In quantum experiments the acquisition and representation of basic experimental information is governed by the multinomial probability distribution. There exist unique random variables, whose standard deviation becomes asymptotically invariant of physical conditions. Representing all information by means of such random variables gives the quantum mechanical probability amplitude and a real alternative. For predictions, the linear evolution law (Schrodinger or Dirac equation) turns out to be the only way to extend the invariance property of the standard deviation to the predicted quantities. This indicates that quantum theory originates in the structure of gaining pure, probabilistic information, without any mechanical underpinning.
In quantum experiments the acquisition and representation of basic experimental information is governed by the multinomial probability distribution. There exist unique random variables, whose standard deviation becomes asymptotically invariant of physical conditions. Representing all information by means of such random variables gives the quantum mechanical probability amplitude and a real alternative. For predictions, the linear evolution law (Schrodinger or Dirac equation) turns out to be the only way to extend the invariance property of the standard deviation to the predicted quantities. This indicates that quantum theory originates in the structure of gaining pure, probabilistic information, without any mechanical underpinning.
In quantum experiments the acquisition and representation of basic experimental information is governed by the multinomial probability distribution. There exist unique random variables, whose standard deviation becomes asymptotically invariant of physical conditions. Representing all information by means of such random variables gives the quantum mechanical probability amplitude and a real alternative. For predictions, the linear evolution law (Schrodinger or Dirac equation) turns out to be the only way to extend the invariance property of the standard deviation to the predicted quantities. This indicates that quantum theory originates in the structure of gaining pure, probabilistic information, without any mechanical underpinning.
As physics searches for invariants in observations, this paper looks for invariants of probabilistic observation without assuming physical structure. Structure emerges from the basic assumption of science that new information shall lead to more accurate knowledge of the invariants. This leads to statistically unique random variables for expressing observed information: Complex probability amplitudes. Predictions are also just random variables computed from observed data, and must become more accurate with more observations as input. This singles out the quantum mechanical superposition principle. The external conditions of a probabilistic experiment can themselves be monitored at the most detailed level, resulting in observation of coicidence probabilities. The invariants of any multi-coincidence experiment are the same as those of a one-event experiment with the same number of possible outcomes. An observable probability turns out to be controllable by two independent experimental conditions, naturally parametrized as a direction in a 3-dimensional space. In summary, the probabilistic paradigm itself defines a unique method of forming concepts and making predictions. The method appears irrefutable within probability, because, whenever a prediction turns out wrong the existence of an as yet unmonitored condition is postulated, and a formal way to incoroporate it is shown. The Hilbert space formalism of quantum theory seems to be isomorphic to this method.
Quantum experiments yield random data. We show that the most efficient way to store this empirical information by a finite number of bits is by means of the vector of square roots of observed relative frequencies. This vector has the unique property that its dispersion becomes invariant of the underlying probabilities, and therefore invariant of the physical parameters. This also extends to the complex square roots, and it remains true under a unitary transformation. This reveals quantum theory as a theory for making predictions which are as accurate as the input information, without any statistical loss. Our analysis also suggests that from the point of view of information a slightly more accurate theory than quantum theory should be possible.
Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, the Born rule, commutation and uncertainty relations, probability density current, momentum operator, and rules for including the scalar and vector potentials and antiparticles can be obtained from the probabilistic description of results of measurement of the space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, the Schrödinger equation, and the Dirac equation are obtained from the requirement of the relativistic invariance of the space-time Fisher information. The limit case of th e δ-like probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Manyparticle systems and the postulates of quantum mechanics are also discussed.
The Question of the Completeness of Quantum Mechanics When a classical system is in a pure state, that is, when its preparation represents maximal information, the value of every physical quantity is uniquely predicted. In other words, classical pure states are dispersion-free, and classical mechanics is deterministic. Only in case the state is nonpure, thus representing nonmaximal information about the system, does probability enter into the theory. Things are different with quantum systems. Even in case the state of the system is pure, it is impossible to predict a definite value of every physical quantity: only the probability distributions of the physical quantities are given. Pure quantum states have dispersion (though, of course, they can be dispersion-free for particular physical quantities), and probability enters into quantum theory at a much more fundamental level. The basic idea of so-called hidden-variable theories is to reject the fundamental role of probability in quantum theory and to argue that probability arises because quantum states, even the pure ones, do not represent the ultimate information about the system. Accordingly, it is conjectured that there are certain hidden variables, not yet subject to experimental detection, which would complete the information carried by the quantum states, thus giving rise to so precise a knowledge of the system that probabilities would disappear and precise values of all physical quantities would be uniquely determined.
Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, commutation and uncertainty relations, probability density current, momentum operator, rules for including the scalar and vector potentials and antiparticles can be obtained from the probabilistic description of results of measurement of the space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrodinger equation and Dirac equation are obtained from the requirement of the relativistic invariance of the space-time Fisher information. The limit case of the delta-like probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Many particle systems and the postulates of quantum mechanics are also discussed.