On Sat, Jul 23, 2011 at 8:45 PM, Stephen P. King <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>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

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