Does the Holographic Principle Need a Quantum Theory of Gravity?
If anyone was to select the most revolutionary ideas in 20th century
physics the Holographic principle would be very high on the list. As it turns
out, based on String -M theory, gravity can only be understood in Holographic
terms. Even many models of gravity, not strictly based on String -M theory
reformulate gravity in terms of the Holographic principle. Most
interestingly, Sakharov's Induced Gravity proposal becomes much better justified in
terms of the Holographic principle , whether in String -M theory or the
various thermodynamic models of Gravity, Verlinde, Jacobson etc. Based on this
we might argue that the answer to the question "Does a Quantum theory of
Gravity Need the Holographic principle would be yes.
However if we turn this question around by asking, does the Holographic
principle need a Quantum Theory of Gravity the surprising answer is no, at
least in a qualified sense. More specifically, we can ask the question,
must space and time be discrete in order to establish information bounds
inside a given volume of space. I will argue the answer is no.
This makes particular sense in terms of String -M theory which does not
quantize space and time, despite numerous claims to the contrary. The actual
nature of space time in String -M theory is still not defined , but does
appear to be an emergent property of the Universe , or in deference to Vic,
not a useful Human invention at the most fundamental scale i.e.. the
Transplanckian energy regime. In String theory we find that we must reformulate
Einstein's equations for gravity as equations with infinite perturbations.
This fact BTW allows String -M theory to model gravity as an Induced force
because it evades the Weinberg Witten theorem.
But even more important , the fundamental duality of String-M theory, the
Anti De Sitter space /conformal field duality describe gravity as an
emergent property due to the duality of degrees of freedom between a bounded
volume D and its D-1 boundary.
I think the answers offered for the above questions can be justified by a
simple calculation, first proposed by Dr Vic Stenger in his book " Has
Science Found God, The Latest Results For the Search for Purpose in the
Universe." This calculation is in Appendix C , "The Entropy of The Expanding
Universe." I modify the calculation slightly and fudge a few constants, but it
nonetheless still illustrates the basic principle I am asserting.
Given any system we can define its entropy by:
S= k*ln[ N]
Where N is the number of micro states that produce the same macro state.
If we describe N is terms of Quantum fields we can write
N= D^M
Where D are the degrees of freedom and M are the number of maximum
number of quanta.
This gives us
S= M*k*ln[D]
We can take
k*ln[D] on the order of unity
So that we have
S= M
M can be defined as;
M= E/E_min
Where E is the total energy of the system and E_min is the lowest energy
quanta possible.
Next we define this system in terms of a duality between the degrees of
freedom on the two dimensional boundary and the three dimensional volume.
This gives us
Lambda_min= 2*pi*R= hbar*c/E_min
Which gives us
S=E/E_min= 2*pi*R*E/(hbar*c)
Which is the well known Bekenstein Bound.
Given that for a black Hole we have;
E= R*c^4/(2*G)
We get
S= {2*piR/hbar*c)} *{R*c^4/(2*G)}
S= pi*R^2*c^3/hbar*G
S= A/4*L_plk^2}
Where L_plk is the Planck length.
This is, of course the Holographic equation.
Note that no where was space or time quantized in this calculation and no
assumptions made as to the scale of information quantization. The finite
bound on information is strictly the result of the quantization of the Quantum
fields. I think this illustrates that we need not quantize space or time
in order to avoid a divergent entropy in any given volume of space contrary
to the assertions of some.