Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
On 04-Sep-02, Hal Finney wrote: > Brent - FYI you sent your comment just to me again. I > don't know if you intended to send it to the list or not. > But I will reply just to you based on how you sent it. Sorry, Hal - my mistake. > You wrote: >> I have always had two problems with the MWI. Initially it >> was measurements that caused the splitting into different >> universes, and that's apparently still the view of people >> who propose tests like Plaga, but later it was realized >> that there was no prinicpled way to distinguish >> measurements from other interactions. But then it seems >> that the universe must split everytime a photon is >> emitted by an atom anywhere in the visible universe. >> Since that atom could have interacted with some atom near >> us - all the visible universe was in interaction before >> inflation - then that split implies a split here & now. >> But to what effect? >> MWI seems to commit us to an essentially infinite rate >> of splitting; yet nobody uses it except in a QM >> measurement analysis. To take it seriously I would need >> to see why all this infinite splitting can be ignored and >> whay I need only pay attention to certain cases. This >> seems very much like Bohr's division into >> classical/quantum realms. > I would make two comments about this. The first, following > the lines of my earlier message, is that the same > consideration applies to the multiverse models we have > discussed on the everything list. We have all possible > universes existing, which means that all possible > alternatives are explored. Whether you think of it as one > universe "splitting" or multiple formerly-identical > universes becoming distinct, the process is the same. And > you have the same problem, that conceptually the smallest > change anywhere in the universe now makes there be two > different universes where formerly there were one. So if > you don't like the MWI, you ought to object to more > general multiverse models (and maybe you do!). Yes, I think I understand this similarity between all-universes-are-computed and the MWI of QM. > But second, I don't think this objection is as bad as it > sounds. As is perhaps made more clear in the multiverse > example, this "splitting" is a somewhat figurative or > subjective phenomon. It's not so much that the universe > splits; there is only one universe, in the largest sense. > We use the word multiverse to describe this grand > ensemble, but in its traditional meaning of "everything > there is" it is fair to call it the universe. In the MWI, > there is only one universe, which evolves > deterministically by following the Schrodinger equation. > That's all there is to the physics. OK, except don't the universes interfere to produce (approximate) diagonalization of the density matrix? So that is why we observe classical physics at the macroscopic level. > The splitting comes in when we study the wave function of > the universe and find that it can be decomposed into > pieces which add together to form the whole, and where the > time evolution of each separate piece is essentially > independent of the others. And then, each of these pieces > can further be interpreted as being themselves made of > separate pieces which become causally independent as time > moves forward. So it's not that the universe splits, it's > that the global wave function can be interpreted as having > causally independent parts. The decomposition though is only an approximation holding for macroscopic observables that average over many "universes" or in other words, projects out onto a subspace of the universes Hilbert space. So the separation and independence have to do with our macroscopic perspective and the fact that our knowledge (or all knowledge) must be encoded at a macroscopic (irreversible) level. >> The second problem has to do with time and casuality. At >> a microscopic level QM is time symmetric. If we say there >> is no real collapse of the wave function - all evolution >> is unitary (and therefore reversible) - then it seems we >> should from each datum or measurement result, compute the >> past as well as the future. All that can be known of the >> past, what happened within our past light cone, is >> determined by the present. So does the universe split as >> we go back in time? If not, why not? Are we stepping >> outside physics and assuming that the experimenter, as an >> agent, sets up initial, but not final, conditions and so >> defines the arrow-of-time by his causal action as an >> agent outside of physics? > That's a good question and one I don't really know the > answer to. It's possible that you're right, it has > something to do with initial conditions. Or it's possible > that universes are merging at the micro-scale even as they > are splitting at the macro-scale, which would also be > fundamentally due to initial conditions. For example, > dual-slit diffraction can be seen as worlds where > particles go through the slits sepa
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments (fwd)
OOPS! I sent this reply only to Hal - instead of the list. So here's the fwd to the list. Brent Meeker On 04-Sep-02, Hal Finney wrote: > I think on this list we should be willing to seriously > consider the many-worlds interpretation (MWI) of quantum > mechanics as the ontology for our universe. In particular, > we should not assume that wave function collapse is > anything more than an illusion caused by decoherence of > formerly interacting components of the universal wave > function. Almost all of the supposed paradoxes of QM go > away if you eliminate wave function collapse. > There are a few objections which I am aware of which have > been raised against the MWI. The first is its lack of > parsimony in terms of creating a vast number of universes. > We gain some simplification in the QM formalism but at > this seemingly huge expense. The second is its > untestability, although some people have claimed > otherwise. And the third is that it retains what we might > call the problem of measure, that is, explaining why we > seem to occupy branches with a high measure or amplitude, > without just adding that as an extra assumption. I have always had two problems with the MWI. Initially it was measurements that caused the splitting into different universes, and that's apparently still the view of people who propose tests like Plaga, but later it was realized that there was no prinicpled way to distinguish measurements from other interactions. But then it seems that the universe must split everytime a photon is emitted by an atom anywhere in the visible universe. Since that atom could have interacted with some atom near us - all the visible universe was in interaction before inflation - then that split implies a split here & now. But to what effect? MWI seems to commit us to an essentially infinite rate of splitting; yet nobody uses it except in a QM measurement analysis. To take it seriously I would need to see why all this infinite splitting can be ignored and whay I need only pay attention to certain cases. This seems very much like Bohr's division into classical/quantum realms. The second problem has to do with time and casuality. At a microscopic level QM is time symmetric. If we say there is no real collapse of the wave function - all evolution is unitary (and therefore reversible) - then it seems we should from each datum or measurement result, compute the past as well as the future. All that can be known of the past, what happened within our past light cone, is determined by the present. So does the universe split as we go back in time? If not, why not? Are we stepping outside physics and assuming that the experimenter, as an agent, sets up initial, but not final, conditions and so defines the arrow-of-time by his causal action as an agent outside of physics? Brent Meeker
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
Dear Tim, Thank you for the reply but ... Well, I did not mean to imply that we should "look to Egan's fictional character for actual theories" or any other novel or fiction... I think that I asked you a similar question before regarding the idea that Egan is discussing using the fictional character of Mosala. I am trying to see if you understood the idea well enough to discuss it with me. It seems that my question below got missed somehow. Scerir mentioned the following paper previously: http://arxiv.org/abs/quant-ph/0110124 Quantum Physics, abstract quant-ph/0110124 From: Antoine Suarez <[EMAIL PROTECTED]> Date: Sat, 20 Oct 2001 13:02:58 GMT (17kb) Is there a real time ordering behind the nonlocal correlations? Authors: Antoine Suarez Comments: 4 pages Latex, 1 eps figure It is argued that recent experiments with moving beam-splitters demonstrate that there is no real time ordering behind the nonlocal correlations: In Bell's world there is no "before" and "after". If you get a chance to read it perhaps my terms "a priori" and "a posteriori" might make some sense. ;-) I have been looking for Smolin's book in my local bookstore. I will try again. Incidentally, I have read books by most of the writters you mention! I too am very fond of Zindell's Neverness. ;-) Kindest regards, Stephen - Original Message - From: "Tim May" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Wednesday, September 04, 2002 12:24 PM Subject: Re: Time as a Lattice of Partially-Ordered Causal Events or Moments > > On Wednesday, September 4, 2002, at 07:47 AM, Stephen Paul King wrote: > > > Dear Tim and scerir, > > > > I am VERY interested in this discussion! ;-) It seems to me that > > fact > > that the amplitudes of observables in QM are complex valued and thus > > do not > > obey trichotomy may be at the root of the difficulty. When we attempt > > to > > make sense of situations such as those we obtain in EPR we have to be > > very > > careful that we take into account the configuration of the experiment > > itself. This implies that the lattice of relations or poset aspect of > > causality is a posteriori and not a priori to the specifics of the > > experiment. This implies, at least to me, that it is a mistake to > > assume the > > a priori existence of a space-time (with a unique light cone > > structure). > > I agree that imposing a space-time structure for quantum events is a > problem. In fact, this is one of the motivations of quantum gravity > work, to get rid of the "events on a background of space and time" > which most QM has been using. > > However, assuming a space-time and with local light cones seems very > reasonable to me. We have no particular evidence that the light cone > formalism isn't still applicable to quantum kinds of events (whatever > EPR "spooky action at a distance" may be, there is certainly no > evidence that faster than light travel of particles or photons is > involved, so there's no reason to throw out the speed of light as a > maximum speed). > > You may be interested in Smolin's "Three Roads to Quantum Gravity." He > argues for the _relational_ view of space-time as being more suitable > than the absolute space-time coordinates in Newtonian and (ironically) > most quantum theories. Relativity of course uses the less absolute > scheme. > > However--and this is very important!--there are no theories given > experimental support today which show real violations of causality. (We > could debate for a few days whether delayed-choice experiments, > Aharonov-Bohm experiments, etc. show violations of causality. While > entangled states show behaviors not found in the macroscopic, > classical, human-scale world, they don't violate causality.) > > This argues for the emphasis placed on causal sets and causal > relations. And hence on posets and lattices. (In my opinion, following > the lead of the several authors I've mentioned a few times here.) > > > One possible solution is to consider space-times strictly from an a > > posteriori point of view. You had mentioned Greg Egan's novels and the > > "All > > Topologies model" (for instance in the novel Distress) in previous > > posts. Do > > you think that the ideas of the character Mosala could be used to "make > > sense" of this? > > Well, I try not to get too many of my theories out of science fiction! > > Not to sound flip or dismissive, but Egan's novels and short stories > are best seen as romps through a landscape of strange and stimulating > ideas. I like his stuf
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
Tim May wrote: > On Wednesday, September 4, 2002, at 10:08 AM, Hal Finney wrote: > > There are a few objections which I am aware of which have been raised > > against the MWI. The first is its lack of parsimony in terms of > > creating a vast number of universes. We gain some simplification in > > the QM formalism but at this seemingly huge expense. The second is its > > untestability, although some people have claimed otherwise. > > The latter is the more important. If, for example, the plurality of > worlds are out of communication with each other, forever and always, > then it means nothing to assert that they "actually" exist. Worlds are never absolutely out of touch. There is a small degree of interference which is always present between worlds in the MWI. Of course this is infinitisimal in practice and can't be observed between macroscopically distinct worlds. But there is a continuum from totally disjoint worlds down to worlds which differ only microscopically and where interference is easily observed, such as an electron double-slit experiment. As our technology improves, we will be able to detect smaller and more subtle degrees of interference, corresponding to worlds which are more and more macroscopically distinct. Assuming the MWI is in some sense "correct", then eventually it will become increasingly difficult to draw the line and say that, for example, small objects can be in distinct superpositions, but large objects cannot. > In weaker forms of the MWI, where it's the early state of the Big Bang > (for example) which are splitting off into N universes, De Witt and > others have speculated (as early as around 1970) that we may _possibly_ > see some evidence consistent with the EWG interpretation but NOT > consistent with other interpretations. I'm not familiar with the details of this. But I know that much of the impetus for increased acceptance of MWI models comes from the cosmologists. > > And the > > third is that it retains what we might call the problem of measure, > > that is, explaining why we seem to occupy branches with a high measure > > or amplitude, without just adding that as an extra assumption. > > What's the problem here? I find it utterly plausible that we would be > in a universe where matter exists, where stars exist, where entropy > gradients exist, etc., and NOT in a universe where the physical > constants or structure of the universe makes life more difficult or > impossible (or where the densities and entropy gradients mean that > evolution of complex structures might take 100 billion years, or more, > instead of the billion or so years it apparently took). The problem is more formal, that if we abandon measurement as a special feature of the physics, there is no longer an axiom that says that probability is proportional to amplitude squared. > > By the metrics we typically use for > > universe complexity, basically the number of axioms or the size of a > > program to specify the universe, the MWI is in fact simpler and > > therefore > > more probable than the traditional interpretation. > > This I'm not convinced of at all. I don't find the Copenhagen (aka > "Shut up and calculate") Interpretation requires any more axioms. So > long as we don't try to understand what is "really" happening, it's a > very simple system. The conventional formulation, as described for example at http://www.wikipedia.com/wiki/Mathematical_formulation_of_quantum_mechanics, has special axioms related to measurement. Systems evolve according to the Schrodinger equation except when they are being measured, when we get wave function collapse. MWI rejects the axioms related to observables and collapse. In the page above we can eliminate axiom 3 and probably axiom 2 as well. You are left with nothing but Hilbert space and the Schrodinger equation. It's a simpler set of axioms. > And, putting in a plug for modal/topos logic, the essence of nearly > every interpretation, whether MWI or Copenhagen or even Newtonian, is > that observers at time t are faced with unknowable and branching > futures. (In classical systems, these arise from limited amounts of > information available to observers and, importantly, in limited > positional information. Even a perfectly classical billiard ball > example is unpredictable beyond a few seconds or maybe tens of seconds, > because the positions and sets of forces (turbulence in the air > currents around the balls, even gravitational and static electricity > effects, etc.) are only known to, say, 20 decimal places (if that, of > course). Because the "actual" positions, masses, sphericities, static > charges, etc. are perhaps defined by 40-digit or even 200-digit > numbers, the Laplacian dream of a suffiicently powerful mind being able > to know the future is dashed. This is true in practice, but I think there is still a significant difference between deterministic systems like classical physics or the MWI, and
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
On Wednesday, September 4, 2002, at 10:08 AM, Hal Finney wrote: > I think on this list we should be willing to seriously consider the > many-worlds interpretation (MWI) of quantum mechanics as the ontology > for > our universe. I remain agnostic on the MWI or EWG interpretation. While I don't strongly believe that the MWI is "reality" (cough cough), I agree with Hal that it's a plausible ontology. Further, I take more seriously than many the "plurality of worlds" ontology of the late philosopher David Lewis. (The guy who argues that we should not give special linguistic treatment to "our" world and should give equal standing to "the world in which World War II was won by Germany," for example. Lewis is sometimes caricaturized by capsule summaries of the sort "David Lewis believes unicorns really do exist," but what Lewis is claiming is fully consistent with modal logic and possible worlds semantics.) > There are a few objections which I am aware of which have been raised > against the MWI. The first is its lack of parsimony in terms of > creating a vast number of universes. We gain some simplification in > the QM formalism but at this seemingly huge expense. The second is its > untestability, although some people have claimed otherwise. The latter is the more important. If, for example, the plurality of worlds are out of communication with each other, forever and always, then it means nothing to assert that they "actually" exist. In weaker forms of the MWI, where it's the early state of the Big Bang (for example) which are splitting off into N universes, De Witt and others have speculated (as early as around 1970) that we may _possibly_ see some evidence consistent with the EWG interpretation but NOT consistent with other interpretations. > And the > third is that it retains what we might call the problem of measure, > that is, explaining why we seem to occupy branches with a high measure > or amplitude, without just adding that as an extra assumption. What's the problem here? I find it utterly plausible that we would be in a universe where matter exists, where stars exist, where entropy gradients exist, etc., and NOT in a universe where the physical constants or structure of the universe makes life more difficult or impossible (or where the densities and entropy gradients mean that evolution of complex structures might take 100 billion years, or more, instead of the billion or so years it apparently took). > > The point is, all of these objections apply equally to the more > ambitious multiverse models we consider here. Our multiverse is even > more profligate than the MWI; it is if anything less observable; and > the problem of measure is at least as acute. I certainly agree with this! Tegmark's and Schmidhuber's and Egan's "all mathematics, all programs" models form supersets of the conventional MWI. > By the metrics we typically use for > universe complexity, basically the number of axioms or the size of a > program to specify the universe, the MWI is in fact simpler and > therefore > more probable than the traditional interpretation. This I'm not convinced of at all. I don't find the Copenhagen (aka "Shut up and calculate") Interpretation requires any more axioms. So long as we don't try to understand what is "really" happening, it's a very simple system. > Quantum randomness does not exist in the MWI. It is an illusion > caused by > the same effect which Bruno Marchal describes in his thought > experiments, > where an observer who is about to enter a duplication device has > multiple > possible futures, which he treats as random. If Schmidhuber would > adopt > this model for the physics of our universe it would improve the quality > of his predictions. And, putting in a plug for modal/topos logic, the essence of nearly every interpretation, whether MWI or Copenhagen or even Newtonian, is that observers at time t are faced with unknowable and branching futures. (In classical systems, these arise from limited amounts of information available to observers and, importantly, in limited positional information. Even a perfectly classical billiard ball example is unpredictable beyond a few seconds or maybe tens of seconds, because the positions and sets of forces (turbulence in the air currents around the balls, even gravitational and static electricity effects, etc.) are only known to, say, 20 decimal places (if that, of course). Because the "actual" positions, masses, sphericities, static charges, etc. are perhaps defined by 40-digit or even 200-digit numbers, the Laplacian dream of a suffiicently powerful mind being able to know the future is dashed. Unpredictability, or randomness, arises even in a fully classical real world. --Tim May "As my father told me long ago, the objective is not to convince someone with your arguments but to provide the arguments with which he later convinces himself." -- David Friedman
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
Hal Finney wrote: >Quantum randomness does not exist in the MWI. It is an illusion caused by >the same effect which Bruno Marchal describes in his thought experiments, >where an observer who is about to enter a duplication device has multiple >possible futures, which he treats as random. > Could somebody incorporate complementarity in a thought experiment in the style of Bruno's duplication experiment? George
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
I think on this list we should be willing to seriously consider the many-worlds interpretation (MWI) of quantum mechanics as the ontology for our universe. In particular, we should not assume that wave function collapse is anything more than an illusion caused by decoherence of formerly interacting components of the universal wave function. Almost all of the supposed paradoxes of QM go away if you eliminate wave function collapse. There are a few objections which I am aware of which have been raised against the MWI. The first is its lack of parsimony in terms of creating a vast number of universes. We gain some simplification in the QM formalism but at this seemingly huge expense. The second is its untestability, although some people have claimed otherwise. And the third is that it retains what we might call the problem of measure, that is, explaining why we seem to occupy branches with a high measure or amplitude, without just adding that as an extra assumption. The point is, all of these objections apply equally to the more ambitious multiverse models we consider here. Our multiverse is even more profligate than the MWI; it is if anything less observable; and the problem of measure is at least as acute. If we're willing to seriously consider the prospect that "everything exists" and specifically "all universes exist", I think it makes most sense to take the MWI as the model for our own universe, when working within the multiverse framework. By the metrics we typically use for universe complexity, basically the number of axioms or the size of a program to specify the universe, the MWI is in fact simpler and therefore more probable than the traditional interpretation. This is one of the things that bothered me in Jurgen Schmidhuber's paper, Algorithmic Theories of Everything, http://www.idsia.ch/~juergen/toesv2/toesv2.html. His "Example 1.1" at http://www.idsia.ch/~juergen/toesv2/node2.html describes a universe like ours that has wave-function collapse, which requires the continual creation of a large number of random bits. He proposes the use of a pseudo-random number generator as the source of this randomness, and later explores the possibility that this would imply that we could detect patterns in quantum randomness, and (if I understood this part right) that quantum computers would not work. IMO this line of analysis is misguided (note, I'm not saying that the rest of the paper is wrong). None of these perhaps-unlikely observations is a true prediction of assuming his Speed Prior as the foundation for a universal computation engine. It would make much more sense, in terms of his overall approach, to assume that we ourselves live in a universe whose physics is based on the MWI. Such a universe has a simpler physical description (so it seems), therefore a smaller program and higher measure. And it requires no information creation, hence no need for a random number generator, neither true nor pseudo. Quantum randomness does not exist in the MWI. It is an illusion caused by the same effect which Bruno Marchal describes in his thought experiments, where an observer who is about to enter a duplication device has multiple possible futures, which he treats as random. If Schmidhuber would adopt this model for the physics of our universe it would improve the quality of his predictions. Hal Finney
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
On Wednesday, September 4, 2002, at 07:47 AM, Stephen Paul King wrote:
> Dear Tim and scerir,
>
> I am VERY interested in this discussion! ;-) It seems to me that
> fact
> that the amplitudes of observables in QM are complex valued and thus
> do not
> obey trichotomy may be at the root of the difficulty. When we attempt
> to
> make sense of situations such as those we obtain in EPR we have to be
> very
> careful that we take into account the configuration of the experiment
> itself. This implies that the lattice of relations or poset aspect of
> causality is a posteriori and not a priori to the specifics of the
> experiment. This implies, at least to me, that it is a mistake to
> assume the
> a priori existence of a space-time (with a unique light cone
> structure).
I agree that imposing a space-time structure for quantum events is a
problem. In fact, this is one of the motivations of quantum gravity
work, to get rid of the "events on a background of space and time"
which most QM has been using.
However, assuming a space-time and with local light cones seems very
reasonable to me. We have no particular evidence that the light cone
formalism isn't still applicable to quantum kinds of events (whatever
EPR "spooky action at a distance" may be, there is certainly no
evidence that faster than light travel of particles or photons is
involved, so there's no reason to throw out the speed of light as a
maximum speed).
You may be interested in Smolin's "Three Roads to Quantum Gravity." He
argues for the _relational_ view of space-time as being more suitable
than the absolute space-time coordinates in Newtonian and (ironically)
most quantum theories. Relativity of course uses the less absolute
scheme.
However--and this is very important!--there are no theories given
experimental support today which show real violations of causality. (We
could debate for a few days whether delayed-choice experiments,
Aharonov-Bohm experiments, etc. show violations of causality. While
entangled states show behaviors not found in the macroscopic,
classical, human-scale world, they don't violate causality.)
This argues for the emphasis placed on causal sets and causal
relations. And hence on posets and lattices. (In my opinion, following
the lead of the several authors I've mentioned a few times here.)
> One possible solution is to consider space-times strictly from an a
> posteriori point of view. You had mentioned Greg Egan's novels and the
> "All
> Topologies model" (for instance in the novel Distress) in previous
> posts. Do
> you think that the ideas of the character Mosala could be used to "make
> sense" of this?
Well, I try not to get too many of my theories out of science fiction!
Not to sound flip or dismissive, but Egan's novels and short stories
are best seen as romps through a landscape of strange and stimulating
ideas. I like his stuff a lot because he's one of the few writers today
able to (or interested in) keep up with modern physics and modern math.
There was a time when SF writers were engineers or scientists (Asimov,
Heinlein, Clarke, Hal Clement, even Larry Niven, who was at Caltech for
a while). Their editors were also science-oriented people, like Hugo
Gernsback and John Campbell, who wanted hard science in the science
ficiton they bought.
In their day, these authors wrote stories and novels involving the
then-weird ideas of hypercubes, Mobius bands, planets with extremely
high gravity, time travel, neutron stars, and black holes. (Larry Niven
was the Greg Egan of his day. Niven still writes, but his recent novels
are less compelling and certainly no longer are exploring cutting-edge
stuff.)
As SF spread in popularity to the Baby Boom generation, more and more
non-scientists and non-engineers started writing. So we got more
"sociological" SF (some of it very good, like Ursula LeGuin's stuff).
And more of the "palace intrigue" kind of novels ("The planet Cthwox
has been ruled by the Klanring for 2000 years. A spaceship from the
Vegan Federation has arrived..."). And then there is fantasy...dragons,
magic, etc.
A few writers have stuck to hard science themes. Vernor Vinge is a good
example. And some of the "cyberpunk" themes have been more or less
based on plausible science, including Gibson, Walter Jon Williams,
Bruce Sterling, Dan Simmons, etc. (I also like a less popular author,
David Zindell, and his quartet of novels set on the planet "Neverness."
Mathematics plays an unusual role.)
Steven Baxter also writes Stapletonian novels about the distant future
and the end of time, though his themes are often depressing (to me at
least).
Greg Egan is one of the few writers today actually _using_ the latest
developments in physics and even math to explore ideas about the nature
of our reality, the anthropic principle, and the colonization of
cyberspaces.
It also turns out that Egan has been doing some Java and Mathematica
programming for so
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
Dear Tim and scerir, I am VERY interested in this discussion! ;-) It seems to me that fact that the amplitudes of observables in QM are complex valued and thus do not obey trichotomy may be at the root of the difficulty. When we attempt to make sense of situations such as those we obtain in EPR we have to be very careful that we take into account the configuration of the experiment itself. This implies that the lattice of relations or poset aspect of causality is a posteriori and not a priori to the specifics of the experiment. This implies, at least to me, that it is a mistake to assume the a priori existence of a space-time (with a unique light cone structure). One possible solution is to consider space-times strictly from an a posteriori point of view. You had mentioned Greg Egan's novels and the "All Topologies model" (for instance in the novel Distress) in previous posts. Do you think that the ideas of the character Mosala could be used to "make sense" of this? Kindest regards, Stephen - Original Message - From: "Tim May" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Tuesday, September 03, 2002 9:30 PM Subject: Re: Time as a Lattice of Partially-Ordered Causal Events or Moments > > On Tuesday, September 3, 2002, at 02:21 PM, scerir wrote: > > > Tim May: > > I don't have a comprehensive theory of time, > > but I am very fond of "causal time." > > > > Sometimes we read papers saying there is now > > experimental evidence that quantum phenomena > > are "a-causal" or "non-causal" or "out-of-time". > > > > See, in example, these recent papers > > http://arxiv.org/abs/quant-ph/0110124 > > http://arxiv.org/abs/quant-ph/0201036 > > > > Now, can lattices capture also those important > > features? > > I haven't read the papers, just the abstracts. I could wait to comment > for a few days or weeks until I've had a chance to absorb the papers, > if ever, or comment now. > > First, it looks like these events are the usual "entangled states," > which can be spacelike (the usual example of particles separated by > light years). > > Second, for such spacelike intervals, they are outside each others' > light cones in the extreme cases, so it would not be expected for any > partial ordering to exist. > > Third, my own idiosyncratic view is to look at entangled particles as a > single system, regardless of separation. > > Fourth, as to the mechanics of lattices: the essence of a > partially-ordered set (poset) is that it does not require trichotomy, > where either a is less than b, a is greater than b, or a is equal to b. > In a chain, a linear form of a lattice, trichotomy holds. So, the > integers obey trichotomy, as one integer is either less than, greater > than, or equal to any other integer. Orders which obey trichotomy are > said to be well-ordered. > > But not all sets are well-ordered. If the ordering relation is set > inclusion, then a series of sets need not obey trichotomy. Some sets > may be disjoint, with one neither including the other, being included > by the other, or equal. > > In terms of causality, not even getting involved with speed of light > issues and light cones, it is quite possible to say "event A neither > caused event B nor was caused by event B nor is the same as event B." > That is, the events A and B are incommensurate, or disjoint...they fail > trichotomy. Clearly, most events all around us are such examples of > incommensurate. They form posets. > > What a lattice does is to formalize the notions of order and to say > there is only one edge between two events, and nothing in between (no > other nodes in between). If two events are separated by many instants > of time, many other events, then the lattice is made up of the smallest > identifiable events. The events look like a lattice. (As I said, the > Web has many nice pictures. No point in my spending 20 minutes drawing > an ASCII lattice here, having it reproduced poorly, when entering > "lattice poset" into Google will turn up nice pictures.) > > So, I would say from reading the abstracts that the Bell example just > fits the ecample of a poset, where two events, which may or may not be > entangled, are spacelike to each other. (This is the essence of the > usual "instantaneous action" of EPR/delayed choice experiments.) > > > --Tim > >
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
On Tuesday, September 3, 2002, at 02:21 PM, scerir wrote: > Tim May: > I don't have a comprehensive theory of time, > but I am very fond of "causal time." > > Sometimes we read papers saying there is now > experimental evidence that quantum phenomena > are "a-causal" or "non-causal" or "out-of-time". > > See, in example, these recent papers > http://arxiv.org/abs/quant-ph/0110124 > http://arxiv.org/abs/quant-ph/0201036 > > Now, can lattices capture also those important > features? I haven't read the papers, just the abstracts. I could wait to comment for a few days or weeks until I've had a chance to absorb the papers, if ever, or comment now. First, it looks like these events are the usual "entangled states," which can be spacelike (the usual example of particles separated by light years). Second, for such spacelike intervals, they are outside each others' light cones in the extreme cases, so it would not be expected for any partial ordering to exist. Third, my own idiosyncratic view is to look at entangled particles as a single system, regardless of separation. Fourth, as to the mechanics of lattices: the essence of a partially-ordered set (poset) is that it does not require trichotomy, where either a is less than b, a is greater than b, or a is equal to b. In a chain, a linear form of a lattice, trichotomy holds. So, the integers obey trichotomy, as one integer is either less than, greater than, or equal to any other integer. Orders which obey trichotomy are said to be well-ordered. But not all sets are well-ordered. If the ordering relation is set inclusion, then a series of sets need not obey trichotomy. Some sets may be disjoint, with one neither including the other, being included by the other, or equal. In terms of causality, not even getting involved with speed of light issues and light cones, it is quite possible to say "event A neither caused event B nor was caused by event B nor is the same as event B." That is, the events A and B are incommensurate, or disjoint...they fail trichotomy. Clearly, most events all around us are such examples of incommensurate. They form posets. What a lattice does is to formalize the notions of order and to say there is only one edge between two events, and nothing in between (no other nodes in between). If two events are separated by many instants of time, many other events, then the lattice is made up of the smallest identifiable events. The events look like a lattice. (As I said, the Web has many nice pictures. No point in my spending 20 minutes drawing an ASCII lattice here, having it reproduced poorly, when entering "lattice poset" into Google will turn up nice pictures.) So, I would say from reading the abstracts that the Bell example just fits the ecample of a poset, where two events, which may or may not be entangled, are spacelike to each other. (This is the essence of the usual "instantaneous action" of EPR/delayed choice experiments.) --Tim
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
Tim May: I don't have a comprehensive theory of time, but I am very fond of "causal time." Sometimes we read papers saying there is now experimental evidence that quantum phenomena are "a-causal" or "non-causal" or "out-of-time". See, in example, these recent papers http://arxiv.org/abs/quant-ph/0110124 http://arxiv.org/abs/quant-ph/0201036 Now, can lattices capture also those important features? s.
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
I'll say a few words on my personal journey in math.
On Tuesday, September 3, 2002, at 08:46 AM, Osher Doctorow wrote:
> One confusing point, I think, is the tendency of many mathematical
> logicians
> to identify with algebra and in fact to claim that their field is a
> branch
> or outgrowth of algebra.
I don't place _too_ much importance on which fields is the parent and
which is the child. My personal journey in physics drifted away from
spacetime stuff almost 30 years ago, as I sought to prepare myself for
a career by moving more into solid state physics. I read the popular
accounts of guys like Kip Thorne on black holes, but thought it was far
removed from anything in my life. And aside from reading Rucker ("Mind
Tools" is one of my favorites), Gardner, Devlin, W.W. Sawyer, Halmos,
and the recent accounts of logic (Hofstadter, Smullyan), I hadn't
looked at math in many years.
This changed half a year ago when I was reading Greg Egan's "Distress,"
with the "All Topologies Model" idea, then John Baez's site, then Lee
Smolin's fabulous book "Three Roads to Quantum Gravity." And I was
starting to learn about category theory, which I had only _heard_ of
before, but had never had the slightest clue about.
As I have said here before, the stuff on category theory and
(especially) topos theory, spoke to me in a way that was compelling.
Whether it is algebra or logic or topology/geometry, it's for me the
right approach. It provides a set of tools and, more importantly, a set
of concepts for unifying a bunch of things. The unification of geometry
and logic via algebra is compelling to me.
But maybe not to others. Wei Dai asked me to recommend just one book,
so I picked Lawvere and Schanuel's "Conceptual Mathematics: A first
introduction to categories." Well, WD reported that he had not found
anything relevant to his interests, so..."different strokes for
different folks."
For me, the categorification approach unifies and makes sense of causal
sets, modal logic, possible worlds semantics, probability, and the
nature of time.
And it's fun. This is by far the most important thing, that it
stimulates me to delve into so many areas.
> There are also many built-in biases in mathematical and theoretical
> physics,
> and one of them in my opinion is the bias toward dissolution of
> geometry at
> the sub-Planck level.Part of this is the pre-quantum computer bias
> toward the discrete and finite or at most countably infinite and the
> digital
> vs analog computer bias (in favor of digital computers). The real
> line and
> real line segments of course are uncountably infinite and connected,
> and
> thee would essentially be no applied mathematics or mathematical
> physics for
> example without it - and not much pure mathematics either.
I'm essentially a constructivist. I believe the abstractions about R^N
(real line, real plane, real space, etc.) are useful, and that even the
Axiom of Choice is useful, but I'm not sure any of these platonic
ideals exist outside of Reality. Brouwerian pragmatism in Cantor's
paradise.
But the views are not contradictory. Topos theory gives us a set of
tools for constructing mathematical universes. Paul Taylor puts it well
in his recent book, "Practical Foundations of Mathematics," when he
says that the apparently conflicting views of Platonism vs. Formalism
can be reconciled through these modern results.
> I hope that we can resist the temptation to go into absolutes. I am
> glad
> to see that you started your reply with a tolerant and compromising
> tone,
> and I will end this posting with a similar tone.
My tone may be an accident of how I was writing at that particular
time! (insert silly smiley here as you wish).
I have no reason to be partisan on these issues. I don't know what time
is, and I doubt anyone does. But I see glimpses, and new concepts,
which are giving me what I think is a more useful understanding than
any I got back when I was studying relativity intensively! (I may have
been too young, or too focussed, or too worried about a career, or just
not exposed to the wonderful mathematics I'm now aware of.)
> The discrete and the
> connected are in my opinion different theories or parts of different
> theories overall, and they are also parts of different
> interpretations. My
> view is that science progresses by tolerating different theories and
> different interpretations for competition and because many supposedly
> wrong
> theories or interpretations end up much later having something useful
> to
> contribute.
I agree. However, this argues for spending some more time on the
"discrete" view, as certainly the _continuous_ view of space and time
has dominated for most of the past century (time lines, time as a
river, the flow of time, Riemannian manifolds, etc.).
In particular, whether space and time "really" are discrete at the
Planck scale or "continuous all the way down" (to 10^-35 c
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
From: Osher Doctorow [EMAIL PROTECTED], Tues. Sept. 3, 2002 8:26AM It also depends on the logic that one chooses (e.g., Lukaciewicz/Rational Pavelka and Product/Goguen and Godel fuzzy multivalued logics - see P. Hajek Metamathematics of Fuzzy Logics, Kluwer: Dordrecht 1998 for an excellent exposition except for his mediocre probability section).. See my contributions to http://www.superstringtheory.com/forum, especially to the String - M Theory - Duality subforum of their forum section (most of which is archived, but membership is free, and archives are accessible to members). Or my paper in B. N. Kursunuglu et al (Eds.) Quantum Gravity, Generalized Theory of Gravitation, and Superstring Theory-Based Unification, Kluwer Academic: N.Y. 2000, 89-97, which has some further references to my earlier work. Analysis including nonsmooth analysis does combine the discrete and the connected/continuous, but in my opinion it generally regards the discrete as an approximation to the continuous/connected or piecewise continuous/piecewise connected (pathwise, etc.). One confusing point, I think, is the tendency of many mathematical logicians to identify with algebra and in fact to claim that their field is a branch or outgrowth of algebra. This was originally claimed by *Clifford Algebra,* but Clifford himself and many of his wisest descendants/followers such as Hestenes of Arizona State U. realized than the opposite true - *Clifford Analysis,* *Spacetime Algebra,* and so on are typical terminology used by the latter and others to indicate that they are really dealing with analysis and geometry and related things. Why do so many mathematical logicians identify with algebra?Largely, in my opinion, because algebra is much more mainstream-accepted than mathematical logic (and popular, and respected, etc.), but also because algebra is abstract and mathematical logic seems to many of its practitioners to be more abstract than concrete. I have cautioned in various places that even in pure mathematics there needs to be a balance between abstractness/abstraction and concreteness/physical application. Analysis historically has had much more of this balance (rough equality of abstraction and concreteness). There are also many built-in biases in mathematical and theoretical physics, and one of them in my opinion is the bias toward dissolution of geometry at the sub-Planck level.Part of this is the pre-quantum computer bias toward the discrete and finite or at most countably infinite and the digital vs analog computer bias (in favor of digital computers). The real line and real line segments of course are uncountably infinite and connected, and thee would essentially be no applied mathematics or mathematical physics for example without it - and not much pure mathematics either.It helps to occasionally look back in mathematical history, especially to Georg Cantor's Contribution to the Theory of Transfinite Numbers, which even Birkhoff and MacLane in their algebra textbooks made sure to include. Of course, Birkhoff ended up in applied differential equations and hydrodynamics largely, but MacLane has never been accused of being Analysis-inclined to my knowledge, and Birkhoff started out at least algebraic. I hope that we can resist the temptation to go into absolutes. I am glad to see that you started your reply with a tolerant and compromising tone, and I will end this posting with a similar tone. The discrete and the connected are in my opinion different theories or parts of different theories overall, and they are also parts of different interpretations. My view is that science progresses by tolerating different theories and different interpretations for competition and because many supposedly wrong theories or interpretations end up much later having something useful to contribute. The majority of scientists (the *Mainstream* as I refer to them) do not subscribe to this view, but consider that science advances in a spiral by *killing off* the wrong theories or by only generalizing (including generalizing in the limit) the partly correct theories - the Law of the Jungle viewpoint of competition by (intellectual) warfare or *cannibalistic* absorbtion as opposed to the Competing Teams idea of competition in which one keeps other teams alive in order to keep competing and for motivation and ultimately because one respects them and regards them as like oneself trying to achieve the *Impossible Dream*. You may find my contributions to math-history (see the Math Forum and epigone sites) to be interesting in this regard. Osher Doctorow - Original Message - From: "Tim May" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Monday, September 02, 2002 11:07 PM Subject: Re: Time as a Lattice of Partially-Ordered Causal Events or Moments > > On Monday, September 2, 2002, at 09:22 PM, Osher Doctorow wrote: > > > From: Osher Doctorow [EMAIL PROTECTED
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
On Monday, September 2, 2002, at 09:22 PM, Osher Doctorow wrote: > From: Osher Doctorow [EMAIL PROTECTED], Mon. Sept. 2, 2002 9:29PM > > It is good to hear from a lattice theorist and algebraist, although I > myself > prefer continuity and connectedness (Analysis - real, complex, > functional, > nonsmooth, and their outgrowths probability-statistics and > differential and > integral and integrodifferential equations; and Geometry). > Hopefully, we > can live together in peace, although Smolin and Ashtekar have been > obtaining > results from their approaches which emphasize discreteness (in my > opinion > built in to their theories) and so there will probably be quite a > battle in > this respect at least intellectually. I'm not set one way or the other about discreteness, especially as the level of quantization is at Planck length scales, presumably. That is, 10^-34 cm or so. Maybe even smaller. And the Planck time is on the order of 10^-43 second. One reason discrete space and time isn't ipso facto absurd is that we really have no good reason to believe that smooth manifolds are any more plausible. We have no evidence at all that either space or time is infinitely divisible, infinitely smooth. In fact, such infinities have begun to seem stranger to me than some form of loops or lattice points at small enough scales. Why, we should ask, is the continuum abstraction any more plausible than discrete sets? Because the sand on a beach looks "smooth"? (Until one looks closer.) Because grains of sand have little pieces of quartz which are smooth? (Until one looks closer.) But, more importantly, the causal set (or causal lattice) way of looking at things applies at vastly larger scales, having nothing whatsoever to do with the ultimate granularity or smoothness of space and time. That is, a set of events, occurrences, collisions, clock ticks, etc. forms a causal lattice. This is true at the scale of microcircuits as well as in human affairs (though there we get the usual "interpretational" issues of causality, discussed by Judea Pearl at length in his book "Causality"). You say you prefer continuity and connectednessthis all depends on the topology one chooses. In the microcircuit case, the natural topology of circuit elements and conductors and clock ticks gives us our lattice points. In other examples, set containment gives us a natural poset, without "points." (In fact, of course mathematics can be done with open sets, or closed sets for that matter, as the "atoms" of the universe, with no reference to points, and certainly not to Hausdorff spaces similar to the real number continuum.) The really interesting things, for me, are the points of intersection between logic and geometry. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
From: Osher Doctorow [EMAIL PROTECTED], Mon. Sept. 2, 2002 9:29PM It is good to hear from a lattice theorist and algebraist, although I myself prefer continuity and connectedness (Analysis - real, complex, functional, nonsmooth, and their outgrowths probability-statistics and differential and integral and integrodifferential equations; and Geometry). Hopefully, we can live together in peace, although Smolin and Ashtekar have been obtaining results from their approaches which emphasize discreteness (in my opinion built in to their theories) and so there will probably be quite a battle in this respect at least intellectually. Osher Doctorow - Original Message - From: "Tim May" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Monday, September 02, 2002 8:32 PM Subject: Time as a Lattice of Partially-Ordered Causal Events or Moments > > On Saturday, August 31, 2002, at 11:31 PM, Brent Meeker wrote: > > > Time is a construct we invented to describe things. Most > > basically we use it to describe our sequence of experiences > > and memories. We feel hot and cold, but we needed to > > quantify hot and cold and give them operational definitions > > in order make definite predictions about them. So we > > invented temperature and thermometers. For mechanics we > > needed a quantified, operational definition of duration - > > so we invented time and clocks. > > > > Besides psychological time,there are at least three > > different possible definitions of time used in physics What > > they all have in common is that they assign numbers to > > different physical states, i.e. they index different states > > into some order so that this sequence of states can be > > compared to that sequence of states. > > I don't have a comprehensive theory of time, but I am very fond of > "causal time." > > Picture events as a series of points in a lattice (a graph, but with > the properties I talked about a while back in a post on > partially-ordered sets). Basically, a lattice of events where there is > at most one edge connecting two points. (There are formal properties of > lattices, which the Web will produce many good definitions and pictures > of.) > > Lattices capture some important properties of time: > > * Invariance under Lorentzian transformations...any events A and B > where B is in the future light cone of A and A is in the past light > cone of B, will be invariantly ordered to all observers. > > * The modal logic nature of time. Multiple "futures" are possible, but > once they have happened, honest observers will agree about what > happened. (Echoing the transformation of a Heyting algebra of > possibilities into the Boolean algebra of actuals...this sounds like it > parallels quantum theory, and Chris Isham and others think so.) > > * Personally, I believe the arrow of time comes from more than > statistical mechanics. (I believe it comes from the nature of subobject > classifiers and the transformation Heyting --> Boolean.) > > > * I am indebted to the books and papers of Lee Smolin, Fotini > Markopoulou, Louis Crane, Chris Isham, and several others (Rovelli, > Baez, etc.) for this interpretation. > > None of us knows at this time if time is actually a lattice at Planck- > or shorter-time-scale intervals. But discretized at even the normal > scales of events (roughly the order of seconds for human-scale events, > picoseconds or less for particle physics-scale events), the > lattice-algebraic model has much to offer. > > * I don't see any conflict with Huw Price, Julian Barbour, and others > (haven't read Zeh yet), though I don't subscribe to all of their > idiosyncratic views. > > > --Tim May >
Time as a Lattice of Partially-Ordered Causal Events or Moments
On Saturday, August 31, 2002, at 11:31 PM, Brent Meeker wrote: > Time is a construct we invented to describe things. Most > basically we use it to describe our sequence of experiences > and memories. We feel hot and cold, but we needed to > quantify hot and cold and give them operational definitions > in order make definite predictions about them. So we > invented temperature and thermometers. For mechanics we > needed a quantified, operational definition of duration - > so we invented time and clocks. > > Besides psychological time,there are at least three > different possible definitions of time used in physics What > they all have in common is that they assign numbers to > different physical states, i.e. they index different states > into some order so that this sequence of states can be > compared to that sequence of states. I don't have a comprehensive theory of time, but I am very fond of "causal time." Picture events as a series of points in a lattice (a graph, but with the properties I talked about a while back in a post on partially-ordered sets). Basically, a lattice of events where there is at most one edge connecting two points. (There are formal properties of lattices, which the Web will produce many good definitions and pictures of.) Lattices capture some important properties of time: * Invariance under Lorentzian transformations...any events A and B where B is in the future light cone of A and A is in the past light cone of B, will be invariantly ordered to all observers. * The modal logic nature of time. Multiple "futures" are possible, but once they have happened, honest observers will agree about what happened. (Echoing the transformation of a Heyting algebra of possibilities into the Boolean algebra of actuals...this sounds like it parallels quantum theory, and Chris Isham and others think so.) * Personally, I believe the arrow of time comes from more than statistical mechanics. (I believe it comes from the nature of subobject classifiers and the transformation Heyting --> Boolean.) * I am indebted to the books and papers of Lee Smolin, Fotini Markopoulou, Louis Crane, Chris Isham, and several others (Rovelli, Baez, etc.) for this interpretation. None of us knows at this time if time is actually a lattice at Planck- or shorter-time-scale intervals. But discretized at even the normal scales of events (roughly the order of seconds for human-scale events, picoseconds or less for particle physics-scale events), the lattice-algebraic model has much to offer. * I don't see any conflict with Huw Price, Julian Barbour, and others (haven't read Zeh yet), though I don't subscribe to all of their idiosyncratic views. --Tim May
