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Classical paradoxes of locality and their possible quantum resolutions in deformed special relativity Lee Smolins Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2J 2Y5, Canada July 5, 2010 Abstract In deformed or doubly special relativity (DSR) the action of the Lorentz group on momen- tum eigenstates is deformed to preserve a maximal momentum or minimal length, supposed equal to the Planck length, to, = VO. The classical and quantum dynamics of a particle prop- agating in ti-Minkowski spacetime is discussed in order to examine an apparent paradox of locality which arises in the classical dynamics. This is due to the fact that the Lorentz transfor- mations of spacetime positions of particles depend on their energies, so whether or not a local event, defined by the coincidence of two or more particles, takes place appears to depend on the frame of reference of the observer. Here we discuss two issues which may contribute to the resolution of these apparent paradoxes. First it may be that the paradox arises only in the classical picture, because it is inconsistent to study physics in which h = 0 but Ir = R ;‘ 0. Second, there may still be an observer independent notion of a local interaction, which slightly extends the usual notion without coming into conflict with the observed locality of interactions in nature. These considerations may be relevant for phenomenology such as observations by the Fermi observatory, because at leading order in 4 x distance there is both a direct and a stochas- tic dependence of arrival time on energy, due to an additional spreading of wavepackets. lsmolineperimeterinstitute.ca 1 EFTA_R1_01441413 EFTA02403023 Contents 1 Introduction 3 1.1 The strategy of this paper 4 1.2 Outline of the argument 5 2 DSR for a free particle in terms of non-commutative geometry 8 2.1 An algebraic approach to DSR 8 2.2 Review of K-Poincare 9 2.3 Correspondence with n-Poincare 9 3 Classical paradoxes of locality and their quantum resolutions 10 3.1 Non-locality 10 3.2 Boosts and events 11 3.3 Transverse length contraction and relativity 12 3.4 A classical locality paradox 12 4 Quantum theory of a free relativistic particle 13 4.1 Bases in Hilbert space 13 4.2 Dynamics in momentum space 14 4.3 Dynamics in position space 14 4.3.1 Propagation and spreading of a wavepacket 14 4.3.2 Wave equation in spacetime 15 4.4 Velocity in the commutative space-time coordinates (xi, T) 16 4.4.1 Phase velocity 16 4.4.2 Classical computation of velocity 16 5 Redefining locality 17 5.1 Demonstration of non-genericity of m-intersection for 3 or more particles in d = 3 spatial dimensions 19 5.1.1 The set up 19 5.1.2 Bringing three or more worldlines to intersect 20 5.1.3 A simplifying assumption 20 5.1.4 Dropping the simplifying assumption 21 5.2 The case of d = 1 spatial dimension 22 5.3 Stars and so forth 23 6 Conclusions 24 2 EFTA_R1_01441414 EFTA02403024 1 Introduction Doubly or deformed special relativity (DSR) is the hypothesis that the Poincar4 group or its action is deformed to take into account the possibility of a maximal momentum or energy for individual elementary particles, without violating the relativity of inertial frames(1, 2, 3]. While this is an attractive idea, not least because it is accessible to investigation by current experiments[4], inter- pretations of and predictions for these experiments have been challenged by several confusions as to the interpretation of DSR in spacetim05, 6]. These involve the notions of locality and veloc- ity. The purpose of this paper is to propose an origin for these confusions which afflict spacetime descriptions of DSR and to investigate three features of DSR theories which may play a role in resolving them. One very physical way to understanding the idea of DSR is to see it as a phenomenological description which arises in a particular limit of some underlying quantum theory of gravity'. In this limit we take, h —. 0, AND G —. 0 (1) in such a way that the dimensional constant2 MP = — constant (2) G is held fixed. (We work here in units where c = 1, since we assume there is still an invariant velocity.) That is, we turn off both quantum mechanics and gravity, but in such a way as to preserve phenomena depending on Mp and c. DSR is then something like the smile that is left of the Chesire cat of quantum gravity. We can call this the classical DSR limit of quantum gravity. Just like the better studied limits in which we take h 0 or G 0 separately, this is a limit that must exist if the quantum theory of gravity is well defined. It is easy to state how physics may be modified in the DSR limit: momentum space becomes curved, with the radius of curvature measured by the invariant Mp. There are then two cases to discuss depending on how momentum space may be curved. Poincare invariance may be broken, in which case the symmetry group of momentum space will have fewer than the ten generators of the Poincar€ algebra. Or, if we want to preserve the existence of a ten parameter symmetry group, then momentum space must have constant curvature, ie it has a deSitter or anti-deSitter geometry. This results in a deformation of Poincare invariance. This is the basic idea of DSR. So long as we stay in momentum space the implementation of this idea is straightforward, but issues develop when we ask for the effect on physics in the complementary spacetime description. These problems can be seen to arise because we are working in a limit in which h has been taken to zero. In this case there is no fixed length or time corresponding to the mass fixed in (2). Instead, tp = —) 0 (3) This means that the classical DSR limit only yields something new in momentum space. When applied to physics in spacetime, the classical DSR limit is ordinary special relativity. Another way to say this is that the relationship between momentum space and spacetime depends on h being non-zero. We need it to make sense of the fourier transform, without it we 'This viewpoint was first proposed by Kowalski-Glikmanl3] 2In d = 3, the number of spatial dimensions. 3 EFTA_R1_01441415 EFTA02403025 could not write &P. Alternatively when h is zero x° and pa commute, so the idea that one generates translations in the other disappears. Hence, DSR can be only understood as a classical theory in momentum space. If we want to translate the physics of DSR into spacetime we need h $ 0 which means we must work with the quantum dynamics. In spite of this, there have been attempts to describe DSR physics in the language of classi- cal dynamics of spacetime. These have given rise to some confusion. Indeed, as pointed out by Schtzhold and Unruh[5] and Hossenfelder[6] there are apparent paradoxes that challenge the con- sistency of the description. They describe thought experiments in which the simple question of whether the world lines of three or more particles coincide at a particular local event in spacetime appears to be observer dependent. This challenges the basic operational definition of a spacetime event proposed by Einstein, according to which an event is defined by the coincide of several particles in space and time. From the point of view just mentioned, it is not surprising if problems appear when one at- tempts to discuss the physics of DSR in terms of classical physics in spacetime, because quantum effects are being treated inconsistently. If we include effects coming from a finite 1p = VrG we must include quantum effects because h is non-zero. This suggests an hypothesis about the para- doxes raised in [5] and [6], which is that they perhaps arise because it is inconsistent to reason about DSR effects in spacetime as if quantum mechanics was turned off, but Ip is still non-zero, because we are being inconsistent about the dependence of observable quantities in h. 1.1 The strategy of this paper It is one thing to suggest an hypothesis about a problem and another to show that it cleanly solves the problem. When seeking to go further, however, we run into a problem, which is that the scenarios described in [5] and [6] are discussed in somewhat heuristic contexts. This is sufficient to convince one there is are issues worth worrying about, but may be insufficient to resolve them, because it is not clear which precise physical theories-if any- correspond to the assumptions that are made there. For better or worse there are several distinct formulations of theories which are motivated by the idea of DSR, and it is not known if they are equivalent; nor is it known in all cases if these formulations are completely self-consistent. There are also two distinct issues that are easily confused in these discussions: 1) Is a particular formulation of DSR internally consistent? 2) If it is consistent, does it lead to predictions that disagree with well confirmed observations such as the locality of physics? Because of these issues we follow a cautious strategy in addressing the apparent paradoxes raised in [5] and [6]: 1. We work within the best defined formulation of DSR, which is physics on the non-commutative manifold K-Minkowksi spacetime. For simplicity we discuss in this paper only the theory of free relativistic particles on rc-Minkowksi spacetime[7, 8, 9, 10, 11]. 2. In this context we formulate a paradox similar to those discussed by [5] and [6]. 3. We discuss in this paper two approaches to resolving that paradox, one having to do with additional quantum effects special to DSR theories, a second having to do with a possible relaxation of the notion of locality. 4 EFTA_R1_01441416 EFTA02403026  4. A third strategy or resolving the paradoxes is discussed in another paper, which is that the apparent non-localities are a coordinate artifact associated with an ambiguity in extending Einstein's procedure for synchronizing clocks to clocks with a finite frequency far from the origin of coordinates of a reference frame[251. The results are tentative, in that we find evidence that both effects may play a role, but we are not able to definitively show the problem is solved. It must also be emphasized that because of the strategy we choose we cannot address directly the paradoxes of Schatzhold and Unruh[51 and Hossenfelder[613. We can here only address similar issues which appear in n-Minkowski spacetime; whether the same insights apply to the cases discussed in [5, 61 can only be resolved by further work. 1.2 Outline of the argument This having been said, let us introduce the basic ideas and results that are discussed below. It is easy to see how an issue with locality arises in K-Minkowski spacetime. The details are in the sections below but for the sake of clarity of argument we can now sketch the key points of the argument. If momentum space is curved then translations on momentum space don't commute with each other. But if the spacetime coordinates are constructed to be complementary to momentum space in the usual way, they are the generators of translations on momentum space, which means that they don't commute. So we have a non-commutative spacetime geometry We have then to investigate whether this non-commutative spacetime can support a consistent framework for physics, which agrees with experiment. The new commutation relations have to be consistent with a ten parameter algebra of symme- tries, inherited from the symmetries of the deSitter geometry of momentum space. It turns out this can be achieved if we take it that the space coordinates do not commute with the time coordinate [t, = itpx. (4) where to = VrIG is the Planck time. This certainly gives rise to some general problems of interpre- tation because the notion of an event, which requires localizing two or more particles at the same point of space and time, appears to be compromised by the inability to make both space and time measurements simultaneously sharp4. The next step is that, in order to preserve the commutation relations (4), the Lorentz transfor- mations of the position and time coordinates of a relativistic particle turn out to depend on its energy and momenturn[3]. (These transformations are reviewed in (37,38) below.) This energy and momentum dependence of the Lorentz transformations leads directly to prob- lems with locality. To see why, consider a scattering event defined by the coincidence of four worldlines of particles of different energy and momentum-two worldines for the incoming parti- cles and two worldlines for the outgoing particles. Suppose that one inertial observer sees them 31t should also be mentioned that Hossenfelderl6l does consider the possibility that quantum effects resolve the problem, but comes to a different conclusion then we do here. Whether that is because the models are different or for other reasons remains unclear. ' Alternative formulations of DSR which involve instead of (4) an energy dependent metric also give rise to similar confusions, but these will not be discussed here. 5 EFTA_R1_01441417 EFTA02403027 as coinciding at a single value of (xi, t). Suppose also we want to use a Lorentz transformation to derive the trajectories of those particles as seen by a second intertial observer. Then it is easy to construct cases in which the four particles no longer coincide in the second frame of reference, be- cause the modified Lorentz transformation takes the positions coincident in one frame to different locations, depending on the energy and momentum of the particles. This certainly sounds bad, as it means an interaction that looks local to one observer involves four separated events in another observer's description. However, we have argued that the clas- sical picture of DSR may not be self-consistent, so this puzzle should be re-examined in the quan- tum theory. To see why quantum effects may help to resolve these apparent paradoxes, note that the commutation relation (4) imply additional uncertainty relations[16, 17, 18, 19], &tax ≥ tplxi (5) This implies immediately that we cannot construct a quantum state in which we can precisely localize a single particle both in x' and t. So the first observer cannot actually be sure that the four worldlines coincided at a single event. All observers must describe the possible interactions amongst the particles in terms of quantum probability amplitudes. Indeed, as is discussed by sev- eral authors(16, 17, 18, 19, 6, 23], this additional uncertainty gives rise to an anomalous spreading of wavepackets due to the modified commutations relations (5). We reproduce this below, and in- vestigate the extent to which this may provide a solution to the problem of non-locality generated by energy and momentum dependent Lorentz transformations. However, we are not able to demonstrate that the spreading of the wavepacket is sufficient to hide the non-locality generated by the Lorentz transformation for all states. Thus, we next inves- tigate another approach to the issue. This is whether there might be a relaxation of the notion of locality which can incorporate the Lorentz dependence of the notion of a localized event, without at the same time leading to non-locality being so generic that it blatently disagrees with known physics. Thus, in section 5 we introduce a notion of n-locality in the context of the classical free relativistic particles, which is defined as follows: A set ofN ≥ 3 events, E', each on a worldline 4 of a free relativistic particle, are n-local if there is an (energy and momentum dependent) Lorentz transforma- tion which takes the N events to a single event 13°'. We can then hypothesize that interactions among particles propagating in n-Minkowski spacetime are n-local rather than locals. We can call N ≥ 3 worldlines that contain mutually n-local events, n-intersecting. This will be acceptable only if generic sets of N ≥ 3 worldlines do not contain any mutually n-local events. Equivalently, it should not be the case that N ≥ 3 sets of worldlines can be brought to intersect by an energy and momentum dependent Lorentz tranformation. For were this pos- sible, the principle that physics is n-local would imply that any set of N ≥ 3 particles could be interacting. Were this the case the notion that physics is local would be entirely lost. Happily, we show that this catastrophe does not occur. In section 5 we find that n-locality is not a generic property of any set of three or more worldlines. To the contrary, the sets of n-intersecting triples of worldlines are of measure zero in the sets of three worldlines, and become even rarer as four or more particles are involved. To investigate whether physics was n-local rather than Note that we restrict consideration to 3 or more particles, because if there is a real interaction amongst two par- ticles, either they exchange energy and momentum, in which case the outgoing worldlines have different energy and momentum than the incoming ones, or they annihilate into a third particle. Coincidences of two particles that do not change either's energy and momentum are not physical interactions. 6 EFTA_R1_01441418 EFTA02403028 local would then take very delicate experiments, with finely tuned initial and final conditions to restrict to the case where the worldlines of the ingoing and outgoing particles are all mutualy +c-intersecting. Hence, the world could be pc-local rather than local, and we would not yet have noticed the disfinctione. There is a third point which may play a role in resolving the paradoxes of locality. Let us return to the time coordinate t. It is possible to choose the phase space description so that t commutes with the components of spatial momentum. Hence t can be used to discuss the dynamics in momentum space. However, as we shall see in the next section, this comes with a cost, which is that the usual position-momentum commutators are deformed to [xi, pi] = f (pi)) (6) where f(p0) is the function J(E) = et00/ 4 (7) This has an interesting consequence, which is that we can introduce a new time coordinate that does commute with the xi. This is7 T=t+ (8) rtI(vo) It is easy to confirm that, [T, xi] = 0 (9) Hence, it is unproblematic to discuss events defined in a spacetime coordinatized by (x1,T), even in the quantum theory. So it is natural to propose that the classical spacetime in which we observers make measurements is defined by clocks that measure T and rulers that measure xi. However, T no longer commutes with pa, instead [T, Pi] = ItyPi (10) Hence there is a good time coordinate, T, to discuss evolution of wavefunctions in position space. And there is a good time coordinate, t, to discuss evolution in momentum space. But they are related to each other by the non-local transform (8). This plays a role in the derivation of the spreading of wavepackets in (x1.71) space. While the aim of this paper is to address the issue of physical consistency of DSR, we note that there are phenomenological implications of the results derived below. By studying the prop- agation of a wavepacket in (x`,T) spacetime we are able to study the question of the energy de- pendence of the speed of massless particles. We find that there is a first order variation of the speed of light with energy. In addition, there is another first order effect, which is the new contri- bution to spreading of wavepackets. This gives a stochastic variation of arrival times of photons proportional to TtpAE, where T is the time traveled and AE is the uncertainty in energy of the 6We may note that this appears to disagree with the claim of Hossenfelder in (6J that in this language can be trans- lated as the assertion that n-local physics is ruled out by experiment. However, it should be stressed that whether this is due to her model of DSR physics being different or to another cause is not clear at this time. What we can assert is that no claim can be made that physics in ic-Minkowski spacetime is grossly in contradiction with the observed locality of physical interactions. 'This is also discussed in Mb 7 EFTA_R1_01441419 EFTA02403029 wavepacket. This is at the same order as the linear dependence of velocity with energy, and so might be observable in current observations by Fermi and other observatories. A new strategy to bound or measure this kind of stochastic effect in the Fermi data needs to be developed. The remainder of this paper is organized as follows. In the next section we describe a general approach to deforming the quantum physics of a free particle in Minkwoski spacetime and then show how it can be specialized to tc-Minkowski spacetime. In section 3 we show how paradoxes of locality can be generated by studying inconsistently classical physics in n-Minkowski space- time. In section 4 we investigate the extent to which these apparent paradoxes of locality may be resolved in the quantum theory of a free relativistic particles. Then in section 5 we introduce the concept of n-locality and n-intersecting and show that they are very non-generic properties of sets of worldlines. We conclude by listing some of the open issues that remain to be resolved before it can be asserted that DSR is either can or cannot be fully realized within quantum physics and hence before its experimental implications can be unambiguously predicted. 2 DSR for a free particle in terms of non-commutative geometry 2.1 An algebraic approach to DSR We begin by examining how the Hilbert space for a single free relativistic particle can be deformed consistently. We start with a set of possible deformed commutation relations. [4P2) = (Po), ft, Po) = tho(ro) (11) [t, xi) = ttprih(po) (12) with the rest vanishing. In particular, we assume, [CA] = 0 (13) because I would like to define the quantum evolution in a time that commutes with momentum so I can evolve states on momentum space in the usual way. By checking the Jacobi relation 0= p + (14) we find that _ 1p (15) T97 It is important to note that the nonvanishing of (12) follows from the deformations in (11) by the Jacobi relations. So a non-commutativity of space and time coordinates is a natural consequence of the deformation of the canonical commutation relations for a relativistic particle. What this means is that we cannot speak of events or evolve position space wavefunctions in the usual way, so long as we use the time coordinate t. 5We note that the uncertainty relations and the resulting spreading of wavepackets have been discussed early in the literature on K Minkowski spacetime and DSRI16, 17, 18, 191. What is new here is only the suggestion that these may be necessary to resolve the apparent paradoxes of locality arising from the dependence of boosts of spacetime coordinates on energy 8 EFTA_R1_01441420 EFTA02403030 2.2 Review of h-Poincare Let us first briefly review physics in ,c-Minkowski spacetime[7, 8, 9, 10, 11, 3]. The basic idea is that momentum space is a deSitter spacetime coordinatized by ko and k; with a metric kp ds2 = —dkg + e 3:P dkidki (16) The commutation relations are [xi,k1] = [t, ko] = (17) [t, xi] = ar yl (18) (t, ki] = -itpki (19) In particular, note that unlike what we have assumed above, the commutator of t with k, is non- zero. The dynamics is defined by a Hamiltonian constraint constructed from the Casimir of the tt- Poincare algebra. This is the invariant length on the curved momentum space, which is invariant under an 5O(1, 3) subgroup of the deSitter group. 7i= 4E2sinh2 — kikiet —'»a2 = 0 (20) 2Ep An integration measure on momentum space, invariant under the non-linear action of the Lorentz group is defined by dw = dko A d3ke (21) 2.3 Correspondence with s-Poincare To construct the quantum theory we prefer to work with our original ansatz according to which = 0 (13). This way we can evolve the eigenstates of momentum p, in the time t. We can see that this corresponds to the solution of the Jacobi relations given by f=et, h=9= 1 (22) with the relation = Po = ko (23) The commutation relations are then [4 Pt] = iheieg", Et,pol =th (24) [t, x'] = stpx' (25) [t, pi] = 0 (26) In terms of these variables the metric on the de Sitter momentum space is dso _ 20 MA) PidPidPo dpidpi (27) dpo — 4:11) Ea 9 EFTA_R1_01441421 EFTA02403031  The Casimir of ,c-Poincare now takes the form, = 4.6P2S/nn2 Ps — rnipie- — rn2 = 0 (28) 2E and the invariant measure is du) = dm A dap (29) Because of (25) we cannot discuss evolving the position of the particle in the time t. However, because of (26), the time t is suitable for evolving the particle in momentum space. To evolve the system in the position coordinates e we define a new lime variable (8) which obeys [T, xi] = 0 (30) However, [T, pi] = ttpm (31) so T cannot be used to evolve wavefunctions in momentum space parameterized by pi. Thus, if we take pi for the spatial momenta, we see the very interesting conclusion that the evolution in position space and momentum space must take place with different time variables, whose relation to each other, as defined by (8), is non-local in both position and momentum space. We can however eliminate (31) by going back to measuring spatial momenta in terms of k1, because we have, [T, kg] = 0 (32) 3 Classical paradoxes of locality and their quantum resolutions We now show how paradoxes of locality can be generated by inconsistently taking h = 0 but tp 0 0. Then we show how they may be resolved when h is turned back on. 3.1 Non-locality We can see just from the algebra of observables that there will be apparent issues with non-locality if we use the wrong set of coordinates. Suppose that we subject our particle to sudden force coming which is local in space so it occurs at a particular x' = a'. Since x' and T commute we can localize the event precisely also in T, so that it occurs at a particular T = Ta. Thus, in the (x' ,T) variables, the force can be modeled as coming from from a potential V(x,T) = 63(x1 — al)(5(T — TO) (33) Note that we could not write a potential local in terms of xi and t because they don't commute. Hence, V'(x, t) =?/53(xl — ai)6(t — to) (34) is undefined as it is beset with operator ordering issues. 10 EFTA_R1_01441422 EFTA02403032  So let us stick with the first event, local in x' and T, but suppose we want to describe when it happens in terms of the other time variable, t. We will have that it takes place at ai but at a different, momentum dependent time, given by tp t = Ta — a M (35) hf0,0) however, we cannot measure T and pi at the same time, since [T, pi] doesn't vanish. Alternatively, since x1 is sharply defined, pi is maximally uncertain. So we cannot predict when the event will take place in the t coordinate. Conversely, if we measure pi we can determine that the event takes place at a definite t — with a particular momentum b,. But then in terms of the spacetime variables, the event will take place at a position dependent time • T = t + —ebi tp (36) but since p; has been measured sharply, xi will be maximally uncertain, so when the event takes place in terms of T will be maximally uncertain. These facts have to be taken into account carefully in any description of events in spacetime. To investigate their effect on propagation of particles we will in the next section construct the quantum theory of a single relativistic particle. But first we see how the paradoxes we referred to in the introduction arise. 3.2 Boosts and events The issues that give rise to the apparent paradoxes of Unruh et al and Hossenfelder become appar- ent when one writes down the Lorentz transformations for position in te-Minkowski spacetime. From [12,13] we find for a boost of magnitude fry denoted by a spatial vector cat, the position and time coordinates transform as = — tpeijkwiLk (37) (5t = —co • x + tpco • N (38) where Li are the spatial angular momentum generators =1 — eijkx;Pk • (39) and Ni are the generators of deformed boost transformations _ kit 4 21 N1= -Pie sP t - xi [—E 2 (1 - e- eq, )+ • pe (40) One cart also check that 6T = • x(1 + tpr(po)) + tou • pe- - (41) where 1 r(po) = (1 - e-24m) + tpp • pe'PP° + (42) 2tp 11 EFTA_R1_01441423 EFTA02403033  To first order in tp we have (5T = —us • x(1 + tppo) + tpco • pr + O(1) (43) We now see how some apparent paradoxes stem from these transformation laws. 3.3 Transverse length contraction and relativity We note first of an, that from (37), directions perpendicular to the direction of the boost can con- tract. This gives rise to an apparent paradox. Consider two inertial observers, Alice and Bob, who are approaching each other along their i axes, each of whom carries a stick along their x axis. Sup- pose that by (37) Alice sees Bob's stick contract. What does Bob see? By the relativity of inertial frames Bob should also see Alice's stick contract. This is what we say with sticks parallel to their relative velocities and this turns out to be consistent. But now let us note that as they pass Alice and Bob can mark where the end of the other's stick passes their stick. This does not yield a prob- lem when they are held parallel because of the relativity of simultaneity but it is a problem now, because the events of marking the ends of the sticks are simultaneous in both observer's frames. Hence, if Alice sees Bob's stick to have contracted relative to hers, Bob must agree that Alice's stick is longer. But this appears to violate the relativity of inertial frames. This is why ordinarily we do not have contraction of directions perpendicular to motion in special relativity. The resolution of this problem is that the contraction of the perpendicular directions is propor- tional to conserved quantities, which in this case is they component of angular momentum of the sticks. So whether it is Alice of Bob who sees the other's stick as shorter than theirs depends on which stick has a larger y component of angular momentum. 3.4 A classical locality paradox There are other more serious apparent paradoxes connected with the fact that the transformations (37) and (38) are dependent on energy, momentum and angular momentum of the objects which are being transformed. Here is a prototype of an apparent paradox involving transformations between the observa- tions of two inertial observers Alice and Bob. Suppose that Alice sees a collision of two particles at a position ai and time T = s in her frame. We can use the transformations (37) and (38) to compute the first order positions and times of the particles at the collision, as they will be seen according to Bob's instruments. Bob will see the two particles to have positions and times given by xi' = + de, = s + ST (44) where de and 6T are given by (37) and (41), respectively. If the two particles have different values of energy, momentum and angular momentum, Bob will see the event that Alice sees as two particles colliding as corresponding to two events, sepa- rated in space by a vector with space and time components Dx' and DT, given by Dx' = tp (e.dakthik — (Wiwi DLk) (45) DT = —tp (co • aDE — sus • Ap) (46) 12 EFTA_R1_01441424 EFTA02403034 where Dpk, DE and DL' are the differences in conserved quantities carried by the two particles, as observed by Alice, ie Dpk = pt — 4 etc. Thus, the two particles that Alice sees coincide need not be seen to coincide by Bob. Indeed, since there is a shift perpendicular to the direction of the boost it is possible that they never collide at all. It is thus easy to construct apparent paradoxes by, for example, supposing that the two particles have a short ranged interaction that scatters them. Suppose that the boost to Bob's frame is large enough that the Dx are larger than the interaction range. Does Alice see them to scatter, but Bob not? 4 Quantum theory of a free relativistic particle The purpose of this section is to investigate whether quantum dynamics can contribute to the resolution of the puzzle we have just discussed. We study the quantum dynamics of a free particle in ,c-Minkowski spacetime. and show that there is an anomalous spreading of the wavepacket proportional to tp x distance. This shows that quantum effects could contribute to the resolution of the locality paradox. 4.1 Bases in Hilbert space We begin our study of quantum physics in n-Minkowski spacetime by constructing the I-filbert space, 11, by constructing eigenstates of complete commuting sets of operators. We will see shortly that a key point of our approach is that the dynamics is defined first in mo- mentum space, then transformed to the commutative spacetime defined by the (xi ,T) operators. One complete commuting set of observables is composed of (E, p;). They define a basis p >= p,IE,p >, .tlE,p >= ElE,p > (47) with completeness relation given by the measure (29) 1 = r dEd3plE,p >< E,pl (48) Another complete commuting set is (t,pi). These are useful for discussing evolution in time in momentum space. These define a basis Alt, p >= p,It,p >, ilt,p >= tlt,p > (49) with completeness relation 1= J dtd3Pit,P >< t,PI (50) We note that because of It,EI = th we have < t, >= 63(p,pf)e:Eifit (51) We note that there is no basis which simultaneously diagonalizes t and x1. 13 EFTA_R1_01441425 EFTA02403035  If we want to discuss evolution in position space we have to use a different time coordinate, T, which is part of another complete set of commuting observables, given by (T, x'). They define a basis by x >= x`IT, x >, TIT,x >= TIT, x > (52) with completeness relation 1 = f dTcf3xIT,x >< T, xi (53) To transform from an energy basis to evolution in position space then requires two steps. First we have to change from energy to the time t by using the relation (51). Next we change from the (t, p) basis to the (T, x) basis through amplitudes < t,pIT,x >. ea 8(T - t - ±t xip-) (54) hf This gives us the non-local transform 41(x', T) = <T, x141 > = 1 d3pdteddE < T,xlt,p > < t,plE > < E, pit > d3pdEe-i -FaE(T—Sx14/ .) 41(E,p) (55) 4.2 Dynamics in momentum space We will impose dynamics by defining a modified dispersion relation in energy-momentum space. We define the dynamical Hamiltonian constraint operator by the quantization of (28). = 4E2sinh2 ) — pipe EP -m 2 =0 (56) ( 2EA We define the physical Hilbert space 7-tphy, to be the subspace of 7i defined by >. 0 (57) In energy-momentum space the solutions of this are given by W(E, = 6(4E:sinh2 ( 12 ) — pipie 4e —1712)x(p). (58) 4.3 Dynamics in position space 4.3.1 Propagation and spreading of a wavepacket We now perform the transform (55) in the case of one spatial dimension with a solution (58) with wl X( 9) = e y= 2Ae (59) 14 EFTA_R1_01441426 EFTA02403036 With Ax = P Pa. T-. 1 the result is proportional to 2x1p(x — T)2 W(x,T) Re• exp ( — [(x T)2 — 8zippo(x — T)1 + 4x p [PO (x T) + &e l I (60) 2Aw2 1 + -g where there is a new width Aw = OX2 + x21;4 2 (61) We see that there is a new effect proportional to 1p in which the width spreads out as the particle propagates. When the particle reaches an x > Art: the width grows as Aw xipAp (62) This is implies there is a stochastic effect on the time of propagation which is linear in /p and of the same order as the linear variation in velocity. It is interesting to ask whether this additional wave packet spreading can address the para- doxes of locality we discussed in section 3.4. We can be sure that the resulting quantum uncer- tainty would dominate over the locality paradox if in every frame of reference Aw > Dx, Aw > DT (63) Neglecting the transverse term proportional to the angular momentum we see that this would be the case if for all particles, AP > (64) This implies, using (45) and (46) that Aw xlpAp > rIpp > klx/pDp = IDxl (65) Thefore, for states where (64) is true in every frame of reference the quantum uncertanty will dom- inate over the apparent non-localities created by the momentum dependence of lorentz transfor- mations. This is encouraging, but not definitive, for one thing because there are states for which (64) is not satisfied. 4.3.2 Wave equation in spacetime We have constructed the wavefunction by transforming from a wavepacket in momentum space, but this should be equivalent to solving a wave equation in spacetime. The problem is that the corresponding wave equation, while linear in W(x, T) is a complicated function of space and time derivatives. To see what it is, we transform the constraint (56) to a wave-equation on spacetime. Using the transform (55) we find that 0 0 —the ---d Par (66) '97" ax. So the wave equation gotten by transforming (56) is = sinh2(-the) ) - e"Pir V2 - m2 W(xi,T) = 0 (67) 15 EFTA_R1_01441427 EFTA02403037  We see that the spacetime wave equation (67) is of infinite order in time derivatives. This means that while every transform of a solu