vol 3: Development
chapter 3: Physics
page 8: Energy, time and proof
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... to restore theology to the mainstream of science
Energy, time and proof
The classical God is eternal, existing all at once outside time. Aquinas 47 The universe, on the other hand, clearly has both a history and a future. If we are maintain that the universe is divine, we must show that time is logically necessary in a consistent model of god. Here we relate energy to the rate of communication, and communication to the execution of algorithms, that is proofs.
The 'standard model'
Over the past century physicists, mathematicians and astronomers have put together a model of the universe which, because it is so successful, is known as the standard model. Standard model - Wikipedia, The standard model is built from general relativity, which describes the large scale structure of the universe, and quantum field theory, which is generally confined to describing microscopic systems. Hobson, Peacock, Zee
The standard model describes the world in terms of fields described by functions whose domain is the space-time described by special relativity. The standard model encounters difficulties at very high energies (the Planck scale) where assumptions about the mathematical continuity of space-time are believed to break down. Planck scale - Wikipedia
Quantum field theory assumes the existence of energy and tells us how this energy is distributed among the various possible structures of the world. We work here on the assumption that the network model can go deeper, explaining the origin of energy. Ultimately, it seems, energy is needed to enable the universe to be consistently both one and many.
A particle exists when a state of the field representing it is excited, that is, has energy. It ceases to exist when this energy goes elsewhere. Creation and annihilation thus correspond to obtaining and losing energy. This idea is quite general. I am a particle. I need energy to live. When my energy goes, I go. Quantum field theory describes birth and death at all scales.
The initial state
From a physical point of view, space-time is the stage in which the game of existence is played. Intuitively space is something that remains remains the same (like the playing field) while other things move around in it (the players). Space orders the positions of the players while time orders their motions.
This intuition is good as far s it goes, but there was a sage in the history of the universe when it had not yet differentiated into space-time and particles. A first step toward creating the present universe is to explain how space-time came to be in the initial singularity. Initial singularity
Let us say that the initial singularity is zero dimensional system with but one state having probability 1 (since it exists). We model the initial singularity as the only point in a zero dimensional Hilbert space. We let the Hamiltonian for the initial singularity be 0, so that (like the classical god) it is pure eternal action without energy.
In logical terms, we imagine the initial singularity as an embodiment of the identity operation, which is equivalent to no operation, nop. We see it as the static root of a dynamical tree which grows within it. This is consistent with our assumption that the universe is divine, creatin occurring within rather than outside god.
Dynamics
Our next step is to introduce dynamics. Dynamics can only have meaning in a space of two or more states. Things cannot move if they have nowhere to go. We thus guess that space-time and motion come into existence at the same time: in some way they are two sides of one coin.
We associate energy with both space and motion. The energy of motion is kinetic energy; the energy of space is potential energy. It is universally observed that energy is conserved, that is in a closed system the sum of potential and kinetic energy remains constant.
General relativity suggests that the total energy of the universe, like the energy of the initial singularity, may be zero. Feynman All the kinetic energy (which includes mass energy) that we see in the universe may be exactly equivalent to the potential energy of the space that has grown out of the initial singularity.
We experience time as a flowing boundary between past and future. It is a one way flow. The past is determined and cannot be revisited, as much as we would sometimes like to go back and change things.
We guess that the past is complete in the mathematical sense developed by Goedel and Turing. Completeness - Wikipedia Experience suggests that the past is eternal and cannot be cahnged. Goedel's incompleteness theorem suggests that no consistent recurvsive universe can be complete. Gödel's incompleteness theorems - Wikipedia This sugests we identify potential energy with the completed, static past, and kinetic energy with the incomplete dynamic future.
Let us guess that the logical operation corresponding to time it not. We can represent this quantum mechanically using the Pauli X operator operating on a qubit:
From the Einstein relation E = hf, we see that the frequency of the X operation is a function of its energy. Taken in itself, this operator gives no direction to time, but merely 'ticks'. The X operator itself we take to be a potential whose kinetic effect is to swap the basis vectors of the qubit. As it is written, the frequency of this operation is arbitrary, and we may assume that the potential energy represented by the operator is exactly balanced by the kinetic energy of the operation.
Space and time
We are aware of the difference between past and future because we live in the current highly evolved universe. At the level we are currently exploring, however, the not operation is not observable because the past is annihilated to create the future and so it is not possible in principle for them to exist simultaneously and come into contact.
Let us place the invisible dynamics of the not operation into correspondence with the invisible dynamics postulated by quantum mechanics. Quantum mechanics postulates two distinct processes. One is the unitary evolution of the wave function in an isolated system. The other is the collapse of the wave function when an isolated system comes into contact with an observer. The universe as we define it is an isolated system ('nothing outside'), so we expect the not operation to be unitary.
Quantum mechanics represents the general two state system by the expression |q> = a |0> + b |1> where |q> is a vector in two dimensional complex Hilbert space with basis vectors |0> and |1> and a and b are complex numbers called amplitudes. These amplitudes are normalized so that |a | 2 + |b |2 = 1. In the case of the not operator, the only values that a and b can take are 0 and 1. The not operator annihilates the state |0 > to give |1> and vice versa.
Energy
Where does energy come from? We can take a hint from cosmology. General relativity suggests that the total energy of the universe is zero, since its kinetic energy may be the exact negative of its potential energy, the energy stored in its structure. Feynman, So we assume that a real space has potential energy equal and opposite to the kinetic energy of the processes within it. At the current level of complexity, however, the most we can say is that potential energy is not kinetic energy
The not operation toggles between two states, one of which is not the other. Let us interpret this physically to mean that it toggles between potential and kinetic energy like a pendulum. Unlike a pendulum, however, this operation is digital rather than continuous.
We want to arrange things so that the total energy of this oscillator is zero. This desideratum brings to mind Newton's third law - action and reaction are equal and opposite. Newton We couple this with the principle of conservation of energy: the total energy of the universe (which we have chosen for heuristic reasons to equate to 0) does not change with the passage of time.
We can develop this picture further by thinking in terms of sources and messages, the elements of networks. The elementary operation in the symmetric network is the transformation of one permutation into another by the exchange of two elements of the permutation. Let these elements be p and not-p. Exchanged, they are not-p and p.
We may imagine this process as requiring two messages, one the inverse of the other. In our network model, a message is a Turing machine, that is a function or mapping. The two messages envisaged above are two functions which are inverses of one another, one sending p to not-p and the other sending not-p to p. Complex entities (like ourselves) are able to send and receive messages while maintaining their identity. At the level of fundamental particles, however, a particle must annihilate itself to communicate.
From the point of view of wave mechanics, this operation is equivalent to a complete circuit, a phase change of 360 degrees or 2π radians. Here we recall that quantum mechanics identifies phase and action so that we can equate a 2π change of phase with one quantum of action, and identify this in turn with an elementary exchange in the symmetric network.
Space and time
We have complicated the initial singularity from a completely static system to one which is a superposition of two states, one with positive, one with negative energy. It is also a superposition of two messages, one which converts
Although we have two sources and two flows of action, the size of the initial singularity remains zero, since the space time interval between its two states is zero. Our little space has a Minkowski metric (1, -1), space being the inverse of time.
What makes the universe go from one state to two? Our answer is that given Cantor's theorem, it would be formally inconsistent not to. We may see this theorem as the formal model of a complexifying force which we might call the Cantor force. Complexification
We may think of this two state system as the ticker of a clock without the accompanying counter. In an ordinary clock the counter gives a name to each tick by mapping it onto the natural line. This structure will come later when the universe acquires memory. Each cycle of the clock is a quantum of action, and we map this setup onto the natural line considered as a numbered and ordered series of ticks of no particular 'distance' apart. This 'line' can be constructed by a computer (a 'Peano machine') that simply adds 1 to its history at each cycle.
Amplitude and wave function
What we have now is an invisible two state process. We assume that this little oscillator is ubiquitous throughout the universe (like the initial singularity), and that it is the first layer in the network above the initial singularity.
Logically we model this with repeated not operations. We guess that this invisible structure is what quantum mechanics calls amplitudes and represents with complex exponentials. Logically, we will let the rate of this operation for the universe be aleph(0). From a quantum mechanical point of view, this rate is measured by the Hamiltonian of the system, that is the rate of change of phase. Since we have yet to establish any particular direction for time, we may see the phase of our oscillator as rotating in two directions, so that we may say that the energy of the universe is aleph(0) - aleph(0) = 0.
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