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Introduction Coherent State Approach to Biophotons The Cancer Problem Acknowledgements References

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Many attempts have been made to trace biophoton emission back to coherent states of the biophoton field [2, 6]. In this paper I would like to add a further description which is based on the well-known form of a Hamiltonian that keeps coherent states coherent. By evaluating the properties of coherent states under the condition of a Hamiltonian that conserves the coherence of the field, one sees that all the properties of biophoton emission can be looked upon as necessary consequences. On the other hand, it is unlikely (if not impossible) to understand the properties of biophoton emission in terms of chaotic emission. An example of an application of Gurwitsch's great idea of mitogenetic activity shall demonstrate the value of this approach.

A coherent state is defined as an eigenstate of the annihilation operator a, i.e.

. (1)

Glauber and Mehta et al.[7, 8] have shown that a coherent state remains coherent at all times when the Hamiltonian takes the form

, (2)

where the terms describe the free energy, the exchange of energies among modes and the interaction of classical currents with the radiation modes. The functions f and B are real, while g(t) are arbitrary complex functions of time t. Without loss of general significance, we simplify the summation by introducing the functions

Thus we write instead of eq.(2) 

Let us now calculate the properties of a coherent state |a> under the influence of the Hamiltonian H(t).

The Schrödinger equation takes the form
 

. (3)

The initial condition is 

The solution can be written as

. (4)

Consequently we obtain

, (5)

where  are abbreviations of , respectively.

Since  is simply a C-value, we can factorise:

(6)
 

where we used the Baker-Hausdorff identity

.

With straight-forward calculation using we arrive at

(7)
 

By use of where D(a) is the displacement operator and  the vacuum state, and by use of the Baker-Hausdorff identity, we obtain

(8)

The last two operators combine again to a displacement operator and - by use of the Baker-Hausdorff identity - we arrive at

The final solution then takes the form

where  . (10b)

The phase factor certainly has the norm 1 since  This holds because  and  are real functions.

The number of photons remain unchanged unless  This is a rather remarkable result since it means that as long as no pumping takes place, sources or sinks of free energy do not influence the photon number of coherent states. Both free energy and classical currents can only influence the complex amplitude which moves along circles in the complex plane. However, as soon as destructive and constructive interference play a role, the free energy may dramatically change the number of photons of the coherent field.

Actually, from (10b) we get

(11)

This shows that n(t) is not necessarily constant even if F and G are not dependent on time. In order to explain this strange result, let us look at the expectation value of the energy.

(12)

If G = 0, then n(t)=const., and the energy is simply the classical expectation value

(13)

For we simply then have , which describes the photon energy of a field with a constant number of photons.

For F = 0 and G = constant, where G and b are real quantities, we have

and (14a)

(14b)

The conservation of energy, i.e. the constant mean value of energy as well as a constant number of photons, is for G ¹ 0 only possible if , which means that an excited coherent state relaxes under ergodic conditions according to an hyperbolic decay function (as has been already demonstrated some years ago) [9].

A sufficient condition of a homeostatic regulation of a coherent field follows from

(15a)

, (15b)

and for  representing again a hyperbolic law.

It is worthwhile to note that B(t) does not influence the number of photons in a coherent state, and F also does not change the photon number unless G is not switched on.

However, we get a remarkable result if one considers the mode coupling between coherent states of different energy (or frequencies).

Since we can write

, (16)

we can include the non-diagonal elements of by the substitution of

(17a)

(17b)

where we used the fact that  is real.is the original operator for  and the k^th mode under study.

Consequently, the mean value n(t) of the photon number  is

(18)

This result points to some rather remarkable features of biological systems as far as they use the capacity of basic homeostatic regulation in terms of the 

Homeostatic regulation can take place in the following way:

1. , which means that the mean value of creating and annihilating photons disappears.

a). This may take place, for instance, for destructive interference.

(19)

for all j and k.

n(t) is then constant and independent of free energy as well as of classical currents. This case is at least partially realised in living matter, since under stationary conditions n(0) is a constant and the mode coupling satisfies the f(w)= constant-rule which is just a consequence of (19).

b). , (20)

where the + accounts for constructive interference. It may take place immediately when (19) happens. This means that there are regions of destructive interference and consequently others of constructive interference in such a way that the total energy is conserved:

, where  is assigned to destructive interference and  to constructive interference.

This case corresponds to "photon sucking" which has been already discussed in former papers [10].

1. One may also consider the case  by oscillatory behaviour of  in a way that remains zero by the lack of correlations (for ) or by the balance of correlations and anticorrelations. However, this case shall not be discussed here.

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