The classical notions of continuity and mechanical causality are left in order to reformulate the Quantum Theory starting from two principles: (I) the intrinsic randomness of quantum process at microphysical level, (II) the projective representations of symmetries of the system. The second principle determines the geometry and then a new logic for describing the history of events (Feynman's paths) that modifies the rules of classical probabilistic calculus. The notion of classical trajectory is replaced by a history of spontaneous, random and discontinuous events. So the theory is reduced to determining the probability distribution for such histories accordingly with the symmetries of the system. The representation of the logic in terms of amplitudes leads to Feynman rules and, alternatively, its representation in terms of projectors results in the Schwinger trace formula.
The classical notions of continuity and mechanical causality are left in order to reformulate the Quantum Theory starting from two principles: (I) the intrinsic randomness of quantum process at microphysical level, (II) the projective representations of symmetries of the system. The second principle determines the geometry and then a new logic for describing the history of events (Feynman's paths) that modifies the rules of classical probabilistic calculus. The notion of classical trajectory is replaced by a history of spontaneous, random and discontinuous events. So the theory is reduced to determining the probability distribution for such histories accordingly with the symmetries of the system. The representation of the logic in terms of amplitudes leads to Feynman rules and, alternatively, its representation in terms of projectors results in the Schwinger trace formula.
A history of the discovery of `new' quantum mechanics and the paradoxes of its probabilistic interpretation are briefly reviewed from the modern point of view of quantum probability and information. Modern quantum theory, which has been developed during the last 20 years for the treatment of quantum open systems including quantum noise, decoherence, quantum diffusions and spontaneous jumps occurring under continuous in time observation, is not yet a part of the standard curriculum of quantum physics. It is argued that the conventional formalism of quantum mechanics is insufficient for the description of quantum events, such as spontaneous decays say, and the new experimental phenomena related to individual quantum measurements, but they have all received an adequate mathematical treatment in quantum stochastics of open systems. Moreover, the only reasonable probabilistic interpretation of quantum mechanics put forward by Max Born was, in fact, in irreconcilable contradiction with traditional mechanical reality and causality. This led to numerous quantum paradoxes, some of them due to the great inventors of quantum theory such as Einstein and Schrödinger. They are reconsidered in this paper from the modern point of view of quantum stochastics and information. The development of quantum measurement theory, initiated by von Neumann, indicated a possibility for resolution of this interpretational crisis by divorcing the algebra of the dynamical generators and the algebra of the actual observables, or Bell's be ables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of the actual histories with the potential future. This rule, together with the self-compatibility of the measurements ensuring the consistency of the histories, is called the nondemolition, or causality principle in modern quantum theory. The application of this rule in the form of dynamical commutation relations leads to the derivation of the von Neumann projection postulate, and also to the more general reductions, instantaneous, spontaneous, and even continuous in time. This gives a dynamical solution, in the form of the quantum stochastic filtering equations, of the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting with Schrödinger and Bohr. It has been recently proved that the quantum stochastic model for the continuous in time measurements is equivalent to a Dirac type boundary-value problem for the secondary quantized input `offer waves from future' in one extra dimension, and to a reduction of the algebra of the consistent histories of past events to an Abelian subalgebra for the `trajectories of the output particles'. This supports the corpuscular-wave duality in the form of the thesis that everything in the future are quantized waves, while everything in the past are trajectories of the recorded particles.
The present article is written by a theoretical physicist who, for some time, has endeavored to trace the origin of the concepts and principles of quantum theory to an empirical background broader than that afforded by delicate optical and mechanical experiments on a microphysical scale involving the microconstant h. In particular he tried to reduce the dominant role of probability in modern physics to irrefutable evidence of a quite general nature. As long as quantum probability is deduced from experience with seemingly uncontrollable micro-mechanisms—think of radioactive disintegration as the most typical example—there will always be the objection that one day we still may find a hidden causality behind those small scale gambling devices called atoms, although one example of an exactly predictable atomic event would make present-day quantum theory untenable. In order to dislodge the remaining classicists from their fortress of doubt one may try to show that quantum probability is not an alibi for a temporary impotence of predicting microphysical events, but is a positive conclusion drawn from experience of a most general type, namely, from the principle of continuity, the basic tenet of the philosophy of Leibniz.
Philosophical analyses of causation take many forms but one major difficulty they all aim to address is that of the spatiotemporal continuity between causes and their effects. Bertrand Russell in 1913 brought the problem to its most transparent form and made it a case against the notion of causation in physics. The issue highlighted in Russell's argument is that of temporal contiguity between cause and effect. This tension arises from the imposition of a spectrum of discrete events occupying spacetime points upon a background of spacetime continuum. An immediate and natural solution is to superpose instead spatiotemporally continuous entities, or processes, on the spacetime continuum. This is indeed the process view of physical causation advocated by Wesley Salmon and Phil Dowe. This view takes the continuous trajectories of physical objects (worldlines) as the causal connection whereby causal influences in the form of conserved quantities are transported amongst events. Because of their reliance on spatiotemporal continuity, these theories have difficulty when confronted with the discontinuous processes in the quantum domain. This thesis is concerned with process theories. It has two parts. The first part introduces and criticizes these theories, which leads to my proposal of the History Conserved Quantity Theory with Transmission. The second part considers the extension of the idea of causal processes to quantum physics. I show how a probability distribution generated by the Schrodinger wavefunction can be regarded as a conserved quantity that makes the spacetime evolution of the wavefunction a quantum causal process. However, there are conceptual problems in the interpretation of the wavefunction, chiefly to do, as I shall argue, with the difference in the behaviours of probabilistic potentials between quantum and classical physics. I propose in the final chapter, the Feynman Path Integral formulation of quantum mechanics (with the Feynman histories) as an alternative approach to incorporating the probabilistic potentials in quantum physics. This account of how to introduce causal processes in quantum mechanics fares better, I claim, than the previous one in dealing with the situational aspect of quantum phenomena that requires the consideration of events at more than one time.
A history of the discovery of quantum mechanics and paradoxes of its interpretation is reconsidered from the modern point of view of quantum stochastics and information. It is argued that in the orthodox quantum mechanics there is no place for quantum phenomenology such as events. The development of quantum measurement theory, initiated by von Neumann, and Bell's conceptual critics of hidden variable theories indicated a possibility for resolution of this crisis. This can be done by divorcing the algebra of the dynamical generators and an extended algebra of the potential (quantum) and the actual (classical) observables. The latter, called beables, form the center of the algebra of all observables, as the only visible (macroscopic) observables must be compatible with any hidden (microscopic) observable. It is shown that within this approach quantum causality can be rehabilitated within an extended quantum mechanics (eventum mechanics) in the form of a superselection rule for compatibility of the consistent histories with the statistically predictable future. The application of this rule in the form of the nondemolition principle leads to the statistical inference of the von Neumann projection postulate, and also to the more general quantum information dynamics for instantaneous events, spontaneous localizations (i.e. quantum jumps), and state diffusions (i.e. continuous trajectories). This gives a dynamical solution, in the form of a Dirac boundary value problem and reduced filtering equations, of the notorious decoherence and measurement problems which was tackled unsuccessfully by many famous mathematicians and physicists starting with von Neumann, Schroedinger and Bohr.
Even though only the classical action explicitly appears in the Feynman path integral, quantum fluctuations show up in the fluctuating nature of Feynman paths. This prevents a classical interpretation of the latter objects. Still, it makes sense to ask whether the Feynman paths have any relation to the classical dynamical trajectories in configuration space-a possibility naturally suggested by the semiclassical approximation. Unfortunately, no such relation exists (in general). Things are quite different for the generalized Feynman paths, which actually contribute in a novel path integral giving an alternative representation of the quantum mechanical propagator. It is shown that these paths fluctuate about a classical dynamical trajectory (in configuration space) because of a certain white noise. That is to say, the generalized Feynman paths are fluctuating curves that collapse on a classical dynamical trajectory as h to 0. This result throws new light on the interplay between classical and quantum mechanics and hints at a new approach to the quantum theory.
A path , in general, is defined by a determinate trajectory in time, from an initial to a final point. The classical trajectory is only one of the possible trajectories, and in quantum mechanics all the possible paths between the initial and final point come into play. Recall that the probability amplitude is a complex number that is assigned to each determinate path. Indeterminate paths are defined as a collection of determinate paths that are experimentally indistinguishable . In the Dirac–Feynman approach, the inherent indeterminacy of the quantum entity is realized by the degree of freedom – in undergoing time evolution – “taking” indeterminate paths [Baaquie (2013e)]. For a quantum degree of freedom evolving from an observed initial state to the observed final state – and with no other observations made – the Feynman path integral is a mathematical construction that computes the probability amplitudes by summing over all the allowed determinate paths of the degree of freedom – discussed in Feynman and Hibbs (1965), Zinn-Justin (1993), Zinn-Justin (2005) and Baaquie (2013e). Probability amplitude and time evolution Recall that the description of a quantum system, at a particular instant,isgiven by its state vector, namely |ψ>. To avoid confusion with the concept of a state vector, the term probability amplitude is used for describing a quantum entity undergoing transitions in time.