Physics and Biology: towards a unified model

Physics and Biology: Towards a Unified Model

Michael Conrad

Department of Computer Science Wayne State University Detroit, Michigan 48202

ABSTRACT

Biological systems have a vertical architecture that allows them to exploit microphysical dynamics for information processing and related macroscopic functions. Macroscopic sensory information in this vertical picture is transduced to increasingly rAcroscopic forms within biological cells and then back to macroscopic form. Process- ing of information can occur at any level of organization, but much of the most powerful processing occurs at the molecular and submolecular level. The open process of Darwinian evolution plays an important role, since this provides the mechanism for harnessing physical dynamics for coherent function. The vertical architecture is anakgous to a quantum measurement system, with transduction of input signals corresponding to state preparation and amplification of the microstate corresponding to measurement. The key point is that the microphysical dynamics is not classically picturable, whereas the macroscopic actions are definite and picturable. If this analogy is taken seriously, it becomes necessary to suppose that irreversible projection processes occur in organisms, despite the fact that the standard equations of motion are reversible. We comtruct a model that embeds such irreversible measurement interactions into interactions that mediate the conventional forces. The idea is that the forces between observable particles depend on the density of negative energy particles in a surrounding Dirac type vacuum. In systems that are not overly macroscopic or overly far from equilibrium this dependence is hidden, and as a consequence the force appears conservative. The model suggests that the irreversible aspect of the dynamics played an important role in the early universe, but became masked in ordinary laboratory situations as the structure of the vacuum and the distribution of mass and charge equilibrated. Organisms are particularly effective at unmasbg the underlying irreversibility due to their sensitive amplification mechanisms. U:%fying measurement and force type interactions makes it possible for physical models to fit more naturally to models of cognition.

1. INTRODUCTION

Evolution by variation and selection is an intrinsically opportunistic process. It is hard to see how such an open search mechanism could refrain from

n:‘.KZED MATHEMATZCS AND COMPUTATZON 32:75-102 (1989)

0 Elsevier Science I’ddishing Co., Inc., 1989

7s

655 Avenue of the Americas, New York, NY 10010 0096-3003/89/$03.50

76 MICHAEL CONRAD

reaching into the subtlest accessible levels of physical reality if these could in any way contribute to more effective function at the level of the organism. The information p mcessing and cognitive capabilities of organisms are per- haps the most dramatic example. If it is admitted that such fnnctional capabilities are molded by variation and selection, it must also be admitted that models of them, to be adequate, cannot in general stand alone, in clean isolation from the unpicturability of the microphysical world and from the exotic physical dynamics that predominate in this world.

But if the evolutionary principle calls for i&sing the models of biology with the ideas and models of physics, it becomes necessary to ask whether the latter are adequate for the task. The problem is in the translation between the picturable and the unpicturable. The modeling methodology of qnantnm mechanics starts from a mathematical description of a classical, picturable situation, uses this to construct a mathematical description of an unpicturable situation, then projects this unpicturable description into some possible picturable one. The projection process is equivalent to measurement. In ordinary physics it is possible to treat the translation between the different descriptions, in part&&r the measnrement process, as a side issue. The physically inclined student of biology does not have this luxury. The sine qua non property of the system of interest (an organism such as himself) is that it makes measurements [23], and choices about what and when to measure [20]. The process of translating an unpicturable to a pictnrable description is irreversible, however, while the equations that standardly govern the time development of both classical and quantum systems are reversible [22,31, 321.

Pt is of course conceivable that mezrurement and decision making are fictions that emerge from the complexity of biological systems, and that it is sufficient to abstract biological models from this complexity in a way that completely isolates them from microphysical processes. We should, however, at least consider the alternative possibility that our very strong subjective sense of control and cognition reflects features of the physical world that are masked in ordinary physical systems, but that are unmasked and brought to macroscopic significance by the mechanism of evolution. The purpose of this paper is to investigate this hypothesis, first by considering how the semitive nonlinear constraints of biological systems might amplify irreversible processes that are too negligible to detect in ordinary physical systems, then by constructing a toy model that embeds measurement processes in conventional force laws, and finally by considering the significance in principle of snch an embedding for modeling the fnnctional and cognit+e capabilities of biological systems. One side of this is that the life process is an expression of measurement-like mteractiOnS (Wave function collapse prOCeSSeS) that Were

ubiquitous early in the histoly of the universe, but that are negligible in an

equilibrium universe. The other side is that systems with a large number of nonstatic interactions undergo some spontaneous wave function collapse, but it is only in the presence of sensitive arrplification mechanisms such as occur in the biological cell that significant macrosc@c effects occur.

2. MODELS IN CQLLISIQN

Let us explicate more formally the dualism in conventional quantum mechanics between motion and measurement. First recall that the wave function may be expressed as a superposition of a complete set of eigenfunctions

where the u*(x) are eigenfunctions and the coefficients C,, may be expressed asen ’ -*nap If a system can be described by a wave function (l), it is a pure case. The past and future of a pure case is uniquely determined by an equation of motion, such as the time dependent S&&linger equation

where Hip is the Hamiltonian operator corresponding to the sum of kinetic and potential energies. In order to relate the wave function to ordinary experience it is necessary to use a projection operatkn of the form

ii = J/*(x, t)A,,Jl(x, t) dx, (3)

where Z is the expectation value of some variable, and A, is the operator corresponding to that variable. The projection operation (3) and the superposition expressed in Eq. (1) are consistent if Ci*C. is interpreted as the probability that the eigenvalue associated with the eigenfunction ai wih be produced by a measurement. The measurement may thus be thought of as producing a randomization of phases such that all the eigenfunctions other than I+(X) disappear. The loss of information about relative phases that accompanies the reduction of the wave function entails an increase in entropy. A measurement therefore transforms a pure case into a mixture, of

78 MICHAELCONRAD

the form

where the k wave functions each have probability wk. The past and hture of a ntiure is Determined uniqudy by the equation of motion, just as in the pure case. But measurement increases the entropy of a mixture just as it does for a pure case.

The conceptual problem is that the time evolution process (2) is unitary and reversible, whereas the projection process (3) is nonunitary and irreversible [22, 321. In principle an irreversible transformation can never be extracted from the equation of motion, since a mixture can never be produced by a superposition of pure cases. As a practical matter, therefore, two distinct types of influences are at work in quantum theory. One is force in the traditional sense, which enters through the potential function in the Hamilto- nian. The second is measurement-like interactions, characterized by phase randomization and macroscopic recording of information. The latter have no counterpart in classical physics, where the values of physical variables could always be thought of as definite :xad picturable. As a consequence measurement interactions could be viewed as just a special case of conventional force- based interactions.

The measurement problem and the various paradoxes to which it gives rise have been a topic of perennial discussion for over fifty years [ 191. Many approaches have been proposed. Some authors view the increase in entropy accompanying measurement as being a consequence of coupling a microphysical system to a macrosystem with a large number of degrees of freedom. The increase in entropy in measurement would then be assigned to the ergodic type loss of information that occurs in classical statistical mechanics. The difficulty remains, however, that we must then either regard the picturability of the classical description as an approximation or the unpicturability of the quantum description as an incomplete description. In the former case we would have to admit that the entropy increase is merely a matter of ignorance, and therefore that there is no fundamental reason why we should not be able to obtain precise values for a complete set of initial conditions. This would of course violate the uncertainty principle, and therefore the concept of unpicturability itself. In the latter case we would have to admit the existence of hidden variables. But according to Bell’s theorem this is possible only if action at a distance is admitted, which implies not only a more microscopic level of microphysical phenomena but a vast departure from conventional physics at that level [2].

Physics and Biology: Towards a Unijbd Model 79

We cannot here review the great variety of interpretitions that have been proposed to meet these difficulties. The best known is the Copenhagen interpretation. The chief idea is complementarity [3]. Quantum and classical descriptions are regarded as con$cii^r&tiy, just as different ways of setting up a measurement are regarded as complementary. This analysis may well be sound. But it does not mean that the biological process of choosing between complementary descriptions is beyond all analysis.

3. THE MESOSCOPIC LINK

NOW let us outline, in a more concrete fashion, how biological systems link function to physics through evolution. In particular, how could the unpicturability of the microphysical world be linked to information and cognitive functions?

Microphysical measurement machinery usually consists of a highly sensitive component (such as a photon detector) linked to a macroscopic recording mechanism through powerful electronic amplifiers. The components of the system are in general either microscopic or clearly macroscopic. This is true even for the amplifying machinery. These comprise statistical aggregates of particles that are clearly macroscopic. The individual events that are recorded are microscopic, and these trigger individual electronic processes that are microscopic. Some middle scale (or mesoscopic) processes occur in the course of amplification, but they are not the critical controlling processes in the system.

The situation is radically different in biology. Mesoscopic processes, on a macromolecular size scale, play a key linking role. Protein enzymes, for example, are large enough to recognize substrate molecules by shape, yet small enough to explore the substrate through Brownian motion. As single molecules proteins and nucleic acids have an inherent stability that eludes statistical aggregates of molecules. As sequences of monomers, alterable through the genetic mechanism, they have enormous combinatorial variety, and hence may be tailored through evolution for performing a vast range of specific functions. These functional capabilities arise from the coordinated action of tens of thousands of atomic nuclei and electrons. Exchanges of electrons, photons, and protons contribute, and various quasiparticles such as solitons, excitons, and polarons probably play a role as well. Interactions between the nuclear conformation and the electronic structure are nearly always important [6, 7, 301. Extended biomolecular structures, such as the lipid membranes that surround and perfuse the cell, elaborate cytoskeletal networks, and the network of mobile protons in bound water provide a larger theater for such interactions. The dynamics of these structures is controlled

80 MICHAEL CONRAD

by proteins associated with the cytoskeleton or proteins bound to the membrane. In general we can picture the cell as a field of particle and quasiparticle interactions that are specifically tailored through evolution by choosing particular DNA sequences and hence particular RNA and protein sequences.

What emerges is a vertical conception of biological information processing, schematically illustrated in Figure 1 [9,10, 131. The environment that affects the organism is macroscopic. Signals, such as photons, represent macroscopic objects and situations. These are transduced to the electrochemical activity of nerve cells, w&h is less macroscopic on a size and energy scale than the original environmental object. The neuron electric activity is transduced to second messenger signals inside the nerve cell, for example cyclic nucleotide signals [21]. The second messengers trigger intracellular target proteins, kinases in the case of cyclic nucleotides, and these in turn trigger effector proteins. The effector proteins may modify DNA activity, trigger activity in the cytoskeleton, or trigger channel proteins that &ectly control neuron electric activity, leading back up the scale to muscle action md macroscopic effects on the environment. This picture is not confined to nerve cells. We could equally well have used immune system cells as an example, or cellular differentiation. Nor is it limited to information processing functions. All of the activities of the organism depend on control and coordination at the subcelhrlar level. In each case the mesoscopic domain of evolutionarily modifiable macromolecules serves as the link between macroscopic function and the potentialities mherent in the microphysical world.

4. A FRAMEWORK FOR EMBEDDING MEASUREMENT IN FORCE

Let us now consider how the principles governing this microphysical realm might be modified so as to allow for measurement-like wave function collapse. The basic idea is that the force between two particles depends on the surrounding vacuum structure, which is really a plenum of unmanifest particles [ll, 141. In a near to static situation the motions of the interacting particles have a negligible effect on the vacuum structure, which therefore enters as a constant. In more dynamic situations, when either large masses or intricate motions of many particles are involved, this is not the case. Under these circumstances the potential function depends on the wave function of the vacuum, thereby introducing a complex potential. Alternatively, we can think of the potential as not being well defined unless the vacuum structure is made definite, which means a collapse of its wave function.

It is evident that this approach departs from the ordinary modeling methodology of quantum physics. In the ordinary methodology the investiga-

Physics and Biology: Towards a Unified Model

r

External World +--

sensory input

81

1 cellular network t

level I

macroscopic signal - - - - - - - --+macroscopic signal

(first messenger) -

I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..~....................... receptor on cell membrane effector on cell membrane

1 1 111m1------11-111 f

intracellular t

mesoscopic signal level ---------------a mesoscopic signal

(second messenger) (second messenger)

I quantum computing /

(macromolecule controlled)

FIG. 1. Vertical scheme of information processing. The signals become increasingly microscopic as the energies involved become increasingly small relative to kT. The dashed arrows represent alternative routes of information flow, such as macroscopic processing at the intracellular level. Microphysical information processing (unpicturable quantum computing) at the intracellular level provides the potentially most powerful contribution to macroscopic function.

82 MICHAEL CONRAD

tor begins with a description of particles and forces in terms of classical variables. This is then converted to a nonclassical description in terms of operators and wave functions. If the potential function depends on the wave function of the vacuum, the initial classical description must itself embed a projection process. As a consequence the initial description is inherently an approximation. The situation calls for a recursive, self-consistent approach. For the present purposes we will follow a more naive, toy model approach, with ‘he goal of eliciting the main features of the concept.

The basis of our model is a Dirac type vacuum, comprising vacuum fermions in negative energy states. Forces between positive energy particles (to be called absorbers) are mediated by exchange of virtual bosons, as in conventional field theories. However, bosons will be identified with virtual pair production processes that propagate in this vacuum (or plenum) in a chainlike manner, rather than with undecomposable field quanta with a continuous range of energies and momenta. Such pair-creating fluctuations can originate or terminate in the neighborhood of positive energy particles, but not in free space, due to the requirement for conservation of momentum. Propagation occurs as a consequence of the fact that macroscopic violations of conservation of energy and momentum should be forbidden-only fluctuations consistent with the uncertainty principle can occur. Force is dependent on the density of vacuum fermions, since the time and space intervals of the mediating transient pair production processes are determined by this density. We shall show that the density is influenced by positive energy mass and by charge, leading to the dependence of the potential on the wave function of the vacuum.

5. A TOY MODEL

Actually we shall start by showing that the inverse square law can be derived from a classical scattering (first order perturbation theory) formula under a linearity assumption that may be thought of as the analog of the classical workdistance relation for virtual processes. The objective is to lend some credence to this linearity assumption, since it leads to the vacuum particle interpretation described above. The applicability of what might appear to be a very approximate formula in an unusual context would not be apparent in the absence of the interpretation; at present we can regard it as a mathematical trick to be justified after the fact.

Consider two positive energy particles separated by a distance r and exchanging momentum through a virtual process. Denote the initial state of the system by i and the final state by 6 According to first order, time dependent perturbation theory, the probability that a system undergoes a

Physics and Biology: Tmtirds a Unified Model 83

transition from state i to state f in a time interval t = r in the presence of a perturbing potential constant over that time interval is given by

P(+f)= 41 (fiW I” sin2( ;(AE/A)r}

AE2 Y (5)

where AE = Ei - Ei. The probability of undergoing a transition increases with T, in accordance with intuition. However, as T increases, the probability of undergoing a transition to an equal energy state increases as TV, consistent with the fact that conservation of energy should hold rigorously over macroscopic times [ 17, 271. This r 2 increase in the probability of satisfying energy conservation is a key feature of the model to be developed. The intuitive idea is that the probability that a viritilal momentum carrier satisfies energy conservation by actually effectuating a transfer of momentum increases as the square of T, interpreted as a characteristic time determined by the density of mediating vacuum particles. T will be called the fluctuation time, and the pertubation V will be viewed as a self-perturbation allowed by a virtual fluctuation of energy compatible with aEat - h, where V = aE and at = T. The term self-perturbation is used because the transition is caused by a virtual (spontaneous) fluctuation of energy, not by an imposed potential.

Equation (5) may be written as

P@-+f)- 0.921 (f pqi) I2

AE2 ’ (6)

This is due to the fact that AE, as the deviation from energy conservation allowed by the uncertainty principle, may be written as

AE - A/r, (7)

where AE corresponds to the standard deviation of the energy, GE, which would be obtained by measurements made at some point in the time interval r. r is the longest fluctuation time compatible with the fluctuation magnitude BE and may be interpreted as the standard deviation of the times at which the measurement is made. Values of AE much larger than ft/~ may be ignored, since the fluctuations represent violations of a conservation law, not all the uncertainty accumulated in an experimental measurement. The unusual feature is that the only causative agency in our model is fluctuation, and as a consequence AE is the same as the potential V.

84 MICHAEL CONRAD

Now let us assume that the energy exchange which must occur over any given interval of time in order to achieve any given degree of coupling increases linearly with distance. The simplest linear dependence is given by

hE, = Kr, (8) where K is a proportionality constant with dimensions of force. In terms of the vacuum particle interpretation A E, represents the fluctuation energy of an individual exchange mukiplied by the number of chains required for a given degree of coupling, and it is this which increases with T (since it is the reciprocal of the chance that the chain will actually be absorbed). Of course the energy exchanged must actually decrease with distance, but this is consistent with the fact that the degree of coupling also decreases.

For values of A E -- AE,, Equation (6) may be written as

(9)

If the only influence the particles can exert on each other is through the exchange of momentum, the force must be proportional to this probability. Thus for an ensemble of exchanges we obtain the form of an inverse square law

where A is a proportionality constant and r/r is a unit vector from the particle interpreted as source to the particle interpreted as test.

The appearance of r2 in the denominator of Equation (10) shows that the inverse square law is simply an expression of the linearity relation (8), independent of the matrix elements in the perturbation formula (5). Taken in the context of scattering theory this perturbation formula is the same as the Born approximation [ 11. A scattering potential V could be any agency that causes a transition from state i to state f in momentum space. But it should be recognized that ordinarily V is taken as the potential, which the present model is intended to derive. Thus V might itself be the potential associated with an inverse square force, such as the Coulomb force or the shielded Coulomb force. Here instead it is a fluctuation quantity. In the next section we will show that it must be a fixed fluctuation quantity associated with each of a sequence of transient vacuum particle excitations that mediate the force.

6. IMPLICATIONS OF LINEARJTY

Since our use of the linearity assumption leads to the form of one important force law, let us tentatively accept it and investigate its conse-

Physics and Biology: Towards a Unified Model 85

q;lences. Li nearity may be written as

we, = Km, 01)

where T = cr. Setting AE,, = AE and using Equation (7) to eliminate AE,

The fluctuation time is thus a highly restricted quantity which depends only on I& c, and K. To express its parametric dependence on K we shall from now on denote the fluctuation time by rK.

Substituting Equation (12) into Equation (IO),

f&C4 I- F

K LT--

h2r2 r’ (13)

where A = Al(flVli) 12. But r can be written as r = Nr,, where rK = CQ (with r;= xi + y: + 2:). rK is the fluctuation length, and N is the number of fluctuation lengths separating two particles. Thus the inverse square law may be expressed as

F Lb; r

--- PM2 r

or as

(14)

(15)

We are now in a position to consider the justification for using first order perturbation theory. Equation (9) and all the force expressions derived from it hold if second order time dependent perturbation theory is taken as a starting point, except that I(fluli) 1 would have to be replaced by ](flVji) 4 c4 +f.i(flvlu)(ulvli) I”* H ere u is an intermediate state. However, the r: dependence suggests that such second order processes are not important, and Equation (11) implies they are incompatible with the linearity assumption. The reason is that the occurrence of an intermediate state would require the fluctuation time to be decomposed into constituent fluctuation times that

86 MICHAEL CONRAD

are smaller. Processes occurring on such smaller time scales occur, but do not make a significant contribution to the coupling between two observable particles. As a consequence the exchange of momentum between the particles is properly treated as consisting of N = r/rx first order processes best described by the first order matrix element, (flvli).

Note that as the fluctuation time increases, the maximum amount of momentum which can be exchanged in each fluctuation decreases, since the fluctuation energy AE must decrease. Nevertheless the resulting forces become larger according to Equation (14). This is possible because the fraction of virtual processes which actually lead to an exchange of momentum increases as the time required for a virtu 9 process to effectuate an exchange of momentum increases (or as the minimum space interval which a virtual particle must traverse increases). This increase in absorption probability is based on the fact that the probability of violating energy conservation decreases as 72 in the original perturbation formula. As a consequence the probability that a virtual particle is exchanged increases as the square of the fluctuation time [as in Equation (14)], whereas the maximum energy which it is likely to exhibit decreases as the inverse of the fluctuation time.

?. FLUCTUATION DENSITIES AND FIELD STRENGTIIS

It is useful to express F h terms of a density of events in a vacuum. Since the number of events per unit volume is given by pK = I/& Equation (9) can be written

Recalling our initial picture of a Dirac vacuum, it is natural to identify pK with the density of negative energy particles and to identify the events with transient pair production processes.

To see how charge and mass affect vacuum density, first equate F in Equation (16) with Fgav in the case of two interacting objects of mass m. This yields

where G is the gravitational constant and pgrav is the fluctuation density of


the gravitational field. Eliminating r,

Similarly equating F with Fcod for two interacting objects each carrying charge 4,

F 92 r Ln r

ccd =----a -

r2 T C2h2pg3T 2 T

where pe.,, is the fluctuation density for the electromagnetic field. Eliminating

r,

Thus as the mass increases, the gravitational fluctuation density decreases. As the charge increases, the electromagnetic fluctuation density decreases.

Since the gravitational force is weaker than the electromagnetic force, Equation (16) implies that pgrav must be larger than P,_.,, that is, the fluctuation times and fluctuation volumes must be smaller for the vacuum processes which mediate the gravitational force than they are for the processes which mediate the electromagnetic force. Since F,,/F,, - lOmU for the forces between two electrons, the density of events in the surrounding gravitational field as compared to the surrounding electromagnetic field is

aPProximatelY pgrav /pCod - 10” (assuming A scales as c2A2py3). This means that the number of events which contribute to the force between two electrons is on the order 10B times greater for the weak gravitational interaction than for the stronger electromagnetic one. For two protons the number of gravitational events is about 10” times greater, since F,,/t;cod is about 1O-37 in this case. This is due to the altering effect of mass on vacuum density featured in Equation (18). We will return to this pint in OUT

discussion of wave function collapse and measurement.

8. PROPAGATZON MECHANISM

The quantization of fluctuation time and fluctuation length implies that virtual particle exchanges in our model occur in a series of discrete steps, or

88 MICHAEL CONRAD

FIG. 2. Skipping model of the interaction between two electrons.

skips. The skipping process is illustrated in Figure 2. The momentum carrier is represented by a series of closed loops, each representing a vacuum excitation whose energy extends over a fluctuation time rK and whose momentum extends over a fluctuation length xx. The series is initiated by absorbing particles which serve as sources or sinks of momentum (or, more accurately, by vacuum fluctuations in the neighborhood of absorbing particles). The exchange process is composed of repeated alternations between a completely localized aspect, in which the energy and momentum fluctuations are quantized, and a completely delocalized aspect, associated with the fact that the fluctuations disappear at the nodal points between skips.

The occurrence of skipping is a consequence of the requirement that microscopic violations of conservation of momentum and conservation of energy shouId be forbidden. When a vacuum fluctuation results in a direct exchange of momentum with an absorbing particle, energy conservation is transiently violated, but not momentum conservation. The momentum change of the absorbing parMe is balanced by the momentum associated with the


fluctuation, independent of the choice of coordinate system. The fluctuation must disappear within a time interval rK; otherwise an impermissible violation of energy conservation will occur. But the disappearance of the energy fluctuation is accompanied by the disappearance of the momentum. Since momentum has been conserved up to this point, its disappearance entails a violation of momentum conservation. This violation is restricted to a space interval xk. Outside this interval the annihilated momentum must reappear, leading to a reappearance of the energy fluctuation. Since the energy fluctuation violates energy conservation, the process repeats until the momentum violation is annihilated by a second absorbing particle. Skipping is due to the fact that the quantization of energy and momentum fluctuations leads to a coupling of energy and momentum violations.

To formalize this argument we first consider the energy and momentum fluctuations which occur at each stage. For simplicity we consider only the x-component of the momentum. Let AE(rK, xK) denote the energy fluctuation during a fluctuation interval, and let A E( tn, x,) denote the energy fluctuation at the nodal points separating the fluctuation intervals. Similarly let Ap(rK, xK) denote the momentum fluctuation during a fluctuation interval, and let Ap&, x,) denote the momentum fluctuation at the nodal points between fluctuation intervals. According to Figure 2 the fluctuation energies and momenta are zero at the instants between intervals and are equal to the values allowed by the relations AE - A/rK and Ap, - A/x, during the intervals. Thus

Ai%+,) - h/rK (at - TV), (214

AE(t,,x,) - 0 (at, --) ao),

AP(L x,) - 0 (uxn + co). (214

According to Equations (21b) and (21d) no tiiie or space points can be assigned to the fluctuation energy or momentum when these quantities equal zero. This is required by the uncertainty principle, but this requirement is consistent, since the absence of a fluctuation cannot be assigned to a restricted set of spatial or temporal coordinates in any case. The delocaliza- tion of tn and x, is compatible with the reasonable requirement that the fluctuation energy should disappear at the same points in space and time as the fluctuation momentum, and is compatible with the correlation of these points through the relation x, = ct,,. The relations (21a-d) hold for all the

90 MICHAEL CONRAD

intervals and nodes, including those directly exchanging the momentum with absorbing particles.

We now use the relations (21a-d) to write down the energy and momentum conservation violations which occur at each stage of the skipping pocess. Denote the en,ergy Gonservation violation which occurs during a fluctuation interval by E(Q, xK) and the momentum conservation violation during the interval by fi(~~, xK). Similarly denote the energy conservation violation at the points by E(t,, x,,) and the momentum conservation violation at the points by fi(t,, x~). Recalling that the energy conservation violations occur during the intervals and that the momentum conservation violations occur at the boundaries,.

These relations hold for violations at all intervals and for all nodes, except for momentum violations at the nodes occurring between an energy violation and an absorbing particle. For ‘these initiating and terminating nodes

where t, and x, indicate the time and space coordinates of an initiating and terminating node.

When the energy and momentum violations are zero, they can occur anywhere in time or space. When they are nonzero, they are restricted to the fluctuation intervals. Thus the momentum violation occurs at nodal points (other than initial and terminating nodal points), but extends over the fluctuation interval iEquation (22d)]. The relations (22a-d) differ from (21a-d) because nonexisting fluctuations of momentum correspond to violations of momentum conservation except in the vicinity of absorbing particles.

The major feature of the relations (22a-d) is that it is impossible to satisfy both conservation of energy and conservation of momentum except at initiating or terminating nodes. During the interval the momentum conservation violation is zero, but the energy violation is ft/rK. During the instants

Physics and Biology: Towards a Unified .Mo&l 91

the energy conservation violation is zero, but the momentum conservation violation is tZ/xK. Thus once the skipping process is initiated by an absorbing particle there is no way of annihilating it except by another absorbing particle.

Note that skipping cannot be directly observable, since any observation would allow the skipping momentum to be transfe to the measuring apparatus, thereby terminating the process. Also, note that the assumption

rK = cTK is consistent. !%nce AE rK - Apx xK holds for all components of the momentum, we can write AE = Apt, corresponding to the required relation between energy and momentmn of a massless particle. The new feature is that AE and Ap can only assume the values h/gk and $ ,kk- rz;scctively. If c were not constant in all inertial frames, &e relation between fluctuation length and fhrctuation time would not be xK = ox+rK. But replacing c by u, in the linearity relation (II) would repllace c hy 0, in Equation (13). This occurrent* of a velocity in the inverse square law (other than a constant c) leads to thermodynamically unacceptable consequences. One way to think about this is to imagine transforming to a coordinate system which alters ?he linkage between energy and momentum fluctuations in such a way as to make one negligible and the other nonnegligjble. This would allow an observer to utilize uncertainty principle violations of conservation of energy and thereby to construct a perpetual motion machine. The arguinent implies that rK and rx are identical in all inertial coordinate systems. It also suggests that four-space descriptions in the large are compatible with the three-space descriptions in the small which which form the starting point of the analysis.

9. FEATURES OF THE MODEL

Our discretized picture of virtual particle exchanges differs radically from the conventional field theory picture [Ml. Could a model of this sort meet all the stringent requirements of a physical theory? This question raises many technical issues which cannot be addressed here. I would, however, like to point out a number of features of the model which do in fact suggest that the general picture is in principle tenable.

9.1. Ir&qw=etation of Bosom Virtual bosons in the skipping model are identified with short lived pair

production processes allowed by the uncertainty principle. For example, virtual photons are identified with short lived excitations of vacuum electrons, leading to a transient particleantiparticle pair. It can be shown that the rest mass of the excited vacuum particle is converted to pure kinetic energy, and

92 MICHAEL CONRAD

therefore that this interpretation satisfies the zero rest mass property for bosons not confined to the neighborhood of mass. Interpretations not requir- ing a sea of negative energy electrons are possible, but less convenient.

Recall from Equations (18) and (20) that mass and charge alter the surrounding vacuum density. This means that the rK and 71( are not on the average uniform throu&out a skipping process. To prevent energy and momentum conservation law violations from occurring it is necessary for the transient vacuum particle excitations to be accompanied by expansions and contractions of the vacuum density. As a consequence the skipping mode of virtual particle propagation is coordinated with contractionexpansion waves of the vacuum density that correspond to the more conventional description. Real bosons may be interpreted as skipping processes in which the accompanying vacuum density wave carries macroscopically observable energy and momentum arising from the acceleration of an absorbing particle.

9.2. Nm-inoerse-SquQre Laws Of the fundamental forces only the electromagnetic and gravitational

appear to obey a simple inverse square law in the nonrelativistic limit. Different force laws or force laws with different transformation properties can arise in at least three ways. First, it is possible to derive the more general force laws of electromagnetism from the Coulomb law by applying relativistic transformations and some additional assumptions, such as charge conservation [18, 26, 291. Second, force laws with a non-inverse-square character would arise from inhomogeneities in the vacuum density. For example, the Yukawa potential may be interpreted as an inverse square law whose inverse square character is hidden by a vacuum inhomogeneity in which the fluctuation length (or fluctuation time) falls off slowly up to a distance R, then falls off rapidly. The opposite type of vacuum inhomogeneity is a bubble of high fluctuation density. The forces between two charged particles inside the bubble would be small, but nevertheless they would be of an inverse square character between a particle inside the bubble and an outside parWe. A third mechanism is stretching and compression of fluctuation lengths. If the fluctuation length increases as two attractively interacting p&ides arc separated, the attraction will increase with distance; if the fluctuation length becomes very small as the particles become closer together, they will behave more like free particles, a property suggestive of quark confinement.

9.3. Nonexistme of Self-Energies The vacuum density model diffdrs from conventional models in that it

excludes infinite selfenergies and excludes selfenergies altogether for a single point particle. The classical self-energy for a sphere of uniform surface distribution is E, = 9 ‘/r [5]. The self-energy in terms of fluctuation quanti-

Physics and Biology: Towards a Unified Mock! 93

ties may be obtained directly from this and from the fluctuation density expression for charge [Equation (ZO)],

E self -

‘emrA

C2A2N ’ em (W

where for clarity N is subscripted by the field with which it is associated. In the classical expression there is no intrinsic feature of the formula which prevents T going to zero, thereby leading to an infinite selfenergy for a point particle. This is not possible in Equation (23), since no interaction can occur at all unless Nem >, 1. The self-energy can go to infinity only if rem + 00, corresponding to the unphysical situation in which the interaction involves an infinite volume of space. If rem is the radius of an electron, the selfenergy would go to zero as rem goes to zero. The term self-interaction is strictly speaking inappropriate to the skipping model, due to the fact that two particles are actually always ~22255~ f for an interaction. This is because the skips always propagate away from their source. The skip may originate as an excitation in the immediate surrounding vacuum (in which case the absorbing particle emits) or in the neighborhood of another absorbing particle.

The absence of an infinite selfenergy is ultimately due to the fact that the quantization of fluctuation energy and momentrun provides a natural cutoff for the momentum carried by virtual parti&s. This cutoff is a highly desirable feature, since the divergences in qumtum electrodynamics derive from the absence of any natural cutoff.

9.4. Gravity and Mass Recall [from Equations (18) and (ZO)] that mass depresses the gravitational

vacuum density and charge depresses the electromagnetic vacuum density. This suggests a two fluid model of the vacuum in which the gravitational force is due ito the altering effect of mass on the density of all vacuum particles and the electromagnetic force is due to the effect of charge on the density of only those vacuum particles that are charged. The attractive feature of the two fluid model is that it provides a way of reconciling the geometrical concept of force of general relativity with the virtual particle exchange concept of quantum field theories. All interactions in the two fluid model are mediated by quantized virtual particle exchanges, either photons or gravitons. However, the density of gravitational vacuum particles is so much larger than the density of electromagnetic vacuum particles (cf. Section 7) that the depressive effect of charge on the latter has negligible effect on the former. Th e reverse is not the case, however, and as a consequence pgcaV will have small effects on electromagnetic phenomena.

94 MICHAIEL CONRAD

We can finally consider, here again only qualitatively, how mass arises in the model. The negative mass of the vacuum particles is a formal property inherited from gravitational interactions to which they are subject when acting as transient absorbers trapped in a fluctuation volume. The origin of gravitational mass can be thought of as a bootstrapping process in which vacuum particles serving as a substrate for shipping exert forces on each other when undergoing trapped skipping. Mass and charge per se are associated with depressions in the gravitational and electromagnetic vacuum densities, respectively. The gravitational interaction is due not to the direct exchange of virtual particles between masses, but to asymmetric effects of propagative skipping processes initiated by trapped shipping processes. These asymmetri- cal effects detive from the structure of the gravitational density, which in turn is controlled by the distribution of mass. We can finally note that the mass assigned to a vacuum particle on the basis of the fluctuation energy necessary for pair creation increases with vacuum density, while the mass as defined by the gravitational interaction between two particles decreases with increasing vacuum density. This leads to delicate selfconsistency relations between vacuum particle mass and vacuum particle density and to the possibility of short lived deviations from an equilibrium situation in the presence of sharp variations in the vacuum density.

10. FORCE AND MEASUREMENT

Now let us return to the issue of measurement and irreversibility. Recall our initial idea that if force is controlled by the vacuum density structure, its classical description should depend on the wave function of the vacuum. To make this idea formal, let us denote the wave function of vacuum particle j at time t by a/+(~~, t), where 9i symbolizes the coordinates of the particle at time t. We suppose that the wave function of the vacuum as a whole, to be denoted by I&( 91,. . . , 9,), can be constructed as a Slater determinant of these vacuum particle wave functions. If we used the interpretative postulate of quantum mechanics [Equation (S)] to write PK = I#,( 9,). . . ,9,, t ) I’, our inverse square law (16) would become

Al CflW I2 r F-

C2A2 J~“(9~V, { 9n, t) )2)4/3*2 ;’

where we expand A^ to emphasize that i and f denote the initial and final states of the vacuum and that V also depends on the vacuum density.


Obviously we are considering a highly isolated situation in which only two absorbing particles are involved. In an equilibrium system the vacuum structure remains essentially static. Thus j&l2 enters as a constant, estab- lished in the initial determination of the force law. As a consequence the wave function collapse (or phase randomization) that must accompany the initial specification of 1 &I 2 will not be pertinent to the description of motion. However, this will not be so if many particles are involved and their relative distribution changes. A precise determination of the force would then require the investigator to perform new measurements.

Our description is still incomplete, since we have not considered how the wave function collapse can enter into the dynamics of the system per se, without bringing in the operations that would have to be performed by an external observer. The projection process as expressed in the interpretative postulate (3) is sudden, without any time dependence. In reality the process of becoming definite should itself have a time development, reflecting the time dependence of the underlying randomization of quantum mechanical phase. The problem is that this process involves the recursive interaction between the macroscopic and microscopic structure of the universe, and between the classical and quantum descriptions of these structures. This calls for a recursive or selfconsistency formalism that extends well beyond the simple, first order model presented here. We can, however, list the key steps:

1. Alterations in the distribution of mass and charge lead to alterations in the density structure of the vacuum, and conversely, until selfconsistency is achieved. The density structure must be a member of an ensemble of microscopic complexions compatible with the macroscopic distribution of mass and charge. Jumps between one complexion and another occur as a result of vacuum fluctuations which are irreducibly spontaneous. Entropy increases in this process because the stable situation is as near as possible to an equilibrium situation.

2. The macroscopic distribution of mass and charge corresponds to the amplifying and recording machinery in a man-made measuring instrument. The vacuum density structure corresponds to the unpicturable microphysical system being measured. The interaction between the macrostructure and the microstructure makes some features of the latter more definite. According to the uncertainty principle this randomization of phase must appear in the increasing indefiniteness of the conjugate features, and as a whole entropy must increase. The phase randomization and entropy increase have their origin in movement to an equilibrium situation via the spontaneous vacuum fluctuations.

3. The above processes are negligible in microphysical systems standing alone, since the mass and charge involved are too small to affect the vacuum

96 MICHAEL CONRAD

density. As a consequence such systems will appear to obey reversible dynamics to a high order of accuracy. As systems become larger and more complex, alterations in the distribution of mass and charge become more significant. Some self-measurement and concomitant irreversibility associated with phase randomization will therefore become increasingly significant. However, this entropy increase will ly be a component of the conventional entropy increase that can be attributed to the asymmetry of the initial conditions of the positive energy particles, and therefore only one source of the increasing ignorance on the part of the observer.

A conventional measurement process, as done in the physics laboratory, is a special case of the above recursive interaction between microscopic and macroscopic in which the iecording and amplifying machinery is highly sensitive and specialized, especially the recording and amplifying mechanisms embodied in the human observer. We should also note that in a more formal, yet phenomenological description of the self-measurement process we could specify pK in terms of some functional of $J, that preserves its complex character rather in terms of a real number obtained from the square law 1 #,I 2. This yields a complex potential function, hence an absorptive process that is irreversible and expressive of the underlying phase randomization.

11. BACK TO LIFE

Let us now reconside, the vertical concept of biological information processing in the light of our measurement embedding model of force. The main idea is that evolution recruits the dynamics of the microphysical world for macroscopic function. The situation is highly analogous to a measurement process. Macroscopic signals are transduced to microscopic form, corresponding to state preparation. The microphysical dynamics proceeds, leading to a microstate appropriate to the external situation as expressed in the input signals. This microstate is then amplified to macroscopic form, leading to the appropriate actions of the cell or organism.

The advantage is that the unpicturability of the microphysical dynamics increases problem solving power. For example, suppose that the problem is to find the global energy minimum on a potential surface with multiple minima. A classical particle, such as a marble, has no chance of doing so if it is initially in the wrong mkimum, and only some chance if it is subject to thermal agitation. The “high ~~arallelism” inherent in the wave function allows for exploration of all the minima through tunneling [14, 151. If the global minimum is capable of absorbing the electron and triggering an appropriate


macroscopic action, the problem of responding correctly to the input sign&s reduces to the problem of using these signals to influence the potential surface. But this is also more easily achieved if the reorganization of the potential surface is controlled by microphysical dynamics. The reorganization, for example, might arise through the self-assembly of proteins released by the input signals [I2, 131. The ultimate organizing principle, tailoring all levels of the dynamics, is the principle of evolution.

The key feature is the translation between the unpicturable microdynam- its and the classical macroscopic actions. The behavior of the organism is macroscopic and definite. So our inclination might at first be to suppose that the controlling dynamics must be definite. But the tunneling example described above shows that unpicturable controlling dynamics yields more effective macroscopic function. The most intelligent, functionally competent systems should embed the most microscopic available elements in a macroscopic architecture, with a series of intermediately scaled components serving to link the pure microscopic and the pure macroscopic, and in addition acting as the evolutionarily controllable elements.

This is where our measurement embedding model of force becomes important. In the Copenhagen interpretation of quantum mechanics we could think of the cell as being governed by unpicturable microdynamics; but there is no room for translating this into definite macroscopic actions apart from the intervention of an external observer. Puttig irreversibility into the fundamental physics allows us to put the observation process into the dynamics of the organism.

Our model suggests a peculiar analogy between the life process and the early universe. We can imagine “&at the early universe exhibited a vast deviation from a self-consistent equilibrium between its macroscopic and microscopic structure. As a consequence it must have gone through a succession of organizations, somewhat like ecological’ succession, in which wave function collapse on a vast scale dominated the time development. An enormous entropy production would have accompanied this evolution to selfconsistency. When the universe came into selfeonsistent equilibrium the wave function collapse and its associated entropy production would have become a much less important factor. Reversible equations of motion, such ns SchrGdinger’s or Dirac’s equations, became highly accurate descriptions of nature at this point, at least at a local level. The origin of biological systems is an exception, though, because of their highly heterogeneous microdynamics and the sensitive amplification mechanisms that allow these to control macroscopic behavior. The origin of such systems unmasked the fundamen- tally irreversible projection dynamics of measurement, thus ushering in an era which in local domains mimicked the global dynamics of the early universe.

98 MICHAEL CONRAD

12. SELF-MEASUREMENT AND COGNITION

A number of authors have argued that the measurement process introduces a mental element into quantum mechanics that was completely absent in classical physics [8,20,23,31,32]. The idea is simply that the observer as a distinct entity must decide what to measure and must actually perform the act of cognition that reduces the wave function. Measurement is thus naturally associated with thought processes such as choice, decision making, intention, and the sense of control. The collapse process associated with measurement, for example, involves a reduction in possibilities highly remi- niscent of the decision process [16,28]. Can the association of such cobtitive concepts with physics processes locate the observer in the physical system? Unfortunately the conventional formulation of quantum mechanics does not provide any help on this point, due to the fact that the act of observation is treated as a distinct process separate from the equations of motion. If the st~&~d equations of quantum mechanics were considered to completely encompass t.hc *material world, the implication would be a dualistic model of ~ai~zl and matter.

Let us consider what the self-measurement model suggests when taken together ~tith the vertical model of biological information processing. The feature necessary for measurement type processes to be unmasked is a hierar&.ical organization in which the actions of macroscopic structures at the upper level control and are controlled by microcomponents that require a quantum description. It is these components whose time development must be described by a superposition of possibilities, and which must undergo collapse in order to trigger definite actions at the macro level. As a first approximation suppose that our subjective experience of the thought process parallels the potentiality laden microphysical dynamics at the subcell&r level of the organism, and that the process of converting these potentialities into actual behavior corresponds to the collapse process. This is a first approximation, since for intelligent behavior it requires an elaborate architecture, such as the architecture of the brain, to set up useful microphysical dynamics within cells and to set up the necessary communication links among cells to orchestrate them in a coherent way. This is where the principle of evolution by variation and natural selection plays its sine qua non role. The collapse process prominent in the early history of the universe would not have occurred in the context of an architecture yielding intelligent behavior. Similarly,the less significant collapse phenomena that might exist in the large scale universe today do not occur in the context of a tailored architecture. Systems in which collapse phenomena are completely masked are in principle excluded from having any subjecti ,ve or conscious experience. A classical computer is an example. Its behavior can be completely described in terms of

Physics and Biology: Towads a Unified Mot&d 99

components whose actions can be specified in a completely classical manner (even if in practice they use quantum electronics principles). The implication is that a classical ma&me, no matter how intricate its behavior, would have no subjective mental aspect comparable to those we privately experience. A second implication is that its public behavior could not be as rich as that of an organism, due to the fact that the parallehsm inherent in the wave function would be unavailable to it as a computational resource.

13. CONCLUSIONS

Two types of interactions are explicit in today’s standard formulations of quantum theory. The first is connected with force laws (such as the electromagnetic force) and reversible equations of motion. The second is associated with measurement and is irreversible. The model presented here combines 5ese two types of influences. The idea is that the forces between observable particles depend on the surrounding vacuum (or plenum) structure, and the proper formulation of a force law should refer to this structure in sufficiently macroscopic and dynamic situations. This introduces measurement type interactions into the force law. Our model is only a first pass description of this concept, since the interaction between microscopic and macroscopic is highly recursive. The evolution of the universe is, in this respect, a bit like the evolution of a biological ecosystem. Change continues until selfconsistency between the macrostructure (environment) and microstructure (biota) is achieved., at which point the system becomes trapped in a basin of attraction. The vacuum structure is stable under these conditions, and as a consequence the irreversible aspect of the dynamics is suppressed. As systems become more macroscopic some irreversibility is unmasked. Measuring instruments, with their sensitive amplifying mechanisms, are a special case in which the function of specifying features external to the system depends on this unmasking of underlying irreversibility.

Biological organisms are another case in which function is largely controlled by measurement type interactions, but unlike laboratory instruments the measurements are most significantly directed to features of the self. Signals from the external world set the microphysical state of biological cells by acting through a downward chain of chemical and molecular events. These unpicturable microdynamics proceed, in effect performing a powerful quantum computation. When the computation is complete, the microstate is amplified to macroscopic actions through an upward chain of molecular and chemical events. In a sense the organism is a bit like our picture of the early universe in that it is dominated by self-measurement. The difference is that constraints tailored through the mechanisms of variation and selection allow

100 MICHAEL CONRAD

the organism to act on influences from an external ‘environment in a highly adaptive manner. The privacy of the organism-the inability of organisms to directly experience each other’s internal state-is due to the high integration and sensitivity of these constraints.

LRt us mention some of the methodological implications of this scheme. The first is that the viewpoint is more functional than the Newtonian style of physics (cf. [4, 251). This is because the whole architecture influences the character of the interactions among the components. The second is that this architecture has a highly vertical (macroscopic to mesoscopic to microscopic) organization, and as a consequence a modeling methodology capable of dealing with biological function should be able to accommodate multiple levels of organization [8]. The third point is that the framework is essentially evolutionary. The selfconsistency between the macroscopic and microscopic levels is crucial for the physical domain, and leads to an evolutionary development. In the biological domain the special mechanism of variation and selection is the key to understanding how microphysical dynamics are exploited for macroscopic function. The fourth point is that the traditional approach of setting up a classical description, moving to a quantum description, and then going back to a classical description is just a first pass. The requirement for selfconsistency between macroscopic and microscopic calls for a much more recursive approach. The simple model that we have constructed to illustrate our framework should be regarded as a first pass.

Finally, I would like to consider the philosophical implications. The vacuum fluctuations that form the central element in our model are random events, for all practical purposes like irreducible, spontaneous mutations. The model may thus be interpreted as indeterministic; but this does not mean that it entails indeterminism. Conceivably a hidden variable theory, with chaotic subquantum fields and action at a distance, could reproduce the same effects [2]. Probably it is undecidable (in the sense of Poinca&‘s conventional- ism) whether we adopt a simpler indeterministic model or a more complex deterministic one. The important point is that deterministic theories have the definite philosophical implications that central biological and psychological concepts are at bottom fictitious or illusory. The “variation” in “variation and selection,” for example, would in principle have a deeper deterministic description, implying that the course of evolution would also have a deeper description. Similarly, our direct experience of choice, control, and decision making would also have to be ilhrsory. Our unified model, with its inbuilt irreversibility, allows these quintessentially biological concepts to be honored with a fundamental status, but it does not require them to be so honored. As a consequence the model protects itself from definitively resolving inherent antinomies of philosophy, an advantage not enjoyed by definitively deterministic models.

Physics and Biology: Towards a Unifaed Model 101

X&s research was supported in part by National Science Fmm&ion Grant lR1 IV-Q2600.

REFERENCES

1 2

3

4

5

6

7

8 9

10

11

12 13

14

15 16

17

18

19

20

D. Bohm, Quantum Thewy, PrenticeHall, Englewood Cliffs, NJ_ 1951. D. Bob, whohs and the Zmplicatc Order, Boutledge and Kegan Paul, London., 1981. N. Bohr, Discussion with Einstein on epistemological problems in atomic physics, reprinted in Aton& Physics and Human Knowledge, Science Editions, New York, 1961. J. Casti, Newton, Aristotle and the modeling of living systems, in Proceedings of the 1987 Abisko Wmkshq~ on System M&ling, to appear. E. U. Condon, Basic ekxtromagnetic phenomena, in Handbook ofPhysics (E. U. Condon and H. Odishaw, Eds.) McGraw-Hill, New York, 1967. M. Conrad, Electron pairing as a source of cyclic instabilities in enzyme catalysis, Phys. Lett. 68A:127-130 (1978). M. Conrad, Unstable electron pairing and the energy loan model of enzyme catalysis, J, Theoret. Biol. 79:137-X!% (1979). M. Conrad, Adaptability, Plenum, New York, 1983. M. Conrad, Microscopic-macroscopic interface in biological information processing, Biosystems 16345-363 (1984). M. Conrad, On design pticiples for a molecular computer, Camm. ACM 28:464-480 (1985). M. Conrad, Beversibility in the light of evolution, Mot&x en Deuebppement 14:111-121 (1986). M. Conrad, The lure of mole&u computing, IEEE Spectrum 23:55-60 (1986). M. Conrad, Molecular computer design: A synthetic approach to brain theory, in Real Brains, Artificial Minds (J. Casti and A. Karlqvist, Eds.) North-Holland, New York, 1987. M. Conrad, Quantum mechanics and molecular computing: ?&tual implications, in Znt. J. Quantum Chem: Quantum BiOl. Synap. 15287-361 (1988). M. Conrad, The brain-machine disanalogy, Biosystems 22:197-213 (1989). M. Conrad, D. Home, and B. Josephson, Beyond quantum theory: A realist psychobiological interpretation of physical reality, in Microphysical Reality and Quantum F~lism, Vol. 1 (A. van der Merwe, F. Selleri, and G. Tarazzi, Eds.), Kluwer Academic Publishers, Amsterdam, 1988. P. A. M. Dirac, 7%~ principles of @..mniam Mechanics, 4th NL, Oxford UP., 1958. D. Frisch and L. Wilets, Development of the Maxwell-Lorentz equations from special relativity and Gauss’s law, A-. J. Phys. 24:574-579 (1969). M. Jammer, The Cweptual Development of Quantum Mechanics, McGraw-Hi& New York, 1966. B. D. Josephson, The art&id intelligence/psychology approach to the study of the brain and nervous system, in Physics and Mathematics of the Ikuous

102 MICHAEL CONRAD

System (M. Conrad, W. Guttinger, and M. Dal Cin, Eds.), Springer-Verlag, Heidelberg, 1974.

21 E. A. Liberman, S. V. Minina, N. E. Shklovsky-Kordy, and M. Conrad, Neuron generator potentials evoked by intraceIIuIar injection of cychc nucleotides and mechanicaI distension, Bruin Res., 33833-44.

22 F. London and E. Bauer, La l%eorie de 1 ‘Obseruation en Mecanique Quantique, Hermann, Paris, 1939.

23 H. H. Pattee, Physica.I problems of decision-making constraints, in Physical Principles of Neuronal and Organismic Behauior (M. Conrad and M. Magar, Eds.) Gordon and Breach, New York, 1973.

24 P. Roman, Introduction to Quantum Field Theory, Wiley, New York, 1969. 25 R. Rosen, On the scope of syntactics in mathematics and science: The machine

metaphor, in Real Bruins, Atifici~l Minds (J. Casti and A. Karlqvist, Eds.), North-Holland, New York, 1987.

26 W. G. V. Rosser, The special Theory of Relutiuity, Butterworth, London, 1964. 27 L. I. Schiff, @u&urn Mechanics, 2nd ed., McGraw-Hill, New York, 1955. 28 H. Stapp, Consciousness and values in the quantum universe, Found&ions of

Physics 1535-47 (1985). 29 J. Tessman, Maxwell-out of Newton, Coulomb, and Einstein, Amer. J. Phys.

34:1948-1055 (1966). 30 M. VoIkenstein, Simple physical presentation of enzymatic catalysis. J. Theoret.

Biol. 89:45-51 (1981). 31 J. von Neumann, Mathemuticul Founduti~ of Quantum IMechunics, ?rinceton

U.P., 1955. 32 E. P. Wigner, Smm&ries and Refictions, Indiana U.P.. Bloomington, 1967.

Physics and Biology: towards a unified model

Documents

Transcript of Physics and Biology: towards a unified model