The premise that physical variables must be viewed as random variables, with observed values, assigned as averages over distributions, is shown to yield the formalisms of thermodynamics, Hamiltonian mechanics, and quantum theory. A general mathematical argument first establishes the validity of an embedding of th space of physical variables in a larger manifold of "hidden" variables and a general structure for the distribution of these variables. Then, by consideration of various ways of specifying the preparation of a system for measurement, the formalisms of physics cited above are deduced. It is also shown that the generalized Langevin equation follows from these premises and is applicable to a wide class of phenomena. This formulation yields a new definition for compatible observables as used in quantum theory and demonstrates, in a natural way, that physical laws should be invariant under orthogonal transformations of certain variables, which in one circumstance include the complex orthogonal transformations of the Lorentz form, as in special relativity. In summary, it appears that most of the formal structure of physical theory is a consequence of the statistical nature of physical variables.
The premise that physical variables must be viewed as random variables, with observed values, assigned as averages over distributions, is shown to yield the formalisms of thermodynamics, Hamiltonian mechanics, and quantum theory. A general mathematical argument first establishes the validity of an embedding of th space of physical variables in a larger manifold of "hidden" variables and a general structure for the distribution of these variables. Then, by consideration of various ways of specifying the preparation of a system for measurement, the formalisms of physics cited above are deduced. It is also shown that the generalized Langevin equation follows from these premises and is applicable to a wide class of phenomena. This formulation yields a new definition for compatible observables as used in quantum theory and demonstrates, in a natural way, that physical laws should be invariant under orthogonal transformations of certain variables, which in one circumstance include the complex orthogonal transformations of the Lorentz form, as in special relativity. In summary, it appears that most of the formal structure of physical theory is a consequence of the statistical nature of physical variables.
This essay begins with considerations of the relational structure of a set of variables taken to represent a conceptual universe. A plausible argument is used to establish the necessity for a statistical description of any nonstatic set of such variables. A particular statistical description is then constructed based on a partition of the set of variables into two statistically independent sets. This statistical description is then shown to yield equations formally equivalent to familiar equations of thermodynamics, quantum mechanics, and classical mechanics. This mathematical development is strongly suggestive of a unified way of looking at several divergent fields of physics.
It is a scientific truism nowadays that quantum mechanical predictions are of statistical or rather of probabilistic nature: a quantum theory assigns a probability distribution to each measurable quantity (observable) of a physical system prepared according to a given preparation prescription (state). Such sets of probability distributions represent the most general quantitative predictions of physical theories. The aim of this study is to determine all theoretical frameworks which yield such a kind of output taking the probabilistic nature of quantitative physical predictions as initial hypothesis. Astonishingly, this project succeeds with a minimal number of additional assumptions, which arise quite naturally in the course of the analysis, and which essentially amount to regularity conditions. The result is that there exist only two classes of physical theories providing quantitative probabilistic predictions: classical theories where observables are represented by real functions over a phase space and states are represented by probability measures upon the phase space, and quantum theories where observables are represented by selfadjoint operators acting on a complex Hilbert space and states are represented by state operators. On empirical grounds, classical theories cannot provide the all-encompassing, fundamental frame for the mathematical description of physical systems. Quantum theories remain, and there is no more complete formal frame for quantitative physical theories. As an additional result, it is shown that there exist physical systems which possess a quantum mechanical but no classical description; a prominent example is the spin system of two spin-1/2 particles. The classical hidden variables model, which J. Bell relates to this system in order to prove his theorem, cannot be a true description of it because no such classical model exists, and Bell's theorem loses its foundation.
Abstract It is well known that a new form of statistical mechanics has been recently developed by Einstein for an ideal gas of structureless mass-points. This starts from a discussion by Bose of the laws of temperature radiation based on the light quantum hypothesis, and has been further analysed by Schrödinger. Yet another new form has been proposed independently by Fermi and Dirac. The latter based his theory on a discussion of lightly coupled systems with the help of Schrödinger’s equation. Combined with Heisenberg’s work on the many-body problem, Dirac’s work forces us to conclude at least that the classical form of statistical mechanics must be changed. It indicates that the true form, which satisfies the laws of the new mechanics, is almost certainly that of Fermi and Dirac, which is the natural generalization of Pauli’s principle of exclusion for electronic orbits in an atom. The work of Heisenberg and Dirac already quoted has shown that Pauli’s principle and its extension are satisfied in the new mechanics by a complete self-consistent solution of the equations of motion. So far as I am aware, the discussions of the new forms have as yet dealt only with the statistics of a gas of structureless mass-points (and, of course, temperature radiation). There has, moreover, been as yet no attempt to define the entropy (and the absolute temperature) in strict analogy with rational thermodynamics by means of the equation dQ = TdS. If another definition is preferred, then this equation must be deduced from it. It has, therefore, seemed worth while to reopen the discussion by examining ab initio a quite general form of statistical mechanics of which the classical form and instein’s and Fermi-Dirac’s are special cases. This is rendered possible by using the powerful method of complex integration already applied to the classical form. The sequence of the argument is then to take the general form, which covers a very large range of ways of assigning possibilities and counting complexions, and construct on that basis exact integral expressions for the number of complexions possible to the assembly and for the average number of systems of the assembly in their various quantum states. We then derive from these the average energies and external reactions and so the form of dQ, deduce from dQ the existence of S and so define S and T. This can be done in the general form for assemblies of ideal systems just as general as can be handled in the classical way—ideal gases of molecules of any structure and crystals and radiation. Such assemblies are in all cases thermodynamical systems.
Every vertex of the magic cube of physics discussed so far introduced a major element in the development of physics and the whole of science. Mechanics introduced the basic concept of equations of motion, or time-evolution of a physical system given certain initial condition and the associated idea of determinism, even with its generalization in the form of chaos, the dependence of the outcomes of the temporal evolution on the initial conditions. These concepts taken for many particles, or elements of a physical system, led to the probabilistic description of many particle systems, in the realm of statistical physics and thermodynamics, whereby the averages over time of various observable quantities were related to the ensemble averages, that involve classical, so-called Kolmogorov, probability measures, at least for certain “equilibrium” situations. The second law of thermodynamics introduced the concept of entropy and its probabilistic increase, the first truly emergent, probabilistic law in physics (temperature and pressure, being other emergent quantities, apart from entropy, the measure of disorder of a statistical system). In the context of Newtonian gravity, one gets, at least in retrospect the geometric description of gravity in non-relativistic, Galilean spacetime. Galilean relativity, based on the asymmetric mixing of space and time, with the absolute, universal time, true for all observers, gets generalized in special theory of relativity to observer dependent space and observer dependent time, viewed as “slices” of a new concept of spacetime, with the new concept of causal structure, that distinguish timelike, null and spacelike events. This in turn is made into a full dynamical geometric structure of spacetime with a relativistic description of gravity stemming from curvature of spacetime, and a dynamical causal structure, tied to gravity. In our next story, the story of quantum theory, we will encounter fundamental, intrinsic probabilities, that can interfere, unlike the classical probabilities of statistical thermodynamics, and which are consistent with equations of motion of individual elements, and not, as in statistical thermodynamics, with temporal averages over the time evolution of many elements. This is the big new lesson of quantum theory, that transcends the lessons of classical spacetime physics.
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.
Both statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic quantum theory.A total parameter space Φ, equipped with a group G of transformations, gives the mental image of some quantum system, in such a way that only certain components, functions of the total parameter φ can be estimated. Choose an experiment/question a, and get from this a parameter space Λa, perhaps after some model reduction compatible with the group structure. As in statistics, it is important always to distinguish between observations and parameters, in particular between (minimal) observations ta and state variables (parameters) λa. Let Ka be the L2 ‐space of functions of ta, and let Ha be the image of Ka under the expectation operator of the model. The measure determining the L2‐spaces is the invariant measure under the maximal subgroup which induces a transformation on Λa.There is a unitary connection between Ka and Ha, and then under natural conditions between Ha and Hb for a and b are different. Thus there exists a common Hilbert space H such that Ha equals UaH for a unitary Ua. In agreement with the common formulation of quantum mechanics, the vectors of H are taken to represent the states of the system. The state interpretation is then: An ideal experiment Ea has been performed, and the result of this experiment, disregarding measurement errors, is that the parameter λa is equal to some fixed number λka. This essentially statistical construction leads under natural assumptions to the basic axioms of quantum mechanics, and thus implies a new statistical interpretation of this traditionally very formal theory. The probabilities are introduced via Born’s formula, and this formula is proved from general, reasonable assumptions, essentially symmetry assumptions.The theory is illustrated by a simple macroscopic example, and by the example of a spin 12 particle. As a last example we show a connection to inference between related macroscropic experiments under symmetry. The result of this last example seems to have potential for being generalized considerably.