It is shown that the probability density satisfies a hyperbolic equation of motion with the unique characteristic that in its many-particle form it contains derivatives acting at spatially remote regions. Based on this feature we explore inter-particle correlations and the relation between the quantum equilibrium condition and the permutation invariance of the probability density. Some remarks with respect to the quantum to classical transition are also presented.
It is shown that the probability density satisfies a hyperbolic equation of motion with the unique characteristic that in its many-particle form it contains derivatives acting at spatially remote regions. Based on this feature we explore inter-particle correlations and the relation between the quantum equilibrium condition and the permutation invariance of the probability density. Some remarks with respect to the quantum to classical transition are also presented.
In (1) it is argued that a quantum particle’s probability P(x,t) satisfies a Nelson stochastic equation: d/dt partial P(x,t) + d/dx partial (b(x,t)P(x,t)) = D d/dx d/dx partial P. ((1)) where b(x,t) = hbar/m d/dx partial S(x,t) + 2D d/dx partial ln(R(x,t)) ((2)) with W(x,t)=wavefunction= R(x,t) exp(i S(x,t). Thus both a quantum wavefunction (satisfying the Schrodinger equation) and a separate P(x,t) are present. If P(x,0)= W*(x,0)W(x,0) ((3)), i.e. Born’s rule, then this rule holds for later times. It is, however, suggested in (1) that if P(x,0) not= W*(x,0)W(x,0) then the system may relax over time into a situation for which ((3)) holds. In this note, we argue that for a free particle moving with constant p in one direction, P(x,t)=constant, but there is an underlying two dimensional wavefunction exp(ipx). We ascribe this to two x-probability distributions associated with motion. In particular: v= (xfinal - xintial) / delta t. If xfinal and xinitial are both distributions such that on average v = constant, one needs to give a probability for both i.e. have two dimensions or an ordered pair. The two probability distributions, however, are not independent as they must describe a constant velocity i.e all points of space being the same. This leads to exp(ipx) such that exp(-ipx) exp(ipx) =1 . In other words, one may constrain two probability distributions to yield a deterministic type result for a single particle which appears as a probabilistic result for a steady stream.Born’s Rule automatically applies to a free particle and is not a “postulate” as argued in (1). The form exp(ipx) follows from the Lorentz invariant -Et+px we argue. We further argue that if one creates a linear superposition of exp(ipx)s i.e. W(x)=Sum over p a(p)exp(ipx) then Integral dx W*(x)W(x) is always linked to a steady stream probability picture ie. Sum over p a*(p)a(p) = 1. This is why given a p, one does not know its x point- it is associated with a steady stream. As a result, we argue that there is always a meaning for Born’s rule W*(x,t)W(x,t). As a result P(x,t=0) not= W*(x,t=0)W(x,t=0) must describe a probability which differs from the steady stream scenario linked with exp(ikpx).
It was recently shown that the Born rule for transition probabilities of a microscopic particle can be derived from the linear Schr{\"o}dinger equation with a Hamiltonian represented by random matrices from the Gaussian Unitary Ensemble, taken to be independent at different time points. Under such an evolution, the particle's state undergoes an isotropic random walk on the projective state space, and the relative frequencies of reaching different regions obey the Born rule. In this work, we extend these results to demonstrate how the same framework accounts for the emergence of Newtonian dynamics for macroscopic bodies on a submanifold of state space, thereby providing a unified geometric account of the quantum-classical transition.
What singles out the Born probability distribution in Bohmian mechanics when the same probability assignment must remain consistent across different physical controls? This paper proves a conditional uniqueness theorem for quantum equilibrium in nonrelativistic interacting systems. Under explicit regularity assumptions, a normalized probability assignment depending only on the current wavefunction must equal the Born distribution if it is consistent with Schrödinger evolution and Bohmian guidance across the admitted local scalar potentials. The pair interactions remain fixed, connect the particles, and have genuine mixed spatial dependence. The proof does not require additional controllable many-body potentials. The required controls can be reduced to translates and real amplitudes of a single suitable spatial profile for each particle. The manuscript also provides extensions to effective spin and identical-particle sectors, a separate characterization based on binary-flag heredity, and continuity conditions extending the results to wavefunctions with nodes. Consequences for apparatus records are stated under explicit preparation and measurement-coupling assumptions. The result characterizes the Born law within the stated statistical framework. It does not derive the control-consistency premise from deterministic dynamics or establish relaxation from arbitrary initial distributions. This revised preprint presents the theorem statements and proofs in a self-contained form for independent scholarly assessment.
It was repeatedly underlined in literature that quantum mechanics cannot be considered a closed theory if the Born rule is postulated rather than derived from the first principles. In this paper the Born postulate is derived from time-reversibility of solutions of quantum equations of motion. The Born postulate is obtained as a solution of simple functional equation that takes into account the properties of probability and time-reversibility of the equation of motion. The form of the equation of motion is not specified, therefore, the result obtained is applicable to any type of reversible quantum equation of motion. In the second part of this article it is emphasized that, due to topological limitations, time-reversibility of quantum equations does not automatically result in reversibility of real physical processes even when a single particle is involved.
An alternative formulation of quantum mechanics is proposed based on a principle that can be considered a generalization of the Born rule. The principle is composed of a mathematical expression and an associated interpretation, and establishes a correlation between the positions of a particle at two different times. Under reasonable conditions for the wave function, this correlation implies that the particles follow quasiclassical trajectories. It is also shown that the Born rule is equivalent to a particular case of the evolved principle. These features allow the principle to provide a unified explanation of the results of the statistical experiments and of the quasiclassical macroscopic evolution. There is a strong analogy between the quantum principle and a probabilistic principle which is necessary to derive empirical predictions from the mathematical formalism of probability theory. This principle is referred to by some authors as Cournot's principle, while other authors use the equivalent notion of typicality. In this paper probability theory and quantum mechanics are formulated in such a way as to explicitly include the two principles and to emphasize the very similar conceptual structure of the two theories.
The Born rule — the postulate that measurement probabilities in quantum mechanics are given by ∣ψ∣2 — is typically introduced as an axiom. We show that the functional form of the Born rule is already present in the semiconjugacy between the doubling map on the circle and the logistic map at r=4: the semiconjugacy h(θ)=sin2(θ/2) is identically the Born probability for a real qubit parameter. The exponent 2 in ∣ψ∣2 is forced by the degree of the doubling map, which is the minimal non-trivial degree for a self-map of S1. The semiconjugacy is the unique continuous function satisfying three conditions: evenness (invariance under the Z/2 quotient that destroys phase), the intertwining relation with the logistic map, and homeomorphism on a fundamental domain. We then show that the full complex Born rule arises from the Hopf fibration S3→CP1, which is the natural complexification of the real quotient, and that the complete U(1)-invariant content of the quotient is the density matrix ρ=∣ψ⟩⟨ψ∣. The invariant measure transitions from the arcsine distribution (Beta(21,21)) in the real case to the uniform distribution (Beta(1,1)) in the complex case, with a geometric explanation in terms of the transition from fold to fibration. These results establish that the Born rule's functional form is a consequence of the degree-2 quotient structure rather than an independent postulate.