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(a) The cosmic background - a red-shifted primal fireball. This radiation separated from matter, as charged plasma condensed to atoms. The fluctuations are smoothed in a manner consistent with subsequent inflation. (b) Fractal inflation model leaves behind mature universes while inflation continues. (c) Darwin in Eden: "Paradise on the cosmic equator. " - life is an interactive complexity catastrophe resulting from force differentiation (Scientific American, King).
To understand how sexual paradox may lie at the foundation of cosmology, it is necessary for us to digress into the spooky world of quantum reality. We exist in a quantum universe, not a classical one. We thus need to set aside the ideas of mechanism, determinism and the mathematical notions of sets made out of discrete points and come to terms with ultimate paradoxes of space-time and complementarities so deep that it may be impossible to determine which are the fundamental constituents and which are composites. To fully understand the cosmoloigcal implications of sexual paradox we need to examine all aspects of the universe in detail from the smallest particles to the universe as a whole and only then to come to a synthesis of the role sexual paradox may play at the cosmological level.
The Quantum Universe
The universe appears to have had an explosive beginning, sometimes called the big bang, in which space and time as well as the material leading to the galaxies were created. The evidence is pervasive, from the increasing red-shift of recession of the galaxy clusters, like the deepening sound of a train horn as the train recedes, to the existence of cosmic background radiation, the phenomenally stretched and cooled remnants of the original fireball. The cosmic background shows irregularities of the early universe at the time radiation separated from matter when the first atoms formed from the flux of charged particles. From a very regular symmetrical 'isotropic ' beginning for such an explosion, these fluctuations, which may be of a quantum nature, have become phenomenally expanded and smoothed to the scale of galaxies consistent with a theory called inflation. Our view of the distant parts of the universe, which we see long ago because of the time light has taken to reach us, likewise confirm a different more energetic galactic early life. We can look out to the limits of the observable universe and because of the long delay which light takes to cross such a vast region, witness quasars and early energetic galaxies, which are quite different from mature galaxies such as our own milky way.
The ultimate fate of the universe is less certain, because it 's rate of expansion brings it very close to the limiting condition between the gravitational attraction of the mass energy it contains ultimately reversing the expansion, causing an eventual collapse, and continued expansion forever. The evidence is now in favour of a perpetual and possibly accelerating expansion and astronomers are seeking an explanation for this apparent lack of mass in dark matter and a dark energy sometimes called quintessence, promoting accelerating expansion. The missing mass is clearly evident in close galaxies, which spin so rapidly they would fly apart if the only matter present was the luminous matter of stars black holes and gaseous nebulae. WMAP data now suggests the universe 's rate of expansion has increased part way through its lifetime and that its large-scale dynamics are governed mostly by dark energy (73%), with successively smaller contributions from dark matter (23%) and ordinary galactic matter and radiation (4%).
The inflationary model explains the big bang neatly in terms of the same process of symmetry-breaking which caused the four forces of nature, gravity, electromagnetism and the weak and strong nuclear forces to become so different from one another. The large-scale cosmic structure is thus related to the quantum scale in one logical puzzle. In this symmetry-breaking the universe adopted its very complex 'twisted ' form which made hierarchical interaction of the particles to form protons and neutrons, and then atoms and finally molecules and complex molecular life possible. We can see this twisted nature in the fact that all the charges in the nucleus are positive or neutral protons and neutrons while the electrons orbiting an atom are all negatively charged. Symmetry-breaking is a classic example of engendering at work. Cosmic inflation explains why the universe seems to have just about enough energy to fly apart into space and no more, and why disparate regions of the universe which seemingly couldn 't have communicated since the big-bang at the speed of light, seem to be so regular. Inflation ties together the differentiation of the fundamental forces and an exponential expansion of the universe based on a form of anti-gravity which exists only until the forces break their symmetry (p 311). Inflation explains galactic clusters as phenomenally inflated quantum fluctuations and suggests that our entire universe may have emerged from its own wave function in a quantum fluctuation.
Our quantum world is very subtle and much more mysterious than a mechanical 'building blocks ' view of the universe with simple separate classical particles interacting in empty space. Many people lead their lives at the macroscopic level as if quantum reality didn 't exist, but quantum reality runs from the very foundations of physics to the ways we perceive. Our senses of sight, hearing, touch and taste/smell are all distinct quantum modes of interaction with the environment. Senses aren 't just biological adaptions but fundamental quantum modes of information transfer, by photons, phonons, solitons and orbital interactions. Quantum processes such as tunnelling are central to the function of our enzymes and to the ion channels and synapses that support our excitable neurons (Walker R724).
The 'correspondence principle ' by which the quantum world is supposed to fade into classical 'reality ' is never fully realized. Many phenomena in the everyday world involve chance events which themselves are often sensitively related to uncertainties at the quantum level. Chaotic, self-critical and certain other processes may 'inflate ' quantum effects into global fluctuations. Conscious interaction with the physical world may likewise depend both on quantum excitations and the loophole of uncertainty in expressing 'free-will '. We need to understand how quantum reality interacts with conscious experience, however in doing so we immediately find the most challenging examples of sexual paradox that lie at the core of the cosmological puzzle. This is the paradox of wave-particle complementarity. A quantum manifests in two complementary ways as a non-local flowing 'wave ' which has a frequency and as a localized 'particle ' which is created or destroyed in a single step. It can manifest as either but not both at the same time - a kind of Janus transvestite hermaphrodite - dyadic sexual complementarity in monadic primal form. All the weird quantum paradoxes of non-locality, entanglement and collapse emerge from this complementary relationship. To understand the full dimensions of this mystery we need to do a little fairly simple maths.
Wave-Particle Complementarity and the Cat Paradox
Supposing we try to imagine how we would calculate the frequency of a wave if we had no means to examine it except by using another similar wave and counting the number of beats that the 'strange wave ' makes against the standard wave we have generated. This is exactly the situation we face in quantum physics, because all our tools are ultimately made up of the same kinds of wave-particle quanta we are trying to investigate. If we can't measure the amplitude of the wave at a given time, but only how many beats occur in a given period, we can then only determine the frequency with any accuracy by letting several beats pass. We then however have let a considerable time elapse, so we don 't know exactly when the frequency was at this value. The relationship between frequencies and the beats is: .
The closer we choose our frequency to get a given accuracy, the longer the beats take to occur. We thus cannot know the time and the frequency simultaneously. The more precisely we try to define the frequency, the greater the time is smeared out.
Measuring a wave frequency with beats has intrinsic uncertainty as to the time, which becomes a smeared-out interval.
Despite gaining his fame for discovering relativity, and the doom equation E = mc2 which made the atom bomb possible, Einstein, possibly in cooperation with with his wife (p 496), also made a critical discovery about the quantum. Einstein's law connects to every energetic particle a frequency
Energy is thus intimately related to frequency - in a sense it IS frequency. Measuring one is necessarily measuring the other. If we apply the above together, we immediately get:
This is the famous Heisenberg uncertainty relation . It tells us something is happening which is impossible in the classical world. We can't know the energy of a quantum interaction and the time it happened simultaneously. Energy and time have entered into a primal type of prisoners ' dilemma catch 22. The closer we try to tie down the energy, the less precisely we know the time. This peculiar relationship places a specific taboo on knowing all the features of a situation and means we cannot predict precise outcomes, only probabilities. The same goes for momentum and position in each of the three spatial dimensions. Notice also that this links energy and momentum, time and space, and frequency and wavelength as three manifestations of one another. The way in which this happens is illuminating.
Each quantum can be conceived as a particle or as a wave but not both at the same time. Depending on how we are interacting with it or describing it, it may appear as either. We are all familiar with the fact that CDs have a rainbow appearance on their underside. This comes from the circular tracks spaced a distance similar to the wavelength of visible light. If we used light of a single wavelength we would see light and dark bands. We can visualize this process more simply with just two slits as in the figure below. When many photons pass through, their waves interfere as shown and the photographic plate gets dark and light interfernce bands where the waves from the two slits reinforce or cancel, because the photons are more likely to end up where their superimposed wave amplitude is large. The experiment confirms the wave nature of light, since the size of the bands is determined by the distance between the slits in relation to the wavelength where c is the velocity of light:
We know each photon passes through both slits, because we can slow the experiment down so much that only one photon is released at a time and we still eventually get the interference pattern over time. Each photon released from the light bulb is emitted as a particle from a single hot atom, whose excited electron is jumping down from a hight energy orbit to a lower one. It is thus released locally and as a single 'particle ' created by a single transition between two stable electron orbitals, but it spreads and passes through both slits as a wave. After this the two sets of waves interfere as shown in the figure to make bands on the photographic plate or the rainbows we see on a CD.
The evolution of the wave is described by an equation involving rates of change of a wave function j with respect to space and time. For example for a massive particle in free space, we have a 1-D differential equation:
This equation emphasizes the relativistic relationship between space and time. The relationship between Schrödinger 's wave equation and Heisenbergs matrix mechanics (p 493) highlights a deeper complementarity in mathematics between the discrete operations of algebra and the continuous properties of calculus which may be expressed in the brain (p 367).
Two-slit interference experiment (Sci. Am. Jul 92)
For the bands to appear in the interference experiment, each single photon has to travel through both slits as a wave. If you try to put any form of transparent detector in the slits to tell if it went through one or both you will always find only one particle but now the interference pattern will be destroyed. This happens even if you use the gentlest forms of detection possible such as an empty resonant maser chamber (a maser is a microwave laser). Any measurement sensitive enough to detect a particle alters its momentum enough to smear the interference pattern into the same picture you would get if the particle just went through one slit. Knowing one aspect destroys the other.
Now another confounding twist to the catch 22. The photon has to be absorbed again as a particle by an atom on the photographic plate, or somewhere else, before or after, if it doesn 't career forever through empty space, something we shall deal with shortly. Where exactly does it go? The rules of quantum mechanics are only statistical. They tell us only that the particle is more likely to end up where the amplitude of the wave is large, not where it will actually go on any one occasion. The probability is precisely the complex square of the wave's amplitude at any point:
Hence the probability is spread throughout the extent of the wave function, extending throughout the universe at very low probabilities. Quantum theory thus describes all future (and past) states as probabilities. Unlike classical probabilities, we cannot find out more about the situation and reduce the probability to a certainty by deeper investigation, because of the limits imposed by quantum uncertainty. The photon could end up anywhere the wave is non-zero. Nobody can tell exactly where, for a single photon. Each individual photon really does seem to end up being absorbed as a particle somewhere, because we will get a scattered pattern of individual dark crystals on the film at very low light intensities, which slowly build up to make the bands again. This is the mysterious phenomenon called 'reduction, or collapse, of the wave packet'. Effectively the photon was in a superposition of states represented by all the possible locations within the wave, but suddenly became one of those possible states, now absorbed into a single localized atom where we can see its evidence as a silver mark on the film. Only when there are many photons does the behaviour average out to the wave distribution. Thus each photon seems to make its own mind up about where it is going to end up, with the proviso that on average many do this according to the wave amplitude 's probability distribution. So is this quantum free-will? It may be.
This situation is the subject of a famous thought experiment by Schrödinger, who invented the wave equation. In Schrödinger 's cat paradox, we use an interference experiment with about one photon a second and we detect whether the photon hits one of the bright bands to the left. If it does then a cat is killed by smashing a cyanide flask. Now when the experimenter opens the box, they find the cat is either alive or dead, but quantum theory simply tells us that the cat is both alive and dead, each with differing probabilities - superimposed alive and dead states. This is counterintuitive, but fundamental to quantum reality.
Cat paradox experiment (King)
In the cat paradox experiment, the wave function remains uncollapsed at least until the experimenter I opens the box. Heisenberg suggested representing the collapse as occurring when the system enters the domain of thermodynamic irreversibility, i.e. at C. Schrödinger suggested the formation of a permanent record e.g. classical physical events D, E or computer data G. However even these classical outcomes could be superpositions at least until a conscious observer experiences them, as the many-worlds theory below suggests. Wigner's friend is a version of the cat paradox in which an assistant G reports on the result, establishing that unless the first conscious observer collapses the wave function, there will be a conscious observer in a multiplicity of alternative states, which is an omnipresent drawback of the many worlds view. In a macabre version the conscious assistant is of course the cat. According to the Copenhagen interpretation, it its not the system which collapses, but only our knowledge of its behavior. The superimposed state within the wave function is then not regarded as a real physical entity at all, but only a means of describing our knowledge of the quantum system, and calculating probabilities.
This clash between subjective experience and quantum theory has lead to much soul-searching. The Copenhagen interpretation says quantum theory just describes our state of knowledge of the system and is essentially incomplete. This effectively passes the problem back from physics to the observer. Some physicists think all the possibilities happen and there is a probability universe for each case. This is called the many-worlds interpretation of Hugh Everett III. The universe becomes a superabundant superimposed set of all possible probability futures and indeed all pasts as well in a smeared out 'holographic ' multi-verse in which everything happens. It suffers from a key difficulty. All the experience we have suggests just one possibility is chosen in each situation - the one we actually experience. Some scientists thus think collapse depends on a conscious observer. Many worlds defenders claim an observer wouldn 't see the probability branching because they too would be split but this leaves us either with infinite split consciousness, or all we lose all forms of decision-making process, all forms of historicity in which there is a distinct line of history, in which watershed events do actually occur, and the role of memory in representing it.
Zurek (R784) describes decoherence as an inevitable result of interactions with other particles. Penrose in 'objective reduction ' singles out gravity as the key unifying force and suggests that interaction with gravitons splits the wave function, causing reduction. Others try to discover hidden laws which might provide the sub-quantum process, for example a particle piloted within a wave as suggested by David Bohm (R69). This has difficulties defining positions when new particles with new quantum degrees of freedom are created. Another approach we will explore, is the transactional interpretation, which has features of all these ideas and seeks to explain this process in terms of a hand-shaking relationship between the past and the future, in which space-time itself becomes sexual. Key here is the fact that reduction is not like any other physical process. One cannot tell when or where it happens again suggesting it is part of the 'spooky ' interface between quantum and consciousness.
In many situations people try to pass the intrinsic problems of uncertainty away on the basis that in the large real processes we witness, individual quantum uncertainties cancel in the law of averages of large numbers of particles. They will suggest for example that neurons are huge in terms of quantum phenomena and that the 'law of mass action ' engulfs quantum effects. However brain processes are notoriously sensitive. Moreover history itself is a unique process out of many such 'unstable ' possibilities at each stage of the process. Critical decisions we make become watersheds. History and evolution are both processes littered with unique idiosyncratic acts in a counterpoint to the major forces shaping the environment and landscape. Chaotic processes are potentially able to inflate arbitrarily small fluctuations, so molecular chaos may 'inflate ' the fluctuations associated with quantum uncertainty.
The Two-timing Nature of Special Relativity
We also live in a paradoxical relationship with space and time. While space is to all purposes symmetric and multidimensional, and not polarized in any particular direction, time is singular in the present and polarized between past and future. We talk about the arrow of time as a mystery related to the increasing disorder or entropy of the universe. We imagine space-time as a four dimensional manifold but we live out a strange sequential reality in which the present is evanescent. In the words of the song "time keeps slipping, slipping, slipping ... into the future ". There is also a polarized gulf between a past we can remember, the living present and a shadowy future of nascent potentialities and foreboding uncertainty. In a sense, space and time are complementary dimensionalities, which behave rather like real and imaginary complex variables, as we shall see below.
A second fundamentally important discovery in twentieth century physics, complementing quantum theory, which transformed our notions of time and space, was the special theory of relativity. In Maxwell 's classical equations for transmission for light, light always has the same velocity, c regardless of the movement of the observer, or the source. Einstein realized that Maxwell's equations and the properties of physics could be preserved under all intertial systems - the principle of special relativity - only if the properties of space and time changed according to the Lorenz transformations as a particle approaches the velocity of light c :
Space becomes shortened along the line of movement and time becomes dilated. Effectively space and time are each being rotated towards one-another like a pair of closing scissors. Consequently the mass and energy of any particle with non-zero rest mass tend to infinity at the velocity of light:
By integrating this equation, Einstein (p 496) was able to deduce that the rest mass must also correspond to a huge energy Eo=moc2 which could be released for example in a nuclear explosion, as the mass of the radioactive products is less than the mass of the uranium that produces them, thus becoming the doom equation of the atom bomb. General relativity goes beyond this to associate gravity with the curvature of space-time caused by mass-energy.
In special relativity, space and time become related entities which form a composite four dimensional space-time, in which points are related by light-cones - signals travelling at the speed of light from a given origin. In space-time, time behaves differently to space. When time is squared it has a negative sign just like the imaginary complex number does.
Hence the negative sign in the formula for space-time distance and the scissor-like reversed rotations of time and space into one another expressed in the Lorenz transformations. Stephen Hawking has noted that, if we treat time as an imaginary variable, the space-time universe could become a closed 'manifold ' rather like a 4-D sphere, in which the cosmic origin is rather like the north pole of Earth, because imaginary time will reverse the above negative sign and give us the usual Pythagorean distance formula in 4D.
Space-time light cone permits linkage of 'time-like ' points connected by slower-then-light communication. In the 'space-like ' region, temporal order of events and causality depends on the observer.
A significant feature of special relativity is the fact that the relativistic energy-momentum equation E2=p2+ m2 has dual energy solutions:
The negative energy solution has reversed temporal direction. Effectively a negative energy anti-particle travelling backwards in time is exactly the same as a positive energy particle travelling forwards in time in the usual manner. The solution which travels in the normal direction (subsequent points are reached later) is called the retarded solution. The one which travels backwards in time is called the advanced solution. A photon is its own anti-particle so in this case we just have an advanced or retarded photon.
Reality and Virtuality: Quantum fields and Seething Uncertainty
We have learned about waves and particles, but what about fields? What about the strange action-at-a-distance of electromagnetism and gravity? Special relativity and quantum theory combine to provide succinct explanations of electromagnetism, in fact they are the most succinct theories ever invented by the human mind, accurate to at least seven decimal places when describing the magnetic moment of an electron in terms of the hidden virtual photons which the electron emits and then almost immediately absorbs again.
Richard Feynman and others discovered the answer to this riddle by using uncertainty itself to do the job (p 495). The field is generated by particles propagated by a rule based on wave spreading. These particles are called virtual because they have no net positive energy and appear and disappear entirely within the window of quantum uncertainty, so we never see them except as expressed in the force itself. This seething tumult of virtual particles exactly produces the familiar effects of the electromagnetic field, and other fields as well. We can find the force between two electrons by integrating the effects of every virtual photon which could be exchanged within the limits of uncertainty and of every other possible virtual particle system, including pairs of electrons and positrons coming into a fleeting existence. However, note that we can 't really eliminate the wave description because the amplitudes with which the particles are propagated from point to point are the hidden wave amplitudes. Uncertainty not only can create indefiniteness but it can actively create every conceivable particle out of the vacuum, and does so sine qua non. Special relativity and the advanced and retarded solutions that arise are also essential to enable the interactions that make the fabric of the quantum field. The advanced solutions are required to have negative energy and retarded solutions positive energy thus giving the correct reults for both scattering and electron-positron interactions within the field so that electron scattering is the same as electron positron creation and annihilation.
Quantum electrodynamics: (a,b) Two Feynman diagrams in the electromagnetic repulsion of two electrons. In the first a single virtual photon is exchanged between two electrons, in the second the photon becomes a virtual electron-positron pair during its transit. All such diagrams are integrated together to calculate the strength of the electromagnetic force. (c) A similar diagram shows how neutron decay occurs via the W- particle of the weak nuclear force, which itself is a heavy charged photon as a result of symmetry-breaking. (d) Time-reversed electron scattering is the same as positron creation and annihilation.
Each more complex interaction involving one more particle vertex is smaller by a factor where e is the electron charge and h and c are as above, called the 'fine structure constant '. This allows the contribution of all the diagrams to sum to a finite interaction, unlike many unified theories, which are plagued by infinities, as we shall see. The electromagnetic force is generated by virtual photons exchanged between charged particles existing only for a time and energy permitted by the uncertainty relation. The closer the two electrons, the larger the energy fluctuation possible over the shorter time taken to travel between them and hence the greater the force upon them. Even in the vacuum, where we think there is nothing at all, there is actually a sea of all possible particles being created and destroyed by the rules of uncertainty.
The virtual particles of a force field and the real particles we experience as radiation such as light are one and the same. If we pump energy into the field, for example by oscillating it in a radio transmitter, the virtual photons composing the electromagnetic field become the real positive energy photons in radio waves entering the receiver as a coherent stream of real photons, encoding the music we hear. Relativistic quantum field theories always have both advanced and retarded solutions, one with positive and the other with negative energy, because of the two square roots of special relativity. They are often described by Feynman space-time diagrams. When the Feynman diagram for electron scattering becomes time-reversed, it then becomes precisely the diagram for creation and annihilation of the electron 's anti-particle, the positron, as shown above. This hints at a fundamental role for the exotic time-reversed advanced solutions.
The weak and strong nuclear forces can be explained in a similar way, but gravity holds out further serious catch-22s. Gravity is associated with the curvature of space-time, but this introduces fundamental contradictions with quantum field theory. To date there remains no fully consistent way to reconcile quantum field theory and gravitation as we shall see.
The Spooky Nature of Quantum Entanglement
We have already seen how the photon wave passing through two slits ends up being absorbed by a single atom. But how does the wave avoid two particles accidentally being absorbed in far flung parts of its wave function out of direct communication?
Wheeler delayed choice experiment: A very distant quasar is gravitationally lensed by an intervening galaxy. We can sample photons either by an interference pattern, verifying they went around both sides of the galaxy, or place separate directional detectors which will detect they went one way around only as particles (which will destroy the interference pattern. Moreover, we can decide which to perform after the photon has passed the galaxy, at the end of its path. Thus the configuration of the latter parts of the wave appear to be able to alter the earlier history (Sci. American).
Just how large such waves can become can be appreciated if we glance out at a distant galaxy, whose light has had to traverse the universe to reach us, perhaps taking as long as the history of Earth to get here. The ultimate size is as big as the universe. Only one photon is ever absorbed for each such wave, so once we detect it, the probability of finding the photon anywhere else, and hence the amplitude of the wave, must immediately become zero everywhere. How can this happen, if information cannot travel faster than the speed of light? For a large wave, such as light from a galaxy, (and in principle for any wave) this collapse process has to cover the universe. When I shine my torch against the window, the amplitude of each photon is both reflected, so I can see it, and transmitted, escaping into the night sky. Although the wave may spread far and wide, if the particle is absorbed anywhere, the probability across vast tracks of space has to suddenly become zero. Moreover collapse may involve the situation at the end of the path influencing the earlier history, as in the Wheeler delayed choice experiment (p 305). In this experiment we can determine whether a photon went both ways round a lensing galaxy, focusing the light from a very distant quasar long after the light has passed across the universe, by either measuring the interference between the paths as in the double slit experiment or by detecting photons from one direction or another.
Because we can 't sample two different points of a single-particle wave, it is impossible to devise an experiment which can test how a wave might collapse. One way to learn more about this situation is to try to find situations in which two or more correlated particles will be released coherently in a single wave. This happens with many particles in a laser and in the holograms made by coherent laser light and in Bose-Einstein condensates. It also happens in other situations where two particles of opposite spin or complementary polarization become created together. Many years ago Einstein, Rosen and Podolsky (EPR) suggested we might be able to break through the veil of quantum uncertainty this way, indirectly finding out more about a single particle than it is usually prepared to let on.
(a) Pair-splitting experiment for photons. (b) Time-varying analysers are added driven by an optical switch to fast for light to cross the apparatus. (c) The results are consistent with quantum mechanics but inconsistent with Bell's inequalities for a locally causal system. (d) The calcium transition (Aspect R25).
For example a calcium atom 's electron excited into a higher orbital sometimes cannot fall back to its original orbital in one step because a photon always turns out to have spin 1 and the spins don 't match. For example you can 't go between two orbits of equal spin and radiate a spin-1 photon or the spins don 't tally. The atom however can radiate two photons thereby cancelling one another 's spins, to transit to its ground state, via an intermediate spin-1 orbit. This releases a blue and a yellow photon, each of which travel off in opposite directions, with complementary polarizations.
When we perform the experiment, it turns out that the polarization of neither photon is defined until we measure one of them. When we measure the polarization of one photon, the other immediately - instantaneously - has complementary polarization. The nature of the angular correlations between the detectors is inconsistent with any locally-causal theory - that is no theory based on information exchanged between the detectors by particles at the speed of light can do the trick, as proved in a famous result by John Bell (R55) and subsequent experiments. The correlation persists even if the detectors ' configurations are changed so fast that there is no time for information to be exchanged between them at the speed of light as demonstrated by Alain Aspect (R25). This phenomenon has been called quantum non-locality and in its various forms quantum 'entanglement ', a name itself very suggestive of the throes of a sexual 'affair '. The situation is subtly different from any kind of classical causality we can imagine. The information at either detector looks random until we compare the two. When we do, we find the two seemingly random lists are precisely correlated in a way which implies instantaneous correlatedness but there is no way we can use the situation to send classically precise information faster than the speed of light by this means. We can see however in the correlations just how the ordinary one-particle wave function can be instantaneously auto-correlated and hence not slip up in its accounting during collapse.
Since this result in the 1980s there have been a veritable conjurer's collection of experiments, all of which verify the predictions of quantum mechanics in every case and confirm all the general principles of the pair-splitting experiment. Even if we clone photons to form quartets of correlated particles, any attempt to gain information about one of such a multiple collection collapses the correlations between the related twins. Furthermore these effects can be retrospective, leading photons to be able to be superpositions of states which were created at different times. It is also possible to 'uncollapse ' or erase such losses of correlation by re-interfering the wave functions so we can no longer tell the difference. This successfully recreates the lost correlations, inducing information about one of the particles and then erase it again by re-interfering it back into the wave function provided we use none of its information - the quantum eraser. In such situations the interference, which would be destroyed had we looked at the information, is reintegrated undiminished.
Quantum erasure (Scientifi American)
Erasing information about the path of a photon restores wavelike correlated behavior. Pairs of identically polarized correlated photons produced by a 'down-converter ', bounce off mirrors, converge again at a beam splitter and pass into two detectors. A coincidence counter observes an interference pattern in the rate of simultaneous detections by the two detectors, indicating that each photon has gone both ways at the beam splitter, as a wave. Adding a polarization shifter to one path destroys the pattern, by making it possible to distinguish the photons ' paths. Placing two polarizing filters in front of the detectors makes the photons identical again, erasing the distinction, restoring the interference pattern.
Quantum teleportation, in which information creating a quantum in a given state is 'teleported ' by another has also become an experimental reality. These experiments give us a broad intuition of quantum reality.
Vlatko Vedral, who showed that entanglement is involved in superconductivity claims it can explain the Meissner effect, in which a magnet levitates above superconducting material. The magnetic field induces a current in the surface of the superconductor, and this current effectively excludes the magnetic field from the interior of the material, causing the magnet to hover. The current halts the photons of the magnetic field after they have travelled only a short distance through the superconductor. For the normally massless photons it is as if they have suddenly entered treacle, effectively giving them a mass. Vedral also claims a similar mechanism may be behind the mass of all particles. The source of this mass is believed to be the Higgs field mediated by the Higgs boson (p 311), thought to exist in a "condensed " state that excludes mediator particles such as gluons in the same way that a superconductor 's entangled electrons exclude the photons of a magnetic field (Quantum quirk may give objects mass New Scientist 24 October 2004).
Quantum teleportation: A quantum (blue left) is combined in an interference measurement with one of an entangled pair (pink left) by experimenter 1, who then sends the result of the measurement as classical information to 2 who applies this to transform the other entangled particle, causing it to enter the same quantum state as the original blue one.
Classical computation suffers from the potentially unlimited time it takes to check out every one of the possibilities. To crack a code we need to check all the combinations, whose numbers can increase more than exponentially with the size of the code numbers and possibly taking as long as the history of the universe to compute. Factorizing a large number composed of two primes is known to be computationally intractable enough to provide the basis for public key encryption by which banks records and passwords are kept safe. Although the brain ingeniously uses massively parallel computation, there is as yet no systematic way to boot strap an arbitrary number of parallel computations together in a coherent manner.
However quantum reality is a superposition of all the possible states in a single wave function, so if we can arrange a wave function to represent all the possibilities in such a computation, superposition might give us the answer by a form of parallel quantum computation. A large number could in principle be factorized in a few superimposed steps, which would otherwise require vast time-consuming classical computer power to check all the possible factors one by one. Suppose we know an atom is excited by a certain quantum of energy, but only provide it a part of the energy required. The atom then enters a superposition of the ground state and the excited state, suspended between the two like Schrödinger 's cat. If we then collapse the wave function, squaring it to its probability, as in , it will be found to be in either the ground state or excited state with equal probability. This superimposed state is sometimes called the 'square root of not ' when it is used to partially excite a system which flips between 0 and 1 corresponding to a logical negation.
To factorize a large number, we could devise a quantum system in two parts. The left part is excited to a superposition. Suppose we have a small array of atoms which effectively form the 0s and 1s of a binary number - 0 in the ground state and 1 in the excited state. If we then partially excite them all they represent a superposition of all the binary numbers - e.g. 00, 01, 10 and 11. The right half of the system is designed to give the factorization remainder of a test number taken to the power of each of the possible numbers in the left. These turn out to be periodic, so if we measure the right we get one of the values. This in turn collapses the left side into a superposition of only those numbers with this particular value in the right. We can then recombine the reduced state on the left to find its frequency spectrum and decode the answer. As a simple example, you are trying to factorise 15. Take the nest number x = 2. The powers of 2 give you 2, 4. 8, 16, 32, 64, 128, 256 ... Now divide by 15, and if the number won 't go, keep the remainder. That produces a repeating sequence 2, 4, 8, 1, 2, 4, 8, 1 ... with period n = 4 we can use this to figure is a factor of 15. The quantum parallelism solves all the computations simultaneously.
In the transactional interpretation, a single photon exchanged between emitter and absorber is formed by constructive interference between a retarded offer wave (solid) and an advanced confirmation wave (dotted). (b) The transactional interpretation of pair-splitting. Confirmation waves intersect at the emission point. (c) Contingent absorbers of an emitter in a single passage of a photon. (d) Collapse of contingent emitters and absorbers in a transactional match-making (King R365). (e) Experiment by Shahri Afshar (see Chown R114). A grid is placed at the interference minima of the wave fronts coming from two slits just below a lens designed to focus the light from each slit into a separate detector. Measurements by detectors (top) test whether a photon (particle) passed through the left or right slit (bottom). There is no reduction in intensity when the grid is placed below the lens at the interference minima of the offer waves from the two slits. The grid does however cause a loss of detector intensity when the dashed left-hand slit is covered and the negative wave interference between the offer waves at the grid is removed, so that the non-interfered wave from the right slit now hits the grid, causing scattering. This suggests both that we can measure wave and particle aspects simultaneously, and that the transactional interpretation is valid in a way which neither many worlds (which predicts a splitting into histories where a photon from the source goes through one slit or other) or the Copenhagen interpretation of complementarity (where detecting a particle forbids the photon manifesting as a wave).
Quantum Match-making: Transactional Supercausality and Reality
For reasons which immediately become apparent, the collapse in the pair-splitting experiment has to not only be immediate, but also to reconcile information looking backwards in time. The two photons we are trying to detect are linked through the common calcium atom. Their absorptions are thus actually connected via a path travelling back in space-time from one detector to the calcium atom and forward again to the other detector. Trying to connect the detectors directly, for example by hypothetical faster-than-light tachyons, leads to contradictions. Tachyons transform by the rules of special relativity, so a tachyon which appears to be travelling at an infinite speed according to one observer, is travelling only at a little more than the speed of light according to another. One travelling in one direction to one observer may be travelling in the opposite direction to another. They can also cause causality violations (King R365). There is thus no consistent way of knitting together all parts of a wave or the detector responses using tachyons. Even in a single-particle wave, the wave function in regions it has already traversed (and those it would subsequently pass through in future) also have to collapse retrospectively (and prospectively) so that no inconsistencies can occur, in which a particle is created in two locations in space-time from the same wave function, as the Wheeler delayed choice experiment makes clear.
In the transactional interpretation (Cramer R136), such a 'backward travelling ' wave in time gives a neat explanation, not only for the above effect, but also for the probability aspect of the quantum in every quantum experiment. Instead of one photon travelling between the emitter and absorber, there are two shadow waves, which superimposed make up the complete photon. The emitter transmits an offer wave both forwards and backwards in time, declaring its capacity to emit a photon. All the potential absorbers of this photon transmit a corresponding confirmation wave. The confirmation waves travelling backwards in time send a hand-shaking signal back to the emitter. In the extension transactional approach to supercausality, a non-linearity now reduces the set of possibilities to one offer and confirmation wave, which superimpose constructively to form a real photon only on the space-time path connecting the emitter to the absorber as shown in the figure. This always connects an emitter at an earlier time to an absorber at later time because a real positive energy photon is a retarded particle which travels in the usual direction in time.
A negative energy photon travelling backwards in time is precisely the anti-particle of the positive energy photon and has just the same effect. The two are identifiable in the transactional interpretation, as in quantum electrodynamics (p 304), where time-reversed electron scattering is the same as positron creation and annihilation. The transactional relationship is in effect a match-making process. Before collapse of the wave function we have many potential emitters interacting with many potential absorbers. After all the collapses have taken place, each emitter is paired with an absorber in a kind of marriage dance. One emitter cannot connect with two absorbers without violating the quantum rules, so there is a frustration between the possibilities which can only be fully resolved if emitters and absorbers can be linked in pairs. The number of contingent emitters and absorbers are not necessarily equal, but the number of matched pairs is equal to the number of real particles exchanged.
In the pair-splitting experiment you can now see that the calcium atom emits in response to the advanced confirmation waves reaching it from both the detectors simultaneously right at the time it is emitting the photon pair. Thus the faster than light linkage is neatly explained by the combined retarded and advanced aspects of the photon having a net forwards and backwards connection which is instantaneous at the detectors. One can also explain the arrow of time if the cosmic origin is a reflecting boundary that causes all the positive energy real particles in our universe to move in the retarded direction we all experience in the arrow of time. This in turn gives the sign for increasing disorder or entropy and the time direction for the second law of thermodynamics to manifest. The equivalence of real and virtual particles raises the possibility that all particles have an emitter and absorber and arose, like virtual particles, through mutual interaction when the universe first emerged. However even if dark-energy, 'quintessence ' causes an increasing expansion, or fractal inflation leads to an open universe model in which some photons may never find an absorber, the excitations of brain oscillations, because they are both emitted and absorbed by past and future brain states could still be universally subject to transactional supercausal coupling.
The hand-shaking space-time relation implied by transactions makes it possible that the apparent randomness of quantum events masks a vast interconnectivity at the quantum level, which has been termed the 'implicate order ' by David Bohm (R70). This might not itself be a random process, but because it connects past and future events in a time-symmetric way, it cannot be reduced to predictive determinism, because the initial conditions are insufficient to describe the transaction, which also includes quantum 'information ' coming from the future. However this future is also unformed in real terms at the early point in time emission takes place. My eye didn't even exist, when the quasar emitted its photon, except as a profoundly unlikely branch of the combined probability 'waves ' of all the events throughout the history of the universe between the ancient time the quasar released its photon and my eye developing and me being in the right place at the right time to see it. Transactional supercausality thus involves a huge catch 22 about space, time and prediction, uncertainty and destiny. It doesn 't suggest the future is determined, but that the contingent futures do superimpose to create a space-time paradox in collapsing the wave function.
Roger Penrose (R535, R536), has suggested that the one-graviton limit of interaction is an objective trigger for wave packet reduction, because of the bifurcation in space-times induced, leading to theories in which the random or pseudo-random manifestations of the particle within the wave are non-linear consequences of gravity. Objective orchestrated reduction or OOR is then cited as a basis which intentional consciousness uses to follow collapse rather than participating in it, as the transactional model makes possible. The OOR model unlike transactional anticipation thus leaves free-will with a kind of orphan status, following, but not participating in, the collapse process itself.
By reducing the energy of a transaction to a superposition of ground and excited states, the transactional approach may combine with quantum computation to produce a space-time anticipating quantum entangled system which may be pivotal in how the conscious brain does its computation. The brain is not a marvelous computer in any classical sense. We can barely repeat seven digits. But it is a phenomenally sensitive anticipator of environmental and behavioral change. Subjective consciousness has its survival value in enabling us to jump out of the way when the tiger is about to strike, not so much in computing which path the tiger might be on, because this is an intractable problem and the tiger can also take it into account in avoiding the places we would expect it to most likely be, but by intuitive conscious anticipation. What is critical here is that in the usual quantum description which considers only the emitter, we have only the probability function because the initial conditions are insufficient to determine the outcome. There is thus no useful way quantum uncertainty can be linked to conscious free-will. Only by completing the sexual paradox of time by including the advanced absorber waves can we see how anticipation might be achieved.
Engendering Nature: Cosmic Symmetry-Breaking and Inflation
The basis of the cosmic inflation concept is symmetry-breaking, in which the fundamental forces of nature, which make up the matter and radiation we relate to in the everyday world gained the very different properties they have today. There are four quite different forces. The first two are well known - electromagnetism and gravity - both long-range forces we can witness as we look out at distant galaxies. The others are two short-range nuclear forces. The colour force holds together the three quarks in any neutron or proton and indirectly binds the nucleus together by the strong force, generating the energy of stars and atom bombs. The weak radioactive force is responsible for balancing the protons and neutrons in the nucleus by interconverting the flavours of quarks and leptons (p 311).
There is a fundamental sexual division among the wave-particles. Particles come in two types, fermions, of half-integral spin, which can only clump in complementary pairs in a single wave function and thus, being incompressible, make up matter, and bosons of integral spin which can become coherent and can all enter the same wave function in unlimited numbers, as in a laser, and hence form radiation and as virtual particles appearing and disappearing through quantum uncertainty, the forces which act between the particles. We thus have another fundamental sexual complementarity manifesting as the relationship between matter and radiation. The half integral spin of electrons was first discovered in the splitting of the spectral lines of electorns in atomic orbitals into pairs whose spin angular momentum corresponded to +/-1/2 rather than the 0, 1 , 2 etc. of atomic s, p - orbitals (p 318). As spin states have to differ by a multiple of Planck 's constant h a particle of spin s has 2s+1 components. A glance at the known wave-particles (p 311), indicates that the bosons and fermions we know are very different from one another in their properties and patterns of arrangement. There is no obvious way to pair off the known bosons and fermions, however there are reasons why there may be a hidden underlying symmetry, which pairs each boson with a fermion of one-half less spin, called super-symmetry, because in super-symmetric theories the infinities that plague quantum field theories cancel and vanish, the negative contributions of the fermions exactly balancing the positive contributions of the bosons. This would mean that there must be undiscovered particles. For example corresponding to the spin-2 graviton would be a spin-3/2 gravitino, a spin-1 graviphoton a spin-1/2 gravifermion and a spin-0 graviscalar.
(a) Sexual paradox in the standard model of the four fundamental forces: The wave-particles are divided into two disparate groups of bosons and fermions. The fermions, which make matter are divided between quarks which experience all the forces including colour and leptons which experience only the electroweak and gravity. The bosons, which mediate the forces have integer spin and freely superimpose, as in lasers and hence also make radiation. Half-integer spin fermions only superimpose in pairs of opposite spin and hence resist compression into one space, thus making solid matter. Each quark comes in three colours(RGB) and pairs of flavours (up, down etc.) Electromagnetism is first united with the weak force ostensibly through the spin-0 Higgs boson, then with the colour force gluons and finally with gravity. (b) The forces converge at high energies. (c) Force differentiation tree, in which the four forces differentiate from a single super-force, with gravity displaying a more fundamental divergence. (d) the scalar Higgs field has lowest energy in the polarized state, (e) the stable atomic nuclei with their increasing preponderance of neutrons are equilibrated by the weak force. This force is chiral, engaging left-handed interactions, for example in neutron decay, as shown. Weak interactions may explain the chirality of RNA and proteins (King R372, R374).
The four fundamental forces appear to converge at very great energies and to have been in a state of symmetry at the cosmic origin as a common super-force. A key process mediating the differentiation of the fundamental forces is cosmic symmetry-breaking. The short-range weak force behaves in many ways as if it is the same as electromagnetism, except the charged W+,W- and neutral Z0 carrier particles corresponding to the electromagnetic photon are very massive. One can of course consider this division of a common super-force into distinct complementary forces as a nd of sexual division, just as the division into male and female is a primary division. In this respect gravity stands apart from the other three forces which share a common medium of spin-1 bosons and broke symmetry first.
A key explanation for this symmetry-breaking is that originally all the particles had zero rest mass like the photon, but some of the boson force carriers like the W changed to mediate a short-range force by becoming massive and gaining an extra degree of freedom (the freedom to change speed) by picking up an additional spin-0 particle called a Higgs boson. The elusive Higgs may also explain why the universe flew apart. The universe begins at a temperature a little below the unification temperature - slightly supercooled, possibly even a result of a quantum fluctuation. In the early symmetric universe empty space is forced into a higher-energy arrangement than its temperature can support called the false vacuum. The result is a tremendous energy of the Higgs field which behaves as exponential anti-gravity, inflating the universe in 10-35 of a second to something already close to its present size. This inflationary phase becomes broken once the Higgs field collapses, breaking symmetry to a lower energy polarized state, rather like a ferromagnet. does, to create the asymmetric force arrangement we experience to form the true vacuum. In this process the Higgs particles, which are zero spin and have one wave function component, unite with some of the particles, such as W+/- and Z0 to give them non-zero rest mass by adding their extra component , allowing the additional longitudinal component of the wave function associated with a varying velocity. Because the true vacuum is at a lower energy than the false one it grows to engulf it releasing the latent heat of this energy difference as a shower of hot particles, the hot fireball we associate with the big bang. Gravity has now reversed to become the attractive force we are familiar with. Two energies which cancelled now became two which add - an insignificant universe - almost nothing became one of almost incalculable proportions. The end result is a universe flying apart at almost exactly its own escape velocity whose kinetic energy almost balances the potential energy of gravitation. Symmetry-breaking can leave behind defects if the true vacuum emerges in a series of local bubbles which join. Depending on whether the symmetries which are broken are discrete, circular, or spherical, corresponding anomalies in the form of domain walls, cosmic strings or magnetic monopoles may form. In addition other weakly-interacting particles may emerge such as the axions which some researchers associate with cold dark matter.
In some models, inflation is a fractal branched structure like a snowflake which is perpetually leaving behind mature universes like ours (p 298). Recently it has become clearer that, even with additional dark matter, possibly comprising neutrinos and other exotic particles, there may not be enough mass to stop the expansion, which may even be accelerating. Various hyperbolic forms of inflation and an additional repulsion called quintessence involving a long-range repulsive dark energy have both been invoked to address this problem.
Rehabilitating Duality: Quantum Gravity and Space-time Structure
Quantum theory is formulated within space-time, but mass-energy, through gravitation in general relativity alters the structure of space-time by curving it. This has made a comprehensive integration of gravity with the other forces of nature difficult to achieve and may indicate a fundamental complementarity between the theories. Something of this paradox can be understood in graphic terms if we consider the implications of quantum uncertainty over very small time intervals, small enough to allow a virtual black hole to form. In this case a quantum fluctuation could give rise to a wormhole in the very space-time in which it is conceived raising all manner of paradoxes of connected universes and time loops into the bargain. This leads to a fundamental conceptual paradox in which space-time is flat or slightly curved on large scales but a seething topological foam of worm-holes on very small scales. These problems lead to fundamental difficulties in describing any form of quantum field in the presence of gravity.
The unification of gravity with the other forces brings new and deeper mysteries into play. Theories which treat particles as points are plagued with infinities the very points themselves imply as infinite concentrations of energy. Point particles may thus on very small scales become string, loop or membrane excitations. The theories broadly called 'superstring ' explain the infinite self-energies associated with a point particle, and the different particles themselves as different excitation on a closed or open loop or string. However none have been found so far which correspond to our own peculiar asymmetric set of particles.
Left: Point particles have infinite self-energies and precise vertices of interaction, strings smooth these. Right: Strings have characteristic harmonic excitations (Wolfson R760, Sci. Am. Jan 96). They can be regarded either as open strings or loops. The different excitations being different particles.
Central to such theories is supersymmetry - a pairing between bosons and fermions of adjacent spin. The idea behind this is based on ground state zero-point fluctuations - the energies that arise through uncertainty when a quantum is considered in its lowest (ground) energy state. Only a perfect balancing of the negative zero-point energies of the fermions against the corresponding positive zero-point energies of the bosons implied by supersymmetry would cancel the potential infinities arising from the arbitrarily short wavelengths that result from the electromagnetic field when quantum gravitation is included in the unification scheme. These would effectively curl space-time to a point (Hawking R303 46, 50). It is possible however that it is the collective contribution of the two groups which balance so that there is not an individual set of boson-fermion pairings but two symmetry-broken groups - bosons and fermions which collectively blanace one another - reflecting the standard model.
Compactification of the 12 or so unseen dimensions leave
only our 4 of space-time on large scales (Sci. Am. Jan 96). Compactification of one dimension to form a tube is a way 11-D M-theory can be linked to 10-D superstrings which are on smaller scales, string-like tubes.
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
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.
The Sexually-Complex Quantum World
We have seen that all phenomena in the quantum universe present as a succession of fundamental complementarities in a shifting vacuum ground-swell of uncertainty, out of which the super-abundance of quantum diversity emerges. In this process we have discovered a multiple overlapping series of divisions: (i) wave-particle complementarity fundamental to the quantum, (ii) the roles of emitters and absorbers, (iii) the advanced and retarded solutions of special relativity, (iv) the fermions comprising matter complementaing the bosons mediating radiation, (v) virtual and real particles distinguishing force fields from positive energy matter and radiation, and the engendered symmetry-breakings between (vi) space and time (reflecting that between momentum and energy) and (vii) between the four fundamental forces of nature, which in turn cause the quantum architecture of atoms and molecules to be asymmetric and capable of complexity of interaction to form living systems (p 317) and finally (viii) duality, which makes it difficult or impossible to determine what is a fundamental particle and what is composite in a sexual paradox between dual descriptions. Sexual paradox may also be manifest in the difficulty of separating the forces from the seething quantum 'ground ' of vaccum uncertainty, which is generative of all types of quantum. To understand conscious anticipation, or free-will, may require the inclusion of advanced waves, forming a paradoxical complement to the positive energy arrow of time.
All these complementarities possess attributes of sexual paradox and are pivotal to generating the complexity and diversity of the universe as we know it. There is no way to validly mount a single description based on only one of these complementary aspects alone. All attempts to define a theory based only on one aspect implicitly involves the other as a fundamental component, just as the propagators of the particles in quantum field theory are based on wave-spreading. Classical mechanistic notions of a whole made out of clearly defined parts, as well as temporal determinism fail. The mathematical idea of a reality made out sets of points or point particle becomes replaced by the excitations of strings, again with wave-based harmonic energies. Just as we have an irreducible complementarity between subjective expereince and the objective world, so all the features of the quantum universe present in sexually paradoxical complementarities. It is thus hardly surprising that these fundamental and irreducible complementarities may come to be expressed as fundamental themes in biological complexity, thus making sexuality a cumulative expression of a sexual paradox which lies at the foundation of the cosmos itself.
Although both the Taoist and Tantric views of cosmology are based on a complementation between female and male generative principles, many people, including a good proportion of scientists still adhere to a mechanistic view of the universe as a Newtonian machine. In this view biological sexuality seems to be barred from having any fundamental cosmological basis, being an end product of an idiosyncratic process of chance and selection, in a biological evolution which has no apparent relation with or capacity to influence the vast energies and forces which shape the cosmological process. The origins of life remain mysterious and potentially accidental rather than cosmological in nature and evolution an erratic series of accidents preserved by natural selection.
However if we reverse this logic and begin with a sexually paradoxical cosmology, the phenomenon of biological sexuality then becomes a natural cumulative expression of physical sexual paradox operating in a new evolutionary paradigm in the biological world, rich with new feedback processes which give it the central role in genetics and organismic reproduction we regard as the signature and raison d 'etre of reproductive sexuality.