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"Quantum Concepts Eds.R.Penrose Clarendon in Space and Time' and C.J.lsham e2Jrivt 15.1. Introduction The rise of quantum physics, very surprisingly, has exerted little influence on our views about space and time. With the exception of quantum gravity, all theories assume that quantum processes occur in the spacetime of classical physics, and even in quantum gravity the modifications are presumably restricted to regions of space-time of the size of the Planck length. In this chapter I argue that the very structure of all quantum theories suggests a revision of the classical notions of space and time. I will present evidence that two copies of space-time, rather than one, are the proper arena for all quantum processes. At the heart of this observation lies the very well known fact that every set of equations and formulae in quantum theory, from which all the transition amplitudes are determined, may always be written in two equivalent forms, differing by complex conjugation. We obtain one set from the other by reversing the sign of the imaginary unit i. This is clearly seen already in the simplest case of the non-relativistic quantum mechanics. The sign of i may be chosen according to the following two conventions: ili at'll = H'II, -ili at'll = H'II, [x, pI = iii, = [F, Hl, [x, p] = -iii, -iliP = [F, HI. eiEtlh. P=A* ·A, Iwo Bialynicki-Birula II. Opposite convention or (15.2) When the two formulations were compared, it became obvious that one of the signs had to be reversed to achieve consistency. Of course, both formulations, I and II, are physically completely equivalent; they give identical transition probabilities. The transition probabilities are defined as squared moduli of transition amplitudes, Transition amplitudes versus transition probabilities and a reduplication of space-time I. Standard convention 227 occur, since in both formulations of quantum mechanics, considered independently from each other, it is natural to describe the time evolution through the harmonic factors Press, Oxford 1986 15 iltP Transition amplitudes versus transition probabilities (15.1) Convention I originated with matrix mechanics (Born and Jordan 1925), while convention II was used originally by Schr6dinger (1926, 1927) in his ftrat papen on wave mechanics. The clash of the signs was bound to (15.3) so that a change of convention results merely in an interchange of A and A*. The interchange of the two formulations can also be effected by the sign reversal of the Planck constant Ii. This is the first indication that the existence of the two formulations may have something to do with space-time properties of quantum mechanics. After all, the Planck constant is a dimensional quantity that depends on the units of length and time. In particular, it changes its sign when the unit of time is reversed. As a result of this, the interchange of the formulations I and II is essentially equivalent to the Wigner time reversal (Bialynicki-Birula 1957). The association of the transition amplitude A with the process evolving forward in time and of the complex conjugate amplitude A * with the time-reversed process enables one to view the transition probability P as describing a process that evolves around a loop in time. Such loops in time were introduced long ago (Schwinger 1960, 1961) as a mathematical device that helps to evaluate expectation values. I believe that these loops in time should be taken more realistically and I shall present here some tentative ideas on this subject. 15.2. Probabilities and two-sided diagrams Let me begin with non-relativistic quantum mechanics. The transition probability for a particle to travel from Xo, to to x, t can be expressed as a product of two path integrals (Feynman and Hibbs 1965): J p(xt; Xoto) = DXl f DX2 exp~ (Sl - Sz)· (15.4) This formula is explicitly symmetric under complex conjugation and under the change of the sign of It, As a result, expression (15.4) is Quantum concepts in space and time 228 Transition amplitudes versus transition probabilities II 229 the new variables, the formula (15.4) reads xl! p(xt; Xo4J) J J = Dx Dx exp[ -i fdt(mX + VV(x» . x + 0(;"2) ] to J = Dx <5(tiix + VV) + 0(;"2), (15.5) where Xl xoT!o Fig. 15.1 invariant under the interchange of the two conventions. However, if expression (15.4) is to be considered as more fundamental than the single path integral representing the transition amplitude, then both trajectories Xl and Xz must be treated on an equal footing. This can be achieved in a natural way if we introduce two copies of space-time and we sum over all trajectories lying in both copies. At the end points both copies are identified (Fig. 15.1). This identification is presumably related to the classical nature of the measuring devices that determine the initial and final co-ordinates of the particle. I shall say more about it later. The perturbation expansion of the transition probability p(xt; Xo4J) further explains my rationale for the reduplication of space-time in quantum mechanics. Upon expanding the transition probability into a power series in the potential V, we can represent each term by a simple two-sided diagram (Fig. 15.2). The total probability is the sum of contributions from all possible combinations of elementary scattering processes occurring in both copies of space-time. In order to clarify a little the role of the classical description in the context of the reduplicated space-time, I will perform a change of variables from Xl and Xz to X and x (Bialynicki-Birula 1971). In terms of + + + PlI·15.2 + + + ... = X + hX12, Xz =X - hX 12. (15.6) Formulae (15.5) and (15.6) show clearly that in the classical limit, when ;,,~ 0, the two copies of space-time collapse into one, and only one trajectory, namely the one that is determined by the Newton equations, contributes to the transition probability. The same concept of a reduplicated space-time can be used also in relativistic theories. As a matter of fact, the description based directly on transition probabilities that makes use of two-sided diagrams of the type shown in Fig. 15.2 has several advantages in quantum electrodynamics and in other gauge theories. This description was introduced in the past (Bialynicki-Birula 1970; Bialynicki-Birula and Bialynicka-Birula 1975) to simplify the study of gauge invariance and renormalization. The twosided diagrams representing the lowest order expressions for the transition probability in the electron-electron (Meller), electron-positron (Bhabha), and electron-photon (Compton) scattering are shown in Fig. 15.3. Two-sided diagrams are similar to the vacuum polarization diagrams-they never have open lines. It is this very feature of the two-sided diagrams that leads to significant simplifications, when the corresponding expressions for the transition probabilities are written down. In particular, it has been possible to eliminate all wave function renormalization constants, which are a source of serious complications in gauge theories, due to their gauge dependence. It is also known (Bloch and Nordsieck 1937; Jauch and Rohrlich 1954; Lee and Nauenberg 1964) that the transition probabilities, but not the transition amplitudes, are free from infra-red divergences in theories with massless particles. I have wondered for some time whether through all these cancellations and simplifications nature is not trying to tell us something quite important. Perhaps the two-sided diagrams representing, as they are, directly observable quantities, are closer to reality. If so, we ought to depart from the conventional approach to quantum scattering and treat all processes as really going around a loop in time in the manner represented by two-sided diagrams. The proper arena for quantum processes would then be not one copy of Minkowski space-time, but rather two copies of it-one copy for each side of the diagram. Once two copies of space-time are introduced, we must face some burning questions: how are these two copies joined together; what is the nature of 230 Quantum concepts in space and time M011erscattering Bhabha scattering Transition amplitudes versus transition probabilities renormalization procedure is adapted (mass-shell renormalization) to the S-matrix, i.e. to infinite time-lapse calculations. The limits of the propagators, when t ~ ±oo, yield the same transition amplitudes as those obtained more conventionally by Fourier transforming the truncated propagators and taking the mass-shell values. This equivalence enables one to interpret the two-sided diagrams as representing processes occurring in space-time. Originally (Bialynicki-Birula 1970) these diagrams were introduced to describe transition probabilities in momentum space, as they appear in the S-matrix calculations when the states are labelled with the momentum vectors of the initial and final particles. Since the space-time interpretation of the S-matrix plays an essential role in my analysis, I shall describe in some detail the space-time description and the momentum description of relativistic scattering processes. The truncated propagators, whose Fourier transforms determine the transition amplitudes in relativistic field theory, are related to the full propagators through the relations. G(Xb ... Compton scattering Fig. 15.3 this join; does this description apply equally well to particles? I must admit that the discussions that I had in Oxford during the conference have shattered some of my naive hopes for a simple solution to these problems. However, I believe that the main idea has not been proven to be worthless. 15.3. Transition probabilities on scri In the full relativistic theory, in contradistinction to quantum mechanics, the two copies of space-time should be joined together along the future and the past infinity. This conclusion can be reached from the analysis of the most promising relativistic theories-gauge field theories. Transition probabilities for finite times in such theories are not gauge invariant and hence devoid of physical meaning. This is because the propagators of particles carrying charges, Abelian or non-Abelian, are gauge dependent; only their mass-shell values are gauge invariant. In addition, the whole 231 ,xn) = (-i)n f d4YI ... d4ynG(xb YI)· .. G(xn' Yn)F(Yb ... ,Yn), (15.7) where I have taken, for simplicity, a theory of only one type of scalar particles. Instead of taking the Fourier transform of F and evaluating it on the mass-shell, one may take the limits of the full propagator G, when its arguments Xi are removed to infinity along the world lines of particles moving freely with velocities VI. This follows from the fact that the (renormalized) one-particle propagator G(x, y) behaves for large times as the free Feynman propagator ~F and from the asymptotic form of the free propagator: J d''y ~F(Vt - r', t - tl)F(r', t'; ... - (Ym/2)( -2ni It I \1'1- )t->±oo ~ VZ)-3/Ze-im 2 ItI vr=v f d4ye±iP"YF(y, ... ), (15.8) where p denotes the four-momentum of the particle, (p 1') = m(1 - vZ)-lIZ(1, v), (15.9) and r' and t' are the space and time components of the four-vector y. For massless particles, the situation is quite different. Due to the absence of dispersion in the propagation of the waves describing these particles, the asymptotic behaviour of the massless propagators is characterized by the first power of It I in the denominator. This factor is common in the ma••ive and in the massless cases. In the massive case, we 232 Transition amplitudes versus transition probabilities Quantum concepts in space and time 233 obtain an additional factor of Itl1/2 in the denominator due to the spreading of the wave packet in the direction of the propagation. The counterpart of formula (15.8) in the massless case has the form: f d'y D F(nt - r', t + to - tl)F(r' , t'; ... )t-+±oo - I) ( -8n2i It -1 x LoodWe=Fiwtfd4ye±ik'YF(y, ... ) (15.10) where k is the four-momentum vector of the massless particle, (klL) = w(l, n). (15.11) Thus, the full information about the scattering of massless particles is contained in the limits of the propagators, when t ~ ±oo, if they are known for all the values of the initial time to. In other words, one must take the limit, when the advanced time v = t + r tends to +00, or the retarded time u = t - r tends to -00 while the retarded time (in the first case), or the advanced time (in the second case) is held fixed. Upon comparing the formulae (15.8) and (15.10), we notice one additional important difference: the limit of the propagator in the massless case does not give directly the scattering amplitude of a particle with a given four-momentum, but its Fourier transform with respect to energy. Still, the limits (15.8) and (15.10) of the propagators with respect to all their arguments determine all the elements of the S-matrix. Of course, prior to taking the limits t ~ ±oo we must remove the kinematical factors in the expressions (15.8) and (15.10): the appropriate powers of It! and the oscillating factor describing the change of the phase of the wave function along its free trajectory. Our analysis has shown that the transition amplitudes in relativistic quantum field theory are determined by the appropriately scaled values of the propagators on the boundary of the Minkowski space-time. Therefore, it seems natural to use in their analysis the concepts introduced by Penrose (1964, 1968) in his 'conformal treatment of infinity'. Such a conformal treatment, however, is not suitable to the description of massive particles (d. also Binegar et al, (1982) for a somewhat different approach). The description of those particles is clearly pathological; their trajectories all begin at one point 1- (past time-like infinity) and they all end at one point 1+ (future time-like infinity). As a result, the limits of the (scaled) propagators at the points 1- and 1+ are ill-defined: they depend on the path along which these points are approached. I shall restrict myself in my analysis to the massless case and make only some speculative remarks about the massive calo at tho ond. Fig. 15.4 Transition amplitudes for massless particles are determined by the (scaled) values of the propagators with their arguments lying in the future null infinity .1> + and the past null infinity .1> -. As I have argued above, in order to accommodate two-sided diagrams we need two copies of the Minkowski space-time joined together along their common boundary, in such a way that closed loops in time are possible. In the conformal framework of Penrose there is essentially only one way of doing it: the .1> + of the first copy is to be identified with the .1> - of the second copy and vice versa (Fig. 15.4). The two copies of Minkowski space-time with their boundaries identified in such a manner form a hypertorus. The hypertorus is divided into two halves by a distinguished light cone made of .1> + and .1> - -the scri. The basic tenets of relativistic quantum theory of massless particles may be stated as follows. Quantum processes take place on the whole hypertorus. The corresponding transition probabilities pick up contributions from trajectories lying in both halves of the hypertorus. The only difference between the contributions from the first and from the second half is that the imaginary unit i is replaced by -i. All transition probabilities 'live' on the scri, where the two copies of the space-time meet each other. These .statements follow from the standard formulation, so that the physical content of the theory remains unchanged. However, the transition probabilities appear here, in a natural manner, as functions of the retarded and advanced times that label the points on the scri, rather than as functions of the energies of the incoming and outgoing particles. Otherwise, 'everything is just translated into an alternative, but equivalent, mathematical framework. But the new framework suggests different directions in which to proceed, from the ones which might seem natural in a more conventional theory'. These words were written (Penrose 1975) in connection with the twistor program, but I believe that they are also applicable here. Additional roalonl for quotina Ponrole Quantum concepts in space and time 234 stem from my expectations that the twistor formalism should prove very useful in the analysis of the transition probabilities on the scri. After all, twistors are a natural tool when the conformal properties playa role, and one is led, nolens volens, to the conformal compactification in order to describe two copies of the Minkowski space-time in 'a compact way'. At the end I should say something about the mass, which is a source of some embarrassment in every conformal theory. The standard treatment of mass does not look very promising. In the conformally compactified space-time all trajectories of massive particles converge at one point, which in turn leads to a pathological behaviour of the transition probabilities treated as functions on the scri. These functions must be highly singular at 1+ and 1- in order to give different values at these points for all different directions along which these points are approached. A possible way out of these difficulties is to assume that at the very fundamental level massive particles always move with the speed of light. The result of the mass term is a series of random 'kicks', which cause discontinuous changes in the direction of the particle velocity (Zitterbewegung). That the Dirac equation admits such an interpretation is shown by Penrose and Rindler (1984) (zigs and zags) and by others in connection with the Feynman path-integral approach to the Dirac equation. In the most recent publication on this subject (Gaveau et al. 1984) the authors write: 'An electron propagates at the speed of light and with a certain chirality (like a two-component neutrino), except that at random times it flips both direction of propagation (by 180°) and handedness: the rate of such flips is precisely the mass, m (in units Ii::::: c :::::1)'. If this analogy with Brownian motion is assumed to be valid also for interacting particles, then we could imagine that a random kick, combined with a kick caused by the interaction, might send the particle across the scri to the other copy of space-time. It is also possible that quantum tunnelling plays a role here. More hard facts, however, will have to be ascertained before we would know whether these speculations can be taken seriously. Acknowledgements I would like to thank Z. Bialynicka-Birula, A. Bialynicki-Birula, C. Fronsdal, J. Kijowski, A. Trautman and S. Woronowicz for many stimulating conservations. Discussions during the Oxford Meeting, most notably those with R. Penrose, have contributed significantly to the evolution of my ideas. References Bialynicki-Birula, I. (1957). Bull. Acad. Polen. Set. a. Ill. 5, 80S. - (1970). Ph". Rlv. D 2, 2877. Transition amplitudes versus transition probabilities 235 _ (1971). Ann. Phys. N.Y. 67,252. __ Bialynicka-Birula, Z. (1975). Quantum electrodynamics. Pergamon, Oxford. Binegar, B., Fronsdal, C., Flato, M., and Salano, S. (1982). Czech. Jour. Phys. B 32, 439. Bloch, F. and Nordsieck, A. (1937). Phys. Rev. 52, 54. Born, M. and Jordan, P. (1925). Z. Phys. 34, 858. Feynman, R. P. and Hibbs, A. R. (1965). Quantum mechanics and path integrals McGraw-Hill, New York. Gaveau, B., Jacobson, T., Kac, M., and Schulman, L. S. (1984). Phys. Rev. Lett. 53, 419. Jauch, J. M. and Rohrlich, F. (1954). Helv. Phys. Acta 27, 613. Lee, T. D. and Nauenberg, M. (1964). Phys. Rev. B 133, 1549. Penrose, R. (1964). In Relativity, groups and topology (ed. C. M. DeWitt and B. DeWitt). Gordon and Breach, New York. __ (1968). In Battelle rencontres 1967 (ed. C. M. DeWitt and J. A. Wheeler). Benjamin, New York. __ (1975). In Quantum theory and the structure of time and space (ed. L. Castell, M. Drieschner, and C. F. von Weizsacker). Carl Hauser, Munich. Penrose, R. and Rindler, W. (1984). Spinors and space-time Vol. 1. Cambridge University Press, Cambridge. Schrodinger, E. (1926). Ann. Phys. 81, 129. (1927). Ann. Phys. 82, 186. Schwinger, J. (1960). In Brandeis University summer institute in theoretical physics, lecture notes. Brandeis University. (1961). J. Math. Phys. 2, 407.