Abstract
I review the holographic theory of quantum space-time and cosmology, and argue that it may yield insight into the Boltzmann-Penrose question of why the universe began with low entropy. In this model, the observed low entropy initial conditions, maybe the most general initial conditions for cosmology, which avoid collapse into a dense black hole fluid phase.
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- 1.
In two dimensions, where the formalism actually leads to a well defined mathematical construction, one is forced to include processes in which the universe splits into disconnected pieces, in order to describe a unitary quantum theory. One obtains the topological expansion of string theory, which only makes sense for a very restricted class of generally covariant second Lagrangians. The observables of these well defined models have no obvious connection to local measurements on a two dimensional world sheet. Similar remarks are valid in one space-time dimension, though the freedom to choose the world-line Lagrangian is not compromised in this case.
- 2.
There is a possible modification of this suggested by string theory. We have evidence in string theory for the existence of naked singularities like orbifolds, orientifolds, conifolds, which are resolved by the quantum theory. For example, singular limits of K3 manifolds are perfectly regular string compactifications. One can certainly imagine that one can start with a classically regular string compactification on K3 and choose initial conditions for the moduli, which send them through a singular point in moduli space, in some localized region in the non-compact dimensions. It is possible, that this does not give rise to a black hole, but simply that quantum effects become important, as new light states have to be included in the effective field theory description of physics near the singularity.
- 3.
That is, the light cone Hamiltonian maybe identified with a time-like generator in an ordinary Lorentz invariant field theory.
- 4.
More precisely, we should discuss foliations of the null boundary by space-like d − 2 surfaces. The claim is that there is a unique maximal area surface which can be chosen as a leaf of a foliation.
- 5.
Actually, observables are only the hermitian elements of the algebra. We will continue to use the phrase algebra of observables, with this implicit understanding.
- 6.
And thus has a unique unitary equivalence class of representations: the algebra of N×Nmatrices in a finite dimensional Hilbert space.
- 7.
Despite the traditional bio-centric language, our observers need not be living creatures. The only requirement is that they be quantum systems with a large number of observables that have very small quantum fluctuations. Observers have neither gender nor consciousness, only approximate local fields.
- 8.
Implicit in the idea that the observer is approximately described by field theory is the assumptionthat the region of space-time in which the observer lives is approximately classical. Later on, we will use the formalism we construct to describe situations in which no such semi-classical observer could exist.
- 9.
In the universal cover of AdS, which has no closed time-like curves.
- 10.
Parenthetically, the analysis above may shed some light on the difficulties of extracting local physics from AdS/CFT. Bulk physics on scales much smaller than the AdS radius should be described by an approximate S-matrix. The connection of boundary time evolution with causal diamonds larger than the AdS radius suggests that the relation between this S-matrix and the boundary Hamiltonian might be complicated. It would be interesting to revisit the proposals of Polchinski [12] and Susskind [13] for extracting the flat space S-matrix from CFT correlators, in view of this insight.
- 11.
λ is real or complex, depending on the character of the minimal spinor representation in dimension d.
- 12.
Recall that for a general pixelation of the function algebra, the label nstands for a single element in the basis of the finite dimensional algebra.
- 13.
We do not yet specify the properties of these functions, with regard to continuity, but they should at least be measurable.
- 14.
I am not sure whether this terminology is standard for the maximal abelian subalgebra of an associative, rather than a Lie, algebra. Instead of the Cartan subalgebra, we could use the whole non-abelian von-Neumann algebra, but insist that inner automorphisms of the algebra are gauge transformations.
- 15.
As noted above we could in fact have required our algebra to be the tensor product of the full II ∞ factor with the algebra of functions on the sphere, subject to this gauge equivalence.
- 16.
This is not precise. The torsion elements of K-theory, which classify stable, non-BPS particles, do not appear in the SUSY algebra.
- 17.
An often overlooked part of this analysis is that the Hamiltonians and Hilbert spaces for different classical solutions are not related to each other in any simple way.
- 18.
It is perhaps worthwhile to point out a linguistic nicety. We have become used to using the word theoryto mean a particular mathematical model, defined by a fixed evolution operator U(t, t 0), and I am using the term in this sense. The general THEORY of quantum gravity is the broader subject of this review, which is illustrated by a variety of particular theories. It’s clear that in a sensible world we would use the word model instead of theory in phrases like “the ϕ4quantum field theory”.
- 19.
In AdS space-time this statement is modified, because the label nrepresenting the area of the causal diamond, becomes infinite for finite proper time separation between tips of the diamond. It is not clear what the infinite sequence of partial S-matrices converges to. The familiar boundary dynamics of these space-times most naturally describes what happens to causal diamonds after they hit the time-like boundary and have infinite area.
- 20.
Once the model is constructed, it will be clear that the resulting universe cannot actually have any observers in it. Nonetheless, it will be describable by the formalism we constructed to model the experience of a normal observer.
- 21.
Recall that for our purposes, observers are large quantum systems, well described by quantum field theory in a classical space-time, which have many semi-classical observables.
- 22.
Note that if w = 1, we do not need this stringent inequality and we satisfy all conditions with L(t) = L, a constant. The point is that here we simply have the p = ρ geometry everywhere and the horizon is outside the artificial sphere we have drawn.
- 23.
Initial refers to the time at which the microscopic discrete time dynamics of the DBHF is well approximated by its coarse grained description as a scale invariant p = ρ FRW universe.
- 24.
We are here assuming the normal region is radiation dominated. This is motivated by an heuristic picture in which a normal region contains a black hole smaller than the horizon size, which decays into radiation. Perhaps a better argument is that, while our definition of normal regions is meant to imply that the system in these regions is described by effective field theory, we are at a high enough energy density to expect the relevant degrees of freedom to be described by some conformal field theory.
- 25.
I am here classifying pre-Big Bang and cyclic universe scenarios among those not universally accepted.
- 26.
So that \({N}_{e} = 10 - 20\)is plausible, N e ∼ 60 − 100 seems to require some explanation, and much larger N e seems terribly unlikely.
- 27.
Note that holographic cosmology has a built in arrow of time, coming from the direction in which the number of degrees of freedom inside the horizon increases. However, in the DBHF solution, there is no thermodynamic arrow of time since the system inside the horizon is always in equilibrium.
- 28.
Inflation theorists often argue that there is no need for homogeneity over our current horizon volume, but only over a small patch of order a few times the inflationary Hubble scale. The argument deals with those degrees of freedom not describable by field theory in that initial patch (but which are so describable today) by invoking the adiabatic theorem and assuming those degrees of freedom are in their ground state. But the adiabatic theorem is only valid for very special states of large quantum systems, so this is tantamount to assuminga particularly low entropy starting point for the universe.
- 29.
Careful readers of previous work will note discrepancies between the present section and those earlier estimates. I believe the current discussion is the more accurate one.
- 30.
And the number of e-foldings of inflation, N e .
- 31.
We have neglected entropy dumps which occur later than the primordial reheating of the universe.
- 32.
Recall that we are trying to find the most probable initial conditions, which can lead to a universe that escapes collapse into the DBHF.
- 33.
In this bizarre context, a localized observer simply means a subsystem of the states available in a causal diamond at any time.
- 34.
The large scale homogeneity and isotropy and flatness of the DBHF also play a role in this argument.
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Acknowledgements
This research was supported in part by DOE grant number DE-FG03-92ER40689.
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Banks, T. (2012). Holographic Cosmology and the Arrow of Time. In: Mersini-Houghton, L., Vaas, R. (eds) The Arrows of Time. Fundamental Theories of Physics, vol 172. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23259-6_5
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