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1 Rescuing the Born Rule for Quantum Cosmology

Joshua H. Cooperman

Published Jan 1, 2011• 2 citations
OpenAlex

Abstract

Page has recently argued that the Born rule does not suffice for computing all probabilities in quantum cosmology. He further asserts that the Born rule's failure gives rise to the cosmological measure problem. Here I contend that Page's result stems from his use of an overly restrictive definition of the Born rule. In particular, I demonstrate that all of the probabilities he wishes to compute follow from the Born rule when generalized measurements are permitted. I also register two comments on Page's theoretical setting, relating respectively to Hilbert space dimensionality and permutation symmetry. These considerations lead me to conclude that the claimed insufficiency of the Born rule is by no means specific to the cosmological context.

Related papers

Rescuing the Born rule for quantum cosmology

Joshua H Cooperman

Page has recently argued that the Born rule does not suffice for computing all probabilities in quantum cosmology. He further asserts that the Born rule's failure gives rise to the cosmological measure problem. Here I contend that Page's result stems from his use of an overly restrictive definition of the Born rule. In particular, I demonstrate that all of the probabilities he wishes to compute follow from the Born rule when generalized measurements are permitted. I also register two comments on Page's theoretical setting, relating respectively to Hilbert space dimensionality and permutation symmetry. These considerations lead me to conclude that the claimed insufficiency of the Born rule is by no means specific to the cosmological context.

arXiv:1010.3395•Journal of Cosmology and Astroparticle Physics•Feb 9, 2011•3 citations

The Born Rule Without a Measurement Postulate

D. Ring

The Born rule governs the probability of outcomes of measurements of quantum systems. Many attempts have been made to derive the Born rule from other postulates. We perform such an analysis in an Everett interpretation without any measurement postulates at all. We explain that probability is an ill defined concept and that an agent who nevertheless wishes to make approximate predictions will have no alternative measure to weigh the alternatives without subjecting herself to a dutch book. Our demonstration is complete for projective measurements of pure states in a finite dimensional Hilbert space and we discuss how it might be applied to generalized measurements. A similar demonstration is impossible for mixed states. Nevertheless, following the standard convention, probability for mixed states has the same validity and issues as it does in classical physics.

•Jan 23, 2026•0 citations

Born in an infinite universe: A cosmological interpretation of quantum mechanics

Anthony N. Aguirre, Max Tegmark

We study the quantum measurement problem in the context of an infinite, statistically uniform space, as could be generated by eternal inflation. It has recently been argued that when identical copies of a quantum measurement system exist, the standard projection operators and Born rule method for calculating probabilities must be supplemented by estimates of relative frequencies of observers. We argue that an infinite space actually renders the Born rule redundant, by physically realizing all outcomes of a quantum measurement in different regions, with relative frequencies given by the square of the wave-function amplitudes. Our formal argument hinges on properties of what we term the quantum confusion operator, which projects onto the Hilbert subspace where the Born rule fails, and we comment on its relation to the oft-discussed quantum frequency operator. This analysis unifies the classical and quantum levels of parallel universes that have been discussed in the literature, and has implications for several issues in quantum measurement theory. Replacing the standard hypothetical ensemble of measurements repeated ad infinitum by a concrete decohered spatial collection of experiments carried out in different distant regions of space provides a natural context for a statistical interpretation of quantum mechanics. It also shows how, even for a single measurement, probabilities may be interpreted as relative frequencies in unitary (Everettian) quantum mechanics. We also argue that after discarding a zero-norm part of the wave function, the remainder consists of a superposition of indistinguishable terms, so that arguably ``collapse'' of the wave function is irrelevant, and the ``many worlds'' of Everett's interpretation are unified into one. Finally, the analysis suggests a ``cosmological interpretation'' of quantum theory in which the wave function describes the actual spatial collection of identical quantum systems, and quantum uncertainty is attributable to the observer's inability to self-locate in this collection.

arXiv:1008.1066•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Nov 3, 2011•55 citations

Born's Rule Is Insufficient in a Large Universe

Don Nelson Page

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 various options for the average density matrix of a region with an observation. The question of what replacement to use is part of the measure problem in cosmology.

arXiv:1003.2419•Mar 11, 2010•12 citations

The Born rule fails in cosmology

Don Nelson Page

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 fails 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.

arXiv:0903.4888•Journal of Cosmology and Astroparticle Physics•Jul 3, 2009•44 citations

The Born Rule as Constraint Measure: Dissolving the Probability Problem in Quantum Mechanics

Ryuhei ISHIBASHI

The Born rulethe prescription that measurement probabilities equal the squared mod- ulus of wave function amplitudeshas resisted satisfactory explanation for nearly a century. We argue that this diculty stems from a hidden assumption: that probability names a causal concept requiring causal explanation. Drawing on causal eliminativism, we reinterpret the Born rule not as describing the probabilistic eects of measurement, but as specifying a measure on the space of constraint-satisfying congurations. From this perspective, Glea- son's theorem does not derive probability from quantum mechanics; rather, it identies |ψ|2 as the unique coherent measure compatible with Hilbert space geometry. The question Why does measurement cause probabilistic outcomes? dissolves: there is no causation, hence no probabilistic causation, hence no puzzle. What remains is a geometric fact about constraint structure

•Jan 15, 2026•0 citations
Page has recently argued that the Born rule does not suffice for computing all probabilities in quantum cosmology. He further asserts that the Born rule's failure gives rise to the cosmological measure problem. Here I contend that Page's result stems from his use of an overly restrictive definition of the Born rule. In particular, I demonstrate that all of the probabilities he wishes to compute follow from the Born rule when generalized measurements are permitted. I also register two comments on Page's theoretical setting, relating respectively to Hilbert space dimensionality and permutation symmetry. These considerations lead me to conclude that the claimed insufficiency of the Born rule is by no means specific to the cosmological context.
Page has recently argued that the Born rule does not suffice for computing all probabilities in quantum cosmology. He further asserts that the Born rule's failure gives rise to the cosmological measure problem. Here I contend that Page's result stems from his use of an overly restrictive definition of the Born rule. In particular, I demonstrate that all of the probabilities he wishes to compute follow from the Born rule when generalized measurements are permitted. I also register two comments on Page's theoretical setting, relating respectively to Hilbert space dimensionality and permutation symmetry. These considerations lead me to conclude that the claimed insufficiency of the Born rule is by no means specific to the cosmological context.
The Born rule governs the probability of outcomes of measurements of quantum systems. Many attempts have been made to derive the Born rule from other postulates. We perform such an analysis in an Everett interpretation without any measurement postulates at all. We explain that probability is an ill defined concept and that an agent who nevertheless wishes to make approximate predictions will have no alternative measure to weigh the alternatives without subjecting herself to a dutch book. Our demonstration is complete for projective measurements of pure states in a finite dimensional Hilbert space and we discuss how it might be applied to generalized measurements. A similar demonstration is impossible for mixed states. Nevertheless, following the standard convention, probability for mixed states has the same validity and issues as it does in classical physics.
We study the quantum measurement problem in the context of an infinite, statistically uniform space, as could be generated by eternal inflation. It has recently been argued that when identical copies of a quantum measurement system exist, the standard projection operators and Born rule method for calculating probabilities must be supplemented by estimates of relative frequencies of observers. We argue that an infinite space actually renders the Born rule redundant, by physically realizing all outcomes of a quantum measurement in different regions, with relative frequencies given by the square of the wave-function amplitudes. Our formal argument hinges on properties of what we term the quantum confusion operator, which projects onto the Hilbert subspace where the Born rule fails, and we comment on its relation to the oft-discussed quantum frequency operator. This analysis unifies the classical and quantum levels of parallel universes that have been discussed in the literature, and has implications for several issues in quantum measurement theory. Replacing the standard hypothetical ensemble of measurements repeated ad infinitum by a concrete decohered spatial collection of experiments carried out in different distant regions of space provides a natural context for a statistical interpretation of quantum mechanics. It also shows how, even for a single measurement, probabilities may be interpreted as relative frequencies in unitary (Everettian) quantum mechanics. We also argue that after discarding a zero-norm part of the wave function, the remainder consists of a superposition of indistinguishable terms, so that arguably ``collapse'' of the wave function is irrelevant, and the ``many worlds'' of Everett's interpretation are unified into one. Finally, the analysis suggests a ``cosmological interpretation'' of quantum theory in which the wave function describes the actual spatial collection of identical quantum systems, and quantum uncertainty is attributable to the observer's inability to self-locate in this collection.
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 various options for the average density matrix of a region with an observation. The question of what replacement to use is part of the measure problem in cosmology.
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 fails 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.
The Born rulethe prescription that measurement probabilities equal the squared mod- ulus of wave function amplitudeshas resisted satisfactory explanation for nearly a century. We argue that this diculty stems from a hidden assumption: that probability names a causal concept requiring causal explanation. Drawing on causal eliminativism, we reinterpret the Born rule not as describing the probabilistic eects of measurement, but as specifying a measure on the space of constraint-satisfying congurations. From this perspective, Glea- son's theorem does not derive probability from quantum mechanics; rather, it identies |ψ|2 as the unique coherent measure compatible with Hilbert space geometry. The question Why does measurement cause probabilistic outcomes? dissolves: there is no causation, hence no probabilistic causation, hence no puzzle. What remains is a geometric fact about constraint structure