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Bohr and Einstein - their debate which sparked the Copenhagen interpretation
eventually led to the discovery of quantum non-locality.
Further Genesis Quantum Chapters
The Exotic Quantum World
We are living in a quantum universe, not a classical one. Most people however are still living in the classical Newtonian world which expired at the beginning of the twentieth century. Our world is very subtle and much more mysterious than the "building blocks" view of the universe would indicate. Many people lead their lives at the macroscopic level as if quantum reality were only true for atoms and somehow not true for larger things like people and thier immediate enivronment, but the correspondence principle by which the quantum world is supposed to fade into the classical world never works out for a host of reasons.
Chaotic, self-critical and certain other processes may inflate quantum effects in unforseen ways to the macroscopic level. The physics underlying conscious interaction with the physical world may likewise depend both on quantum effects, criticality and chaos in its functioning. The entire universe itself may be a self-consistent interconnected whole which has emerged from a single quantum wave function, therefore it is non-classical in its entire description. For these reasons it is necessary for us to understand how the quantum works and how it may differ from our classical view of order, solidity, determinism and mechanism.
Uncertainty of Timing Wave Beats
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 beats that it makes against a standard wave we generate. This is exactly the situation we face in quantum physics, because all our tools are ultimately made up of the same 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.
Fig 1.1:Uncertainty of time in wave beats
The relationship between the frequencies and the beats is: [1.1]
The closer we choose our frequency to get a given accuracy, the longer the beats take to occur. From this basic relationship we cannot know the time and the frequency simultaneously.
Now things start to get interesting. Despite gaining his fame for discovering relativity, and the doom equation [1.9] which made the atom bomb possible, Einstein's law is actually a fundametal equation of quantum mechanics. It says: [1.2]
That is the energy of any particle is proportional to its frequency as a wave by a factor h called Planck's constant - the fundamental unit of quantumness. This was found out from the photo-electric experiment. If you shine light on a plate in a vacuum valve and vary the voltage required to stop the resulting current flow, you find the more light, the more current, but no more voltage. The voltage turns out to depend only on the frequency. That is, the energy doesn't change, just the flow rate. This makes no sense with a classical wave, because a bigger wave has both more flow and more energy. The answer is that a given frequency of light contains particles called photons. The more photons, the more excited electrons cross the vacuum by gaining this energy, but there is no change in the energy because each photon has the same energy for a given colour (frequency), regardless of how bright the light.
Why we don't Burn to a Crisp
There is another very famous experiment, the other of Kelvin's two small dark clouds on the horizon of classical physics which together plunged it into the quantum-theoretic age. This was black-body radiation, named after the thermal radiation from a dark cavity and also from bright thermal objects like the sun. We know the sun has some ultra-violet and can burn us, but not as much as the peak of visible light. If classical physics were true it should have more ultra-violet and even more x- and gamma rays - a situation called ultra-violet catastrophe.
The solar spectrum Fraunhoffer 1814, Planck's radiation law [1.3] both have a peak about 5,000 oC
Planck eventually solved the problem by quantizing the radiation into little packets proportional to h called quanta. The particles responsible for this packeting are now identified as the photon. The answer to the problem is this. In the classical view energy distribution should increase endlessly into the high frequencies, but in the quantum view, to release a particulate photon of a given frequency, there has to be an atom somewhere with an energized enough electron to radiate the photon, so the energy is limited by the temperature of the thermal body. Thus because the photons come in quanta, or packets, the radiation cannot go endlessly up into the ultra-violet. Planck's equation is displayed in the figure above [1.3]. It starts out growing for small energies but falls off exponentially after the peak corresponding to the exponentially rarer thermodynamic excitations at a given temperature.
Now let's return to the problem of uncertainty. When we examine Einsteins equation 1.2, we see that energy is somehow intimately related to frequency - in a sense it IS frequency. Measuring one is necessarilty measuring the other. If we apply 1.1 and 1.2 together, we immediately get:
This tells us something is happening which is impossible in the classical world. We can't know the energy of anything and the time it happened simultaneously. 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 precies outcomes, only probabilities. The way in which this happens is illuminating. There is a similar uncertainty relation connecting momentum and distance (wavelength): [1.4b]
Complementarity and Interference
Each quantum can be conceived as a particle or as a wave but
not both at the same time - this is called complementarity. 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 instead of many as illustrated below, and a very faint
source consisting of light of just one wavelength made by an atomic
transition. When lots of photons pass through their waves always
interfere as shown and the photographic plate gets dark bands
where the waves from the two slits reinforce, because the photons
are more likely to end up where their total wave is large and
minimal exposure where they cancel. The experiment as a whole
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 [1.5],where
c is the velocity of light.
Fig 1.3: Interference experimentwave peaks are green and troughs are black. Where troughs or peaks
coincide the oscillation is large wher peaks meet troughs they cancel giving light and dark bands.
Now when just one photon is released from the bulb it is emitted as a praticle from a single hot atoms electron jumping orbits, but it passes through both slits as a wave. After this the two sets of waves interferee as shown in the diagram. The photon has to be absorbed again as a particle by an atom somewhere. Where does it go? The rules of quantum mechanics are only statistical. They tell us only that it is more likely to end up where the amplitude of the wave is large, in fact the probability is the square of the wave's amplitude: [1.6]
Unlike classical causality, quantum theory 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, because of the limits imposed by quantum uncertainty.
Really 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 somewhere, because we will gett a scattered pattern of individual dark crystals on the film at very low light intensities.. This is the mysterious phenomenon called reduction of the wave packet. Only when there are lots of photons does the behaviour average out to the wave distribution. So is this quantum free-will? Nobody knows.
All particles, such as the electrons, protons and neutrons which make up the atoms of our chemical elements and molecules all exist as both particles and waves. The orbitals of the electrons around atoms and those linking each molecule together occur only at the energies and sizes which correspond to a perfect standing wave, forming a set of discrete levels like the layers of an onion. These in turn determine the chemical properties of each substance. Because the molecular orbitals formed between a pair of atoms have lower energy than their individual atomic counterparts, the atoms react to form a molecule releasing the spare energy as heat. The characteristic energy differences between the levels of a given atom can be seen, both on earth and in the universe at large, as emission or absorbtion lines in the electromagnetic spectrum.
(a) s and p orbitals have spin 0 and 1 respectively. Each occurs in a series, forming the shells of the atom.. (b) Two s orbitals form a lower energy molecular orbital as well as a higher energy anti-bonding orbital. (c) Six p orbitals combine to form a single delocalized pi molecular orbital in the benzene ring. (d) Absorbed or emitted photons cause electron transitions between orbitals in hydrogen, giving rise to the signature of the hydrogen spectrum (e)
The Cat Paradox and Quantum Reality
The situation is the subject of a famous thought experiment by Schrödinger, called the cat paradox. In the 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 tothe left. If it does then the 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 prababilities - superimposed alive and dead states. This is very conterintuitive but fundamental to quantum reality.
Fig 1.4: Schrödinger cat paradox
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. Some people, following Hugh Everitt III, think all the possibilities happen and there is a probability universe for each case. This is called the many-worlds interpretation. The cosmic wave function is thus in effect giving life to all the contingencies of cintingencies as probability universes each with equal validity and there is no need for the wave function ever to collapse. It suffers from one difficulty which there are always elaborate explanations to circumvent. All the experience we have suggests just one possibility is chosen. The one we actually experience. Some scientists thus think collapse depends on a conscious observer. Others try to discover hidden laws which might provide the sub-quantum process which chooses one possibility rather than another, for example Bohm's idea of a particle piloted within a wave. This also has certain difficulties but we will examine a theory called the transactional interpretation which has features of all these ideas.
In many considerations 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. A modern expression of this is decoherence theory. However history is a unique process out of many such 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
A second fundamentally important discovery in 20th century physics was the special theory of relativity. In the nineteenth century, the wave equations for light were developed by Clerk Maxwell. In these equations, light always has the same velocity, c regardless of the movement of the observer or the source. When waves travel through a medium, the velocity of the wave is the sum of its velocity through the medium and the velocity of the medium through space. However no one was able to find the medium called the aether, and all experiments confirmed the velocity of light was precisely c, regardless of the detector's motion in space.
Einstein realized that Maxwell's equations and the properties of physics could be preserved under all intertial systems [principle of relativity] only if the properties of space and time changed according to the Lorenz transformations:
as a particle approaches the velocity of light. Space becomes shortened along the line of movement and time becomes dilated. Effectively space and time are bieng 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 was able to deduce that the rest mass must also correspond to a huge energy[1.9] 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 by just such an amount, thus becoming the doom equation.
In 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. It turns out that the space-time interval
is preserved under special relativistic transformations.
Fig 1.5: 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.
Another significant feature of special relativity is the fact that the relativistic energy-momentum equation [1.11] has dual energy solutions [1.12]. The negative energy solution has reversed temporal behaviour in space-time. 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 advanced.
Richard Feynman with his diagram
Relativistic Quantum Field Theories
We have seen about waves and particles, but what about fields? What about the strange action-at-a-distince of electro-magnetism and gravity? Special relativity and quantum theory combine to provide succinct explanations of force fields, in fact they are the most succinct theories even invented by the human mind, accurate to at least seven decimal places. Richard Feynman and others discovered the answer to this riddle by using uncertainty itself to do the job. The field is generated by particles propagated by a rule based on wave spreading. However these particles are called virtual because they have no net positive energy and appear and disappear entirely through quantum uncertainty, so we never see them except as expressed in the force itself.
Fig 1.6: (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.
(d) A time reversed electron scattering is the same as positron creation and anihilation.
The electromagnetic force is generated by virtual photons exchanged between charged particles, fig 1.6(a). They exist only for a time and energy permitted by the uncertainty relation [1.4]. The closer the two electrons, the larger the energy fluctuation possible over the consequently shorter time taken to travel between them and hence the greater the force on 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. These often have pivotal effects. By integrating all possible particle interactions over every possible path, force fields can be completely explained in terms of exchanged virtual particles, fig 1.6(a,b). The results predict details such as the magnetic moment of the electron (the electron, despite being a fundamental particle, still behaves like a tiny spinning magnet) with astonishing accuracy. All the four forces of nature can in principle be explained in this way, although the others have additional complexities which we shall examine later. In particular, each more complicated diagram in electrodynamics is 127 (ee/hc) times smaller in contribution, so the infinite set of diagrams converges to a finite result. This does not happen with some of the other forces, notably the strong nuclear force.
Relativistic equations in quantum electrodynamics always have both advanced and retarded solutions because of [1.12]. When the Feynman diagram for electron scattering becomes time-reversed, it then becomes precisely the diagram for creation and anihilation of the electron's anti-particle, the positron, as shown in fig 1.6(d). This hints at a fundamental role for the exotic time-reversed advanced solutions.
As a simple example, the wave equation [1.13]
for a zero spin particle with mass m has two solutions: [1.14], where .
Collapse of the Wave-packet and Pair-splitting Experiments
We have already seen how the photon wave passing through two slits ends up being absorbed by a single atom. 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. The ultimate size of the wave of such a photon is almost 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 in space-time. How can this happen if information cannot travel faster than the speed of light? For a large wave, such as the light from the galaxy, this collapse process has to cover a large swathe of the universe. 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.
However it is possible to devise a single wave with two particles in it, just as it is possible to get many particles into a common wave, for example in a laser. If we can make a wave with precisely two particles, we can see how the collapse of one particle's wave effects the other. For example a calcium atom with an excited zero-spin s orbital can radiate two photons, each of spin one, to transit to its zero-spin s ground state, via an intermediate spin-one p-orbital state. This releases a blue and a yellow photon, each of which may travel off in opposite directions, with complementary polarizations.
Fig 1.7: (a) The 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.
Now it turns out that the polarization of neither photon is defined until we measure one of them. The same ting goes for the spin of a pair of correlated elcetrons or other particles. When we measure the polarization of one photon, the other immediately has complementary polarization. The probability of a given polarization varies sinusoidally with the relative angle between the detectors in a manner inconsistent with any locally-causal theory based on information travelling at the speed of light from one particle or detector to the other, as proved in a famous theorem by Bell. The phenomenon is thus called quantum non-locality. Moreover the effect persists, even when the detectors are switched so fast that there is no time for information to pass across the apparatus at the speed of light.
There have been since this resultin the 1980s a veritable conjurer's collection of experiments, all of which verify the predictions of quantum mechanics in every case. It is even possible to induce information about one of the particles and then erase it again by re-interfering it back into the wave function. In such situations the interference which would be destroyed had we looked at the information is reintegrated undiminished. Quantum teleportation has also become an experimental reality.
A variety of experiments elucidate and confirm all the general trends of the pair-splitting experiment. Even if we clone photons to form quartets of correlated particles, any attempt to gain information about one branch of such a multiple branching collapses the correlations on all the related twins. Furthermore these effects are retrospective leading photons to be able to be superpositions which were created at different points in time. It is also possible to 'uncollapse' or erase such losses of correlation by reinterfering the wave functions so we can no longer tell the difference. This successfully recreates the lost correlations. Taken all in all these experiments give us a broad intuition of quantum reality.
The Transactional Interpretation
For reasons which immediately become apparent, this collapse has to not only be immediate, but also has to travel backwards in time. Since the two photons are linked only through the common calcium atom, their absorbtions are 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 particles called tachyons, has significant problems. 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. Theyalso cause wierd causality violations. There is thus no consistent way of knitting together all parts of a wave using tachyons. Even in a single-particle wave, regions the wave has already traversed also have to collapse retrospectively.
Fig 1.8: (a) 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. (c) Collapse of several related transactions.
In the transactional interpretation, 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. 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. 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. 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.
If you wish, you can think of a negative energy photon travelling backwards in time as the antiparticle of the positive one and it will have just the same effect. The two are thus identifiable in the transactional interpretation, just as in quantum electrodynamics.
The hand-shaking space-time relation implied by the transactional interpretation 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. 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 mechanical determinism.
1.16 Transcausality and the Implicate Order
Because quantum mechanics cannot predict which outcome will occur when any quantum choice is made, several physicists have suggested that there may be a deeper theory which expalins the principle of choice. One such theory is the pilot wave theory of Bohm, in which the wave pilots a real localized particle via a quantum potential which is designed to return the same statistics as quantum mechanics. However there are several situations in which quantum interactions give rise to new degrees of freedom, for example when matter and anti-matter anihilate. The theory does not explain how such new particles gain their trajectories.
The space-time hand-shaking of the transactional interpretation however provides a possible mechanism to connect all wave-particles in the universe in a single system. This system does not have to be random, but could consist of a complex non-linear relationship. Many complex systems display pseudo-random behaviour because of the interaction of many degrees of freedom. Transactional hand-shaking would provide a space-time mechanism for non-local interaction which also cannot be decoded at the point of emission, because, unlike classical causality in which past conditions fully determine future ones, the transactional boundary conditions include future states of the universe, and these will not be fully determined until the entire transaction has occurred. I will call such an extended form of causality transcausality. It allows the universe to display correlations between seemingly unrelated events united in a greater global order, the synchronicities of Carl Jung. It also provides a pivotal role for consciousness as a means to anticipate unpredictable changes in the environment through the manifestation of space-time handshaking in the brain.
The Hidden Realm
The transcausal model is one of several theories dealing with collapse of the wave function and its alternatives.
These theories give a good comparison with the transactional principle. In my opinion the transactional principle is a central and inevitable consequence of three things:
The critical issue about quantum transactions is that they address directly the informational decision problem to resolve which emitters and absorbers are actually connected in the measurement problem. It is possible that a full explanationof transactions may require the evolution to full dimensional string and membrane theories and it is notable that some such theories do postulate an additional time-like dimension which might be needed to portray the evolution of transactions which are already space-time spanning events.
The interface between chaos and quantum mechanics has become an area of challenge and interest because of the new ideas about the nature of chaos,introduced by quantum smoothing. It turns out that confined quantum systems from the nucleus to magnetically excited atomic orbitals which should display chaos demonstrate a variety of subtle forms of repression of chaos which separates the energy levels, and converges in probability to the periodic repelling orbits called hidden in any chaotic system. Orbits with time thus tend to end on these periodic solutions. This phenomenon is called quantum-'scarring' of the wave function.
These constraints however begin to evaporate as soon as we leave confined systems and begin to enter the domain of unbound systems, such as electrons traversing a free molecular medium. This raises the distinct possibility that quantum chaos expressed in biological systems is right at the transition between the classical and quantum worlds. Chaotic systems may thus be able to amplify quantum effects into global fluctuations.
Some of the most challenging aspects of quantum entanglment arise when we consider the question of quantum computation. Classical computation has a problem which is the potentially unlimited time it takes to check out every one of a series of possibilites. E.g. 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.
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 states in such a computation, collapse of the wave packet might give us the answer by a form of parallel quantum computation.
A key idea here is the "square root of not". Suppose we know an atom is exicted by a certain amount of energy, but only shine a laser on it for half the time needed to provide this energy. Then the atom is in a superposition of the ground state and the excited state. 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.
We can use this uncertainty to perform a quantum calculation in the following way. Suppose we have a collection of such 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. for two atiom 'bits' - 00, 01, 10 and 11..
We now devise a problem to solve - decrypting by factorizing a large number. We have a number register in two parts. The left part L is excited to asuperpositon. The right half R is designed to give the results of a quantum factorization remainder of each of the possible numbers in L. These turn out to be periodic, so if we measure R we get one of the values.
Choose a random number x between 0 and n, then raise it to the power of the number in the L register. Divide by n, and place the remainder in a second register. It turns out that for increasing powers of x, the remainders form a repeating sequence. Because the number in the first register is different in each universe the result varies from universe to universe.
As an example, take the very simple case where the number you
are trying to factorise is 15, and 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 ...
This in turn collapses L into a superposition of only those numbers with this particular value in R. We then do a cunning trick with the L register which is consistent with wave functions, we recombine them by interference to produce the Fourier transform, replacing the wave by the frequency spectrum. When we now look in L we find a number which is the periodicity of the possible solutions in R. We can now use this quickly to find a factor of the number we were seeking.
The final observed value, the frequency f, has a good chance of revealing the factors of n from the expression x^(f/2)-1. in the simple example above, the repeat sequence is the four values 2, 4, 8, 1, so the repeat fiequency is 4. Thus Shor's algorithm produces the number: 2^(4/2) -1 = 3 which is a factor of 15.
The essential principles of this calculation may pass over into a general problem solving paradigm - (1) make a superimposition of problem states, (2) find a representation of the solution which is periodic, (3) collapse this solution to one of its states, (4) transform the new superposition (5) measure the periodicity to get the answer.
Key is the idea that measurement of part of an entangled system may enable the whole system to collectively solve a problem connecting its entangeld parts.
The first three chapters look at chaos and its possible relationships with quantum mechanics and how these may interactively combine to evoke the paradoxes of conscious subjectivity and perception. The following articles underpin these ideas because they deal with ideas about brain function or consciousness which are compatible with the idea.
The transactional interpretation 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.