The auxiliary rules of quantum mechanics have always included the Born rule that connects probability with square modulus. This need not be the case, for it is possible to introduce probability into the theory through probability current alone. When this is done, other rules can provide for stochastically triggered measurements within a system of any size, microscopic or macroscopic; and solutions to the Schrodinger equation can be consistently applied to individual trials, not just to ensembles of trials. Other advantages appear. The rules can then resolve the paradox associated with the Schrodinger cat experiment, and remove the possibility of the many world thesis of Everett. As a result, the system can accommodate any conscious observer, including the principal investigator who cannot otherwise be included in a quantum mechanical system.
The auxiliary rules of quantum mechanics have always included the Born rule that connects probability with square modulus. This need not be the case, for it is possible to introduce probability into the theory through probability current alone. When this is done, other rules can provide for stochastically triggered measurements within a system of any size, microscopic or macroscopic; and solutions to the Schrodinger equation can be consistently applied to individual trials, not just to ensembles of trials. Other advantages appear. The rules can then resolve the paradox associated with the Schrodinger cat experiment, and remove the possibility of the many world thesis of Everett. As a result, the system can accommodate any conscious observer, including the principal investigator who cannot otherwise be included in a quantum mechanical system.
Abstract I provide a simple derivation of the Born rule as giving a classical probability, that is, the ratio of the measure of favorable states of the system to the measure of its total possible states. In classical systems, the probability is due to the fact that the same macrostate can be realized in different ways as a microstate. Despite the radical differences between quantum and classical systems, I show that the same can be applied to quantum systems, and the result is the Born rule. This works only if the basis is continuous (an eigenbasis of observables with continuous spectra), but all known physically realistic measurements involve a continuous basis (the position basis). The continuous basis is not unique, and for subsystems it depends on the observable. But for the entire universe, there are continuous bases that give the Born rule for all measurements, because all measurements reduce to distinguishing macroscopic pointer states, and macroscopic observations commute. This allows for the possibility of a unique ontic basis for the entire universe. In the wavefunctional formulation, the basis can be chosen to consist of classical field configurations, and the coefficients Ψ[ϕ] Ψ [ ϕ ] can be made real by absorbing them into a global U(1) gauge. For the many-worlds interpretation, this result gives the Born rule from micro-branch counting.
Traditionally, some version of Born’s rule is always considered to be an indispensable part of any interpretation of quantum mechanics, either as a postulate, or as a result derived-not always on the basis of convincing reasoning-from other postulates. In this section, we have a close look at the possible forms of Born’s rule and discuss the limits of its validity. All traditional foundations of quantum mechanics heavily depend on the concept of (hypothetical, idealized) experiments-far too heavily. This is one of the reasons why these foundations are still unsettled, over 90 years after the discovery of the basic equations for modern quantum mechanics. No other theory has such controversial foundations. The main reason is that the starting point of the usual interpretations is an idealization of the measurement process that is taken too seriously, namely as the indisputable truth about everything measured. But in reality, this idealization is only a didactical trick for the newcomer to make the formal definitions of quantum mechanics a bit easier to swallow. Except in a few very simple cases, it is too far removed from experimental practice to tell much about real measurements, and hence about how quantum physics is used in real applications. In experimental physics, measurement is a very complex thing, far more complex than Born’s rule (the usual starting point) suggests. To measure the distance between two galaxies, the mass of the top quark, or the Lamb shift, just to mention three basic examples, cannot be captured by the idealistic measurement concept used there, nor by any of the refinements of it discussed in the literature. In each of the three cases mentioned, one assembles a lot of auxiliary information and ultimately calculates the measurement result from a best fit of a model to the data. Clearly, the theory must already be in place in order to do that. We do not even know what a top quark should be, whose mass we are measuring, unless we have a theory that tells us this! The two most accurately determined observables in the history of quantum physics, namely the anomalous magnetic moment of the electron and Lamb shift, are not even q-observables! To present the stage for the criticism of Born’s rule in Section 14.3, we first need to clarify the meaning of the term “Born’s rule”. To distinguish different useful meanings, we look in Sections 14.1-14.2 at the early history of Born’s rule.
In order to make the quantum mechanics a closed theory one has to derive the Born rule from the first principles, like the Schroedinger equation (SEq), rather than postulate it. The Born rule was in certain sense derived in several articles, e.g. in [1] and [2]. In this work some arguments of previous authors are simplified and made more “physical”. It is shown how to derive the Born rule using the conservation of quantum state norm (Ψ|Ψ) that is the unitary evolution property determined by the SEq. It is this property that makes the probability equal to the square of the amplitude modulus. Possible modification of the Born rule is briefly discussed in the case when the evolution of the system is described by a nonlinear Schroedinger equation. We also present arguments in the spirit of the Many-World Interpretation to explain the origin of probabilistic behavior. Simply speaking, the randomness appears as a result of representing the wave function by using a detector of discrete nature that is found only in one state at a time, out of two or more possible states.
In this treatise I introduce the time dependent Generalized Born's Rule for the probabilities of quantum events, including conditional and consecutive probabilities, as the unique fundamental time evolution equation of quantum theory. Then these probabilities, computed from states and events, are to be compared with relative frequencies of observations. Schrodinger's equation still is valid in one model of the axioms of quantum theory, which I call the Schrodinger model. However, the role of Schrodinger's equation is auxiliary, since it serves to help compute the continuous temporal evolution of the probabilities given by the Generalized Born's Rule. In other models, such as the Heisenberg model, the auxiliary equations are quite different, but the Generalized Born's Rule is the same formula (covariance) and gives the same results (invariance). Also some aspects of the Schrodinger model are not found in the isomorphic Heisenberg model, and they therefore do not have any physical significance. One example of this is the infamous collapse of the quantum state. Other quantum phenomena, such as entanglement, are easy to analyze in terms of the Generalized Born's Rule without any reference to the unnecessary concept of collapse. Finally, this leads to the possibility of quantum theory with other sorts of auxiliary equations instead of Schrodinger's equation, and examples of this are given. Throughout this treatise the leit motif is the central importance of quantum probability and most especially of the simplifying role of the time dependent Generalized Born's Rule in quantum theory.
The basic idea of quantum mechanics is that the property of any system can be in a state of superposition of various possibilities (or eigen states).This state of superposition is also known as wave function and it evolves linearly with time in a deterministic way in accordance with the Schrodinger equation.However, when a measurement is carried out on the system to determine the value of that property (say position), the system instantaneously transforms to one of the eigen states and thus we get only a single value as an outcome of the measurement.Quantum measurement problem seeks to find the cause and exact mechanism governing this transformation.In an attempt to solve the above problem, in this paper, we will first define what the wave function represents in real-world and will identify the root cause behind the stochastic nature of events.Then, we will develop a model to explain the mechanism of collapse of the quantum mechanical wave function in response to a measurement.In the process of development of model, we will explain Schrodinger cat paradox and will show how Born's rule for probability becomes a natural consequence of the measurement process.
The Born rule asserts the probability distribution of eigenstates observed in unbiased quantum measurements, but the reason it holds remains elusive. This manuscript discusses how the Born rule might be explained by Schrodinger equation dynamics, if a measurement comprises a system responding to random fluctuations until it is within an arbitrarily small tolerance of a measurement eigenstate. We describe the random walk dynamics that produce this behavior in terms of a class of time-dependent, stochastic unitary matrices U(t). We also discuss the class of stochastic potential energies in the Schrodinger equation that is equivalent to this class of unitary matrices. This analysis raises some questions worth considering, including how to determine if any measurements actually follow the predicted random walk mechanism and whether a reliable measurement apparatus could be designed that deviates from Born rule probabilities. Interestingly, if any measurements do follow this random walk mechanism, then exposing a quantum system to stochastic 'noise' is an intrinsic part of such a measurement, not merely an unwanted side effect. This characteristic would have implications for reducing noise in quantum sensing and quantum computing. This is a draft of a work in progress. Questions and suggestions are welcome.