On Wed, 22 Sep 2004, Daniel Elander wrote:

While it would indeed be very interesting for a theory to, in
principle, allow the number of dimensions to vary and have 4 come out
as a prediction, I think it is pretty unreasonable to claim that if
the number of dimensions is instead fixed to 4, this is an assumption
with *no experimental justification*!

Dear Daniel, I understand where you're coming from. ;-) Nevertheless if we
talk about the Planckian physics - and loop quantum gravity tries to - I
think that you are not quite right. Theories with extra dimensions *can*
agree with all observed phenomena much like the 4D theories - in fact they
agree better than the simple 4-dimensional GR; the latter cannot be
quantized.

The only reason that can lead someone to say that she prefers 4 dimensions
as the universal answer is a notion of "simplicity". While simplicity is
good, I am not the only one who is convinced that it is not really the
exact key idea that can be applied in physics. At the end, Nature does not
care how much paper you need to understand a physical system. In physics,
we have something similar to simplicity, but not quite the same thing: it
is symmetry.

In fact, I think that Gell-Mann is the discoverer of what is called the
"totalitarian" (or alternatively, "anarchic") principle that states that
everything that is not forbidden can happen, which also means that we must
always think about the most general (equally consistent) theory that is
compatible with the same (local and global) symmetries. Compactified
higher-dimensional theories can respect all the known symmetries of the
simple 4D theories. The interesting theories are, at the end, the
constrained ones.

Therefore we must include them as possibilities (otherwise we are using
some sort of random selection process) together with purely 4D theories
(well there are no pure 4D quantum theories of gravity, but let me not
repeat this point too many times).

From this perspective, and I think that it is the most rational
application of the principles we learned from Renormalization Group and
elsewhere, having exactly four dimensions at the Planck scale is a form of
fine-tuning that does not have much justification. Of course, I don't have
any unique algorithm to calculate "how much" fine-tuning it is and what is
the probability measure for the size of the extra dimensions; very small,
Planckian dimensions do not really "exist" (or their existence is not
sharp), and large hidden dimensions require a kind of fine-tuning
themselves. Does someone think that there is a "natural" measure or
argument that determines how likely different sizes should be considered?
Such a measure probably has to know everything about the dynamics and
cosmology etc. But even if you know everything, will you have a recipe to
decide which sizes are "natural"?

In the words of Kaluza-Klein decomposition, requiring 4 dimensions at very
high energies is a constraint about which fields cannot occur. No KK tower
can occur. I don't know how strong constraint it really is, but string
theory intuition suggests that it is a significant constraint.

However what we do have are consistent theories with extra hidden
dimensions that not only can agree with the predictions of the older 4D
theories, but the lead to much more meaningful quantum results. If one
construct something equally functioning but having 4 dimensions only, that
will be interesting. However, at the present, the ensemble of quantum
theories with gravity which only has 4 dimensions contains 0
representatives, so of course my measure is dominated by theories with
extra dimensions that are more or less hidden.

Moreover, you cannot work with a circular argument. If you say that pure 4
dimensions are natural, and therefore you should study pure 4-dimensional
theory, and then you say that a pure 4-dimensional theory should be still
tried and considered a solution even if it does not work, because four
dimensions are natural - then you are a subject of a circular argument.
Circular arguments are great if it least one of the elements in the circle
works - then it guarantees that all of them work and they often support
each other. But in this case, none of the statements has an independent
justification ("meat"), and therefore it is a vacuous unjustified circular
argument.

Independently, the quantum theories of gravity in purely 4 dimensions
simply do not work.
__________________________________________________ ____________________________
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Lubos Motl wrote in message ...
On Wed, 22 Sep 2004, Daniel Elander wrote:

Hello all, this is my first post your new NG, (please be kind).
Regarding the post below, (no I'm really not top posting),
is there a general concensus on the definition of "dimension"?

I have freedom to move in x,y,z but not in t, indeed we
can only occupy a near point in t. I have read that we may
regard ourselves in a 4D space, but moving at "c" parallel
to the 4th and *Lorentz* contracts that 4th spatial dimension
to nil, limiting our freedom to excersize movement in that
4th spatial dimension, which we designate time.

While it would indeed be very interesting for a theory to, in
principle, allow the number of dimensions to vary and have 4 come out
as a prediction,

Pardon the dated reference, Dover's "Principle of Relativity",
see Weyl's "Gravitation and Electricity", pg 211, following his
Eq. (14), I quote, "the integral has significance only when the
number of dimensions n=4."

I think it is pretty unreasonable to claim that if
the number of dimensions is instead fixed to 4, this is an assumption
with *no experimental justification*!

Dear Daniel, I understand where you're coming from. ;-) Nevertheless if we
talk about the Planckian physics - and loop quantum gravity tries to - I
think that you are not quite right. Theories with extra dimensions *can*
agree with all observed phenomena much like the 4D theories - in fact they
agree better than the simple 4-dimensional GR; the latter cannot be
quantized.

The 4 dimensional GR is nonorthogonal. The nonorthogonal effect
on spacetime may be described by KK's (Kaluza...) finite 5th
dimension embedded onto a Minkowski metric which basically
corresponds to a charge in units of length. That same finite
dimension may be created by a 4D curl field that Einstein
employed when using nonsymmetrical metrics.
For example part of his g_14 = - g_41, call that part a_14,
and produces a unit vector perpendicular to orthogonal
spacetime.

The only reason that can lead someone to say that she prefers 4 dimensions
as the universal answer is a notion of "simplicity". While simplicity is
good, I am not the only one who is convinced that it is not really the
exact key idea that can be applied in physics. At the end, Nature does not
care how much paper you need to understand a physical system. In physics,
we have something similar to simplicity, but not quite the same thing: it
is symmetry.

In fact, I think that Gell-Mann is the discoverer of what is called the
"totalitarian" (or alternatively, "anarchic") principle that states that
everything that is not forbidden can happen, which also means that we must
always think about the most general (equally consistent) theory that is
compatible with the same (local and global) symmetries. Compactified
higher-dimensional theories can respect all the known symmetries of the
simple 4D theories. The interesting theories are, at the end, the
constrained ones.

But we need to deliver to experimentalists predictions
in 4D, additional dimensions seem to be operators.

Therefore we must include them as possibilities (otherwise we are using
some sort of random selection process) together with purely 4D theories
(well there are no pure 4D quantum theories of gravity, but let me not
repeat this point too many times).

From this perspective, and I think that it is the most rational
application of the principles we learned from Renormalization Group and
elsewhere, having exactly four dimensions at the Planck scale is a form of
fine-tuning that does not have much justification. Of course, I don't have
any unique algorithm to calculate "how much" fine-tuning it is and what is
the probability measure for the size of the extra dimensions; very small,
Planckian dimensions do not really "exist" (or their existence is not
sharp), and large hidden dimensions require a kind of fine-tuning
themselves. Does someone think that there is a "natural" measure or
argument that determines how likely different sizes should be considered?
Such a measure probably has to know everything about the dynamics and
cosmology etc. But even if you know everything, will you have a recipe to
decide which sizes are "natural"?

In the words of Kaluza-Klein decomposition, requiring 4 dimensions at very
high energies is a constraint about which fields cannot occur. No KK tower
can occur. I don't know how strong constraint it really is, but string
theory intuition suggests that it is a significant constraint.

However what we do have are consistent theories with extra hidden
dimensions that not only can agree with the predictions of the older 4D
theories, but the lead to much more meaningful quantum results. If one
construct something equally functioning but having 4 dimensions only, that
will be interesting. However, at the present, the ensemble of quantum
theories with gravity which only has 4 dimensions contains 0
representatives, so of course my measure is dominated by theories with
extra dimensions that are more or less hidden.

Moreover, you cannot work with a circular argument. If you say that pure 4
dimensions are natural, and therefore you should study pure 4-dimensional
theory, and then you say that a pure 4-dimensional theory should be still
tried and considered a solution even if it does not work, because four
dimensions are natural - then you are a subject of a circular argument.
Circular arguments are great if it least one of the elements in the circle
works - then it guarantees that all of them work and they often support
each other. But in this case, none of the statements has an independent
justification ("meat"), and therefore it is a vacuous unjustified circular
argument.
+
Independently, the quantum theories of gravity in purely 4 dimensions
simply do not work.

You may have a proof of that last +claim, that notwithstanding
there are reasonable grounds to base QT on GR in 4D.

Regards
Ken S. Tucker