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

( 2 ) quantum kosmolojy.

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Shrödinger's stationary wav equation.

Julian Barbour's idea of a tImles univers has to do with turning Scrödinger's quantum mekaniks into a quantum kosmolojy. But to do that, he first has to relativis the remaining klasikal absolut tIm and spas fraim-work, within wich Schrödinger's equation is expres'd. Barbour's popular explanation of al this givs non-fysisists a nw in-sIt into thys mysterys.

Barbour maks the point, that only the quantum mekaniks of a singl partikl taks plas in the ordinary thry dimensions. Quantum fenomena kreat nw puzls in sertain tw-partikl staits, or mor. Thys multi-partikl efekts tak plas in a konfiguration spas, wich Schrodinger kal'd 'Q'. ( Nothing to do with the angl, Q, yus'd on the previus web paj. )

In Platonia or relativ konfiguration spas ( deskrib'd in the web pajes about the trytment of klasikal fysiks, in 'The End Of Time' ) the simplest Platonia kal'd Triangl Land konsisted of ych posibl aranjment of thry partikls. This requirs thry dimensions for the lenths of the thry sIds of ych triangular konfiguration, wich has its own point in a 'konfiguration spas'.

But Schrödinger's Q, for triangl land, wud not myrly relI on the relativ positions betwyn the thry partikls. Q also depends on an external or absolut fraim-work. This lokaits the senter of mas of ych triangl in absolut spas, requiring thry mor numbers. Ych triangl's orientation in absolut spas also requirs thry mor numbers.
The Q, of triangl land is a nIn-dimensional konfiguration spas. In fakt for any number of partikls, Q always has six mor dimensions than Platonia.

The Schrödinger equation koms in a tIm-dependent and a tIm-independent form. Barbour sujests, kontrary to konventional wisdom, that the later is the mor fundamental. The wav equation, that finds al the posibl stationary staits of a system, hints at a universal stait of afairs in wich super-positions of stationary wavs kreat a variation in tIm of the probability density.

The stationary staits, such as in Bohr's model of the atom, korespond to a fix'd enerjy level, betwyn quantum 'jumps' with the emision or absorption of a foton. The probability density of finding the atom in thys staits is konstant, wIl the komplex or komposit valus of the wav funktion osilat with a fix'd frequensy. But ading tw such solutions, with ther respektiv frequensys, maks them interfyr. The osilations sys to be regular. Ading thys tImles solutions yet maks the probability density vary with tIm.

Barbour givs a piktorial deskription of the Schrödinger wav funktion in a stedy stait.
At ych point of the konfiguration spais Q, imajin a child swinging a bal in a vertikal sirkl, on a string of konstant lenth, the 'amplitud', denoted by Greek smal leter phi. Its squar'd valu stands for the konstant probability density.

The swinging bal's kontinuusly chanjing hIt, abov or below the senter, stands for uon of the tw order'd pairs of numbers, that mak a komplex variabl. Its other komponent is the distans to the rIt ( positiv ) and to the left ( negativ ).

The stationary stait is lIk swinging such bals at the saim rait, every-wer in Q, and al perfektly in feiz or ryching the top of the sirkl together. In the momentum eigen-stait, phi is the saim every-wer. But jeneraly it varys akording to a kondition, impos'd at ych point of Q, by the equation of the stationary stait. Barbour deskribs this kondition as: Kurvatur number plus Potential number equals Enerjy number.

The kurvatur number is komplikated. For a quantum system of thry bodys, ych point in Q koresponds to a konfiguration of the thry bodys in absolut spas. Holding tw of the bodys fix'd and moving the third, along a lIn in absolut spas, movs on a lIn in Q.
Phi, the string lenth kan be ploted as a kurv mor or les abov the lIn. ( In kalkulus, this kurvatur is the sekond derivativ. A thry-partikl Q has thry tIms thry dimensions of movment. So ther ar nIn such kurvaturs at ych point of Q. The 'kurvatur number' is 'the sum of thys nIn kurvaturs after ych has byn multiply'd by the mas of the partikl for wich it has byn kalkulated.'

The sekond number, the Potential is deriv'd by multiplying phi by the potential. The potential enerjy depends on a given konfiguration of bodys and ther natur, such as ther mases.
The third number, the Enerjy is found by multiplying phi by the previusly mention'd quantum enerjy relation, E = hf. The frequensy, f, is the number of rotations of the 'bals' in a sekond.

Schrödinger kompar'd the stationary stait of the hydrojen atom to a vibrating string, wich is fix'd at either end and must always hav a houl number of wavs ( kounted in haf wav-lenths ) lIk the harmoniks of a musikal instrument. The hIer harmoniks kompar to the atom's hIer enerjy levels. The fundamental nout, wen the string is just uon over-arching and under-arching vibration ( that is, uon haf wav-lenth ) kompars to the lowest enerjy level of the atom, its 'ground stait'.

This analojy suplIs a boundary kondition for the solution of Schrödinger's stationary stait equation, as an explanation of the diskryt enerjy levels, posited in Bohr's quantum model of the atom. This kondition is that the ends of the vibrating string are fix'd, therfor the amplitud of phi must tend to zero at larj distanses.

Wer the enerjy, E, minus the potential, V, is mor than zero, phi osilats. Wer E - V is les than zero, phi tends to zero, only in sertain wel-behav'd solutions ( the eigen-funktions ) for special valus of E ( the enerjy eigen-valus ). The eigen-funktion of the system, with the lowest enerjy valu, is the ground stait. HIer enerjy staits ar kal'd exIted staits.

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.

Relativis'd Schrödinger equation of the kosmos.

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Barbour folows much the saim plan, to dispens with the remaining Newtonian fraim-work in quantum mekaniks, as he did with klasikal fysiks. ( This is diskus'd in my first tw web pajes on 'The End Of Time.' ) The Schrödinger wav funktion, of a given system of partikls, chanjes with ther relativ konfiguration, senter of mas, orientation and tIm.
Barbour dispenses with the later thry, as he did for klasikal dynamiks, sins the relativ konfiguration of the houl universe is its own absolut spas and tIm, deriving them independently of an external fraim-work. This aplIs Mach's prinsipl to Schrödinger's equation for a quantum kosmolojy.

In aplIing quantum ruls to a klasikal theory of kosmolojy, 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 tImles system, the over-al enerjy is zero. So, in the stationary Schrödinger equation, at every point of Q, the sum of the kurvatur number and the potential number is zero. As in klasikal fysiks, the potential alredy is deriv'd from relativ konfigurations of the bodys that mak up a ( Machian ) system, independently of absolut spas and tIm.

As for kurvatur, that is the rait at wich a kurv's sloup chanjes, with respekt to a distans in absolut spas, in ordinary quantum mekaniks. Barbour sujests replasing thys distanses with the Machian best matching distanses in relativ konfiguration spas, as he did to eliminat absolut spas from klasikal fysiks.

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' wav funktions ar the Schrödinger eigen-funktions, hws eigen-valus hav zero angular momentum, wich was the kais for the Machian trytment of klasikal dynamiks.

On platonia or relativ konfiguration spas, only the potential and best matching distans govern the statik wav funktion's variation from point to point. This tImles 'topografy' determins wer the 'mist' of the probability density gathers.

This predikts how probabl al the inkonsyvably many permutations of atomik and molekular strukturs, and ultimatly, Barbour syms to argu, how the most probabl konfigurations of the univers best 'resonat' ych other, in a sort of kompetition for the apyrans of historikal reality.

Quantum theory of rekords.

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Barbour imajins how history emerjes from wat he sys as the esentialy tImles arena of quantum mekaniks. He relIs on John Bell's analysis of how rekords are mayd, in the kontext of radio-aktiv dekay in a kloud chamber, wer an alfa partikl lyvs a trak of ionis'd atoms.

Bell givs tw interpretations of this fenomenum depending on wen it is asum'd a mesurment is taken, that suposedly 'kolapses the wav funktion' of posibl plases the partikl wil be found. The simpler interpretation asums that atom ionisation is the 'klasikal external mesuring instrument' for revyling the alfa partikl, and suksesivly kolapsing the wav funktion, with gradual los of partikl enerjy and inkrysing deflektion, that kan be statistikly predikted.

Bell's sekond interpretation tryts not only the alfa partikl but the houl system in quantum mekanikal terms, that is to say bilions of potentialy detachabl elektrons, from ther hydrojen atoms, al given thry dimensions ych ( together with the alfa partikl's thry dimensions ).
Given tIm for the ionization of, say, a thousand atoms, a foto taks a mesur of the komplyt system, kolapsing the wav funktion onto a komplyt trak, not onto uon position of uon partikl.

In the sekond senario, the wav funktion has a vastly inkrys'd konfiguration spas to serch-out. But this land-skaip is also vastly mor struktur'd and the the wav funktion, lIk a mist setls mor densly, akordingly, determining the points most probably mesur'd. Ych point, in this biger platonia's path, 'lwks lIk a history of the thry dimensional trak up to som point along it.'

DespIt the diferent viw of wen the wav funtion kolapses, the results ar much the saim, bekaus the experiment is a hIly organis'd situation, wer-by a hIly regular Hamiltonian wav funktion produses a semi-klasikal solution that givs the apyrans of a path taken in tIm or a history.
But, in quantum mekanikal terms, it is the wav funktion's probabilistik serch thru a tImles topografy of al posibl historys, that mesurment, 'kolapsing the wav funktion', rylIses as uon history.

Reviwer's koments:

Jeometrys and mesurment theory.

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Barbour belyvs konsistensy demands that a konsiusnes of motion must be in uon konfiguration. He geses that the brain kan tak in several 'snap-shots' at uons and 'play the movy.'

He folows Bell's belyf that 'the past' is inaksesibl and therfor irelevant. We hav only rekords of an ilusory history. Bell, however, didnt denI the reality of tIm, watever his belyfs about it.

This reviwer kan't help but think that rekords ar of som-thing. So, to denI that som-thing is a kontra-diktion. Supos tIm has a komparabl reality to that of spas. It kan be thot of in the saim way. For instans, special relativity tryts tIm, as wel as spas, as having spyd.

Maps ar a rekord of a topografy, to a greiter or leser skail. As Charles Dodgson pointed out, the map kan be inkrys'd to ful skail til the land-skaip is a map of itself. Jeolojikal strata kontaining fosils mIt be konsider'd naturaly skail'd-down tIm-maps of evolution. The strata ar a spatial map of the ajes. Uon kud say they had lost most of ther temporal dimension, rather lIk the Minkowski Interval maks for an inter-chanj of temporal with spatial dimensions, in spas-tIm.

In denying the reality of tIm, Barbour finds himself at ods with other fysisists wishing to develop the spas-tIm fraim-work of relativity theory. I wud lIk to sujest that the Minkowski Interval has an independent ryson for being taken as a basik struktur. Naimly, it syms to fit in the lojik of mesurment, as devis'd by S S Stevens.

Stevens distinguish'd 4 skails of mesurment, the nominal or klasifikatory, the ordinal or ranking, the interval and the ratio skails, 'on the basis of the prinsipl of invarians under transformations' -- how-ever that aplIs. But I hav ilustrated how thys skails aplI to lojik of chois, on my first web paj about 'Sientifik method of elektions'. ( A problem with both natural and social siens is that they dont emfasis the dynamik of nolej of frydom and frydom of nolej. I'v diskus'd this in my tw web pajes on the ethiks of sientifik method and a short web paj on fysiks and frydom. )

Without being abl to giv any sort of expert prwf, I wud lIk to mak som points of komparison betwyn Stevens' mesurment theory and som basik fyturs of fysikal theory. Newton's laws of motion apyr to hav tw out of the for mesurment skails. Relativity theory kan be konsider'd as suplIing the other tw, so that fysiks employs a mor komplytly lojikal system of mesurment.

Ther ar quIt som diferenses in formulation of Newton's laws. I tak Allan M Munn's ( in 'From Nought To Relativity' ). Munn didnt think the sekond law, postulating every aktion has an equal and oposit re-aktion, was operational, bekaus forses other than the mutual uons wil always be present.

Law uon staits: 'Every body tends to kontinu in a stait of rest or of uniform motion in a strait lIn, unles it is kompel'd by an externaly aplI'd fors to chanj that stait.'

Law uon ilustrats Stevens' klasifikatory skail of mesurment. A body's stait is put in tw mutualy exklusiv kategorys, rest or uniform motion. This reflekts Newton's belyf that ultimatly the univers did hav a spatial fraim-work, wich was absolutly at rest, in relation to al motion.

The ordinal skail of mesurment is a lojikal refinment or progresion from dualistik klasifikation. Insted of saying rest is rest and motion is motion and never the twain shal myt, rest is not konsider'd an absolutly distinguish'd stait. Ther is but uon ranj or order of motion by wich observers relat. Ther is only relativ motion. Yst and west may be mor akuratly represented in an order'd kultural kontinuum than as a sharp dikotomy.

Zero motion is not a tru rest but the result of an arbitrary chois of ko-ordinats betwyn observers in relativ motion. Ther is an absolut zero temperatur, wich dos not nesesarily myn molekular motion syses, only that it kan not transfer to other systems. LIk lIt's absolut maximum spyd, this absolut minimum spyd is a limit that kan only be aproch'd.

Hens, temperatur skails also hav no tru zero. Ther is no such thing as uon temperatur being, say, twIs as hot as an other. ( Temperatur is therfor not a ratio skail, karakteris'd by a tru zero, wich is the next lojikal step in the for skails of mesurment. ) The varius skails, naim'd after Celsius, Fahrenheit, etc ar arbitrary but they do hav the property of being in proportion to ych other and translatabl by formula. This is karakteristik of interval skails.

Ther may be a jeometrik sens in wich Minkowski's Interval is aktualy an interval skail. ( The yus of the saim term, interval, is, as far as I no, not deliberat. ) The Interval is akin to arbitrary temperatur skails, wich kan translat betwyn ych other, bekaus it alows arbitrary ko-ordinat systems in relativ motion, to translat betwyn ych other, akording to a komon spas-tIm formula.

The Interval is Euclid's jeometry of thry-dimensional flat spas extended to a for-dimensional spas-tIm. Jeneral relativity adopts a Riemannian jeometry of kurv'd spas-tIm, to alow observers in relativ akseleration to translat ko-ordinats. It is a jeneralisation from Minkowski spas-tIm konsider'd as of zero kurvatur, and thus konstituts a ratio skail jeometry of kurvatur.
The metrik of jeometry myts the theory of mesurment. Einstein's jeometrising endevor in fysikal theory kud as wel be kal'd a mensural endevor.

Of kors, Newtonian fysiks is rationality 'par excellence', as in the third law, wich says 'Rait of chanj of momentum of a body is proportional to the fors akting upon it and it has the saim direktion.'
Never the les, a konsept such as 'fors at a distans' was kritisis'd as mysterius. Newton himself said it was an aparent nonsens, apart from the fakt that his law of gravity work'd. Jeneral relativity replases fors with jeodesiks to determin a body's path.
The point being mayd is that progres in klasikal fysiks was mayd by its bekoming a mor fuly integrated mesurment struktur, born-out by the lojik of progresiv skails in mesurment theory.

TIm dependens on konfigurations.

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Barbour shows how tIm emerjes from chanjing konfigurations of bodys, ultimatly of the houl univers. This raises the question of the natur of the univers in its first ten thousand yers befor mater was form'd. Barbour denIs a temporal development or evolution.

But konsider the konventional viw that befor mater koagulated, ther was only radiation. The kosmik bak-ground radiation is konsider'd a fosil remnant of the big bang. In the sublIm words of Genesis, 'And God said, Let there be light'.
On the basis of Minkowski's Interval, lIt has the fastest motion thru spas, so that it has no spyd left for motion thru tIm. Ther is no passaj of tIm at lIt spyd. A foton has never aj'd from the big bang til the tIm it is employ'd in the dubl-slit experiment.

In this tw-hol experiment, lIt dosnt distinguish tIm. Fotons mak wav interferens regardles of wether yu bym them together or send them uon by uon. Supos this fenomenum is a 'fosil' remnant of the radiativ univers befor material konfigurations kud distinguish tIm.
TIm wud not be distinguish'd from spas. Wen fysisists trI'd to relat the theorys of the quantum and of gravity, they try'd to match ther respektiv thry degrys of frydom, or dimensions, ych. As deskrib'd on the previus web paj, this involv'd a quandry of wether a tIm dimension kud arbitrarily tak over uon of thr spatial dimensions.
The Wheeler-DeWitt equation defer'd this problem of chwsing wich spatial dimension to signify tIm. A justifikation for this desision mIt be that, in a univers yet laking material konfigurations to defin ( efemeris ) tIm, tIm was not yet distinguish'd from spas.

Also, this ryson may not exklud David Deutsch's many-worlds interpretation of the dubl-slit experiment ( in 'The Fabric Of Reality' ). He argyus that the fenomenum of a foton interfering with itself is ryly evidens of an other foton, we kan not persyv -- and hens from 'another univers' -- interakting with the persyv'd foton in our univers. Deutsch's lojik is that just bekaus we kan not direktly persyv the other foton dos not nesesarily myn that it dos not exist. And we shud not ignor the evidens of interferens for its presens.

A temporaly distinkt univers prety much exkluds any kind of 'interferens' from alternativ realitys. Much of Deutsch's bwk is about thot experiments with tardis-lIk tIm macyns that wont result in paradoxes. But the dubl-slit experiment as a fosil from an era, that had not temporaly rigidify'd, mIt indikat a kondition alowing universes to be slItly les exklusiv.

Stephen Hawking's no-boundarys solution, to the jeometry of the univers, also sys no distinktion betwen spas and tIm. This maks yus of Feynmann's quantum elektro-dynamiks.
Richard Feynman ( as in his popular klasik, QED ) givs a kompeling quantum theoretikal explanation of the tw-hol experiment and indyd jeneral lIt fenomena. Barbour dosnt refer to his work and, in jeneral, popular rItings dont sym to relat Feynmann's 'sum over historys' aproch to konventional quantum mekaniks.

In his popular lektur, Feynmann himself didnt yus his own frais, for the fakt that a klasikal partikl has uon history, but a quantum partikl has to hav al posibl historys, with ther asociated wav sIz and feiz, taken into akount. So, for instans, a foton's jurny is kalkulated by suming al posibl paths it kud tak in spas-tIm.

In 'A Brief History Of Time', Stephen Hawking mentions his yus of the 'sum over historys' quantum theory towards his no-boundarys solution to the kreation, or rather being, of the univers.
In doing such sums, wen tIm is mesur'd as an imajinary number ( being multiply'd by the operator, i, wich myrly equals the squar rwt of minus uon ), 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 folowing is a, no dout, mislyding glos on it. ) Uon kud arbitrarily start off tIm, say, at a position akin ( in fwer dimensions ) to the erth's jeometrikal pol and chart the univers's expansion along the lIns of longitud til ryching a maximum expansion ( suposing the univers is not infinitly expanding ) at the equator.
The amount ( and kinds ) of gravitational mas in the univers is not yet akuratly nown. Estimats ar yus'd to ges how much it is slowing down the big bang.

The amount of gravitational mas determins the kurvatur of the spas-tIm path of the univers. This, tw, kan be kalkulated yusing Feynmann's suming teknyk, that is in quantum, rather than in singular klasikal terms.
Penrose and Hawking show'd that the determinism of jeneral relativity ment 'the beginning of time would have been a point of infinite density and infinite curvature of space-time.'
That myns the klasikal laws brok down. But the quantum rul of the unsertainty relation of position and momentum ofer'd a way out, tho uon not yet fuly achyv'd.

The no-boundarys solution wud remov the problem of a begining of the univers and the paradox of wat hapen'd befor kreation. It ofers a self-kontain'd solution that a univers, by definition, lojikly requirs.
Hyr is the saim goul, but by a diferent aproch to tIm, that Barbour syks.

Fysisists, lIk the wav funktion, explor al avenus.

Richard Lung.

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