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Dynamical Reduction Models: Present Status and Future Developments Dynamical reduction models: present status and future developments Angelo Bassi Dipartimento di Fisica Teorica, Universit`adegli Studi di Trieste, Strada Costiera 11, 34014 Trieste, Italy. Mathematisches Institut der Ludwig-Maximilians Universit¨at, Theresienstr. 39, 80333 M¨unchen, Germany. E-mail: [email protected], [email protected] Abstract. We review the major achievements of the dynamical reduction program, showing why and how it provides a unified, consistent description of physical phenomena, from the microscopic quantum domain to the macroscopic classical one. We discuss the difficulties in generalizing the existing models in order to comprise also relativistic quantum field theories. We point out possible future lines of research, ranging from mathematical physics to phenomenology. 1. Quantum Mechanics, measurements and environment Standard Quantum Mechanics is known to talk only about the outcomes of measurements, but it has nothing to say about the world as it is, independently of any measurement or act of observation. This is a source of serious difficulties, which have been clearly elucidated e.g. by J. Bell [1]: It would seem that the theory is exclusively concerned about ‘results of measurements’, and has nothing to say about anything else. What exactly qualifies some physical systems to play the role of ‘measurer’? Was the wavefunction of the world waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little bit longer, for some better qualified system ... with a Ph.D.? arXiv:quant-ph/0701014v2 8 Feb 2007 Measuring devices, like photographic plates and bubble chambers, are very sophisticated and highly structured physical systems, which anyhow are made up of atoms; we then expect them to be ultimately described in quantum mechanical terms by means of the Schr¨odinger equation. What else should we do, taking into account that people are trying to describe also the entire universe quantum mechanically? But if we describe measurements in this way, then we do not get any outcome at the end of the process. The Schr¨odinger equation is linear, the superposition principle enters into play and it does it in such a way that all possible outcomes are there simultaneously in the wave function, but none of them is selected as the one which occurs physically. Yet, if we perform a measurement, we always get a definite outcome. So we have a problem with Quantum Mechanics. In recent years, a not so new idea is gaining more and more credit: measuring devices are different from microscopic systems: they are big objects which unavoidably interact with the surrounding environment. Such an interaction turns out to be very peculiar because it destroys the coherence between different terms of a superposition and seems to reduce a pure state, where all terms of the superposition are there simultaneously, into a statistical mixture of the states, and moreover it does so with the correct quantum probabilities. What else do we need? This idea, that the environment somehow naturally guarantees the emergence of definite properties when moving from the micro to the macro, by destroying coherence among different terms of a superposition, is very appealing. But wrong. I will not spend much time on this issue, since many papers already have appeared on the subject, starting from those of Bell [2] to very recent ones [3, 4, 5]. I note here that the division between a system and its environment is not a division dictated by Nature. Such a division is arbitrarily set by the Physicist because he or she is not able to solve the Schr¨odinger equation for the global system; he or she then decides to select some degrees of freedom as the relevant ones, and to trace over all other degrees. This is a very legitimate division, but not compelling at all. Such a division is more or less equivalent to the division between a quantum system and a measuring device: it’s artificial, just a matter of practical convenience. But if the physicist were able to analyze exactly the microscopic quantum system, the macroscopic apparatus and the surrounding environment together, i.e. if he or she used the Schr¨odinger equation to study the global system, he or she would get a very simple result: once more, because of linearity, all terms of the superposition would be present at the same time in the wave function, no one of them being singled out as that which really occurs when the measurement is performed in the laboratory. The so called measurement problem of Quantum Mechanics is an open problem still waiting for a solution. Dynamical reduction models, together with Bohmian Mechanics, up to now are, in my opinion, the most serious candidates for a resolution of this problem. 2. The dynamical reduction program Continuing quoting Bell: If the theory is to apply to anything but highly idealized laboratory operations, are we not obliged to admit that more or less ‘measurement-like’ processes are going on more or less all the time, more or less everywhere? Do we not have jumping then all the time? The basic idea behind the dynamical reduction programm is precisely this: spontaneous and random collapses of the wave function occur all the time, to all particles, whether isolated or interacting, whether they form just a tiny atom or a big measuring device. Of course, such collapses must be rare and mild for microscopic systems, in order not to disturb their quantum behavior as predicted by the Schr¨odinger equation. At the same time, their effect must add up in such a way that, when thousands of millions of particles are glued together to form a macroscopic system, a single collapse occurring to one of the particles affects the global system. We then have thousands of millions of such collapses acting very frequently on the macro-system, which together force its wave function to be very well localized in space. The aim of the dynamical reduction programm is then to modify the Schr¨odinger evolution, by introducing new terms having the following properties: They must be non-linear, as one wants to break the superposition principle at the • macroscopic level and assure the localization of the wave function of macro-objects. They must be stochastic because, when describing measurement-like situations, one needs • to explain why the outcomes occur randomly; more than this, one needs to explain why they are distributed according to the Born probability rule. There must be an amplification mechanism according to which the new terms have negligible • effects on the dynamics of microscopic systems but, at the same time, their effect becomes very strong for large many-particle systems such as macroscopic objects, in order to recover their classical-like behavior. If we look carefully at these requirement, we realize that they are very mandatory: there is no assurance at all beforehand, that they can be consistently fulfilled. I think that one of the greatest merits of the GRW proposal [6] is to have shown that they can be implemented in a consistent and satisfactory model. 3. The GRW model Let us consider a system of N particles which, only for simplicity’s sake, we take to be scalar; the GRW model is defined by the following postulates: States. The state of the system is represented by a wave function ψ(x1, x2,... xN ) belonging to the Hilbert space 2(R3N ). L Evolution. At random times, each particle experiences a sudden jump of the form: L (x)ψ (x , x ,... x ) ψ (x , x ,... x ) n t 1 2 N , (1) t 1 2 N −→ L (x)ψ (x , x ,... x ) k n t 1 2 N k where ψt(x1, x2,... xN ) is the statevector of the whole system at time t, immediately prior to the jump process. Ln(x) is a linear operator which is conventionally chosen equal to: 3 4 α α 2 Ln(x) = exp (qn x) , (2) s π − 2 − where α is a new parameter of the model which sets the the width of the localization process, and qn is the position operator associated to the n-th particle; the random variable x corresponds to the place where the jump occurs. Between two consecutive jumps, the statevector evolves according to the standard Schr¨odinger equation. The probability density for a jump taking place at the position x for the n-th particle is given by: p (x) L (x)ψ (x , x ,... x ) 2, (3) n ≡ k n t 1 2 N k and the probability densities for the different particles are independent. Finally, it is assumed that the jumps are distributed in time like a Poissonian process with frequency λ, which is the second new parameter of the model. The standard numerical values for α and λ are: λ 10−16 sec−1, α 1010 cm−2. (4) ≃ ≃ Ontology. Let the mn be the mass associated to the n-th “particle” of the system (I should say: to what is called “a particle”, according to the standard terminology); then the function: (n) 3 3 2 3 2 ρ (x ) m d x ...d x − d x ...d x ψ (x , x ,... x ) (5) t n ≡ n 1 n 1 n+1 N | t 1 2 N | Z represents the density of mass [7] of that “particle” in space, at time t. These are the axioms of the GRW model: as we see, words such as ‘measurement’, ‘observation’, ‘macroscopic’, ‘environment’ do not appear. There is only a universal dynamics governing all physical processes, and an ontology which tells how the physical world is, according to the model, independently of any act of observation.
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