It is argued (Part A) that quantum mechanics can be derived as a principle-based dynamical framework, the basic equation of which is an alternative form of the Hamilton-Jacobi equation. Schrodinger's equation obtains as a result of linearising that equation, and so-called wave functions can be given no straightforward physical interpretation. It is suggested, partly in relation to a theorem by Gromov, that a finite action quantum would make it practically inevitable, for purposes of prediction, to resort to a probabilistic formulation. The structure of the space of square-integrable solutions of the Schrodinger equation happens to lend itself to the introduction of the appropriate kind of predictive scheme. Investigating the nature and scope of such a scheme is the subject of Part B. It is shown that basic features of the formalism of quantum theory, like composition rules for 'amplitudes' or the 'Born' probability rule, can be derived independently of any physical assumptions. A generalisation of the basic formalism using tensor product composition appears to be required if all correlations are to be extracted from locally accessed data. A detailed discussion of quantum teleportation leads to the conclusion that a 'one-shot' account leads to a distorted picture of what is actually achieved. An analogy with classical cryptography is made and the statistical significance of the 'transfer', which does not require introducing any novel form of 'quantum information', is emphasised. Results obtained over the last decade using the extended formalism of positive operator-valued measures are reviewed and discussed. These lend further support to the idea that the set of basic 'quantum' rules functions as a general kind of probabilistic scheme for prediction, the structural features of which are not constrained in any direct way by the underlying physics. On the other hand, the very existence of such a predictive framework hinges on selecting a particular class of solutions of the Schrodinger equation, which selection has been incorrectly interpreted as reflecting a physical necessity.
Some recent experiments claim to show that any model in which a quantum state represents mere information about an underlying physical reality of the system must make predictions which contradict those of quantum theory. The present work undertakes to investigate the issue of reality, treading a more fundamental route from the Hamilton-Jacobi equation of classical mechanics to the Schrodinger equation of quantum mechanics. Motivation for this is a similar approach from the eikonal equation in geometrical optics to the wave equation in electromagnetic theory. We rewrite the classical Hamilton-Jacobi equation as a wave equation and seek to generalise de Broglie's wave particle duality by demanding that both particle and light waves have the freedom of being described by any square-integrable function. This generalisation, which allows superposition also for matter wave functions, helps us to obtain the Schrodinger equation, whose solution can be seen to be as much objective as the classical mechanics wave function. Several other equations which one writes in quantum mechanics, including the eigenvalue equations for observables, series expansion of energy states in terms of eigenstates of observables other than energy, etc., can be written in the classical case too. Absence of any collapse of the wave function, entanglement, etc. in the classical realm have their origin in the nonlinearity of the classical wave equation. These considerations indicate that many of the puzzles in quantum mechanics are present also in classical mechanics in a dormant form, which fact shall help to demystify quantum mechanics to a great extent.
Sec. I. This paper inverts certain quantum mechanical ideas by constructing a "probability of presence" from first principles. If
xi=fi(s,a1,a2) represents a 2-parameter family of curves in 3-dimensional space, the density of the functions
fi with respect to the parameters
a1a2 is defined by means of an expression for the normal cross section of a sheaf of curves. This density plays the role of a probability. Its derivative along a curve of the family is independent of the parameters
a1a2. This circumstance permits a special density to be constructed, called the "characteristic" or "smoothest" density, which is independent of the parameters and which can be found by solving a linear second-order partial differential equation. The extension to
n-dimensional space and ($n\ensuremath{-}1$) parameters is given. Sec. II. The concept of orbital density is the dynamical equivalent of the density of Sec. I. Dividing the orbital density by the velocity of the representative point gives the quantity we call kinematical probability
P, which satisfies the fundamental equation (2.8), and which has the same force as any other probability of presence. Equation (2.8) is used to derive from the Hamilton-Jacobi equation a "wave" equation for $\ensuremath{\surd}P$ for a system of
n planets moving around a sun under the condition that the space average of the mutual interactions of the planets is a minimum. This provides a new approach to planetary dynamics and yields an equation that is very much like Schroedinger's equation for the stationary states of
n particles, and has the same basic interpretation. Sec. III. A non-conservative system of
n particles is assumed to be in a state such that the action function can be written $W=\ensuremath{-}Et+R({x}_{i},t)$ in which
R still contains the time explicitly. Starting with the Hamilton-Jacobi equation, using Eq. (2.8), and imposing the condition that the space average of $\frac{\ensuremath{\partial}R}{\ensuremath{\partial}t}$ is a minimum, an equation [viz., (3.10)] is derived which differs from Schroedinger's equation for stationary states only in certain respects that cannot be regarded as fundamental. Despite this basic equivalence of Schroedinger's equation and our equation (3.10), conflicts in interpretation and formalism arise which are briefly discussed. Sec. IV. Complex
U' s are introduced into Eq. (3.10) in a simple way which fails to bridge the gap between our formalism and that of quantum mechanics. The kinematical probability defined by Eq. (3.10) is shown to have the "smoothest" property. To account physically for the stationary states an electromagnetic wave system is proposed which has the property that a particle moving in it may be in "phase-mechanical" resonance with it. Macroscopic examples of this kind of resonance are given. This concept represents an inversion of physical ideas of quantum mechanics corresponding to the mathematical inversion with which we began.
Applications of quantum mechanics have led to many successful predictions and explanations of puzzling phenomena, and we now apply quantum mechanics to gain, process, and communicate information in novel ways. We can understand quantum mechanics by understanding how we have applied it. We should not seek agreement on the nature of the world it represents, because this theory does not itself represent the physical world (though its applications do help us to represent it better). When applied to a quantum state, quantum mechanics yields probabiities for physical events: both state and probability are objective--not because they represent elements of phyiscal reality, but because each exerts norrmative authority over the beliefs of anyone who accepts quantum mechanics and applies it relative to a physical situation they may (but need not) occupy. These events may be described by statements that are meaningful in an appropriate environmental context, and quantum mechanics can help one to say when that is. Measurement creates an appropriate context, so here the Born rule indirectly yields probabilities of measurement outcomes. The quantum state of a system does not "collapse" on measurement: a new state must be assigned relative to a physical situation in which information about the outcome is accessible. Understood this way, there is no measurement problem, and violations of Bell inequalities does not demonstrate "spooky" non-local action. Quantum field theories have no physical ontology of their own: a quantum field is a mathematical object in a model whose application helps us to improve and extend our descriptions of the world in other terms. We cannot realise the scenario of Wigner's friend and its recent extensions: but the data that provide overwhelming evidence for quantum mechanics are objective in the same sense as the relative measurement outcomes described in those scenarios.
Standard quantum mechanics—the theory that is made use of in practice and is, because of this, taught in university courses— is a theory that is weakly objective only. For this reason it is most conveniently and neatly expressed by means of a set of basic rules of calculation that enable physicists to make predictions concerning what will be observed. Classical and quantum physics both make considerable use of the expressions "physical system" and "state of a physical system." Quantum mechanics is essentially a probabilistic theory. Except in special cases it makes no predictions that bear on individual systems. Rather, it predicts statistical frequencies. In other words, it predicts, as a rule, the number n of times that a given event will be observed when a large number N of physical systems of the same type and satisfying specified conditions are subjected to a measurement process.