On the Environment of a Quantum System and an Associated Form of Incoherence

One turns now to one of these openings: Long ago, Heisenberg called briefly attention on a possible determining action of environment on the state of a quantum system (Heisenberg 1958).

An environment can perturb the wave functions of a system and these pertur­bations grow with the number of particles, which are in contact with environment. The quantum state of a macroscopic system is therefore very sensitive to instability, except under very special conditions. In the present case of a Geiger detector, one can think of the environment as involving the atmosphere around it, and eventually a table below.

One advocates the existence of a strong influence of the nearby environment on the quantum state of a macroscopic measurement device. More than a direct action on the measuring system itself, this influence appears rather as a conditioning of its quantum state. More explicitly, one wants to show that the action of environment on a well-defined macroscopic object (namely one for which the notion of a specific quantum state makes sense), brings incoherence into the wave functions of that state. Still more explicitly, one will show that this incoherence is not associated with the bulk (or average) action of the environment, but with fluctuations in this action. The nature of this incoherence is closely linked with local entanglement, which is why it appears new, and why one looks at it now.

One will deal again with the example of a Geiger counter, denoted by B. The measurement is concerned with a particle, denoted by A. The counter is initially standing alone, with nothing to measure. One can think of it as consisting essen­tially of an atomic gas, enclosed in a solid box.

Nothing could look apparently simpler that this object in that situation, but one will see that local environment maintains a permanent invisible turmoil in the gas, and also in the box. This unrest is much less obvious than the thermal motion of atoms, because it is contained in wave functions and one needs several hints for getting at it, as follow:

The direct action of environment consists in a multitude of collisions by atmospheric molecules on the box. One looks first at a unique collision, in a mathematical model where the environment would be an empty universe containing a unique molecule M, which hits the box at some time. The theory of local entanglement, which one sketched previously, implies that a wave of LE starts from the place where the molecule hits the box (solid state physics shows that this LE is carried through the box by phonons, before it reaches the gas). This is the kind of local entanglement, which one described before, now meant as LE with the out­going state of the external molecule.

One found earlier that the associated LE wave moves at the velocity of sound. It takes some time A? before it fills up the whole gas with its local entanglement, and thus realizes a state of complete algebraic entanglement. In a counter with size 10 cm, this time At is of order 10^{— 4} s, which is quite significant.

One looks next at average effects of environment. One knows well enough how to deal with them and a famous formula, which extends one by Boltzmann to the quantum domain, gives an explicit formula for the average state of the gas, Z^—1 exp( — H/k_BT), where H is the total energy, T the temperature, k_B Boltzmann’s constant and Z a normalization coefficient. No local entanglement is associated with this average so that the effects, in which one is presently interested, must be associated with fluctuations.

When considering an effect of fluctuations on the quantum state, one would rather speak of an “influence” rather than of an action, as one did earlier when introducing LE. Several features must be considered then and one begins with the most formal one, which is mathematical and as such can orientate towards the concepts that must be used.

One already wrote down the average state of the system, which one now denotes by < p >. Its main character is simplicity, when compared with the actual “mixed” state p of B (its density matrix). One may think of p as involving an extremely large number of wave functions expressing the thermal disorder in the gas.

This p is a matrix, with a tremendous number of lines and columns, but which is nevertheless a square matrix. The difference Ap = p — < p > is also a matrix, but one will refine a little more on its algebraic aspects by splitting it into a part, p ₊, which involves its positive eigenvalues, and a part—p_ involving the negative ones.

I make acknowledgements to readers considering that these mathematical manipulations are puzzling. Physicists would probably prefer hearing of positive fluctuations in the external flux of molecules as being above average, and of negative fluctuations below. The two present matrices p ₊ and p _ are not exactly in one-to one correspondence with these positive and negative fluctuations however, although they are closely linked with them. To cut short a story, which could become long, one will say only that these two matrices express the occurrence of fluctuations in the present case, and that their size (which mathematicians call their trace) is determined by the strength of fluctuations.

One found that every external collision by a molecule leaves during a time At a mark of its occurence in the quantum state of the gas. One can find such marks also in the matrices p ₊ and p, as a presence of so many LE waves in every one of their eigenfunctions.

One pays then more attention to the random character of the fluctuations. One cannot identify individually external collisions, as belonging to excesses above average or lacks below. One can only know their number and assert that their properties must be random. This randomness has crucial consequences, requiring considerations of quantum mechanics and which one makes now.

Randomness of the collisions by molecules belonging to fluctuations implies that the place x where the molecule hits the box, its momentum p and its time of arrival are random. Quantum mechanics tell us then that a phase a = Ap.x/h is present in the wave functions of atoms in the gas (Ap is the momentum transfer in the collision). An essential consequence of this effect is that every wave of local entanglement, associated with a molecule belonging to fluctuations, carries a def­inite phase, which is random. But in optics and in quantum mechanics, an occur­rence of random phases is synonymous with incoherence! One can then sum up this first part of the analysis by the following.

Proposition 1 Fluctuations in the influence of environment on the quantum state of the macroscopic system, generate incoherence in that state.

More precisely, one may say that this incoherence is present in the matrices p ₊ and p, which one can use when testing the influence of environment.

Proposition 1 is new and plays a central part in the present theory. One may consider it as the exact opposite of the main assumption in Everett’s interpretation of quantum mechanics, which is an infinitely exact coherence of a wave function of the universe.

One will leave out this aspect and rather turn to an exercise in geometry. When closing eyes and concentrating, one can see in the state of the matrix p ₊ (for instance) a complex pattern of LE waves, which move everywhere along all directions, showing fronts with small width A, the fronts crossing each other under their motion and their expansion.

A substantial value of the number N_x of fronts, which overlap over a point x in the gas, implies an action of as many random phases of the previous type a, in the wave functions at that place. But every subset of these phases can add up together and yield another random phase fl as their sum. These other various phases are also random and also mutually independent, so that their local number is about as large as the factorial N_x!. It shows that Proposition 1 does not only mean that there is some amount of randomness in the state of the system, but that there is a very strong incoherence.

A further comment is suggested by a cogent question, which is: If there is such a high amount of incoherence in most macroscopic systems, if not all of them, how could it be that nothing of that kind has never been observed?

The answer is twofold. It says first one that this incoherence is associated with local entanglement and, therefore, is inaccessible to observables. As a conse­quence, an observation, whatever it can be, needs absolutely projection operators for expressing its outcomes, so that it cannot be sensitive to this very special kind of incoherence. A second answer, which enforces the first one, is that the amounts of incoherence and its distribution are exactly the same in the matrices p + and p _, so that their observable effects, if they had some, would cancel each other anyway.

Coming back then to the matrix p ₊, for instance, one can give a look at one of its instantaneous eigenfunctions which is defined mathematically as representing one eigenvectors of the time-dependent matrix p + (t) at a sharp time t. Because of the very high number of composite phases fl everywhere, and of the non-negligible width (of order A) of the front where a phase a is active, one finds that the disorder in wave functions is huge and their correlations are of short range (of order A). This situation can be expressed by another proposition, in which one denotes by N_c the number of atoms in a cell of space with size A and which is:

Proposition 2 Every eigenvector (or wave function) / of p_B + (or p_B_i splits under incoherence into a sum of independent components

y = Z /n, ⁽¹⁾

n

where every component /_n carries a distinct random phase, extends in space over a distance of order A and involves a limited number of atoms, of order N_c, with

N_c = n_eA³, Z (2)

( n_e denoting the number density of atoms in the macroscopic system).

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