Physical measurement consists of transferring information from one place to another. At some stage during measurement energy is transferred from the system being measured to the apparatus involved in the measurement process and this energy transfer, or the trace of the transfer in the form of records, becomes the 'information'.

The records or energy can be analysed in terms of information theory. If the event being measured has two states, A and B, then it can be represented by the presence or absence of an arbitrary object. If the object is present the state might be A and if absent the state is, by default, B. The arbitrary object is known as a 'bit'. Two bits can represent four states (00, 01, 10, 11, see Note 5). An information system is a set of channels through which information flows and is transformed prior to observation.

It is the arrangement of the bits that encodes the measurement. Suppose the length of objects between 0 and about 100 cm are being measured. A seven bit encoding of the lengths could be used in which case 1 cm would be 0000001 and 7 cms 0000111. The arrangement of the bits would consist of a string of ones and zeroes on paper, on magnetic tape, in a wire etc. Notice that information consists of arbitrary objects and a space-time in which these objects are arranged. But what is an 'arrangement' of bits?

Arrangements are sets of relative positions, for example, '10010' is an object, two object spaces, another object and a space, all determined from the position of the first object. In science external observers with their own coordinate system are required to determine arrangements because without them there is no way to specify which arrangement is under consideration or which direction is to be taken when specifying the relative positions.

Suppose the scientific observer itself was an information system that was running a process that performs experiments to support Pythagoras' theorem. It measures a triangle with sides of 3, 4 and 5 metres and the measurement results in three stores with 00011, 00100 and 00101 as contents. The stores are boxes with five subdivisions and the presence or absence of steel balls represents the bits (although the stores could equally well be transistors and areas of charge in a computer). The information system checks Pythagoras' theorem as follows:

What do we have after all this processing? We have eight sets of metal balls in boxes that have five subdivisions. How does the system 'know', without further processing, which set of boxes is the answer and how is, say, one metal ball in a box an 'answer'? How is the answer '00001' to be distinguished from the arrangement '10000'? Even if further processing occurred to investigate these questions the result of this would be just another metal ball in a box.

A partial answer to these questions might be to propose that the external observer is not a simple information system but is capable of some sort of remarkable simultaneous observation of things at a point in a coordinate system.

The nature of the observer will be considered again after the discussion of quantum theory below.

Information can be moved from place to place to perform functions. As an example, an abacus might be constructed so that the arrangement of balls on rods can be used to perform the functions of simple addition or subtraction. In fact it is possible to take sets of bits and rearrange them according to almost any mathematical rule and this property of bits is used in the construction of electronic digital computers. In an electronic digital computer the steel balls in the example above are replaced by electrical charges. Long strings of charges called 'programs' are used to specify simple operations such as adding, subtracting, comparing etc and the computer's central processor takes in these strings and performs the operations. At the end of reading and processing a program a set of charges in one part of the computer is transformed into patterns of phosphorescence on a screen or ink marks on paper. Electronic computers have become very fast and sophisticated but they are no more than devices that use bits to encode information and a small set of transformations of these bits that can be repeated as desired.

In fact it is possible to construct a computer using steel balls in boxes connected by tubes with mechanical actuators to move the balls around. In principle the steel balls could perform any function that can be performed with electrons. Electronic computers are very fashionable at the moment and commentators occasionally compare the human mind to a computer. This is absurd because anything can comprise a 'bit' in an information system. Turnips could replace the charges. Given appropriate mechanisms we could construct a computer using conveyer belts and turnips but would it be a mind?

Although the mind is not a computer much of the non-conscious brain is indeed like a computer. The non-conscious part of the brain carries pulses of electrical activity and processes these pulses. It is quite likely that the non-conscious brain is an information system that performs functions that could also be done by a computer (with charges, steel balls or even turnips as bits). Experience is different however. Our experience consists of things arranged in at least two coordinate axes in space, one time axis and has an observation point. This arrangement is not found in computers and is about geometry rather than bits. The brain contains a phenomenon that is different from a simple information system.

Modern Quantum Theory starts with the concept that the location and other physical properties of things can be uncertain. This is most evident with very small things such as electrons. If the position of a moving electron is determined at one instant its position a few seconds later will be uncertain and the probability of finding the electron in a given volume of space varies in a wavelike manner with distance from the original position. These waves of probability are stranger than a simple likelihood of finding an electron however, because they seem to act like real waves of electrons extended in space. This is seen most clearly in the famous Two Slit Experiments where electrons are sent through a two slit apparatus ONE AT A TIME (see illustration below).

In the case of a water wave a disturbance on the surface of the water occurs at both slits, there are two slits and two disturbances. In the case of an electron (or ionised atom etc) there is only one particle and two slits. How can the particle be a single thing yet go through both openings? There seem to be two possible answers, either the 'particle' is distributed in space and comes together again at the screen or the particle can somehow 'feel out' the alternative paths and then take the appropriate route based on this.

If a detector is placed in one of the paths the interference pattern disappears and is replaced by two distributions of electrons. This is what would be expected if the electron were a simple particle.

The transfer of information from the apparatus via the detector removes the uncertainty about the position of the electron. But how can simple uncertainty give rise to wave-like behaviour and the removal of uncertainty abolish the waves?

According to Max Born the wave behaviour can be traced to the way that the probability that a particle is in a particular region of space can be calculated in terms of 'complex numbers' (see Penrose(1989) for an excellent introduction to this and a simplified treatment). When Penrose's simplification is applied to the two slit apparatus the chance that a particle will go through a particular slit can be expressed as the squared norm of a complex number, ie:

Probability of the electron going through top slit to a point on the screen = | e + if | ²

Where i is the square root of minus one and e and f relate to the chance of passing through the slit. In the same way, the probability of the electron going through the bottom slit to the point on the screen = | g + ih | ²

The wave-like properties derive from the overall probability of the electron reaching the point on the screen. This is the sum of the probabilities of going through each slit and, because complex numbers are involved, this results in a wave-like distribution of the probability of the join event, ie:

T = e + if

B = g + ih

Joint probability = | T + B | ²

Which, because of the rules for combining complex numbers becomes:

Joint probability = |T|² + |B|² + 2 |T| |B| cos θ

This analysis provides a mathematical foundation for quantum theory but it leaves a huge unanswered question: why is the probability of finding a particle at a particular place proportional to the squared norm of a complex number?

In the 'Transactional Interpretation' of Quantum Mechanics (Cramer(1986)) these squared norms that calculate probabilities are conceptually related to the squared norms that are presumed to underlie the metric of relativity theory (see Note 4).

If a detector is placed at one of the slits the probability of finding an electron in a particular zone on the screen reduces to a simple real number (0.5 if the zones are defined as the two areas where the electrons strike the screen). In other words, in the Two Slit Experiment, observation removes or ignores the possibility of arrangements of the electron that might be described by imaginary numbers.

The wave nature of the probability of finding an electron can be expressed by a mathematical formula called a 'wavefunction' and hence the idea that observation can remove the wave nature of a particle is called the "collapse of the wavefunction". On the other hand, if the observation somehow ignores the arrangements that involve imaginary numbers it might be selecting certain states or have some sort of access to the final position of the particle rather than 'collapsing' anything. The term "state vector reduction" avoids any assumptions about the mechanism or phenomenon that causes the wave nature to disappear. 

The probability of finding a free particle in a particular part of space is described by the squared norm of the "wave function" symbolised as: |y (r,t)| ². The wave function, y(r,t), is a complex function of the three coordinates in space and the time coordinate. The probability of finding a particle in a small volume element dx dy dz at point r is:

dP(r,t) = C |y (r,t) |² dr³

Where dr³ = dx dy dz and C is known as a "normalisation constant" that ensures that the probability of finding the particle in the small volume element is a fraction of unity (the probability of finding the particle somewhere in the universe being unity ie: one).

The wave function, y(r,t), describes the amplitude of the wave. Why should an 'amplitude' be related to a probability of finding a particle?

If light waves are used as an analogy, the probability of finding a particle of light, or 'photon', in a given area is proportional to the ratio of the number of photons hitting the area to the total number of photons that are emitted. The number of photons hitting an area per second is proportional to the intensity of the light. This intensity of light and hence the number of photons hitting the area is proportional to the square of the amplitude of the electromagnetic field (see Note 6). In fact, in the case of most waves the intensity at a particular location is proportional to the square of the amplitude of the wave. If this intensity is due to particles then the square of the amplitude can be used to calculate the probability of finding a particle.

Schrodinger worked out how to calculate the wave function of a particle with a definite mass that is subject to a potential V(r,t), that varies with position and time:

iћ(δy(r,t)/ δt) = -(ћ²/2m)6 ²y(r,t) + V(r,t) y(r,t)

Where 6 ² is the Laplacian operator = δ²/δx² + δ²/δy² + δ²/δz²

This is a difficult equation to understand if you are not a mathematician. Fortunately it can be simplified for various applications. The Schrodinger equation is a form of wave equation (see Note 6). The objective of the Schrodinger Equation is to calculate the wave function y(r,t) which is the amplitude of the wave at any time and place. Once the amplitude is known the probability of finding the particle can be calculated.

The solution of the Schrodinger Equation for a free particle in a time independent potential is:

y(r,t) = Ae^{i(k.r - ωt)}

Where A is a constant and ω and k are related by: ω = ћk²/2m. This equation describes a plane wave and corresponds to the sort of wave that occurs in the Two Slit Experiment.

The Schrodinger equation is linear and homogenous in y, these are mathematical properties which mean that if y₁, y₂, y_3.... are solutions of the equation then the sum a₁y₁ + a₂y₂ + a₃y₃ is also a valid solution (a_1.. etc are constants). This adding together of wave functions is called superposition. The possibility of superposition means that the probability of finding a particle in a particular volume of space can be derived from the the sum of numerous probability waves. When many waves are added together it is possible to obtain a vast range of forms from square steps to sharp peaks. The probability of finding a fast moving electron at any moment has a peaked distribution in space so the electron in motion is conceptualised as a wave packet rather than as a simple particle.

The existence of superpositions means that it is possible to specify a wave function for an entire system of particles, or even perhaps, the entire universe.

When a particle is described by an extended wave function the system is said to have a coherent state. A system in a coherent state behaves as a single entity in the sense that it's wave function will change form in its entirety if observed. This is the most general definition of a coherent state in quantum physics but there are also other usages of the term "coherent" such as that used by Schrodinger where a system is in a coherent state if it minimises the uncertainty of both momentum and position for a particle.

Particles are entangled if a measurement on one affects a measurement on the other; it is then said that the properties of the particles are correlated with each other. Relatively few ambient particles are entangled and entanglement usually requires a special apparatus to generate the effect (see Bell's Theorem below).

The Schrodinger Equation is an approximation and is of limited application. In 1928 Dirac produced a Lorentz Invariant form of the Schrodinger equation that accounted for 'spin'. This merger of Special Relativity (eg: symmetry and invariance) and Quantum Mechanics has been extended and now forms an area of physics called Quantum Field Theory.

It is often believed that Quantum Theory, with its apparently instantaneous action at a distance, is a replacement for, or a competitor of Special Relativity (see Bell's Theorem below). This is not true, the two theories are actually merged and the view that one will replace the other is a misunderstanding of the problems encountered with reconciling the description of gravity in General Relativity with Quantum Field Theory.

It was noted above that the probability of finding a particle in a particular place at a particular time is dP(r,t) = C |y (r,t) |² dr³. In other words, the probability depends on the square of the amplitude. The amplitude is composed of a function involving complex numbers such as (a+ib) but the required probability is expressed in real numbers, this means that the squared norm is used to evaluate |y (r,t) |² ie: y^* (r,t) times y (r,t) where y^* (r,t) is the complex conjugate of y(r,t). The probability of finding the particle over all space can be calculated as:

∫y^* (r,t) y (r,t) dr³ = 1

The wave function y(r,t) represents a quantum state of a particle, it can be represented as a state vector with a length related to the amplitude. The two wave functions y^*(r,t) and y(r,t) can be represented by two vectors, one the complex conjugate of the other. The vector corresponding to y(r,t) is given the symbol |y> and is known as a ket, the vector corresponding to y^*(r,t) is given the symbol <y| and is known as a bra. As we have seen, for every ket there is a corresponding bra (however there can be bras without corresponding kets). The point of defining kets and bras is to simplify the calculation of quantum states that are superpositions of multiple states and of the states of many particles. In these complicated circumstances there can be many vectors pointing in all different directions. The space that contains all these vectors is known as the state space, it is a particular sort of mathematical space that has the general properties of a Hilbert Space (see Bibliography 12). The bras and kets are interrelated by the scalar product. The scalar product of the bra <Ф| with the ket |y> is symbolised as <Ф|y> and in general:

<Ф|y> = ∫Ф^*(r,t) y(r,t) dr³

See Bibliography 13

The state space can have numerous dimensions that are each based on particular solutions of the wave function (these solutions are the position eigenstates corresponding to a set of eigenvalues (see maths appendix ). The state space is not like an ordinary space in which single, material things can be arranged in a variety of directions. It is a mathematical construction to aid the manipulation of the possible superpositions of amplitudes. It is very important to realise that the Hilbert Space of Quantum States is not continuous with, nor part of the same vector space as vectors in space-time such as lengths, time intervals etc. (see section on Physical and Mathematical Spaces below).

For many years quantum theorists described the reduction of the state vector in terms of an event that occurs when a quantum system is 'in principle' observable by an external observer. As an example, when a detector is placed on one of the slits in the Two Slit Experiment the path of the electron becomes observable 'in principle' so the electron exhibits localised properties rather than the properties of a wave. The external observer does not need to actually look at the detector whilst it is doing the detecting.

Although a detector can appear to affect the state vector of an electron it is also apparent that the apparatus as a whole is having an effect because the slits are drastically changing the form of the probability distribution of the electron. Decoherence theory discusses the entire system of the electron, apparatus, detector, screen and surrounding environment. All of these are considered to be quantum systems with the proviso that the bulk of the environment has preferred states that become embedded in measuring devices, slits etc. We call the preferred states the "classical universe". The apparatus and measuring devices that are in equilibrium with this classical universe are only sensitive to a subset of all the positions and properties possessed by the quantum system under study.

The appearance of only certain states from the many possible states in a quantum system is known as 'superselection'. The combined state of a measuring device and a quantum system results in the appearance of only a limited number of alternatives that can become measurements. It is as if the coherent quantum system 'decoheres' onto the measuring device. Decoherence theory was first developed by Zeh(1970).

"Decoherence is a process which, through interaction of the system with external degrees of freedom often referred to as the environment - singles out a preferred set of states sometimes called the pointer basis. This emergence of the preferred basis occurs through the "negative selection" process (environment induced superselection) which, in effect, precludes all but a subset of the conceivable states in the Hilbert space of the system from behaving in an effectively classical, predictable manner." Zurek(1994).

Zurek(1981) argues that any apparatus that is used to measure a quantum state has certain preferred states called a "pointer basis" that correspond to classical states of the events being observed. As an example, an apparatus that is used to measure spin will have states that correspond to particular values of spin and the environment will continuously monitor these states so that only one state is present in the apparatus. When a measurement of spin is made there is a "one to one correlation between the state of the apparatus and the state of the measured system" (Zurek,1981) which results in the value of spin that is measured being correlated with one of the preferred states in the apparatus.

Zurek further points out that when 'spin' is being measured the apparatus makes a coordinate-independent measurement. Some property of the apparatus may change state but at this stage the changes of state are not related to any coordinates or even to the nature of the event. The measurement finally acquires position data when the measuring apparatus becomes correlated with the observer. Zurek considers that the observer is correlated with environment and that the environment can dissipate information to such an extent that it sets the states of both the apparatus and the human observer. The sudden change in state that occurs at a detector shows that decoherence can be extremely rapid.

The decoherence approach of Zurek et al is similar to the "Consistent Histories" approach of Griffiths and of Gell-Mann and Hartle (see Gell-Mann 1994). In the consistent histories approach the "coarse grained histories" correspond to Zurek's pointer basis and the evolution of the quantum system becomes dependent on the pre-existing state of the system

The Operational Interpretation of Quantum Theory and the Consistent Histories Interpretation both conceive of the universe as being composed of many sub-universes which are created whenever a major branching of the pointer basis occurs (eg: such as might occur when there is a superposition of two horses winning a race). Both theories limit the probability of such major branching events.

Decoherence Theory and The Operational Interpretation of quantum theory is also compatible with Everett's "Many Worlds" theory with the modification that minor superpositions are unlikely to form viable alternative "Worlds". According to Zeh(2002) Decoherence Theory is deeply intertwined with consciousness theory:

"This approach, which avoids a collapse as a new dynamical law, is essentially identical with Everett's "relative state interpretation" (so called, since the worlds observed by these observer states are described by their corresponding relative factor states). Although also called a "many worlds interpretation", it describes one quantum universe. Because of its (essential and non-trivial) reference to conscious observers, it may more appropriately be called a "multiconsciousness" or "many minds interpretation" (Zeh 1970, 1971, 1979, 1981, 2000, Albert and Loewer 1988, Lockwood 1989, Squires 1990, Stapp 1993, Donald 1995, Page 1995)." Zeh(2002)

The conscious observer that exists in a given strand of the universe has a classical view of this strand. Why such an observer exists is not explained by the theory:

"Decoherence can certainly explain why and how within quantum theory certain objects (including fields) appear classical to "local" observers. It can, of course, not explain why there are such local observers at all." Kiefer & Joos(1998)

Zeh(1979) theorises that the observer of the classical world could be a geometric point within the brain that observes the brain and goes on to state that:

"However, since no precise "localisation of consciousness" within the brain has been found yet, the neural network (just as the retina, say) may still be part of the "external world" with respect to the unknown ultimate observer of the system." Zeh(1979)

Cartesian Dualists are attracted by the concept of instantaneous communication. The idea that a vast number of processes could occur in no time at all somewhere in the brain appears to provide a physical basis for a res cogitans. The phenomenon of entanglement seems to provide encouragement for dualists but what is entanglement and how does it work?

When the quantum state of an object involves the quantum states of other objects they are said to be 'entangled'. In the case of two objects, if the state of one object is observed it is possible to deduce the state of the other object even if the objects are widely separated spatially.

Einstein maintained that entanglement means that there is some sort of action at a distance between the objects (Einstein's creation, General Relativity, had by this time become a theory of action through intimately connected points). Einstein's misgivings are summarised in the Einstein-Podolsky-Rosen Paradox, known as the EPR Paradox.

Suppose two entangled particles are generated. These particles might be, for example, an electron and a positron generated by an energetic photon. The two particles would have an entangled state because they originate from a single particle. The spin of the photon is zero and it gives rise to an electron and a positron that both have a spin of 1/2. These spins would be in opposite directions to give a net spin of zero.

If the state of the positron were measured by itself the probability of finding a given direction of spin would be described by a wave function that allows any orientation of spin. The spin state would evolve after the creation of the positron, being defined at the moment of creation but then becoming subject to the laws of quantum probability with time. The positron would be like any other free positron in this respect. Similarly the electron would also be like any other free electron with a spin that could be in any direction with increasing time after it is generated. If the particles were not entangled they would start with opposite spins and then their spins would rapidly cease to be related. Entanglement changes this however.

The entangled positron and electron started as a single object and continue to be described by a conjoint wave function as well as their individual wave functions. In the conjoint wave function the net spin is zero so if one particle has a spin in one direction the other must have a spin in the opposite direction. There seems to be a paradox, how can the two particles evolve like free particles in general and yet have a state that is determined by their twin?

Experiments have shown that the conjoint wave function exists and the state of one particle appears to determine the state of the other. There have been attempts to explain this apparently instantaneous effect of one particle on another using "local hidden variable" theories whereby the state of both particles is somehow encoded as they are created. In 1964 John Bell analysed "local hidden variable" theories and worked out that if the state of particles was encoded at the time of creation then the probability of finding spins in particular directions would be less than or equal to the sum of the probabilities of finding the spins in other directions. The derivation of Bell's 'Inequalities' is considered in depth in "The EPR Paradox" in the bibliography (11). Bell demonstrated that Quantum Theory would give a different result and soon afterwards experiments were performed that showed that "local hidden variable" theories were invalid.

Although entangled states can lead to instantaneous changes between widely separated particles this does not violate Special Relativity theory. Special Relativity theory does not demand that the speed of light is the maximum velocity for massless communication media. Instantaneous communication may, however, violate causality and cannot occur if massive particles carry the signal.

It is difficult to see how the instantaneous state change of entanglement could provide a res cogitans, a place where many processes can be performed in no time at all. All that seems to be happening is that an observer of one particle can tell another observer about the state of another particle without the need to observe it.

A quantum system has several properties that might be used to build an information system that can perform computations. A single system of particles could, in principle, have a superposition of spins or some other property that can be used to encode information. This information can be extracted from the system non-destructively because properties can be entangled between particles. If the particles are entangled then inspecting the state of one can provide information about the state of the others without disturbing the system. It is possible to perform operations on a set of particles that can be used to perform computations whilst maintaining the superposition of states. All of these features of quantum systems can be combined to make an artificial computing device. These devices are difficult to construct because decoherence occurs rapidly in most systems.

In practice particles such as photons are produced in entangled pairs by passing them through suitable optical devices or generating them with special lasers (see for instance Lamas-Linares et al (2003)). Quantum operations can then be performed on the photons (or other particles) using devices such as the Quantum Phase Gate that can change the phase of the quantum amplitude of the wave. Changing the phase can have a profound effect because it can lead to parts of the system interfering with other parts. This property of one part affecting others is where quantum systems are different from, say, a set of steel balls. Although it is possible to set some, or all of a group of steel balls spinning in one direction or another this does not lead to a system of interacting steel balls with predictable, reliable, mathematical properties. In contrast, in an entangled quantum system operations such as changing phase can have a predictable effect on the whole system that depends on the initial states of individual components.

One of the most famous quantum computing algorithms is that given by Grover (1996). This algorithm describes the quantum system and operations required for a database search. A brief outline of the algorithm will now be given but readers should consult Grover (1996) for full details. The first step is to choose a property such as spin that can be defined as one of two states and hence be an information 'bit' (see above). In a quantum system the property actually occurs in a superposition of states and becomes what is known as a "qubit" (see bibliography 14). The system of particles is then encoded with a set of binary numbers to form the database. This database is a superposition of N states and the objective is to find whether one particular state is present in the database. The system is then placed in an apparatus that rotates the phase of the binary number being sought by pi radians and leaves the others unaltered, a Walsh-Hadamard transformation is then applied, a phase rotation and then another Walsh-Hadamard transformation. This is done O(√N) times after which, if the number being sought is in the database, there is a unique state of the system that will occur with a probability of at least 0.5 (where N is the number of items in the database and the unique state can be calculated).

The "big O" notation, such as O(N), signifies that the number of steps depends on a constant times the variable plus a constant, if an algorithm requires O(N) steps it actually takes AN+B steps where A and B are constants. A quantum computer could, in principle, search a database in about O(√N) steps whereas a classical computer requires O(N) steps. If a classical computer needs 1000000 steps to search a database the quantum computer would need roughly 1000 steps. But what has this got to do with consciousness?

Ordinary quantum computing is probably unrelated to consciousness but other features of quantum computation such as the simple additive property of the wave function are definitely of interest. The additive property of the wave function means that if there is a system in a superposition of states and then another, reference state, that already has a counterpart in the superposition is added, then this reference state will increase in amplitude and hence probability. On decoherence by the aqueous medium of the brain the observed state is more likely to be the reference state than the other states. If the observed state were fed back to the system that produces the superposed system then in a few iterations the reference state could be, very nearly, the only state. A mixed classical/quantum system such as this might be useful in the brain where there is a possibility that the cortex might contain a superposition of models of the world.

Everett (1957) proposed that the probabilistic wave functions of quantum theory exist and describe the actual universe. Prior to Everett it was proposed that the apparent uncertainty of the positions and momentum of things did not apply to the world at large because the mere possibility of observation was supposed to cause the wave function to 'collapse' and disappear.

From the 1970's onwards physicists (especially Zeh et al and Zurek et al) have developed ways of calculating the way that wave functions interact with the universe in general. They discovered that when a small particle encounters a large object the wave descriptions of both are combined almost instantly to create a new wave description. This combination of waves involves no 'collapse' and suggests that the wave equations of Quantum Theory apply to all systems whether large or small. (See Tegmark and Wheeler's Scientific American article: Tegmark and Wheeler (2001)). What seems to happen is that the wave function of the particle and the wave function of the apparatus combine (this is called decoherence - see section above).

Decoherence Theory shows that the environment selects a limited set of states that can be observed. This selection can lead to several, equivalent states and it is proposed that the world, including the observer, branches to contain all of the possible, parallel outcomes. Each outcome is unable to interfere with any other outcome and only one outcome is observed in a particular branch.

One of the major issues in research on the quantum physics of the brain is whether environmental decoherence is always faster than the time needed to activate a single neuron. Tegmark (1999) calculated that decoherence of free ion movements in the warm, wet environment of the brain would occur within 10^-19 seconds, neurons require about 10^-3 seconds to fire so it is concluded that quantum superpositions would not, according to Tegmark's assumptions, occur at the level of neural firing. Unfortunately these assumptions are difficult to define because of the complexity of the brain. For example, Tegmark does not consider the quantum superposition of sodium channels and their effect on 'threshold' events in neurone firing. A neuron can be depolarised to within a few microvolts of firing and then the opening of a single sodium channel may tip the balance so that the neuron fires. The local sodium concentration would then be in a superposition of firing/not firing if the opening of the channel depends, as it does, on quantum physics (See for instance Chancey (1992). Calcium channels, being more sensitive, are even more likely than sodium channels to set the states of local ion concentrations into a superposition of those suitable for firing/not firing.

According to decoherence theory the neuron that fires and the neuron that does not fire may well constitute separate branches of the world. In each of these branches the ion channel would rapidly cease to be in a mixture of states and the neuron would fire in the first branch and not fire in the second. The two branches would not interfere with each other and each would contain a separate observer.

Tegmark's calculations suggest that it is extremely unlikely that an entire neuron can be in mixed state of firing/not firing but considerations of 'trigger' events such as channel activation suggests it is quite plausible that the brain could contain millions of branches of the world. If the brain itself is branching then which observer is in which branch? The branching would be caused by the state of the water or membrane next to a particular channel, gate or pump or due to the state of motion of synaptic vesicles etc. This would damage the brain as a processor because the brain should be responding to the state of the environment beyond the cranial cavity, not the vagaries of its aqueous medium.

Tegmark's calculations of 'decoherence times' in the brain have been challenged. Hagan et al (2002) re-examined Tegmark's calculations of microtubule decoherence times and came to the conclusion that microtubules might have a decoherence time of 10^-5 to 10^-4 seconds or might even be resistant to or screened from decoherence. The debate over neuron decoherence times looks like it will only be resolved by direct experiments.

The Consistent Histories Theory of Hartle and Gell Mann is a version of decoherence theory that emphasises the existence of 'records'. It postulates that the brain is an "Information Gathering and Utilising System" (IGUS) which contains its own set of records (memories). The IGUS would branch every time a branching of the world occurs. This theory would be most useful if long term superpositions of events were possible (such as going to Paris and not going to Paris) in which case all records for the event that did not occur would disappear once the superposition was resolved into a single event.

Mathematicians take a very broad view of the word 'space'. The mathematical term 'vector space', which is often shortened to simply 'space', can be applied to any system of linear equations. A set of elements is said to constitute a vector space if they satisfy the algebraic operations of vector addition and multiplication by a scalar (see maths appendix). Vectors do not all occupy the same vector space, for instance, the vector space of all lines on a sheet of paper is separate from the vector space of gravitational force on particles because there is no vector addition of a Newton with 2 cm of ink on paper.

In normal usage 'space' means the space between things that allows things to adopt various shapes and arrangements. Normal space is a type of vector space. This physical space is what experimental scientists are describing when they use the term 'space' without a qualifier.

Mathematicians also use the term 'state space' and might talk of the state space of stock prices or the state space of quantum amplitudes. A 'state' might be a position of a switch (this has two states: on and off) or the places a counter might be/not be on a board (2 x 64 for one counter on a checkers board) etc. A 'state space' is the set of all possible states. State spaces can be discrete like the switch or continuous like the possible positions of a person in a room. Almost any process can have a state space. State spaces are useful in probability theory.

An 'inner product space' is a vector space that has the property that for every pair of vectors there is a quantity called the 'inner product' (denoted (a,b)) that conforms to particular mathematical definitions of linearity, symmetry, positive definiteness, orthogonality etc. Most importantly the space contains a concept of vector length, known as the norm of a vector, which is defined as the square root of the inner product of a vector with itself. Euclidean space and the space of all continuous functions in a given interval are examples of inner product spaces.

State spaces that are also inner product spaces are known. The state space of all quantum amplitudes for a given system is an inner product space known as a Hilbert space. Notice that quantum amplitudes are not positions and cannot be added to positions using vector addition. Amplitudes and positions are not combined in the same vector space in any simple fashion. The use of a mathematical state space to describe the amplitude of quantum events does not mean that this 'space' is continuous with or geometrically related (by an inner product for instance) to the vector space that describes the positions of things in normal space and time.

King (1991) has written the definitive text on fractals and chaos in the nervous system (see Bibliography 15). Chaos and fractals may be involved wherever there is a system that is non-linear (where the output is not directly proportional to the input). Chaotic behaviour can give rise to remarkably ordered states over periods of time or across large groups of things. Chaotic systems can generate features that repeat regularly and which contain small copies of themselves which contain small copies of themselves. These features are known as fractals; the Mandlebrot Set is the most famous example of fractal formation. Chaotic systems can also be highly sensitive to small changes, in fact so sensitive that changes that cannot in principle be measured can create huge changes in the form of the overall system. The combination of unpredictability (a weak form of "free will") and the possibility of specifying a huge system on the basis of some tiny original form has led to proposals that chaos underlies consciousness (cf: King 2003).

Non-linearity and the potential for chaos are found ubiquitously in biological systems. The electrical characteristics of nerve cells (neurons) have non-linearities at almost every level. As an example, when the voltage across a neuron is made more positive compared with the surrounding fluid there is a potential difference at which the neuron suddenly admits sodium ions and the neuron 'fires'. A wave of positive potential difference then spreads along the cell membrane as a nerve impulse called an 'action potential'. The relationship between the flow of ions across the membrane and the applied electrical potential during the generation of an action potential is highly non-linear and has potential for chaotic behaviour. The ion channels that mediate the action potential also operate non-linearly as do synapses etc. Studies of the olfactory bulb (that mediates the sense of smell) have shown that different chemical stimuli produce different spatial patterns of ordered activity that can be predicted from chaos theory.

Bibliography