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The physical principles that govern the microworld, as provided by Quantum Mechanics (QM), are profoundly different from the ones that the macrocosmos obeys.

The "microworld" here denotes anything at and below the molecular level: molecules, atoms, electrons, nuclei, protons, neutrons, quarks.

As Linus Pauling taught us, chemistry is nothing else but applied quantum mechanics at the atomic and molecular level.

Interestingly enough, Molecular Biology holds a very intriguing position between the macro and micro worlds in the following sense: ab initio, molecular biology is concerned with the structure and function of the cell [10], which is mainly composed of macromolecular structures (DNA, RNA, proteins, ...) and as such, most [8] of the time and for many purposes, are suffciently and accurately described by classical physics.

Nevertheless, we should not be carried away and discard QM from the picture by interpreting most of the times as implying at all times! After all, as Watson and Crick [18] taught us, the double helical structure of DNA, which is the source of DNA's fundamental genetic properties is due to the quantum mechanical H-bonds between purines (A,G) and pyrimidines (T,C): always a double H-bond for A=T and a triple H-bond for G = C. It is in the stability and universality of these H-bonds, as verified experimentally by Chargaff [19], that the secret of the genetic code lies!

Since my central thesis here, as emphasized earlier, is that quantum mechanics plays also a very fundamental role in the emergence of the mental world from the physical world, i.e., in the brain-mind relation, I will discuss very briefly some elements of QM, that I will need later.

The central dogma of Quantum Mechanics is the particle-wave duality: it depends on the particular circumstances if a quantum state is going to express itself as a particle or as a wave [20].

Consider for example a particle travelling in spacetime. Its quantum state is described by a wavefunction (Greek letter psi in eq.3) obeying a Schroedinger-type equation of the form

(3)

where h ( 1 in natural units) is the Planck constant (divided by 2pi), and H is a system-dependent operator, called the Hamiltonian of the system. It provides the unitary, time-evolution of the system, and with eigenvalues identifiable with the different energy levels of the system.

A fundamental, and immensely crucial for us here, property of the quantum equation (3) is its linearity.

Imagine that psi1, psi 2 ... psi N are different solutions of (3), then clearly the linear superposition

(4)

with C(t)i arbitrary complex numbers, is also a solution of (3). This is the mathematical statement of quantum superposition.

Let us discuss next its physical meaning. Suppose that we would like to describe quantum mechanically the following "history" of a particle, say an electron: it starts at some initial point around (xo; to), it goes through a wall that contains N slits, say 1; 2; : : : ; N, without us knowing which specic one, and it ends up at some final point around (xf; tf). Let psi1; psi2; : : :; psiN denote the wavefunctions of the electron, referring to the case that the electron went through the slit 1; 2; : : : ; N, respectively. Since we don't know the specific slit that the electron went through, we are obliged to take as the wavefunction of the electron, a linear superposition of the N states,i.e., (4).

The physical meaning then of the ci's becomes clear: | c(t)i |^ 2 is the probability that the electron went through the slit i, and thus c(t)i is referred to as the the probability amplitude.

Notice that conservation of probability entails that at any time t

(5) [9]

The probability density to find the electron at some specific point (xa; ta), after it has passed through the slits and before it ends up at (xf ; tf) is given by

(6)

Clearly, this is a standard wave-like behavior and (4) may be interpreted as describing a quantum state evolving in a coherent way, or obeying the fundamental quantum mechanical principle of quantum coherence, the physical meaning of linear superposition.

Imagine now, that we would like to find out through which specific slit the electron went through. Then, we have to make a "measurement" or "observation", i.e., to concentrate on those aspects of the quantum system that can be simultaneously magnified to the classical level, and from which the system must then choose.

In other words, we have to disturb the system (electron in our example) with the magnifying device, which results in decoherence, thus (6) is replaced by

(7)

In other words, we get classical probabilities, highly reminiscent of a standard particle-like behavior.

The "measurement"/"observation" process has caused decoherence of the wavefunction and thus led to its collapse to a specific state.

Note also there are non-disturbing "negative" or "Renninger" measurements in addition to the ones that Nanopoulos describes. For example, a nucleus gamma decays at the center of a hollow sphere whose inner walls are gamma photon sensitive. A hollow hemisphere of similar construction but smaller radius is also placed with the same center as the outer sphere.

If the inner surface of the hemisphere does not flash, then we know that the opposite hemispheric inner surface part of the outer sphere is going to flash (assuming 100% efficiency of detection). Therefore, the initial wave function collapsed before any disturbing measurement was made. In other words, retarded causality has been violated. The future flash at the outer sphere has sent a message back in time to collapse the wavefunction. We are able to decode that message even in orthodox quantum mechanics in this very special situation. To repeat, negative "non-disturbing" Renninger measurments indicate decodable backwards in time precognitive signalling effects in apparent violation of Eberhard's theorem which only appears to apply to "disturbing" measurements.

Here are then, in a nutshell, our basic quantum mechanical rules, that constitute quantum reality:

(i) A quantum system can, in principle, be in many states simultaneously ... and its quantum state in Eq. 4 is a pure state which evolves coherently and according to the quantum equation (3), as long as we don't disturb it. This is quantum linear superposition or quantum parallelism, leading to wave-like behavior.

(ii) A "measurement"/"observation" forces the quantum state to decide what it wants to be, with probability | ci(t) |^2 that the quantum state will turn out to be the i-th state ..., after the "measurement"/\observation". This is the "collapse of the wavefunction", leading to classical particle-like behavior.

Incidentally, the famous Heisenberg uncertainty principle [21] is nothing else but a quantitative expression of our intuitive statement above that a "measurement"/"observation" disturbs the system in an uncontrollable way, entailing always uncertainties in the outcome. e.g.,

(rootmean square deviation in x)(rootmean square deviation in p) > h (8)

Clearly, (8) indicates the fact that it is impossible to "measure" simultaneously, at a desirable level, both the position and the momentum of a particle. Notice that this is a fundamental principle, and has nothing to do with the potentially diffcult and practical problems that face experimentalists. Whatever she does, she cannot beat the uncertainty principle.

Introducing back-action transcends the Heisenberg principle even at the statistical level because there is a new element of control from the feedback-control loop between particle and wave that is absent in ordinary quantum mechanics. Intention or free will require this. The effect of quantum foam at the Planck scale in which the "global states" of string theory influence complex many-particle systems at low energy must correspond to the back-action if the two theories are consistent with each other.

The endemic, in the quantum world, wave-particle duality is responsible for the necessity of the two-step approach to quantum dynamics discussed above. This kind of approach is very dierent from the deterministic approach of classical dynamics and, in a way, it creates a schism in our understanding of the Universe.

There is the classical world and there is the quantum world, each following its own laws which, frankly, do not seem to have much common ground. It may even, sometimes, lead to some embarrassments [12], like e.g., the Schroedinger's cat paradox, a peculiar situation where a quantum event may oblige us to treat a cat as 50% alive and 50% dead!

Furthermore, in the passage from the quantum to the classical world it is not clear at all who is there to decide that we crossed the quantum-classical border! This dualistic view of the world (classical versus quantum) is reminiscent of the ancient needs for heavenly-terrestial dynamics, abolished by Galileo and Newton for universal dynamics, or for space and time dynamics, abolished by Einstein for spacetime dynamics, or for electromagnetic and weak interactions, abolished recently for electroweak interactions.

It looks to me that this classical versus quantum dualistic view of the world cries out, once more, for a unified approach which for many practical purposes would effectively look like two separate worlds (classical and quantum).

Any resemblance with the unified approach I discussed in the Introduction for the brain versus mind problem is not accidental!

A unified approach for classical and quantum dynamics will be attempted in section 5, but let me prepare the ground here by generalizing a bit the notion of quantum state and the likes. What we are really after is some kind of formalism that enables us to express, at least in principle, the two-step process of quantum dynamics in a more uniform language.

The following is under construction. The full equations are available now in the pdf version.

Let us represent a given quantum state ... by a state vector ... while.... denotes the complex conjugate state vector ..., and let us assume that this state vector has "length" one: ..... Consider now a complete set of orthonormal state vectors ..... , implying that any pure state can be written as ... with ci complex numbers obeying the conservation of probability condition ......(see (4),(5)). Then the scalar product.... expresses the probability amplitude that starting with the state vector .... iwe end up in the state ..... Actually, we can consider all the tensor products ...... with the understanding that .......... It is very convenient tointroduce the notion of the density matrix..... j with matrix elements ...... and such that..... ,i.e., the conservation of probability condition.

Notice that, in the case of a pure state, the description of a quantum system by the state vector ..... or by the density matrix ....... is equivalent.

For example, the measurable quantities .... correspond to Trace [density matrix A]......, with A denoting the quantum operator representing the "measurable quantity", etc. The quantum equation (3) becomes in the density matrix approach ..... (9)11 which is nothing else but the quantum analogue of the classical statistical mechanics Liouville equation, describing the time evolution of the phase-space density function.

The great advantage of the density matrix approach is its ability to describe not only pure states, but also mixed states.

Imagine that for practical reasons it is impossible to know the exact pure state of our quantum system, i.e., we only know that we have a combination of different pure states ... each with classical probability pi. Clearly, in this case we cannot use the quantum equations (3) or (9) because it is only applied for single pure states, but we can still use the density matrix approach.

Write the density matrix of the system as a mixed state .... (10) then the probability that a "measurement"/"observation" will find our system in some pure state .... (11) which is a sum of products of classical and quantum probabilities!

Notice that in the case of a single pure state, say .... all pa =b =\ 0 and pb = 1 in (10), and a "measurement"/"observation" causes the "collapse of the wavefunction" ...., that implies turning a pure state .... into a mixed state .... which is nothing else but (7)!

Of course, in the case of a "measurement"/\observation" we open the system under consideration, and clearly (9) needs modification, i.e., addition of extra terms that represent the "disturbances".

On the other hand, since the "collapse of the wavefunction" implies loss of quantum coherence, there is no way to use a wave equation like (3), or possible modifications, to represent the "disturbances". The notion of description of a quantum state by state vectors or wavefunctions really gives in to the density matrix approach, which is thus the correct approach for a unification of classical and quantum dynamics.

Usually, when we deal with realistic quantum systems, composed of different independent or loosely interacting parts, it helps to express the quantum state of the system as the product of different independent components.

Imagine, for example, a spinless particle .... decaying into two photons 1 and 2. Since the particle has no spin, the most general description of the system of two photons is given by

where the subscripts indicate the polarizations of the two photons, always opposite, such that the whole system has angular momentum zero, corresponding to the spinless particle . Imagine that a "measurement"/"observation" is done on the system by measuring say the polarization of 2 and found to correspond to the the quantum world!

Experiments done in the mid-80's have confirmed [23], beyond any shadow of doubt, the non-local nature of quantum mechanics, and the failure of classical spacetime notions to describe quantum reality.

1)contrafactual definiteness

2)locality

The many-worlds theory violates contrafactual definiteness. However, it is possible that one can have a nonlocal many-worlds theory because of conditions that Bell's theorem does not take into account. Indeed, Bohm's theory shows that this is indeed the case. Gell-Mann is simply wrong on this.

The macroscopic quantum states (MQS), mentioned in section 2, correspond here to something like

...... (13)

where... refers to the quantum state of the k-th fundamental constituent in the i-th macroscopic quantum state. Of course, for a MQS N is O (NAvogadro 10^ 23 ), a rather larger number and in several occasions the index i can also run into large numbers.

For example, in the case of a ferromagnet, the ordered state would be described by (13), and if ....indicates the spin polarization of the k-th electron, then only one ci =\ 0. While in the case of quasicrystals, describable also by (13), not only is N large ( O (NAvogadro)), but also the linear combinations may involve ahuge number of alternatives, i.e., the i-index can be also large.

Quasicrystals are rather intriguing physical structures that may need quantum mechanics in an essential way for their understanding. According to Penrose [12], the quasicrystal assembly cannot be reasonably achieved by the local adding of atoms one at a time, in accordance with the classical picture of crystal growth, but instead there must be a non-local essentially quantum mechanical ingredient to their assembly.

Instead of having atoms coming individually and attaching themselves at a continually moving growth line (standard classical crystal growth), one must consider using something like (13), an evolving quantum linear superposition of many different alternative arrangements of attaching atoms. There is not one single thing that happens, many alternative atomic arrangements must coexist! Some of these linearly superposed alternatives will grow to very large conglomerates, and at certain point the "collapse of the wavefunction" will occur and thus more specific arrangements will be singled out, and so on, until a good-sized quasicrystal is formed.

But why is Nature employing such an intriguing mechanism? Penrose claims [12] that maybe "energetics" is the answer. Usually, crystalline configurations are configurations of lowest energy, and the correct arrangement of atoms can be discovered simply by adding one atom at a time, and solving its own minimizing problem, etc.

In quasicrystal growth, finding the lowest energy state is a very complicated and difficult problem, because it involves a large number of atoms at once, and thus, we have a global, nonlocal problem to solve. Clearly, a quantum mechanical description, a la (13), seems appropriate where many different combined arrangements of atoms are being "tried" simultaneously, and eventually collapsing, through physical environment entangling, to the "energetically" and "enviromentally" appropriate arrangements, the observable quasicrystal.

It should be stressed that the QM rules have been in place and in successful use for about 70 years now, and have led to a most deep understanding of the microworld. Nevertheless, the fundamental mechanism triggering the "collapse of the wavefunction" has escaped us, until I believe recently, when string theory enabled us to put a definite proposal on the table, to be discussed in section 5.

Intriguingly enough, Molecular Biology and Neurobiology in particular, lies just in the classical-quantum interface and thus very interesting phenomena may occur. So, let us turn our attention now to the detailed structure of the brain.