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Unit for HPS, Faculty of Science
University of Sydney, Australia
Centre for Time, SOPHI
University of Sydney, Australia
Three possible implications
of spacetime discreteness
Shan Gao
Foundations of Physics Seminar
University of Sydney, 10 November 2010
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
G
Lp 
3
c
G
Tp 
5
c

Max Planck (1899). “On
irreversible radiation processes”,
in Proceedings of the Prussian
Academy of Sciences in Berlin.
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
The discreteness of space and time

Modern history since 1930s
Popular concept in quantum gravity
Various formulations and meanings

No experimental evidence


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.
Its three possible implications

In discrete space and time ( L p , T p )

There is an invariant speed c (SR)
LP / TP  c

Motion is discontinuous and random (QM)
Lp Tp


Matter curves spacetime (GR)
8LPTP /   
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)
 x 
1
  
1  Kv 2
 t 


K  0:
 v  x 
 1

 
  Kv 1  t 
Galileo transformations
K  1 / c 2 : Lorentz transformations
Why there is an invariant speed (SR)
Why
K 0 ?
Why is there an invariant speed c?

The answer may lead us to a deeper
understanding of spacetime and relativity.
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.
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 c  LP / TP,
then it will move more than LP during TP , and thus
moving LP will correspond to a time interval shorter
than TP during the motion.

This contradicts the discreteness of spacetime.
 TP is the minimum time interval in discrete space and time,
and the duration of any real change cannot be shorter than it.
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.
vmax  c  LP / TP
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.
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.
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.
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.
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)
(2)
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.
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?
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 c  LP / TP ,
then it will move less than LP during TP .
This also contradicts the discreteness of spacetime.
 LP 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.
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.
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.
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 LP during TP 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.
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 TP 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.
Why the quantum? (QM)

Why the quantum?
Why the quantum? (QM)
Random discontinuous motion (RDM) in CST

 ( x, t )
j ( x, t )

The wave function in QM provides a complete
description of the RDM of particles in continuous
space and time (CST).
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.
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.
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?
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?
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).
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.
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 spacetime, International Journal of Theoretical Physics 45, 1965.
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….)
Why matter curves spacetime (GR)

According to GR, matter curves spacetime.

But why?
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.
Why matter curves spacetime (GR)

Does spacetime discreteness really imply gravity?
  8LPTP / 


This formula itself seemingly suggests that gravity
originates from the discreteness of spacetime (together
with the quantum principle that requires   0 ).
In continuous spacetime where TP  0 and LP  0,
we have   0 , and thus gravity does not exist.
Why matter curves spacetime (GR)

A detailed argument
Heisenberg’s uncertainty principle in QM:

x 
2 p

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.
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

x 
2m0 c

The reason is that when the energy uncertainty exceeds two
times of the rest energy of the particle, it will create a particle antiparticle pair from the vacuum and make the position of the original
particle invalid.
Why matter curves spacetime (GR)

The minimum position uncertainty of a moving particle is
then

x 
2mc
or
c
x 
2E
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.
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:
x  2LP
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
2
c 4 LP E
x 

2E
2c

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 2LP .
Why matter curves spacetime (GR)

How to understand the new term demanded by the
discreteness of spacetime?
2 LPTP E
L 


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.
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
influence of matter on spacetime.
The energy E contained in a region with size L will change
the proper size of the region to
2 LPTP E
L  L 

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 stressenergy 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.
Why matter curves spacetime (GR)
R

The left thing is to determine the value of the Einstein
gravitational constant k.


1
 g  R  T
2
It is usually derived by requiring that the weak and slow limit of the
Einstein 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.
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
ds 2  (1 
rS 1 2
r
) dr  r 2 d 2  r 2 sin  2 d 2  (1  S )c 2 dt 2
r
r
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
R
L  2 (1 
0
rS 1/ 2
E
) dr  2 R 
R
4
The change of the proper size due to the contained energy E is
By comparing with the previous formula
2 LPTP E
L 

LPTP
  8

L 
we find
E
4
.
Why matter curves spacetime (GR)

To sum up, the
discreteness of spacetime implies that
gravity as a geometric property of spacetime is
fundamental.


In particular, the dynamical relationship between matter and
spacetime 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 ( L p , T p )

There is an invariant speed c (SR)
LP / TP  c

Motion is discontinuous and random (QM)
Lp Tp


Matter curves spacetime (GR)
2
1 / LP
8LPTP /   8

3
EP / LP
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.