This is a research project in the foundations of quantum mechanics that focuses on the concept of spontaneous collapse of the wave function, and more specifically on a specific theory that characterizes the collapse that was first put forward by Ghirardi, Rimini, and Weber (GRW) in 1986. Although Quantum mechanics underlies most of modern physics, it is beset with foundational problems; it is very successful in predicting observable phenomena, ye it does not actually say what kind of microscopic events and mechanisms lead to these phenomena. Theories that attempt to complement quantum mechanics and close this gap are called "quantum theories without observers." They do so by replacing Schroedinger's equation by a stochastic time-evolution law for the wave function that implements the concept of collapse of the wave function in a precise way; more to the point, it also implies that macroscopic superpositions such as Schroedinger's cat quickly decay, while the Schroedinger equation remains an excellent approximation for any system comprising only a small number of particles.
This project is an investigation into philosophical aspects of the GRW theory. The first goal, on which the PI and a graduate student will work together, is to mathematically prove statements like the following: It is impossible for inhabitants of a GRW universe to measure the number of collapses that occur in a given system in a given time interval, although, according to the GRW theory, this is a well-defined number. More generally, the statements studied describe limitations to what inhabitants can know about their world. The researchers in this project do epistemology by means of mathematical proofs. Another goal, on which the PI will work with collaborators from physics and philosophy, is to explore and explain the philosophical framework of the GRW theory, in particular in the light of a recent technical development in which the PI was involved: the derivation of an operator formalism, analogous but not identical to the operator formalism of quantum mechanics, encoding the empirical predictions of the GRW theory including small deviations from standard quantum mechanics. The topics to be discussed include the need for a precise ontology in space and time (such as the flash ontology) and the idealizations involved in both operator formalisms.
This project concerns the philosophy of quantum physics. Quantum physics allows for several interpretations as to how nature works, one of which is the Ghirardi-Rimini-Weber (GRW) theory of wave function collapse. The main outcome of the project is a detailed analysis of the limitations to knowledge that the GRW theory entails (joint work of PI with C. W. Cowan); grant money was used particularly to support graduate student C. W. Cowan during this work. By a "limitation to knowledge" we mean that certain facts about the world cannot be discovered or confirmed in an empirical way, no matter how big our effort, including possible future technological advances. For example, a limitation to knowledge is in place if a quantity cannot be measured although it has a well-defined value. The very idea of a limitation to knowledge may seem to go against the principles of science; in particular, it conflicts with (what may be called) the positivistic view of science. However, the existence of limitations to knowledge is a fact (indeed a consequence of quantum mechanics, independently of which interpretation of quantum mechanics we prefer). The specific limitations to knowledge that we have investigated concern the possibility of detecting the objective fact (according to GRW theory) whether or not a collapse has occurred. Our main finding is that it is impossible to detect reliably (or even, in many cases, with any useful probability) whether a collapse has occurred. Our results also apply to interaction-triggered collapse in other interpretations of quantum mechanics. Our results can be expressed as mathematical theorems and yield a rather rich and complex picture about the possibility of detecting collapse. We have found that if a quantum system is in a known pure state psi and, with probability 0<p<1, undergoes collapse relative to some known basis, then no experiment can distinguish with perfect reliability between psi and the collapsed state (i.e., the diagonal matrix with the same diagonal entries as the pure state). We have been able to compute the maximal possible reliability and identify the optimal experiment (as a function of psi, p, and the basis). If psi is random with all possibilities equally probable, then no experiment can yield any information at all about whether collapse has occurred. Concerning the case of unknown psi, we have found that in dimension 2 or for p<0.146, no experiment is more reliable than blind guessing on more than 50% of all quantum states psi. There are other cases in which more than 50% can be reached, but never (we have reason to expect) more than 64%. Further results show that the configuration of the primitive ontology of GRW theories can be measured with macroscopic but not microscopic accuracy. Further outcomes of the project: Another work (jointly of PI with D. Bedingham, D. Duerr, G.C. Ghirardi, S. Goldstein, and N. Zanghi) shows how the law governing the "matter density ontology" (a particular choice of "primitive ontology" in GRW theory) can be made Lorentz invariant. So far, only the "flash ontology" could be made relativistic. Another work (jointly of PI with W. Feldmann) appraises and compares the proposed empirical tests between GRW theory and no-collapse theories; none of the tests that can presently be carried out are decisive, but their relative strength can be quantified in terms of the so-called parameter diagram. These tests are perhaps the most meaningful tests of quantum mechanics because they actually test between two different interpretations of quantum mechanics.
