The most general mathematical law for summing bounded quantities is not the arithmetic law, but a composition law of which the summation law for velocities in special relativity is only one particular example. We believe that this composition law, should also be used in quantum theory. We present it with a few examples, we discuss its physical meaning and we show how it can be useful in quantum theory. We show in particular that its use may open the way for a new interpretation of the use of probabilities in quantum theory.
The most general mathematical law for summing bounded quantities is not the arithmetic law, but a composition law of which the summation law for velocities in special relativity is only one particular example. We believe that this composition law, should also be used in quantum theory. We present it with a few examples, we discuss its physical meaning and we show how it can be useful in quantum theory. We show in particular that its use may open the way for a new interpretation of the use of probabilities in quantum theory.
Mathematical methods in optics show that the overall amplitude resulting from the infinite summation over all the various alternative paths that the light can take inside a Fabry-P\'erot interferometer can directly be obtained by using a complex generalization of Einstein's composition law for parallel velocities. Moreover, this result appears to be quite general. In fact, the reflection coefficient for any number of interfaces or quantum wells can be easily obtained by using the composition law. This suggests that a more general way to obtain the probability amplitude for an event which usually needs infinite summation on virtual paths (as in the Fabry-P\'erot interferometer) is to compose amplitudes following Einstein's addition law. I explore this problem and show how the composition law avoids divergences and guarantees the modulus of any total wave function to be less than (or equal to) unity. By examining its physical meaning in the light of probability theory, I also show that it provides an interesting expression for nonlocality and nonseparability.
This paper gives a brief description of the derivation of a composition law for the propagator of a relativistic particle, in a sum over histories quantization. The extension of this derivation to the problem of finding a composition law for quantum cosmology is also discussed. It has been argued that the sum over histories should be regarded as a more generally applicable method of quantization than the canonical framework, in the sense that the former method may be applied to a broader variety of theories. 1 A case in point is general relativity, which admits, at least formally, a sum over histories quantization, but may not admit a canonical one. The existence of a composition law is closely related to the derivation of a canonical formalismfrom the sum over histories. This observation provides the motivation for our discussion of the derivation of a composition law for the relativistic particle propagator. Although we know that in this particular case, composition laws, of the fo...
In modern physics only relative quantities are considered to have physical significance. For example, position assigned to a system depends on the choice of coordinates, and only relative distances between different systems have physical relevance. On the other hand, in quantum theory the scenario where one system, A, is localised around certain position while another system B is in a spatial superposition is considered to be physically different from the scenario where A is in a superposition, while B is localised. Different physical effects are anticipated in the two scenarios especially when the two systems have widely different masses. Here we show that for any superposed amplitudes that are related by a symmetry transformation, the above scenarios are physically equivalent: probabilities for any measurement are the same whether we assign the superposition state to system A or to system B. More generally, linearity of quantum theory guarantees that if a theory is invariant under some symmetry transformations it is necessarily also invariant under their arbitrary `superpositions'. Thus the notion of a superposition turns out to be relative to the choice of coordinates, once it is realised that relations between coordinates do not need to be classical.
In this paper, we discuss the importance of measurement in quantum mechanics and the so-called measurement problem. Any quantum system can be described as a linear combination of eigenstates of an operator representing a physical quantity; this means that the system can be in a superposition of states that corresponds to different eigenvalues, i.e., different physical outcomes, each one incompatible with the others. The measurement process converts a state of superposition (not macroscopically defined) in a well-defined state. We show that, if we describe the measurement by the standard laws of quantum mechanics, the system would preserve its state of superposition even on a macroscopic scale. Since this is not the case, we assume that a measurement does not obey to standard quantum mechanics, but to a new set of laws that form a “quantum measurement theory”.
Quantum mechanics is basically a mathematical recipe on how to construct physical models. Historically its origin and main domain of application has been in the microscopic regime, although it strictly seen constitutes a general mathematical framework not limited to this regime. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics. The interpretational problems of quantum mechanics are also connected with the interpretation of the concept of probability. In this thesis the use of probability theory as extended logic, in particular in the way it was presented by E. T. Jaynes, will be central. With this interpretation of probabilities they become a subjective notion, always dependent on one's state of knowledge or the context in which they are assigned, which has consequences on how things are to be viewed, understood and tackled in quantum mechanics. For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion, meaning, e.g., that two observers will in general assign different statistical operators to the same phenomenon, which is demonstrated in the papers of the thesis. It is also presented an alternative and conceptually clear approach to the problematic notion of "probabilities of probabilities", which is related to such things as probability distributions on statistical operators. In connection to this, we consider concrete numerical applications of Bayesian quantum state assignment methods to a three-level quantum system, where prior knowledge and various kinds of measurement data are encoded into a statistical operator, which can then be used for deriving probabilities of other measurements. The thesis also offers examples of an alternative quantum state assignment technique, using maximum entropy methods, which in some cases are compared with the Bayesian quantum state assignment methods. Finally, the interesting and important problem whether the statistical operator, or more generally quantum mechanics, gives a complete description of "objective physical reality" is considered. A related concern is here the possibility of finding a "local hidden-variable theory" underlying the quantum mechanical description. There have been attempts to prove that such a theory cannot be constructed, where the most well-known impossibility proof claiming to show this was given by J. S. Bell. In connection to this, the thesis presents an idea for an interpretation or alternative approach to quantum mechanics based on the concept of space-time.