On Sat, Jul 23, 2011 at 8:45 PM, Stephen P. King
<[email protected] <mailto:[email protected]>> 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
<[email protected] <mailto:[email protected]>> 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
--
