The subjective and the objective aspects of probabilities are incorporated in a simple duality axiom inspired by observer participation in quantum theory. Transcending the classical notion of probabilities, it is proposed and demonstrated that all probabilities may be fundamentally quantum mechanical in the sense that they may all be derived from the corresponding amplitudes. The classical coin-toss and the quantum double slit interference experiments are discussed as illustrative prototype examples. Absence of multi-order quantum interference effects in multiple-slit experiments and the Experimental tests of complementarity in Wheeler's delayed-choice type experiments are explained using the involvement of the observer.
The subjective and the objective aspects of probabilities are incorporated in a simple duality axiom inspired by observer participation in quantum theory. Transcending the classical notion of probabilities, it is proposed and demonstrated that all probabilities may be fundamentally quantum mechanical in the sense that they may all be derived from the corresponding amplitudes. The classical coin-toss and the quantum double slit interference experiments are discussed as illustrative prototype examples. Absence of multi-order quantum interference effects in multiple-slit experiments and the Experimental tests of complementarity in Wheeler's delayed-choice type experiments are explained using the involvement of the observer.
I show that the classical coin tossing experiment involves two distinct definitions of probability, one ontological (the relative frequencies of initial deterministic states) and another empirical (the relative frequencies of observations). In quantum theory, I argue, only the latter definition can be invoked, since a single superposition state can give rise to multiple observation experiences. This difference explains why the present statistical quantum mechanics is an ontological dead end, despite its enormous pragmatic success. To get at the ontological content of quantum theory, I propose that the observations themselves can be interpreted on a different footing, without reference to determinate detector states.
Classically, probability is considered in terms of the number of possible outcomes. For example, a die toss has six possible outcomes and a coin toss, two. Probability calculations make use of probabilities and hence of the total number of possible outcomes. Thus, they are not deterministic calculations which follow “what happens” as in Newtonian mechanics. We suggest that if free particle quantum mechanics is probabilistic, it must also involve the notion of possible outcomes in an experiment and not represent a deterministic approach. We suggest, however, that something interesting applies to quantum mechanics because we have suggested that free particle quantum mechanics arises from the Lorentz invariant A = -Et+px. We note that there exist x,t trajectory points which follow x=vt. We also note, however, that one may introduce the idea of intervals dx, dt (because Newton’s dx→0, dt→0 is an idealization) and that A is unchanged for x,t and x+hbar/p , t+hbar/E. In other words, we suggest that there are intervals with probability about each x,t trajectory point. A trajectory point is associated with deterministic physics, but probability about this point is not. In other words, one has a kind of hybrid probability scheme. Thus, exp(ipx) and exp(-iEt) really represent fluctuation probabilities about a trajectory point x and t. These ranges of possible x and t fluctuations about a trajectory point x,t allow for further probability to occur in certain interactions. We consider two examples. The first is 2-slit interference which involves a deterministic particle moving along a straight line towards a 2-slit mechanism perpendicular to its path. At the 2-slit apparatus, one considers a probabilistic calculation as the uncertainty in x, i.e. dx=hbar/p allows for possible interaction with both slits. This involves considering all possible outcomes (i.e. motion through one slit or the other which is a classical situation), but exp(i p dot r1) and exp(i p dot r2) “sit on top” of these trajectories at the 2-slit apparatus and one has interference of the probabilities linked with each classical possible outcome. Thus, there are two sets of probabilities which are intertwined, the probability range hbar/p for each path and the two possible paths which have equal probability. One must consider the possible outcomes for both sets together. We suggest that a similar analysis applies to the case of 1-D reflection-refraction for x=0 at an n1-n2 index of refraction junction. Thus, we suggest that a deterministic free particle with x=vt may encounter probabilistic interaction situations, such as a two-slit apparatus or an n1-n2 index of refraction junction. These are probabilistic only because there is a probability fluctuation scheme in x associated with each deterministic x point. It is this probability which allows for a second probability to arise due to interaction with the slits or at the 1-D n1-n2 junction. In such a case, one must consider all possible outcomes which means that one does not only consider the exp(ipx) type probabilities, but also those linked with the physical interaction, i.e. two possible paths, one from each slit, or an incident, reflected and refracted photon at x=0 (n1-n2 junction). It is a combination of all possible paths which must be used in calculations. We thus suggest that even though there is a sense of determinism, i..e free particles move as x=vt and these exist in the 2-slit case and the 1-D reflection-refraction case, in a probabilistic calculation one must consider all probabilistic outcomes consistent with these. Thus, quantum mechanical calculations are true probabilistic calculations in the sense that they involve all possible outcomes in an analysis.
An Axiom is presented and justified which (a) Explains duality in interference without complementarity or “which way” (welcher-weg) observation (b) Shows the equivalence: Coherence and alignment ≡ Interference ≡ No “which way” observation; No coherence or alignment ≡ No interference ≡ “which way” observation (c) Explains Wheeler’s delayed choice thought experiment (d) Explains results of experimental implementations of Wheeler’s experiment which show retro-causality with and without entanglement (e) Explains non-local action at a distance, and (f) Rephrases Albert Einstein’s unanswered question “Is quantum mechanics complete?” at a more fundamental level than just duality and non-locality. The explanation given does not require that the particle (photon) somehow “know” about the test setup or “which way” observation or change its behavior from particle to wave and vice versa as required by currently accepted explanation based on Niels Bohr’s complementarity principle. No new assumptions are made, only a new complete interpretation of probability which is already a fundamental assumption of quantum mechanics.
The concept of probability was prominent in the original foundations of quantum mechanics and continues to be so today. Indeed, the controversies regarding objective and subjective interpretations of probability have again become active. I argue that, although both objective and subjective probabilities have domains of relevance in QM, their roles are quite distinct. Even where both are legitimate, the objective and subjective probabilities differ, both conceptually and numerically. There are quantum probabilities that have no useful subjective interpretations, and there are subjective probabilities that cannot be realized as quantum probabilities.
The principle of complementarity is one of the cornerstones of quantum theory. The aim of this study was to advance our understanding of complementarity by analyzing the role of delayed choice and quantum erasers in two-slit experiments, and by proposing experiments for verifying the analysis. The analysis is based on models consisting of measurable spaces and probability measures involved in the experiments. The main findings are as follows: (a) The complementarity principle manifests itself in such a way that wave and particle behaviors cannot be simultaneously observed almost surely with respect to any single, fixed measure. (b) Described by different measures, complementary properties can coexist in the same experimental setup. (c) Which-way information will not preclude or erase interference fringes. (d) Delayed choice and quantum erasers are irrelevant to testing complementarity. (e) It is possible for us to know through which slit each quantum object passed almost surely with respect to the measure corresponding to the slit while the interference pattern is intact. Based on the experiments analyzed, realizable experiments are proposed for verifying the above results.
At the intersection of the foundations of statistics and the foundations of quantum theory, one comes across a hotly-debated question: what are the probabilities in quantum theories? Do they all reflect the subjective opinions of various agents, or do some of them reflect objective features of the world? One answer—Lewisian dualism, a view that takes some quantum probabilities to be subjective and others to be objective—has gained near-consensus among philosophers of science. However, our most sophisticated defense of this view (due to Deutsch, Wallace, Saunders, Myrvold, Greaves, and others) has two significant problems. First, it requires a very particular approach to the measurement problem (namely, the Everett interpretation). Second, it relies on the pragmatic goals of scientists (e.g., their aim to avoid losses in bets) rather than their epistemic goals (e.g., their aim to accurately describe the world). I give a new defense of dualism that rectifies both of these problems. I generalize the Deutsch-Wallace argument to defend the objective probabilities in a host of realist approaches to measurement, and I modify the accuracy-dominance argument for probabilism to defend the subjective probabilities in these approaches. I also critically examine several previous attempts to interpret all quantum probabilities subjectively.