Julian Barbour's The End Of Time - in quantum mechanics:

( 1 ) making waves.

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Quantum energy waves of particles.

Relativity is based on the limiting maximum velocity of the speed of light, c for constant. Quantum theory is based on a limiting minimum of energy transfer in 'lumps' or 'quanta'. Max Planck invoked this near infinitessimal quantity, h, the 'quantum' in relation to a problem with infinitely continuous waves of radiation, thrown up by the theory of ovens or black boxes.

Light had been shown to have wave-like effects of interference patterns, such as are seen in water waves. But introducing light quanta was like saying waves are fundamentally made up of particles. It is as if one were to change one's belief, that water waves form a continuous flow, to considering them as really more like dunes or waves of sand, that on very close inspection are made up of tiny grains.
However, this simple analogy glosses over the deep puzzles encountered on an altogether smaller scale of physical phenomena.

Einstein took up Planck's idea in the form of 'light quanta' or photons, to explain the 'photo-electric effect'. It was found that bombarding the surface of metal, with certain beams of light, dislodged electrons from its surface. But the effect didnt depend on the intensity or brightness of the light used. If the light was ultra-violet, no matter how dim the beam, it still succeeded in knocking off electrons.

Einstein explained that higher frequencies of light, or more generally electro-magnetic radiation, such as violet or ultra-violet light were, in effect, faster 'bullets' or quanta of light energy. Dimming this more energetic light only meant that fewer bullets were being fired. But when they hit an electron, bound to a metal atom on the surface of a metal plate, they would still dislodge it.
In contrast, it didnt matter how bright you made longer wave, red or infra-red light; in other words, it didnt matter how many red bullets were fired, they were all too low energy to ionise the metal surface.

An analogy to the photo-electric effect might be two walkers, walking into someone. They are both going at the same speed. But one walker is taking long slow strides and so not using much energy. He hardly disturbs you, passing by. But the other walker is taking short fast steps, and knocks you out of the way, bustling by.
These two walkers going the same speed compare to light speed as being constant. The velocity of light equals its wavelength times its frequency. ( Or, v = l x f. Texts normally use Greek small letters lambda for wave-length and nu for frequency. ) Red light is just as fast as violet but the former makes up for its lower frequency with longer wavelength. Red light's longer wave-length is like the longer strides of the walker making up for a lower frequency of steps.

Einstein's 1905 paper on the photo-electric effect is summed up in the formula, E = hf. Energy, E, equals Planck's constant, h, times the radiation frequency, f ( usually denoted by the Greek small letter nu ).
The formula E = hf suggests a compromise between the wave and particle theories, of matter in sub-atomic physics, measured as continuous and discrete quantities. The quantum, h, is a minimum discrete energy quantity, of which continuous quantities of wave-lengths must always be an exact multiple. The frequency, f, is known in the classical physics of continuous circular motion as the time rate of change of an angle -- call it angle Q. ( Again the typical Greek small letter theta is not generally available for use. )

Linear and rotary co-ordinate systems.

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Wave motion is related to circular motion, which is conveniently expressed in polar co-ordinates.

When I had to draw graphs at school, a long time ago, they were always rectangular shaped. That is to say in rectilinear or Cartesian co-ordinates. You have an x-axis which is the horizontal co-ordinate and a y-axis which is the vertical co-ordinate, both measured from a zero starting point or origin, by convention, from the bottom left-hand corner of your graph paper.

But an equally important type of co-ordinate system is polar co-ordinates. Their origin is the 'pole' or centre of a circle. One, of their two ordinates, is the radius, which sweeps round like the minute hand of a clock. ( Tho, the convention is that the radius turns anti-clockwise. ) The other ordinate is the angle Q that the radius sweeps out, like an angle between two lines of longtitude from pole to equator, if you are only seeing a sphere two-dimensionally from the north or the south pole.

The two co-ordinate systems may be matched so that the x-axis stands at '3 o'clock' and the y-axis stands at '12 o'clock', so to speak, on the circular 'face' of the polar co-ordinate system. This allows one to match or synchronise the two ways of drawing the same graph. Suppose the radius starts off at 3 o'clock and turns anti-clockwise to the vertical '12 o'clock position. In other words, the radius moves a ninety degree angle from an x-axis position to a y-axis position.

Placed beside this combined graph, you can have an ordinary rectilinear graph, which follows or maps the changing vertical height of the radius, where it touches the circumference, above or below the horizontal diameter, and traces a path up to the crest of a smooth hill.

The ascent's shape is matched by the descent from the crest back to the 'surface', level with the origin. This is the half wave-length point, corresponding to the radius having reached the '9 o'clock' position on the polar 'clock face': that is, its negative x-axis.

Then, the ordinary rectilinear graph's trace descends into the troff of the wave, which is as deep as the crest is high. ( Crest and troff correspond to the full height of the positive and negative y-axis positions on the polar circle, at '12 o'clock' and 'six o'clock' respectively. ) Finally, the rectilinear trace climbs out of the troff to the surface to complete the wave-length, which is equivalent to the radius completion of a full circle and returning to the positive x-axis or '3 o' clock' position.

The wave-length, l, such as from crest to crest or troff to troff, on the rectilinear x-axis is the same length, two pi times r, as the Circumference, C, of the circle in the polar system: C = 2πr = l. As such, the wave-length is a spatial wave, measured by the circumference distance.

But it is also possible to have a wave-length in time, called a period, P. This has been implicit in comparing the polar system to a 'clock'. Given clocks with 24 hours on the dial, one revolution of the hour hand marks out a period of revolution of the earth on its axis. The usual 12-hour clocks mark out a half period of revolution.

The number of revolutions a radius completes usually goes by the letter, n. On the corresponding rectilinear graph, the wave-length, is repeated n times like a series of perfectly uniform ripples. This can be done for wave-lengths in space or in time. Hence, the distance, s, covered by n wave-lengths is: s = ln = 2πrn.
And the time, t, over n temporal wave-lengths or periods is: t = Pn.

Distance, in a straight line, traveled, continuously, over time, equals velocity or speed in a straight line. Long ago, school made me familiar with this. But I dont remember learning the related formulas for rotary motion, as distinct from linear motion.
The distance covered in circular motion is given by the number, n, of revolutions thru a complete circle, whose angle is 360 degrees or 2 pi. This is called the 'angular distance', say, Q. So, Q = 2πn.
Therefore, s = Qr. Or, the linear distance equals the angular distance times the radius.

Similarly, for ( linear ) velocity, v, and angular velocity, w. Angular velocity equals angular distance divided by time: w = Q / t = ( 2πn / Pn ) = 2π / P.
And v = s / t = ( 2πrn / Pn ) = wr.
( In turn, a further similar relation exists between acceleration and angular acceleration. )

But we've seen, above, that velocity also equals wave-length times frequency: v = l x f.

And v = s / t = ln / Pn = l / P. Therefore, the frequency is inversely equal to the period: f = 1/P. Twelve hours on the clock face account for half the period of revolution of the earth. Therefore, the ( angular ) frequency is two turns per period.

A radius sweep which starts at '3' on the polar 'clock' produces a so-called sine wave from its equivalent starting point in a separate graph of rectilinear co-ordinates at their origin, where x- and y-axis cross.
Starting the radius sweep at '12' on the polar clock, starts a so-called cosine wave from that full height ( the maximum amplitude ) on the rectilinear y-axis, directly above the origin.
The sine wave starts at zero amplitude.The cosine wave starts at full amplitude or the crest of the wave. If you draw them both on the same graph, the x-axis always marks the sine wave 90 degrees or one-half pi ahead of the cosine wave. This is just as the polar graph shows the 3 o'clock position is 90 degrees ahead of the 12 o'clock position, as the radius counter-clockwise sweep is made to go by mathematical convention.

This ninety degrees out of step between the sine and cosine wave is called a difference of phase angle ( call it angle q ). Suppose that the sine and the cosine waves have both gone the same length of wave, corresponding to an angle of size Q in the polar system. Given that the phase, q is 90 degrees, then sine ( Q + q ) = cosine ( Q ).
This follows from school trigonometry of sine Q equals the right angled triangle sides' ratio of opposite over hypotenuse. Cosine Q is adjacent over hypotenuse ( triangle sides ). In the polar system, a radius is considered as the hypotenuse of triangles it forms by dropping a vertex ( for a y-axis reading ), from the point the radius touches the circumference, onto the circle's horizontal diameter ( for an x-axis reading ). One can check on one's calculator, for example, if Q = 0 degrees and q = 90 degrees, then sine ( 0 + 90 ) = cosine ( 0 ); or, sin ( 30 + 90 ) = cos ( 30 ).

The double slit experiment.

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Julian Barbour uses the standard introduction, to the many mysteries of wave mechanics or quantum mechanics, with the double slit experiment. Shine a beam of light thru a single slit onto a wall. Most of the light will go straight thru the slit and form the densest target area on the wall. The rest of the light will get more or less deflected from the edges of the slit and scatter about the densest area.

Replacing one slit with two slits, close together, produces a surprisingly different picture. On the wall, a series of bars form, the middle bars being most densely lit. There is an absence of light strikes between the bars.
The single slit experiment could have been explained as either a particle or wave activity of light. But the double slit experiment has to be explained in terms of light being in the form of both particles and waves.
The bars of light are characteristic of interference patterns found in water waves. When two radiating circles of waves, such as from two stones dropped in a pond, collide, the crests of the two rings may reinforce each other creating higher crests. The troffs may reinforce each others' depth. When one ring's crest coincides with the other ring's troff they neutralise each other to surface level.

The snag is that even when one photon at a time is sent at the double slit, the photon hits on the wall build up the same pattern, as if they were sent in a steady stream that waved into each other. The basic paradox of quantum theory's wave-particle duality of light ( and electrons etc ) is that a single particle can interfere with itself like a wave.

The double slit experiment's pattern of photon hits is given a probabilistic prediction by Schrödinger's equation of the wave function ( denoted by the Greek letter psi ). The wave function that gives the best interference effects, in the experiment are so-called momentum 'eigenstates' ( German for 'proper' or 'characteristic' states ).
Eigenstates of position or momentum are the only ones that can be measured with complete or unit probability of matching prediction.

It turns out that the wave function of a particle with a definite momentum ( the momentum eigenstate ) is two super-posed plane waves out of phase by a quarter of a wave-length. By definition, sine and cosine waves are out of phase by 90 degrees or a quarter of a wave-length ( as was discussed in the previous section ).
The quantum mechanical wave function is a complex function. Roger Penrose, in 'The Emperor's New Mind', explains how complex numbers are used in this function.

Suffice it to say, a horizontal x-axis could represent the direction of the two waves. Backward or forward direction is related to which of the two waves comes first. A y-axis could give a back-ground dimension to the sine and cosine waves, turning them from just undulating lines into planar waves. A z-axis could measure the height or amplitude of these waves. But the y and z axes are treated as composite or complex numbers, which are ordered pairs of numbers. These represent two intensities. The sum of their squares is the 'probability density' of the ( complex ) wave function, psi, of the x-variable. Notably, this gives the probability that a trial measurement will find a particle at x.

A particle has a definite momentum because its wave function has a regular and definite wave-length. At the same time, the particle's position is completely indefinite. Its probability density is uniform thru-out space, because the sum of the squares of two sinusoidal waves, one-quarter wave-length out of phase, is always one, given that they have unit amplitude.
This comes from Pythagoras' theorem in trigonometric form: sin²Q + cos²Q = 1.

Fourier showed that adding or super-posing harmonic waves of different wave-lengths can produce any curve, even down to a spike, that characterises the position eigenstate. ( Mathematics imitates nature's wave-particle duality. The same wave pattern can be regarded as super-posed waves of different wave-lengths or super-posed spikes, with different coefficients. )

These are the extremes between complete and null information we can have between complementary pairs of quantities, such as momentum and position, or energy and time. Heisenberg's uncertainty relation measures the extent, that more accurately measuring one of these pairs of quantities, is always at the expense of precisely measuring the other.

The experimenter can measure one or other of the complementary pairs, all implicit in the wave function. So a lack of complete knowledge is offset by a range of choice as to what can be known.

The double slit experiment can be considered in terms of two similar plane light waves super-posed or merged at a slight angle ( of five degrees ) to each other. At right angles to the mid-line of the five-degree angle, Barbour's book shows, as a computer-generated probability density, the result, in a concertina-like series of light ridges, corresponding to the light 'fringes' that show the interference effect on the wall.

This result relates to William Hamilton's classical work in optics. His wave theory showed that regular wave patterns reproduce light rays, without particles, yet corresponding to the older particle theory of light, and explaining more than it could.

Hamilton found an analogy to this in Newtonian dynamics, for only one value of energy allowed. Depending on an equation, his 'principal function' has a varying intensity, like the 'mist', at each point of configuration space. This equation is like that for his wave optics, but in multi-dimensional configuration space, instead of the ordinary three dimensions. When the intensity forms regular wave patterns, their respective families of paths, at right angles to their crests, are Newtonian dynamics' histories, having the same energy.

This path-making property of regular wave patterns has given them the name 'semi-classical', which is also the name of a physicists' program, to show how apparent paths in time may 'funnel' out of a timeless geometric structure underlying the universe.

Quantum entanglement of a two-particle system.

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The Einstein, Podolsky, Rosen paradox was a thought-experiment designed to reveal that quantum mechanics, compared to classical mechanics, was incomplete in the scope of its explanations. Two particles, such as photons sharing polarisation, or electrons whose total spin was conserved, would be correlated. Changing the state of one would have a correlative effect on the other.
For example, a system of two electrons, with a total of zero angular momentum, implies that if one electron has spin up, the other must have an equal and opposite spin down.

The EPR paradox was that quantum mechanics predicted that if you moved these correlated particles far apart, then changed the state of one, there would be an instant conservative response from the separated particle. But Einstein's special relativity forbids any signal passing, at more than light speed, from one particle to influence the lawful adjustment of the other particle.

Bell's theorem showed how quantum correlations must surpass any relations attributable to classical causes. Roger Penrose has further refined these distinctions, especially in his second popular book, 'Shadows Of The Mind'.

The EPR team believed the law of local causes would be upheld against quantum correlation. But by the 1980's, Alain Aspect's experiments were proving a super-luminal connection between correlated particles. This did not mean signals could be sent faster than light; it did not violate the foundations of special relativity. But it did mean the experimenter could bring about a known, faster-than-light change on a distant particle, by a certain change on its correlated particle.

With the help of half a dozen diagrams, Julian Barbour gives readers a feel for quantum correlations and 'entanglement' in the simplest possible two-particle system. Two particles moving on a line combine their one-dimensional configuration spaces to make a two-dimensional Q.

The wave function's value, for a single particle or a duet, as here, varies with time at each point in Q, which carries information about both particles, as to their positions or other quantities. These predictions are comprehensive, if often mutually exclusive, and refer to the system rather than its parts.

To find the relative probability of configurations of the two-particle system at some point, representative averages are found from the mid-points of a grid on Q. The probability density gives the relative numbers of these representative configurations likely to be found by repeated trial measurements.

Barbour likened this process of prediction and measurement, to giving the predicted numbers of configurations a proportionate number of tags in a bag and then drawing them out at random, as a trial confirmation of the predicted proportions.

These configurations are, in effect, ranked by their greater or lesser probabilities, which is how the Schrödinger equation configures atomic and molecular structure from the configuration space of all possibilities.
( In the simplest 'platonia' or relative configuration space of Triangle land -- discussed on the first web page of this review -- a probability ranking may be established thru best matching all possible triangles, each represented by a point in their platonia. )

Measuring the two-particle quantum system for, say, the position of one of the particles reduces the two-dimensional grid on Q to one dimension. This so-called 'collapse of the wave function' yields the only possible positions of the unmeasured particle, as relative probabilities of being somewhere on the remaining grid line.

Hugh Everett assumed that the wave function is 'the basic physical entity'. Its unimaginably huge numbers of possibilities are taken to constitute 'many worlds'. Tho we are only conscious of one world being realised, this does not necessarily mean that is all there is. Everett defended this possibility by the linearity or super-position principle of wave mechanics. Waves can split and combine, to create interference effects, but they remain themselves, essentially unaffected by it.

According to Barbour, 'To save the appearances, we do not have to create a unique history: we need only explain why there seems to be a unique history. That was Everett's insight.'

Barbour sees the essence of things in platonia, the relative configuration space, or a completely relativised version of Schrödinger's Q. This geometric landscape, of all possible configurations of reality, gathers, like a more or less dense mist, the wave function's probability density. Following Boltzmann, Barbour assumes: only the probable is experienced.

Richard Lung.

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Barbour on quantum cosmology.

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