EXPERIMENTAL PROPOSAL

            We are currently into our second year of work, and are now in position to develop an experimental component with Philip Stamp at the University of British Columbia/Pacific Institute of Theoretical Physics, who has just joined our group as co-Investigator. Our goal is to develop an experiment that attempts to differentiate between precessional / spin bath decoherence and 'intrinsic' decoherence, so that the latter can be further elucidated.
            Our work will build upon recent groundbreaking experiments on the mechanisms of decoherence in molecular magnets.  In Bertaina et al., published last year (Nature 453: 8 May 2008--see also in the same issue P.C.E. Stamp, “Quantum Information: Stopping the Rot” p.167, a discussion of the Bertaina experiment) three decoherence mechanisms were accounted for: Spin bath, lattice vibrations, and dipolar pairwise decoherence.  The observed coherence time was 800 ns for the spin ½ transitions of the Vanadium molecule.  It is proposed that if the experiment were performed at liquid He temperatures, with spin-free molecules, sufficiently separated by a surfactant to restrict dipolar pairwise decoherence, coherence time could be extended to 100 microseconds.  Our own experimental work will proceed to test this and related hypotheses.
            This experimental investigation is crucial for the relational realist interpretation of quantum mechanics, and its conception of logical causality, because spin-bath / precessional decoherence involves no transfer of energy. It is entirely non-dissipative, unlike the more common oscillator bath models that have been used in previous experiments. Non-dissipative decoherence is arguably something 'deeper' than dissipative environmental decoherence via simple efficient physical causality, which is the framework environmental decoherence has typically been fitted to.  Our goal, then, is to develop an experiment that attempts to differentiate between precessional / spin bath decoherence and other possible sources of 'intrinsic' decoherence, so that the latter can be further elucidated.
            Our proposed experimental work will help shed light on, 1. how decoherence can be examined as a potential challenge to (or refinement of) Leggett's thesis of macrorealism (which would be a much needed blow to the conventional mechanistic-materialistic paradigm); and 2. how decoherence—especially non-dissipative decoherence seen in the spin bath, 3rd party, and 'intrinsic' conceptions—can be interpreted as a physical exemplification of 'logical causality' via ‘logical-relational decoherence.’  In this regard, decoherence might be seen as implying a logically governed conditioning of physical causality at the level of microphysical reality—one that could account for the tacitly presupposed correlation of the causal and logical orders at the classical level as an emergent phenomenon.
            Since the correlation of the causal and logical orders has been a centuries-old problem in natural philosophy, the obvious relevance of the logical order to decoherence warrants careful study. Possible implications include new understandings of: 1. the quantum measurement problem, 2. the supposed quantum-classical boundary, 3. relating the asymmetrical logical order to the asymmetrical cosmological arrow, thermodynamic arrow, and quantum arrow (to the extent that the latter is asymmetrical, for example, via Von Neumann’s Process 1); and 4, the conception of classical physics as ‘emergent’ from quantum physics, via the decoherence effect.
            With respect to the latter, one might focus on the more physical question of how classical quasi-deterministic behavior emerges for large systems, and/or how quasiclassical stochastic behavior emerges, even for small systems.  Co-investigator Philip Stamp writes, “a proper answer to this question requires understanding the real decoherence mechanisms operating in Nature and that these are not so simple, or necessarily completely understood. Thus we do not yet have a theory which derives classical physics from quantum physics solely using ideas from decoherence, even though we do have some derivations of classical behaviour within certain models. It is important to note that in some other models one can actually find non-classical behavior emerging in the large-scale dynamics, because of decoherence (this happens, for example, when one is dealing with a spin bath environment (Prokof’ev & Stamp, 2006)). Thus there is nothing inevitable about classical behaviour!” 

            The idea of intrinsic decoherence in Nature is implicit in some theories of hidden variables, and also in the 'non-linear' adaptations of quantum mechanics by Ghirardi, Pearle, et al. (1986, 1990). More recently it has been widely appreciated that any sign of 'intrinsic decoherence' in Nature, or for that matter any breakdown of quantum mechanics, is almost certainly going to happen in systems with large numbers of degrees of freedom. More recent ideas therefore focus on this. The recent ideas of     't Hooft (1999, 2001) are very interesting in this respect; he points out that if one believes that the holographic principle really does apply (and it is now a pretty central part of quantum gravity), then genuine entanglement in Nature cannot involve more than 10^{120} degrees of freedom, since this is the size of the Hilbert space allowed by this principle. However any experiments to check out ideas like this are dogged by the fact that almost all physical systems in Nature show far more decoherence than theory predicts (often at least 3 orders of magnitude more). The reason for this is pretty obvious: there are many sources of 'junk decoherence,' coming from defects, nuclear spins, dislocations, charge fluctuators, etc., in either physical systems of interest, or in surrounding parts of the experimental cell, or even in the measuring system. This 'junk' is typically described by models of '2-level systems', and the decoherence from them is described by the 'spin bath' model of environmental decoherence (Prokof'ev and Stamp, Rep Prog Phys 69, 669 (2000)).
            The best systems to initially do decoherence tests in are thus the ones  (i) in which one has a large number of degrees of freedom, and (ii) in which decoherence processes are understood best. The only systems that have so far shown agreement between experiment and theoretical predictions, and which involve large numbers of degrees of freedom, are large-spin magnetic molecules. Those that have been tested are the V-15 molecule (Bertaina et al., Nature 2008), and the Fe-8 molecules (Takahashi et al., preprint about to be issued); in both cases, experiment agrees with theoretical prediction to within a relatively small experimental error. The Fe-8 molecule has over 10^{50} degrees of freedom in the nuclear bath; the low-T quantum dynamics of this system is entirely governed by the coupled dynamics of the electron spins with these nuclear spins, with the phonon bath, and with other molecules (with photons playing a secondary role).
            In unpublished work the Bertaina-Barbara group has now also checked out the role of dipolar interactions between the V-15 molecules, by varying the distance separating the molecules. Theory and experiment still agree. In both of these systems (and also in related systems like the Erbium tungstenates, or the LiHoF system, where one deals with a large Rare Earth spin, and one also has a very good handle on the experimental systems [here the number of degrees of freedom is much smaller however]), one can therefore look for a key signature of ‘intrinsic decoherence,’ where it is most likely to show up: in the dynamics of decoherence once entanglement has been set up between 2 or more of these molecules.
            There as yet are no theoretical works, with or without predictions, exploring this important hypothesis. So the first step would be to give a quantitative analysis of what one would expect in the coupled dynamics of such entanglement experiments. This would require a calculation of the coupled dynamics of the 2 molecules; these couple to each other purely via magnetic dipolar interactions (very well understood) and each couples to nuclear spins in their surroundings (all the nuclear species are known, as are all the relevant hyperfine couplings, to very high accuracy--better than one part in 10^3 in most cases). There is also the coupling to photons (understood exactly) and to phonons (we know this very well for V-15, for Fe-8, and for the LiHo system, because of existing very thorough studies).
            These calculations would predict the time-correlations of the coupled molecules, and then the prediction of what would be seen in Rabi experiments, ESR experiments, and in spin echo experiments. At the same time it would be terribly useful to do NMR on the nuclear spins, to check that they were behaving as expected (the nuclear spins cause almost all the decoherence, and while they are doing this their dynamics is strongly modified).
            The experimental testing of ideas of intrinsic decoherence in Nature requires measuring the time-dependent entanglement between the molecules or rare-earth spins. The main problem here is that no single experimental group possesses all the apparatus to do these experiments, which work at quite different frequencies. What typically happens in such a situation is that the different experiments are done in different research centers by one collaboration, with different students or postdocs working in the different centers. Thus the ESR and Rabi experiments could be done in one center, and the spin echo and NMR experiments in another).
            More detail on the experimental work will follow once the theoretical studies have made it clear exactly what has to be measured. However it is already clear that separate experiments will be necessary at microwave (1-20 GHz) and ESR (100-300 GHz frequencies), as well as measurements of T_1 and T_2 on the nuclear spins. These may have to be supplemented by measurements of spin-phonon couplings. No equipment costs are anticipated, although sometimes if large facilities are being used—for example, to do ESR and spin echo experiments at low T (below 50 mK) and high magnetic field (around 10T)—one may need to use large facilities, and there may be extra costs involved if the group does not have user privileges.

PHILOSOPHICAL IMPLICATIONS

holding between any set

regarded as defining a predicate or class of integers, and any subset s’of s (Finkelstein, 2002).

            The quantum theory of Bohr and Heisenberg was the first physical theory to transcend the mechanical notion of absolute truth implicit in mathematics; Heisenberg called it ‘non-objective.’ Like an integer, physical systems seem to have maximal descriptions. Unlike an integer, these are incomplete; every predicate has complementary ones. Bohr and Von Neumann respectively renounced and revised classical logic in formulating quantum theory. We see this as renouncing ontology (theory of being) for a praxiology (theory of acting); this is not an ‘interpretation’ of quantum theory but rather a ‘paraphrase.’ Since all actual observations make changes in the system beyond our control, we do not assume that a system has an absolute ontology, except as a singular limit

but only an absolute praxiology, a network of quantum processes represented by an operator algebra associated with the system.
            Most quantum theory to date has retained the classical theory of the causal order and an absolute space-time, projecting these pre-quantum concepts into the quantum microcosm in what has long been recognized as unphysical and probably a failure of the theorist's imagination. Such mixed quantum/classical field theories are structurally unstable and singular as well as false to actual practice. They challenge us to reconstruct the physical theory of the causal order based on explicit linkages with the logical order.
            Just as classical unition ι (taken with union) generates classical set theory, a quantum
unition operator ι  generates a quantum set theory rich enough for field theory. Like classical set theory, this quantum set theory comes with no manual for building a physical theory with its tools. This is provided by a correspondence principle generalizing Bohr’s, which is implicit in a suggestion by Irving Segal: Present-day singular physical theories are singular limits of a regular physical theory. Slight errors in the commutation relations have converted simple Lie algebras into nearby non-simple ones.
            This suggests that a suitable Lie algebraic (i.e. the algebra characterizing a physical theory’s transformations, as idealized in an infinitesimal limit) simplification can restore the hypothetical regular theory (i.e., a theory devoid of singularities which are denotive of a theory’s failure to give definite information.) This is an extension of canonical quantization; one may call it simplification quantization. One way to implement it is to move physical theories from their singular foundations in classical set theory onto the regular logical foundations of quantum set theory, which provides the necessary variety of simple Lie algebras. ‘Simplification quantization’ regularizes singularities that have eluded canonical quantization, while maintaining agreement with experiment. Simplification quantizations of gauge theory in general and of the gravitational theory of the causal order in particular are underway by Finkelstein and consultant Mohsen Shiri-Garakani.
            This hypothesis provides an origin for the important Lie algebras of quantum physics, including the Lorentz, Heisenberg, Poincaré, and unitary ones. The basic Lie algebras define the statistics of quantum aggregates. These then generate kinematical algebras—i.e., algebras characterizing all possible dynamical outcomes of a quantum system, as underwritten by the theory. Finally, the operators in kinematical algebras that are symmetries of organized modes like condensates make up the symmetry groups and Lie algebras. In this approach, there are no truly fundamental symmetries in nature (i.e., dynamics that do not fundamentally presuppose logical asymmetry). Empirical symmetries tell us about the symmetry of some organized substratum and are contingent upon that organization. Space-time curvature and classical gravity can now be regarded as residual effects of the quantum non-commutativity of a simple space-time-energy-momentum Lie algebra near the singular limit of classical space-time.

 The Dipolar Relational Realist Interpretation of Quantum Mechanics
(see Epperson and Zafiris, 2013)

            This interpretation begins with Heisenberg’s suggestion that apart from its epistemic significance, there is a sense in which the wave function must be understood to entail at least some ontological significance; for the potentia it describes, while not physically 'actual,' are nevertheless ‘real’ to some extent given that their probability valuations yield ‘real’ physical outcomes. And indeed, Heisenberg would later insist upon the fundamental reality and function of potentia in this regard. For him, potentia are not merely epistemic, statistical approximations of an underlying veiled reality of predetermined facts; potentia are, rather, ontologically fundamental constituents of nature.  They are things “standing in the middle between the idea of an event and the actual event, a strange kind of physical reality just in the middle between possibility and reality.”  (1958, 41)  Elsewhere, Heisenberg writes that the correct interpretation of quantum mechanics requires that one consider the concept of “probability as a new kind of ‘objective’ physical reality. This probability concept is closely related to the concept of natural philosophy of the ancients such as Aristotle; it is, to a certain extent, a transformation of the old ‘potentia’ concept from a qualitative to a quantitative idea.” (1955, 12)
            The relational realist interpretation of quantum mechanics begins with this conception of ontologically significant potentia—‘real’ apart from their actualizations, but ‘non-actual’ until actualized quantum mechanically. Within that context, this interpretation incorporates foundational features of various theoretical programs that explicitly explore the phenomenon of quantum decoherence as a process integral to quantum mechanical actualization of potentia (i.e., ‘wave function collapse’): In particular, the work of Wojciech Zurek, Roland Omnès, Robert Griffiths, and Murray Gell-Mann & James Hartle. A great deal of experimental work has been committed to the study of quantum decoherence in recent years, and it has become a robust topic of inquiry in physics.  While there is as yet no empirically validated ‘generic’ model for decoherence, the basic features of the theory have been sufficiently established to warrant careful philosophical exploration.  The dipolar relational realist interpretation is one such exploration.
            By this interpretation, the quantum mechanical actualization of potentia is defined as a decoherence-driven process by which each actualization (in ‘orthodox’ terms, each measurement outcome) is conditioned both by physical and logical relations with the actualities conventionally demarked as ‘environmental’ or external to that particular outcome. But by the relational realist interpretation, the actualization-in-process is understood as internally related to these ‘environmental’ data per the formalism of quantum decoherence. The concept of ‘actualization via wave function collapse’ is accounted for solely by virtue of these relations, and thus requires no ‘external’ physical-dynamical trigger—the Gaussian hits of GRW, acts of conscious observation, etc.  By the relational realist interpretation, it is the physical and logical relations among quantum actualities (quantum ‘final real things’) that drives the process of decoherence and, via the latter, the actualization of potentia. In this regard, the relational realist interpretation of quantum mechanics is a praxiological interpretation; that is, these physical and logical relations are ontologically active relations, contributing not just to the epistemic coordination of quantum actualizations, but to the process of actualization itself.
            The wave function / density matrix represents the totality of potential relations for that actualization-in-process—a coherent superposition of both logical and illogical potential relations, represented by the diagonal and off-diagonal terms of the density matrix, respectively. The latter, for example, would represent relations violating the Principle of Non-Contradiction (PNC), the theoretical actualization of which has been famously represented by the concept of an ‘actual’ Schrödinger Cat, both alive and dead at the same time.
             As described via von Neumann’s ‘Process 1’ the process of quantum mechanical actualization/state evolution entails a ‘reduction’ of this density matrix whereby the ‘illogical,’ off-diagonal terms are eliminated from the coherent superposition, rendering it ‘decoherent.’  The diagonal terms which survive this logical conditioning each represent a potential outcome actualization as before, but with an important new qualification: They are now valuated as probabilities and as such satisfy a second presupposed logical desideratum—the Principle of the Excluded Middle (PEM).  Thus, the probable outcome states of the reduced density matrix are both mutually exclusive in satisfaction of PNC (i.e., at most, one will be actualized), and exhaustive in satisfaction of PEM (i.e., at least one will be actualized), since the probabilities for each possible outcome state must sum to unity.
          Von Neumann’s non-unitary ‘Process 1’ evolution, as a formal aspect of environmental decoherence (Zurek, 2003), reflects a fundamental logical asymmetry underlying the symmetrical unitary evolution.  In von Neumann’s non-relativistic formulation the density operator ρ evolves in time, and ρ(t) represents the state of the system S at time t. The state S at time t is considered to exist over the subset of the set of space-time points (x’, y’, z’, t’) for which t’= t. The quantum dynamical law of evolution asserts that for an isolated system S, ρ(t’) = exp –iH(t’-t) ρ(t) exp iH(t’-t), where H is the Hamiltonian operator, here assumed to be time independent.  This dynamical law holds except at a discrete set of times at which Process 1 decoherence occurs, when the density matrix is reduced to a set of logical bivalent truth propositions derived from the logically conditioned relations between the system, detector, and environment (as reflected by the preferred/pointer basis). Each such proposition is associated with a pair of projection operators P and P’ = (1-P), and is represented mathematically by the Process 1 action ρ à ρ’ = PρP + P’ρP’. The probability that a proposition PρP is actualized is Trace PρP/ Trace ρ.  Again, the probable outcome states of the reduced density matrix are both mutually exclusive in satisfaction of PNC and exhaustive in satisfaction of PEM, since the probabilities for each possible outcome state must sum to unity.         
            Perhaps most important, the explicit acknowledgement of PEM as a logical desideratum of quantum mechanics, presupposed by the very concept of probability as absolutely fundamental to the mechanics, obviates the need for some physical dynamical ‘mechanism’ that actualizes the final, unique outcome state from among the reduced matrix of probable states.  The actualization of one probable state is presupposed by the logical, probabilistic nature of the mechanics and the mathematics by which the mechanics are described; the fact of a unique outcome need not (and indeed, logically cannot) be accounted for by the mechanics which necessarily presuppose it. Likewise, quantum mechanics cannot account for the existence of unique, actual, system-detector-environment states prior to measurement, similarly presupposed by the theory. In the same way that we easily stipulate the actual existence of the system-detector-environment as a presupposition for the very possibility of measurement, a unique actual outcome subsequent to measurement is similarly presupposed by quantum mechanical measurement.  The fact that such measurement yields a set of probability-valuated outcome states alone guarantees the actualization of one of those states.
            Put another way, quantum mechanics can no more account for the existence of unique actuality via some purely efficient ‘cause of wave function collapse’ than classical mechanics can account for the existence of matter via descriptions of motion and inertia. Unique actual existence characterized by logically conditioned causality—explicitly acknowledged in the relational realist interpretation—is a presupposition of both quantum and classical mechanics.  The difference is that classical mechanics provided philosophers of the early modern period no readily discernable indication of how the order of physical causal relation correlated with the order of logical implication.  The resulting philosophical proposals either attempted to reduce the logical order to the physical causal order (e.g., the philosophy of Locke), to reduce the causal order to the logical order (e.g., Spinoza), to depict the correlation of the logical and causal orders as a schematization by which we experience the otherwise transcendent ‘things in themselves’ (e.g., Kant), or to deny the necessary presupposition of either order (e.g., Hume).
            Like classical mechanics, quantum mechanics raises the same issue and has generated analogous philosophical approaches: The GRW approach generally comports with the philosophy of Locke, the transcendent realist approaches of Bohr and Everett generally comport with the philosophy of Kant, etc. The advantage of the relational realist interpretation and its reliance upon the concept of logically conditioned decoherence is that it offers a more detailed proposal for how, precisely, the logical order correlates with the physical-causal order as exemplified by modern quantum mechanics. It does so by way of describing quantum actualities, quantum ‘things in themselves,’ as fundamental becomings rather than fundamental beings.  Quantum mechanics, when interpreted via those approaches that explicitly acknowledge the process of decoherence, exemplifies three primary features of these quantum actualizations:

1) They are fundamentally relational, not just in ad hoc epistemic terms of ‘measuring apparatus’ and ‘measured system’ but also at the ontological/substance level of quantum actualities, such that quantum mechanical actualities both internally and externally ‘environmental’ to ‘system’ and ‘detector’ are involved. In this way, all actualities within the closed system of the universe are considered in logical relation quantum mechanically, thus allowing for a coherent ontological (universal) interpretation.

2) These quantum mechanical relations are logically governed potential integrations. It is the explicit incorporation of these relations into logical equivalence classes, relative to a particular actualization, with its particular preferred basis, that provides sufficient degrees of freedom for a logical negative selection process. This process is represented mathematically by the trace over, elimination of off-diagonal terms, and by reduction to the reduced density matrix with its mutually exclusive and exhaustive probability-valuated potential outcome states.

3) The logical integrations of coherent, superposed potentia into the decoherent, mutually exclusive and exhaustive probable outcome states represented by the reduced matrix can be correlated with von Neumann’s Process 1; and likewise, the physical fact of unitary evolution to a unique outcome state can be correlated with his Process 2.  Thus the logical features of Process 1 and the physical features of Process 2 are mutually implicative in every quantum mechanical actualization, and it is in this sense that every such actualization is a dipolar (logical and physical) relational ‘becoming.’

            There have been two major criticisms of interpretations of quantum mechanics grounded in the concept of decoherence.  The first is that they do not ‘physically account for’ wave function collapse—that is, the actualization of a unique outcome state from among the menu of probable outcome states represented by the reduced density matrix.  One answer to this criticism was given above: namely that such actualization is a logical presupposition inherent in the fact that the potential outcome states represented by the reduced density matrix are probability-valuated outcomes, mutually exclusive (in satisfaction of PNC) and exhaustive (in satisfaction of PEM).  It is because of these two logical desiderata that some actual outcome is guaranteed (by PEM) and that (by PNC) it will not be some actualized superposition—i.e., an actualized Schrödinger Cat.  Thus the fact that quantum mechanics terminates in probabilities rather than unique actualities is no theoretical deficiency or evidence of ‘incompleteness.’  Nevertheless, the inability of decoherence-based interpretations to ‘account for actuality’ remains a source of criticism.  This is likely driven in large measure by the fact that the ‘problem’ of wave packet collapse is an infamous one, and its solution by simple denial on logical grounds rings of sophistry for many physicists.
            Regardless, the argument remains: First, it is ultimately unreasonable to suppose that quantum mechanics could ever ‘account’ for either 1) the fact of ‘actuality’ which it necessarily presupposes; or 2) the fact that this actuality is logically conditioned, which the theory also necessarily presupposes.  Physicist Roland Omnès, whose approach to quantum mechanics can be said to fall within the dipolar relational realist category, puts it thus: “One may consider that the inability of the quantum theory to offer an explanation, a mechanism, or a cause for actualization is in some sense a mark of its achievement. This is because it would  otherwise reduce reality to bare mathematics.” (494) Indeed, one could argue that it is typically when fundamental physics reaches beyond its proper task of describing nature and presumes to explain nature—i.e., account for its existence—that it encounters problems of philosophical incoherence. 
            Second, capitalizing on the logical features of quantum mechanics in the manner proposed by the various decoherence-based interpretations is in no way a sophistic maneuver given that these same logical features are unavoidably implicit in every aspect of the quantum theory—both the mathematics by which the theory is described, but also, equally important, in the logical nature of physical causality.  Advocates of decoherence-based interpretations of quantum mechanics simply make use of logical principles that are already presupposed by the quantum theory and universally evinced in every scientific theory of physical causality. The only difference is that rather than merely stipulating the logical nature of physical causality (or worse, pretending that the scientific enterprise is not founded upon this stipulation as an article of faith), the decoherence-based interpretations explicitly acknowledge and capitalize upon the presupposition of logical causality, applying it mathematically to the ‘problem’ of wave function collapse.
            Wojciech Zurek (2003), for example, referring to a paper by Richard Cox (Cox, R. T., 1946, Am. J. Phys. 14 1-13) emphasizes that the theory of probability as it is employed in quantum mechanics presupposes the laws of Boolean logic.  Zurek writes:

Intuitively, this is a very appealing demand. Probability emerges as an extension of the two-valued logic into a continuum of the “degrees of certainty.” The assumption that one should be able to carry classical reasoning concerning “events” and get consistent estimates of the conditional degree of certainty leads to algebraic rules which must be followed by the measure of the degree of certainty.

This implies that an information processing observer who employs classical logic states and classical memory states which do not interfere will be forced to adopt calculus of probabilities essentially identical to what we have grown accustomed to. In particular, likelihood of c and b (i.e., “proposition c · b”) will obey a multiplication theorem:

 μ (c · b|a) = μ (c|b · a) μ (b|a) .

Above μ(b|a) designates a conditional likelihood of b given that a is
certain. Moreover, μ should be normalized:

μ (a|b) + μ (∼ a|b) = 1 ,

where ~ a is the negation of the proposition a. Finally, likelihood
of c or b
(c ∪ b) is:

μ (c ∪ b|a) = μ(c|a) + μ(b|a) − μ(c · b|a) ,

which is the ordinary rule for the probability that at least one of two events will occur.

In short, if classical Boolean logic is valid, then the ordinary probability theory follows. We are halfway through our argument, as we have not yet established the connection between the μ’s and the state vectors. But it is important to point out that the assumption of the validity of Boolean logic in the derivation involving quantum theory is nontrivial. As was recognized by Birkhoff and von Neumann, (Birkhoff, G., and von Neumann, J., 1936, Ann. Math. 37, 823-843) the distributive law a · (b∪c) = (a · b) ∪ (a · c) is not valid for quantum systems. Without this law, the rule for the likelihood of the logical sum of alternatives, Eqs. (26), (27) would not have held. The physical culprit is quantum interference, which, indeed, invalidates probability sum rules (as is well appreciated in the examples such as the double slit experiment). Decoherence destroys interference between the einselected states [states ‘superselected’ via their environmental relations]. Thus, with decoherence, Boolean logic, and, consequently, classical probability calculus with its sum rules are recovered… Thus, starting from an assumption about the validity of classical logic (i.e., absence of interference) we have arrived, first, at the sum rule for probabilities and, subsequently, at the Born’s formula. (1998, 12-13)

            The second major criticism of the decoherence-based approaches is that although the phenomenon of quantum decoherence can be and has been rigorously studied in the laboratory setting, it remains unclear what, precisely, generates decoherence—that is, what are the physical relata that generate the required degrees of freedom necessary for logical integration and negative selection of the off-diagonal terms? Typically, decoherence is understood to be ‘induced’ by phase correlations between system and ‘environment.’ Spin baths and oscillator baths are two types of experimental arrangements that have been employed to test environmental decoherence.  The difficulty is that a number of experiments have yielded decoherence rates that far exceed those predicted by the usual environmental decoherence models—i.e., those restricted to ‘external’ physical interactions.
            This has led some theorists to explore the notion of intrinsic or ‘internally environmental’ relata that, when added to externally environmental relations, might account for the observed decoherence rates.  Philosophically, this idea of ‘intrinsic’ sources of decoherence implies the possibility of a substance-definitive conception of decoherence such as that proposed by the relational realist interpretation of quantum mechanics—sources that can be described ontologically as Nature’s ultimate constituent ‘final real things.’ The relational realist depiction of the latter as quantum mechanical ‘becomings,’ discussed above, is certainly compatible with the physical theory of ‘intrinsic decoherence.’
 
Co-Investigator P.C.E. Stamp writes:

By ‘intrinsic’ sources [of decoherence in Nature] is meant sources which are inevitable in the world as it is, not arising from dissipative processes and perhaps even arising as part of the basic structure of the universe.

Such intrinsic sources of decoherence in Nature, operating even at T=0, would not only provide a way of explaining the ‘emergence of classical physics’ in fields ranging from quantum cosmology to condensed matter physics; they would also place a fundamental limit on the observability of quantum phenomena. This would limit the possibility of seeing macroscopic quantum phenomena, and also place fundamental limits on the superpositions required for quantum computing.

Possibilities for intrinsic decoherence mechanisms have already emerged from both low- and high-energy physics. From low-energy physics there has been a suggestion that zero point modes of continuous quantum fields (in particular, the photon field) could cause T=0 decoherence. This has, for example, been suggested as an explanation of the decoherence saturation at low T in mesoscopic conductors (Mohanty et al., 1997)… (490)

            To the extent that the decoherence-based interpretations of quantum mechanics are at least sufficiently developed to have generated rigorous experimental testing, likewise the philosophical implications of the general principles of the se interpretations are equally worthy of careful exploration.  In Epperson 2004, I proposed that the philosophical innovations inherent in the decoherence-based interpretations correlate closely with those proposed by Alfred North Whitehead in Process and Reality.  It must be emphasized, however, that my exploration of these correlations was and is not intended as an ‘explanatory’ scheme by which either Whitehead’s metaphysics or the quantum theory can be ‘properly’ understood, each in terms of the other. What I explore, rather, is simply the proposition that the fundamental features of Whitehead’s speculative metaphysical scheme and his conception of dipolar actuality can be seen as exemplified in those interpretations of quantum mechanics that make use of the decoherence effect.  Circularity of validation—whether the physics is to be understood as validating the metaphysics or vice versa—is avoided by acknowledging, in a speculative philosophical context, that the first principles common to both are necessarily presupposed within the proposition and thus incapable of simple proof by deduction or demonstration.  These first principles, both logical and ontological, constitute the starting point for exploration, not the end point of explanation.   
            But beyond merely exploring this proposition, I do attempt to argue for its fitness to the task of interpreting quantum mechanics. I offer this argument in the same speculative philosophical spirit in which Whitehead argued for the fitness of his metaphysical scheme to the task of understanding (though not ‘explaining’) nature—not by the ‘provability’ of his first principles via deduction or demonstration, but by their evaluation against the metrics of coherence and empirical adequacy.
            In the most general terms, both decoherence-based interpretations and relational realist philosophy depict the following:

1. Fundamental substance defined as quantum facts whose actualization is predicated upon internal relations with an environment of antecedently settled quantum facts—the data of the settled world.  These relations are both causal-physical, and logical-conceptual in that the physical relations are, via the process of quantum decoherence, logically conditioned and thus productive of an integrated objectification. Thus each quantum actualization-of-fact, when considered as a substantial ‘thing-in-itself’ evinces both a physical-causal pole and a logical-conceptual pole.  The term ‘dipolar’ signifies the fact that the physical and logical features of a quantum mechanical becoming are mutually implicative in quantum mechanical actualizations.

2. The internal relations with the dative environment are logically integrated, first into a ‘pure state’ comprising superdenumerable potential outcome states.  These integrations are then further integrated into coarse-grained equivalence classes. A negative selection process conditioned by the logical principle of Non-Contradiction and fuelled by the massive degrees of freedom yielded by environmental relations (both physical-dissipative and logical non-dissipative), eliminates potentia logically incapable of integration. This is represented by the cancellation of off-diagonal terms and the evolution of the density matrix to the reduced state.

3. This integration, described by von Neumann’s Process 1, results in a probability ‘valuation’ of the remaining potential outcome states. By the Principle of the Excluded Middle, one of these probability-valuated potential outcome states/subjective forms will be actualized, in accord with its valuation—i.e., its probability amplitude.

            When actualized, the occasion has its causal efficacy upon subsequent actualizations; and likewise, its process of actualization is affected by both 1) its causal physical relations with the actualities ‘physically’ prior to it in terms of spatiotemporal extensiveness—i.e., within its backward lightcone, per the restrictions of relativity theory. Physical antecedence is reflected, for example, in the statement ‘p causes q’; and 2) its logical relations with those actualities ‘logically’ prior to it, per the restrictions of the Principle of Non-Contradiction, among other logical restrictions.  Logical antecedence is reflected in the statement ‘p then q.’
            Each actual occasion is thus fundamentally a unit of relation.  In the process of actualization, the becoming occasion is internally related to its dative world of antecedently actualized quantum facts, and these relations are both causally and logically integrated; and since the actualization is internally related to its dative world, the relations form potential histories subsuming both the becoming quantum fact and its dative world.
            Thus, another philosophically problematic conceptual pair, ‘unity’ and ‘diversity’ are also brought into coherence via mutual implication.  “The many,” writes Whitehead, “become one, and are increased by one. In their natures, entities are disjunctively ‘many’ in process of passage into conjunctive unity.” (21) These integrations are always logically governed, such that upon actualization via decoherence, the quantum fact is related to the world in a manner free of violations of the Principle of Non-Contradiction, or any other such violation of the logical order.  Writ large, the universe is thus described as a networked system of serial routes or ‘histories’ (Griffiths, 1984 and 2002) of quantum facts; the integrative internal relations of every actualization constitutive of each route are logically governed such that the orders of causal relation and logical implication are correlated (again, this is a speculative presupposition grounded in the concept of the dipolar actual occasion).
            Those interpretations of quantum mechanics that make use of the decoherence effect are unique in their reliance upon the explicit operation of two categories of first principles, ontological and logical.  The ontological first principle is that which categorizes actuality and potentiality as two species of reality—a long overdue rehabilitation of Aristotle’s signature improvement of the Parmenidean worldview, which depicted ‘becoming’ as pure illusion.  The logical first principles are PNC and PEM, without whose presupposition the correlation of causal relation and logical implication would remain wholly unfounded. Science requires that these conclusions follow both causally and logically from premises and not merely by conjunction, either random or constant; quantum mechanics, by the Heisenberg uncertainty principle, requires that the final outcome state be not only subsequent to the evolution but causally and logically consequent of it.

From the Emergence of Classicality
To the Emergence of Complex Adaptive Systems

            In collaboration with Co-I and CPNS Fellow Stuart Kauffman, Professor of Biological Sciences and Physics, and Director, Institute for Biocomplexity and Informatics (IBI), University of Calgary, we will examine the function of bivalent propositional logic in describing intrinsic decoherence at the micro-scale, and explore the implications for complex adaptive systems theory at the macro-scale (e.g., the ‘emergence’ of classical mechanics from quantum mechanics).
            We will investigate the ways in which Kauffman’s work on biological adaptive complexity might find a robust philosophical and physical foundation in the relational realist approach to quantum logical causality—a modern rehabilitation of the process event-ontology of Whitehead.  Among the topics to be explored will be the relationship between the Whiteheadian spacetime mereotopology described by the relational realist interpretation, and possible topological models describing Kauffman’s conception of evolving selective complexity in nature.
            More broadly, we will undertake the development of a rigorous philosophical-physical model that accommodates both 1. the phenomenon of natural self-organization (cf. the Kauffman Models);  and 2. the phenomenon of natural selection. Though there are interpretations of quantum mechanics that can be applied to the task of modeling self-organization (i.e., complexity) in nature, there are as yet no interpretations that can adequately accommodate the phenomenon of selective complexity exemplified by Darwinian evolution. And likewise, though Darwinism offers a robust, empirically validated model for natural selection, it does not accommodate the phenomenon of self-organization. Kauffman thus writes that we must seek to “broaden evolutionary theory to describe what happens when selection acts on systems that already have robust self-organizing properties. This body of theory simply does not exist.”
            Epperson suggests that the logical-causal, praxiological formalism of the relational realist interpretation of quantum mechanics might provide a basic framework for such a theory—particularly when explored in the context of Whitehead’s Philosophy of Organism. The latter entails a rigorous schematization of both natural self-organization and natural selection in terms of fundamental relational, event-ontological processes; and recent work (Epperson 2004) has shown that the formalism by which Whitehead describes these processes is precisely reflected in modern, decoherence-based interpretations of quantum mechanics (cf. Robert Griffiths, 1984, 2002; Roland Omnès, 1994, and Murray Gell-Mann, 1997.)  A key goal of the relational realist program is to develop a descriptive scheme by which both these quantum mechanical features of nature and its self-organizing selective complexity are accommodated--both in terms of philosophical coherence and empirical adequacy.

APPLICATIONS

A Category-Theoretic Mereotopological Model of Quantum Spacetime

See the following project publications:  

E. Zafiris, A. Mallios, “The Homological Kähler-De Rham Differential Mechanism, Part I: Application in General Theory of Relativity,” Advances in Mathematical Physics (2011)

E. Zafiris, A. Mallios, “The Homological Kähler-de Rham Differential Mechanism, Part II. Sheaf-Theoretic Localization of Quantum Dynamics,” Advances in Mathematical Physics (2011)

M. Epperson, E. Zafiris, Foundations of Relational Realism: A Topological Approach to Quantum Mechanics and the Philosophy of Nature. Lexington Books / Rowman & Littlefield, New York (2013)

          The quantum decoherent histories formalism, in its path-integral form, provides a spacetime formulation of quantum theory that is highly compatible with the mereotopological model of spatiotemporal extension proposed by Alfred North Whitehead. This is crucial because the mathematical rigidity of the Hilbert space (as a topological vector space) does not allow a relativization analogous with the one of classical relativity theory on smooth manifolds. By contrast, a mereotopological/category-theoretic reformulation of quantum logic in terms of Boolean localization systems (Booelan sheaves, per the work of Co-I Elias Zafiris) achieves precisely such an objective by generalizing the smooth manifold construction in generic algebraic/categorical terms.
            This makes it possible to formulate a framework of local/global or part/whole relations without the intervention of a spacetime classically conceived. On the contrary, the usual spacetime manifold and its metrical relations (imposing extra conditions like the light-cone causality relations) appear as emergent at a higher level than the more fundamental level of logical and algebraic part-whole relations.  Epperson’s distinction between a) mereological/topological/logical relations, and b) physical/metrical/causal relations, is reflected precisely in Zafiris’ distinction between these two different levels--the algebraic part-whole relations and the metrical spacetime relations, respectively-- where the latter can be seen as an emergent/metrical specialization of the former. The key issue to be explored is the problem of localization and the problem of passing from the local to the global in an extensive continuum.
            Note that these problems are not metrical but mereotopological. The emergent metrical relations of classical relativity theory presuppose a point-wise localization on a smooth manifold and the metric is a (local) expression of the simultaneity conventions imposed by the constancy of the speed of light. But in the quantum theoretic level, the assumption of point-wise localization of events is not feasible unless a measurement context has been specified and also a specific type of measurement has been performed.
            In Whitehead's language this is reflected in the distinction between genetic division (presupposing a generic/algebraic type of localization) and coordinate division (like the one of the classical spacetime manifold endowed with a semi-Riemannian metric and dictating the light-cone causality relations). Since now we can prove that the latter is an emergent form of the more fundamental former division, there is no tension between the fundamental mereological order of the extensive continuum (modeled in terms of Co-I Elias Zafiris’s sheaf-theoretic topos
model) and the more special spatio-temporal causal extensive order pertinent to a point-wise localized and metricized continuum. This has fundamental consequences for the interpretation of the mechanism of differential extensive analysis, which is crucial for physics since it captures the dynamical relations in an extensive continuum of events.
            In Process and Reality and other works, Alfred North Whitehead developed the basic conceptual framework for this mereotopological understanding of spacetime, which he termed ‘The Theory of Extensive Abstraction.’  The formal categorical/algebraic formulation we propose herein, however, permits us to refine and integrate these ideas into modern quantum theory via, in particular, the decoherent histories formalism.

A Topological, Sheaf-Theoretic Explication of Quantum Geometric Phases
By Analysis of Experimental Data on the Aharonov-Bohm Effect, the Pancharatnam Phase, and the Quantum Hall Effect, Toward a Unified Interpretation

White Paper

Current Publication: E. Zafiris, "The Global Symmetry Group of Quantum Spectral Beams and Geometric Phase Factors," Advances in Mathematical Physics (2015)

            Foundations of Relational Realism (Lexington Books / Rowman & Littlefield, 2013) represents the capstone of our initial project, along with a number of published papers and conference presentations. With this foundational work in place, our next step is to demonstrate the experimental applicability of this sheaf theoretic quantum formalism and its philosophical conceptual framework—viz. its advantages both in terms of prediction and interpretation of data. To that end, for 2013-15, we propose to apply the relational realist framework to the task of explicating the well-known but poorly understood problem of quantum geometric phases. We will do so by analysis of experimental data on the Aharonov-Bohm Effect, the Pancharatnam Phase, and the Quantum Hall Effect, toward a unified interpretation of all three.

            

"Knowledge of the classical electromagnetic field acting locally on a particle is not sufficient to predict its quantum-mechanical behavior... For a long time it was believed that A [the vector potential] was not a ‘real’ field... There are phenomena involving quantum mechanics which show that in fact A is a ‘real’ field in the sense that we have defined it... E and B [electromagnetic fields] are slowly disappearing from the modern expression of physical laws; they are being replaced by A [the vector potential] and [the scalar potential].”

Richard Feynman
The Feynman Lectures on Physics

"We believe that the concept of  a geometric phase, repeating the history of the group concept, will eventually find so may realizations and applications in physics that it will repay study for its own sake, and become part of the lingua franca."

A. Shapere and F. Wilczek

            In a groundbreaking paper published in the mid-1980’s, Sir M. Berry discovered that a quantum system undergoing a slowly evolving (adiabatic) cyclic evolution retains a “memory” of its motion after coming back to its original physical state. This “memory” is expressed by means of a complex phase factor in the wave-function of the system, called Berry’s phase or the geometric phase. The main idea is that a quantum system in a slowly changing environment displays a history dependent geometric effect: When the environment returns to its original state, the system also does, but for an additional phase.
            The Berry phase is a complex number of modulus one and is experimentally observable. The two most important properties of the geometric phase are [1] that it is a statistical object, and [2] it can be measured only relatively. Thus it becomes observable by comparing the evolution of two distinct statistical ensembles of systems through their interference pattern. The Berry phase is called ‘geometric’ because it depends solely on the topology of the pathway along which the system evolves and neither on the temporal metric duration of the evolution nor on the dynamics that is applied to the system.
            After Berry’s initial experimental discovery of the geometric phase, it has been demonstrated that it also exists for non-adiabatic evolutions and even for sequences of measurements. All these experimental observations of the geometric phase phenomenon, including the Aharonov-Bohm, Pancharatnam, and Quantum Hall effects—as well as features of molecular and nuclear spectra, vortices occurring in superfluid helium, and even chemical reactions—help to emphasize its purely topological nature. In this way, the geometric phase observable initiated a tremendous surge of experimental research across a diverse range of physics disciplines establishing surprising connections among a number of apparently disparate phenomena—again, viz. the Aharonov-Bohm effect, its molecular physics analogue by Mead and Truhlar, polarization optics effects, and the Quantum Hall effect. Thus, we expect that a unified topological approach to the study, description and interpretation of all these experimentally  well-established  geometric phase phenomena will shed light on their nature, lead to predictions of new effects of a similar type, and even provide a paradigm for applications in areas beyond the strict domain of natural sciences.
            The novel topological approach to quantum mechanics we present in Foundations of Relational Realism, in the context of our overall Fetzer research project publications, provides a promising candidate for evaluating consistently the experimental data of geometric phases observed in all the above experiments. Such an evaluation would demonstrate our work’s applicability to diverse quantum phenomena, explicated via the lens of a unifying topological theoretical framework, which is based on the powerful modern mathematical methods of category theory and sheaf theory.
            The application of our topological formalism to the analysis of experimental data, carrying with it an associated unifying realistic interpretation of geometric phase phenomena, thus constitutes a natural continuation and experimental substantiation of our previous research program. The basic principle underlying the crucial explanatory role of this approach in relation to geometric phases is the following: Whenever a global physical system is partitioned into local parts and one attempts to describe a subsystem in isolation, the influence and connectivity of the others is manifested through geometric phase effects of a pure topological origin.
            The pervasiveness throughout the natural sciences of this part-whole methodology accounts for the ubiquity of the geometric phase effect, but not its explanation. We propose to solve this problem via our realistic mereotopological formalism and conceptual framework. To that end, the precise mathematical notion which captures the topological origin of geometric phases is the anholonomy of a connection on a sheaf modeling the structure of events. The notion of a sheaf connection provides the technical means for describing the process of parallel transport of the state vector of a system along a curve on the space of the control variables. The particular transformation undergone by the state vector when it is parallel-transported along a closed curve is called the anholonomy of the sheaf connection. Thus, the anholonomy describes the state vector transformation induced by cyclic changes (loops) in the controlling variables. A simple intuitive example from macroscopic geometry is provided by the rotation of a vector when parallel-transported along a closed path on a curved surface.  The topological nature of the effect is realized by the fact that if the parallel transport of a vector takes place along a closed path on a plane, the vector comes back unchanged; whereas if it takes place along a closed path on a sphere it gets rotated by an angle, which is measured precisely by the anholonomy of the connection.            

Sample of Experiments Analyzed

Yano, S. and Yamada, H. (1986). “Evidence for Aharonov-Bohm Effect with Magnetic Field Completely Shielded from Electron Wave” Physical Review Letters, 56 (8): 792-795

Osakabe, N; et al. (1986). “Experimental confirmation of Aharonov–Bohm effect using a toroidal magnetic field confined by a superconductor”. Physical Review A 34 (2): 815–822.

Webb, RA; Washburn, S; Umbach, CP; Laibowitz, RB (1985). “Observation of h/e Aharonov–Bohm Oscillations in Normal-Metal Rings”. Physical Review Letters 54 (25): 2696–2699

Schönenberger, C; Bachtold, Adrian; Strunk, Christoph; Salvetat, Jean-Paul; Bonard, Jean-Marc; Forró, Laszló; Nussbaumer, Thomas (1999). “Aharonov–Bohm oscillations in carbon nanotubes”. Nature 397 (6721): 673.

van Oudenaarden, A; Devoret, Michel H.; Nazarov, Yu. V.; Mooij, J. E. (1998). “Magneto-electric Aharonov–Bohm effect in metal rings”. Nature 391 (6669): 768.

Pancharatnam, S. Proc. Indian Acad. Sci. A44, 247−262 (1956); reprinted in Collected Works of S. Pancharatnam (Oxford Univ. Press, 1975)

Anandan, J. “The Geometric Phase,” Nature 360, 307 - 313 (26 November 1992)

Tomita, A. & Chiao, R. Y. “Observation of Berry's Topological Phase by Use of an Optical Fiber” Phys. Rev. Lett. 57, 937−940 (1986)

Hsieh, D.; Qian, D.; Wray, L.; Xia, Y.; Hor, Y. S.; Cava, R. J.; Hasan, M. Z. (2008). “A topological Dirac insulator in a quantum spin Hall phase”. Nature  452 (7190): 970–974.

Baumgartner, A.; Ihn, T.; Ensslin, K.; Maranowski, K.; Gossard, A. (2007). “Quantum Hall effect transition in scanning gate experiments”. Physical Review B 76 (8)

References