Chapter 13: The emergence of quantum mechanics

Synopsis

Classical mechanics was developed by Isaac Newton in the seventeenth century. It is expressed in terms of forces and velocities in Euclidean geometric space and universal time, independent of space. Quantum mechanics required the invention of a new space. It works in Hilbert space which may be imagined as a space of musical notes spanning an infinite range of frequencies. The transformation from quantum mechanics to classical mechanics, which includes special and general relativity, is called measurement or observation. The principal feature of quantum mechanics is that it executes all the communication and computation in the Universe. Classical mechanics is rather like the screen and keyboard of a computer which handle input and output to the hidden processes inside. This chapter provides an impression of what the quantum orchestra does.

13.1: An introduction to quantum theory

13.2: From functions to operators

13.3: Complex numbers, periodic functions and time

13.4: The double slit experiment

13.5: How does it work?

13.6: Superposition

13.7: Which slit does the particle go through?

13.8: Wigner's mystery

13.1 An introduction to quantum theory

Modern physics seeks to construct a theory of everything in Minkowski space and its child, general relativity. Here we see Hilbert space and quantum mechanics as the foundation of a universal theory that unites physics and theology. The mathematical foundation of quantum mechanics are explained by John von Neumann. This foundation is built with linear operators operating on the Hilbert space defined in Chapter 11. This space is independent of and underlies the Minkowski space of special relativity described in Chapter 17: Gravitation + particles = Minkowski space. John von Neumann (2014): Mathematical Foundations of Quantum Mechanics

13.2: From functions to operators

Isaac Newton built on the work of Galileo and Kepler to give us classical dynamics and the universal law of gravitation that explains the orbit of the Moon around Earth. A key to his success was the invention of differential and integral calculus which he called the method of fluxions. Functions establish relationship between two numbers, like spatial position and time. The differential of this function with respect to time yields a new function, velocity. The differentiation of velocity with respect to time gives us acceleration. Going the other way, integration of acceleration yields velocity; integration of velocity gives position. Differentiation and integration, are called operators. Functions relate numbers to one another. Operators relate functions to one another.

A function is just like a vector, list of numbers paired with one another. So operators operate on vectors. A Hilbert space is a function space, which means every point in the space represents a function, f. By establishing an origin in this space we may imagine every point in it as a vector from the origin (that is the zero) to the point representing the function.

In Chapter 11 we imagined a Hilbert space whose maximum number of dimensions is a countable infinity. This which means that each vector or function in the space comprises at most a countable infinity of pairs of numbers. The linear operators of quantum mechanics operate on these vectors

One of the beauties of quantum theory is that the rules are independent of the dimension of the space. As we go along we can learn almost all we want to know about quantum mechanics by studying it in a two dimensional Hilbert space. We exploit this property in Chapter 20: Measurement: the interface between Hilbert and Minkowski spaces.

13.3: Complex numbers, periodic functions and time

The secret power of quantum mechanics lies in the complex numbers. Complex numbers are themselves two dimensional vectors which are periodic. They rotate on the complex plane as described in Chapter 5: Eternity, time and Hilbert space. They are ideal for describing waves and the passage of time.

At the end of §3 in Chapter 11 we noted that:

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.

Quantum mechanics seems very strange because it works in counterintuitive space more like an orchestra playing together rather than a football game played by individual actors trying to control a ball in four dimensions of space and time.

13.4: The double slit experiment

A common simple illustration of the difference between the behaviour of microscopic quantum particles and macroscopic classical particles is provided by a simple thought experiment which can be made real with the right equipment. Double-slit experiment - Wikipedia

If we spray real bullets at random at a barrier with two holes, we see that some bullets go through one or other hole and strike a screen behind the holes in line with the holes. Quantum particles, on the other hand, appear to go through both holes and interact with themselves to generate an interference pattern on a screen behind the holes. In his lectures on the double slit experiment Richard Feynman summarizes the quantum experiment in three propositions:

1. The probability of an event in an ideal experiment is given by the square of the absolute value of a complex number ψ which is called the probability amplitude:  P = probability, ψ = probability amplitude,   P = |ψ|².

2. When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference: ψ = ψ₁ + ψ₂;  P = |ψ₁ + ψ₂| ².

3. If an experiment is performed which is capable of determining whether one or another alternative is actually taken, the probability of the event is the sum of the probabilities for each alternative. The interference is lost:  P =  P₁ +  P₂.

From this we might conclude that interference is something which occurs in the invisible complex Hilbert domain rather than the observable real classical domain. The fact that the particles appears to go through both holes indicates that this process occurs in a domain other than Minkowski spacetime. Feynman, Leighton & Sands: Quantum Behaviour

13.5: How Does it Work?

Feynman then goes on to ask:

“How does it work? What is the machinery behind the law?” No one has found any machinery behind the law. No one can “explain” any more than we have just “explained.” No one will give you any deeper representation of the situation. We have no ideas about a more basic mechanism from which these results can be deduced.

13.6: Superposition

This statement seems to be a bit pessimistic. The explanation, devised in the first decades of the twentieth century, is quantum mechanics. The key idea, identified by Paul Dirac in his treatise on quantum mechanics, is superposition, which simply means the addition of the vectors representing physical states in Hilbert space. Paul Dirac (1983): The Principles of Quantum Mechanics (4th ed)

We meet a simple classical version of superposition in the bathroom. We have two sources of water, one painfully hot, the other painfully cold. We adjust the taps until the temperature if just right by adding cold water and hot water.

We understand the amplitudes added by quantum superposition are waves represented by the complex numbers described in section 13.3 above. We can get an idea of the linear addition of moving waves by throwing stones into smooth pond.

When two stones fall simultaneously close by we see that when the expanding circles of waves spreading from each impact meet they add and subtract from one another to form a complex patterns which seem to pass through one another unaffected.

The phenomenon is also clear in sound. We can usually distinguish voices of different instruments or people sounding together. In Hilbert space these sounds are represented by vectors. Each instrument generates complex overtones related to the fundamental frequency of each note. These overtones make the difference between C on a piano and C on a trumpet. Listening to an orchestra we can often pick out different instruments even though they are playing the same note.

Each slit in the two slit experiment emits a quantum wave. Because the slits are separated, the paths taken by the waves to any point on the screen, except the exact centre, have different lengths. When the waves at a given point are in phase they add giving a high probability of observing a particle (Feynman's proposition 1 above). When they are out of phase and cancel one another, the probability is low and it is unlikely that a particle will appear. The observed pattern of particle impacts on the screen is the result.

The kinematic behaviour of vectors in Hilbert space has an effect on the dynamic behaviour of events in Minkowski space. We might see a similar effect at the movies. Although the moving pictures and sounds may be purely kinematic and artificially created, they can nevertheless have the real emotional effects on the audience that the movie makers seek. The movie is a signal carrying information which relies on the intelligence of the viewers to be realized as physical emotion.

13.7: Which slit does the particle go through?

It seems intuitively obvious that a real particle would go through one slit or the other. If we block one slit, or devise a way to decide which slit the particle goes through, however, the interference pattern is lost. (Feynman's proposition 3). How can this be?

The answer proposed here is that the formalism of quantum mechanics operates at a level that lies beneath the spacetime familiar to us in everyday life. From an intuitive point of view, we can say that Hilbert space is the realm of the imagination of the Universe, just as our own minds are the realm of our imagination. Minds, like quantum mechanics, offer a spectrum of options to act.

This idea becomes more plausible as we progress. It lies at the heart of cognitive cosmology, the idea that we can produce a comprehensive theory of everything by interpreting the Universe as a mind.

13.8: Wigner's mystery

What does this remarkable collusion between mathematics and physics mean? After a critical discussion, Eugene Wigner writes:

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. Eugene Wigner (1960): The Unreasonable Effectiveness of Mathematics in the Natural Sciences

I feel that there is a theological reason for this gift which was expressed by Aquinas in his explanation of the limits to God's omnipotence:

. . . God is called omnipotent because He can do all things that are possible absolutely;

. . . For a thing is said to be possible or impossible absolutely, according to the relation in which the very terms stand to one another, possible if the predicate is not incompatible with the subject, as that Socrates sits; and absolutely impossible when the predicate is altogether incompatible with the subject, as, for instance, that a man is a donkey. Aquinas, Summa I, 25, 3: Is God omnipotent?

The formalist approach to mathematics proposed by Hilbert, which justifies the existence of Cantor's Paradise, puts a similar bound on the omnipotence of mathematics: every mathematical statement is acceptable as long as it does not involve a contradiction. God and mathematics are playing the same game, and this may be why, in a cognitive universe, mathematics, physics and theology have a lot in common. Cantor's paradise - Wikipedia

Ultimately we will conclude that Hilbert space is the scene in which the Universe observes itself, that is where it is conscious of itself, talking to itself. This is analogous to the way we consciously talk to ourselves to decide what to do. Hilbert space is the imagination of the Universe, the source of the variation which makes its evolution possible.

Our next step is to explain how quantum mechanics picks the music out from the noise. This is achieved by the world solving the eigenvalue problem which is also the problem that physicists have to solve to get answers from quantum mechanics. Its ability to solve this problem is the foundation of the idea proposed in this book that the Universe is creative and intelligent, creating itself within the initial singularity, what I like to call cognitive cosmogenesis.

Chapter 13: The emergence of quantum mechanics

Synopsis

Contents

13.1 An introduction to quantum theory

13.2: From functions to operators

13.3: Complex numbers, periodic functions and time

13.4: The double slit experiment

13.5: How Does it Work?

13.6: Superposition

13.7: Which slit does the particle go through?

13.8: Wigner's mystery

Notes and references

Further reading

Books

Links