The Nature of Consciousness

Piero Scaruffi

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These are excerpts and elaborations from my book "The Nature of Consciousness"

Loop Quantum Gravity

The ultimate goal of Loop Quantum Gravity (LQG) is still the "quantization" of general relativity, but the way it approaches the problem is very different: it is purely geometric.

Roger Penrose (“Applications of negative dimensional tensors “, 1971) toyed with the notion of a "spin network" (derived from Louis Kauffman's "knot theory") in an attempt to explain the structure of three-dimensional space. A spin network is a graph whose edges are labeled by integers, corresponding to the possible values of the angular momentum. Penrose sensed that these could be the simplest geometric structures  to describe space. The revolutionary assumption was that space and time are not primitive entities but are themselves constructed out of something more primitive. In quantum terms, space and time can fluctuate, as long as casual sequences are preserved.

The Canadian physicist Bill Unruh (“Notes on black-hole evaporation”, 1976) discovered that an accelerating observer must measure a temperature (a black-body radiation) where an inertial observer observes none, the temperature being proportional to the acceleration. You see vacuum in a field that is at rest with you where an accelerating observer sees a thermal bath. This means that the very concept of "vacuum" depends on the state of motion of the observer: an accelerating observer will never observe any vacuum. (This also proved that Einstein's principle of equivalence was slightly incorrect: a constantly-accelerating observer and an observer at rest in a gravitational field are not equivalent, as the former would observe a temperature and the latter would not). Every accelerating observer has a hidden region (all the photons that cannot reach her because she keeps accelerating, getting closer and closer to the speed of light) and a horizon (the boundary of her hidden region).

Jakob Bekenstein’s theorem implies that every horizon separating an observer from her hidden region has an entropy. That entropy turns out to be proportional to the information that is hidden or trapped in the hidden region (the missing information). According to Bekenstein's theorem, the entropy of the radiation that the accelerated observer experiences is proportional to the area of her horizon.

Lee Smolin and the Italian physicist  Carlo Rovelli ("Knot theory and quantum gravity", 1988) put Unruh and Bekenstein together and realized something that is built into any theory of "quantum gravity" (into any quantization of relativity): the volumes of regions in space must come in discrete units, just like energy comes in discrete units. If energy comes in discrete units, then space must come in discrete units. Just like matter is made of discrete particles, space itself must be made of discrete units. A volume cannot be divided forever: there is an elementary unit of volume.

Smolin used Bekenstein’s and Unruh’s theorems to prove that spacetime must be discrete. If spacetime were continuous, then a volume of spacetime (no matter how small) would contain an infinite amount of information. But for any volume of spacetime an accelerating observer would observe a finite entropy (finite because it is proportional to the surface of the volume) and therefore a finite amount of missing information. The amount of information is finite because the surface of the horizon is finite and therefore entropy is finite. The amount of information within a volume of spacetime must be finite, therefore spacetime cannot be continuous. Spacetime must have an “atomic” structure just like matter has an atomic structure. (This conclusion had been reached independently by Jacob Bekenstein in his studies on the thermodynamics of black holes).

Kenneth Wilson had first hypothesized that space was a discrete lattice (“Confinement of Quarks”, 1974). What Smolin did was to make Wilson's discrete lattice also change dynamically, able to evolve in time, as General Relativity requires. In his formulations the inter-relationships among Wilson's structures (the "loops") define space itself. Smolin used the work of two Indian scientists. Abhay Ashtekar (“New Variables for Classical and Quantum Gravity “, 1986) came up with the "loop-space model", based on a theory by Amitaba Sen (“Gravity as a Spin System”, 1982) that splits time and space into two distinct entities subject to quantum uncertainty (analogous to momentum and position). The solutions of Einstein's equations would then be quantum states that resemble loops. Smolin's theory was simply a theory of loops and how they interact and combine. (The Uruguayan physicist Rodolfo Gambini had independently reached similar conclusions).

In this way Einstein’s theory of gravitation (General Relativity) is reformulated to resemble Maxwell's theory of electromagnetism, with loops playing the role of field lines.

Loop-quantum gravity has dramatic implications on cosmology. For example, the Big Bang turns out to be a “Big Bounce” from an imploding universe to an expanding universe. The world as we know it with all its galaxies, stars and moons was created by tiny ripples in spacetime (otherwise the universe would be a boring homogeneous lattice). The Big Bounce contains quantum fluctuations that would explain those tiny ripples and therefore our galaxy, sun, planet and, ultimately, myself typing these words.

Ashtekar ("Quantum Nature of the Big Bang", 2006) showed that the Big Bang would not come from a singularity but from a previous universe: you canÎéÎ÷t retrace the universe all the way back because the contraction eventually causes a repulsive force that starts a new expansion. The Uruguayan physicists Rodolfo Gambini and Jorge Pullin reached a similar conclusion for a black hole: there is no singularity in the center of the black hole, but instead a tunnel to another space-time ("Loop Quantization of the Schwarzschild Black Hole", 2013).

Loop-states turned out to be best represented by Penrose's spin networks. The lines of a spin network carry units of area. The structure of spin networks generates space.

The space that we experience is continuous. Spin networks, instead, are discrete.  They are graphs with edges labeled by spins (that come in multiples of 0.5) and with three edges meeting at each vertex.  As these spin networks become larger and more complex, they "yield" our ordinary, continuous, smooth 3-dimensional space.  A spin network, therefore, "creates" geometry. It is not that a spin network yields a metric (the metrics being what define the geometry of a region of space) but that each vertex of a spin network creates the volume of a region of space.

An evolving spin network (a "spin foam") is basically a discrete version of Einstein's spacetime. Spin-foams are four-dimensional graphs that describe the quantum states of spacetime, just like spin networks describe the quantum states of space. Spin foams describe the quantum geometry of spacetime (not just space). A spin foam may be viewed as a quantum history. Spacetime emerges as a quantum superposition of spin foams (topologically speaking, it is a two-dimensional “complex”).

The way spin networks combine to form space is not clear, as there seems to be no "natural law" (no equivalent of gravitation or of electromagnetism) at work.  Spin networks "spontaneously" combine to form space. The formation of space resembles the Darwinian process that creates order via natural selection of self-organizing systems.  Space appears to be the result of spontaneous processes of self-organization à la Stuart Kauffman. 

The hypothesis that space is discrete also helps remove some undesired "infinites" from Quantum Theory. For example, charged particles interact with one another via electromagnetic fields. The electromagnetic field gets stronger as one gets closer to the particle. But a particle has no size, so one can get infinitely closer to it, which means that the field will get infinitely strong. If space is discrete instead of continuous, the paradox is solved: there is a finite limit to how close to a particle one can get.

Spin networks thus solve "quantum gravity" in three dimensions. Spin networks describe the quantum geometry of space. In order to introduce the (fourth) temporal dimension, a concept of “history” has been added by some researchers.

Basically, Einstein’s great intuition is that spacetime is not just a stage but an actor in the story of the universe, a story that evolves from its interaction with matter, and Loop Quantum Gravity shows that it is made of its own atoms just like matter is made of its own atoms.

The US physicist Martin Bojowald (“Loop Quantum Cosmology”, 2008) deduced that a true singularity cannot exist, as each space atom can only contain a finite amount of energy-matter. Therefore he revised Big Bang cosmology: the Big Bang was a moment of maximum density that must have come from a previous Big Crunch. In other words, the story of the universe should be roughly symmetric (or, better, a mirror image) before and after the Big Bang: there was a collapsing universe before the Big Bang created an expanding universe.

The Greek physicist Fotini Markopoulou showed that spin networks evolve in time in discrete steps: at every step, the change of each vertex of the spin network only depends on its immediate neighbors.  This is reminiscent of Von Neumann's cellular automata and of algorithm-based thinking, as opposed to the traditional formula-oriented thinking of Physics.

Markopoulou ("The Internal Description of a Causal Set", 1999) introduced causality in Loop Quantum Gravity.  In her view, time is not an illusion, just an approximation. She compares it to the river that seems to flow in a smooth way even though the motion of its water molecules is chaotic. Causality does exist at a very fundamental level, although it may not be the one that we perceive in our daily life.

The idea of Loop Quantum Gravity was further expanded by Causal Dynamical Triangulation, a theory introduced in the 1990s by Renate Loll (Dutch), Jan Ambjorn (Danish) and Jerzy Jurkiewicz (Polish) that constructs spacetime from elementary structures called “four-simplexes”. A four-simplex is the four-dimensional equivalent of a tetrahedron.

Yet another approach to quantum gravity is purely mathematical. For example, Markopoulou  noticed similarities between the “categories” used by General Relativity and those used by Quantum Theory. These have little in common with the traditional category of Physics whose objects are sets and whose morphisms are functions. In her view Quantum Theory is better understood as a theory of spacetime.

 


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