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January 7 - January 20, 2019
Think now of Maxwell’s electromagnetic field. We have noted that it carries energy. By E = mc2, it must also have mass. Thus, Maxwell’s field is also matter! This must now certainly be accepted since Maxwell’s field is intimately involved in the forces which bind particles together. There must be a substantial contribution28 to any body’s mass from the electromagnetic fields within it.
Material reality according to classical theory, let alone in the quantum theory that we are about to explore, is a much more nebulous thing than one had thought.
QUANTUM MAGIC AND QUANTUM MYSTERY DO PHILOSOPHERS NEED QUANTUM THEORY?
Physical reality is taken to exist independently of ourselves; and exactly how the classical world ‘is’ is not affected by how we might choose to look at it. Moreover, our bodies and our brains are themselves to be part of that world. They, also, are viewed as evolving according to the same precise and deterministic classical equations. All our actions are to be fixed by these equations–no matter how we might feel that our conscious wills may be influencing how we behave.
One tends to think of the discrepancies between quantum and classical theory as being very tiny, but in fact they also underlie many ordinary-scale physical phenomena. The very existence of solid bodies, the strengths and physical properties of materials, the nature of chemistry, the colours of substances, the phenomena of freezing and boiling, the reliability of inheritance–these, and many other familiar properties, require the quantum theory for their explanations.
Many physicists, taking their lead from the central figure of Niels Bohr, would say that there is no objective picture at all. Nothing is actually ‘out there’, at the quantum level. Somehow, reality emerges only in relation to the results of ‘measurements’. Quantum theory, according to this view, provides merely a calculational procedure, and does not attempt to describe the world as it actually ‘is’.
Where an argument remains unclear, I advise that you press on, and try to gain a flavour for the structure as a whole. But do not despair if a full understanding proves elusive. It is in the nature of the subject itself!
How do we know that classical physics is not actually true of our world? The main reasons are experimental. Quantum theory was not wished upon us by theorists. It was (for the most part) with great reluctance that they found themselves driven to this strange and, in many ways, philosophically unsatisfying view of a world.
Yet classical theory, despite its superb grandeur, has itself some profound difficulties. The root cause of these is that two kinds of physical object must coexist: particles, each described by a small finite number (six) of parameters (three positions and three momenta); and fields, requiring an infinite number of parameters. This dichotomy is not really physically consistent.
THE BEGINNINGS OF QUANTUM THEORY
(Planck’s constant is very tiny by everyday standards, about 6.6 × 10–34 Joule seconds.)
Thus, according to de Broglie’s proposal, the dichotomy between particles and fields that had been a feature of classical theory is not respected by Nature! Indeed, anything whatever which oscillates with some frequency ν can occur only in discrete units of mass hvlc2. Somehow, Nature contrives to build a consistent world in which particles and field-oscillations are the same thing! Or, rather, her world consists of some more subtle ingredient, the words ‘particle’ and ‘wave’ conveying but partially appropriate pictures.
Fig. 6.7. With a purely wave picture we can understand the pattern of bright and dark bands at the screen (although not the discreteness) in terms of interference of waves.
The reader may recall my warning in Chapter 3 that complex numbers are ‘absolutely fundamental to the structure of quantum mechanics’. These numbers are not just mathematical niceties. They forced themselves on the attentions of physicists through persuasive and unexpected experimental facts. To understand quantum mechanics, we must come to terms with complex-number weightings.
A probability amplitude represented as a point z within the unit circle in the Argand plane. The squared distance |z|2 from the centre can become an actual probability when effects become magnified to the classical level.
THE QUANTUM STATE OF A PARTICLE
Let us try to think of a single quantum particle. Classically, a particle is determined by its position in space, and, in order to know what it is going to do next, we also need to know its velocity (or, equivalently, its momentum).
This collection of complex weightings describes the quantum state of the particle. It is standard practice, in quantum theory, to use the Greek letter Ψ (pronounced ‘psi’) for this collection of weightings, regarded as a complex function of position – called the wavefunction of the particle.
These functions are called Fourier transforms of one another – after the French engineer/mathematician Joseph Fourier (1768-1830).
Fig. 6.13. The two-slit experiment analysed in terms of the corkscrew descriptions of the photon momentum states.
THE UNCERTAINTY PRINCIPLE
Most readers will have heard of Heisenberg’s uncertainty principle. According to this principle, it is not possible to measure (i.e. to magnify to the classical level) both the position and the momentum of a particle accurately at the same time. Worse than this, there is an absolute limit on the product of these accuracies, say Δx and Δp, respectively, which is given by the relation Δx Δp ≥ℏ.
To get some feeling for the size of the limit given by Heisenberg’s relation, suppose that the position of an electron is measured to the accuracy of the order of a nanometre (10–9m); then the momentum would become so uncertain that one could not expect that, one second later, the electron would be closer than 100 kilometres away!
In other descriptions one learns that the uncertainty is a property of the particle itself, and its motion has an inherent randomness about it which means that its behaviour is intrinsically unpredictable on the quantum level.
HILBERT SPACE
It will be convenient to adopt a notation (essentially one due to Dirac) according to which the elements of the Hilbert space – referred to as state vectors - are denoted by some symbol in an angled bracket, such as |ψ〉, |x〉, |φ〉, |l〉, |2〉, |3〉, |n 〉, |↑〉, |↓〉, |→〉, |↗), etc. Thus these symbols now denote quantum states.
After measurement, the state of the system jumps to one of the axes of the set determined by the measurement – its choice being governed by mere probability. There is no dynamical law to tell us which among the selected axes Nature will choose. Her choice is random, the probability values being squared moduli of probability amplitudes.
Despite the fact that we are normally only provided with probabilities for the outcome of an experiment, there seems to be something objective about a quantum-mechanical state.
We have seen that even the quantum state for a single particle, namely a wavefunction, has the sort of complication that an entire classical field has.
There is no set of prepared answers which can produce the quantum-mechanical probabilities. Local realistic models are ruled out!14
But we saw in the last chapter that, so long as relativity holds true, the sending of signals faster than light leads to absurdities (and conflict with our feelings of ‘free will’, etc., cf. p. 273).
Recall that ℏ is Dirac’s version of Planck’s constant (h/2 π) (and i = √–1) and that the operator ∂/∂t (partial differentiation with respect to time) acting on |ψ〉 simply means the rate of change of |ψ〉 with respect to time. Schrödinger’s equation states that 〈‘H|ψ〉’ describes how |ψ〉 evolves.
But what is ‘H’? It is the Hamiltonian function that we considered in the previous chapter, but with a fundamental difference! Recall that the classical Hamiltonian is the expression for the total energy in terms of the various position coordinates qi and momentum coordinates pi, for all the physical objects in the system.
In fact, if |ψ〉 describes the state of a single photon, then it turns out that Schrödinger’s equation actually becomes Maxwell’s equations!
Sometimes people take the line that complicated systems should not really be described by ‘states’ but by a generalization referred to as density matrices (von Neumann 1955). These involve both classical probabilities and quantum amplitudes. In effect, many different quantum states are then taken together to represent reality.
For another viewpoint, we may take note of the fact that the only completely clear-cut discrepancy with observation, in the Schrödinger cat experiment, seems to arise because there are conscious observers, one (or two!) inside and one outside the container.
Perhaps the laws of complex quantum linear superposition do not apply to consciousness!
He suggested that the linearity of Schrödinger’s equation might fail for conscious (or merely ‘living’) entities, and be replaced by some non-linear procedure, according to which either one or the other alternative would be resolved out. It might seem to the reader that, since I am searching for some kind of role for quantum phenomena in our conscious thinking – as indeed I a...
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WHERE DOES ALL THIS LEAVE US?
Where does all this leave us? I believe that one must strongly consider the possibility that quantum mechanics is simply wrong when applied to macroscopic bodies – or, rather that the laws U and R supply excellent approximations, only, to some more complete, but as yet undiscovered, theory.
Newton’s elegant and powerful theory of universal gravitation owed much to the fact that the forces of the theory add up in a linear way. Yet, with Einstein’s general relativity, this linearity was seen to be only an (albeit excellent) approximation – and the elegance of Einstein’s theory exceeds even that of Newton’s!
Our journey will start close to home, but we shall be forced to travel far afield. It turns out that we shall need to explore very distant reaches of space, and to travel back, even to the very beginning of time!
I have taken for granted that any ‘serious’ philosophical viewpoint should contain at least a good measure of realism. It always surprises me when I learn of apparently serious-minded thinkers, often physicists concerned with the implications of quantum mechanics, who take the strongly subjective view that there is, in actuality, no real world ‘out there’ at all!
The past was one thing and can (now) be only one thing. What has happened has happened, and there is now nothing whatever that we, nor anyone else can do about it! The future, on the other hand, seems yet undetermined.
Yet physics, as we know it, tells a different story. All the successful equations of physics are symmetrical in time. They can be used equally well in one direction in time as in the other. The future and the past seem physically to be on a completely equal footing.
According to relativity, there is not really such a thing as the ‘now’ at all. The closest that we get to such a concept is an observer’s ‘simultaneous space’ in space–time, as depicted in Fig. 5.21, p. 259, but that depends on the motion of the observer! The ‘now’ according to one observer would not agree with that for another.
Our physical understanding actually contains important ingredients other than just equations of time-evolution – and some of these do indeed involve time-asymmetries. The most important of these is what is known as the second law of thermodynamics.
THE INEXORABLE INCREASE OF ENTROPY
The laws of mechanics are time-reversible; yet the time-ordering of such a scene from the right frame to the left is something that is never experienced, whereas that from the left frame to the right would be commonplace.
The reader may perhaps be asking where the energy comes from, which raises the glass from floor to table. That is no problem. There cannot be a problem with energy, because in the situation in which the glass falls from the table, the energy that it gains from falling must go somewhere. In fact the energy of the falling glass goes into heat.
