Quantum mechanics may be formulated as Sensible Quantum Mechanics (SQM) so that it contains nothing probabilistic except conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued awareness operators. Ratios of the measures for these sets of perceptions can be interpreted as frequency- type probabilities for many actually existing sets. These probabilities gener- ally cannot be given by the ordinary quantum “probabilities” for a single set of alternatives. Probabilism, or ascribing probabilities to (...) unconscious aspects of the world, may be seen to be an aesthemamorphic myth. (shrink)
Probabilities in quantum theory are traditionally given by Born’s rule as the expectation values of projection operators. Here it is shown that Born’s rule is insufficient in universes so large that they contain identical multiple copies of observers, because one does not have definite projection operators to apply. Possible replacements for Born’s rule include using the expectation value of various operators that are not projection operators, or using vari-.
A simple proof is given that the probabilities of observations in a large universe are not given directly by Born’s rule as the expectation values of projection operators in a global quantum state of the entire universe. An alternative procedure is proposed for constructing an averaged density matrix for a random small region of the universe and then calculating observational probabilities indirectly by Born’s rule as conditional probabilities, conditioned upon the existence of an observation.
The Born rule may be stated mathematically as the rule that probabilities in quantum theory are expectation values of a complete orthogonal set of projection operators. This rule works for single laboratory settings in which the observer can distinguish all the different possible outcomes corresponding to the projection operators. However, theories of inflation suggest that the universe may be so large that any laboratory, no matter how precisely it is defined by its internal state, may exist in a large number (...) of very distantly separated copies throughout the vast universe. In this case, no observer within the universe can distinguish all possible outcomes for all copies of the laboratory. Then normalized probabilities for the local outcomes that can be locally distinguished cannot be given by the expectation values of any projection operators. Thus the Born rule dies and must be replaced by another rule for observational probabilities in cosmology. The freedom of what this new rule is to be is the measure problem in cosmology. A particular volume-averaged form is proposed. (shrink)
The Jarzynski equality equates the mean of the exponential of the negative of the work (per fixed temperature) done by a changing Hamiltonian on a system, initially in thermal equilibrium at that temperature, to the ratio of the final to the initial equilibrium partition functions of the system at that fixed temperature. It thus relates two thermal equilibrium quantum states.
A motivation is given for expressing classical mechanics in terms of diagonal projection matrices and diagonal density matrices. Then quantum mechanics is seen to be a simple generalization in which one replaces the diagonal real matrices with suitable Hermitian matrices.
Here I shall call elements (1)-(3) the quantum state (or the “state”), since they give the quantum state of the universe that obeys the dynamical laws and is written in terms of the kinematic variables, and I shall call elements (4)-(6) the probability rules (or the “rules”), since they specify what it is that has probabilities (here taken to be the results of observations, Oj, or “observations” for short), the rules for extracting these observational probabilities from the quantum state, and (...) the meaning of the probabilities. What I shall write below is largely independent of the meaning of the probabilities, though personally I view them in a rather Everettian way as objective measures for the set of observations with positive probabilities. Usually it is implicitly believed that the observational probabilities depend strongly upon the quantum state. (Sometimes the Everett interpretation  is taken to mean that all of physical reality is determined purely by the quantum state, without the need for any additional rules to extract probabilities, but this extreme view seems untenable  and will not be adopted here. Instead, I shall discuss the opposite view, that the probabilities are independent of the quantum state.) However, some advocates of inflation[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] often claim that our observations do not depend upon the quantum state at all, but rather that inflation acts as an attractor to give the same statistical distribution of observations from any state. In this note, I shall use the framework of state plus rules to discuss this possibility that observational probabilities might be independent of the quantum state. I shall show that this indeed is logically possible, but apparently only if the probability rules are rather ad hoc. If indeed the rules are this ad hoc, so that the probabilities of our observations do not depend upon a quantum state at all, it would seem to leave it mysterious why many of our observations can be simply interpreted as if our universe really were quantum.. (shrink)
Often it is assumed that a quantum state or a phase-space distribution must be normalizable. Here it is shown that even if it is not normalizable, one may be able to extract normalized observational probabilities from it.