On The Flow of Time by George F R Ellis

Essay Abstract

Current theoretical physics suggests the flow of time is an illusion: the entire universe just is, with no special meaning attached to the present time. This paper points out that this view, in essence represented by usual space-time diagrams, is based on time-reversible microphysical laws, which fail to capture essential features of the time-irreversible nature of decoherence and the quantum measurement process, as well as macro-physical behaviour and the development of emergent complex systems, including life, which exist in the real universe. When these are taken into account, the unchanging block universe view of spacetime is best replaced by an evolving block universe which extends as time evolves, with the potential of the future continually becoming the certainty of the past; spacetime itself evolves, as do the entities within it. However this time evolution is not related to any preferred surfaces in spacetime; rather it is associated with the evolution of proper time along families of world lines. The default state of fundamental physics should not be taken to be a time irreversible evolution of physical states: it is an ongoing irreversible development of time itself.

Author Bio

George Ellis is Professor Emeritus of applied mathematics at the University of Cape Town. He has written or co-authored many books and papers on relativity theory and cosmology, including On the Large Scale Structure of Space Time with Stephen Hawking.

Download Essay PDF File

Dear Prof. Ellis,

It seems to me that, in order to talk about 'the flow of time', the past should not be sufficient to determine the present (cf. Conway-Kochen 'Strong Free Will Theorem', arXiv:0807.3286v1 [quant-ph]), and we have to consider the possibility that the Aristotelian 'final cause' may complement the relativistic causality, as elaborated here.

A penny for your thoughts!

Dear Prof. Ellis,

Thank you for the beautiful and accurate picture of an Evolving Block Universe. I agree with you that the standard view of the block universe leaves to little room for the interesting phenomena occurring at higher levels. Please, consider that there may be two levels of the Physical World:

1. One of the possible solutions of the time evolution equation (forming a sheaf).

2. Another of the conditions, including an incomplete set of initial conditions, and "delayed initial conditions" caused by the quantum measurements.

It is possible to account for the quantum collapse without recurring to discontinuities, if we appeal to the entanglement of the system with the preparation device, and to the "delayed initial conditions". At the level 1, the things happen deterministically, but there is a sheaf of such deterministic worlds. The level 2, containing the observers, and an incomplete set of initial conditions, can provide the evolution required. Being an incomplete set of constrains for the sections of the sheaf, it can be extended in multiple ways. There is room for the free-will, and the only uncertainty is provided by the yet-to-be-determined initial conditions. Our (temporal) choices allow us to select the section of the sheaf of solutions, in a non-local and trans-temporal way. I present briefly this idea in the section "Smooth Quantum Mechanics" in my essay, and in more detail in the attached document. Sorry for this intrusion, I wouldn't dare if I haven't thought that I can bring a small contribution to the Evolving Block Universe idea.

Cristi Stoica

"Flowing with a Frozen River",

http://fqxi.org/community/forum/topic/322Attachment #1: smooth_qm.pdf

Appreciated Prof. Ellis,

After reading your nice essay, I think that in mine I agree with you in most of your thougs but at a different level of argumentation.

I would be proud if you take a look at my essay Time Traveling by simuverses

Good luck in the contest.

Dear George,

I'm sorry but your I have to stress that when you ask whether time is a macroscopic coarse grained phenomenon and you answer "No, because the quantum measurement process is not time reversible" unfortunately you make a common mistake.

You are not able to define the words "time reversible" and "time irreversible" if you don't implicitly use the notion of "time", so using the concept of "time reversibiltiy" when trying to saying something about the concept of time itself is a very obvious loop which implies that the consequent conclusions are wrong.

Obviously, since your work is based on this wrong argument, all the conclusions are probably wrong.

John

Curious logic yours, Mr.John Smith. With a self made law about how to define thinks and a couple of sentences without conection you concludes that an entire job is not valid.

Be serious, please.

Prof. Ellis, while I agree in general with your overall view of the evolution of the world, with determined past states and indeterminate future states described by quantum mechanics (QM), I disagree that the world lines of matter particles should be given primary importance. World lines are classical concepts that have no physical significance in QM, except as mathematical devices in e.g. path integrals. A continuous world line is not physically observable in QM, not even in principle, and not even for the past portion of world lines. What we can observe are at best discrete points on an imagined world line, which should not be taken as the primary concept in a theory of Becoming of the world. It seems that both in QM and General Relativity, 3-D space-like hypersurfaces are the best candidates to carry the world state at each moment of Becoming, thus they should be used as primary elements in a physical theory of Becoming of the world, at least in first approximation. But of course we then face the question of which space-like hypersurface to choose as the carrier of each moment of Becoming, as you pointed out, for which I don't have a definitive answer, though I did offer a toy model in my essay...

Hello George!

Great essay!

You write, "This paper points out that this view, in essence represented by usual space-time diagrams, is based on time-reversible microphysical laws, which fail to capture essential features of the time-irreversible nature of decoherence and the quantum measurement process, as well as macro-physical behaviour and the development of emergent complex systems, including life, which exist in the real universe. When these are taken into account, the unchanging block universe view of spacetime is best replaced by an evolving block universe which extends as time evolves, with the potential of the future continually becoming the certainty of the past; spacetime itself evolves, as do the entities within it."

Space-time diagrams are a human constructs, that arise because people forgot that Einstein never stated that time is the fourth dimension, but rather he wrote x4=ict.

You would enjoy my paper! Time as an Emergent Phenomenon: Traveling Back to the Heroic Age of Physics by Elliot McGucken

"In his 1912 Manuscript on Relativity, Einstein never stated that time is the fourth dimension, but rather he wrote x4 = ict. The fourth dimension is not time, but ict. Despite this, prominent physicists have oft equated time and the fourth dimension, leading to un-resolvable paradoxes and confusion regarding time's physical nature, as physicists mistakenly projected properties of the three spatial dimensions onto a time dimension, resulting in curious concepts including frozen time and block universes in which the past and future are omni-present, thusly denying free will, while implying the possibility of time travel into the past, which visitors from the future have yet to verify. Beginning with the postulate that time is an emergent phenomenon resulting from a fourth dimension expanding relative to the three spatial dimensions at the rate of c, diverse phenomena from relativity, quantum mechanics, and statistical mechanics are accounted for. Time dilation, the equivalence of mass and energy, nonlocality, wave-particle duality, and entropy are shown to arise from a common, deeper physical reality expressed with dx4/dt=ic. This postulate and equation, from which Einstein's relativity is derived, presents a fundamental model accounting for the emergence of time, the constant velocity of light, the fact that the maximum velocity is c, and the fact that c is independent of the velocity of the source, as photons are but matter surfing a fourth expanding dimension. In general relativity, Einstein showed that the dimensions themselves could bend, curve, and move. The present theory extends this principle, postulating that the fourth dimension is moving independently of the three spatial dimensions, distributing locality and fathering time. This physical model underlies and accounts for time in quantum mechanics, relativity, and statistical mechanics, as well as entropy, the universe's expansion, and time's arrows."

http://fqxi.org/community/forum/topic/238

Dear Dimi Chakalov

I have puzzled over the Conway-Kochen 'Strong Free Will Theorem' paper, without really understanding what if anything it has to do with free will [the wikipedia entry on the theorem is interesting in this regard]. However from the viewpoint of my own paper, that is not important: what matters is that their paper appears to reinforce the view that the outcome of quantum events is unknown until they happen. That is a key feature on which I build my proposal; so there is no conflict.

Dear Cristi Stoica

thak you for the comments. I concur that your and my views may be regarded generally as in agreement; you elaborate the issues in interesting ways. A few comments: as to delayed measurement experiments, I agree with the view that you give in your paper: these experiments do not really measure the system, but rather its entanglement with the preparation device. The outcome is determined when the measurement is made, and is unknown and indeed unpredictable before then; this agrees with my view. I appreciate your emphasis on the preparation of the quantum system; this is the part of the whole package that is usually ignored, and which is just as mysterious as the measurement process. And finally you are concerned about free will in this overall context; please see, in this regard, my comments on causation in complex systems [the last reference in my posted paper]. I emphasise there the important role played by top down causation in the hierarchy of complexity, and of adaptive selection as a top down process. This does not solve the problem, but does indicate some ways whereby there can be a relaxtion of the iron grip of lower level causation.

Dear Venerando

Your essay is very imaginative, and in some ways congruent to mine, but I find it diffcult to accept that "At the end, it will result on that, speaking about time travels, the virtual ones are more possible, more "real", than the physical ones." In my own essay I have tried to carefully deal with only the real ones. And the very real problem with incredibly complex simulations such as those you (and others) envisage is that it is inconceivable they would run for more than a few microseconds without crashing. The task of designing them to actually work for a long time would be unfeasible. That is why biological systems do not work in an digitally-based algorithmic way; rather they work by adaptive selection.

Dear John Smith,

you state `You are not able to define the words "time reversible" and "time irreversible" if you don't implicitly use the notion of "time", so using the concept of "time reversibiltiy" when trying to saying something about the concept of time itself is a very obvious loop which implies that the consequent conclusions are wrong.'

You can't talk about time at all without using the concept of time. My paper is based on how standard quantum theory in fact implies the flow of time in an ireversible way. This is one of the best tested theories in physics. If one is banned from using concepts such as time reversibility when talking about time, then no sensible discussion of the topic wil be possible. All the essays submitted to fqxi will be null and void.

Dear Chi Ming Hung

the points you raise are interesting. One needs to discuss th eissue at the classical, semi-classical, and quantum gravity levels. I have focused on the first, where the use of timelike world lines is I believe valid. However at the quantum gravity level, one wants a full spacetime quantisation in my view (and as you suggest in your essay). Then my concept of development taking place point by point makesz sense. Still it would be interesting to develop further your proposal that "both in QM and General Relativity, 3-D space-like hypersurfaces are the best candidates to carry the world state at each moment of Becoming, thus they should be used as primary elements in a physical theory of Becoming of the world, at least in first approximation". It might be better in some contexts.

Dear Dr. E (The Real McCoy)

I agree with the spirit of what you do in your essay, which as you point out is going in the same direction as mine. The main point where I differ is in the use of the imaginary time coordinate. I prefer to see it all done with a real time coordinate (see e.g. my book with Ruth Williams entitled "Flat and Curved Spacetimes"). So a key element is how proper time relates to coordinate time as we move to the future, which is what your key equation establishes; I think one can do this without introducing the imaginary quantity "i".

I thought your paper was pretty interesting. I does seem to me that an evolving block universe is hard to make consistent with the concept of the block universe. We might think of the worldlines of particles or observers as having a fibration of spatial surfaces. Then a path integral of such world lines, on some base space of support, as related to each other by a fibration with an so(3) symmetry for spatial sheets.

The problem I see is how is it that time in general relativity and quantum mechanics can ever be made to agree with each other about time. In general relativity time is a symmetry of the theory, or in the su(1,1) part of the sl(2,C) group. One can in ADM relativity consider spatial surfaces as foliating a spacetime. How these spatial surfaces link together is chosen by the analyst in a manner similar to a gauge choice. This is of course curious, for we can think of time as some one dimensional space with a fibration given by spatial surfaces. So "choosing" how time acts is equivalent to choosing a section in a fibration of spatial surfaces.

Quantum mechanics in the other hand treats time as more concrete. The Schrodinger equation holds time as an evolutionary parameter which is not a symmetry of the Hamiltonian. In relativistic quantum theory one must assign fields on a spatial surface of simultaneity (equal time commutators of fields etc), and from there time acts as a parameter which fixes a Hamiltonian, but is not a symmetry of the Hamiltonian. Things get somewhat odd when the space or spacetime is curved. In particular the distinction between a vacuum and a particle state breaks down.

General relativity does not derive a Schrodinger equation per se, but in the "space plus time" ADM approach the canonical quantization of variables give HY[g] = 0. This Wheeler-DeWitt equation is then not a wave equation of evolution, but a constraint type of equation which specifies a wave functional on metric configuration variables.

This dichotomy between how general relativity and quantum mechanics treat time still obtains. The big theory of unification is string theory, which is more particle based based. There general relativity is treated with a background, which adulterates some aspect of gravitation. The other theory called Loop Quantum Gravity is more general relativity oriented, but this theory has difficulty in deriving particle or quantum physics in a workable manner. At the core of this problem is that GR and QM simply regard time in basically different ways.

In euclidean gravity time is related to a temperature as

t = hbar/kT

which indicates that on a microscopic level time is the phase description of spinor fields (or whatever substructure there is) to gravitation. There may then be a time, or equivalently a temperature, where the physics is scale invariant, which will occur for some very low temperature (T goes to zero) or equivalently a long time parameter. This is I think a quantum critical point, similar to the "breakdown" of electrons or quasi-particle Fermions in a Landau fluid.

Lawrence B. Crowell

Dear Prof. Ellis,

Thank you for your kind comments. Also thank you for pointing me your article about causation in complex systems, I read it and found it very interesting. I agree with the view you presented there, and highly recommend it to everyone interested in causation.

Cristi Stoica

“Flowing with a Frozen River”,

http://fqxi.org/community/forum/topic/322

Dear Professor Ellis,

In your reply to Elliot McGucken alias Dr.E (The real McCoy), essay 238, you denied the necessity of an imaginary time. Because my essay 369 claims having revealed improper interpretation of complex quantities, and I feel anyway obliged to react to some mistakes in several discussions, I suggest you might read my comment tomorrow there.

Sincerely, Dr.-Ing. Blumschein

Dear Lawrence B. Crowell

thank you for your thoughtful comments. The problem lies in the interplay between spatial constraint equations and timelike evolution equations, the former being conserved by the latter. One needs to distinguish here between a generic formalism that applies in all circumstances, and a useful formalism that tells what happens in real world situations. In the recent universe on an astronomical scale it is only scalar perturbations that matter, with (in the zero-pressure case) characteristics that are timelike world lines. Spatial variation is a secondary consideration (one has what is called a `silent universe' where spatial derivatives can be neglected in the dyamical equations). In complex systems such as networks it is arrays of ordinary differential equations that best describe what is happening. How does that arise out of the underlying unification of quantum gravity and particle physics? So as well as the dichotomies you mention, there is the dichotomy between micro-physics and effective theories at macro scales. One can try to force the macro phyiscs to look more like the microphysics, which is in effect what happens when theoretical physicists suggest time is an illusion at the macro scale; or one can suggest the micro physical descriptions are missing some aspect that is apparent at the macro scale, which is in effect my suggestion here. In detail terms that means taking the measurement/collapse of the wave function issue seriously. Dealing with the Wheeler-de Witt equation alone, or any similar equation, won't suffice. The Hamiltonian development by itself is not the whole story.

Dear Dr.-Ing. Blumschein,

I am not against complex numbers and complex functions in general; indeed Roger Penrose has a very nice discussion of their merits in his book The Road to Reality. I am against their use in the coordinates and/or space-time metric in General Relativity, except sometimes as complex conjugate pairs where they are a just shorthand way of descrbing two real quantities. The reason is that general coordinate transformations then become very difficult to handle, as they irretrievably mix up real and imaginary quantities, and what started off as a 4-dimensional spacetime begins to look like 8 an dimensional entity, but this is an illusion brought on by use of unsuitable notation.

Thanks for the reply.

This makes me wonder if there is not some dualism of time. As a toy I might imagine the following. In four dimensions there is the dualism on chains 4 = m n. We have the ADM model of spacial surfaces and a Hemiltonian defined on them so that HY[g] = 0. So there is a foliation of M^3 spaces, we choose a coordinate condition and then time is given according to the diffeomorphisms between these surfaces (laspe function etc). So this is what Wheeler called the many fingered time: you can push time in any direction you want, choose one and the "time" is a symmetry of the theory. We might then consider the dual of this approach. Suppose there exists a set of one dimensional spaces (lines or curves) These then have a fibration given by M^3 spaces, where one is free to choose any space possible at each point on a fibre. Pick a section in the fibre and local changes between coordinate on each line gives tranformations between M^3's.

So a theory of M^3 and you choose time, and a theory M^1 and you choose M^3. One is a constraint theory, the other a purely dynamical theory.

Something to ponder,

L. C.

Dear Professor Ellis,

Thank you for your reply. You are quite right. Nonetheless I do not share your aversion towards complex numbers. My point is:

Complex plane is in general assumed to contain much more data as compared to the real line. That's why quantum mechanics run into the trouble of Schroedinger's cat, entanglement, decoherence, etc.

If I am correct then only past time relative to any process of concern counts and analytical continuation just creates redundant data. Is there nobody who either refutes or supports this?

Sincerely, Eckard Blumschein

Dear Prof. Ellis,

So far, I assume that Hilbert and his pupil J. v. Neumann were at least among the first who declared the flow of time an illusion. Do you know earlier ones?

Sincerely,

Eckard Blumschein

Dear Eckard Blumschein

The idea is very much older than Hilbert. According to the Wikipedia article on Plato, "Plato, like so many other Greek philosphers, was stymied by the question of change in the physical world. Heraclitus had said that there is nothing certain or stable except the fact that things change, and Parmenides and the Eleatic philosophers claimed that all change, motion, and time was an illusion. Where was the truth? How can these two opposite positions be reconciled? Plato ingeniously combined the two" [in his theory of forms].

Sincerely

George Ellis

Thank you for the reminder that toy models of reality are just that -- toys -- and that any theory which hopes to explain the real world must reflect that reality.

Thanks for that, Tevian. I think your point is crucial: theoretical physicists tend to get totally caught up in their very simplified models of reality, and then start to confuse them with reality. We have attained a huge sucess in terms of our progress in fundamental physics (apart from the open question of quantum gravity). The problem now is to get a more realistic relation of this body of foundational theory to physical and biological reality, specifically the nature and functioning of truly complex systems. They too provide data about the nature of the real world; if necessary, the fundamental theories must be modified to take this data into account. And whatever one says about microphysics, it seems clear that the flow of time is not an illusion at the macro scale. I suggest, for example, that it is not yet determined who will win the fqxi essay prizes. This will be determined eventually, as part of the overall historical process, as the flow of time rolls inevitably on.

Prof. Ellis: "I suggest, for example, that it is not yet determined who will win the fqxi essay prizes."

Perhaps also not after the prizes are awarded and you forgot who won?

About your essay, do you think that one can do experiments to find deviations of standard quantum mechanics, e.g. decoherence in isolated systems that are not due to interactions with the environment?

't Hooft has worked on models in which quantum mechanics is not fundamental. He is led to models in which information loss has to be present at the fundamental level, see e.g. here:

http://arxiv.org/abs/gr-qc/9903084

http://arxiv.org/abs/hep-th/0105105

http://arxiv.org/abs/quant-ph/0212095

http://arxiv.org/abs/quant-ph/0604008

http://arxiv.org/abs/quant-ph/0701097

Dear Saibal Mitra

Your essay starts "Assuming the validity of the Many Worlds Interpretation of quantum mechanics (MWI), we examine the possibility of changing the past by memory erasure". I have problems with the many worlds assumption, as I have for example never had an answer from those I consulted as to how often the wave function branches, and what decides when it happens. I do not grasp the utility of the idea, which appears to be untestable. But maybe I will be persuaded some day. Your essay then carries on "In the future our descendents will probably be machines". I cannot concur with this assumption, indeed I am not sure that it makes sense. Then I cannot see any reality in your proposals about reloading and erasing memory: I do not believe this is remotely practicable. So I do not think there is any reality to the proposal that the identity of those who won the prizes can be altered after they have been won. I cannot see that this is a profitable line of speculation to pursue. However T'Hooft's ideas on quantum mechanics are surely worth consideration. Yes those kinds of experiments are certainly worth considering.

Dear Prof. Ellis,

thanks for your comments. I'll write a detailed reply to your criticism on my essay on the comments page of my essay:

http://www.fqxi.org/community/forum/topic/354

to prevent the discussion here from going off topic.

I think the following points are appropriate to mention here:

1) It is legitimate to investigate the consequences of purely unitary time evolution. As of yet, there is no evidence that time evolution does not proceed in a unitary way. In my essay, I just considered a hypothetical (machine) observer who can erase his/her/its memory, I didn't speculate about new fundamental physics.

2) If it is proposed that time evolution is not unitary, then this has to be supported by experimental evidence. Because then a more complicated theory is proposed and, as of yet, it hasn't been shown that the purely unitary time evolution is not sufficient to account for all experimental data.

Dear Professor Ellis,

Thank you for pointing me to Plato (427-348) and Aristoteles (384-322) who claimed: Numbers are basic to anything.

Ancient mathematicians did not have negative numbers. They certainly did not have the notion negative time.

After Descartes and Fourier necessarily obeyed the usual notion of what clocks putatively show, maybe, it was Minkowski who invented world-lines thought to range from minus infinity to plus infinity, Hilbert denied the arrow of time, and Einstein called the division between past and future an illusion. They all did so for mixed mathematical, religious, and speculative reasons, which are still dominating fund-raising including its reflection in the votes of this contest.

Mathematics claimed to generalize anything, and scientists are still considering most abstract and most general models as the deepest ones. Benefits for science, technology and economy seem to be more or less occasional spin-offs.

Regards,

Eckard Blumschein

Dear Saibal Mitra

every time a quantum measurement is made the evolution does not proceed in a unitary way. Please see for example Roger Penrose: The Emperor's New Mind, pages 250-251, which makes explicit aspects of what is for example in Isham's book Lecturs on Quantum Theory, pages 71-73 and 82-85. You state "It is legitimate to investigate the consequences of purely unitary time evolution. As of yet, there is no evidence that time evolution does not proceed in a unitary way." You are talking of quantum theory without any measurements taking place, and hence no outcomes of any experiments and no evidence of anything. The many worlds picture does not change the abundant experimental proof that what we verify by experiment in the unverse domain we actually experience is measurements that proceed in a non-unitary way.

Dear Prof. Ellis,

I think we would need evidence that a completely isolated system undergoes non unitary time evolution. If you are locked up in a hypthetically perfectly isolated box and you perform measurements in tat box then, if you are right, the wavefunction of the entire box would not evolve according to the unitary time evolution. Then, since you are a large collection of particles, one should expect that unitary time evolution is violated in general (apart form the, wavefunction collapse associated with measurements, which in the MWI is explained as the observer becoming entangled with the system).

Dear Sabal Mitra

You say "I think we would need evidence that a completely isolated system undergoes non unitary time evolution." The whole point of my article is that there do not exist any completely isolated systems in the real world (except perhaps the universe itself). They are a fiction. The best you can do is partially isolate particular systems for limited times. But for example, the matter of which they are now composed was not isolated 14 billion years ago when they were part of the primeval soup. And the whole point about living systems is that they cannot function if isloated: they are open systems. The "hypothetically perfectly isolated box" you invoke is just that: hypothetical. It cannot exist in physical reality.

Dear Prof. Ellis,

Yes, I agree with you here. I think 't Hooft also argued in a similar way in defense of his ideas. You have to assume that the fact that you cannot isolate a (macroscopic) system from the rest of the universe is (also) due to fundamental physics in such a way that assuming you could do so would lead to flawed conclusions.

Suppose that quantum gravity is given by noncommutative geometries (plural) according to unitarily inequivalent groups. Further let there be associator maps between these groups. This would then be nonassociative QM, which is nonunitary. However, it can preserve quantum bits if described by E_8 or some sporadic group with an error correction capacity. A coarse grained description of this system, which eliminates nonassociative graphs, will then give a thermal description similar to Bogoliubov transformations used in black hole radiation.

This system might then be the entire universe, maybe more compactly described in it earliest fractions of femptoseconds of existence. Any supposedly closed system an experimenter might set up suffers from a number of problems. The system can't be kept completely away from the boundaries of the box, except for a black hole in the AdS spacetime. So there is no procedure for arriving at a perfect fine grained description for a system. Hence all systems we observe are ultimately open on some scale of frequencies, time or space.

Lawrence B. Crowell

Dear Lawrence B. Crowell

thanks for that interesting elaboration.

Dear George,

In response to my first posting from Dec. 2, 2008 @ 07:02 GMT above, on Dec. 3, 2008 @ 16:47 GMT, you wrote:

"... the outcome of quantum events is unknown until they happen. That is a key feature on which I build my proposal; so there is no conflict."

I'm glad to notice that you can surf the Web, so please check out the link to my essay 'Quantum Mechanics 101', in my first posting above. The crucial issue is *not* that the outcome of quantum events is "unknown until they happen".

We aren't talking epistemology here. The puzzle is know since 1935. To quote from Erwin Schrödinger's "Die gegenwärtige Situation in der Quantenmechanik":

"... measuring it does not mean ascertaining the value that it (the quantum system - D.C.) has."

As to whether there is conflict in your reasoning, I think it is too early to say anything conclusive. You haven't yet elaborated on the so-called Dynamic Dark Energy -- the driving force of 'the flow of time'.

I wish you best of luck in placing this perfectly smooth "dark stuff" on some Cauchy surface.

Dimi

Dear Dimi Chakalov

you correctly note that my statement "... the outcome of quantum events is unknown until they happen. That is a key feature on which I build my proposal; so there is no conflict." is not as strong as it could be, and indeed should be. It must have been penned in haste, or late at night. I should have made it much stronger, as follows: "... the outcome of quantum events is unknown and indeed undetermined until they happen." Then I am no longer talking just epistemology: I am talking about the way things actually occur, as emphasized for example by Feynmann. This is what I in fact intended, amd is meant throughout my paper.

I have not related my paper to dark energy, as usually understood, because in my view that is a completely different puzzle from what is considered here. I do indeed have views on that issue, but see no reason to post them here.

Dear George,

Good to see you (and your essay) here.

"The most important property of time is that it unfolds. The present is different from both the past and future, which in turn are completely different from each other, the past being fixed and the future changeable. The present is the instant of transition between these two states. The time that is the present at this instant will be in the past at the next instant."

I feel like I may be begginning to repeat myself a little in this essay contest, but it can be shown that if instants (and instantaneous magnitudes) existed, motion and change would be impossible. Furthermore, as instants would constitute the building blocks of time, if they can't/don't exist, neither can time.

"But one thing is clear: both the entire Darwinian process of evolution through which we come into existence, and the processes by which we read this article, depend on the flux of time. You would not exist and have the ability to read this article if the view proposed here (and expounded in more detail in Ref. [6]) was not a correct description of the way things are."

That's not the case. Indeed, it would only apply if instants (and time) did exist. I agree that it does apply to the standard interpretation of block time, but this is also because of its assumption of the physical existence of instants, spatial points and space-time points (and as such, the resulting assumption of the existence of time, space, and space-time). As soon as one recognizes that instants, instantaneous magnitudes, space-time points etc, do not exist, however, motion and change become possible, and can be seen to be completely compatible with gr and the block view, with all times (those shown by a clock) sharing equal footing.

Also, in order for change to be possible, one does not need time to exist, as one only need motion. Indeed, I think that if one thinks carefully about it, it becomes apparent that it is motion that enables the hands of a clock to rotate and for one to represent an interval, rather than the other way around (the existence of time enabling motion). My essay (and notes) go into this further. I naturally agree that certain processes are not time reversible, but this is completely compatible with a timeless block view (if interpreted correctly). Moreover, it does not suggest at all that time flows.

Best wishes

Peter

Dear Peter Lynds,

Thank you for your response. You state "it can be shown that if instants (and instantaneous magnitudes) existed, motion and change would be impossible. Furthermore, as instants would constitute the building blocks of time, if they can't/don't exist, neither can time." I presume you are referring to the Zeno's paradox argument, presented in your own fqxi essay and elsewhere. However I do not accept that argument, for the following reason: many discussions of quantum gravity conclude that at the very smallest scale, space-time will be discrete, for this is how the basic quantum theory principle will manifest itself in the context of the nature of space-time (quantized area and volume operators are one of the main deductions from loop quantum gravity, and there are 701,000 web-pages identified by a Google search for "discrete space time").

I concur in this view both because of quantum principles, and because this rids physics of the profound problem of the uncountable infinities of numbers that occur in the real line of mathematics. Indeed David Hilbert in a famous passage [D Hilbert (1964), ``On the Infinite". In Philosophy of Mathematics, ed. P Benacerraf and H Putnam (Englewood Cliff, N. J.: Prentice Hall), pp 134-151] stated that "Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought . . . The role that remains for the infinite to play is solely that of an idea . . . which transcends all experience and which completes the concrete as a totality." On this view, infinities are mathematical entities that never occur in physical reality; this may be taken as applying to the nature of space in a profound way. Incidentally, at a stroke this gets rid of many of the divergences that plague theoretical physics.

If space-time at its foundations is discrete, then the Zeno's paradox argument falls away. In this case it is not time that is an illusion, it is space-time continuity that is an illusion, generated by the coarse graining that underlies all macroscopic physics and hence also chemistry and biology. Furthermore, you state that most people believe that Zeno's paradox is resolved by calculus; but this ignores that fact that one of the great advances in mathematics was the transition from calculus to analysis, where the taking of limits is replaced by the use of bounding methods (utilising the famous `epsilons' and `deltas' of analysis) and the associated characterisation of solutions in terms of Sobolev spaces. Mathematical physics got rid of those troublesome limiting processes decades ago.

So my position is that continuity is an illusion, and your argument does not apply to the real physical world. Finally you state "in order for change to be possible, one does not need time to exist, as one only needs motion. Indeed, I think that if one thinks carefully about it, it becomes apparent that it is motion that enables the hands of a clock to rotate and for one to represent an interval, rather than the other way around (the existence of time enabling motion)." I cannot understand how you conceptualise motion without the flow of time to underpin it. Motion is identified by something being in a different position at a later time than at an earlier time. Perhaps your view is related to a relative rather than absolute view of the nature of space and time; that is a further interesting argument, which I will not enter here, except to comment that I believe there may be a fundamental duality between absolute and relative views of the nature of space and time. But adopting a relative view of space or time will still leave one with change taking place as relative time flows.