Glenn Morton writes, "I would agree that Nietzche's assumption would be
violated by an infinite universe. But Nietzche's assumption was that there was
a finite amount of energy, which would imply a finite space. As I recall,
Minkowski space is flat and thus not capable of a boundary. Is this correct?
It may be that Nietzche knew that it is easier to limit energy rather than
space and thus hid his problem inside of his assumption. Do any philosophers
on this list know if this would be the case?"
A finite amount of energy does not imply a finite space. In the
absence of gravity, it could be Minkowski spacetime, which by definition is
flat AND infinite in all directions (both spatially and temporally). With
gravity, it could be a small distortion of Minkowski spacetime, but still
infinite in all directions and asymptotically flat at large spatial distances
from the matter. (Generically the matter would eventually disperse, so the
spacetime would also presumably be asymptotically flat at sufficiently large
past and future times as well.)
However, flatness does not imply no boundary, for one could imagine a
flat spacetime that had a boundary or edge. E.g., it might exist only for t >
0 (so that one might say it was created at t = 0, though I would prefer to say
it is the whole spacetime that is created, and not just its beginning), or it
might exist only for t < 0 (so that one might say it is destroyed at t = 0), or
it might exist only for 0 < x < 1, 0 < y < 1, 0 < z < 1, so that it has edges
in space. Also, flat spacetime could be finite without boundary, say if it
were a four-torus so that if one imagined going in the x, y, or z directions at
infinite speed (physically impossible), one would return to where one started
after a finite didtance, and if one went forward in time t, one would also
return after a finite time to the spacetime event at which one started. (I.e.,
this universe, finite in all directions, would be periodic in time as well as
space.)
When one adds matter and classical (i.e., not quantized) gravity, it is
very difficult to have a universe that is finite in all spacetime directions
and yet has no edge anywhere. One can easily imagine one periodic in space, or
forming a three-dimensional sphere, so that spatially it is finite, but then
gravity tends to pull things together so that there is an edge in time, at a
big bang in the past and/or at a big crunch in the future.
Thus Nietzsche's assumptions would have been consistent with (part of)
physics as we know it if he had assumed that space were finite (so that there
would not be an infinite phase space even with finite energy) and if gravity
did not exist, but with (classical) gravity, such a finite universe generically
would have a big bang and/or big crunch, when gravity pulls things together
into a singularity or edge of spacetime, where the curvature usually goes to
infinity, so that one generically cannot get a classical universe periodic in
time when gravity is present.
Actually, I realize one could also gets Nietzsche's requirement of a
finite phase space (when the energy is limited) without finite space if one had
particles rather than fields, and if the particles had forces between them that
did not permit them from being pulled indefinitely far apart with the available
finite energy, and if the system could not radiate away any energy in fields.
It might be more plausible that Nietzshe assumed something like that, rather
than assuming that space is finite (Does anyone know?), but now we believe that
also is pretty unrealistic; virtually everything we know of can decay (except
massless particles, such as photons, gravitons, and maybe neutrinos, and the
lightest particles with a certain conserved charge, such as electrons and
positrons and presumably magnetic monopoles and the lightest supersymmetric
particles, if they exist) or disperse (everything that we know of; even a black
hole can disperse its energy by Hawking radiation).
George Murphy noted, "Bear in mind that, as Tolman showed >60 years
ago, a strictly cyclic universe is impossible if there are dissipative
processes."
However, if there were no gravity, and if the universe could have a
static space of finite volume or some other way of getting a finite phase space
that continued forever, one would have at least an approximate repetition after
the Poincare recurrence time (which is crudely referred to in the saying,
"Until the cows come home," and is in practice extremely long), and over such
enormous times dissipation does not continue, and the second law of
thermodynamics breaks down.
Glenn writes further, "After a couple of readings I think I finally
understand what you are suggesting. Why is it not more likely that the
universe has to be described as a superposition of the states of the component
particles rather than having the entire mass of the univers be in a single
state. I am expressing this poorly. let me try again. The second suggestion
appears to require the many-world's hypothesis. Our universe be in one state,
but other universes are in another. It seems to me that since we can only
really observe one universe at the moment, that the eigenstate should be the
superposition of the states of the individual particles. I don't see how
Christianity could survive the confirmation of the many-world's hypothesis.
Furthermore in describing a system of particles don't we treat the system as
the superposition of the individual particle states? Why wouldn't the
universe be the same?"
No, in quantum theory the state of a system is not treated as a
superposition of the quantum states of its parts, and neither is it usually a
product of the states of the parts. The quantum state of a composite system
having many component parts (e.g., different particles, or different field
modes) can be given by a wavefunction which is a function of the configurations
of all of the component parts (technically a functional when the component
parts themeselves form a field configuration, which is itself a function, or
set of functions, over space). If the states of the components are
uncorrelated in the quantum sense (i.e., not entangled, to use a word common in
quantum theory), then the wavefunction is a product of wavefunctions for each
of the component parts, but this is a very special case, and in general the
wavefunction for the whole system is not just a product of wavefunctions of the
parts. This implies that there usually is more information in the whole than
the sum of the information about each part separately.
For example, take two coupled harmonic oscillators, with position
variable x for the first and y for the second. The ground state wavefunction
is proportional to exp(- a x^2 - b y^2 - 2c x y) with appropriate constants a,
b, and c. If the coupling is zero, then the ground state has c = 0, and the
wavefunction is a product of wavefunctions for each oscillator, proportional to
exp(- a x^2)exp(- b y^2), but if c is nonzero, the wavefunction is not a
product, and the quantum state of the two oscillators is said to be entangled.
Even though Glenn later retracted his statement that "I don't see how
Christianity could survive the confirmation of the many-world's hypothesis," I
would like to comment (because I do believe in something like many worlds as
the simplest version of quantum theory, since it avoids all fundamental
randomness in the theory) on what he said in an intermediate posting, "The
problem I have with the many-world hypothesis is that it would have a universe
(ours) in which I am a Christian. There would be others in which I am a
Christian. But there would be still others in which I decided to become an
atheist during my crisis of faith a few years ago. Thus, in the multiverse (the
existence of all universes) there is are many non-christian Glenn's and many
Christian Glenn's. What then does redemption mean? I don't know the answer."
I have thought of this before, and once Jim Hartle also pointed it out
to me. I even thought of paraphrasing Matt. 22: 23-29 thus:
The same day atheistic quantum theorists came to Him, who say that
there is no resurrection, and they asked Him a question, saying, "Teacher,
Everett said, `If quantum theory allows two possibilities, both will occur.'
Now there was a man who was faced with the choice of accepting You. Quantum
theory predicted that there would be one set of worlds in which he would choose
Heaven by accepting You, and another set of worlds in which he would choose
Hell by rejecting You. Then in all these Everett worlds he died. In the
resurrection, therefore, will he go to Heaven or to Hell? For he chose both."
But Jesus answered them, "You are wrong, because you know neither the
scriptures nor the power of God."
Unfortunately, I don't know how to continue the parable. Can anyone
help me? My guess might be that the versions of the man that have a memory of
accepting Christ go to Heaven, and the versions of the man in the other worlds
that have a memory of rejecting Christ go to Hell. If there is no problem with
different people going to different final destinations, I would see no problem
with different versions of what had started out to be the same person going to
different final destinations. It is somewhat counter-intuitive, but I don't
regard that as evidence against many worlds. As Euan Squires (whom I learned
in his obituary was a devoted Christian) noted in _The Mystery of the Quantum
World_, 2nd edition (Institute of Physics Publishing, Bristol and Philadelphia,
1994), p. 72, "It is probably fair to say that much of the `unease' that most
of feel with the Everett interpretation comes from our belief, which we hold
without any evidence, that our future will be unique."
Finally, Glenn wrote, "Would the name Sisyphus mean anything?
Eternally condemned by Zeus to push the rock uphill each day only to have it
fall back. What sort of purpose would that serve? That is why I think the
eternal return would lead to a purposeless universe."
I would think that a major part of the horror of Sisyphus is that he
presumably would remember more and more times of trying to get the rock up, so
no doubt he would be getting more and more frustrated (and maybe bored). But
if the whole universe repeated, including his memories, he wouldn't know that
he was condemned to try forever, and then I don't see how it would be any worse
than having to push the rock up once and have no other future experiences
(which also wouldn't be that great). In other words, I think it would matter
more what occurred within the universe, rather than whether or not it repeated.
I think I mostly agree with the postings of Andre Bressan, Keith
Miller, Gladwin Joseph, Jim Taggart, Ted Davis, George Andrews, and Loren
Haarsma on this subject, at least if I understand them, so I won't comment in
detail on them right now, though I might say in response to Haarsma's that I do
suspect that our decisions are indeed strongly influenced by quantum mechanics
and would be different in different Everett worlds.
Best wishes,
Don Page