We formulate a discrete two-state stochastic process with elementary rules that give rise to Born statistics and reproduce the probabilities from the Schrödinger equation under an associated Hamiltonian matrix, which we identify. We define the probability to observe a state, classical or quantum, in proportion to the number of \textit{events} at that state--number of ways a walker may materialize at a point of observation at time t through a sequence of transitions starting from known initial state at t=0. The quantum stochastic process differs from its classical counterpart in that the quantum walker is a pair of qubits, each transmitted independently through all possible paths to a point of observation, and whose recombination may produce a positive or negative event (classical events are never negative). We represent the state of the walker via a square matrix of recombination events, show that the indeterminacy of the qubit state amounts to rotations of this matrix, and show that the Born rule counts the number qubits on this matrix that remain invariant over a full rotation.
We formulate a discrete two-state stochastic process with elementary rules that give rise to Born statistics and reproduce the probabilities from the Schrödinger equation under an associated Hamiltonian matrix, which we identify. We define the probability to observe a state, classical or quantum, in proportion to the number of \textit{events} at that state--number of ways a walker may materialize at a point of observation at time t through a sequence of transitions starting from known initial state at t=0. The quantum stochastic process differs from its classical counterpart in that the quantum walker is a pair of qubits, each transmitted independently through all possible paths to a point of observation, and whose recombination may produce a positive or negative event (classical events are never negative). We represent the state of the walker via a square matrix of recombination events, show that the indeterminacy of the qubit state amounts to rotations of this matrix, and show that the Born rule counts the number qubits on this matrix that remain invariant over a full rotation.
Abstract It was recently demonstrated that the Born rule for transition probabilities can be modeled using the linear Schrödinger equation with a Hamiltonian represented by a random matrix from the Gaussian unitary ensemble. The matrices representing the Hamiltonian at different time points throughout the observation period are assumed to be independent. Instead of suppressing superpositions, such Schrödinger evolution makes the state perform an isotropic random walk on the projective space of states. The relative frequency of reaching different eigenstates of an arbitrary observable in the random walk is shown to satisfy the Born rule. Here, we apply this methodology to investigate the behavior of a particle in the context of the double-slit experiment with measurement, offering new insights into its potential underlying mechanisms.
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.
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.
A physical experiment comprises along the time trajectory a start, a time evolution (duration), and an end, which is the measurement. In non relativistic quantum mechanics the start of the experiment is defined by the wave function at time 0 taking into account the starting conditions, the evolution is described by the wave function following the Schrödinger equation and the measurement by the Born rule. While the Schrödinger equation is deterministic, it is the Born rule that makes quantum mechanics statistical with all its consequences. The nature of the Born rule is thereby unknown albeit necessary since it produces the correct ensemble averaged measures of the experiment. Here, it is demonstrated that the origin of the Born rule is the projection from the quantum frame (i.e. wave description) to the classical mechanics frame (i.e. particle description) described by a Ehrenfest theorem-oriented Fourier transformation. The statistical averaging over many measurements is necessary in order to eliminate the unknown initial and end time coordinate of the experiment in reference to the beginning of the universe.
Abstract In terms of its time trajectory, a physical experiment is comprised of a preparation time, a start time, a time evolution (duration), and an end time (point at which the measurement actually occurs). In non-relativistic quantum mechanics the start of the experiment is described by the wave function at time t = 0 taking into account the starting conditions, the evolution is described by the wave function according to the Schrödinger equation and the measurement is described by the Born rule. While the Schrödinger equation is deterministic, it is the Born rule that renders quantum mechanics stochastic with all its consequences it entails. Still, the nature of the Born rule is unknown yet it is useful since it produces the correct ensemble averaged measures of the experiment. Here, it is demonstrated that the origin of the Born rule is the projection from the quantum frame (i.e. wave description) to the classical mechanics frame (i.e. particle description) described by a Ehrenfest theorem-oriented Fourier transformation. The statistical averaging over many measurements is therefore only necessary in order to account for the unknown preparation time and thus starting time of the experiment because only the experimental duration (time between start time and end time) can be known.
The Born rule is a rule that a probability we observe a small particle like an electron is proportional to the square of the absolute value of the wave function. In this paper, we try to derive the Born rule from the many-worlds interpretation. Many researchers have tried to derive the Born rule (also called Born's rule, Born's law, or probability interpretation) from Many-Worlds Interpretation (MWI). However, no one succeeds. Thus, the derivation of Born’s rule had become an important issue for MWI. We try to derive Born’s rule by introducing an elementary event of probability theory to the quantum theory as a new method. We interpret the wave function as a manifold like a three-dimensional sphere, and interpret the absolute value of the wave function as the surface area of the manifold. We suppose that the manifold exists in the discrete space that has lattice points. We interpret a point on the surface of the manifold as a state that we cannot divide any more, an elementary state. We draw an arrow from any point to any point. We interpret an arrow as an event that we cannot divide any more, an elementary event. Probability is proportional to the number of elementary events, and the number of elementary events is the square of the number of elementary states. The number of elementary states is proportional to the surface area of the manifold, and the surface area of the manifold is the absolute value of the wave function. Therefore, the probability is proportional to the square of the absolute value of the wave function.