Born Rule Research Articles

Lindbladian-Induced Alignment in Quantum Measurements

An expression of the Lindbladian form is proposed that ensures an unambiguous time-continuous reduction of the initial system-pointer wave-packet to one in which the readings and the observable's values are aligned, formalized as the transition from an outer product to an inner product of the system's and apparatus' density matrices. The jump operators are in the basis of the observables, with uniquely determined parameters derived from the measurement set-up (thereby differing from S. Weinberg's Lindbladian resolution of wave-packet formalism) and conforming to Born's probability rules. The novelty lies in formalising the adaptability of the surroundings (including the measuring device) to the mode of observation. Accordingly, the transition is of finite duration (in contrast to its instantaneousness in the von Neumann's formulation). This duration is estimated for a simple half-spin-like model.

New considerations on a spiritual difficulty for public health, including questions about quantum probabilities and comprehensibility of worlds

In a previous study a spiritual difficulty for public health was considered, that many people say that there would be definitely no afterlife with Judgment Day, and that some such persons might sometimes try to harm innocent others secretly to obtain advantages, e.g., financial, with negative impact on public health. Considering additional knowledge, new thought experiments are conducted, including arbitrary infinite bitstrings as programs, run, e.g., on theoretical Turing machines, and mathematical structures providing generalized findings, including results for questions of quantum probabilities, e.g., the Born rule - which is experimentally confirmed. In quantum processes, randomness instead of pseudorandomness is more probable than within the less general structures of the previous study. This is achieved designing generalized structures, e.g., programs, which do not function when they are “alone”. Additional structures show that pseudorandomness remains relevant. A paradox of the extreme variety of incomprehensible universes is described. It could be solved here considering further structures, e.g., the extremely large number of, e.g., world part copies simulated by various short programs, providing a large probability for comprehensible worlds. In all these mathematical structures, aliens’ pains emerge naturally.

Can Pragmatist Quantum Realism Explain Protective Measurements?

According to Healey’s pragmatist quantum realism, the only physical properties of quantum systems are those to which the Born rule assigns probabilities. In this paper, I argue that this approach to quantum theory fails to explain the results of protective measurements.

Beyond the Born Rule in Quantum Gravity

We have recently developed a new understanding of probability in quantum gravity. In this paper we provide an overview of this new approach and its implications. Adopting the de Broglie–Bohm pilot-wave formulation of quantum physics, we argue that there is no Born rule at the fundamental level of quantum gravity with a non-normalisable Wheeler–DeWitt wave functional $$\Psi$$ . Instead the universe is in a perpetual state of quantum nonequilibrium with a probability density $$P\ne \left| \Psi \right| ^{2}$$ . Dynamical relaxation to the Born rule can occur only after the early universe has emerged into a semiclassical or Schrödinger approximation, with a time-dependent and normalisable wave functional $$\psi$$ , for non-gravitational systems on a classical spacetime background. In that regime the probability density $$\rho$$ can relax towards $$\left| \psi \right| ^{2}$$ (on a coarse-grained level). Thus the pilot-wave theory of gravitation supports the hypothesis of primordial quantum nonequilibrium, with relaxation to the Born rule taking place soon after the big bang. We also show that quantum-gravitational corrections to the Schrödinger approximation allow quantum nonequilibrium $$\rho \ne \left| \psi \right| ^{2}$$ to be created from a prior equilibrium ( $$\rho =\left| \psi \right| ^{2}$$ ) state. Such effects are very tiny and difficult to observe in practice.

Quantum Theory of the Classical: Einselection, Envariance, Quantum Darwinism and Extantons.

Core quantum postulates including the superposition principle and the unitarity of evolutions are natural and strikingly simple. I show that-when supplemented with a limited version of predictability (captured in the textbook accounts by the repeatability postulate)-these core postulates can account for all the symptoms of classicality. In particular, both objective classical reality and elusive information about reality arise, via quantum Darwinism, from the quantum substrate. This approach shares with the Relative State Interpretation of Everett the view that collapse of the wavepacket reflects perception of the state of the rest of the Universe relative to the state of observer's records. However, our "let quantum be quantum" approach poses questions absent in Bohr's Copenhagen Interpretation that relied on the preexisting classical domain. Thus, one is now forced to seek preferred, predictable, hence effectively classical but ultimately quantum states that allow observers keep reliable records. Without such (i) preferred basis relative states are simply "too relative", and the ensuing basis ambiguity makes it difficult to identify events (e.g., measurement outcomes). Moreover, universal validity of quantum theory raises the issue of (ii) the origin of Born's rule, pk=|ψk|2, relating probabilities and amplitudes (that is simply postulated in textbooks). Last not least, even preferred pointer states (defined by einselection-environment-induced superselection)-are still quantum. Therefore, unlike classical states that exist objectively, quantum states of an individual system cannot be found out by an initially ignorant observer through direct measurement without being disrupted. So, to complete the 'quantum theory of the classical' one must identify (iii) quantum origin of objective existence and explain how the information about objectively existing states can appear to be essentially inconsequential for them (as it does for states in Newtonian physics) and yet matter in other settings (e.g., thermodynamics). I show how the mathematical structure of quantum theory supplemented by the only uncontroversial measurement postulate (that demands immediate repeatability-hence, predictability) leads to preferred states. These (i) pointer states correspond to measurement outcomes. Their stability is a prerequisite for objective existence of effectively classical states and for events such as quantum jumps. Events at hand, one can now enquire about their probability-the probability of a pointer state (or of a measurement record). I show that the symmetry of entangled states-(ii) entanglement-assisted invariance or envariance-implies Born's rule. Envariance also accounts for the loss of phase coherence between pointer states. Thus, decoherence can be traced to symmetries of entanglement and understood without its usual tool-reduced density matrices. A simple and manifestly noncircular derivation of pk=|ψk|2 follows. Monitoring of the system by its environment in course of decoherence typically leaves behind multiple copies of its pointer states in the environment. Only pointer states can survive decoherence and can spawn such plentiful information-theoretic progeny. This (iii) quantum Darwinism allows observers to use environment as a witness-to find out pointer states indirectly, leaving systems of interest untouched. Quantum Darwinism shows how epistemic and ontic (coexisting in epiontic quantum state) separate into robust objective existence of pointer states and detached information about them, giving rise to extantons-composite objects with system of interest in the core and multiple records of its pointer states in the halo comprising of environment subsystems (e.g., photons) which disseminates that information throughout the Universe.

Possibilities for probabilities

In ordinary situations involving a small part of the universe, Born's rule seems to work well for calculating probabilities of observations in quantum theory. However, there are a number of reasons for believing that it is not adequate for many cosmological purposes. Here a number of possible generalizations of Born's rule are discussed, explaining why they are consistent with the present statistical support for Born's rule in ordinary situations but can help solve various cosmological problems.

An atomistic entropy based finite element multiscale method for modeling amorphous materials

A new concurrent multiscale method based on the maximum entropy statistical method is proposed for the analysis of amorphous materials. In addition to reducing the number of degrees of freedom, any irregular structure of amorphous materials can be accurately analyzed. The amorphous structure is generated from a solid crystalline structure by a heating/cooling process without the need for any specific independent technique to create such a random structure. The method is expected to perform efficiently because of its entropic and irregular intrinsic. Regions with moderate conditions are discretized by the entropy-based finite element method while the severe parts are simulated by the present atomistic-based multiscale technique. The new proposed approach allows for accurate analysis of amorphous structures across multiple scales and does not suffer from conventional complications such as the standard Cauchy Born rule and consistency of the molecular structure with the standard finite element geometries.The conventional Cauchy-Born rule cannot be directly used due to the non–crystalline microstructure of the material. A remedy is proposed based on the meshfree techniques by constructing a continuous atomic deformation field from the imposed macro deformation gradient. The resultant deformation gradient and the stress field remain consistent in micro scales. In addition, a genetic algorithm-based method, which has less sensitivity to the choice of initial point and number of parameters, is adopted for the maximization of the entropy function.The silicon amorphous structure is considered for MD simulations. It is obtained by quenching from a melted sample. The MD-obtained structure is further analyzed and the predicted displacements and stress contours, as well as the density and radial distribution functions are examined to assess the state of the material. Then, the proposed meshfree technique is applied to construct the continuous form of the deformation gradient on the MD model to improve the accuracy of the solution. The proposed concurrent multiscale method is verified and then employed to simulate an amorphous silicon specimen. Finally, the effects of sample size, strain rate and quenching speed on rupture stress and strain in different 3D tensile simulations are investigated by the proposed multiscale method.

A Unifying Theory for Quantum Physics, Part 1:

Is the quantum world as strange as they say? If this were an unsolved mathematics question, we might try a new angle of attack. We know quantum mechanics (QM) is the most accurate and productive science humans ever had, meaning its probability predictions are accurate. Every probability has two square roots. The Born rule says either would produce the same probability. Assume nature uses the negative of QM’s equations. What could that mean? We’d need to revise Feynman’s path-integrals and Schrödinger’s equation. If waves travel in the opposite direction as what QM believes, that could produce the negative equations. No wave-particle duality. Free particles would follow backwards zero-energy waves coming from detectors. This, surprisingly, gets rid of quantum weirdness. Our proposal is that nature uses the negative of QM’s equations because particles follow zero-energy waves backwards. Considerable evidence fits this model, including a neutron-interferometer and the Davisson-Germer experiments, a quantum-eraser experiment, Wheeler-gedanken and double-slit experiments, Bell-test experiments, Stern-Gerlach, and high-energy scattering experiments. Finally, we propose a plan for how to motivate students to want to study quantum technologies, thereby addressing the most prominent problem in QM today: the shortage of an educated workforce, the scarcity of aspiring students.

Extending Kolmogorov's Axioms for a Generalized Probability Theory on Collections of Contexts.

Kolmogorov’s axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Aufféves and Grangier, which in turn are inspired by Gleason’s theorem.

Quantum-Mechanical Description of Physical Reality Shall Be Considered Complete

After the deduction and comparison of wave motions of classical particles and quantum particles, we prove that there are two restrictions applied to both classical particles and quantum particles. Both restrictions are just to ensure energy conservation principle of the wave motion. The Heisenberg’s famous inequality was nothing to do with the measurements of particle’s position or mentum, the restriction of the inequality is just the lower limit of the wave motion. His uncertainty principle was not correct from both mathematical and physical perspective. Based on the mathematic analysis and logic reasoning process review, we conclude that all the Heisenberg’s uncertainty principle, Born rule and Bohr’s complimentary principle were not correct. Even for a quantum particle, at any time, it must be classically localized instead of non-local. Without the Copenhagen interpretations, the quantum mechanics shall still be considered complete according to EPR’s criteria which are used to verify the completeness of a physics theory. The particle-wave duality of quantum particles comes from their inner features of electric vector(s).

Quantum conditional probabilities

We investigate the consistency of conditional quantum probabilities. This is whether there is compatibility between the Kolmogorov–Bayes conditional probabilities and the Born rule. We show that they are not compatible in the sense that there are situations where there is no legitimate density matrix that may reproduce the conditional statistics of the other observable via the Born rule. This is to say that the Gleason theorem does not apply to conditional probabilities. Moreover, we show that when this occurs the joint statistics is nonclassical. We show that conditional probabilities are not equivalent to state reduction, so these results do not affect the validity of the Lüders expression.

A Unifying Theory for Quantum Physics, Part 2:

The quantum world is allegedly strange. But is it? What if there’s a simple mathematical explanation, and a simple solution? The success of quantum mechanics (QM) arises from the accuracy of its probability predictions, which are obtained by squaring amplitudes (the Born rule). Suppose for a moment that nature uses the negative of QM’s equations. When squared they would yield the same probabilities, confirmed by the same experiments and technological triumphs. If that were true, if nature uses the negative of QM’s equations, then the quantum world would become transparent, easy to understand. No more Schrödinger’s-cat. No quantum-eraser. No backwards-in-time cause-and-effect. No paradoxes nor enigmas. But, what’s a negative quantum equation? It could mean that particles follow zero-energy waves backwards, instead of forwards. That’s still an eccentric idea, a residual strangeness. Overall, it’s a bargain. We could swap one odd idea for another, because wave-particle duality is odd. We are accustomed to wave-particle duality. But we’ll show that experiments support the other arrangement: quantum particles follow zero-energy waves backwards. How could that possibly be true? We present substantial experimental evidence, new mathematics, and six dozen colorful illustrations. This is the Theory of Elementary Waves (TEW).

Securing the objectivity of relative facts in the quantum world

This paper compares and contrasts relational quantum mechanics (RQM) with a pragmatist view of quantum theory (DP). I first explain important points of agreement. Then I point to two problems faced by RQM and sketch DP?s solutions to analogous problems. Since both RQM and DP have taken the Born rule to require relative facts I next say what these might be. My main objection to RQM as originally conceived is that its ontology of relative facts is incompatible with scientific objectivity and undercuts the evidential base of quantum theory. In contrast DP?s relative facts have all the objectivity we need to accept quantum theory as scientific knowledge. But a very recent modification to RQM has successfully addressed my main objection,bringing the two views into even closer alignment.

A modified applied element model for the simulation of plain concrete behaviour

A modified applied element model to simulate the behaviour of plain concrete continuum structures including discrete cracking is proposed in this study. In the classical applied element model, Poisson effects are fully ignored. To remediate this issue, diagonal elements are introduced to include the Poisson effect, and the constitutive parameters are rigorously determined using the Cauchy–Born rule and the hyper-elastic theory. The formulation is validated for linear elastic problems and the consistency and convergence behaviour of the numerical approach is shown. Tensile softening formulation using the concept of fracture energy is utilised for the nonlinear range. In this range, the approach is validated using the classical benchmark tests with pure tensile, split-tensile, combined shear-tensile and bending dominated push-over loading. The load–displacement behaviour and crack response were captured successfully, showing the proposed methodology can be used to quantify discrete cracks on large systems, such as dam monoliths, from initiation to significant damage levels.

Epistemic Separability and Everettian Branches: A Critique of Sebens and Carroll

Abstract We discuss the proposal by Sebens and Carroll to derive the Born rule in Everettian quantum mechanics from a principle they call ‘ESP-QM’. We argue that the proposal fails: ESP-QM is not, as Sebens and Carroll argue, a ‘less general version’ of an independently plausible principle, ESP, and can only be motivated by the empirical success of quantum mechanics, including use of the Born rule. Therefore, ESP-QM cannot have the status of a meta-theoretical principle of reasoning and provides no viable basis for deriving the Born rule.

A computational framework to obtain nonlinearly elastic constitutive relations of special Cosserat rods with surface energy

We present a computational framework to obtain nonlinearly elastic constitutive relations of special Cosserat rods with surface energy. The framework allows the rod’s cross-section to have arbitrary shape and the rod material to obey arbitrary three-dimensional material model for its bulk volume and arbitrary surface energy model for its lateral surface. The kinematics of the Helical Cauchy–Born rule is used to first construct a family of six-parameter (corresponding to the six strain measures of rod theory) helical rod configurations in which the generated three-dimensional strain field is uniform in the rod’s arc-length coordinate. This uniformity allows the three-dimensional nonlinear equations of elasticity to reduce to just the rod’s cross-sectional plane wherein the cross-section’s boundary also experiences traction due to the inherent surface energy in the rod’s lateral surface. The solution of this cross-sectional problem yields the induced in-plane displacement and out-of-plane warping in the cross-section for arbitrarily prescribed strain measures of the rod. We then derive expressions for the induced internal contact force, moment and stiffnesses of the rod in terms of the solution of the deformed cross-section and the prescribed strains of the rod. A finite element formulation is presented to solve the nonlinear cross-sectional deformation problem and further obtain the induced internal contact force, moment and stiffnesses. The presented framework will be useful for modeling of nanorods where surface energy plays a dominant role. Several numerical examples are presented using the presented framework to illustrate the effect of surface energy parameters on the deformation of the nanorod’s cross-section and the nanorod’s bending, torsional, extensional and shearing stiffnesses. We also derive improved analytical formulas for the extensional and bending stiffnesses of isotropic rectangular nanorods in their reference state. These formulas reduce to the existing widely used formulas for a special choice of material parameters, i.e., when the surface Poisson’s ratio and the bulk Poisson’s ratio match thus highlighting the limitation of the existing formulas.

Bohmian Mechanics is Not Deterministic

I argue that Bohmian mechanics (or any similar pilot-wave theory) cannot reasonably be claimed to be a deterministic theory. If one assumes the “quantum equilibrium distribution” provided by the wave function of the universe, Bohmian mechanics requires an external random oracle in order to describe the (Kolmogorov–Levin–Chaitin) algorithmic randomness properties of typical outcome sequences of long runs of repeated identical experiments (which provably follow from the Born rule). This oracle lies beyond the scope of Bohmian mechanics (or any deterministic extension thereof), including the impossibility of explaining the randomness property in question from “random” initial conditions. Thus the advantages of Bohmian mechanics over other interpretations of quantum mechanics, if any, must lie at an ontological level, and in its potential to derive the quantum equilibrium distribution and hence the Born rule.

Probabilities and Certainties Within a Causally Symmetric Model

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.

Positron position operators. I. A natural option

By “position operators,” we mean here a POVM (positive- operator-valued measure) on a suitable configuration space acting on a suitable Hilbert space that serves as defining the position observable of a quantum theory, and by “positron position operators,” we mean a joint treatment of positrons and electrons. We consider the standard free second-quantized Dirac field in Minkowski space–time or in a box. On the associated Fock space (i.e., the tensor product of the positron Fock space and the electron Fock space), there acts an obvious POVM Pobv, but we propose a different one that we call the natural POVM, Pnat. In fact, it is a PVM (projection-valued measure); it captures the sense of locality corresponding to the field operators Ψs(x) and to the algebra of local observables. The existence of Pnat depends on a mathematical conjecture which at present we can neither prove nor disprove; here we explore consequences of the conjecture. We put up for consideration the possibility that Pnat, and not Pobv, is the physically correct position observable and defines the Born rule for the joint distribution of electron and positron positions. We describe properties of Pnat, including a strict no-superluminal-signaling property, and how it avoids the Hegerfeldt–Malament no-go theorem. We also point out how to define Bohmian trajectories that fit together with Pnat, and how to generalize Pnat to curved space–time.

When Greenberger, Horne and Zeilinger Meet Wigner’s Friend

A general argument is presented against relativistic, unitary, single-outcome quantum mechanics. This is achieved by combining the Wigner’s Friend thought experiment with measurements on a Greenberger–Horne–Zeilinger (GHZ) state, and describing the evolution of the quantum state in various inertial frames. Assuming unitary quantum mechanics and single outcomes, the result is that the Born rule must be violated in some inertial frame: in that frame, outcomes are obtained for which no corresponding term exists in the pre-measurement wavefunction.