On 7/23/2011 11:25 PM, Jesse Mazer wrote:


On Sat, Jul 23, 2011 at 8:45 PM, Stephen P. King <stephe...@charter.net <mailto:stephe...@charter.net>> wrote:

    Hi Jesse,


    On 7/22/2011 8:03 PM, Jesse Mazer wrote:


    On Fri, Jul 22, 2011 at 4:54 PM, Stephen P. King
    <stephe...@charter.net <mailto:stephe...@charter.net>> wrote:


        Hi Jason,

            None of those papers address the concern of narratability
        that I am considering. In fact they all assume narratability.
        I am pointing out that thinking of time as a dimension has a
        big problem! It only works if all the events in time are
        pre-specifiable. This also involves strong determinism which
        is ruled out by QM. See
        http://plato.stanford.edu/entries/determinism-causal/#StaDetPhyThe
        for a general overview



    But the link notes that strong determinism is *not* ruled out by QM:

    "So goes the story; but like much popular wisdom, it is partly
    mistaken and/or misleading. Ironically, quantum mechanics is one
    of the best prospects for a genuinely deterministic theory in
    modern times! Even more than in the case of GTR and the hole
    argument, everything hinges on what interpretational and
    philosophical decisions one adopts. The fundamental law at the
    heart of non-relativistic QM is the Schrödinger equation. The
    evolution of a wavefunction describing a physical system under
    this equation is normally taken to be perfectly deterministic.[7]
    If one adopts an interpretation of QM according to which that's
    it—i.e., nothing ever interrupts Schrödinger evolution, and the
    wavefunctions governed by the equation tell the complete physical
    story—then quantum mechanics is a perfectly deterministic theory.
    There are several interpretations that physicists and
    philosophers have given of QM which go this way. "

    The many-worlds interpretation, which many on this list are
    presumably sympathetic to, is an example of a deterministic
    interpretation of QM. In fact many-worlds advocates often argue
    that not only is it deterministic, but it's also a purely local
    interpretation, which doesn't violate Bell's theorem because the
    theorem makes the assumption that each measurement yields a
    single unique result, something that wouldn't be true in the
    many-worlds interpretation. For more on how MWI can be local, see
    these papers:

    http://arxiv.org/abs/quant-ph/0103079
    http://arxiv.org/abs/quant-ph/0204024


    Umm, did you notice the words "non-relativistic" in the paragraph?



I think that's only because they're talking about the Schrodinger equation, but relativistic QM (quantum field theory) also has its own deterministic evolution, look at the second of the two papers I linked to which is specifically about relativistic field theory, specifically at p. 3.

     Even MWI does not help because in it one is taking constructions
    that exist in Hilbert space and assuming that they are ordered a
    priori.


Not if you're talking about MWI of quantum field theory, again see the paper I linked to.


            The idea that time is a dimension assumes that the events
        making up the points of the dimension are not only isomorphic
        to the positive Reals but also somehow can freely borrow the
        well order of the reals.


    Not sure what you mean by this, events at a spacelike separation
    aren't "well-ordered" in time, are they? Only if one event is in
    the light cone of the other (a timelike or lightlike separation)
    will all frames agree on the time-ordering, that's just a
    consequence of the relativity of simultaneity.


    Sure, let us focus on the space-like surfaces. Does a unique order
    of them exist? No. This is the foliation problem that I mentioned
    below.


Depends what sort of foliation and what spacetime you're using, if you have a "globally hyperbolic" spacetime (which excludes certain weird conditions like closed timelike curves, see http://en.wikipedia.org/wiki/Globally_hyperbolic_manifold ) then it should be possible to introduce a global time function such that the time parameter increases continuously along every timelike curve, see p. 18 of http://www.uco.es/geometria/documentos/OMuller.pdf for example. Naturally this time function corresponds to a foliation, with each hypersurface corresponding to a set of points that all have the same value of the time parameter. There will be many different such time functions and thus many different foliations with different definitions of simultaneity, but for any *specific* set of space-like surfaces of this nature, it seems to me that a unique order of them does exist. As for non globally hyperbolic spacetimes, I think it's an open question whether they will even be possible in a theory of quantum gravity (there is much speculation that quantum gravity will rule out the possibility of closed timelike curves, see for example Kip Thorne's discussion of the "chronology protection conjecture" towards the end of http://plus.maths.org/content/time-travel-allowed ).


    Also, that situation where "all frames agree on the time-order"
    assumes that the ordering already exists. I am asking about how it
    got there in the first place.


The choice about which direction along a single timelike curve to label as "increasing proper time" and which to label "decreasing proper time" is arbitrary thanks to the time-symmetry of general relativity (see http://en.wikipedia.org/wiki/T-symmetry ) but once you have picked a convention, in a globally hyperbolic spacetime I believe this would uniquely determine the two directions along every other timelike curve. See the concept of "time-orientability" at http://en.wikipedia.org/wiki/Causal_structure#Time-orientability and p. 2 of http://arxiv.org/pdf/gr-qc/9704075 which says a globally hyperbolic manifold is by definition time-orientable: "Here we recall (see e.g. [6, 7]) that a spacetime is said to be globally hyperbolic if it is time-orientable and has a Cauchy surface – i.e. an achronal spacelike hypersurface which is intersected exactly once by every inextendible
causal curve."


        The block universe idea assumes a unique and global ordering
        of events, the actual math of SR and GR do not!


    Why do you think the block universe idea assumes a unique
    ordering? It doesn't, not for pairs of events with a spacelike
    separation. For such events, the question of which event occurs
    at a later time depends entirely on what coordinate system you
    use, with no coordinate system being preferred over any other.
    Similarly, on a 2D plane the question of which of two points has
    a greater x-coordinate depends entirely on how you orient your x
    and y coordinate axes, even if you restrict yourself to Cartesian
    coordinate systems. And the whole idea of block time is that time
    is treated as a dimension analogous to space, so it's not
    surprising that there could be situations where different
    coordinate systems disagree about which of two events has a
    greater t-coordinate, with no coordinate system's answer being
    more "correct" than any other's.

    So the block universe does or does not assume a "time dimension"?
    If it does then that "time dimension" is equivalent to a unique
    ordering of events such that events are labeled with values in the
    positive Real numbers, other wise known as the Real line.


No, having a "time dimension" does not imply a unique ordering, that's a non sequitur. It just means there is a clear distinction between curves which are "timelike" and those which are "spacelike" (or "lightlike").

    Consider the famous words of Laplace:

    "We may regard the present state of the universe as the effect of
    its past and the cause of its future. An intellect which at a
    certain moment would know all forces that set nature in motion,
    and all positions of all items of which nature is composed, if
    this intellect were also vast enough to submit these data to
    analysis, it would embrace in a single formula the movements of
    the greatest bodies of the universe and those of the tiniest atom;
    for such an intellect nothing would be uncertain and the future
    just like the past would be present before its eyes."
    —Pierre Simon Laplace, /A Philosophical Essay on Probabilities/

        This is the block universe idea. Given that we now known, per
    QM, that the positions, momenta and other observables cannot be
    simultaneously given for 'all items' in the universe, how can we
    still think that the universe just exists as a fixed and eternal
    3,1 dimensional 'block'?



Not sure why you think not being able to assign unique values of position and momentum is in any way essential to the block universe idea. In the many-worlds interpretation of relativistic quantum field theory, the local "state" of each point in spacetime wouldn't be about position and momentum, it would be about field operators as discussed on p. 3 of http://arxiv.org/abs/quant-ph/0204024 and as I mentioned the values of these field operators is said to evolve in a local and deterministic way. So Laplace's quote would still be valid if you just replaced his talk of "positions" and "forces" with talk of the values of local field operators in the many-worlds interpretation.


            My claim is that the idea that time is a quantity like
        space only works in the conceptual sense where we are
        assuming that all events are chained together into continuous
        world lines.



    Not really, just as you can have a collection of points on a 2D
    plane without continuous lines joining them, so you could
    potentially have a collection of events in spacetime that are
    causally related but don't have a continuous series of similar
    events between them. Sort of like if you took vertices on a
    Feynman diagram to be events, and understood the lines joining
    them to just express causal relationships, not worldlines.

    No, that requires that a basis be chosen for that particular
    Feynmann diagram.


I was just imagining something "sort of like" a Feynman diagram, my point was to counter your very broad claim that time can't be a dimension like space unless events are chained together by continuous worldlines. To counter this claim it would be sufficient to come up with a toy model of a hypothetical set of physical laws (that need not resemble our own) which involve discrete collections of events in spacetime that aren't connected by continuous worldline, that's what I was getting at with the "sort of like" comment. That said, I think loop quantum gravity does propose something like a set of discrete events with causal connections between them, not continuous worldlines.

         It is impossible to define a unique Cauchy hyper-surface of
        initial (final) data that completely determines all of the
        world lines in the space-time block in a way that is
        consistent with QM.


    What specific source are you getting that claim from? I checked
    the first link you posted after it:

    I was trying to not write a book.... What is a Cauchy surface?
    http://en.wikipedia.org/wiki/Cauchy_surface

    "a plane in space-time <http://en.wikipedia.org/wiki/Space-time>
    which is like an /instant/ of time; its significance is that
    giving the initial conditions
    <http://en.wikipedia.org/wiki/Initial_conditions> on this plane
    determines the future (and the past) uniquely.

    More precisely, a Cauchy surface
    <http://en.wikipedia.org/wiki/Hypersurface> is any subset of
    space-time which is intersected by every non-spacelike
    <http://en.wikipedia.org/wiki/Spacelike>, inextensible curve
    <http://en.wikipedia.org/wiki/Curve>, i.e. any causal curve
    <http://en.wikipedia.org/wiki/Causal_curve>, exactly once."

    Since we know from QM that the values of the initial conditions
    are subject to the Uncertainty principle


No, the uncertainty principle doesn't apply to the evolution of quantum states, it's just about how those quantum states are projected onto basis vectors of different observables. So if you don't treat "measurements" as randomly collapsing the quantum state onto one of the basis vectors of the observable being measured (this is something called the "Born rule" or "projection postulate", see my post #2 on the thread at http://www.physicsforums.com/showthread.php?t=490919 ), as MWI advocates do not, there is no conflict with determinism here.


    Wave functions do not exist 'in' spacetime.


As stated at http://arxiv.org/abs/quant-ph/0204024 in quantum field theory you can replace the notion of a wave function with a physically equivalent description of a set of local field operators at each point in spacetime.

    Additionally, the diffeomorphism invariance (also known as general
    covariance) of GR does not allow any particular meaning to "a
    point in space"


But you can assign a physical meaning to "a point in spacetime", diffeomorphism invariance is only about the possibility of using different coordinate systems on the selfsame spacetime geometry.


    , so the idea that 'a time-evolution is determined by local
    differential equations..." fails because the mapping that one has
    to use to identify a particular set of field operators with each
    point of spacetime is not invariant with respect to diffeomorphisms.


The value of the field operators might depend on the coordinate system, but there is a unique truth about which point in spacetime as described in one coordinate system constitutes the "same event" as which point in a different coordinate system, and I presume in the MWI you could derive some coordinate-invariant facts about what's happening at a given point using the field operators in whatever coordinate system you chose, perhaps something like the ratio of different "Everett copies" of measurement outcomes at that point as described on p. 2 of http://arxiv.org/pdf/quant-ph/0310186

This is similar to how in GR, although the value of the curvature tensor at each point depends on your choice of coordinate system, there is a coordinate-invariant spacetime geometry in the sense that if you pick a continuous series of points making a curve, all coordinate systems agree on the "length" of that curve (which can be found using the correct metric equation for each coordinate system).

Jesse

--
Hi Jesse,

We seem to be talking past each other. I am thinking about the notion of time as a dimension and its origin and implications. You seem to just assume its existence. I ask "why?".

Onward!

Stephen

--
You received this message because you are subscribed to the Google Groups 
"Everything List" group.
To post to this group, send email to everything-list@googlegroups.com.
To unsubscribe from this group, send email to 
everything-list+unsubscr...@googlegroups.com.
For more options, visit this group at 
http://groups.google.com/group/everything-list?hl=en.

Reply via email to