 6= Pf (τ) if and only if for some values of τ the Fourier transform Furthermore, given that both initially and finally the ˜Ω ˜γ energy distributions of the opener and photon are uncor- P (τ) is zero and the changes in Pi (τ) are confined to these τ values. related, and that the distributions of total energy and To understand the significance of the above results, we opener energy do not change, one can easily derive the first note the general meaning of the Fourier transform of fact that all the moments of the photon energy distribu- n the energy distribution. Consider a particle prepared in tion hHγ i are unchanged. the state |Ψ(t)i and evolving according to a hamiltonian We thus arrive at another remarkable conclusion: the H. Then (see Supplementary Information 3) energy distribution of the photon changes although none of its moments change. At first it seems that something must be wrong - indeed P˜(τ) = hΨ(t)|Ψ(t + τ)i. (22) it is generally assumed that the moments of a distribution completely define it. This however is not so. It is actu- Note that since the hamiltonian is time independent, ally perfectly possible for a distribution to change with- P˜(τ) is independent of t; indeed hΨ(t)|Ψ(t + τ)i is in- out any of its moments changing. In fact, in quantum dependent of t. mechanics this behavior characterizes some of the most In our case, we want the opener-photon interaction to basic phenomena (such as momentum conservation in the take place only for a finite time: the box must be opened, two-slit experiment [10–14]); and many of the “myster- the mirror inserted, then extracted and the box closed, ies” of quantum mechanics have this mathematical effect all before information from remote places in the box can at their core. This behavior generally stems from deep 8 reasons connected with causality and non-locality - as In concluding this section we would like to emphasize our present example illustrates. that the whole issue of exchange of modular energy (or So, if none of the moments of the photon’s energy dis- momentum) without any (significant) exchange of any tribution change, what changes? It is the average of ob- of its moments (or where the exchange of the moments servables that we call “modular energies” [12, 13], as we plays a trivial role) is a general characteristic of phenom- now show. ena in which a localized probe interacts with a system Consider the operator eiHτ ; we call this operator in an extended wave function. At the same time, the “modular energy” since it depends only on the energy particular phenomena described in this paper (the high modulo 2π/τ. Each τ defines a different modular energy. energy of the photon that emerges from the box) depend Since H is a conserved operator, so are its associated on the particular form (1) of the extended wave function. modular energies: The possibility of exchange of modular energy without changes in any of the moments of the energy distribution iHτ iHτ he ii = he if . (25) simply opens a window of opportunity, through which the phenomena described here can manifest themselves. As noted in the previous section, long before the interac- In other words, the exchange of modular energy with- tion and long afterwards, we can replace replace the full out changes in the moments of the energy distribution Hamiltonian by the free part, so that is just a necessary but not a sufficient condition for the phenomena we describe here.

i(Hγ +HΩ)τ i(Hp+HΩ)τ he ii = he if . (26)

Since there are no correlations between the states of the IX. DISCUSSION photon and the opener either before the interaction or To summarize, in our paper we present an effect that after it, from (26) we obtain that raises questions about what we actually mean by conser- vation laws. The standard approach is statistical and it iHγ τ iHΩτ iHγ τ iHΩτ he iihe ii = he if he if . (27) is good as far as it goes. However, our effect begs the question of what happens in individual cases. A pho- And since the energy distribution of the opener doesn’t ton, prepared in a superposition of low energy states and change, the averages of its modular energy for every τ with no high-energy component whatsoever, comes out iHΩτ iHΩτ don’t change either, i.e. he ii = he if . Hence, the of a box with great energy. It is legitimate to ask where only changes in the energy distributions of the photon are the energy comes from. We showed that the mechanism those of averages of the modular energy corresponding to used for extracting the photon - the only other system in those values of τ for which the average of the correspond- the problem- did not provide this energy, so we are left ing modular energy of the opener is zero. (This statement with a puzzle. is just Eq. (23) stated differently.) Here in particular we Our example concerned energy conservation. It is, have however, clear that one can construct examples involv- ing conservation of other quantities such as momentum

iHΩτ or angular momentum. The phenomenon is therefore a he i = 0 (28) general one. Thus we argue that the conservation laws of quantum mechanics must be revisited and extended. for every τ > T . This means that for τ > T the modular Without doing this, we will be missing a large part of the energy of the photon may change. message quantum mechanics is telling us. Note the interesting way in which the conservation of modular energy works. The total modular energy is con- served, so if one of two interacting systems changes its Acknowledgements Y.A. thanks the Israel Science modular energy this must be accompanied by changes in Foundation (Grant No. 1311/14) for support, and also the modular energy of the other, yet the average mod- the ICORE Excellence Center Circle of Light and the ular energy of the photon (for some τ) changes while German-Israeli Project Cooperation (DIP). S.P. thanks the corresponding modular energy of the opener doesn’t. the ERC AdV Grant NLST. D.R. thanks the John Tem- This is possible due to the fact that, as opposite to en- pleton Foundation (Project ID 43297) and the Israel Sci- ergy, which is an additive , its modular ence Foundation (Grant No. 1190/13) for support. The part is non-additive but multiplicative, and the modular opinions expressed in this publication are the authors and energy of one of the systems, namely the opener, is com- do not necessarily reflect the views of any of the support- pletely uncertain, which makes its average zero. ing foundations.

[1] This is not contradicted by the existence of the well- Bohm effect [2] and violations of Bell-inequalities [3] as known non-local quantum effects such as the Aharonov- well as Aharonov-Bohm type which are local in this sense. 9

[2] Y, Aharonov and D. Bohm, “Significance of electromag- 213 (1969). netic potentials in quantum theory”. Physical Review [13] Y. Aharonov and D. Rohrlich, Quantum Paradoxes: 115: 485491 (1959). Quantum Theory for the Perplexed, (Weinheim: Wiley- [3] J.S. Bell, “On the Einstein Podolsky Rosen Paradox”. VCH), 2005, Chaps. 5-6. Physics 1, 195200 (1964). [14] A. J. Short “Momentum changes due to quantum local- [4] Y. Aharonov, S. Popescu and D. Rohrlich, TAUP ization”, Fortschr. Physik 51, 498 (2003). preprint TAUP 1847-90, unpublished. [5] S. Popescu Ph.D. Thesis Tel Aviv University, 1991. [6] M. V. Berry and S. Popescu, “Evolution of quan- tum superoscillations and optical superresolution with- Supplementary Information 1 out evanescent waves”, J. Phys. A: Math. Gen. 39, 6965 (2006). To compute the time evolution we note that the [7] M. V. Berry, “Faster than Fourier”, in Quantum Coher- Schr¨odingerequation of a particle in a box (square po- ence and Reality; in celebration of the 60th Birthday of tential well) is identical to that of a free particle, except Yakir Aharonov, eds. J. S. Anandan and J. L. Safko (Sin- for the constraint that the wave function is zero at the gapore: World-Scientific) (1994), pp. 55-65. walls (at xL and xR). Hence, if one finds a free parti- [8] Y. Aharonov, F. Colombo, I. Sabadini, D. C. Struppa, cle solution that happens to be zero at all times at xL J. Tollaksen “Superoscillating sequences as solutions of and xR, and at t = 0 its restriction between xL and xR is generalized Schr¨odingerequations,” J. Math. Pures Appl. identical to the initial box wave function, then its restric- 103, 522 (2015). [9] E. H. Lieb, D. W. Robinson, “The finite group velocity of tion between xL and xR is also a solution for the particle quantum spin systems”, Commun. Math. Phys. 28, 251 in the box problem for all times. The wave function (5) (1972). with −∞ < x < ∞ is zero at all times at xL,R = ±πN [10] S Popescu, “Dynamical quantum non-locality”, Nature as one can immediately see by expanding the binomials, Physics 6, 151 (2010). so it obeys the above conditions. We will thus now solve [11] Y. Aharonov, “Nonlocal Phenomena and the Aharonov- this free evolution, using the free propagator method. Bohm Effect”, in Foundations of Quantum Mechanics in Light of New Technology [Proceedings of the Interna- tional Symposium, Tokyo, August 1983], eds. S. Kame- Z ∞ 0 2 1 0 1 i (x−x ) 0 fuchi, H. Ezawa, Y. Murayama, M. Namiki, S. Nomura, Ψ(x, t) = √ Ψ(x , 0)√ e 2t dx . (29) Y. Ohnuki and T. Yajima (Tokyo: Physical Society of 2πi −∞ t Japan), pp. 10-19. (1984) The integral can be computed by using complex integra- [12] Y. Aharonov, H. Pendleton and A. Petersen, “Modular tion and changes of contour in the complex plane. Eval- Variables in Quantum Theory”, Int. J. Theor. Phys. 2, uating the first term in the superposition (5) we obtain

1 Z ∞ 1 + α x0 1 − α x0 N 1 (x − x0)2 f(x, t) = √ exp(i ) + exp(−i ) √ exp(i )dx0 2πi −∞ 2 N 2 N t 2t 1 Z ∞ 1 + α x0 + x 1 − α x0 + x N 1 x02 = √ exp(i ) + exp(−i ) √ exp(i )dx0 2πi −∞ 2 N 2 N t 2t 0 1+i 0 1+i !N 1 Z ∞ 1 + α x ( √ ) + x 1 − α x ( √ ) + x 1 x02 = √ exp(i 2 ) + exp(−i 2 ) √ exp(− )dx0. (30) 2π −∞ 2 N 2 N t 2t

√ Here the last equality stems from rotating the integral Consider now t . N. Then√ the only contributions to line from the real axis to the π/4 line. the integral come from x0 N so we can expand the . 0 exponentials to first order of x . Furthermore, since we N √ To evaluate the integral, note that the absolute√ value of are interested only in the region x N we can also 0 . the expression in brackets increases as exp(x / 2) but expand the exponentials to first order of x . We then 02 √ √ N it is multiplied by exp(− x ), a term that for large x0 2t obtain for x . N and t . N decreases faster. For x0 > t the later term dominates and hence the contribution to the integral from these values of x0 are negligible. 10

0 1+i 0 1+i !N 1 Z ∞ 1 + α x ( √ ) + x 1 − α x ( √ ) + x 1 mx02 f(x, t) = √ exp(i 2 ) + exp(−i 2 ) √ exp(− )dx0 2π −∞ 2 N 2 N t 2t

0 1+i 0 1+i !N 1 Z ∞ 1 + α x ( √ ) + x 1 − α x ( √ ) + x 1 mx02 ≈ √ (1 + i 2 ) + (1−i 2 ) √ exp(− )dx0 2π −∞ 2 N 2 N t 2t

0 1+i !N 1 Z ∞ x ( √ ) + x 1 mx02 = √ 1 + iα 2 √ exp(− )dx0 2π −∞ N t 2t 1 Z ∞ 1 + i 1 mx02 α2 ≈ √ exp(iαx0 √ + iαx)√ exp(− )dx0 = exp(iαx) exp(−i t). (31) 2π −∞ 2 t 2t 2m √ In√ a similar way we can evaluate the second term of the superposition (5) and thus obtain that for x . N and t . N

α2 Ψ(x, t) ≈ sin(αx) exp(−i t). (32) 2

Supplementary Information 2 In other words, the particle is released if it is in the region |x| ≤ L and it remains trapped in the box otherwise. Consider the following Hamiltonian to model our ex- Let us concentrate on the released particles. We obtain periment:

2 pˆ 1 + σz π Θ (q, x) = Hˆ = + V (x) +p ˆq + g(x)δ(q)σx. (33) ↓ 2 2 2 2 2 −i pˆ (T −τ) −i( pˆ +V (x))τ −δ(q + τ − T )e 2m ig(x)e 2 ψ(x). (In this section, to avoid confusions, we will put hats over (37) the hamiltonian and momentum of particle and opener, to indicate that they are operators.) Note the effect of the spin flip: for time 0 < t < τ the Let the particle be prepared in the state Ψ(x, 0) = particle was confined in the box and evolved according to ψ(x), with spin ↑ (i.e. confined inside the box) and the the Hamltonianp ˆ2/2 + V (x) and for τ < t < T the par- opener in state φ(q), localised in the region −T ≤ q ≤ 0. ticle was free and evolved according to the Hamltonian We now calculate the time evolution of the particle- pˆ2/2. opener system By construction g(x) = 1 around the center of the box, i.e. for |x| < L and zero outside this region. As −iHTˆ √ e φ(q)ψ(x)| ↑i = Θ↑(q, x)| ↑i + Θ↓(q, x)| ↓i. (34) noted before, in this region for times τ < T < N the fake high-energy photon behaves like a true high-energy Let us first suppose that φ(q) = δ(q + τ), where 0 ≤ eigenstate, so we get (up to normalisation) τ ≤ T . The opener advances from t = 0, and at t = τ it reaches q = 0. When it crosses q = 0, the opener- π ig(x) σx particle state acquires a non-abelian phase e 2 , as Θ↓(q, x) ≈ we see by integrating the Schr¨odinger equation around 2 2 −i pˆ (T −τ) −i α τ the time t = τ. Thereafter, from t = τ to t = T it δ(q + τ − T )e 2 ig(x)e 2 sin(αx). evolves according to the free propagator. At time t = T , (38) therefore, we have the two-particle state A more realistic initial wave function φ(q) for the −iHˆ (T −τ) −ig(x) π σ −iHˆ τ δ(q + τ − T )e 0 e 2 x e 0 ψ(x)| ↑i, (35) opener would have support over a finite, extended region −T < q < 0. To compute the effect of such a trigger, we 2 just fold the previous result with this wave function. We where Hˆ0 =p ˆ /2 + V (x)(1 + σz)/2. The effect of the trigger is to flip the spin in the region write |x| ≤ L where g(x) = 1 and leave it undisturbed where Z g(x) = 0: φ(q) = φ(−τ)δ(q + τ)dτ (39)

−i π g(x)σ e 2 x | ↑i = (1 − g(x))| ↑i − ig(x)| ↓i. (36) and we obtain for the released particles 11

Θ↓(q, x) ≈ Θ↓(q, x) 2 α2 pˆ α2 −i T −i( − )q Z ∞ 2 2 2 φ(q − T )e 2 e 2 2 g(x) sin(αx). −i α T −i( pˆ − α )q ikx ≈ φ(q − T )e 2 e 2 2 h(k)e dk, (40) −∞ Z ∞ 2 2 2 −i α T −i( k − α )q ikx An essential thing to note about (38) and (40) is that = φ(q − T )e 2 e 2 2 h(k)e dk.(43) they are identical (up to normalisation) to what would −∞ have happened had we started with the particle in the energy eigenstate sin(αx) instead of the “fake” state ψ(x) of eq. (5). To analyze the energy distribution of the emitted par- Supplementary Information 3 ticles we write the final photon-opener state using the ˜ momentum representation for the particle. Using To compute P (τ):

R iEτ Z ∞ P˜(τ) = e P (E)dE ikx g(x) sin(αx) = h(k)e dk (41) = R eiEτ hΨ(t)|EihE|Ψ(t)idE −∞ = hΨ(t)|eiHτ |Ψ(t)i = hΨ(t)|Ψ(t + τ)i (44) with Z ∞ where |Ei denotes an eigenstate of energy and where we 0 0 −ikx0 0 h(k) = g(x ) sin(αx )e dx used the fact that −∞ L 1 Z 0 0 = dx0(ei(α−k)x − ei(−α−k)x ) (42) 2i −L Z eiEτ |EihE|dE = eiHτ . (45) and where in the second equality we used the definition of g(x), we have (up to normalisation)