Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2112.10022 · Foundations of Physics

Probabilities and Certainties Within a Causally Symmetric Model

Roderick I. Sutherland

Published Jul 7, 2022• 6 citations
arXivHTMLPDF

Abstract

Abstract This paper is concerned with the causally symmetric version of the familiar de Broglie–Bohm interpretation, this version allowing the spacelike nonlocality and the configuration space ontology of the original model to be avoided via the addition of retrocausality. Two different features of this alternative formulation are considered here. With regard to probabilities, it is shown that the model provides a derivation of the Born rule identical to that in Bohm’s original formulation. This derivation holds just as well for a many-particle, entangled state as for a single particle. With regard to “certainties”, the description of a particle’s spin is examined within the model and it is seen that a statistical description is no longer necessary once final boundary conditions are specified in addition to the usual initial state, with the particle then possessing a definite (but hidden) value for every spin component at intermediate times. These values are consistent with being the components of a single, underlying spin vector. The case of a two-particle entangled spin state is also examined and it is found that, due to the retrocausal aspect, each particle possesses its own definite spin during the entanglement, independent of the other particle. In formulating this picture, it is demonstrated how such a realistic model can preserve Lorentz invariance in the face of Bell’s theorem and avoid the need for a preferred reference frame.

DOI
10.1007/s10701-022-00573-x

Related papers

Causally Symmetric Bohm Model

Roderick I. Sutherland

A version of Bohm's model incorporating retrocausality is presented, the aim being to explain the nonlocality of Bell's theorem while maintaining Lorentz invariance in the underlying ontology. The strengths and weaknesses of this alternative model are compared with those of the standard Bohm model.

arXiv:quant-ph/0601095•Jan 15, 2006•7 citations

Realism and causality. I. Pilot wave and retrocausal models as possible facilitators

Eliahu Cohen, Marina Cortês, Avshalom C. Elitzur +1 more

Of all basic principles of classical physics, realism should arguably be the last to be given up when seeking a better interpretation of quantum mechanics. We examine the de Broglie-Bohm pilot wave theory as a well-developed example of a realistic theory. We present three challenges to a naive reading of pilot wave theory, each based on a system of several entangled particles. With the help of a coarse graining of pilot wave theory into a discrete system, we show how these challenges can be answered. However, this comes with a cost. In the description of individual systems, particles appear to scatter off empty branches of the wave function as if they were particles and conversely travel through particles as if they were waves. More generally, the ``particles'' of pilot wave theory are led by the guidance equation to move in ways no classical particle would, involving apparent violations of the principles of inertia and momentum conservation. We next argue that the aforementioned cost can be avoided within a retrocausal model. In the proposed version of the pilot wave theory, the particle is guided by a combination of advanced and retarded waves. The resulting account for quantum physics seems to have greater heuristic power, demands less damage to intuition, and moreover provides some general hints regarding spacetime and causality. This is the first of two papers. In the second [E. Cohen, M. Cort\^es, A. C. Elitzur, and L. Smolin, Realism and causality. II. Retrocausality in energetic causal sets, Phys. Rev. D 102, 124028 (2020).], we show that, in the context of an explicit model, retrocausality, with respect to an effective, emergent spacetime metric, can coexist with a strict irreversibility of causal processes.

arXiv:1902.05108•Physical review. D/Physical review. D.•Dec 9, 2020•25 citations

Is there really “retrocausation” in time-symmetric approaches to quantum mechanics?

Ruth E. Kastner

Time-symmetric interpretations of quantum theory are often presented as featuring “retrocausal” effects in addition to the usual forward notion of causation. This paper examines the ontological implications of certain time-symmetric theories, and finds that no dynamical notion of causation applies to them, either forward or backward. It is concluded that such theories actually describe a static picture, in which the notion of causation is relegated to a descriptor of static relationships among events. In addition, these theories lead to an epistemic rather than ontologically referring, realist view of quantum states.

arXiv:1607.04196•AIP conference proceedings•Jan 1, 2017•7 citations

Conditional expectations in Quantum Mechanics and causal interpretations: the Bohm momentum as a best predictor

Raymond Brummelhuis

Given a normalized state-vector $ψ$, we define the conditional expectation $\mathbb{E }_{ψ} (A | B ) $ of a Hermitian operator $A $ with respect to a strongly commuting family of self-adjoint operators $B $ as the best approximation, in the operator mean square norm associated to $ψ$, of $A $ by a real-valued function of $B . $ A fundamental example is the conditional expectation of the momentum operator $P $ given the position operator $X $, which is found to be the Bohm momentum. After developing the Bohm theory from this point of view we treat conditional expectations with respect to general $B $, which we apply to non-relativistic spin 1/2-particles. We derive the dynamics of the conditional expectations of momentum and spin with respect to position and the third spin component. These dynamics can be interpreted in terms of classical continuum mechanics as a two-component fluid whose components carry intrinsic angular momentum. Interpreting the joint spectrum of the conditioning operators as a space of beables, we can introduce a classical-stochastic particle dynamics on this space which is compatible with the time-evolution of the Born probability, by combining the de Broglie-Bohm guidance condition with a Markov jump process, following an idea of J. Bell. This results in a new Bohm-type model for particles with spin. A basic problem is that such auxiliary particle dynamics are far from unique. We finally examine the relation of our conditional expectations with the conditional expectations of the theory of $C^* $-algebras and, as an application, derive a general evolution equation for conditional expectations for operators acting on finite dimensional Hilbert spaces. Two appendices re-interpret the classical Bohm model as an integrable constrained Hamiltonian system, and provide the details of the two-fluid interpretation.

arXiv:2411.08532•Nov 13, 2024•0 citations

Invalidity of the Standard Locality Condition for Hidden-Variable Models of EPR-Bohm Experiments

James L. Beck

John Bell and others used a locality condition to establish inequalities that they believe must be satisfied by any local hidden-variable model for the spin probability distribution for two entangled particles in a Bohm-EPR experiment. We show that this condition is invalid because it contradicts the product rule of probability theory for any model that exhibits the expected property of perfect correlation. This breaks the connection between Bell inequalities and the existence of any local hidden-variable model of interest. As already known, these inequalities give necessary conditions for the existence of third/fourth-order joint probability distributions for the spin outcomes from three/four separate Bohm-EPR experimental set-ups that are consistent with the second-order joint spin distributions for each experiment after marginalization. If a Bell inequality is violated, as quantum mechanics theory predicts and experiments show can happen, then at least one third-order joint probability is negative. However, this does not imply anything about the existence of local hidden-variable models for the second-order joint probability distributions for the spin outcomes of a single experiment. The locality condition does seem reasonable under the widely-applied frequentist interpretation of the spin probability distributions that views them as real properties of a random process that are manifested through their relative frequency of occurrence, which gives conditioning in the probabilities for the spin outcomes a causal role. In contrast, under the Bayesian interpretation of probability, probabilistic conditioning on one particle's spin outcome in the product rule is viewed as information to make probabilistic predictions of the other particle's spin outcome. There is nothing causal and so no reason to develop a locality condition.

•Oct 1, 2019•1 citation

Is There Really "Retrocausation" in Time-Symmetric Approaches to Quantum Mechanics?

Ruth E. Kastner

Time-symmetric interpretations of quantum theory are often presented as featuring "retrocausal" effects in addition to the usual forward notion of causation. This paper examines the ontological implications of certain time- symmetric theories, and finds that no dynamical notion of causation applies to them, either forward or backward. It is concluded that such theories actually describe a static picture, in which the notion of causation is relegated to a descriptor of static relationships among events. In addition, these theories lead to an epistemic rather than ontologically referring, realist view of quantum states.

•Jul 14, 2016•7 citations
Abstract This paper is concerned with the causally symmetric version of the familiar de Broglie–Bohm interpretation, this version allowing the spacelike nonlocality and the configuration space ontology of the original model to be avoided via the addition of retrocausality. Two different features of this alternative formulation are considered here. With regard to probabilities, it is shown that the model provides a derivation of the Born rule identical to that in Bohm’s original formulation. This derivation holds just as well for a many-particle, entangled state as for a single particle. With regard to “certainties”, the description of a particle’s spin is examined within the model and it is seen that a statistical description is no longer necessary once final boundary conditions are specified in addition to the usual initial state, with the particle then possessing a definite (but hidden) value for every spin component at intermediate times. These values are consistent with being the components of a single, underlying spin vector. The case of a two-particle entangled spin state is also examined and it is found that, due to the retrocausal aspect, each particle possesses its own definite spin during the entanglement, independent of the other particle. In formulating this picture, it is demonstrated how such a realistic model can preserve Lorentz invariance in the face of Bell’s theorem and avoid the need for a preferred reference frame.
Of all basic principles of classical physics, realism should arguably be the last to be given up when seeking a better interpretation of quantum mechanics. We examine the de Broglie-Bohm pilot wave theory as a well-developed example of a realistic theory. We present three challenges to a naive reading of pilot wave theory, each based on a system of several entangled particles. With the help of a coarse graining of pilot wave theory into a discrete system, we show how these challenges can be answered. However, this comes with a cost. In the description of individual systems, particles appear to scatter off empty branches of the wave function as if they were particles and conversely travel through particles as if they were waves. More generally, the ``particles'' of pilot wave theory are led by the guidance equation to move in ways no classical particle would, involving apparent violations of the principles of inertia and momentum conservation. We next argue that the aforementioned cost can be avoided within a retrocausal model. In the proposed version of the pilot wave theory, the particle is guided by a combination of advanced and retarded waves. The resulting account for quantum physics seems to have greater heuristic power, demands less damage to intuition, and moreover provides some general hints regarding spacetime and causality. This is the first of two papers. In the second [E. Cohen, M. Cort\^es, A. C. Elitzur, and L. Smolin, Realism and causality. II. Retrocausality in energetic causal sets, Phys. Rev. D 102, 124028 (2020).], we show that, in the context of an explicit model, retrocausality, with respect to an effective, emergent spacetime metric, can coexist with a strict irreversibility of causal processes.
Time-symmetric interpretations of quantum theory are often presented as featuring “retrocausal” effects in addition to the usual forward notion of causation. This paper examines the ontological implications of certain time-symmetric theories, and finds that no dynamical notion of causation applies to them, either forward or backward. It is concluded that such theories actually describe a static picture, in which the notion of causation is relegated to a descriptor of static relationships among events. In addition, these theories lead to an epistemic rather than ontologically referring, realist view of quantum states.
Given a normalized state-vector ψψψ, we define the conditional expectation $\mathbb{E }_{ψ} (A | B ) $ of a Hermitian operator $A $ with respect to a strongly commuting family of self-adjoint operators $B $ as the best approximation, in the operator mean square norm associated to ψψψ, of $A $ by a real-valued function of $B . $ A fundamental example is the conditional expectation of the momentum operator $P $ given the position operator $X $, which is found to be the Bohm momentum. After developing the Bohm theory from this point of view we treat conditional expectations with respect to general $B $, which we apply to non-relativistic spin 1/2-particles. We derive the dynamics of the conditional expectations of momentum and spin with respect to position and the third spin component. These dynamics can be interpreted in terms of classical continuum mechanics as a two-component fluid whose components carry intrinsic angular momentum. Interpreting the joint spectrum of the conditioning operators as a space of beables, we can introduce a classical-stochastic particle dynamics on this space which is compatible with the time-evolution of the Born probability, by combining the de Broglie-Bohm guidance condition with a Markov jump process, following an idea of J. Bell. This results in a new Bohm-type model for particles with spin. A basic problem is that such auxiliary particle dynamics are far from unique. We finally examine the relation of our conditional expectations with the conditional expectations of the theory of $C^* $-algebras and, as an application, derive a general evolution equation for conditional expectations for operators acting on finite dimensional Hilbert spaces. Two appendices re-interpret the classical Bohm model as an integrable constrained Hamiltonian system, and provide the details of the two-fluid interpretation.
John Bell and others used a locality condition to establish inequalities that they believe must be satisfied by any local hidden-variable model for the spin probability distribution for two entangled particles in a Bohm-EPR experiment. We show that this condition is invalid because it contradicts the product rule of probability theory for any model that exhibits the expected property of perfect correlation. This breaks the connection between Bell inequalities and the existence of any local hidden-variable model of interest. As already known, these inequalities give necessary conditions for the existence of third/fourth-order joint probability distributions for the spin outcomes from three/four separate Bohm-EPR experimental set-ups that are consistent with the second-order joint spin distributions for each experiment after marginalization. If a Bell inequality is violated, as quantum mechanics theory predicts and experiments show can happen, then at least one third-order joint probability is negative. However, this does not imply anything about the existence of local hidden-variable models for the second-order joint probability distributions for the spin outcomes of a single experiment. The locality condition does seem reasonable under the widely-applied frequentist interpretation of the spin probability distributions that views them as real properties of a random process that are manifested through their relative frequency of occurrence, which gives conditioning in the probabilities for the spin outcomes a causal role. In contrast, under the Bayesian interpretation of probability, probabilistic conditioning on one particle's spin outcome in the product rule is viewed as information to make probabilistic predictions of the other particle's spin outcome. There is nothing causal and so no reason to develop a locality condition.
Time-symmetric interpretations of quantum theory are often presented as featuring "retrocausal" effects in addition to the usual forward notion of causation. This paper examines the ontological implications of certain time- symmetric theories, and finds that no dynamical notion of causation applies to them, either forward or backward. It is concluded that such theories actually describe a static picture, in which the notion of causation is relegated to a descriptor of static relationships among events. In addition, these theories lead to an epistemic rather than ontologically referring, realist view of quantum states.