Genesis of EdenBosons 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.
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.
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.
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:
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.
