I
recently started re-reading a biography of Paul Dirac(Graham Farmelo
*The
Strangest Man)*,
and realised the book alluded to quite a lot of physics I’d
forgotten, and also to a lot of Mathematics I’d never known, so
I’ve been trying to fill in some of the gaps. In the course of
that I’ve had some thoughts about scientific theories and
scientific discovery

What follows is a series of anecdotes and observations, loosely linked by a connection to Dirac and Quantum theory.

The biography concentrated on Dirac’s personality and eccentricities; I shall pay more attention to his Physics

I suspect there are people who know more about the subject than I do, so I apologise in advance if I’m telling you things you already know.

__Presentation__

I’m, experimenting with a new format. Instead of arranging pages in a linear sequence I’m using a tree and branch model, with one root page, containing links to each of the other pages.

__Background__

In
the late nineteenth century, most scientists seemed to have assumed
that theoretical science has gone as far as it could. That is odd
considering how much was unexplained in 19^{th}
century science.

The periodic table classified the elements brilliantly, but chemists had no explanation of the patterns displayed by the table.

Why is sodium a metal and sulphur a non metal? Why does sodium chloride have one atom of chlorine for each atom of sodium, while aluminium chloride has three atoms of chlorine for every atom of aluminium, and why does copper forms two chlorides, while zinc forms only one?

Why are metals good conductors of electricity? What determines the colours, densities, melting points and other properties of various materials? All those matters were just treated as brute facts.

Some chemical reactions produce a mixture of different products. What determines their proportions?

__Atoms__

Answers to such questions require a theory of the atom, and, apart from some speculation that heavier atoms might be made of hydrogen, there was no significant theory about atomic structure until the very end of the nineteenth century.

Atoms were often pictured as little balls with arms representing the ‘valence’, and such models are still found useful, but they do not represent serious theories about atomic structure; they are just convenient ways of visualising chemical formulae.

Experiments with cathode rays and radioactivity showed that various bits and pieces sometimes pop out of atoms, raising the question of atomic structure because people guessed that whatever came out of an atom must be there in the first.

JJ Thomson suggested what has been called a ‘currants in a bun’ structure, in which negatively charged electrons were embedded in positively charged matter. The positively charged matter accounted for most of the mass of the atom. That theory did offer an explanation of how charged particles might co-exist in a stable structure, but the theory did not survive some experiments conducted by Eddington.

Eddington used Alpha particles to bombard thin gold foil (perhaps about 3000 atoms thick) and found that most particles pass straight through, but a few experience very large deflections, sometimes bouncing back. This suggested that atoms are mostly empty space, with most of the mass concentrated in very small regions, leading to the Bohr atom in which negatively charged electrons were pictured as orbiting a positive nucleus.

That picture too was problematic. In terms of classical physics, there was nothing to hold together positive charges in the nucleus, and orbiting electrons should have lost energy as electromagnetic radiation and spiralled into the nucleus.

Bohr applied early quantum theory to account for the stability of electron orbits. Explaining the stability of the nucleus involved postulating a ‘strong force’ that appears to decrease exponentially with distance.

Bohr's success in accounting for the lines in the hydrogen spectrum seems to have been good luck. The picture of an electron in orbit in a sort of micro-solar system, works only for hydrogen.

__Evidence for quanta__

Spectral lines

Black Body Radiation (distribution of frequencies at various temperatures)

Photo-electric effect , energy of ejected electrons defends only on the frequency of the radiation, with a threshold frequency below which no electrons are ejected. Changing the intensity of the radiation affects only the number of electrons ejected, not their energy.

Later, wave properties of electrons

__Heisenberg__

Proposed that, instead of speculating about the structure of atoms, we should concentrate on what we actually observe. We observe atoms emitting radiation of various frequencies, each frequency corresponding to the transition from one energy state to another. Heisenberg presented that information in two dimensional arrays of numbers, each number representing the probability of a transition from one energy state to another. He combined his arrays of probabilities in various ways. Without knowing it, he’d re-invented matrix algebra, which was not widely known in the early 1920’s.

Heisenberg was worried that matrix multiplication is not commutative, but when Dirac saw Heisenberg’s work the non commutativity set him thinking about other non commutative operations he’d encountered, and he alighted on the Poisson Bracket. My understanding of the Poisson bracket is limited to the page gleaned from the Internet (see the link), because it’s beyond my mathematical competence, but I can say roughly where it fits into mathematical Physics.

William Rowan Hamilton reformulated mechanics in terms of a quantity now called a Hamiltonian, which measures the total energy of a system. Poisson brackets appear in Hamiltonian mechanics, and give the difference between two operations (partial differentiations) performed in different orders

Schrödinger formulated quantum mechanics in terms of a partial differential equation. Schrödinger's Heisenberg's and Dirac's formulations eventually turned out to be equivalent.

Paradoxes of quantum theory. The mixture of particle and wave properties prevents our forming a plausible picture of the particle world. That may be because there is no picture. Our ideas of what makes a plausible picture are taken from the macroscopic world of everyday life . Whatever underlies that world does not obey the same rules.

__Theories__

Dirac chose theories for mathematical appeal.

That may sound strange to someone who thinks a theory is supposed to describe how the world really is, by revealing a hidden reality usually concealed behind the veil of appearance, but by no means all scientists take such a realist view

(1) Instrumentalism; the model is just a mnemonic; it helps us to remember what happens.

(2) Operationalism: the atoms and waves in the model can be defined in terms of patterns in our observations.

(3) Realism: there really are waves and particles just as the theory says there are.

Operationalists think all the terms used in science should be defined in terms of observable. Eddington once said that Physics is the study of pointer readings

Others concede that theories go beyond what is directly observable, but think that the additional content is a mnemonic picture to help us assimilate the equations.

__ Norman
Campbell__
(1880-1949), a physicist who as a young man worked with J. J. Thomson
in the Cavendish Lab in Cambridge, put forward a ‘dictionary’
theory of science. He distinguished three components of a scientific
theory.

(1)
A __collection
of ideas and of statements using those ideas __these
make up the theory proper.

(2)
A __dictionary__,
equating certain of the formulae of (1) to experimentally observable
quantities

(3)
An __analogy__,
a way of picturing the components of (1) so they resemble some
familiar macroscopic physical system.

Applied to the Kinetic Theory of Gases

(1) would be the various formulae and equations (to be explained in (3) below)

(2) would be statements like:

- N*m = total mass of the gas
- mean value of (mv
^{2}) = k*(Absolute temperature), where k would be a constant depending on the units used

(3) The analogy would say that (1) describes a set of N small perfectly elastic
particles each with mass m, and each having as velocity some value,
v_{i},
of a random variable V [The formulae of part (1) would include a
specification of the probability distribution of V]

The analogy not only makes it easier for us to remember the equations and to manipulate them, it sometimes seems to stimulate the brightest of us to think of extensions to the theory and of additional entries for the dictionary, so extending the scope of the theory.

Perhaps there is room for choice in what theories we use.

Think of the formulation of a theory a like fitting a line or curve to a collection of points on a graph. The set of possible curves is infinite, and we usually choose the one we consider the simplest, though it is hard to say precisely what comprises simplicity. Karl Popper tried to construct a precise criterion, but that is questionable.

Do we need a picture when we have the equations? A picture can certainly help us remember the mathematics, so it works as a mnemonic, but perhaps that is all it is.

In Campbell's terms, Heisenberg and Dirac wanted to develop quantum theory without an analogy. That may be why the text book on Quantum Theory that Dirac later wrote contained no diagrams.

__Duhem
__compared
the advance of science with the incoming tide on the beach. The most
conspicuous aspect of the tide is the formation and breaking of the
waves, but that is misleading.

“But under the superficial to-and-from motion, another movement is produced, deeper, slower, imperceptible to the casual observer; it is a progressive movement continuing steadily in the same direction and by virtue of it the sea constantly rises. The going and coming of the waves is the faithful image of those attempts at explanation which arise only to be crumbled, which advance only to retreat; underneath there continues the slow and constant progress whose flow steadily conquers new lands, and guarantees to physical doctrines the continuity of a tradition.” (op cit p 39)

That ‘slow and constant progress’ was the accumulation of experimental data and of low level universal generalisation (such as values of physical constants) supported by that data.

__Realism
and scientific discovery__.

Supporters of a realist interpretation of theory claim that only realism can account for the fruitfulness of theories in suggesting new lines of investigation. Because, they argue, a picture of the world can’t point to new discoveries unless it’s true.

However if that argument were valid it would be possible to prove theories true by citing discoveries inspired by them. That cannot be correct because fruitful theories have often been superseded. For instance one of the central ideas of thermodynamics is the Carnot cycle, but when Carnot developed that, he was working in the context of the caloric theory of heat, a theory subsequently abandoned.

Electromagnetism. Maxwell’s equations expressed Faraday’s discoveries in mathematical form. The mathematical formulation predicted electromagnetic radiations, something which Faraday himself had not predicted. That suggests that the mathematical formulation might be more powerful than the original theory. Use of calculus implicitly assumes continuity and infinite divisibility, and we now believe that assumption to be false, yet it was fruitful.

Dirac seems to have attributed the fruitful of a theory to the elegance of its Mathematics.

__Has Scientific
Discovery Slowed Down?__

I graduated in 1959, 54 years ago.

Compare the 54 years after I graduated with the 54 years before.

Before: relativity, quantum theory, DNA, nuclear energy, transistors, computers, radio, television earth satellites

After: apart from quarks, developments have mainly been doing better what we already knew how to do: decoding the genome, nuclear power stations better transistors combinbed together in integrated circuits, better computers, radios, mobile phones and earth satellites.