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Online Book 2

3 March 2012: ONLINE BOOK AVAILABLE ON AMAZON: The sites online book 'Consciousness, Biology and Fundamental Physics' is now available on Amazon both as a paperback and as a kindle book. New paperbacks currently priced from £9.05 and kindle books from £2.63. Text remains free on this site.

2.9: Information theory
Shelley Goldstein, who spans maths, physics and philosophy, criticises information based theories for their failure to deal with the two-slit experiment. He asks how the different paths of the wave function in the two-slit experiment could lead to a wave interference pattern if nothing physical was involved. He thinks that the wave function is objective and physical, and neither some form of purely subjective experience, nor something that is simply the information that we happen to have. He sees the notion of information more in terms of a brain state connected to human needs and desires, rather than as an objective aspect of the external world.

Goldstein discusses a refined version of the double-slit experiment, in which the quanta are sent into the system one-by-one and an interference pattern gradually emerges. He sees the emerging pattern as an entirely objective phenomena not resulting from a limitation on our knowledge of the system. Tim Maudlin appears to agree with this, arguing that in the two-slit experiment, the sensitivity to whether one or two slits are open indicates the response of something physical, rather than just the experimenters ignorance about the location of a particle. Maudlin points to the holistic nature of the two-slit experiment, and suggests that the same thing is apparent in non-locality.

The philosopher, David Wallace, also takes the view that states in physics are facts about the physical world, and not just our knowledge of the physical world. He rejects views that see the quantum as a mixture of our information and our ignorance, because in practise physicists measure particular physical processes. The physicist, Antony Valentini, poses the question as to how the definite states of classical physics arise from the indefinite states of quantum physics. He argues that it is impossible to have a continuous transition or emergent process moving from one to the other. The problem of measurement or reality at the quantum level is therefore argued to be a real problem, and requires some physical theory such as pilot waves or collapse theories to explain it.

The physicist, Ghirardi, a member of the trio of physicists responsible for the GRW collapse theory, views information theory as having played a negative role in terms of evading the need to deal with foundational problems in quantum theory. He sees it as a backward step to go from being concerned about what exists, to merely considering our limited information. John Bell, whose inequalities theorem sparked off the modern interest in entanglement, asked in response to this approach what the information was about. Proponents of information theory denounce this as a metaphysical question, which seems illogical for physicists who are themselves apparently withdrawing from the attempt to produce a physical description of the universe. As in some reaches of consciousness studies, we seem to be seeing the modern mind retreating into a mysterian view, possibly as a last ditch way of defending classical physics, or perhaps we should say metaphysics. Tim Maudlin similarly finds the notion of information theory puzzling. In his view the physical reality exists before we start to get information about it, and it is not meaningful to reverse this process.

2.10: Decoherence theory
Tim Maudlin uses reductio ad absurdum to argue against decoherence theory. Buckyballs (a molecule of 60 carbon atoms) and some biomolecules have been put into superposition, and the line of decoherence argument suggests that larger and larger superpositions are possible without limit, so that decoherence never occurs and superpositions remain hidden in macroscopic objects.

From this Maudlin argues the solid macroscopic objects such as bowling balls should be capable of being put through a two-slit experiment and produce an interference pattern. I suppose the counter argument might be that the superposition is too small to produce that type of separation and interference, at least that's what decoherence arguments appear to be talking about. However, quantum superposition as in a normal two-slit experiment looks to imply complete separation, and this should be achievable by bowling balls or even iron cannon balls if decoherence theory is to be validated.

Similarly Ghirardi says that he would be willing to give up his collapse interpretation of quantum theory, if macroscopic superpositions could be demonstrated. Other philosophers also object to residual approximation and lack of explanation for superposition in the decoherence approach. Ghiradi is also critical of the theoretical basis of decoherence theory, where macroscopic objects are deemed to remain in superpositions, although these are superpositions that cannot be detected by existing technology. He sees this as reversing the normal process of science, and attempting to move from our definite knowledge of macroscopic objects to an approximation.

2.11: The pilot-wave model
The 'pilot-wave' model developed by de Broglie in the 1920s, and was revived by Bohm in the 1950s. This can be argued to be the simplest solution to the measurement or reality problem. Particles existing in reality are argued to be guided by the wave function equation. There are perceived problems with this approach. The wave function itself is supposed to be real as well as the particle, although there is no evidence for this, and the evolution of the wave function without collapse is argued to have implications for a many worlds outcome. The limitation on the Bohmian approach is that it is impossible to know the initial conditions that would give control over the otherwise random outcome of a quantum experiment.

2.12: Collapse models
Wave function collapse models developed by Ghirardi and others are yet another interpretation of quantum theory. These theories require a modification of the Schrödinger equation, so that the evolution of the wave function described by the Schrödinger equation can collapse to the outcome of a particle with a particular position and other properties. P. In Ghirardi's theory of wave function collapse, the wave function can be viewed as the quantity that determines the nature of our physical world and the spatial arrangement of objects. The wave function governs the space and time of the localisation of elementary particles. He prefers collapse theories that assume a process for random localisation of particles operating alongside the standard Schrödinger quantum evolution. Such localisation occur only rarely for quanta, but in a rigid body a localisation in one place will lead to localisation of the whole body. This is seen as defining the distinction between quantum and classical processes. Tim Maudlin defends the Ghiradi-Rimini-Weber (GRW) interpretation of quantum theory against the criticism that it violates conservation of energy. His approach is to accept that the theory does imply a small amount of heating up in the universe over time, but argues that this is not out of line with observation and preserves the general principle of conservation.

As of November 2011 the idea of the reality of the quantum wave function has been given some support by researchers at Imperial College London, Pusey, M., Barrett, J. & Rudolph, T. (arXiv: 1111.3328v1 [quant-ph] 14 Nov 2011 and nature.com) (6.). The authors claim to have shown by their theorem that the view that quantum states are only mathematical abstractions (referred to as the statistical interpretation) is inconsistent with the predictions of quantum theory, and that therefore quantum states are real physical states.

The theorem indicates that quantum states in an experiment must 'know' what state they have been prepared in, i.e. they must be physical systems, or an experiment will have results not predicted by quantum mechanics. They also claim that it is feasible for the theorem to be tested by experiment. Against this it should be noted that some commentators on nature.com argue that there are errors in the authors' work.

Schrödinger originally conceived the wave function as a physical state, but others soon argued that the wave function was not physical, or was merely a convenient fiction, a calculational procedure, or an encoding of experimenters limited information.

The view that the wave function was only a mathematical abstraction was the basis of the Copenhagen interpretation of quantum theory that dominated thinking through much of the 20^th century. Support for the Copenhagen interpretation was eroded in the latter part of the 20^th century, but the idea of the wave function as a mathematical abstraction has more recently been given a new lease of life by quantum information, which views the wave function as abstract information.

If the authors' theorem was to be vindicated it would not merely discredit quantum information theory but conclude the debate of nearly a century as to whether the quantum wave function was a physical reality. Clearly, this has crucial implications for Penrose and similar theories that look to the reality of the wave function to open physical access to a fundamental level of the universe.

2.13 Deeper levels
Lee Smolin (7.&.8.) himself takes a 'tragic' view of modern science, in thinking that it does not even have a toehold on why the universe appears quantum or on the nature of consciousness. Smolin meanwhile looks for a more fundamental theory. He thinks that space and the concept of locality should be thought of as emerging from a network of relationships between the quanta that are regarded as fundamental. He argues that the existence of non-locality shows that spacetime is not fundamental, in that non-locality does not accord with the conventional view of spacetime. He proposes that spacetime emerges from a more connected structure, with the non-local connections observed in quantum theory being a remnant of this more connected system. Lucien Hardy, another physicist at the Perimeter Institute, argues that while quantum theory has probabilities evolving against a fixed spacetime background, relativity is deterministic but has a dynamic version of spacetime. He thinks that some deeper theory of quantum gravity will need to combine the probabilistic and dynamic aspects of these two theories.

There is no easy conclusion to this debate, although the combination of the two-slit and EPR experiments seems to point to the quanta being bound into a non-local structure. A possible solution is that randomness, and the arrangement of the particles in the single-emission two slit experiment are alternative faces of entanglement, which describes the communication of quantum properties across spacetime. The approaches of many modern physicists tend to view spacetime as a discrete network or web, and this in turn hints at a structure which could support some form of pattern or code that could be related to subjective experience and understanding, although not one that could be used to transport energy, matter or any information instantiated in energy or matter.

2.14: SPACETIME
In considering spacetime some modern researchers are keen to stress the existence of spacetime as a reality rather than an abstraction, thus establishing it as something that can interact with the quanta and influence the development of the universe. Below, we look at the arguments of several researchers. They do not agree with one another, and it is beyond our scope here to discuss the merits of the different approaches. Rather there is an overall impression of spacetime as something which exists, has a discrete rather than continuous structure, and thus may contain coding, pattern or even decision making processes that influence the rest of the universe.

2.15: Spacetime – Continuous or discrete
A century ago, Einstein showed that spacetime was not a fixed absolute background, or a rigid theatre within which life is acted out. Instead, it is conceived of as dynamic in response to changes in matter. In particular, general relativity described the functioning of gravity as the curvature of spacetime in response to massive objects. In contrast to the other forces of nature, which have been shown to be intermediated by force carrying particles, the intermediation of the gravitational force or quantum gravity remains elusive. To understand quantum gravity would involve removing the conflicts between relativity and quantum theory.

Relativity was satisfactory for describing gravity and the behaviour of the universe on large scales, while quantum theory was satisfactory for describing the behaviour of particles at a scale where gravity could be ignored. However, although both have been exhaustively tested, the two theories are not compatible, with several conflicting features. The smooth continuous curvature of spacetime in relativity conflicts with the discreteness of the quanta. The dynamic nature of spacetime in general relativity contrasts with the fixed spacetime background in quantum theory. Finally, the determinism of relativity contrasts with the probabilistic and random nature of quantum theory. In mathematical terms, equations that try to link the two theories end in infinities, indicating that something is wrong. Further problems arise in situations such as black holes and the Big Bang where matter is compressed down to a scale that is normally described by the quantum. General relativity implies that matter would be infinitely compressed at the singularity of a black hole, but quantum theory does not allow infinite compression.

2.16: LOOP QUANTUM GRAVITY
One attempt to resolve the incompatibility of relativity and quantum theory is loop quantum gravity (LQG). This approaches the nature of spacetime from the angle of quarks, the fundamental particles making up the protons and neutrons of the atomic nucleus. The force that binds the quarks together is known as the colour force and has field lines connecting charges on the quarks analogous to the electromagnetic field, in which field lines connect electrically charged particles.

However, the action of the colour force is the reverse of the electromagnetic force that falls off with distance. When quarks are close together there is little constraint on their movement, and the force between them seems quite weak. However, if the quarks are pulled apart the force holding them together increases, until it reaches a constant value that does not fall off with distance. This force that is weak at close quarters and strong at a distance is analogous to a piece of string. Of course, this cannot be understood literally as a string or a piece of material. We know from elsewhere in quantum theory that the forces of nature are quantised, with photons conveying the electromagnetic force. In this case, the connection between the quarks is suggested to be quantised, and is viewed as comparable to the quantised magnetic flux in superconductors that allow the dissipationless transfer of energy.

One possibility following from this is that space has some of the energy transmission properties of a super conductor. This is plausible in relation to the ideas of space/the quantum vacuum as being full of oscillating particles. The vacuum fluctuations are in this view seen as the transmitters of force. An alternative view is that the strings binding the quarks are themselves the fundamental entities and have no requirement for a field-type theory. A further view favours a kind of duality in which the strings and the field are dual aspects of an underlying reality. The idea of a non-continuous structure for spacetime helps to deal with the incompatibility of relativity and spacetime.

It is claimed that the most successful approaches to quantum gravity involve three basic ideas, [1.] that space is an emergent property, [2.] that it is based on something that is discrete rather than continuous, and [3.] that causality is fundamental to this. Loop quantum gravity is based on the idea of describing a field in terms of its field lines. The field lines are viewed as describing the geometry of space. Areas and volumes come in discrete units. There is a suggestion that knots and links in this network code for particles. Different ways of knotting or braiding the network are suggested to be different elementary particles. When coupled to the standard model of particle physics there are no infinities in the system. When applied to black holes and the Big Bang there is a suggestion that singularities are eliminated.

Loop quantum gravity emerged out of this thinking, with the geometry of spacetime expressed in terms of loops. These are the loops of colour force flux, but instead of being related to any fixed background, it is their interrelation that defines space. The theory depends on the particular connections between the loops. The theory allows the area of any surface to come not as any value but only in discrete multiples of units. The smallest units is about the Planck area, which is the square of the Planck length. Surfaces are made of discrete parts each of which comprises a finite amount and similarly for volumes. The geometry of spacetime is not fixed, but changes as a consequence of the movement of matter. Geometry here describes the relationships of lines or edges, areas and volumes with respect to space. The laws of physics govern how the geometry of spacetime evolves, but spacetime emerges from the laws rather than providing a stage for them to act on.

With loop quantum gravity, a position in space can only be defined in relation to other objects in space. Following this line of reasoning, it is the objects that create space. If there were no objects or only one object in the universe, there would be nothing that could be identified as space. Similarly, the motion of objects in space can be defined only in relation to other objects. The geometry of space, or the measurement of its areas and volumes, is seen as changing, when the position of objects alters relative to one another. Physicists refer to this approach as ‘background independence’, with objects evolving and creating their own spacetime, rather than operating in relation to a fixed background spacetime. P. Most of the information needed to construct the geometry of spacetime comprises information about its causal structure. The fact that the universe is a causal structure means that even terms such as ‘things’ or ‘objects’ or ‘objects in space’ are not strictly correct, because causal structure means that things or objects are constantly developing, so that they are really processes rather than things, and as such causal structures that are creating spacetime geometry through their dynamic change.

2.17: PENROSE SPIN NETWORK: It was found that the discrete units of loop quantum gravity related to the spin network theory developed by Roger Penrose a generation earlier. Penrose had also considered that space was purely relational. The spin network was the version of quantum geometry that Penrose came up with. The spin network is a graph labelled with integers, with the spins that particles have in quantum theory. The spin networks provide a possible quantum state for the geometry of space. The edges of the network correspond to units of area. The nodes, where edges of the spin network meet, correspond to units of volume.

Further study suggested that the spin network picture follows from combining quantum theory with relativity. The spin networks are not set in space, they generate space, with relationships in space determined by how the edges come together at the nodes. The spin networks can evolve in time in response to changes. Each event in spacetime is seen as a change in the quantum geometry of space. The causal evolution of the spin networks can be described by the development of light cones through time. In Penrose's approach to conscious theory, consciousness and understanding is seen as being embedded in the geometry of spacetime as described in spin networks.

2.18: BLACK HOLES AND HIDDEN REGIONS: The extreme conditions of black holes are viewed as a way of examining the physics necessary to an understanding of quantum gravity, and may point ultimately to a connection between randomness and entanglement. A black hole has a horizon from within which even light cannot escape, because of the strength of the hole’s gravitational force. This creates a region that is hidden from observers anywhere outside of the horizon of the black hole. The entropy of the horizon of a black hole is given as a quarter of the area of the horizon divided by h bar times the gravitational constant. Perhaps more helpfully, the horizon can be conceived as a computer screen with one pixel for every four Planck areas. The amount of information hidden in the black hole is equivalent to the number of notional pixels.

Quite apart from conditions around black holes, all observers have hidden regions, which are comprised of all those regions of the universe from which they will never receive light signals, because of their distance and the continuing expansion of the universe. The boundary between the part of the universe an observer can see and the part they cannot see is called an horizon.

In relation to this the situation of an observer in a spaceship accelerating towards, but not reaching, the speed of light can be envisaged. It is emphasised that as that as long as the ship continues to accelerate, there will be a region behind it from which light does not catch up with the ship, and that this will constitute a hidden region for an observer on the ship. Working from the equivalence of gravity and acceleration, researchers Paul Davies and Bill Unruh also think that if an observer near a black hole saw heat radiation coming from the black hole, it means that an observer accelerating through the quantum vacuum would see heat radiation coming from in front of them. The significance of this is that from the point of view of an accelerated observer, the quantum vacuum is a real thing capable of having an effect.

This relates to uncertainty principle, a key concept within quantum theory, which prevents quantum particles from having a precisely defined position and momentum at the same time. The consequence of this is that even when a system is cooled to a point at which it has no energy, it will still have an intrinsic random motion, because if this ceased its position and momentum would both be precisely defined. This is known as zero point energy. Because of the lack of ordinary energy in the system, detectors do not register this motion. However, if the detector is accelerating, as would be the case of detectors placed on our observer’s spaceship, the accelerated ship/detector is itself a source of energy that allows the zero point energy to be detected.

Uncertainty principle does not only apply to position and momentum, but also to the electric and magnetic fields that permeate space. One cannot simultaneously know the precise position of both the electric and the magnetic field in a particular region of space. Even when a region is cooled so as to contain zero energy, there will be randomly fluctuating electric and magnetic fields, referred to as quantum fluctuations of the vacuum. These, however, would also be detected by the accelerating detector on our observer’s spaceship.

A recent experiment by Chris Wilson et al at Chalmers University, Gothenburg serves to substantiate the prediction that energy quanta, in the form of real photons, could be derived from empty space. This prediction was based on uncertainty principle, which does not allow a permanent state of zero energy, but requires a fluctuation between zero and a small amount of energy. This comes in the form of virtual photons that jump in and out of existence, but can be promoted to real photons if they absorb energy.

The existence of these photons had already been inferred from the Casimir effect. In this, when there is a very small space between metal plates, some longer wave photons are excluded, and therefore pressure from outside the plates is greater than the pressure inside, pushing the plates together.

The Gothenburg experiment has gone beyond inference to the actual production of photons from the vacuum. To achieve this Wilson used a superconducting electrical circuit with an oscillator, which resulted in alterations in the distance that an electron had to travel through the circuit. The alteration meant that the electron was doing the equivalent of travelling at a quarter of the speed of light. This proved sufficient for the kinetic energy of the electron to turn some of the virtual photons into real photons.

This is significant in terms of how we conceive of spacetime, indicating that it is a reality rather than an abstraction, and also that it relates to discrete elements such as virtual quanta. This in turn makes it more plausible to think in terms of spacetime have a fundamental measurement or geometry that can be related to consciousness as a fundamental property.

2.19: RANDOMNESS AND ENTANGLEMENT
It is possible to go a step further, and to try to explain where the randomness in the fluctuations of the electric and magnetic fields comes from. Going back to the example of the spaceship accelerating towards the speed of light, it is claimed that the photons that constitute the electric and magnetic fields are non-locally correlated, with each photon detected by the ship non-locally correlated with a photon in the ship’s hidden region. The observed randomness is a measure of the observer’s lack of information about the hidden region. The entropy represented by this randomness is related to the size of the hidden region. In turns out that the entropy of the particles detected by the ship is proportional to the area of the horizon of the ship’s hidden region. This is referred to as Bekenstein’s law, after the physicist of that name, stating that with the horizon of a hidden region there is an associated entropy that indicates the amount of information hidden by the region. Such things or processes are viewed as being finite in number, and therefore discrete from one another. Given that it is these events and processes that create space, it is therefore also possible to view spacetime as discrete. By contrast, the smooth continuous space implied by general relativity would require an infinite number of relationships. The discreteness of space is one of the few areas in speculative physics where there appears to be something of a consensus among physicists.

2.20: STRING THEORY
In the more popular rival to loop quantum gravity, string theory, the quanta are viewed as one-dimensional strings extending into higher dimensions, beyond the normal four dimensions. The extra dimensions are usually deemed to have been rolled up very small in the Big Bang, which accounts for them never having been detected. The manner in which the strings vibrate determines the nature of the particle involved. The analogy is that of the strings of a violin, where the vibration of the string determines the nature of the note. This has the advantage of being described by mathematics that would allow quantum theory and relativity to be compatible. It can also be speculated that spacetime as structured in string theory could support a discrete web or pattern that undergirded conscious experience and understanding, but this has not been extensively developed.

2.21: SPACETIME AS A FUNDAMENTAL: Other views favour the ideas of spacetime and the energy it contains as fundamental, while quantum particles are suggested to be less fundamental, existing as distortions or disturbances of the underlying spacetime.

Inertia is the built in resistance of objects to being moved if they are stationary, or having their motion changed if they are already moving. This kind of inertial mass is the most familiar form of mass. The associated concept of weight represents the force of gravity acting on the mass, and for this reason weight varies according to the local strength of the gravitational field. This is referred to as gravitational mass, as opposed to the constant of inertial mass.

Higgs field: One possible mechanism for endowing quanta with mass is the proposed Higgs field. The Higgs field is suggested to provide the 'rest mass' that is intrinsic to the particle rather than any mass associated with the energy of its movement. Fields such as the electromagnetic field and the Higgs field are here viewed as being fundamental, with quantum particles being less fundamental, because they are just local excitations of a field.

Still others think that mass comes from interaction between a quantum particle and the quantum vacuum, as the particle moves through the vacuum. The fundamental particles are seen as localised knots in the quantum fields. In this idea, quantum behaviour is traced back to the oscillation of photons jumping in and out of existence in the quantum vacuum.

This idea was developed in relation to Hawking radiation. Hawking proposed that the strong gravity near a black hole distorts the quantum vacuum so that virtual photons that normally pop in and out of existence here receive enough energy to become permanent particles. It is suggested that these permanent photons would to an external observer look like the radiation from a hot furnace, or the hot particles seen by the observer in the spacecraft discussed above.

The electric and magnetic fields flowing through space can be argued to constantly oscillate, as a function of the uncertainty of their position and momentum. The name 'zero-point field' refers to the fact that this is the lowest possible energy state that persists even when the heat/movement of molecules has ceased. Because electromagnetic radiation permeates the whole of space, this adds up to an enormous amount of energy. It is argued that there is no such thing in the universe as a void, and that this lowest energy state is still full of this zero point energy. This quantum vacuum is viewed as a sea of energy fluctuations and force perturbations jumping in and out of existence.

The zero point energy can be treated as a real thing, and concentrates attention on what effect this has. The existence of the zero point energy has long been demonstrated by the Casimir force. At distances smaller than a millimetre metal can be forced together, because long-wave radiation is suppressed between the plates, so more pressure is exerted on the metal sheets from outside than inside. The nearer the plates are brought together, the more radiation is excluded and the greater the external pressure.

The assumption since Newton has been that the mass of an object, which is in effect a measure of its inertia, was an innate property of the object itself. However, this has been recently challenged by an opposite proposal that the inertial resistance to acceleration came not from the object itself, but instead from a contrary force exerted by the surrounding zero-point field. One suggestion is that the oscillation of the virtual particles of the vacuum interact with objects so as to produce inertial mass. Photons are seen as being exchanged between the virtual particles of the quantum vacuum and the quarks and electrons that are most fundamental in matter. This accords with the idea that inertial force comes from outside the body, from the quantum vacuum and from the interaction between the particles of matter and the virtual particles of the quantum vacuum.

In this approach that the fundamental thing is not mass, but the quantum vacuum. In this view Higgs field is relegated to producing rest mass, while inertial mass comes from the vacuum. Photons can be exchanged between the quantum vacuum and the quarks and electrons that make up matter. Although an electron is regarded as a point particle, it behaves as if it had a certain size, and this is viewed as an oscillation that reflects the oscillation of the quantum vacuum around it.

Inertial and gravitational mass: It is further suggested that inertial and gravitational mass share a common origin, which is that they both arise from the interaction of electron charges with the quantum vacuum. With this concept the electric charge in matter distorts the quantum vacuum in their vicinity, attracting or repelling virtual particles with the same or opposite charges. This distortion interacts with the charges in other matter creating a force of attraction between the two pieces of matter. One bit of mass only pulls on another via the quantum vacuum. The bending of light that is seen as a proof of the warping of space in general relativity is here explained in terms of a distortion of the quantum vacuum. Acceleration through the quantum vacuum results in resistance from the vacuum and this is seen as explaining inertia. According to the theory of general relativity spacetime is warped by energy, with mass being categorised as a form of energy. In the quantum theory approach to this concept virtual photons that jump in and out of existence in the vacuum warp spacetime around themselves. The source of the energy that warps space in general relativity is the energy density of space or the amount of energy in a unit volume of space.

2.22: Spacetime and consciousness: What is the significance of all this for consciousness studies? 'Fundamentalist' theories try to explain consciousness in terms of fundamental quantum features, which ultimately involves the nature of the quantum vacuum/spacetime. An understanding of this therefore becomes central to an understanding of the physical basis of consciousness. If spacetime/the quantum vacuum has discrete structure, as these proposals discussed above suggest, it becomes the more plausible that it could provide a network, pattern or code that underlies consciousness.

SECTION 3

QUANTUM BIOLOGY

In this section we introduce recent research indicating the existence of quantum coherence in organic matter and look at the possible implications of this for our understanding of neurons. First, however we need to make an excursion into the physics underlying some long-established organic chemistry, which is relevant to the systems we discuss in this section. Once again no apologies for bringing you to a hard place.

3.1: Π ELECTRONS: We start here by discussing the role of electrons around atoms. The overlap of the atomic orbitals forms bonds between atoms, and thus creates molecules, and also determines the shape of a molecule. The same atoms held in a different shape can result in a different compound. The term ‘n’ is used to describe the energy level of each orbital. Each value of ‘n’ can represent a group of orbitals at different energy levels known as a shell. The first shell, n = 1, can only contain one orbital, the second shell, n = 2, can contain two orbitals, the third shell, n = 3, can contain three orbitals and so on.

Angular momentum: Another quantum number ‘L’ relates to the angular momentum of an electron in an orbital. The value of ‘L’ is at least one less than the value of ‘n’. The values for ‘L’ are conventionally given by letters. For our purposes here we need only deal with the values of 0 and 1, which are labelled ‘s’ and ‘p’. So an electron can be labelled 2s, denoting an orbital energy of 2 and an angular momentum of 0, or it can be labelled 2p with an orbital energy of 2 and an angular momentum of 1.

Electron wave function: The electron orbital is viewed as being a wave function. With a wave, the wavelength or frequency is related to the energy level of the individual quanta, but the amplitude (the height of the wave) squared is the strength of the signal, or in other words the number of quanta involved. With a photon, the quanta of light, frequency determines the colour of visible light, but the square of the amplitude, signifying the number of quanta, determines the brightness.

Spheres and lobes: It is possible to chart the probability of an electron being present at a particular point in space, and this can be referred to as a density plot. For an ‘s’ orbital (see angular momentum para. Above) the density plot is spherical, but with ‘p’ electrons, the shape of the density plot is two lobes with a nodal area in between, where there is no electron density. The wave functions of these two lobes are out-of-phase.

A further quantum number m_L relates to the spatial orientation of the orbital angular momentum. This gives a value of L- or L+ for ‘p’ orbitals, while ‘s’ orbitals have a 0 because a sphere does not have orientation in space. For ‘p’ orbitals there are three possibilities of -1, 0 and +1 that can be related to the mutually perpendicular x, y, and z axes in geometry, and are written as p_x, p_y and p_z.

Structure of an atom: The structure of an atom involves having two electrons in the lowest energy orbital and working up from there. Hydrogen has one electron located in the lowest energy orbital, and helium has two electrons placed in this orbital. Two electrons render an orbital full. An orbital can be full (2 electrons), half-full (one electron) or empty. With lithium which has three electrons, the third electron has to be located in a second orbital. With carbon there are six electrons, with two in the ‘n’ = 1, first shell. In the second, ‘n’ = 2 shell, there is one full orbital with two ‘s’ electrons and two half-full orbitals each with one ‘p’ electron.

Continued Book 2a