Three possible implications of spacetime discreteness Shan Gao Foundations of Physics Seminar...

Three possible implications of spacetime discretenessShan Gao

Foundations of Physics SeminarUniversity of Sydney, 10 November 2010

Unit for HPS, Faculty of ScienceUniversity of Sydney, Australia

Centre for Time, SOPHIUniversity of Sydney, Australia

Three possible implications of spacetime discreteness

The postulate of spacetime discreteness 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.

The discreteness of space and time

Max Planck (1899). “On irreversible radiation processes”, in Proceedings of the Prussian Academy of Sciences in Berlin.

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GLp

5c

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The discreteness of space and timeIhre 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


Modern history since 1930s Popular concept in quantum gravity Various formulations and meanings

No experimental evidence


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.

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)

cTL PP /

/8 PPTL

pTpL

pTpL

Why there is an invariant speed (SR)

Special Relativity

The principle of relativity Constancy of the speed of light

Homogeneity and isotropy of spacetime

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).

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

t

x

Kv

v

Kvt

x

1

1

1

12

2/1 cK 0K


Why ?

Why is there an invariant speed c?

The answer may lead us to a deeper understanding of spacetime and relativity.

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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.

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.

PP TLc /

PLPL

PT

PT

PT


Thus, there is a maximum signal speed in discrete space and time, which equals to the ratio of minimum length to minimum time interval.

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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.

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.


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.

Why there is an invariant speed (SR) Invariance of c: Another argument

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 speed is 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.


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.

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.

Why there is an invariant speed (SR) Relativity in discrete space and time (RDST)

(1) the principle of relativity;

(2) 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.

Why there is an invariant speed (SR) How about the speed of light?

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).

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)

Why the quantum? (QM)

The wave function and its equation

But why?


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.

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PL PT

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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.


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.


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.

PL PT


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.

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Why the quantum?


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).

),( tx),( txj


Equation of motion for RDM in CST The equation of motion can be derived by resorting to space

time 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.


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.


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?


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?


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).


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.


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.


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….)

Why matter curves spacetime (GR)

According to GR, matter curves spacetime. But why?

Why matter curves spacetime (GR) How about the implication of spacetime disc

reteness 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.

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.

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0


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.

px

2

A detailed argument

Why matter curves spacetime (GR) There is a more strict limitation on the localization in relati

vistic 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.

cmx

02

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.

mcx

2

E

cx

2

Why matter curves spacetime (GR) The above limitation is valid in continuous spacetime.

When the energy and energy uncertainty of 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:

PLx 2

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 .

c

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E

cx P

2

4

2

2

PL2

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.

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L PP2

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.

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).

Why matter curves spacetime (GR) We can further estimate the strength of the influenc

e of matter on spacetime.

The energy E contained in a region with size L will change the proper size of the region to

ETL

LL PP2

Why matter curves spacetime (GR) Thus the discreteness of spacetime requires that

matter curves spacetime. This may provide a deeper basis for Einstein’s equivalence principle.

On the basis of the equivalence principle, there are some common steps to “derive” the Einstein field equations (i.e. the concrete relation between the geometry of spacetime and the stress-energy density contained in that spacetime) in terms of Riemann geometry and tensor analysis as well as the conservation of energy and momentum etc.

Why matter curves spacetime (GR) For example, it can be shown that there is only one

symmetric second-rank tensor that will satisfy the following conditions:

(1) Constructed solely from the spacetime metric and its derivatives;

(2) Linear in the second derivatives;

(3) The four-divergence of which is vanishes identically (this condition guarantees the conservation of energy and momentum);

(4) Is zero when spacetime is flat (i.e. without cosmological constant).

These conditions will yield a tensor capturing the dynamics of the curvature of spacetime, which is proportional to the stress-energy density, and we can then obtain the Einstein field equations.


The left thing is to determine the value of the Einstein gravitational constant k. It is usually derived by requiring that the weak and slow limit of the Einste

in field equations must recover Newton’s theory of gravitation. In this way, the gravitational constant is in fact determined by experience.

If our argument is valid, the Einstein gravitational constant can also be determined in terms of the minimum size of discrete spacetime.

TRgR 2

1

Why matter curves spacetime (GR) Consider an energy eigenstate in a region with radius R.

The spacetime outside the region is described by the Schwarzschild metric

By assuming that the metric tensor inside the region R is the same order as that on the boundary, the proper size of the region is

The change of the proper size due to the contained energy E is .

By comparing with the previous formula we find

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rdrdrdr

r

rds SS

4

2)1(20

2/1 ERdr

R

rL

RS

4

EL

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Why matter curves spacetime (GR) To sum up, the discreteness of spacetime implies that gr

avity as a geometric property of spacetime is fundamental. In particular, the dynamical relationship between matter and spacet

ime holds true not only for macroscopic objects, but also for microscopic particles.

Moreover, the fundamental existence of gravity as argued above may have further implications for a complete theory of quantum gravity (see Gao 2010).

DST may provide a deeper basis for GR.

Summary

The discreteness of spacetime might provide a deeper logical foundation for the special and general relativity and quantum theory.

Three fundamental questions

Why there is an invariant speed (SR) Why the quantum? (QM) Why matter curves spacetime (GR)

Summary In discrete space and time ( , )

There is an invariant speed c (SR)

Motion is discontinuous and random (QM)

Matter curves spacetime (GR)

cTL PP /

3

2

/

/18/8

PP

PPP

LE

LTL

pTpL

pTpL

Selected publications S. Gao (2004) Quantum collapse, consciousness and superluminal

communication, Foundations of Physics Letters 17(2), 167-182. S. Gao (2006) A model of wavefunction collapse in discrete space-time,

International Journal of Theoretical Physics 45, 1965. S. Gao (2006) Quantum Motion: Unveiling the Mysterious Quantum

World. Bury St Edmunds: Arima Publishing. S. Gao (2008) God Does Play Dice with the Universe. Bury St

Edmunds: Arima Publishing. S. Gao (2010) On Diósi-Penrose criterion of gravity-induced quantum

collapse, International Journal of Theoretical Physics 49, 849–853. S. Gao (2010) Meaning of the wave function, to appear in International

Journal of Quantum Chemistry. S. Gao (2010) The wave function and quantum reality, to appear in the

Proceedings of the Conference Advances in Quantum Theory 2010.

Thank you!

This work was supported by the Postgraduate Scholarship in Quantum Foundations provided by the Unit for HPS and Centre for Time (SOPHI) of the University of Sydney.

I am very grateful to Dean Rickles and Huw Price for helpful discussions.

Three possible implications of spacetime discreteness Shan Gao Foundations of Physics Seminar...

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