One attractive interpretation of quantum mechanics is the ensemble interpretation, where Quantum Mechanics merely describes a statistical ensemble of objects and not individual objects. But this interpretation does not address why the wave-function plays a central role in the calculations of probabilities, unlike most other interpretations of quantum mechanics. We prove: 1) the wave-function is a parametrization of any probability distribution of a statistical ensemble: there is a surjective map from an hypersphere to the set of all probability distributions; 2) for a quantum system defined in a 2-dimensional real Hilbert space, the role of the (2-dimensional real) wave-function is identical to the role of the Euler's formula in engineering, while the collapse of the wave-function is identical to selecting the real part of a complex number; 3) the collapse of the wave-function of any quantum system is a recursion of collapses of 2-dimensional real wave-functions. The wave-function plays a central role because it is a good parametrization that allows us to represent a group of transformations using linear transformations of the hypersphere. It is precisely the fact that the hypersphere is not the phase-space of the theory that implies the collapse of the wave-function. Without collapse, the wave-function parametrization would be inconsistent.
One attractive interpretation of quantum mechanics is the ensemble interpretation, where Quantum Mechanics merely describes a statistical ensemble of objects and not individual objects. But this interpretation does not address why the wave-function plays a central role in the calculations of probabilities, unlike most other interpretations of quantum mechanics. We prove: 1) the wave-function is a parametrization of any probability distribution of a statistical ensemble: there is a surjective map from an hypersphere to the set of all probability distributions; 2) for a quantum system defined in a 2-dimensional real Hilbert space, the role of the (2-dimensional real) wave-function is identical to the role of the Euler's formula in engineering, while the collapse of the wave-function is identical to selecting the real part of a complex number; 3) the collapse of the wave-function of any quantum system is a recursion of collapses of 2-dimensional real wave-functions. The wave-function plays a central role because it is a good parametrization that allows us to represent a group of transformations using linear transformations of the hypersphere. It is precisely the fact that the hypersphere is not the phase-space of the theory that implies the collapse of the wave-function. Without collapse, the wave-function parametrization would be inconsistent.
One attractive interpretation of quantum mechanics is the ensemble interpretation, where Quantum Mechanics merely describes a statistical ensemble of objects and not individual objects. But this interpretation does not address why the wave-function plays a central role in the calculations of probabilities, unlike most other interpretations of quantum mechanics. On the other hand, Classical Statistical Field Theory suffers from severe mathematical inconsistencies (specially for Hamiltonians which are non-polynomial in the fields, e.g. General relativistic statistical field theory). We claim that both problems are related to each other and we propose a solution to both. We prove: 1) the wave-function is a parametrization of any probability distribution of a statistical ensemble: there is a surjective map from an hypersphere to the set of all probability distributions; 2) for a quantum system defined in a 2-dimensional real Hilbert space, the role of the (2-dimensional real) wave-function is identical to the role of the Euler's formula in engineering, while the collapse of the wave-function is identical to selecting the real part of a complex number; 3) the collapse of the wave-function of any quantum system is a recursion of collapses of 2-dimensional real wave-functions; 4) the wave-function parametrization is key in the mathematical definition of Classical Statistical Field Theory we propose here. The same formalism is applied to Quantum Yang-Mills theory and Quantum Gravity in another article.
It is shown that the wave function describes the state of the statistical ensemble E[S] of individual particles, or the statistical average particle . This result follows from the fact that in the classical limit h=0 the Schroedinger equation turns to the dynamic equations for the statistical ensemble of classical particles. The statement that the wave function describes the state of an individual particle is incompatible with the quantum mechanics formalism. It is shown that the statistical interpretation of quantum mechanics is a corollary of the fact, that the QM formalism is the technique of the statistical ensemble description, restricted by constraints of the dynamic equation linearity.
Whether wave function collapses or not is a major remaining question in the theory of quantum measurement. This difficulty stems from following two facts. First, it has not been recognized that single-particle quantum mechanics and many-particle quantum mechanics must be treated separately. Second, quantum jump (QJ) and wave function collapse (WFC) need clearer definitions. We define a QJ as a process of selecting a set of system eigenvalues (SEVs) of an observable and a WFC as a process of determining the probability distribution (PD) of SEVs, both from a single measurement. The goal of quantum observation is to obtain the PD, which is determined from an ensemble of SEVs. The wave function becomes an observable when the PD is determined. In single-particle quantum mechanics, a single measurement results in only one set of SEVs and the PD is not observable. Therefore the WFC does not happen. In many-particle quantum mechanics, we focus on the occupation number of a singe quantum state. The wave function does not collapse in general, but there are exceptions. The occupation number can be huge and macroscopic for photons or for Bose-Einstein condensates. In such a case, the PD is determined from a single measurement of a real ensemble and the WFC occurs. We call it a macroscopic quantum jump, which effectively is a measurement of a classical observable.
A new type of representation of the wave function of quantum mechanics, and a new form of the statistical interpretation of the wave function, are presented. The variable in the new representation is a mapping of points of differential space, closely related to Hilbert space, but having an associated measure for suitably defined sets of points in it. In the new statistical interpretation, an eigenvalue of every observable defined in the system is associated with each point in differential space. This can be done in such a way that, for each observable, the measure (=probability) of the set of all differential-space points belonging to a given set of eigenvalues is equal to the quantum-mechanical probability, calculated in the usual way (Born statistical postulate), that an experiment will yield an eigenvalue in this set. Thus we obtain an interpretation of quantum mechanics in terms of probability densities or probabilities instead of probability amplitudes.
Whether the wave function collapses or not is a major remaining question in the theory of quantum measurement. This difficulty stems from the following two facts. First, it has not been recognized that single-particle quantum mechanics and many-particle quantum mechanics must be treated separately. Second, quantum jump (QJ) and wave function collapse (WFC) need clearer definitions. We define a QJ as a process of selecting a set of system eigenvalues (SEVs) of an observable and a WFC as a process of determining the probability distribution (PD) of SEVs, both from a single measurement. The goal of quantum observation is to obtain the PD, which is determined from an ensemble of SEVs. The wave function becomes an observable when the PD is determined. In single-particle quantum mechanics, a single measurement results in only one set of SEVs, and the PD is not observable. Therefore, the WFC does not happen. In many-particle quantum mechanics, we focus on the occupation number of a single quantum state. The wave function does not collapse in general, but there are exceptions. The occupation number can be huge and macroscopic for photons or for Bose-Einstein condensates. In such a case, the PD is determined from a single measurement of a real ensemble, and the WFC occurs. We call it a macroscopic quantum jump, which effectively is a measurement of a classical observable.
Whether the wave function collapses or not is a major remaining question in the theory of quantum measurement. This difficulty stems from the following two facts. First, it has not been recognized that single-particle quantum mechanics and many-particle quantum mechanics must be treated separately. Second, quantum jump (QJ) and wave function collapse (WFC) need clearer definitions. We define a QJ as a process of selecting a set of system eigenvalues (SEVs) of an observable and a WFC as a process of determining the probability distribution (PD) of SEVs, both from a single measurement. The goal of quantum observation is to obtain the PD, which is determined from an ensemble of SEVs. The wave function becomes an observable when the PD is determined. In single-particle quantum mechanics, a single measurement results in only one set of SEVs, and the PD is not observable. Therefore, the WFC does not happen. In many-particle quantum mechanics, we focus on the occupation number of a single quantum state. The wave function does not collapse in general, but there are exceptions. The occupation number can be huge and macroscopic for photons or for Bose-Einstein condensates. In such a case, the PD is determined from a single measurement of a real ensemble, and the WFC occurs. We call it a macroscopic quantum jump, which effectively is a measurement of a classical observable.