Chapter 11: The axioms of abstract Hilbert space

Synopsis

We do physics by imagining spaces where anything can happen and then looking for the rules that make what actually happens look just like our world. Isaac Newton set up classical physics in a combination of Euclidean space and steadily flowing time. He then described the rules obeyed by the solar system. Quantum mechanics began when people began carefully measuring the light spectra emitted by matter. The questions that had to be answered is what sort of space do these spectra come from? What are the rules in this space? The answer is that each line of a spectrum is like a musical note, so we expect the space to be the space of all the possible musical notes and the rules we are looking for are John von Neumann's axioms for abstract Hilbert space.

11.1: Continuous creation

  11.2: Formalism  

11.3: Von Neumann’s axioms of Hilbert space

  11.4: The creation of the Hilbert space of the universe  

11.1: Continuous creation

Before it was replaced by the big bang model, the continuous creation or steady state model of the Universe was popular. As the Universe expands, the theory goes, new matter is created to fill the new space. Locally, the Universe has always been the same, and may be eternal as many of the ancients thought. Steady-state model - Wikipedia

The evolutionary big bang model attributes a beginning to Universe. It assumes that the Universe began from a structureless initial singularity and exploded at some point 14 billion years ago to begin the evolution the Universe as we know it. The Universe, measured in energy, has become less dense as its volume increases. At the moment it averages about 5 hydrogen atoms per cubic metre, less dense than the best vacuum we can create on Earth.

Here we are seeking to model the Universe as the physical mind of god. The language we are using to describe it is a cross between the ancient theological discussion of divinities and angels and the modern understanding of quantum mechanics as the theory of communication and the associated computation between particulate sources. Quantum theory works at all scales from the quantum of action to the Universe as a whole. The creation of mental and cultural space is unbounded, reflected in Cantor's transfinite numbers and the exceedingly complex structures around us like the living Earth and ourselves. Jeffrey Nicholls (1987): A theory of Peace

11.2: Formalism

There are two difficulties with the big bang story. First, where did the initial singularity, which is considered to be prior to space and time, come from? We must say that it is eternal since time does not exist for it. The second problem is that we associate energy and momentum with space and time, so we may wonder how all the energy of the Universe can be contained in an entity existing prior to space and time.

An alternative approach may be to consider the initial singularity as a quantum object analogous to Aristotle's unmoved mover and Aquinas's model of the Christian God, an entity of pure actuality, actus purus, completely simple, omnino simplex, a single particle and so effectively quantized,

Cosmic cosmogenesis works the same way as human creativity (or really, vice versa). First we have an idea, for brain and Universe the work of quantum mechanics. Then we make the idea come true, as the Wright brothers did with their idea for a flying machine. It takes energy to make an idea come true: in Greek to convert a kinema to a dynamis, in English, to turn a plan into a reality.

Later, beginning Chapter 15: Potential + kinetic = zero energy universe and continued Chapter 16 : Gravitation and the creation of dynamic particles we shall explore the idea that gravitational potential supplies the energy necessary to convert the formal kinematic structures of Hilbert space and quantum theory into dynamic physical realities, real particles which will become the source of Minkowski space in Chapter 17: Gravitation + particles = Minkowski space.

The Christian idea of God developed in the New Testament is descended from Yahweh, the God of the Hebrew Bible. A parallel development was the introduction of the Trinity. We may identify the the quantum initial singularity with the Christian God. This is possible because these two entities are formally identical: eternal, absolutely simple and the source of the world. Like the God of Aquinas, we take the initial singularity to be pure activity.

In Chapter 4: A new beginning I introduced the idea of naked gravitation which is simply the singularity predicted by Einstein’s theory of gravitation deprived of the spacetime structure developed by quantum mechanics which it assumes from Minkowski space.

In Chapter 5: Eternity and time I took another step toward creating the world by endowing the initial singularity with an analogue of the idea that Augustine applied to God in his explanation of the Trinity: that God the Father’s conscious image of themself is in fact divine and identical to the Son of God.

There I applied the Brouwer fixed point theorem that shows that under certain conditions mapping a set onto itself, in effect making it conscious of itself, identifies a particular point which can only be mapped onto itself, a fixed point. If we call the points x and the mapping f(x), then the theorem predicts that there must be an x where f(x)= x. Brouwer fixed point theorem - Wikipedia

In Chapter 5 I also noted that AlanTuring identified a countable infinity of functions that can be computed by a mechanical computer, a Turing machine. If we imagine that all of these functions can be applied to mapping a set onto itself, we may arrive at a countable infinity of fixed points. Turing machine - Wikipedia

We will now take the next creative step by identifying these fixed points as the basis states of a Hilbert space, thus laying a foundation for the introduction of quantum mechanics into the initial singularity. We are exploiting the Brouwer fixed point theorem to force the logical creation of structure.

11.3: Hilbert space

Quantum mechanics began with Max Planck's realization that quantization prevented the ultraviolet catastrophe associated with treating black body radiation as a continuum. It took another thirty years to reach its final form. In 1930 Dirac published a treatise which was to become the standard reference, based on his transformation theory. John von Neumann then perfected the mathematical presentation of Dirac's work, locating it in an abstract Hilbert space.

Hilbert space is a complex linear vector space analogous to Cartesian space, with a metric defined by an inner product. Physical states are represented by rays in this Hilbert space. John Neumann defines an abstract n dimensional dimensional Hilbert space (Cⁿ) with three axioms. John von Neumann (2018): Mathematical Foundations of Quantum Mechanics

α) A “scalar product”, i.e., the product of a (complex) number a with an element f of Hilbert space: af;

β) Addition and subtraction of two elements f, g of Hilbert space: f ± g;

γ) An “inner product” of two elements (f, g )in Hilbert space. Unlike α and β this operation produces a complex number which is not an element of Hilbert space but a metric, a measure of "distance".

The theory of quantum computation and quantum information operates in n dimensional Hilbert space, where n may run from 0 to a countable infinity, ℵ₀. Nielsen & Chuang (2016): Quantum Computation and Quantum Information

Each element, f, g, . . . defines the orientation of a complex plane in Hilbert space. It is called a ray, with the property that e^iθf = f where e^iθ is a complex number of modulus 1. The basis vectors of a Hilbert space f and g are normalized by the property (f, f) = 1 and are said to be orthogonal when (f, g) = 0.

Von Neuman points out that:

The noteworthy feature of the operations af, f ± g, (f, g) is that they are exactly the basic operations of the vector calculus: those which make possible the establishment of length and angle calculations in Euclidean geometry or the calculations with force and work in the mechanics of particles.

Although Hilbert space bears a close analogy to Euclidean space, the fact that it is built using complex numbers and complex numbers are periodic makes it very different (Chapter 5). It is not a space of geometric distance, but more a space of musical distance. The vectors of Hilbert space are formed by the addition of basis vectors of different frequencies which in music distinguish the sounds of different instruments by the superposition of overtones.

We may imagine each basis state of a Hilbert space to be a pure tone. A countable infinity of such states add up to a vector which would sound like noise with an infinite range of frequencies. The difference, as we shall see, between this noise and the music of the Universe is the work of quantum mechanics and evolution.

11.4: The Hilbert space of the initial singularity

The theological model of the Trinity is based on two ideas, the procession of the divine persons and their distinction. Augustine, Aquinas and Lonergan all based the procession of the Son and the Spirit on the conception and expression of an idea in the human mind.

All three of these authors base the distinction of the persons on the idea of relationship, relying on the fact that the persons are distinguished by their relationships to the source from which they proceed, the Son to the Father and the Spirit to the Son and the Father together. Here we rely on the definition of Hilbert space rather than relationships. All the basis states (dimensions) of Hilbert space are understood to be distinguished by orthogonality - their pairwise inner products are zero.

We can now apply these ideas to the creation of the Hilbert space defined by von Neumann. We assume that the initial singularity is pure action identical to the traditional God, and that it has the power to reproduce itself indefinitely free of the dogmatic limitations derived from the interpretation of the Trinity in the New Testament. We assume that a sequence of orthogonal actions creates Hilbert space, dimension by dimension.

We may identify the simplest Hilbert space with the initial singularity, a zero dimensional Hilbert space which we may represent by the symbol |0⟩;. We might imagine it to be formally identical to the empty set ∅, with no internal content or structure. This is consistent with the traditional idea that God is absolutely simple, omnino simplex. We can represent the first new vector created by the action of the initial singularity by the symbol |1⟩. This vector is orthogonal to |0⟩, that is distinct from it, but nevertheless within the initial singularity, as the persons of the Trinity are within God.

The qubit and all subsequent superpositions of newly created vectors a|0⟩ + b|1⟩ + c|2⟩ . . . are normalized by the requirement that a² + b² + c². . . = 1. This normalization, which is maintained by the unitary operators of quantum systems, establishes that any quantum system obeys the probabilistic structure of a communication source, to be described on Chapter 19: Quantization: the mathematical theory of communication. Quantum mechanics acts as the computational mind behind all the event which take place the four dimensional spacetime in which we live. It is acting as a communication source, receiving and sending messages. We have more to say about this in Chapter 20: Measurement, the interface between Hilbert and Minkowski.

On the polar complex plane the basis states of a Hilbert space may be thought of as points on a circle with unit radius representing the absolute value of each state and a rational argument θ defining the position of the point on the circle.

It may feel counterintuitive to identify the quantum of action, the smallest entity in the Universe, with the divinity, the largest entity imaginable. The formal approach makes this identification possible by abstracting from all concrete size and detail and standing upon a simple definition alone: an act is an event that changes the world just as a divinity is an event that changes the world. A quantum of action, like a divinity, is a discrete agent limited in its omnipotence by consistency alone.

In its simplest incarnation, we may consider the quantum of action as a not operator, an activity which may annihilate something existing and create something new. This operator changes some situation p into some situation not-p. In the binary realm of propositional logic, we understand that not-p annihilates p and creates not-p. On the other hand, not-not-p = p reverses this operation, annihilating not-p and recreating p. In the real world the not operator has many more options. There are about 8 billion not-mes on Earth.

We may think of this operation in a physical way as a pendulum. A pendulum oscillates between kinetic and potential energy. At the top of its swing, the bob has gravitational potential; as it swings down this potential energy is converted into kinetic energy. On the upswing, the opposite happens. In an ideal frictionless world, a pendulum would swing forever. Then we think of it in formal mathematical terms as an harmonic oscillator.

Action, as we conceive it here, exists before space and time, just like the traditional divinity and the initial singularity. Our interpretation of action in terms of logic may be understood to be purely formal. Logic, like formal mathematics, is kinetic. A puppet without agency. Hilbert, Whitehead and Russell's formal mathematics may be considered to exist outside spacetime. Nevertheless it does have spacetime representation in the mathematical literature and becomes dynamic in the minds of mathematicians, students, users of mathematics and the world.

Chapter 11: The axioms of abstract Hilbert space

Synopsis

Table of contents

11.1: Continuous creation

11.2: Formalism

11.3: Hilbert space

11.4: The Hilbert space of the initial singularity

Notes and references

Further reading

Books

Links