In the standard interpretation of quantum mechanics, the state is described by an abstract wave function in the representation space. Conversely, in a realistic interpretation, the quantum state is replaced by a probability distribution of physical quantities. Bohm mechanics is a consistent example of realistic theory, where the wave function and the particle positions are classically defined quantities. Recently, we proved that the probability distribution in a realistic theory cannot be a quadratic function of the quantum state, in contrast to the apparently obvious suggestion given by the Born rule for transition probabilities. Here, we provide a simplified version of this proof.
In the standard interpretation of quantum mechanics, the state is described by an abstract wave function in the representation space. Conversely, in a realistic interpretation, the quantum state is replaced by a probability distribution of physical quantities. Bohm mechanics is a consistent example of realistic theory, where the wave function and the particle positions are classically defined quantities. Recently, we proved that the probability distribution in a realistic theory cannot be a quadratic function of the quantum state, in contrast to the apparently obvious suggestion given by the Born rule for transition probabilities. Here, we provide a simplified version of this proof.
Position probabilities play a privileged role in the interpretation of quantum mechanics. The quantum wave function (usually designated “Ψ(r)’) cannot itself represent a physical quantity, because it is complex valued. Its square — |Ψ(r)|2 — however, is real; and it is |Ψ(r)|2 which is the fundamental interpreted quantity of the theory. Schrödinger, who originated wave mechanics, interpreted |Ψ(r)|2 as the density of a charged particle; but this suggestion was inadequate. A principal reason is that, for most physical systems, |Ψ(r)|2 spreads out in time, whereas particles themselves do not. Schrödinger's interpretation was replaced by Born who introduced probabilities into quantum mechanics. Born urged that |Ψ(r)|2 represents not the density of the system in space, but rather a probability density. This is almost universally how quantum mechanics is taught today: |Ψ(r)|2 gives the probability that a system in state Ψ(r) is located at r.
The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born's rule and recently suggested method of dequantizer-quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical-like equations for the probability distributions determining the quantum system states. Relations to phase-space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.
The description of quantum states by probability distributions of classical-like random variables associated with observables is presented. An invertible map of the wave functions and density matrices onto the probability distributions is constructed. The relation of the probability distributions to quasidistributions like the Wigner function is discussed. The interference phenomenon and superposition principle of pure quantum states are given in the form of nonlinear addition of the probabilities identified with the quantum states. The probability given by Born’s rule is expressed as a function of the probabilities describing the system states. The suggested probability representation of quantum mechanics is presented using examples of harmonic oscillators and qubits.
Even after a hundred years of quantum theory, the true meaning of the quantum state has not yet been clearly understood. Does a pure quantum state, described by a wave function φ, represent a real physical quantity, comparable to position and momentum in classical mechanics, or does it, even as a pure state, only have a statistical significance, similar to a probability distribution? In this paper, we show that there is no contradiction between these two interpretations. Two concepts must be distinguished here: on the one hand, physical objects, which have a finite lifetime, and on the other hand, subsystems (i.e., tensor factors) of the Hilbert space of the universe. A complete description of an object at a point in time t is given by a linear subspace A of its Hilbert space, which, in the case of dim A = 1, can be specified by a wave function φ as A = [φ]. However, for every subsystem Q, for each point in time t, and for every condition B, the conditional quantum state W = W(Q,t|B) can be defined as a statistical operator on the Hilbert space of Q. Using W, certain (conditional) probabilities can be calculated. In the case of rank W = 1, the operator W can also be specified by a wave function φ, namely as W = |φ><φ|. It is the subsystems of the Hilbert space that play the role of "quantum systems" in the Copenhagen formalism, and the quantum states defined for them behave exactly as described by this formalism. If a wave function φ represents the description of a (quantum) object, then we are dealing with a real physical property, that is, φ plays an "ontic" role. However, if φ represents a quantum state, then its role is merely "statistical". If, in this case, condition B corresponds to the knowledge of a subject, φ plays an "epistemic" role. Additional Remarks The reasoning in this paper is largely based on a new systematic approach to solving the interpretation problem of quantum theory, which was originally published in 2018 (in German) under the title "Realistische Quantentheorie". A comprehensive account of this approach (in English) was published in February 2026 under the title "Realistic Quantum Theory: Solving the Interpretation Problem". It is available in the Zenodo repository (zenodo.18644694, see link below under "Related works" / "References"). The present document provides an English version of an earlier paper in German under the same title, which is also available in the Zenodo repository (zenodo.10694813, see link below).
The basic notion of physical system states is different in classical statistical mechanics and in quantum mechanics. In classical mechanics, the particle system state is determined by its position and momentum; in the case of fluctuations, due to the motion in environment, it is determined by the probability density in the particle phase space. In quantum mechanics, the particle state is determined either by the wave function (state vector in the Hilbert space) or by the density operator. Recently, the tomographic-probability representation of quantum states was proposed, where the quantum system states were identified with fair probability distributions (tomograms). In view of the probability-distribution formalism of quantum mechanics, we formulate the superposition principle of wave functions as interference of qubit states expressed in terms of the nonlinear addition rule for the probabilities identified with the states. Additionally, we formulate the probability given by Born’s rule in terms of symplectic tomographic probability distribution determining the photon states.
We argue that quantum mechanics makes sense without such controversial postulates as the wave function collapse, the quantum probability rule and the observable postulate. We only need the existence of a wave function as a representation of a state, its dynamical evolution rule and another rather trivial postulate proposed here (Read-Off Postulate). The wave function collapse and the probability rule are replaced with intrinsic properties of the theory, i.e. decoherence and relative frequency, respectively. Interpretation of the wave function as a faithful representation of the reality naturally emerges, although quantum reality has a peculiar feature which distinguishes it from classical reality.