Three possible implications of spacetime discreteness

1. Three possible implications of spacetime discreteness Shan Gao Foundations of Physics Seminar University of Sydney, 10 November 2010 Unit for HPS, Faculty of Science University of Sydney, Australia Centre for Time, SOPHI University of Sydney, Australia

2. Three possible implications of spacetime discreteness • The postulate of spacetimediscreteness • Its possible implications for SR, QM and GR • Why there is an invariant speed (SR) • Why the quantum? (QM) • Why matter curves spacetime (GR) Note: These arguments are very speculative. More criticisms are welcome.

3. The discreteness of space and time • Max Planck (1899). “On irreversible radiation processes”, in Proceedings of the Prussian Academy of Sciences in Berlin.

4. The discreteness of space and time Ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können. These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as “natural units”. ---- M. Planck, 1899

5. The discreteness of space and time • Modern history since 1930s • Popular concept in quantum gravity • Various formulations and meanings • No experimental evidence

6. The discreteness of space and time A minimum definition There exist a minimum time interval and a minimum space interval. • It is only a restriction on spacetime intervals. • A spacetime interval shorter than the minimum size of spacetime is physically meaningless, and it cannot be measured in principle either. • It does not necessarily imply a fixed or random lattice structure of spacetime.

7. Its three possible implications • In discrete space and time ( , ) • There is an invariant speed c (SR) • Motion is discontinuous and random (QM) • Matter curves spacetime (GR)

8. Why there is an invariant speed (SR) • Special Relativity • The principle of relativity • Constancy of the speed of light • Homogeneity and isotropy of spacetime

9. Why there is an invariant speed (SR) • SR is an incoherent mixture (Einstein 1935; Stachel 1995) • the first principle is universal in scope • the second is only a particular property of light Drop the light postulate from SR: Ignatowski (1910, 1911a, 1911b); Frank and Rothe (1911, 1912); Pars (1921); Kaluza (1924); Lalan (1937); Dixon (1940); Weinstock (1965); Mitavalsky (1966); Terletskii (1968); Berzi and Gorini (1969); Gorini and Zecca (1970); Lee and Kalatos (1975); Lévy-Leblond (1976); Srivastava (1981); Mermin (1984); Schwartz (1984, 1985); Singh (1986); Sen (1994); Field (1997); Coleman (2003); Pal (2003); Sonego and Pin (2005); Gannett (2007); Silagadze (2007); Certik (2007); Feigenbaum (2008).

10. Why there is an invariant speed (SR) • Relativity without light • The principle of relativity • homogeneity of space and time • isotropy of space (standard convention of simultaneity) • : Galileo transformations • : Lorentz transformations

11. Why there is an invariant speed (SR) Why ? Why is there an invariant speed c? • The answer may lead us to a deeper understanding of spacetime and relativity.

12. Why there is an invariant speed (SR) • Relativity without light implies: • c is not (merely) the speed of light, but a universal constant of nature, an invariant speed. • the existence of an invariant speed partly results from the properties of space and time, e.g. homogeneity of space and time and isotropy of space. • A further conjecture: • The finiteness of the invariant speed may originate from another property of spacetime, its discreteness.

13. Why there is an invariant speed (SR) • Consider continuous transmission of a signal(e.g. light signal in vacuum) If the signal moves with a speed larger than , then it will move more than during , and thus moving will correspond to a time interval shorter than during the motion. • This contradicts the discreteness of spacetime. • is the minimum time interval in discrete space and time, and the duration of any real change cannot be shorter than it.

14. Why there is an invariant speed (SR) • Thus, there is a maximum signal speed in discrete space and time, which equals to the ratio of minimum length to minimum time interval.

15. Why there is an invariant speed (SR) • c is the maximum speed in every inertial frame • According to the principle of relativity, the minimum time interval and the minimum length should be the same in all inertial frames. • If the minimum sizes of space and time are different in different frames, then there will exist a preferred Lorentz frame. This contradicts the principle of relativity. • Thus, c will be the maximum speed in every inertial frame.

16. Why there is an invariant speed (SR) • The maximum speed c is invariant • Suppose a signal moves in the x direction with speed c in an inertial frame S1. • Its speed will be not smaller than c in another inertial frame S2 with velocity in the -x direction relative to S1. • c is the maximum speed in every frame. • Therefore, the speed of the signal in S2 can only be c.

17. Why there is an invariant speed (SR) • This result also means that when a signal moves in the x direction with speed c in the frame S2, its speed is also c in the frame S1 with velocity in the x direction relative to S2. • Since the inertial frames S1 and S2 are arbitrary, we can reach the conclusion that if a signal moves with speed c in one frame, it will also move with the same speed c in all other frames. • This demonstrates the invariance of c in discrete space and time.

18. Why there is an invariant speed (SR) • Suppose a signal moves in the x direction with speed c in an inertial frame S1. Its speed can only be either c or smaller than c in another frame S2 with velocity in the x direction relative to S1. • If its speedis smaller than c, say c-v, then there must exist a speed larger than c-v and a speed smaller than c-v that correspond to the same speed in S1. • Invariance of c: Another argument

19. Why there is an invariant speed (SR) • This means that when the signal moves with a certain speed in frame S1, its speed in frame S2 will have two possible values. This is impossible. • Thus the signal moving with speed c in S1 also moves with speed c in S2. This result also means that when a signal moves in the x direction with speed c in a frame S2, its speed is also c in the frame S1 with velocity in the -x direction relative to S2. • Since the inertial frames S1 and S2 are arbitrary, we also demonstrate that the maximum speed c is invariant in all inertial frames.

20. Why there is an invariant speed (SR) • In summary • The discreteness of spacetime may explain the existence of an invariant and maximum speed. In this way, it may provide a deeper logical foundation for SR. • On the other hand, the existence of an invariant speed c may be regarded as a firm experimental evidence of the discreteness of spacetime, in which the ratio of the minimum length to the minimum time interval is c.

21. Why there is an invariant speed (SR) • Relativity in discrete space and time (RDST) • the principle of relativity; • the constancy of the minimum size of spacetime. RDST  SR as a limit (the constancy of c is a consequence) Galileo’s relativity is a theory of relativity in continuous space and time. Einstein’s relativity is a theory of relativity in discrete space and time.

22. Why there is an invariant speed (SR) • The speed of light in vacuum (defined as the group speed of a photon wavepacket) may be exactly c or smaller than c but very close to c. • The group speed of photons can be larger than c in some sort of abnormal media. This does not contradict the discreteness of space and time. • What the discrete spacetime really restricts is the speed of any (apparently continuous) causal influence, which cannot be larger than c. • Note that the speed of discontinuous causal influence such as quantum nonlocality may be larger than c (Gao 2004). • How about the speed of light?

23. Why there is an invariant speed (SR) • More consequences of RDST…to be studied • GUP (Generalized Uncertainty Principle) • No tachyons • Revised Lorentz transformations? (no length contraction of Lp) • Revised energy-momentum relation? • Noncommutative spacetime? • Doubly special relativity (two invariant scales, the speed of light c and a minimum lengthλ) • Triply special relativity (three invariant scales, the speed of light c, a mass κ and a length R)

24. Why the quantum? (QM) • The wave function and its equation • But why?

25. Why the quantum? (QM) • One loophole in our argument? • A particle cannot move faster than c. • A particle cannot move slower than c either? If a particle moves with a speed smaller than , then it will move less than during . This also contradicts the discreteness of spacetime. • is the minimum space interval in discrete space and time. Obviously this result contradicts experience, as particles can move with a speed smaller than c.

26. Why the quantum? (QM) • There are some possible ways to avoid the contradiction. • It can be conceived that the particle moves with c during some time, and stays still during other time. Then its average speed can be smaller than c, and thus its motion can be consistent with the existing experience.

27. Why the quantum? (QM) • Problems of such continuous jumps • The speed change of the free particle during such motion can hardly be explained. • Next, this motion will also contain some kind of unnatural randomness (e.g. during each time the speed of the free particle will assume either c or 0 in a random way). It seems that such randomness has no logical basis.

28. Why the quantum? (QM) • Another way out • If motion is essentially discontinuous and random and continuous motion is only an approximate average display, then the apparent continuous motion with a speed smaller than c will not be prohibited. • The reason is that a particle undergoing such motion can move a distance larger than during in a discontinuous way. • Moreover, since the direction of each discontinuous movement may be forward and backward, the average velocity of the particle can still be smaller than c.

29. Why the quantum? (QM) • Is there still a maximum speed limitation? If the average velocity of the particle is larger than the maximum speed c, then we can detect a time interval shorter than by measuring the average moving distance of the particle. But this contradicts the discreteness of spacetime. • Thus, although the motion of particles is discontinuous, the maximum speed of apparent continuous motion is still c in discrete space and time.

30. Why the quantum? (QM) • Why the quantum?

31. Why the quantum? (QM) • Random discontinuous motion (RDM) in CST • The wave function in QM provides a complete description of the RDM of particles in continuous space and time (CST).

32. Why the quantum? (QM) • Equation of motion for RDM in CST • The equation of motion can be derived by resorting to spacetime translation invariance and relativistic invariance, and it turns out to be the Schrödinger equation in QM (Gao 2010). • Spacetime translation defines momentum and energy, and spacetime translation invariance entails that the state of a free particle with definite momentum and energy assumes the plane wave form exp{i(px-Et)}. • Besides, the relativistic invariance of the free state further determines the relativistic energy-momentum relation, which nonrelativistic approximation is E=p2/2m.

33. Why the quantum? (QM) • RDM: A realistic alternative to the orthodox view • The wavefunction gives not the density of stuff, but gives rather (on squaring its modulus) the density of probability. Probability of what exactly? Not of the electron being there, but of the electron being found there, if its position is ‘measured’. Why this aversion to ‘being’ and insistence on ‘finding’? The founding fathers were unable to form a clear picture of things on the remote atomic scale. ---- J. S. Bell,1990 • According to RDM, the modulus square of the wave function not only gives the probability of a particle being found in certain locations, but also gives the objective probability of the particle being there.

34. Why the quantum? (QM) • BUT… The transition process from “being” to “being found”, which is closely related to the notorious quantum measurement problem, still needs to be explained. • How about the collapse of the wave function?

35. Why the quantum? (QM) • RDM in discrete spacetime (DST) • Although motion is discontinuous and random, the discontinuity and randomness are absorbed into the motion state (defined during an infinitesimal time interval) in continuous space and time. • We see no randomness and discontinuity in the state of RDM and its evolution law; the wave function is continuous, and the Schrödinger equation is also continuous and deterministic. • Where do the randomness and discontinuity go? They do appear in the actual experiments on microscopic particles. • How can the random motion present itself?

36. Why the quantum? (QM) • RDM in discrete spacetime (DST) • In CST, a durationless instant cannot present itself. Thus it is natural that the randomness of motion, which exists at individual instants, cannot emerge through observable physical effects. • In DST, since instants as finite intervals can have physical effects and be measured in principle, the inherent randomness of motion, which exists at such instants, may emerge (e.g. through the collapse of the wave function).

37. Why the quantum? (QM) • RDM in discrete spacetime (DST) • Concretely speaking, for the RDM of a particle in DST, the particle randomly stays in a position during a finite instant, and the finite stay may have a tiny effect on the continuous evolution of the wave function. • Then during a much longer time interval, such tiny random effects may continually accumulate to generate the observable random phenomena, e.g. the dynamical collapse of the wave function.

38. Why the quantum? (QM) • A model of wavefunction collapse in DST • Indeed, we can give a concrete model of wavefunction collapse in discrete spacetime (by assuming that the source to collapse the wave function is the inherent random motion of particles described by the wave function). Moreover, it can be shown that the minimum size of spacetime also yields a plausible collapse criterion consistent with experiments and macroscopic experience. S. Gao (2006) A model of wavefunction collapse in discrete space-time, International Journal of Theoretical Physics 45, 1965.

39. Why the quantum? (QM) • A model of wavefunction collapse in DST • A serious objection to the dynamical collapse models (e.g. GRW, CSL etc) is that they violate the principle of energy and momentum conservation even at the statistical level. • Recently, I show that my model of wavefunction collapse, when precisely formulated, may be consistent with energy conservation even for individual processes. (But I am still checking this result….)

40. Why matter curves spacetime (GR) • According to GR, matter curves spacetime. • But why?

41. Why matter curves spacetime (GR) • How about the implication of spacetime discreteness for the nature of gravity? • I will argue that spacetime discreteness might imply the fundamental existence of gravity as a geometric property of spacetime described by GR.

42. Why matter curves spacetime (GR) • Does spacetime discreteness really imply gravity? • This formula itself seemingly suggests that gravity originates from the discreteness of spacetime (together with the quantum principle that requires ). • In continuous spacetime where and , we have , and thus gravity does not exist.

43. A detailed argument Why matter curves spacetime (GR) Heisenberg’s uncertainty principle in QM: • The momentum uncertainty of a particle will lead to the uncertainty of its position. • This poses a limitation on the localization of a particle in nonrelativistic domain.

44. Why matter curves spacetime (GR) • There is a more strict limitation on the localization in relativistic domain. A particle at rest can only be localized within a distance of the order of its reduced Compton wavelength, namely • The reason is that when the energy uncertainty exceeds two times of the rest energy of the particle, it will create a particle anti-particle pair from the vacuum and make the position of the original particle invalid.

45. Why matter curves spacetime (GR) • The minimum position uncertainty of a moving particle is then or where m is the relativistic mass, and E is the total energy of particle. This means that when the energy uncertainty of a particle is of the order of its (average) total energy, it has the minimum position uncertainty. Note that the above formula also holds true for particles with zero rest mass such as photons.

46. Why matter curves spacetime (GR) • The above limitation is valid in continuous spacetime. • When the energy and energy uncertaintyof a particle both become arbitrarily large, the uncertainty of its position can still be arbitrarily small. • However, the discreteness of spacetime will demand that the localization of any particle should have a minimum value, namely there should exist a limiting relation:

47. Why matter curves spacetime (GR) • In order to satisfy this limitation, the r.h.s of Heisenberg’s uncertainty relation should at least contain another term proportional to the energy of the particle, namely in the first order of E it should be • This new uncertainty relation can satisfy the limitation imposed by the discreteness of spacetime. It means that the total uncertainty of the position of a pointlike particle has a minimum value .

48. Why matter curves spacetime (GR) • How to understand the new term demanded by the discreteness of spacetime? • Obviously it indicates that the energy uncertainty of a particle results in an inherent position uncertainty proportional to the energy uncertainty. • The problem is how the energy uncertainty generates the position uncertainty.

49. Why matter curves spacetime (GR) • First, the new position uncertainty cannot originate from the quantum motion of the particle, as it is very different from the usual quantum uncertainty of position, which is inverse proportional to the energy uncertainty. • Next, since there is only one particle, the new position uncertainty cannot result from any interaction between the particle and other particles (e.g. electromagnetic interaction) either.

50. Why matter curves spacetime (GR) • Therefore, there is only one possibility left, namely that the energy uncertainty of the particle affects the spacetime where it moves and then results in its position uncertainty. • This further implies that the energy of a particle will change the geometry of its background spacetime (e.g. in each momentum branch of a quantum superposition).