We formulate a quantum theory of the Universe based on Bayesian probability. In this theory, the probability of the Universe is not a frequency probability, which can be obtained by observing experimental results several times, but is a Bayesian probability, which can define a probability of an event that occurs just once. As an example, by applying the quantum theory of the Universe to an action of a scalar field theory in the four dimensions as a toy model for the theory of the Universe, we explicitly obtain the probability of the Universe and the action of matters in the Universe.
We formulate a quantum theory of the Universe based on Bayesian probability. In this theory, the probability of the Universe is not a frequency probability, which can be obtained by observing experimental results several times, but is a Bayesian probability, which can define a probability of an event that occurs just once. As an example, by applying the quantum theory of the Universe to an action of a scalar field theory in the four dimensions as a toy model for the theory of the Universe, we explicitly obtain the probability of the Universe and the action of matters in the Universe.
How can we use quantum theory to model other phenomena of interest to researchers in cognition and decision making? Quantum theory is not easy for researchers in cognition and decision making to accept. In fact, quantum mechanics was not easy for physicists to accept either, but it was forced on them by several paradoxical findings that could not be explained using classical physics. We have a similar problem in cognition and decision making – there are numerous paradoxical findings that just seem irrational according to classic probability theory. For example, under some conditions, people judge the probability of event A and B to be greater than the probability of event B , which is called the conjunction fallacy (Tversky & Kahneman, 1983). Also, under the same conditions, they judge the probability of A or B to be less than the probability of event A (Carlson & Yates, 1989), which is called the disjunction fallacy. In this chapter we examine how quantum probability theory explains these and other puzzling results from human probability judgment research. This chapter has two main parts. In the first part, we use a quantum model to derive qualitative predictions for conjunction errors, disjunction errors, and other closely related findings. The first section provides a general set of predictions that do not depend on specific assumptions about the features used to represent events, and the predictions are parameter free. In the second part, we examine the quantitative predictions of the quantum model for a Bayesian inference task, which we use to explain order effects on inference.
We give an explicit, rigorous framework for calculating quantum probabilities in a model theory of quantum gravity. Specifically, we construct the decoherence functional for the Wheeler-DeWitt quantization of a flat Friedmann-Robertson-Walker cosmology with a free, massless, minimally coupled scalar field, thus providing a complete decoherent histories formulation for this quantum cosmological model. The decoherence functional is applied to study predictions concerning the model's Dirac (relational) observables; the behavior of semiclassical states and superpositions of such states; and to study the singular behavior of quantum Wheeler-DeWitt universes. Within this framework, rigorous formulas are given for calculating the corresponding probabilities from the wave function when those probabilities may be consistently defined, thus replacing earlier heuristics for interpreting the wave function of the universe with explicit constructions. It is shown according to a rigorously formulated standard, and in a quantum-mechanically consistent way, that in this quantization these models are generically singular. Independent of the choice of state we show that the probability for these Wheeler-DeWitt quantum universes to ever encounter a singularity is unity. In addition, the relation between histories formulations of quantum theory and relational Dirac observables is clarified.
Abstract This paper reviews arguments for the claim that all probabilities arising in quantum mechanics are Bayesian degrees of belief. The starting point is the observation that there is a one-to-one map from quantum states to points on the probability simplex. Bayes’s rule remains valid in quantum theory, provided it is kept in mind that the outcomes of quantum measurements have no preassigned values. The probabilities of quantum measurement outcomes depend explicitly on a quantum prior whose general form will be discussed. Similar to the classical case, the quantum de Finetti theorem simplifies the analysis of repeated quantum trials if the prior is exchangeable. The Bayesian approach to quantum theory described here solves the problem of the ‘collapse of the wave function’. It also provides almost trivial explanations of a number of phenomena in quantum information theory, including the no-cloning theorem and quantum teleportation.
Beable-guided quantum theories (BGQT) are generalizations of quantum theory, inspired by Bell's concept of beables. They modify the quantum probabilities for some specified set of fundamental events, histories, or other elements of quasiclassical reality by probability laws that depend on the realized configuration of beables. For example, they may define an additional probability weight factor for a beable configuration, independent of the quantum dynamics. Beable-guided quantum theories can be fitted to observational data to provide foils against which to compare explanations based on standard quantum theory. For example, a BGQT could, in principle, characterize the effects attributed to dark energy or dark matter, or any other deviation from the predictions of standard quantum dynamics, without introducing extra fields or a cosmological constant. The complexity of the beable-guided theory would then parametrize how far we are from a standard quantum explanation. Less conservatively, we give reasons for taking suitably simple beable-guided quantum theories as serious phenomenological theories in their own right. Among these are the possibility that cosmological models defined by BGQT might in fact fit the empirical data better than any standard quantum explanation, and the fact that BGQT suggest potentially interesting nonstandard ways of coupling quantum matter to gravity.