ON THE COHERENCE OF ULTRAWEAK PHOTON EMISSION FROM LIVING TISSUES

Fritz-Albert Popp

Technology Center

Opelstrasse 10

6750 Kaiserslautern 25

C.W. Kilmister (ed.), Disequilibrium and Self-Organisation, 207-230. 1986 Reidel

ABSTRACT. At present it is generally accepted that all living systems exhibit a very weak photonemission of a few up to some hundred photons per second and square centimeter of surface area, ranging at least from ultraviolet to infrared. At a first view it appears likely that this "low-level luminescence" corresponds to a chaotic, spontaneous chemiluminescence. However, its temperature dependence and the manifold correlations to physiological and biological functions, as, for instance, radical reactivity, oxygen consumption, stress, cell proliferation and differentiation, biological rhythms, even DNA conformations, point to a regulatory activity of these "biological"' photons ("biophotons").

Moreover, a careful analysis of the decay behaviour of photonemission after exposure of the living tissue to external light-illumination indicates that "low level luminescence" originates from an electromagnetic field with a surprisingly high degree of coherence, as compared to that of technical fields (laser). The basis of this conclusion, namely the results of photo count statistics as well as the relaxation dynamics within the framework of unstable quantum systems under ergodic conditions, is extensively discussed in this paper.

INTRODUCTION

Today it is well accepted that photons of different wavelengths trigger certain biological functions, as, for instance, photorepair (1), phototaxis (2), photoperiodic clocks (3), cell divisions (4), and multiphoton events (5). For a long time, however, it was on the contrary less accepted that all living tissues themselves emit a quasi-continuous photo radiation. This phenomenon of "ultraweak" photon emission from living cells and organisms (see e.g.Fig.1) which is different from bioluminescence, exhibits an intensity of a few up to some hundred photons per second and per square centimeter of surface area. Its spectral distribution ranges at least from infrared (at about 900 nm) to ultraviolet (up to about 200 nm). Recently, the most modern aspects of "biological luminescence" were extensively discussed (6).

Fig 1a "low-level luminescence" of cucumber seedlings in counts per second (corresponding to approximately photons per second) during time to from 100 to 200 s.

Fig 1b reference of measurement of Fig. 1a, namely the photon emission from the quartz cuvette without cucumber seedlings. The intensity is again presented in counts per second from 100 s (after putting in the sample) to 200 s.

Mainly the weakness of this radiation, however, corresponding to the intensity of a candle at a distance of about 10 kilometers, provoked the opinion that low-level luminescence can only be understood in terms of metabolic "imperfections" (7), originating from spontaneous chemiluminescence (8), and being associated with the permanent trial of the living state to return after metabolic excitation to thermal equilibrium. Actually, some correlations between the intensity of low-level luminescence and quasi-spontaneous biochemical reactivity, mainly oxidative radical reactions, have been found (9,10).

On the other hand, there are distinct correlations of photon intensity and conformational states of DNA (11), or DNase activity during meiosis (12). At present there exists no further doubt about the physiological character of biophoton emission, since it exhibits just the same temperature dependence as it is characteristic for most of the physiological functions (13,14). Despite its obvious weakness, the well known hysteresis effects of low-level luminescence (15,16) indicate the non-linear character of this emission. The comparison of the "ultraweak" intensity of these biological photons with the expected intensity of black-body radiation at physiological temperatures within the same wavelength range (from infrared to ultraviolet) gives evidence for optical couplings within living tissues "far away" from thermal equilibrium. Actually, the spectral intensities of biophotons amount to magnitudes that are up to about times higher than those of thermal equilibrium at physiological temperatures (14). There is no question that, in view of the Arrhenius factor, this fact alone provides the capacity of a much higher chemical reaction rate than is possible in thermal equilibrium systems. Thus, biophotons have a much more powerful potency of regulating biochemical reactivity, already by considering the availability of the necessary activation energy, than enzymes alone which enhance biochemical reactivity by means of lowering the activation energy due to complex binding to the substrate (17).

A further characteristic property of this biological radiation turns out to provide a biological basis of distinctive importance, namely the rule

constant

what is probably generally valid for living systems (18,19). is the probability that photons of energy occupy the (vacuum) phase space.

This rule does not only mean that biological "matter" is far away from thermal equilibrium which, as is well known, in contrast is governed by the law

where kT represents the mean thermal energy (see Fig.2).

Fig.2:

Compared to thermal equilibrium (f) the occupation of the electronic levels of living biological systems is several orders of magnitude higher. At the same time, the "biological" occupation (examples f1,f2 and f3 according to measurements on cucumber seedlings) does not considerably depend on the energy.

= constant indicates an "ideal" open system, not subjected to any constraint. It is supplied with always sufficient energy (20), hence representing an absolute maximum of entropy. This does not mean that the entropy of biological systems has to be extremely high. On the contrary, it may become relatively low (theoretically even zero) by the only possible way in those systems, namely the reduction of the degrees of freedom. Such a system has thermodynamically a variety of important properties, for instance noiseless behaviour, which is easy to understand when one optimizes the signal/noise-ratio with respect to the frequency, delivering just = constant. Under definite circumstances this system can even amplify the incoming signals (17). Moreover, the rule = constant corresponds to a multimode-laser at threshold, since the probability of absorption always equals that of emission between any two excited energy levels, giving thus rise to a theoretically vanishing absorbance or amplification. Variations around this state allow the use of both amplification and absorption of field amplitudes. That this assumption is a realistic one, is demonstrated, for instance, by the surprising results of Mandoli and Briggs (21). They have shown that biological material can guide coherent light without significant loss over distances of some centimeters. In supporting these results, our own transparency experiments (22) indicate that the surprisingly high optical transparency of tissues has not only to be associated with the state of the material, but has to be assigned to a considerably high degree of coherence of the "biophotons" themselves, at least in just our experiments.

In order to show that the coherence of low-intensity laser light is not essential when biological objects are affected, V.V. Lobko et al. (23) assumed the biological matter is in thermal equilibrium and only larger macroscopic entities are relevant. These suppositions are not valid with respect to the real non-equilibrium state, the natural biological distances in significant units of cell-diameters or even smaller units and, in particular for our case, in view of the quasi-stationarity of biophoton-fields.

In fact, recent results of W.B. Chwirot et al. (24) have given evidence of periodic oscillations of low-level luminescence after exposure of synchronized larch microsporocytes to weak quasi-monochromatic light irradiation. The amplitudes and frequencies, which are of the order of a few minutes, depend in these experiments on the wavelength of the irradiated light. Chwirot et al. emphasize that these results are predicted by the electromagnetic model of differentiation (25) that is based on coherent biophoton emission.

There is, in addition, a molecular model which explains the phenomena of low-level luminescence. It is based on metabolically "pumped" exciplex formation in the DNA (27-29).

All these findings and considerations concentrate more and more

to an essential question: Are biological systems paradigms of coherent in such a sense that evolution has naturally "selected" them just according to a definite coherence-rule (for instance =constant), or does coherence, if at all, play only a very limited role?

According to my knowledge, the idea of coherence in biology is commonly neglected or even rejected mainly due to the general concept of biochemistry and linked disciplines, including more local approaches of the living state. On the other hand, Fröhlich's model (30), as well as the concept of dissipative structures (31) and recent papers on the role of Bose condensation in biology (32-35) point to a fundamental importance of coherence in living systems.

In this paper the question of optical coherence in biological systems will be discussed

(1) from a more speculative point of view, considering some principles of the physical background, e.g. the problem of coherence at very low radiation intensities, and

(2) by presenting experimental results or the relaxation dynamics that indicate a surprisingly high degree of coherence within the living state under ergodic conditions.

Some biological and medical consequences have been discussed elsewhere (25, 35-37).

THE QUESTION OF OPTICAL COHERENCE IN BIOLOGICAL SYSTEMS

The degree of coherence has been defined, for instance, by Glauber (38). Coherence to n-th order can be expressed in terms of the expectation value of the operator

in the actual state of the electric field.

where represents the density operator of this state. are the mutually adjoint electric field operators at space-time point . If <O(n)> is completely factorizable such that

|> = |>

satisfy the coherence condition. They are called "coherent states".

This fundamental dependence of coherence on factorization of field operators shows that the degree of coherence is neither a question of the intensity of emitted photons nor of the narrowness of their spectral bands. Coherence may occur at all levels of field amplitudes and for all bandwidths including a continuous spectrum (39).

equ.1

Before presenting sufficient experimental results and their discussion, let us recall that active biological systems actually contain manifold excited states that are really populated. Considering a stationary "white noise" it appears evident that, besides allowed states, also "forbidden" states will be occupied according to the well known equ.2:

equ.2

Although this argumentation does certainly not prove the coherence for weak effects, it shows that low intensities can never be a basis to reject a realistic possibility of even high coherence. In contrast, a quasi-stationary state far away from thermal equilibrium exhibits a natural tendency for coherence.

It is, however, principally difficult to examine coherence at low intensities experimentally, for the following reasons:

(1) The usual methods of interferometry must fail in view of the low amplitudes,

(2) the signal/noise ratio may become rather low,

(3) from a classical point, at low intensities only long coherence times make sense, but it is difficult to keep stationary conditions during the consequently long observation times, in particular for living systems. Apart from these technical problems, the photo count statistics (PCS) that provide the only possible tool for examining coherence of low-level luminescence, fail even for stationary states, namely, if the number of the degrees of freedom is not known. This becomes evident by the following considerations.

The photo count distribution , which represents the probability of registering n counts in a preset time interval , takes for a chaotic field the form

equ. 3a

while a fully coherent field is subject of

where <n> represents the average number of registered photons during . For a derivation of equs.3 see, for instance, ref. (40).

A consequence of (3) is that the variances associated with equs.3a

and 3b are

equ. 4a)

and

equ. 4b)

respectively.

To take an example, let us consider the count rate of cucumber seedlings shown in Fig.1 (case 1) and those which are poisoned by acetone, in order to achieve a high count rate (case 2). After the seedlings reached a maximum emission, they remain for some minutes at a quasi-stationary state, where approximately the following count rates (in cps = counts per second) were registered:

TABLE I

Case <n>

reference 50 ± 10 cps 60 ± 20 cps

1 150 ± 30 cps 180 ± 70 cps

2 45,400 ± 100 cps 45,700 ± 500cps

The preset time interval was chosen in case 1 and case 2 as . The errors have been estimated by taking into account the dark count rate of about 50 ± 20 cps including unavoidable deviations from stationarity.

Comparing this result with equs.4a and 4b, respectively, it appears that only a fully coherent field can account for the experimental values, while a chaotic field could be significantly rejected.

However, equ.4a is only valid for a one-mode field. In the case that multiple different and independent modes contribute to the total intensity, equ.4a has to be considerably modified, while equ.4b remains valid. If M represents the number of modes that may participate (for simplicity, with always the same intensity), we obtain instead of equ.4a

equ. 4a*)

For a derivation of equ.4a* see, for instance, ref. (41).

There are two ways of solving this crucial problem, namely

(1) an extended spectral analysis of PCS with the aim to limit M to its actual boundaries, and

(2) an examination of non-stationary states, e.g. the relaxation dynamics under external stationary conditions (ergodicity).

Actually, by analysing the decay behaviour of different spectral modes of biophotons after exposure of plant seedlings to light illumination we found an extremely strong mode-coupling indicating that M is of order 1 (14). In this case, low-level luminescence corresponds evidently to a fully coherent field. However, in order to confirm these indications of coherence of "low-level luminescence" it is at this state of discussion important to investigate the relaxation dynamics of biophoton emission in more detail. This includes the point of view that "coherence" of biological systems may become only evident under non stationary conditions.

RELAXATION DYNAMICS OF LOW-LEVEL LUMINESCENCE

Let us imagine an unstable system in which a definite part of kinetic energy of the decay products (e.g. photons) is restored again by rescattering within the source system (e.g. excited molecules). By confining ourselves at first to a classical oscillator model we have consequently

, equ. 5

where x represents the amplitude of the oscillator, and accounts for the usual chaotic rescattering, where the average values of kinetic and potential are equal, while is due to the re-storage effect of coherent rescattering (42,43). Of course, for the solution of equ.5 is exponential (equ.6a), whereas in case of coherent rescattering the solution of equ. 5 takes the form of a hyperbolic function (equ.6b).

Equ.6a

is a (generally complex) constant.

Equ.6b

Hence, under the same (ergodic) conditions, an exponential decay turns into a hyperbolic one as soon as a chaotic rescattering is substituted by a coherent rescattering of the decay products to their source.

Unfortunately, the theory of unstable quantum systems, which could confirm this result generally, is not developed to such an extent that coherent rescattering can be described already without some puzzling problems. At present, even the generally accepted experimental evidence of an exponential decay in the case of chaotic rescattering is not yet clearly established theoretically.

However, as Ersak (42) has demonstrated, possible deviations from exponential decay are always due to a coherent rescattering of the decay products to their source. Clearly, by separating the time evolution of the actual state of the system according to

equ.7

where H is the Hamilitonian of the system between any two observation points, and represents the dynamical state of the decay products, we have consequently

equ.8

After multiplication of equ.7 with the bra /, the relation

equ. 9

is obtained.

Since a semi-group law A(t + t’) = A(t).A(t’) is a necessary and sufficient condition of exponential decay, equ.9) indicates that a non-random rescattering of the decay products to their source suffices for deviations from exponential decay. As Fonda et al.(43) and Davies (44) have shown, the exponential decay can be generally derived from

equ. 10

where is the density operator associated with of equ. 7 constant that describes the frequency of randomly distributed rescattering processes of the decay products with respect to their source. Hence, the quantum description of rescattering refers to measurement processes where the are the projectors onto the eigenmanifolds of the corresponding observables.

rescattering according to

is chosen, where is a constant () and is the probability amplitude of the excited state of the source associated to a projector

again a hyperbolic decay is obtained. At the same time, the uncertainty

is then minimized, in accordance to the general property of coherent states (45).

A possibly more interesting approach to the problem originates from the fact that rescattering depends on the number of reductions that occur during decay (43). From this an apparent Zeno's paradoxon arises: the more reductions (observations) are taken into account, the more improbable it becomes that the unstable state decays at all. This problem has been investigated by several authors (46-48). However, as Bunge and Kalnay (49) have shown, one cannot hypothesize that measurements which lead to the reduction of the state under investigation can be carried out in infinitely short time intervals.

This interesting result is confirmed also by the following considerations which deliver a further possibility of differentiating chaotic and coherent fields.

Take the identity

If =1 for all t, the RHS of equ.11 vanishes. Consequently, the LHS should vanish, too. Since according to the Schrödinger equation

we then obtain after substitution into equ.11

equ. 12)

In case of a Hermitian operator' this is obviously correct for the real part. However, for the imaginary part the uncertainty relation comes into play

such that for times

a deviation from equ.12 is allowed.

For an unstable system, on the other hand, the real part of the LHS of equ.11 cannot vanish. Consequently, for time intervals

where is the coherence time, the real part of the RHS of equ.11 cannot vanish, too. This means that an uncertainty in evaluating the RHS of equ.11 has to be taken into account. For cannot be identical to

This leads to a fundamental difference of evaluating equ.11 for chaotic and coherent fields, respectively. Providing ergodic conditions, for both coherent and chaotic rescattering, the value of the RHS increases proportionately to the observation time. For a chaotic field we then have

such that

equ. 13a

This leads obviously to an exponential decay.

In the case of a fully coherent field, on the other hand, we have

and consequently

= const. equ. 13b

Equ.13b) is generally valid if, and only if

Hence we obtain the general result

| equ. 13b*

that describes again the expected hyperbolic decay law.

In order to confirm the argumentation, the PCS theory can be extended to an ergodic unstable quantum system.

By definition we have

equ. 14

where is the count rate of the i-th measuring interval of finite and constant length. Let us imagine that by suitable choice of the number of ensembles under investigation can always be kept at a value of the order of the coherence time of the field.

In order to determine , we may either keep constant and register for all t, or we change at all t slightly the length of and register the alteration of . Since t and are independent quantities for an ergodic field, it doesn't matter what method is preferred.

Noting that the derivation of does not contribute to

we obtain from equ.14

equ. 15

An ergodic system provides the homogeneity of t and . Hence, the relation between and dt for constant

must be a linear one, if the time average can always be represented by the ensemble average. Consequently, we have

equ. 16

is a (generally complex) constant for a preset time interval , for definite coupling parameters and a fixed number of ensembles, including =0 for a stationary system that represents a special case of an ergodic field.

It is well known that a Gaussian field obeys the relation

while a coherent field is subject of

equ. 17b

These relations are valid at any instant for any . The degrees of freedom do not play a decisive role, as we will see later. Hence, let us confine at first to a single-mode field. By calculating

of equ.17a, substituting the general relation equ.15 into these derivations and taking into account equ.16, we then arrive after straightforward calculations at

equ. 18a

for chaotic fields and

equ. 18b

for coherent fields, respectively.

Again the differences of equ.18a and equ.18b would disappear for increasing number M, since the term in the denominator of equ.18a vanishes for in the case of a multimode field or , respectively (41). However, this does not bother the remarkable difference: of the relaxation dynamics of a chaotic and a coherent field. In fact, in case of a chaotic field we have

This means that after taking into account

from equ.18a a relation

equ. 19a

is obtained that delivers in view of = constant an exponential decay law. However, in case of a coherent field it is allowed to extend as long as t is smaller than the coherence time . Consequently, we then have

equ. 19b

that yields again the hyperbolic decay law.

EXPERIMENTAL BACKGROUND AND AN APPLICATION IN CANCER RESEARCH

In a previous paper (14) it has been shown that living tissues exhibit significant deviations from exponential decay after exposure to light illumination, while the agreement to a hyperbolic decay law is excellent, even, and in particular, for the decay of single modes that can be observed by using interference filters.

Fig.4 displays a further example, where the total emission from cucumber seedlings after exposure to a 10-second illumination of a Halogen lamp (150 W) at a distance of 20 cm was observed, subjected to the same technique as referred in (14).

Fig.4a: Photonemission from cucumber seedlings after exposure to weak white-light illumination (in counts per 0.5s) for 300 measuring intervals (150 s).

Fig.4b: Logarithmic scale for the ordinates of the measurements of Fig.3a, where the measured values (000…) were approximated by a hyperbolic law and an exponential one, comparably. The abscissa displays the time in arbitrary units.

Chwirot et al.(50) have demonstrated that synchronized cell cultures at meiosis exhibit a more or less hyperbolic decay after exposure to weak white-light illumination. The agreement to the hyperbolic law is there correlated to the cooperativity within the different stages of the cell cycle, appraised from the biological point of view.

In ref. (14) it was already shown that the relaxation dynamics of normal and corresponding malignant tissues display significant differences, which can be associated to diminished cooperativity in tumours.

Recently, Schamhart et. al. (51) have shown that the total number of counts which are emitted by cell cultures after exposure to white-light illumination

(1) increases with increasing cell densities for malignant liver cell

cultures, and

(2) decreases with increasing cell densities for the corresponding normal ones.

Thereby, they confined themselves to a definite first part of the decay curves immediately after irradiation. Fig.5 displays these results (courtesy of Dr. Schamhart).

Fig.5: Total counts within the first seconds after exposure of cell suspensions to white light illumination. With increasing cell density, HTC cells (000) that are malignant and the corresponding normal hepatocytes (· · · ) show principally different behaviour. The H35 cells (D D D ) are only weakly malignant.

At the same time Schamhart has shown that the relaxation dynamics of the normal cells agree better with the hyperbolic decay than that of the corresponding malignant ones, which display more rapid decay.

Before presenting our own recent results on human cell cultures, the experiments shown in Fig.5 should be discussed.

This system consists of an ensemble of radiating cell layers in a cubic quartz cuvette within a colourless nutrition fluid. The total surface area of the cuvette is 6F, the diameter .

If p is the contributed photo count rate of one cell, and is the cell density, we then obtain an increase di of the measured photon intensity by the contribution of a cell layer of thickness dx at a distance x from the counter according to

equ. 20

W(x) is the probability that the radiation is absorbed within the system on the way of length x between the layer and the counter that is located at point O. First of all, there is no reason to expect for W(x) a value deviating from the Beer-Lambert law:

equ. 21

is a constant absorption coefficient of the device that is always the same for all the experiments. ~ represents the absorption coefficient per unit of cell density for the cells within the medium. It is expected to be or order .

where i(O) is the measured radiation intensity.

that exhibits a principally different behavior for normal and malignant tissues. From equ.22 we obtain

equ. 23

Since both terms of the RHS of equ.23 are positive definite, this model can never explain, firstly

which is observed at higher cell densities of normal cells.

Secondly, it is not possible to explain

for malignant cells at higher cell densities, too, since for we have according to equ.23

Fig.6 demonstrates the differences between the theoretical model due to the most reasonable assumptions and the real behaviour.

Fig.6: Theoretical calculation of the dependence of the photon intensity I(0) on the cell density for the cases that

1. no interactions between the cells play a role(¾ )
2. the interactions become aggregative (---)
3. the interactions become disaggregative (-.-.-.).

There are in principle only two possibilities to explain these significant deviations from expected results.

The first is a dependence of p on the cell density. This would mean that the production of photons alters very sensitively with mutual long range interactions between cells. Malignant cells would produce more photons with decreasing mutual distance. The contrary would be valid for normal cells. This interpretation is supported by reports according to which tumour tissues may show a higher count rate than normal ones. We could not confirm this so far. However, this argumentation is supported by our own observations of a dependence of photon intensity on differentiation. We generally observe a lower count rate of the unperturbed tissue with increasing differentiation.

However considering this argument one should realize that the experiments of Fig.5 are based on a count rate p of the order of about one photon per hour (or even less). If one prefers to envisage photochemical reactions as the source, for instance the alteration of enzymatic activity with the change of mutual distance of the cells, an explanation of even nonlinear (!) effects in terms of those in this case extremely rare events would be quite fantastic.

Hence, we prefer the second possible interpretation of this effect namely the alteration of with varying cell density. Although the first possibility

is not excluded by this and may really play a role, it appears more likely to explain the effect of Fig.5 in terms of

Since the cell densities used in the experiments are rather low compared to that of a solid tissue, a change of would indicate a very sensitive dependence of optical properties of living cells (as entities) on mutual long-range interactions. Since from Fig.5 we have consequently

for normal cells, and

for malignant ones,

normal cells exhibit an increasing absorption of weak mutual photons with increasing density, while malignant ones increase the reflection probability.

Roughly speaking, while normal cells improve the basis of mutual communication in the tendency of forming cell colonies by means of photon interaction with decreasing distance, the contrary holds for malignant cell populations.

Although it is principally impossible to decide whether the effect of Fig.5 is due to

or, alternatively, to

or possibly due to both of these alternatives, in any case there has to be concluded that

(1) there is a sensitive dependence of biophoton emission from living cells on mutual long-range interactions at a distance from at least about ten cell diameters on,

(2) in view of the very low intensities (p << 1 ) and very large distances between the single cells (, where v is the volume of a cell), these non-linear effects corresponding actually to "stimulated emission and absorption of photons at very weak intensities" can only be explained in terms of coherence properties of the interacting photons. Taking into account the interaction distance of at least 10 cell diameters, the maximum coherence volume of biophotons is, according to these results, at least thousand times the cell volume.

At the same time, this test provides a powerful tool of differentiating normal and malignant tissues on the decisive level of intercellular interaction.

Since from the "most reasonable" model of equ.22 one would expect that the decay behaviour of the single cell (p(t)) corresponds exactly to the population, a further examination of these coherence effects concerns the characteristics of the decay functions. Therefore, we studied recently the relaxation dynamics of human cells after white-light illumination under the same conditions as Schamhart et al. have chosen. We compared human amnion cells with corresponding malignant ones, namely wish cells.

Fig.7 shows a typical example, where the decay functions of amnion cells and wish cells at a cell density of cells/ml have been observed under the same conditions.

Fig.7: The decay parameter of the hyperbolic approximation that is adjusted to the relaxation dynamics of photon emission of different cell suspensions after exposure to weak white-light illumination in dependence on the cell density. The lower curve displays the behaviour of normal amnion cells. The opposite behaviour is shown by the corresponding malignant wish cells (upper curve). The three measurements at the right side of Fig.7 correspond to the nutrition medium alone.

These measurements were carried out with different cell densities where was altered unsystematically, in order to avoid systematic errors. Then the best fitting of the hyperbolic decay according to equ.24 was calculated by use of a computer program.

equ. 24)

T is a constant corresponding to the time delay between the first measuring point and the end of excitation. It is 3 seconds and was kept constant for all the measurements. The values A and 1/ were determined for all the decay curves. () is plotted in Fig. 7.

Again it can be seen that normal cell populations display a behaviour controversal to the malignant ones. While in the case of normal cells increases with increasing cell density (see, for clearness also equ.5)), decreases nonlinearly with increasing density for malignant cells, approaching more and more the chaotic exponential rescattering.

These results do not only confirm the importance of coherence in biology even, and in particular, in the case of weak effects: they are obviously fundamental in solving at least one of the most crucial problems and related questions. A more profound discussion of this last topic has been presented elsewhere (e.g. 25,37).

Finally, it may be worthwhile to note that non-exponential decay is also observed sometimes in condensed matter physics. As has been shown by Fain (52), however, in those cases non-random processes, including instabilities of the environmental conditions, are significant. This confirms the opinion that nature exhibits some tendency to coherence at low intensities. As Ngai et al. (53) emphasized, in these cases of coherent rescattering, a single linear exponential form is as unphysical as a superposition of them.

The connection to the usual exponential decay on the basis of perturbation expansions has been presented in ref. (54), while a summary of the given arguments appears in ref.(55).

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