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

( 2 ) quantum cosmology.

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Shrödinger's stationary wave equation.

Julian Barbour's idea of a timeless universe has to do with turning Scrödinger's quantum mechanics into a quantum cosmology. But to do that, he first has to relativise the remaining classical absolute time and space frame-work, within which Schrödinger's equation is expressed. Barbour's popular explanation of all this gives non-physicists a new insight into these mysteries.

Barbour makes the point, that only the quantum mechanics of a single particle takes place in the ordinary three dimensions. Quantum phenomena create new puzzles in certain two-particle states, or more. These multi-particle effects take place in a configuration space, which Schrodinger called 'Q'. ( Nothing to do with the angle, Q, used on the previous web page. )

In Platonia or relative configuration space ( described in the web pages about the treatment of classical physics, in 'The End Of Time' ) the simplest Platonia called Triangle Land consisted of each possible arrangement of three particles. This requires three dimensions for the lengths of the three sides of each triangular configuration, which has its own point in a 'configuration space'.

But Schrödinger's Q, for triangle land, would not merely rely on the relative positions between the three particles. Q also depends on an external or absolute frame-work. This locates the centre of mass of each triangle in absolute space, requiring three more numbers. Each triangle's orientation in absolute space also requires three more numbers.
The Q, of triangle land is a nine-dimensional configuration space. In fact for any number of particles, Q always has six more dimensions than Platonia.

The Schrödinger equation comes in a time-dependent and a time-independent form. Barbour suggests, contrary to conventional wisdom, that the latter is the more fundamental. The wave equation, that finds all the possible stationary states of a system, hints at a universal state of affairs in which super-positions of stationary waves create a variation in time of the probability density.

The stationary states, such as in Bohr's model of the atom, correspond to a fixed energy level, between quantum 'jumps' with the emission or absorption of a photon. The probability density of finding the atom in these states is constant, while the complex or composite values of the wave function oscillate with a fixed frequency. But adding two such solutions, with their respective frequencies, makes them interfere. The oscillations cease to be regular. Adding these timeless solutions yet makes the probability density vary with time.

Barbour gives a pictorial description of the Schrödinger wave function in a steady state.
At each point of the configuration space Q, imagine a child swinging a ball in a vertical circle, on a string of constant length, the 'amplitude', denoted by Greek small letter phi. Its squared value stands for the constant probability density.

The swinging ball's continuously changing height, above or below the centre, stands for one of the two ordered pairs of numbers, that make a complex variable. Its other component is the distance to the right ( positive ) and to the left ( negative ).

The stationary state is like swinging such balls at the same rate, every-where in Q, and all perfectly in phase or reaching the top of the circle together. In the momentum eigenstate, phi is the same every-where. But generally it varies according to a condition, imposed at each point of Q, by the equation of the stationary state. Barbour describes this condition as: Curvature number plus Potential number equals Energy number.

The curvature number is complicated. For a quantum system of three bodies, each point in Q corresponds to a configuration of the three bodies in absolute space. Holding two of the bodies fixed and moving the third, along a line in absolute space, moves on a line in Q.
Phi, the string length can be plotted as a curve more or less above the line. ( In calculus, this curvature is the second derivative. A three-particle Q has three times three dimensions of movement. So there are nine such curvatures at each point of Q. The 'curvature number' is 'the sum of these nine curvatures after each has been multiplied by the mass of the particle for which it has been calculated.'

The second number, the Potential is derived by multiplying phi by the potential. The potential energy depends on a given configuration of bodies and their nature, such as their masses.
The third number, the Energy is found by multiplying phi by the previously mentioned quantum energy relation, E = hf. The frequency, f, is the number of rotations of the 'balls' in a second.

Schrödinger compared the stationary state of the hydrogen atom to a vibrating string, which is fixed at either end and must always have a whole number of waves ( counted in half-wave-lengths ) like the harmonics of a musical instrument. The higher harmonics compare to the atom's higher energy levels. The fundamental note, when the string is just one over-arching and under-arching vibration ( that is, one half wave-length ) compares to the lowest energy level of the atom, its 'ground state'.

This analogy supplies a boundary condition for the solution of Schrödinger's stationary state equation, as an explanation of the discrete energy levels, posited in Bohr's quantum model of the atom. This condition is that the ends of the vibrating string are fixed, therefore the amplitude of phi must tend to zero at large distances.

Where the energy, E, minus the potential, V, is more than zero, phi oscillates. Where E - V is less than zero, phi tends to zero, only in certain well-behaved solutions ( the eigenfunctions ) for special values of E ( the energy eigenvalues ). The eigenfunction of the system, with the lowest energy value, is the ground state. Higher energy states are called excited states.

Finally, if E is large enough for E - V to be positive everywhere, the eigenfunctions oscillate everywhere, though more rapidly where the potential is lowest. The negative eigenvalues E form the discrete spectrum, and the corresponding states are called bound states because for them phi has an appreciable value only over a finite region. The remaining states, with E greater than zero, are called unbound states, and their energy eigenvalues form the continuum spectrum.

Relativised Schrödinger equation of the cosmos.

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Barbour follows much the same plan, to dispense with the remaining Newtonian frame-work in quantum mechanics, as he did with classical physics. ( This is discussed in my first two web pages on 'The End Of Time.' ) The Schrödinger wave function, of a given system of particles, changes with their relative configuration, centre of mass, orientation and time.
Barbour dispenses with the latter three, as he did for classical dynamics, since the relative configuration of the whole universe is its own absolute space and time, deriving them independently of an external frame-work. This applies Mach's principle to Schrödinger's equation for a quantum cosmology.

In applying quantum rules to a classical theory of cosmology, Barbour says:

The central insight is this. A classical theory that treats time in a Machian manner can allow the universe only one value of its energy. But then its quantum theory is singular -- it can only have one energy eigenvalue. Since quantum dynamics of necessity has more than one energy eigenvalue, quantum dynamics of the universe is impossible. There can only be quantum statics. It's as simple as that!

In a timeless system, the over-all energy is zero. So, in the stationary Schrödinger equation, at every point of Q, the sum of the curvature number and the potential number is zero. As in classical physics, the potential already is derived from relative configurations of the bodies that make up a ( Machian ) system, independently of absolute space and time.

As for curvature, that is the rate at which a curve's slope changes, with respect to a distance in absolute space, in ordinary quantum mechanics. Barbour suggests replacing these distances with the Machian best matching distances in relative configuration space, as he did to eliminate absolute space from classical physics.

We then add curvatures measured in as many mutually perpendicular directions as there are dimensions in that timeless arena, and set the sum equal to minus the potential number.

The 'Machian' wave functions are the Schrödinger eigenfunctions, whose eigenvalues have zero angular momentum, which was the case for the Machian treatment of classical dynamics.

On platonia or relative configuration space, only the potential and best matching distance govern the static wave function's variation from point to point. This timeless 'topography' determines where the 'mist' of the probability density gathers.

This predicts how probable all the inconceivably many permutations of atomic and molecular structures, and ultimately, Barbour seems to argue, how the most probable configurations of the universe best 'resonate' each other, in a sort of competition for the appearance of historical reality.

Quantum theory of records.

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Barbour imagines how history emerges from what he sees as the essentially timeless arena of quantum mechanics. He relies on John Bell's analysis of how records are made, in the context of radio-active decay in a cloud chamber, where an alfa particle leaves a track of ionised atoms.

Bell gives two interpretations of this phenomenum depending on when it is assumed a measurement is taken, that supposedly 'collapses the wave function' of possible places the particle will be found. The simpler interpretation assumes that atom ionisation is the 'classical external measuring instrument' for revealing the alfa particle, and successively collapsing the wave function, with gradual loss of particle energy and increasing deflection, that can be statistically predicted.

Bell's second interpretation treats not only the alfa particle but the whole system in quantum mechanical terms, that is to say billions of potentially detachable electrons, from their hydrogen atoms, all given three dimensions each ( together with the alfa particle's three dimensions ).
Given time for the ionization of, say, a thousand atoms, a photo takes a measure of the complete system, collapsing the wave function onto a complete track, not onto one position of one particle.

In the second scenario, the wave function has a vastly increased configuration space to search-out. But this land-scape is also vastly more structured and the the wave function, like a mist settles more densely, accordingly, determining the points most probably measured. Each point, in this bigger platonia's path, 'looks like a history of the three dimensional track up to some point along it.'

Despite the different view of when the wave function collapses, the results are much the same, because the experiment is a highly organised situation, whereby a highly regular Hamiltonian wave function produces a semi-classical solution that gives the appearance of a path taken in time or a history.
But, in quantum mechanical terms, it is the wave function's probabilistic search thru a timeless topography of all possible histories, that measurement, 'collapsing the wave function', realises as one history.

Reviewer's comments:

Geometries and measurement theory.

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Barbour believes consistency demands that a consciousness of motion must be in one configuration. He guesses that the brain can take in several 'snap-shots' at once and 'play the movie.'

He follows Bell's belief that 'the past' is inaccessible and therefore irrelevant. We have only records of an illusory history. Bell, however, didnt deny the reality of time, whatever his beliefs about it.

This reviewer cant help but think that records are of something. So, to deny that something is a contradiction. Suppose time has a comparable reality to that of space. It can be thought of in the same way. For instance, special relativity treats time, as well as space, as having speed.

Maps are a record of a topography, to a greater or lesser scale. As Charles Dodgson pointed out, the map can be increased to full scale till the land-scape is a map of itself. Geological strata containing fossils might be considered naturally scaled-down time-maps of evolution. The strata are a spatial map of the ages. One could say they had lost most of their temporal dimension, rather like the Minkowski Interval makes for an inter-change of temporal with spatial dimensions, in space-time.

In denying the reality of time, Barbour finds himself at odds with other physicists wishing to develop the space-time frame-work of relativity theory. I would like to suggest that the Minkowski Interval has an independent reason for being taken as a basic structure. Namely, it seems to fit in the logic of measurement, as devised by S S Stevens.

Stevens distinguished four scales of measurement, the nominal or classificatory, the ordinal or ranking, the interval and the ratio scales, 'on the basis of the principle of invariance under transformations' -- however that applies. But I have illustrated how these scales apply to logic of choice, on my first web page about 'Scientific method of elections'. ( A problem with both natural and social science is that they dont emphasise the dynamic of knowledge of freedom and freedom of knowledge. Ive discussed this in my two web pages on the ethics of scientific method and a short web page on physics and freedom. )

Without being able to give any sort of expert proof, I would like to make some points of comparison between Stevens' measurement theory and some basic features of physical theory. Newton's laws of motion appear to have two out of the four measurement scales. Relativity theory can be considered as supplying the other two, so that physics employs a more completely logical system of measurement.

There are quite some differences in formulation of Newton's laws. I take Allan M Munn's ( in 'From Nought To Relativity' ). Munn didnt think the second law, postulating every action has an equal and opposite reaction, was operational, because forces other than the mutual ones will always be present.

Law one states: 'Every body tends to continue in a state of rest or of uniform motion in a straight line, unless it is compelled by an externally applied force to change that state.'

Law one illustrates Stevens' classificatory scale of measurement. A body's state is put in two mutually exclusive categories, rest or uniform motion. This reflects Newton's belief that ultimately the universe did have a spatial frame-work, which was absolutely at rest, in relation to all motion.

The ordinal scale of measurement is a logical refinement or progression from dualistic classification. Instead of saying rest is rest and motion is motion and never the twain shall meet, rest is not considered an absolutely distinguished state. There is but one range or order of motion by which observers relate. There is only relative motion. East and west may be more accurately represented in an ordered cultural continuum than as a sharp dichotomy.

Zero motion is not a true rest but the result of an arbitrary choice of co-ordinates between observers in relative motion. There is an absolute zero temperature, which does not necessarily mean molecular motion ceases, only that it cannot transfer to other systems. Like light's absolute maximum speed, this absolute minimum speed is a limit that can only be approached.

Hence, temperature scales also have no true zero. There is no such thing as one temperature being, say, twice as hot as another. ( Temperature is therefore not a ratio scale, characterised by a true zero, which is the next logical step in the four scales of measurement. ) The various scales, named after Celsius, Fahrenheit, etc are arbitrary but they do have the property of being in proportion to each other and translatable by formula. This is characteristic of interval scales.

There may be a geometric sense in which Minkowski's Interval is actually an interval scale. ( The use of the same term, interval, is, as far as I know, not deliberate. ) The Interval is akin to arbitrary temperature scales, which can translate between each other, because it allows arbitrary co-ordinate systems in relative motion, to translate between each other, according to a common space-time formula.

The Interval is Euclid's geometry of three-dimensional flat space extended to a four-dimensional space-time. General relativity adopts a Riemannian geometry of curved space-time, to allow observers in relative acceleration to translate co-ordinates. It is a generalisation from Minkowski space-time considered as of zero curvature, and thus constitutes a ratio scale geometry of curvature.
The metric of geometry meets the theory of measurement. Einstein's geometrising endeavor in physical theory could as well be called a mensural endeavor.

Of course, Newtonian physics is rationality par excellence, as in the third law, which says 'Rate of change of momentum of a body is proportional to the force acting upon it and it has the same direction.'
Nevertheless, a concept such as 'force at a distance' was criticised as mysterious. Newton himself said it was an apparent nonsense, apart from the fact that his law of gravity worked. General relativity replaces force with geodesics to determine a body's path.
The point being made is that progress in classical physics was made by its becoming a more fully integrated measurement structure, borne out by the logic of progressive scales in measurement theory.

Time dependence on configurations.

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Barbour shows how time emerges from changing configurations of bodies, ultimately of the whole universe. This raises the question of the nature of the universe in its first ten thousand years before matter was formed. Barbour denies a temporal development or evolution.

But consider the conventional view that before matter coagulated, there was only radiation. The cosmic back-ground radiation is considered a fossil remnant of the big bang. In the sublime words of Genesis, 'And God said, Let there be light'.
On the basis of Minkowski's Interval, light has the fastest motion thru space, so that it has no speed left for motion thru time. There is no passage of time at light speed. A photon has never aged from the big bang till the time it is employed in the double slit experiment.

In this two-hole experiment, light doesnt distinguish time. Photons make wave interference regardless of whether you beam them together or send them one by one. Suppose this phenomenum is a 'fossil' remnant of the radiative universe before material configurations could distinguish time.
Time would not be distinguished from space. When physicists tried to relate the theories of the quantum and of gravity, they tried to match their respective three degrees of freedom, or dimensions, each. As described on the previous web page, this involved a quandary of whether a time dimension could arbitrarily take over one of three spatial dimensions.
The Wheeler-DeWitt equation defered this problem of choosing which spatial dimension to signify time. A justification for this decision might be that, in a universe yet lacking material configurations to define ( ephemeris ) time, time was not yet distinguished from space.

Also, this rationale may not exclude David Deutsch's many-worlds interpretation of the double slit experiment ( in 'The Fabric Of Reality' ). He argues that the phenomenum of a photon interfering with itself is really evidence of an other photon, we cannot perceive -- and hence from 'another universe' -- interacting with the perceived photon in our universe. Deutsch's logic is that just because we cannot directly perceive the other photon does not necessarily mean that it does not exist. And we should not ignore the evidence of interference for its presence.

A temporally distinct universe pretty much excludes any kind of 'interference' from alternative realities. Much of Deutsch's book is about thought experiments with tardis-like time machines that wont result in paradoxes. But the double-slit experiment as a fossil from an era, that had not temporally rigidified, might indicate a condition allowing universes to be slightly less exclusive.

Stephen Hawking's no-boundaries solution, to the geometry of the universe, also sees no distinction between space and time. This makes use of Feynmann's quantum electro-dynamics.
Richard Feynman ( as in his popular classic, QED ) gives a compelling quantum theoretical explanation of the two-hole experiment and indeed general light phenomena. Barbour doesnt refer to his work and, in general, popular writings dont seem to relate Feynmann's 'sum over histories' approach to conventional quantum mechanics.

In his popular lecture, Feynmann himself didnt use his own phrase, for the fact that a classical particle has one history, but a quantum particle has to have all possible histories, with their associated wave size and phase, taken into account. So, for instance, a photon's journey is calculated by summing all possible paths it could take in space-time.

In 'A Brief History Of Time', Stephen Hawking mentions his use of the 'sum over histories' quantum theory towards his no-boundaries solution to the creation, or rather being, of the universe.
In doing such sums, when time is measured as an imaginary number ( being multiplied by the operator, i, which merely equals the square root of minus one ), then: 'This has an interesting effect on space-time: the distinction between time and space disappears completely.'

( I dont properly understand Hawking's argument. The following is a, no doubt, misleading gloss on it. ) One could arbitrarily start off time, say, at a position akin ( in fewer dimensions ) to the earth's geometrical pole and chart the universe's expansion along the lines of longtitude till reaching a maximum expansion ( supposing the universe is not infinitely expanding ) at the equator.
The amount ( and kinds ) of gravitational mass in the universe is not yet accurately known. Estimates are used to guess how much it is slowing down the big bang.

The amount of gravitational mass determines the curvature of the space-time path of the universe. This, too, can be calculated using Feynmann's summing technique, that is in quantum, rather than in singular classical terms.
Penrose and Hawking showed that the determinism of general relativity meant 'the beginning of time would have been a point of infinite density and infinite curvature of space-time.'
That means the classical laws broke down. But the quantum rule of the uncertainty relation of position and momentum offered a way out, tho one not yet fully achieved.

The no-boundaries solution would remove the problem of a beginning of the universe and the paradox of what happened before creation. It offers a self-contained solution that a universe, by definition, logically requires.
Here is the same goal, but by a different approach to time, that Barbour seeks.

Physicists, like the wave function, explore all avenues.

Richard Lung.

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