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Cause and Effort



This paper is the third of a series [in need of much correction and editing]. In the first paper of this series, I sketched out the philosophical basis and conceptual structure of space and time. In the second, I attempted to give an account of Quantum Mechanics. In this paper [still under construction], I shall explore: the Principle of Least Action; Efficient and Final Causes; Singularities in Newtonian Mechanics; Quantum Indeterminacy; and human Free-Will. My purpose is to develop an understanding of what Free Will might be and to indicate how it might be compatible with causality. These papers are much more physics based and mathematical than any I have previously posted on my Web Site. I beg the indulgence of those readers for which these fields are foreign. I shall attempt to make the issues intelligible. I have appended a Bibliography so that those who are interested in acquiring a deeper familiarity with the issues here discussed may have some idea how to set about doing so. I wish to acknowledge helpful discussions with my mathematician friend: Simon James and philosopher friend: Paul James. Of course, they should not be held in any way accountable for the speculations and proposals made here!

The Principle of Least Action

Lagrangian Mechanics

In classical Physics, Lagrangian Mechanics is derived from Newton's Second Law. In brief, this integral formulation of mechanics states that: if it is known that a particle (or more generally a system: a set of particles) starts at a place x1 at some time t1, and ends at a place  x2 at some time t2, then the trajectory that it follows x(t) between these times is given by:

                                    [x2, t2]
        x(t)    :     d S L (x, dx/dt, t)  . dt  =   0
                                    [x1, t1]


         L (x, dx/dt, t)   =    T (x, dx/dt, t)  - U (x, dx/dt, t)

and, obviously:

                  [x2, t2]
        S dx/dt  . dt   =  x2 - x1
                  [x1, t1]

L is the "Lagrangian", T is the familiar Kinetic Energy and U is the equally familiar Potential Energy.

In other words, the physical trajectory, x(t), either minimizes or maximizes the path integral of the Lagrangian. In colloquial terms, any system is lazy: it evolves in such a way as involves the "Least Action", where the action is defined as the path integral of the Lagrangian. The following points should be stressed:

  1. The Lagrangian is the difference between the Kinetic and Potential Energies, not their sum.
  2. The system is not presumed to conserve energy.
  3. The path integral of the velocity is, of course, fixed at x2 - x1, whatever the path.
  4. Note that the integral is with respect to time. This becomes the "proper time" in the Relativistic Formulation.
  5. All pairs of (x1, t1) and (x2, t2) are valid. Even if a large potential barrier impedes the trajectory, an initial velocity sufficient to overcome this barrier can always be conceived. The start and end conditions of the trajectory do not determine T(0).
  6. As the two times get closer, the pair (x1, t1) and (x2, t2) become equivalent to an initial position and velocity.
While Lagrange's Principle of Least Action can be shown to be entirely equivalent to Newton's Mechanics, the Action is a far from intuitive concept. The questions arise:
  1. Why should the Action be minimized?
  2. What if more than one path exists which minimizes the Action?
  3. How does the system know how to move at the start of the trajectory?

The Path Integral formulation of Wave Mechanics

A better basis for the Principle of Least Action is obtainable within Wave Mechanics. The Green's Function, Impulse Response or Propagator for the Wave Equation with a constant potential, U, can easily be written down. It is composed from a set of outward bound spherical wave or Hankel function. In the non-relativistic case, these are:
h+0(qr-wt)  = (ei (qr-wt) ) / (qr - wt) .
Because it is the impulse response, it must satisfy the property that.
The sum,
taken over a wavefront surface at some time, G(t),
of all the impulse responses
to the original impulse response
is equal
to the original impulse response:
       G0(U;r',t';r,t)  = S G0(U;r',t';r",t") . G0(U;r",t";r,t)  d2r"      :  for any and all t"
                                |r"| = vt"

Where v is the wave velocity, here assumed a constant. This is nothing other than Huygen's principle, which may be familiar from Optics. More generally:

       G(r',t';r,t)[V]  =  S G(r',t';r",t")[V] . G(r",t";r,t)[V]  d2r"      :  for any and all t"

Where G(t") is the closed wavefront surface at t".  Now, if V(r) is sensibly constant within G(t"), then:

       G(r',t';r,t)[V]  =  S G(r',t';r",t")[V] . G0(U:r",t";r,t)  d2r"

and the following formula for G(r',t';r,t)[V] be obtained:

       G(r',t';r,t)[V]  = S       S  G(r',t';r2,t2)[V] . G0(U(r1);r2,t2;r1,t1) . G0(U(r);r1,t1;r,t) . d2r2,1 . d2r1
                                  4p;            4p;
                                r1 = v.t1    r2,1 = v.(t2-t1)

Where r2,1 = r2 - r1. Note that the integral with respect to d2r2,1 is taken over many different spherical surfaces as r1 varies. This process of expanding the Full Propagator in terms of integrals of constant potential propagators leads fairly directly to Feynman's "Path Integral" Formulation of Wave Mechanics [R.P. Feynman "QED, the Strange Theory of Light and Matter" (1985)]. This says that the Full Propagator is the simple sum of an infinite set of partial propagators, each of which is calculated as a series of propagations along an arbitrary trajectory. From the form of the constant potential propagator, for any frequency component, each step contributes a phase factor of  ei (q-w/v)r, where r is the length of the step and v(w) is the local group velocity, dw/dq. This latter quantity depends, as do both q and w, on the local potential. As already remarked, each of the G0 factors has to be constituted as an energy integral of Hankel functions. The magnitude of each factor in the partial propagator is determined by its denominator, which can become very large when the phase does not change much over a step.

This prescription for obtaining the Full Propagator is at first sight confusing. Moreover, it might seem to be a recipe that could never be made to work! An infinite sum of path integrals is not an attractive proposition. However, this is where the Principle of Least Action comes to our aid. Only a tiny subset of all the partial propagators are at all significant. Most are incoherent and cancel out with each other: exactly as do the phasors from the edges of an aperture in a Fresnel Integral. Only those partial propagators that are calculated along bundles of paths for which the total phase factor is stationary with respect to variation of those paths make any sensible contribution to the Full Propagator. These partial propagators align with each other and add up coherently. All others have random phases and so tend to "curl up" and cancel out.


In the case of light, minimum phase always means nil phase, as the proper time interval along any real light trajectory is always zero. All deviations from "the light cone" involve light travelling faster than its observed speed in free space, and although such travel is not impossible, the partial propagators that these trajectories give rise to add up incoherently and can be ignored for all practical purposes.

An additional insight arises from observing that the argument of the phase factor, (q-w/v)dr  = 2p( p.v - E )dt/h. For the overall phase factor to be stationary, so must the time-line integral of the term "p.v - E", where p is momentum, v velocity and E the total energy. Now, in particle terms: p = mv, E = T + U, and T = (p.v)/2.


               (p.v - E)  =  2T - ( T + U )
                              =    T - U

the Action! So if one could only somehow link the propagation of a wave with the trajectory of a particle, one would have a coherent rationale for Lagrange's Principle of Least Action!

Liouville's Theorem: Causality and Determinism

Phase Space

This is a generalization of ordinary space. Just as single particle's position can be represented as a three component vector function of time: so the configuration of an N-particle system can be represented by a single 3N-component vector function of time. This vector exists in a 3N-Dimensional Vector Space: R3N. Moreover, the N-momenta of the many particle system can also be represented by a second 3N-component vector. This vector exists in a 3N-Dimensional Vector Space: P3N.  Finally, it turns out that often integrals in mechanics have to be taken over "all space" (or some specific volume of space) and "all momenta" (or some specific volume in momentum space). The concept then arises of the 6N-component State Vector, which exists in the product space Q6N = R3N x P3N.
The concept of adjacent is more sophisticated in Phase Space than in Geometric Space. If a system in a state Q has a momentum, P, then (excepting the possibility of some collision taking place) its sole next state is Q'  =  Q  +  dt . M-1 P . This is forced on the system by the very meaning of momentum: the temporal rate of change of position. Of course, next is ambiguous as to the sign of the time increment. Hence, although a state Q may have a myriad of nearest neighbours, generally speaking (in a Newtonian system) only two of these will be accessible to it. These are its determinate past and determinate future. Note, however, that in order to deduce this we have had to involve the inverse Mass operator. If this turned out to be a more subtle object than the diagonal matrix implied here, then a formalism is available that is open to behaviours richer than conventional Newtonian mechanics. As we have already seen, a discrete space-time is incompatible with particulate matter, so it is implausible that  M-1 is of Newtonian form.
Phase Space is a particularly important concept because of Liouville's Theorem. This states that a set of identical systems launched with similar initial conditions: such that all lie within a compact and simply connected region of phase space, evolve in such a way that their state vectors always lie in a simply connected region of phase space with the same volume.

Mechanistic Determinism

This would seem to implies that the uncertainty in the final state of a system is always exactly equal to the uncertainty in its initial or launch conditions, and therefore that the accuracy of a prediction can always be improved by tightening the tolerance of the initial conditions. This would amount to Mechanistic Determinism. However, this conclusion is not quite valid. Consideration of the case of a single particle is sufficient to elucidate the point at issue.

Scrambling Space

Initially, the simply connected region of phase space might be a 6D hyper sphere. This is maximally compact: it has the minimum hyper surface for the hyper volume that it contains. Now, as time progresses, this hyper volume might become at first oblate then greatly elongated until it eventually became a hyper tube. This is entirely compatible with Liouville's Theorem. Although its hyper volume at all times remains the same as that of the initial hyper sphere, the length of the shortest path (never leaving the hyper volume) between some pairs of points within such a hyper tube might tend to infinity. The uncertainty in the momentum and position of the single particle would then diverge: in principle to infinity!

The behaviour of a system can be more subtle than this. The hyper tube might become entangled with itself, such that the total volume that it threads through doesn't grow with time: it remains more or less as compact as the original hyper sphere. This would represent a "scrambling of phase space". Points that were adjacent at one time becoming fairly remote from each other at a later time.

Causal Indeterminism

Liouville's Theorem sets no limit on the degree or rate at which "Phase Space Scrambling" can occur. In fact, for linear systems it doesn't occur at all, and they exhibit the following behaviour:

            Lim      { dQ(t) }  = 0

for any value of evolution time, t. In words: "it is always possible to improve the accuracy of a prediction by tightening up on the specification or tolerance of the initial conditions".

On the contrary, non-linear systems can grossly violate this kind of limiting behaviour. They can exhibit causal Indeterminism. While every step of their trajectories is governed by a definite objective non-probabilistic causal law, the final outcome of this lawful behaviour cannot be inferred from initial conditions: even exactly specified initial conditions! Two initial conditions that are indefinitely similar can give rise to two indefinitely dissimilar outcomes. I have mentioned elsewhere the case of the magnetic pendulum, in which infinitesimal variation of initial conditions can result in macroscopic variation of outcome! This corresponds to an infinitely rapid scrambling of phase space, if only on an infinitely small scale length.

As pointed out by Prof. Landsberg  in his book "Seeking Ultimates", this scrambling of phase space is associated with the general increase in "coarse grain" Entropy required by the Second Law of Thermodynamics.

Discrete Space
If space is not continuous, but rather discretized into some kind of grid: as has been presumed in these discussions up to now, then it would seem that determinism is recovered. This is because there is then a definite limit to the variation in initial conditions. In Newtonian Mechanics, it would seem that each definite grid point in phase space must have a definite trajectory: an inevitable sequence of uniquely accessible points.

However, as I have already remarked, a quantized space-time-momentum-energy grid is incompatible with a particle theory of matter. The only particles are particles of space-time. For matter to have co-ordinated motion on a quantized grid, it must either be "guided by Angels" or be constituted of waves. If one rejects the former hypothesis, one is left with the conclusion that even if one started off with a "particle of matter" this would disperse in accordance with the wave propagation laws of the space-time grid. In other words, M-1 does not have Newtonian form.  Moreover, and more important, it is misleading to focus on such an unrealistic case. In general, a single-particle initial condition would have to be presented as a set of amplitudes: one for every space-time grid node over some 3D hyper-surface! The freedom to independently determine these amplitudes recovers the continuity of initial conditions that was at first lost by the discretization of space-time . In summery:

Efficient and Final Causes

St Thomas Aquinas enumerated a number of types of cause of an effect: in particular, the efficient and final causes. The efficient cause is the influence that pushes or pulls the effect into being or from where it obtains its being. The final cause is the purpose, objective, rationale or motivation in view of which an effect is brought into being. So, the efficient cause of a child is the sexual congress of its parents and the final cause of that sexual congress might well have been their desire to become parents. What, then, caused what?

Final causes are not difficult to delineate in biology. Evolution is driven by them. A gene becomes prevalent in a population if it facilitates the survival of a species. The final cause of that gene's persistence is the good of the species. It is almost as if it came into being in order to help. The fact that its origination was entirely fortuitous and without any motivation does not contradict the idea that, from the very moment of its existence, the continuance of its being was supported and caused by this finality.

Elementary physics deals only with efficient causes. These are called forces, and are conceived to have effect only forwards in time. Hence, when I first came across St Thomas' concept of "final cause" I reacted to it with antagonism. Subsequently, I have come to understand my error. As should already be clear from the beginning of this paper, the more advanced Lagrangian formulation of Classical Mechanics is couched in "final cause" language. All trajectories are determined by the finality that their Action is stationary: minimal or maximal. Moreover, when an intense field decays into matter (as in Hawking radiation: with its rough semiconductor physics analogue: thermal generation of mobile charge carriers in a depletion region) those virtual fluctuations that gain substance are selected for by the finality that they become real. This is a rough physics analogy to the type of final causation I have just sketched out for biological systems. Similarly, non-linear systems can exhibit quasi-repetative behaviours that other trajectories tend to converge on. These are called "strange attractors", and play the role of formal objectives that material realities feel their way towards.

Causality and Time

These examples demonstrate that it is legitimate to speak in terms of finality even in the context of physical science, so long as one clearly keeps in mind what one means. Indeed, the idea of efficient causality is not entirely devoid of problems. Consider the interaction between two things, preceded and followed by free space propagation. It is natural to say that the deviation of the post interaction trajectories from the pre interaction trajectories is caused by the interaction. This is compatible with a normal understanding of cause and of force. However, one could just as easily say that the deviation of the pre interaction trajectories from the post interaction trajectories is caused by the interaction. This is just as true. It suggests that the pre interaction trajectories were necessitated by the interaction rather than the interaction necessarily followed from these trajectories.

Mathematically, these and all similar statements are equivalent. They amount to no more than the re-arrangement of elementary equations. The ambiguity I have highlighted is associated with the fact that both Classical and Quantum Mechanics are time reversible. There is no place for temporal succession in microscopic mechanics. In this context, causality is local, instantaneous and reversible. It does not admit of before and after, but only of difference.

Causality and Contingency
Contingency: the "it could have been otherwise" of everything, has little to do with time sequencing. St Thomas' proof of God's Being - the idea that all being and becoming (or movement, as he designates it) originates in The Unmoving Prime Mover, The Perfect and Self-sufficient Being is not dependent either on the Cosmos having a "first time" or being a system that is "running down" from a low entropy state to condition of ultimate disorder. Prof. Landsberg is quite mistaken here. St Thomas' proof is much more an interpretation or inference from the theme of incompleteness that runs through the book "Seeking Ultimates".
Cause and Will
Our naive concept of cause is time bound, being based on our experience of memory,  personal decision making and willing:
Temporal succession
In fact, our experience of temporal succession is not as simple as this. Only today, 8th July 2003 AD, I experienced a deep and somewhat unsettling uncertainty as to whether a certain idea: "my partner going for a dental check-up", was a memory of an event  that had taken place in the recent past (last Saturday) or an expectation that it was about to occur (next Saturday).  Only after careful consideration of other ideas with which it was related, in particular that this appointment affected "when a friend could be invited to visit us" did I conclude that it was a memory from the past: and only then because I found that my mind contained other vivid ideas relating to occurrences associated with the visit of said friend. Only if these ideas were all fantasies about what might happen during the visit could it be that this visit had not yet taken place, and believe me the ideas were not the kind of thing that any fairly normal person fantasizes over!
Memory, Information and the Null State
I suggest that the directionality of our memory and hence our experience of the flow of time is exactly equivalent to the statement that at one end of our life history our embryonic brain is "pristine": its neural structure and form largely determined by innate genetic and physiological developmental forces. These have nothing to do with experiential sense data or the epistemological development of concepts and models to represent the outside world of objective reality. They are characteristic of a system which is intellectually closed and interacts with its environment only by the physical exchange of chemicals and heat, rather than at the synthetic level of sensory stimulus, still less the abstract level of mental conceptualization.

When a person dies (I except those who suffer degenerative brain conditions), their brain is altogether more interesting and difficult to characterize: it contains more information. Whereas it is easy to unpick a sequence if the first term is known: when one has a firm foundation, it is simply impossible to interpret a complex of information when the reference point, the "null state" is mysterious. It is like trying to understand a coded message when one does not have access to the code. This is why we can recall what happened before a certain moment in time but not (generally, at least) recall what happens after that same moment.

The brain/mind changes as the person experiences and learns in such a manner that each modification builds on those temporally adjacent to it.  That statement was carefully phrased so as to be temporally ambiguous. The ambiguity is resolved when one notes that  in one temporal sense the complexity of the brain's neural form decreases and in the other it increases. Somehow, I suggest, memories are stored in the brain/mind as differences/changes in pattern/form/organization. At the extremity of a person's cognitive life that corresponds to physiological conception, the first memory is easily distinguished from the pristine neural state. All others overlie it, tending to complicate, distort and obscure it. If at this initial extremity of life one had knowledge of the state of one's brain/mind at the other extremity of one's life, then memory all that is going to happen in the future would be possible. This would be achieved by a process of unpicking all the changes between the present state of one's mind and its terminal state. However, no such knowledge is possible. By contrast, immediate and ready knowledge of the initial state of one's brain/mind is possible at all points in life: it is the null, uninformed state corresponding to a blank sheet of paper. Hence, memory can readily work backwards in time (towards one's conception), but forwards in time (towards one's death) not at all.

This account explains why when one understands something for the first time it often feels as if one has remembered something that one always knew but had somehow forgotten. One has simply attained a mental state that one was always going to attain: it is simply part of the life sequence that makes up one's life. What is before and after any present, from an objective point of view, is symmetric: only from the partial and subjective point of view is it asymmetric. It is this fact, and the psychological experience that weakly mirrors this fact that gave rise to Plato's theory that all learning and discovery and mental growth was a kind of remembering or recovery of what had been lost at birth.

Singularities in Newtonian Physics

A perfect pin subject to a gravitational field, even balanced exactly on its point, is always falling. The slightest deviation from the vertical grows exponentially with time. However, if it is exactly balanced, it will take an infinite time to topple over: equal times produce equal mutiplications of the angle of inclination. Of course, in the real world, exact balance is impossible. Even were it to be achieved, thermal fluctuations would disrupt it. Even at the absolute zero of temperature quantum fluctuations would provide an impetus. A similar analysis applies to a particle sliding without friction on a surface  y = - ½ x2.  The accelerating force is given by  f  =  -g . dy/dx  =  g x,  where g is the local strength of the gravitational field. Hence  d2x/dt =  g.x   and   x = x0 eg.t.
Continuous Discontinuity
A more interesting case is provided by the surface:
y  =  (2/g) . x2 . ( ln |x| )3    :   x e (0,1)
A particle sliding without friction on this surface from rest at  x = 0  follows the trajectory:
          x  =   exp( - 1/t2 )
This can be shown as follows:
          t2 =  - 1/ln(x)
    dx/dt  =   v  =  2.x / t3
½ m v2  =    2m.x2 / t6  =  - mg.y
             =    2m.x2 . ( ln x )3
This surface is interesting. It is smooth (everywhere continuous in all its derivatives) for real x, although not analytic for complex x. While its gradient along the real axis is zero at x=0, it nevertheless deviates from zero indefinitely quickly over an infinitesimal interval next to the origin. This means that the exponential growth constant for an infinitesimally small deviation from zero is infinitely large. This in turn means that a particle sliding without friction on such a slope would be seen to move a macroscopic distance in a finite time, even if  x|t=0 =  dx/dt|t=0  =  0.

The trajectory is more interesting still. As I have already implied, it has the property that  dnx/dtn|t=0  =  0, "n.  Hence, it smoothly continues the trajectory x(t<0)  =  0. This means that the particle could remain stationary at  x = 0  for an indeterminate period of time: after all, the force acting upon it in this position is exactly zero, then without warning, impetus or efficient cause, accelerate towards x = 1.

Here we have the familiar paradox of the toppling pin writ large. Left to itself, when will the particle begin to slide?  Its eventual, and it would seem inevitable, motion is without efficient cause.  Nothing in its past indicates or predisposes its future motion. The trajectory we have delineated, though causal, is fundamentally spontaneous.

Of course, this is no practical problem. For the reasons given above in connection with the balanced pin, the particle can never be at exact rest at the brow of this mathematically curious hillock. Nevertheless, it shows that even causal Newtonian mechanics can exhibit spontaneous Indeterminism in the simplest of circumstances. Any system which had similar characteristics is free to exhibit spontaneous behaviour. It is not difficult to believe that complex systems might have such "complementary functions" (a term I have borrowed from the theory of differential equations: a behaviour that requires no external stimulus) and hence manifest singularities of causality: suddenly, for no reason acting in a surprising manner. In fact, all of us are familiar with such behaviour. Engineers call such manifestations "gremlins" or "glitches", and dismiss them as "one of those things".


Incidentally, this kind of singular causality offers a clue to the problem as to how one thing can become another. For an Aristotelian, all change is "transubstantiation" and involves the discontinuous destruction of one substance, immediately followed by the creation of another. To a Physicist, accustomed to the ubiquity of differential equations and continuity, such language is offensive. Even the singularities of catastrophic transformations and phase transitions are smoothed out by inertial or statistical effects. The Platonic analysis of a continuous change in a thing's participation in a variety of forms: "a state's projection onto a Hilbert basis" is much more congenial.

Nevertheless, the problem of how any radical change can take place (e.g. from living to dead or vice versa) if all change is gradual still niggles. The best examples of such discontinuous changes arise in connection with the sacraments. In all except matrimony, an identifiable objective change is effected. In the case of the Eucharist, bread and wine become the Body and Blood of The Lord; in Baptism, Confirmation and Ordination an indelible character is imprinted on the soul of the recipient of the sacrament; in Penance and Unction forgiveness and healing are effected. Equally, if consecrated bread decays sufficiently as to cease to represent food, then its participation in the Divine Form ceases and it becomes rotting starch and gluten.

Such radical transitions can be reconciled with continuity if the degree of participation in a form has a temporal dependence similar to exp(-1/t2). As the priest starts the process of consecration, the participation of the Eucharistic Elements in the Divine Form starts to increase from zero, but with no discontinuity. If the consecration is successfully concluded, this participation converges to precisely unity: if the consecration is aborted, it falls back to zero. Similarly, the participation of a sick person in the form of "health" suddenly starts to improve, but with no discontinuity, when an effective medicine is ingested.

Classical and Quantum Systems

There are two types of objects. The first is identified by the matter that composes it. The second is delineated either by the neighbourhood that it occupies, or the form that it represents. In common experience specification by matter and form are interchangeable: certain matter occupies a certain neighbourhood with a certain pattern. In fact they are not at all equivalent. To make this clear, I shall now consider the two possibilities in some detail.

Materially closed or "What" systems

A system might be a definite set of particles, with a certain set of spatial co-ordinates and momenta. Quantum mechanically, these particles may be indistinguishable (in the way that two reef knots tied on the same piece of string are indistinguishable: they can only be told apart when they are spatially remote) but still their total number is known and their wavefunction has a definite set of (interchangeable) arguments. Such a system can not have any spatial bounds. It has a non-vanishing amplitude associated with its constituents being located indefinitely remote from the supposed location of the system. Any Quantum system that is conceived of or expressed in terms of a definite set of co-ordinates (a Wave Function: the analogy of the State Vector for a Classical system) is necessarily non local. In my experience, Quantum systems are always conceived of  in this way: as sets of particles. Sometimes the number of particles is allowed to vary, but even so the system consists of exactly however many particles that it does: wherever each of these may be.

Spatially closed or "Where" systems

Alternately, a system might be a definite neighbourhood of space. In which case it cannot be described in terms of a finite set of arguments. This can be just as true classically as Quantum Mechanically: though it is perhaps more unavoidably true once Quantum Mechanics is taken account of.
A classical example
Consider an iceberg floating in the Arctic. The concept of iceberg relates to those water molecules that compose the continuous solid mass of ice. Over time, this is not a fixed set. Many water molecules break away from the burg and become constituents of the liquid sea, while other molecules from the sea attach themselves to the crystalline ice. Although these processes should properly be described in quantum terms, they are not quantum processes. Over time the iceberg may shrink in volume, grow or remain static. It will certainly change its shape. At all times, its identity will be clear as "the mass of solid ice". Obviously, it may accidentally split or merge with another floe, but such events are exceptional and can always be dealt with by redefining the object we are considering to be the totality of ice floating in the Arctic. Nevertheless, it cannot be described in terms of a definite set of co-ordinates and momenta, because the molecules composing it are not a stable set. Even if, by chance, the iceberg always had exactly the same number of molecules, it would not be legitimate to specify a many-body wavefunction in terms of that number of position vectors. Manifestly, every-day objects are always conceived of and dealt with in this way.
A quantum example
For systems where dynamical interchange of material is insignificant, the distinction is still valid. When considering the small disk of copper nickel alloy in my hand, I would quite properly at any time exclude from the concept of "penny" all electrons and nuclei that were then observed to be remote from the apparent surface of the coin. Any simple wavefunction that I might write down in an attempt to describe the penny can at best be a description of a larger world in which a penny exists. It will necessarily  describe a world in which, as well as the penny, occasionally stray electrons and other material objects are observed remote from the penny. It is quite impossible to write down a simple wavefunction for a penny.
A fundamental difficulty
The difficulty encountered here in specifying a spatially definite system is profound. While it is not characteristic of Quantum Mechanics, it is nevertheless exacerbated by its intrinsic non-locality. One would like to be able to say something like "the iceberg is the sum total of all the matter within a certain periphery". Even ignoring the difficulty of how the the periphery is to be established (which problem I do not think to be important) this stipulation is problematic.

Classically, the material parts of the iceberg continually change: so the concept solid iceberg is in fact fluid! Any prediction for an observable of  the iceberg has to be evaluated over a conditional sum. In this sum, specific terms have to be either included or omitted, depending on whether the molecules to which they relate happen at the time in question to belong to that solid mass. So instead of a convenient sum such as:

Z = Sj=1...N  zj
where "z" is some property of the jth molecule, one has to evaluate sums of the form:
Z = Sj  zj d(rj : W(t))
where "rj" is the position of the jth molecule, W(t)  is the region of space occupied by the iceberg at time "t" and d(rj : W(t))
is unity when:  rj e W(t).
Classical and Quantum Mechanics compared
To progress to a Quantum Mechanical prescription it would first necessary to reformulate this unwieldy classical formula in terms of a state vector in 6N-Dimensional Phase Space. Sadly, this is not in order to make it less unwieldy, but rather because non-relativistic Quantum Mechanics is a prescription for associating a continuous complex amplitude (the Wave Function: whose time evolution is governed by Schrodinger's Equation) with each point in Configuration (not Phase) Space. This contrasts with Classical Mechanics, in which the Wave Function is a 6N-Dimensional Phase Space Dirac Delta-function. The form of this function does not change with time. Instead, it only undergoes local motion: a change in that point in its domain which corresponds to its sole non-zero (infinite) value. This local motion is governed by Newton's Laws of motion, or (equivalently) the Lagrangian principle of Least Action. It is well known that for macroscopic objects (with large masses and small de Broglie wavelengths), Schrodinger's Equation reduces to Newtonian mechanics, as it must.

Unfortunately, it is far from clear how to proceed. In non-relativistic Quantum Mechanics, once any point in 3N-Dimensional Configuration Space has been stipulated, a single complex amplitude, Y(r1,...rN;t), is immediately obtained: for all the matter being considered together en mass and as a single composite entity. It is neither possible to deconstruct this amplitude, nor to obtain anything additional to it. Y(r1,...rN;t), the Wavefunction, is the beginning and the end of Quantum Mechanics. In particular, it is not possible to distribute Y(r1,...rN;t)  between the matter lying within some boundary and other matter lying beyond it. Y(r1,...rN;t) relates to all the matter that is being considered as a single thing. Note that Y(r1,...rN;t) is a function of (r1,...rN) only, not (r1,...rN;p1,...pN). Its doman is 3N-Dimensional Configuration Space. The momenta, p, are not arguments of  Y, but can rather be obtained from it as  pj = dY/drj. This type of relationship is foreign to classical mechanics: as we have already noted, the classical "Wavefunction" is a delta-function in 6N-Dimensional Phase Space.

Really, it is illegitimate to consider Y(r1,...rN;t)  for any sub-set of matter, because as soon as matter other than that governed by any partial formulation of Schrodinger's Equation is conceived of, the question arises "how do you know that this other matter is not somehow subtly crucial to your calculations?" I believe this to be the basis of "The Collapse of the Wavepacket", which is always associated with the interaction of a definite "experimental system" with another distinct and indefinite system: typically called "the observer". If I am correct, this most confusing phenomenon will be revealed as of no deep epistemological significance, but only an artefact of our current formulation of Quantum Mechanics.

Of course, if it is possible to represent some Y(r1,...rN;t)  in terms of determinant(s) of single-particle wavefunctions (as is done for the ground state in Density Functional Theory, and more generally in the various Determinental Expansion techniques of Quantum Chemistry), then each of these can be spatially truncated, and the many-particle wavefunction for a region of space be written down. This would have to include an infinite number of (vanishingly?) small contributions from particles that are "normally remote" from the region, however!

It is now clear that the "wavefunction for a spatially defined object" is of the form:

Y(r1,r2,r3.....roo;t)    ;   rj e W
i.e. a single complex amplitude defined on a domain of an infinite number of three-dimensional spatial coordinates. If all of these coordinates lie outside W, Y takes the value zero.

The number density for such a wavefunction is:

        n(r1)    =      S | Y(r1,r2,r3.....roo;t)  |2  d3r2 ....d3rN

Note that it is neither possible to calculate n(r1) within W; nor to normalize Y, without making reference to its behaviour indefinitely remote from W.

Quantum Indeterminacy

I do not wish to dwell on this topic. Some have proposed that the probabilistic character of conventional Quantum Mechanics represents a certain openness to Free Will and so the Human Spirit and a relaxation of the rigidity of Newtonian Mechanics. I think that this is misconceived:
  1. Newtonian Mechanics is not deterministic, even though it is causal,
  2. Our notion of cause is linked in a confused way with that of temporal succession.
  3. Lagrangian Physics features final causes as much as efficient causes.
  4. Randomness (which appears to be all that is on offer from Quantum Mechanics) is not at all the same thing as Free Will.
I suspect that the idea of virtual or evanescent events, that are "selected for" on the basis of some extrinsic finality will have a part to play in the theory of Free Will, but as yet this is unclear.

Conscious Free-Will

What then are the supposed characteristics of conscious Free-Will? Can they be expressed in a non-contradictory manner? Can they be allowed for or even incorporated in Physical Theory?

It seems to me that the Free Will is characterized by some obscure admixture of spontaneity and deliberation [D.J. O'Conner "Free Will" (1971)]. Spontaneity; for else the will would not be free: but constrained by coercion, circumstance or knowledge. Deliberation; for else the will is not rational: but at the mercy of chaos, prejudice or ignorance. I have sketched out elsewhere how I think these seeming incompatibles can be reconciled.

Knowledge and Ignorance

Spontaneity and Deliberation

Mind and Matter

Adopting the hypothesis that the mind (but not the spirit, person, ego or consciousness) is nothing more than the internal states and processes of the brain, one can immediately identify the mind of any observer within the Minkowskian Cosmos as an aspect of  the organization of potentialities within a certain four-volume. Note that as soon as this is done the problem of a diffuse and indefinite consciousness evaporates. Although individual particles can be thought of as exploring every possibility, and having diffuse life-lines, this is not true of a composite object defined in terms of a macroscopic neighbourhood. The mind is an aspect of the mass of all the potentialities that do objectively exist within such a neighbourhood. The mind is not at all like a particle. It does not have a potentiality to be something or somewhere. The mind is what it is: a composite of potentialities. The mind is not subject to "observation". It can never be forced to adopt some specific Eigenstate. In as far as my mind has a wavefunction, it never collapses. Better, my mind is an aspect of the many-body wavefunction for the matter in a certain region of space. The definite state of my mind is an objective state of that uncollapsed wavefunction. There is no motive for analysing its state as being a superposition of other "more definite" mental states in which the state of every particle is well defined.

My definite objective mental state is itself mixed, in the sense that it is the combination of all the objective potentialities of all the matter that compose my brain. Though the particles that compose my brain have small potentialities for being in positions remote from its neighbourhood, they are only part of my brain in as far as they are not remote from it. Equally, a certain electron that generally speaking forms part of the Sea of Tranquility on Earth's Moon has a tiny potentiality for being part of my brain. To this limited extent, it is part of my brian: and its minute contribution to the physical state of my brain is a minute contribution to the state of my mind.

This account of affairs relies on the idea that the Wavefunction (and so the potentialities that it represents) is objective and so represents episteme, not just a projection of subjective knowledge: doxa. It should be noted that no account has been given of
consciousness here. Moreover, the phenomenon of the "Collapse of the Wavepacket" now seems even more difficult to account for. My observation of a particle now seems to be objectively describable entirely within the Minkowskian framework, and one would expect all subjective knowledge to be of uncollapsed wavefunctions: contrary to all experience!

Free Will Revisited

We have seen that Free Will is not incompatible with causality, but rather that even Newtonian mechanics is open to sponteneity. Moreover, we have seen that Mind may have a holographic relationship with Matter: being the sum total of all the material potentialities in a neighbourhood. We have also seen that final causality has a proper place in Physics. It remains only to put these three ideas together.

When the mind conceives of some complex purpose, how does it will this objective to occur? In some cases (for example "let's make a pot of tea") it is barely conceivable that the subconscious analyzes each practical step that must be carried out to acheive this objective and then organises muscular activity to effect it. In other cases (for example "let's solve this puzzle") it is more plausible that any number of spontaneous trials are run, with a check being made as to the success of each trial. Those trials which fail leave no trace in the mind. They are evanescent, rather like virtual particles in the vacuum. They may even be quantum trials in a quantum-computational neural net! When a successful attempt is identified, it is accepted as valid and is memorized.  The "cause" of the persistence of this trial is that it succeeded: its finality.

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