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Left: Signature of the Zo particle. The Zo, W+ and W- bosons transmit the weak force; their existence was predicted by the unified theory of the weak and the electromagnetic interactions, and their discovery vindicated the theory. The tracks depicted correspond to particles detected following the high-energy collision of a proton and antiproton. The two tracks in white above and below are an electron and a positron, the decay products of the Zo, which disintegrated soon after it materialized in the collision. Right: Top quark candidate at Fermilab 1989. This is the sole piece of the standard model yet to be fully confirmed.

To view my latest review of the cutting edge of quantum reality and quantum cosmology see:
Quantum Cosmology, Complementarity and Sexual Paradox

Unified Field Theories, Symmetry-Breaking and the Inflationary Universe

This vintage article which contains work done in the earlier part of the 2000s has been updated to include the discovery of the Higgs particle and Garrett Lisi's E8-based theory. This still gives quite good coverage of the field as no supersymmeric particles have yet been discovered and no real progress has been made in string theory.

Symmetry and local symmetries are believed to underlie the fundamental forces. Top left: 60 degree rotational geometric symmetry of a snowflake, charge symmetry of electromagnetism and isotopic spin symmetry between a neurton and a proton illustrate symmetries in nature. Right: The electromagnetic force can be conceived of as an effect required to make the global symmetry of phase change local. A global phase shift does not alter the two-slit interference of electron waves (which usually have one light band in the centre), but a phase filter which locally shifts the phase through one slit has precisely the same effect as applying a magnetic field between the slits. The local phase shift causes the centre peak to become split in both cases. Gravitation can likewise be conceived as a symmetry of the Lorenz transformations of relativity, usually referred to as Poincare invariance.

The Four Fundamental Forces and Symmetry-breaking

Four distinct force fields have been discovered in nature. Two are familiar, gravity on the one hand and electromagnetism, comprising many effects including electricity, magnetism, light and the electromagnetic spectrum, and the indirect effects seen in chemical reactions. The other two are distinct forces which occur in the nucleus of the atom.

The nucleus consists of protons and neutrons. These are bound together tightly by the strong nuclear force. This force is responsible for the great energy of nuclear fusion and fission. It is a secondary product of a more inaccessible force which comes in threes, called the colour force which binds the constituent quarks of protons and neutrons.

(a) Unificationof the strong and weak nuclear forces with electromagnetism is suggested by the convergence of the strength of the three non-gravitational forces at high energy (b) The four forces of nature appear to differentiate from a common super-force. (c) Broken electro-weak symmetry has a lower energy than the symmetrical state. This explains how nature has become asymmetric with positive and neutral charges in the nucleus and negative electrons outside. The universe is like a ferromagnetic material which is magnetised in its lowest energy state. The energy released by cosmic symmetry-breaking may represent the hot shower of particles in the big bang. Just as a magnet can be polarised in many directions in space, so symmetry break may be. This could result in anomalies such as cosmic strings, domain walls or magnetic monopoles. Below: The four particles of the unified electro-weak force. The photon, the two charged W particles and the Z0. While the photon is massless the others all inherit a lage mass from symmetry-breaking causing the weak force to be very short-range. The four particles of the unified electro-weak force. (d) The stable nuclei and radioactive decay of the neutron. The balance between neutrons and protons is mediated by the weak force. Too many protons and the electromagnetic repulsion becomes unstable. Too many of either kind and the nuclear energy levels likewise become unstable. Notably the weak force is chiral. The emitted electron is left-spinning.

A second, quite different force, the weak nuclear force, is responsible for radioactive decay. If a nucleus has too many neutrons, one neutron can decay into a proton, an electron and an antineutrino [above figure] This reaction and its reverse act to keep the balance of protons and neutrons, which is roughly 50-50 to keep each nuclear particle in the lowest possible energy states under the strong force, but becomes biased toward neutrons in heavier elements, because of instability caused by the accumulated repulsive positive charges of the protons. Significantly, the reaction does not preserve mirror symmetry, as it gives rise only to left-handed electrons, the anti-neutrino involved in beta decay also has a spefic handedness.

Scalar and vector fields illustrate the classical behaviour or potential functions and electrostatic fields and fluid flows. A scalar is a single quantity wheras a vecotr field in 3 dimensions has 3-dimensional vectors as well. Quantum fields likewise can have differing dimension depending on their spin. Spin-0 fields have one degree of freedom and are scalar. Spin-1 fields have three degrees of freedom and are vectors. Photons, because they are massless have lost the longitudinal mode and have only two degrees of freedom (polarisation). The one additional degree of freedom contributed by the Higg's boson gives back to the weak bosons the degree of freedom they need to be massive and have a varying velocity. Spin 1/2 fermions have two-component wave functions which turn inside out upon a 360 degree revolution leading to the Pauli exclusion principle.

Weak neutral currents mediated by the Z0 can resutl in interactions between uncharged neutrinos and other form of matter.

The weak force is mediated by three types of particle, the charged W+, W-, and the neutral Zo, each of which has a very high rest mass, but otherwise behaves very much as a photon. The heaviest of these was predicted by Stephen Weinberg in a theory which unites electromagnetism and the weak force in a single super-force in which three of the particles have gained a large rest mass by a process called symmetry-breaking, but are otherwise sister particles of the photon. This is basically extending quantum electrodynamics to the weak force.

The very short range of the weak force is immediately explained by this high rest mass. The exchanged virtual particle has to gain at least the energy of the rest mass to exist at all, so there is a very short time and a very small distance beyond which the force cannot occur through uncertainty. By contrast, the photon, which has zero rest mass has only to deal with its energy of transmission and thus can occur with decreasing probability over larger and larger distances. Electromagnetism and gravity are thus long-range forces which fall off gradually with distance.

This new mass is explained through an additional field the Higgs field which has a scalar particle the Higgs boson contributing an extra degree of freedom to the wave function to make it able to change velocity and behave as a non-zero rest mass particle.

Evidence for quarks and gluons: Left: two narrow jets of particles emerge from the collision and mutual annihilation of an electron and an anti-electron, or positron. The detected particles have a variety of masses, charges and spins. If the particles arose directly from the annihilation, they would be expected to follow widely divergent paths. The focused character of the nets suggests instead that each jet developed from a single precursor: a quark or an antiquark. Centre: Three-jet event confirms the existence of the gluon, the mediating particle of the color force. An electron and a positron collided at high energy, creating a quark and an antiquark, as in the previous event. In this case one of the quarks radiated a gluon. The quarks and the gluon diverged; each promptly gave rise to a shower of particles, which preserved the trajectory of the original entity. The event reveals the asymptotic freedom of quarks and gluons: their ability to move independently within a very small region in spite of the enormous strength of the color force across larger distances.

Resonances in particle collision experiments suggest there are three families of fermions.

An important feature of such theories is that a given force can be identified with a local symmetry possessed by the universe. In the case of electromagnetism this is the phase of the wave and in gravity it is the relativistic transformations of space-time. Changing the phase of all wave-particles in the universe has no net effect, but changing the phase of some results in an electromagnetic interaction which makes up the difference by applying a force which changes the phase of only those particles involved. When the larger symmetry between the weak and electromagnetic forces is broken, by some of the particles gaining a non-zero rest mass, the two forces gain their distinctive character. Because the broken-symmetry state has lower energy the universe is no longer in the symmetrical state.

The Particle Menagerie:

Fermions (spin = 1/2):

 lepton  mass  symbol  charge  quark  mass  symbol  charge
 electron neutrino  < 16 eV    0  up  310 MeV  u u u   2/3
 electron  0.5 MeV    e  1  down  310 MeV  d d d  -1/3
 muon neutrino   < 65 eV    0  charm  1500 MeV  c c c    2/3
 muon  106.6 MeV    1  strange  505 MeV  s s s  -1/3 
 tau neutrino  < 65 eV    0  truth  >89 GeV  t  t  t   2/3 
 tau  1784 MeV    1  beauty  5000 MeV  b b b  -1/3 

Neutrino masses and family numbers

Neutrinos may have a small mass. This is consistent with the idea that the neutrino types may be able to interconvert by a resonance similar to that of the Ko meson (see below). This would explain the small observed flux of neutrinos from the sun which is only about 1/3 what it should be for the nuclear energy required to keep it at current luminosity.

In the early universe there was a sea of protons and neutrons constantly interacting with electrons, neutrinos of every type and their anti-particles through weak interactions. Because neutrons are slightly more massive (939.5 MeV) than the proton (938.2 MeV), there are fewer of them. As the expansion separates, these the weak interactions cease, leaving about a 1:5 n:p ratio at 1 second. The neutrons begin to decay with a half-life of 15 minutes. After 3 minutes deuterium becomes stable and is rapidly converted to helium. At this point neutron decay has reduced the n:p ratio to 1:8. These flush out another 1/8 of the particles (protons) leaving a 1:4 ratio of helium to hydrogen. More families of neutrinos than four would cause a faster the expansion rate, and the faster reaction would produce more helium than observed.

Experiments on the supernova 1987A limits electron neutrino mass to less than 16 eV. All neutrinos must have a mas less than 65 eV or the universe will be closed and collapse and moreover the expansion rate would be slower than observed. The observed mass is only 10 to 20 percent of the closing mass. Neutrinos cannot be too massive. There are even more than photons, several billion for every proton, electron and neutron.


 force  range  strength at 10^ -13 cm  particle  mass  spin  charge  status
 gravity  infinite  10^ ­38  graviton  0  2  0  conjectured
 electromagnetism  infinite  10^ ­2  photon  0  1  0  observed
 weak < 10^ ­16 cm  10^ ­13  W+  81 GeV  1  +1  observed
       W­  81 GeV  1  ­1  observed
       Zo  93 GeV  1  0  observed
 strong < 10^ ­13 cm  1  gluons  0  1  0  confined

The electroweak unification occurs at around 10^11 eV. Current limits on collider energy are around 10^13 - 10^14 eV. The hypothetical targets for unification with the colour force have ranged upward from 10^15 to 10^24 eV. Complete unification with gravity requires the Planck scale requires energies of 10^28 eV, requiring a collider 1000 light years around to probe. This makes it difficult for supercolliders to push our knowledge of unification further. It is likely we will have to develop theoretical insight more deeply and use indirect evidence from the universe at large.

The standard model of particle physics: Particles are divided into half-integer spin fermions which obey the Pauli exclusion principle and can only exist in a single wave-function in pairs, thus becoming matter, and integer-spin particles called bosons, which can clump in any number (lasing) and thus form the radiation and force fields. The fermions form two types, leptons which are light and do not experience the colour force ( electrons and their neutrino partners) and the quarks which make up protons and neutrons. Each quark comes in three colours (red, green, blue) and two flavours (up and down). Decay of the neutron into a proton, electron, and anti-neutrino is mediated by the weak nuclear force. This force also mediates flavour in the nucleus. The strong nuclear force, which binds the protons and neutrons together, is a secondary effect of the coulour force which binds quarks together inside these particles. By exchanging the three colours, just as the electromagnetic field mediates charge, gluons bind the quarks together with a force which is relaxed at close distances but becomes stronger as they are pulled apart. This causes quarks to be confined within nuclei and protons and neutrons. The antiquarks are also shown top left to illustrate their anti-colours. There are likewise anti-particles of the leptons such as the anti-electron (positron). Both quarks and leptons come in three (or possibly four) series of rapidly increasing mass. The bosons mediate each of the forces. The photon and graviton are the only massles bosons. The other three electro-weak W and Z particles have inherited a large rest mass (and an extra degree fo freedom) by coupling to a hypothetical spin-0 particle called the Higg's boson, as shown. The photon is exchanged only between charged particles, but the Zo interacts even between neutral particles. The W particles are themselves charged, so emit and absorm photons. There are 8 independent degrees of freedom in the colour field mediate by gluons. Like the quarks these remain confined. The Higgs particle was confirmed at the LHC in 2013.

A similar mechanism is believed to unify these two forces with the strong nuclear force. Every proton and neutron is itself believed to consist of three subparticles called quarks as follows: n=udd, p+=uud. Neutron decay is thus actually the transformation of a down quark into an up [see fig above]. The three quarks are bound together by a force, called the colour force because each quark comes in one of three colours, just as electric charges come in two types, positive and negative. Each neutron has one up and two down quarks and each proton two up and one down. To balance the charges each up must have charge 2/3 and each down -1/3. However, regardless of their up or down flavour, there is always one of each colour, so that the proton and neutron are colourless.

Top left: Mesons (quark-antiquark0 and baryons (three quarks) mediate their color by exchanging gluons of appropriate color-anticolour combinations. Top right: The electromagnetic field reduces effective charge by forming virtual electron-antielectron pairs. The colour force also does this by forming quark-antiquark pairs, but in addition the gluons have a colour charge (unlike the uncharged photon) which increases the effective charge towards infinity at great distances, while remaining relaxed at short distances, allowing the quarks to move freelywithin a confined space. This phenomenon, which is known as camoflage is also illustrated in the lower series of diagrams where electromagnetism has only shielding while colour has shielding and camoflage. The effect of quark and gluon confinement is that individual particles cannot be isolated. When they are drive apart in a very energetic collision, a shower of particles results which eventually neutralizes the colour charge.

The colour force is mediated by particles called gluons, which like the electroweak family are vector particles of spin 1. The colour force and its secondary effect, the strong nuclear force, are believed to be unified with the weak and electromagnetic forces in a similar manner. In fact the strengths of these three forces converge to the same value at a high temperature called the unification temperature for this reason. Because the colour force generates quark-antiquark pairs as any quark is pulled apart from the rest, a quark cannot simply escape, but instead generates a whole jet of excited particles.

Decay of the Ko meson is a parallel to photon polarization. The K1 component (see below) by decay is similar to vertical polarization removing the horizontal component from circularly polarized light. However there is a small amplitude for the K2 to go into resonance back into the K1 form, just as dextrose rotates the polarization of light, allowing it to subsequently decay again, similarly to detecting horizontal polarization in the rotated light. Lower left Feynman diagram for quark flavour mixing. Just as classical chirality requires three dimensions, CP-violation of the Ko requires at least three families of fermions. Recent investigations of the B meson containing a b (beauty) quark indicate flavour mixing suggesting a fourth family is possible. Then can be no more than four or the extra neutrino types would cause an unrealistic expansion rate of the universe.

The weak force is known to be chiral (see above), but the asymmetry of nature runs even deeper. In 1964 the principle of charge-parity conservation was oveerthrown by the neutral K meson, which usually decayed into 3 pions but once in 500 times was found after a strange delay to decay into only two. The neutral Ko meson, and its antiparticle both decay into a pair of mesonsThe rapid decay of the component into pi-mesons, subsequently leaves the remaining component which does not follow the same decay. However subsequently there is a small amplitude for conversion of some of the K2 back to K1 resulting in a KL which is not matter-antimatter symmetric, since it contains differing components of Ko and its anti-particle. Thus the reaction is preferred over the mirror-image Since the Ko has quark constituents (d, anti-s) and its anti-particle (anti-d,s), this implies that the reaction should be directed in time. Similar considerations are used to explain the preponderance of matter over anti-matter. It is suggested that the one part in 10^8 of matter to radiation could have come from a similar process resulting in a slight differential in the stability of matter an anti-matter with respect to time.

Right: The SU5 theory, an attempt to make an immediate extention of the ideas of the electroweak unification to unification with the colour force. The principal difficulty with this theory is that its prediction that the proton should also be unstable, like the neutron, has not proven to be validated, despite major experimental efforts of underground scintillation detectors. Right: Proton decay. The diagram shows one of several proposed decay routes of quarks into leptons. The proton's constituent u quarks combine to form an X particle, which disintegrates into a d antiquark and a positron (a lepton). The d antiquark combines with the remaining quark of the proton, a d quark, to form a neutral pion. Because pions are composed of matter and antimatter, they are short-lived; the mutual annihilation of their constituents will release energy in the form of two photons. The failure to detect proton decay has pushed up the energy of possible unification by orders of magnitude. A more fundamental symmetry proposed is supersymmetry, a hypothetical symmetry between fermions and bosons which identifies each with a partner of one half spin less. The spin-2 gravitonwould thus have a series of partners, as shown as left. Supersymmetric theories are usually developed in higher dimensional spaces in the form of string theories, in which particles become harmonically excitable strings of perhaps 10 to 12 dimensions at very small distance scales.

Bosons and Fermions : Unification with Gravity

There are more problems however when an attempt is made to unify gravity with the other three forces. This requires an explanation of another important feature of wave-particles, their spin. All quanta behave rather like a spinning top. For example an electron in orbit round an atom can have spin 0, 1 etc. giving the s, p orbitals shown above. However individual wave particles have a permanent spin which can either be an integer or a half-integer.

In superstring theories the infinities associated with point vertices and instantaneous interactions become smoothed into continuous tubes with no precise interation moment.

The particles of integer spin are called bosons. They generate radiation and are responsible for each of the force fields we have mentioned, the photon for electromagnetism, the graviton for gravitation etc. Spin-0 particles generate a one-component scalar field, a pure intensity distribution. Spin-1 particles, such as the photon, generate a three-component vector field just like a three dimensional flow in space and mediate attractive-repulsive forces as we see in twos in electromagnetism with like and unalike charges and in threes in the colour force with like and unalike colours. Spin-2 particles have even more degrees of freedom and generate a univerally attractive force as we see in gravity. All bosons are distinguished by their capacity to behave like simple harmonic excitations and to clump together in a single wave to form a coherent excitation as we see in the laser in the case of light.

The many modes of string excitation could explain the many different types of particle and force.

However all the fermions, with half-integer spin, behave in a completely different manner. They can only admit two particles with opposite spin into any given state. This means that they will not clump together in more than a single pair. They consequently form matter, which appears solid because atoms cannot be compressed into the same space, as light or other radiation can. Each filled electron orbit around an atom thus contains precisely two electrons. The electron, proton and neutron are all 2-component spin-1/2 fermions (as are their constituent quarks) and comprise the matter we see around us. All fermions display counter-intuitive properties, such as having to be spun through two whole revolutions to return to their original state. One revolution takes them to the 'minus' of their state.

While the bosons come in several types making up each of the force fields, the fundamental fermions form a relatively simple series of six quarks, three electron-like particles, and three neutrinos, which are like electrons but carry no charge and no rest mass or a very much smaller one than even that of the electron. The interaction strengths of the forces is consistent with just three series of fermions.

Of course we also have the anti-particles of each of these as well! Some of the more exotic bosons [e.g. the neutral K meson, which is its own anti-particle] decay in a manner which breaks the symmtery between matter and anti-matter. A similar mechanism involving very slow decay of the proton may explain the preponderance of matter in our universe.

Compactification of many dimensions could occur on very small scales of space-time.

Two theories which attempt to unify gravity with the other forces are supergravity and its cousin superstring theory. Supersymmetry associates each particle of a given spin with a particle of spin 1/2 more or 1/2 less, and thus represents a hidden transformation which could generate all the particles out of pairs of associated bosons and fermions. For example the spin 2 graviton might be associated with a spin 3/2 gravitino, a spin 1 graviphoton ans a spin 0 graviscalar. Unfortunately at this point such theories cannot account for all the known particles properly.

A second extension, superstring theory, gets round the infinities associated with point particles by describing each particle as a very tiny string or loop which can become excited if we make the energy too high or the time or distance too small. This would mean that particles are not points of infinite energy after all, but tiny loops and interacting particles are a little like tubes joining or separating. Hidden within the loops are also believed to be between 10 and 26 dimensions of space-time which extend our familiar 4-dimensional space-time. Superstring theories have however had some difficulty making precise enough predictions to be testable, and it remains unclear just how the higher dimensions compactify into four. There are a very large variety of ways the higher dimensions can compactify into our usual four-dimensional space-time. It is significant that the four-dimensions of space-time prove an apex of complexity for topological manifolds, which cannot be computably classified (Penrose Shadows of the Mind 1994 378).

Traditionally string theorists have examined manifolds or their generalizations with singulatities - orbifolds, however more recently the topological approach has been opened up to allow for discontinuities in form.

A major breakthrough has recently occurred through the generalization of string theories to higher dimensions, something called p-brane or membrane theory. A new concept called duality (a common notion throughout quantum mechanics and mathematics generally). This type of duality associates certain string or m-theories as dual. This makes it possible to solve the infinities in one string theory by examining its dual because the dual of a non-convergent theory is generally convergent. This idea can be well pictured by contrasting quantum electrodynamics with nuclear interactions. In quantum electrodynamics, each more elaborate Feynman diagram contributes a term only 1/127 (ee/hc) as strong, so the infitinite series of calculations converges with stunning accurcay to seven decimal places. It is such infinities or probability non-conservations (probabilities that don't add to unity) that plague the unification schene. The strong force for example is difficult to calculate as a series because every type of excited exchange is equally probable. However duality may provide a way out but providing a dual convergent theory in which such calculations can be done in these difficult cases.

"It's all string theory to me" NY 28 Sep 98

A variety of dualities have begun to emerge which show some promise of condensing the many candidate string theories into one scintillating jewel of dualities. This may oead to a new view of particles in which large composites such a magnetic monopoles which were previously thought of as a composite of a vast collection of quarks and bosons of every sort as fundamental in the dual theory, while apparently fundamental particles such as quarks become dual composites. This would provide a fundamental change of reference from our atomic search for the fundamental constituents. Something a little more like Fritjof Capra's original view of the "Tao of Physics."

Supersymmetric theories generally require over 10 dimensions to converge, all but four of which are 'compactified ' - curled up on sub-particulate scales, leaving only our four dimensions of space-time as global dimensions. Such 'theories of everything ' or TOEs have not yet fully explained how the particular arrangements of particles and forces in our universe are chosen out of the millions of possibilities for compactification these higher dimensional theories permit when supersymmetry is broken to produce the particles and forces we experience at low energies.

The internal symmetry dimensions of existing particles come close to the additional number required, suggesting the key can be found in the known particles. If we take 1 for the Higgs, 1 for the neutrino, 2 for the electroweak, 3 for colour, and 4 for space-time we have 11. Four-dimensional space-time is optimal mathematically for complexity. In some unification theories, one of the compactified dimensions might be much larger. Duality, in which fundamental particles in one description may become composite in another and vice versa may also enable apparently divergent theories to be understood through a convergent dual.

Relation between M-theory and dualities between string theories (ex Hawking R303, Duff R170).

Recently a possible unification of several theories including 10 dimensional superstring theories and 11 dimensional supergravity have been proposed in the form of M-theory -for membrane, or according to its proponents, magic. The essential idea is that 11-dimensional membrane theory looks like 10-dimensional string theory if one of the two membrane dimensions are rolled up into a tiny tube along with one of the 11-dimensions. In this point of view several of these theories are actually complementary mathematical formulations of the same object. This brings in a second mysterious concept - the 'holographic principle ', in which a theory in a multidimensional region can be equivalent to a theory on the boundary of the region, one dimension lower (Duff R175).

A possible key to the higher dimensional theories is the 8-dimensional number system called the octonians. Just as complex numbers form a two dimensional plane, for which the second component is a multiple of i, the square root of -1, octonians form a system of 8-components. Associated with the octonians are the exceptional symmetry groups such as G4 and E8. Internal symmetries such as that of colour, and of charge, as well as the well-know Lorentz transformations of special relativity are already the basis for explaining the standard model.

Another key to a possible unraveling of the Gordian knot of the theory of everything comes from dualities. Electromagnetism is renormalizable because by adjusting for the infinite self energy of a charge we arrive at a theory like quantum electrodynamics where each more complicated diagram with more vertices makes a contribution 137 times smaller to the interaction and it is then possible to correctly deduce the combined effects without infinities creeping in. Essentially the idea is as follows:

Octonians and the Fano plane: Just as complex numbers have two components a + bi with i2 = -1, so the octonians have eight components 1, e1, ..., e7 such that ei2 = -1. Multiplication of coordinate vectors is determined by the 'Fano plane '. Any ei , ej, ek connected by arrows multiply in the manner ei x ej = ek. Those connected in the reverse direction inherit a minus sign. Each line also loops back to the first coodinate in a cyclic manner.

Particles can come in two types, one vibrational states of strings (vibrating particles) and the other topological - how many times a string wraps around the compactified dimension (winding particles). The winding particles on a tube of radius R are identical to the vibrational particles on a tube of radius 1/R. Duality is a sexually-paradoxical concept in which there is a natural relationship between theories which continue to have strong interactions and the perturbation theory fails with dual theories whose interaction strengths are the reciprocals of the originals and hence converge nicely. The nemesis comes if we end up having to deal with a TOE whose interactions are mid rage so that neither the original nor the dual can be unraveled.

Duality between string theories. Winding particles in one have the same energetics as vibrational particles in the other and vice versa (Duff R175). The concept of duality may solve intractable infinities by finding a dual theory which is convergent. In the dual theory, particles like magnetic monopoles, which are a composite of quarks and other particles, become fundamental and electrons and quarks become composites of these. No particle is thus truly fundamental, each locked in sexual paradox with its dual.

Another dimensional issue is that the only spheres which will admit a vector field without singularities, so-called 'hairy ball 's, are S1 the circle, and S3 , S7 the 3-D and 7-D spheres. Our two-sphere S2 always gets places where one hair stands on end like the crown of your head. Thus the status of the unit octonians has a dual 7-D coincidence between algebra and topology, which may be essential in establishing for example a uniform time flow.

Stephen Hawking, who has been a consistent champion of the TOE quest, has lamented that although the connections implied by M-theory dualities are so convincing that to not think they are on the right track "would be a bit like believing that God put fossils into the rocks in order to mislead Darwin about the evolution of life " (Hawking R290 57), he now worries (R291) that the search for a consistent theory may remain beyond reach in a single theory because of the implications of Godel 's theorem (p 491), which proves that any logical system containing finite arithmetic admits formally undecidable propositions. If the search for a TOE runs up against this nemesis, the description of the universe may become undecidable. An indication of the possible complexity of a TOE uniting gravity and quantum field theories comes from superfluid helium 3. At close to absolute zero, helium 3 remains superfluid, and as the temperature rises fractionally a number of bound quantum excitations rather like quasi-molecules, form in the medium. Many of the known properties of unified field theories can be modeled using superfluidity and these bound structures as equivalents of gravitational and the other quantum fields. This indicates that the theory sought may not just be a limit of gravitation and quantum fields, but a deeper theory in which both of these are merely stability states hinting again at the implications of Godel 's theorem.

Above: A depiction in Garrett Lissi's Exceptionally Simple Theory of Everything. Below: Superstring theories suffer from having many different forms of compactification during symmetry breaking. Here an attempt is made to find why our universe has an optimal configuration among the millions of possibilities.

In late 2007 Garrett Lisi published "An Exceptionally Simple Theory of Everything" (R790) setting out a possible scheme for a theory uniting gravity with the other forces based on root vector systems generating E8 "via a superconnection described by the curvature and action over a four dimensional base manifold". Although this theory remains speculative, it brings together an ingenius utilization of the internal symmetries of E8 with the dynamical topology of the underlying manifold, retaining an intrinsic complementarity between discrete and continuous aspects, despite its manifestly algebraic basis.

2011 LHC Results Narrow the Window: Not much room left for Higgs or Supersymmetry

In 2012 - 2103 the Higgs boson discovery was confirmed at the LHC completing the Standard model of physics.

One particularly siginificant prediction of this model is that the universe may be algebraically symmetry-broken so that the bosons and fermions give a balanced positive and negative contriution to the mass-energy of the universe, but collectively rather than in supersymmetric pairs, while each have different numbers and arrangements of particles, as is the case in the standard model. In the 240 dimensional root system of E8, there are 112 'bosonic' root vectors with integer coordinates and 128 'fermionic ones with half integer coordinates as shown in the figure below. Both types are 8-D vectors with Pythagorean length 2 and coordinates adding to an even number.

The possibilities remain open between our universe having unique laws derived from fundamental symmetries or being one of many types of universe whose laws happen to support complexity and life - a 'many-universes ' perspective. Some theories (Smolin R649) even suggest the laws of nature might be capable of evolution from universe to universe, resulting in one containing observers. The anthropic principle asserts that the existence of (conscious) observers is a constraint delimiting what laws of nature are possible. Anthropic arguments (Barrow and Tipler R45) may enable a form of self-selection in the sense that simple universe which could not sustain life or observers would never be observed, guaranteeing our universe has dimensionalities, symmetry-breakings giving rise to fundamental constants consistent with the interactive fractal complexity (p 317). Regardless of these uncertainties in the final TOE, the general features of force unification, symmetry-breaking and inflation are likely to remain part of our understanding of the cosmic origin.

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1.10 General Relativity, Black Holes and Space-time Foam

The general theory of relativity goes on to establish that the force of gravitation, which is indistinguishable from the inertial effects of acceleration, also changes the structure of space-time, curving it so that light no longer travels in straight lines. At an extreme, this makes possible the bizarre changes in which black holes can collapse matter and trap radiant light in the intense curvature of space-time.

A black hole is formed when a star forms a supernova, after burning all its nuclear fuel and running out of energy. Because the binding energy of the heavier elements starts to rise after iron, the last nuclear reactions produce a great deal of neutrinos but negative net energy. The excess neutrinos blow the outer gas off the star, leaving a cold core which collapses under gravity, to a brown dwarf or neutron star. If the mass is large enough, gravity overwhelms all other forces and space-time collapses to form a region which traps even light.

However certain aspects of quantum theory remain at odds with relativity. Relativity is a classical deterministic theory in which space and time are united in a four-dimensional space-time description. Causality is preserved over time-like intervals, but not however over space-like ones, where causality may be reversed by a Lorenz transformation. In quantum theory, causality emerges only from the pattern of many events, each of which has a random non-causal character from uncertainty.

When an attempt is made to reconcile gravity and quantum mechanics, the situation becomes complicated. A virtual particle can have arbitrarily high energy (or momentum) over very short times (distances). This means that instantaneous virtual black holes become possible. Relativity then asserts that the structure of space-time becomes like a tangled foam of worm holes on a very small scale, raising serious questions about causality as well. Recently a version of string theory may explain why black holes cannot radiate and thus erase the quantum history of all particles it traps on its event horizon. In effect the inflation of strings at a black hole may bring new effects into play which preserve this history.

The Big-bang and Cosmic Inflation

When astronomers examined examined the spectral lines of hydrogen in distant galaxies they discovered that they were shifted to longer wavelengths (reddened). The wavelength of the light is stretched by the Doppler effect just as a horn from a train sounds lower as it races away. More distant galaxies are thus travelling apart from one another at velocities which increase steadily with the distance apart. An explanation had to be found for the expanding universe.

The big-bang model proposes that the universe exploded from a singularity of very high density and temperature around ten thousand million years ago. The model predicts that shortly after the explosion, when neutral atoms formed from a charged plasma, the thermal radiation would have become decoupled from the matter [photons are exchanged only between charged particles] and should still exist, phenomenally stretched, having reduced in apparent temperature from several thousands of degrees to only about 4 degrees above absolute zero. The confirmation of the cosmic backgroud became a key confirmation of the big-bang theory.

However the big-bang may be less than a massive explosion. One of the most succinct explanations describes it more as a mere fluctuation arising from quantum uncertainty amplified by the same rules which cause the differentiation of the forces of nature. In this model the forces start out being in their symmetric state and the force-mediating particles have zero rest mass.

The photon has three degrees of freedom, which can be thought of as three modes of oscillation, two transverse to the direction of motion and one longitudinally along it. However, because its rest mass is zero, the photon travels only at the speed of light and consequently the longitudinal component is thus missing. One way the heavy sister particles which mediate the weak force could gain their non-zero rest mass is to pick up another particle of spin-0 which would add the missing single component. A particle called the Higgs-particle is believed to have this role.

Suppose the universe starts out very small [in fact it could be a quantum fluctuation of the vacuum], with the three non-gravitational forces in a symmetrical state, at a temperature which is high in our terms, but cooler than the unification temperature. Because the non-symmetrical arrangement of the forces we see in the universe today has a lower energy than the symmetrical state, it would now be possible for the forces to break symmetry and collapse into a lower energy non-symmetrical state.

Inflationary model solves the horizon and flatness problems. The universe is uniform because ofthe inflation of features, areas can be some so separated that light could not have crossed between them since the origin and even quantum fluctuations would become expanded to features the order of magnitude of the uiverses own size, consistent with the features of the COBE picture ofthe cosmic background.

While they remain in the symmetrical state, the vacuum contains a very high energy, which is concentrated in the Higgs field. Because the universe is below the unification temperature, it doesn't actually have enough energy to support this field, which consequently represents a large negative energy density. Consequently general relativity causes a large anti-gravity effect, in which the global curvature of space-time declines and the vacuous universe grows exponentially in a manner not limited by the velocity of light, growing close to its present size in a fraction of a second.

At some point in this expansion, the supercooled universe did just what super-cooled water does. It froze, breaking the symmetry of the forces of nature into one of many possible non-symmetric orientations, just as a piece of iron can be magnetized in many possible directions. This halted the inflation and released the latent heat of fusion in a shower of high energy particles. From this point on, the universe cools and expands just as if it had come from an explosion.

Recently doubts have begun to emerge that there is enough matter in the universe to stop its continued expansion and suggestions even that the rate of expansion is increasing this has led to two alterantive ideas. Firstly that there is a very slight but non-zero repulsive cosmological constant causing long-range repulsion in tha vacuum of space. Secondly open inflation a two-phase inflation process which makes the universe appear hyperbolic from inside although still appearing a bubble universe from without. These are discussed below.

Quantum Cosmology

New inflation theories to account for the universes mass and expansion:

Dark Matter / Tachyons

The fractal inflation process may bud off new baby universes which then dominate space-time.

However, because gravity is now attractive, rather than repulsive, energy has not been conserved. The kinetic energy of the hot particles and the expansion, instead of being cancelled by the gravitational potential energy, now adds to it. Suddenly the universe has far more mass-energy than it started out with, having been bootstrapped from a fluctuation of nothing at all! Such a universe is predicted to be right on the edge between collapsing again and expanding forever. This would mean that there must be nearly ten times as much mass hidden away in forms of dark matter than the amount in visible galaxies. Such dark matter is currently under investigation. If the neutrino has a very small mass, this could partly be reponsible for the dark matter, as could other exotic particles associated with unification and small dark star-like aggregations.

When the cosmic background is examined more closely, it has very small fluctuations which are of a size consistent with the inflation model, as shown by the famous pictures from the Cobe satellite. These quantum fluctuations once inflated may have become the seeds of the galaxies such as our own milky way.

Even in an isolated closed system, entropy occasionally fluctuates to a low point. The singular nature of the big bang or inflationary origin may likewise result in a massive global lowering of entropy due to inherent symmetries and uniformities of the pre-cursor state.

The Arrow of Time and the Dimensions of Space

We live in a universe which apparently has three symmetric dimensions of space, and one dimension of time in which symmetry has been broken and time elapses, flowing in one direction.

The arrow of time can be attributed to one of several possible causes:

  1. The direction of increasing entropy. This may however be an effect rather than a cause.
  2. A direction resulting from an inflated quantum fluctuation to a positive energy retarded universe.
  3. A consequence of symmetry-breaking.
  4. A consequence of the big-bang being a transactional mirror which reinforces only retarded particles.
  5. The selection of a universe with a high degree of initial symmetry and a heat death as the only possible configuration containing observers.
  6. A manifestation of the equations of chaos.

Superstring theories are couched in higher numbers of dimensions, but neither the real arrangement of the fundamental particles, nor the form of compactification into from say 10 dimensions into 4 is explained. Since compactification arises through some dimensions being confined within strings, it is logical that the process of compactification is identifiable with the arrangement of the bosons and fermions. The SU5 model combines the three dimensions of colour and the two dimensions of charge into a five-fold symmetry with the scalar higgs field added we have six with four dimensions of space-time makes our ten! Gravity would be associated with space-time.

The Ultimate Fate of the Universe

If the universe expands forever, it will end up degrading into a collection of black holes travelling ever further apart. If it eventually collapses again, it could end up in a sea of radiation and infalling matter returning to a singular state. In some theories space-time becomes curved without boundary so that the singularity is like the north pole of the earth, a place much like any other except that time and space look alike (time has become imaginary), as Stephen hawking has suggested. You could visualize the big-bang in space-time starting with the south pole, with the circles of longitude the expanding universe and the north pole the big crunch. Other ideas are that the big crunch might give birth to a new cycle of creation. In other theories, it can evolve to form a new universe with slightly different laws of nature. Both black holes and the big crunch could lead to baby universes or tube-like links with another universe. Another possibility is that inflation is a fractal, like a snowflake, which spawns non-inflating regions like our current universe as it grows. The difficulty of conceiving of the generative beginnings and the possible ends is that the laws of nature are pushed to such an extreme that we know little about it except that uncertainty appears to be able to utlilze every conceivable possibility.

The hope of the unified field theories approach is that many or all of the features of the laws of nature will be found to be determined by mathematical symmetries or uniqueness theorems which determine the structure and strengths of the forces and the masses of material particles. However the form of the universe is very dependent on such parameters as the fine structure constant determining the relative strength of the strong nuclear and electromagnetic interactions, the relative masses of the proton and electron and so on. A contradictory approach, called the anthropic principle, is to explain the values of such such parameters in terms of the fact that for a universe with conscious or even biological observers, to exist significant constraints must be imposed on these parameters. A universe which deviates significantly from our laws of nature might exist in a rudimentary sense, but no one would be aware of its existence, because the condititions for biological life and conscious observation are lacking.

  1. The Number of Families of Matter Gary Feldman and Jack Steinberger Sci. Am. Feb 91
  2. Elementary Particles and Forces Chris Quigg Sci. Am.
  3. Gauge Theories of the Forces between Elementary Particles Gerard 't Hooft Sci. Am.
  4. Beyond Truth and Beauty: A Fourth Family of Particles David B. Cline Sci. A. Aug 88
  5. Gravty and Antimatter Terry Goldman, Richard Hughes, Michael Nieto Sci. Am.
  6. Particle Metaphysics John Horgan Feb 94

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