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Basil James Hiley (15 November 1935 – 25 January 2025) was a British physicist and professor emeritus of the University of London.
Long-time colleague of David Bohm, Hiley is known for his work with Bohm on and for his work on algebraic descriptions of quantum mechanics in terms of underlying symplectic and orthogonal Clifford algebras. Basil Hiley, website of Maurice A. de Gosson, 2005, accessed on 1 September 2012 Hiley co-authored the book The Undivided Universe with David Bohm, which is considered the main reference for Bohmian mechanics.
The work of Bohm and Hiley has been characterized as primarily addressing the question "whether we can have an adequate conception of the reality of a quantum system, be this causal or be it stochastic or be it of any other nature" and meeting the scientific challenge of providing a mathematical description of quantum systems that matches the idea of an implicate order.
Hiley performed undergraduate studies at King's College London. In 1962 he obtained his PhD from King's College in condensed matter physics, more specifically on cooperative phenomena in Ferromagnetism and long chain polymer models, under the supervision of Cyril Domb and Michael Fisher.
Hiley first met David Bohm during a week-end meeting organized by the student society of King's College at Cumberland Lodge, where Bohm held a lecture. In 1961 Hiley was appointed assistant lecturer at Birkbeck College, where Bohm had taken the chair of Theoretical Physics shortly before. Hiley wanted to investigate how physics could be based on a notion of process, and he found that David Bohm held similar ideas. He reports that during the seminars he held together with Roger Penrose he
Hiley worked with David Bohm for many years on fundamental problems of theoretical physics.See for example the characterization of their work together by Joseph Jaworski in Jaworksi's book Source: The Inner Path of Knowledge Creation, Berrett-Koehler Publishers, 2012 Initially Bohm's model of 1952 did not feature in their discussions; this changed when Hiley asked himself whether the "Einstein-Schrödinger equation", as Wheeler called it, might be found by studying the full implications of that model. They worked together closely for three decades. Together they wrote many publications, including the book The Undivided Universe: An Ontological Interpretation of Quantum Theory, published 1993, which is now considered the major reference for Bohm's interpretation of quantum theory.
In 1995, Basil Hiley was appointed to the chair in physics at Birkbeck College at the University of London. Basil Hiley (short CV), Scientific and Medical Network He was awarded the 2012 Majorana Prize in the category The Best Person in Physics for the algebraic approach to quantum mechanics and furthermore in recognition of "his paramount importance as natural philosopher, his critical and open minded attitude towards the role of science in contemporary culture". Department Fellow wins Majorana Prize, Birkbeck College (downloaded 12 June 2013) The Majorana Prize, www.majoranaprize.com (downloaded 12 June 2013)
Hiley died on 25 January 2025, at the age of 89.
By performing numeric computations on the basis of the quantum potential, Chris Philippidis, Chris Dewdney and Basil Hiley used computer simulations to deduce ensembles of particle trajectories that could account for the interference fringes in the double-slit experiment and worked out descriptions of scattering processes. Their work renewed the interests of physicists in the Bohm interpretation of quantum physics.Olival Freire jr.: A story without an ending: the quantum physics controversy 1950–1970, Science & Education, vol. 12, pp. 573–586, 2003, p. 576 In 1979, Bohm and Hiley discussed the Aharonov–Bohm effect which had recently found experimental confirmation. They called attention to the importance of the early work of Louis de Broglie on , emphasizing his insight and physical intuition and stating that developments based on his ideas aimed at a better understanding than mathematical formalism alone.David Bohm, Basil Hiley: The de Broglie pilot wave theory and the further development and new insights arising out of it, Foundations of Physics, volume 12, number 10, 1982, Appendix: On the background of the papers on trajectories interpretation, by D. Bohm, ( PDF ) They offered ways of understanding quantum non-locality and the measurement process,David J. Bohm, Basil J. Hiley: Some Remarks on Jack Sarfatti's Proposed Connection Between Quantum Phenomena and the Volitional Activity of the Observer-Participator. Psychoenergetic Systems 1: 173–179, 1976David J. Bohm, Basil J. Hiley: Einstein and Non-Locality in the Quantum Theory. In Einstein: The First Hundred Years, ed. Maurice Goldsmith, Alan Mackay, and James Woudhugsen, pp. 47–61. Oxford: Pergamon Press, 1980 the limit of classicality, interference and quantum tunneling.See also the citation of Bohm and Hiley's article Unbroken Quantum Realism, from Microscopic to Macroscopic Levels by David Hestenes: "Bohm and Hiley, among others, have argued forcefully that the identification of bicharacteristics of the Schrödinger wave function with possible electron paths lead to sensible particle interpretations of electron interference and tunneling as well as other aspects of Schrödinger electron theory." David Hestenes: On decoupling probability from kinematics in quantum mechanics, In: P.F. Fougère (ed.): Maximum Entropy and Bayesian Methods, Kluwer Academic Publishers, 1990, pp. 161–183
They showed how in the Bohm model, introducing the concept of active information, the measurement problem and the collapse of the wave function, could be understood in terms of the quantum potential approach, and that this approach could be extended to relativistic quantum field theories. They described the measurement process and the impossibility of measuring position and momentum simultaneously as follows: "The ѱ field itself changes since it must satisfy the Schrödinger equation, which now contains the interaction between the particle and apparatus, and it is this change that makes it impossible to measure position and momentum together".With reference to Bohm's publication of 1952, cited from Basil J. Hiley: The role of the quantum potential. In: G. Tarozzi, Alwyn Van der Merwe: Open questions in quantum physics: invited papers on the foundations of microphysics, Springer, 1985, pages 237 ff., therein page 238 The collapse of the wave function of the Copenhagen interpretation of quantum theory is explained in the quantum potential approach by the demonstration that information can become inactive Interview with Basil Hiley conducted by M. Perus, downloaded February 15, 2012 in the sense that from then on "all the packets of the multi-dimensional wave function that do not correspond to the actual result of measurement have no effect on the particle".Basil J. Hiley: The role of the quantum potential. In: G. Tarozzi, Alwyn Van der Merwe: Open questions in quantum physics: invited papers on the foundations of microphysics, Springer, 1985, pages 237 ff., therein page 239
Summarizing Bohm's and his own interpretation, Hiley has explained that the quantum potential "does not give rise to a mechanical force in the Newtonian sense. Thus while the Newtonian potential drives the particle along the trajectory, the quantum potential organises the form of the trajectories in response to the experimental conditions." The quantum potential can be understood as an aspect of "some kind of self-organising process" involving a basic underlying field.B. J. Hiley: Active Information and Teleportation, In: Epistemological and Experimental Perspectives on Quantum Physics, D. Greenberger et al. (eds.), pages 113–126, Kluwer, Netherlands, 1999, p. 7For a "point of view that goes beyond mechanicm", see also Chapter V. of D. Bohm's book Causality and Chance in Modern Physics, 1957, Routledge, The quantum potential (or information potential) links the quantum system under investigation to the measuring apparatus, thereby giving that system a significance within the context defined by the apparatus.B. J. Hiley: Information, quantum theory and the brain. In: Gordon G. Globus (ed.), Karl H. Pribram (ed.), Giuseppe Vitiello (ed.): Brain and being: at the boundary between science, philosophy, language and arts, Advances in Consciousness Research, John Benjamins B.V., 2004, , pp. 197–214, see p. 207 and p. 212 It acts on each quantum particle individually, each particle influencing itself. Hiley cites the wording of Paul Dirac: " Each electron only interferes with itself" and adds: "Somehow the 'quantum force' is a 'private' force. It thus cannot be regarded as a distortion of some underlying sub-quantum medium as was originally suggested by de Broglie".B. J. Hiley: Nonlocality in microsystems, in: Joseph S. King, Karl H. Pribram (eds.): Scale in Conscious Experience: Is the Brain Too Important to be Left to Specialists to Study?, Psychology Press, 1995, pp. 318 ff., see p. 326–327 It is independent of field intensity, thus fulfilling a precondition for non-locality, and it carries information about the whole experimental arrangement in which the particle finds itself.
In processes of non-signalling transmission of in a system consisting of multiple particles (a process that is generally called "quantum teleportation" by physicists), active information is transferred from one particle to another, and in the Bohm model this transfer is mediated by the non-local quantum potential.B. J. Hiley: Active Information and Teleportation, In: Epistemological and Experimental Perspectives on Quantum Physics, D. Greenberger et al. (eds.), pages 113–126, Kluwer, Netherlands, 1999 ( PDF)
The Bohm–Hiley approach to relativistic quantum field theory (RQFT) as presented in Bohm and Hiley's book Undivided Universe and in the work of their co-worker Kaloyerou was reviewed and re-interpreted by Abel Miranda, who stated:
As of 1980, Hiley and his co-worker Fabio A. M. Frescura expanded on the notion of an implicate order by building on the work of Fritz Sauter and Marcel Riesz who had identified with Minimal ideal of an algebra. The identification of algebraic spinors with minimal left ideals, which can be seen as a generalization of the ordinary spinorBasil Hiley: Algebraic quantum mechanics, algebraic spinors and Hilbert space, Boundaries, Scientific Aspects of ANPA, 2003 ( preprint) was to become central to the Birkbeck group's work on algebraic approaches to quantum mechanics and quantum field theory. Frescura and Hiley considered algebras that had been developed in the 19th century by the mathematicians Hermann Grassmann, William Rowan Hamilton, and William Kingdon Clifford.F. A. M. Frescura, B. J. Hiley: Geometric interpretation of the Pauli spinor, American Journal of Physics, February 1981, Volume 49, Issue 2, pp. 152 ( abstract) As Bohm and his colleagues emphasized, in such an algebraic approach operators and operands are of the same type: "there is no need for the disjoint features of the present mathematical formalism of, namely the operators on the one hand and the quantum state on the other. Rather, one uses only a single type of object, the algebraic element"., and its introductory note More specifically, Frescura and Hiley showed how "the states of quantum theory become elements of the minimal ideals of the algebra and .. the projection operators are just the idempotents which generate these ideals". In a 1981 preprint that remained unpublished for many years, Bohm, P. G. Davies and Hiley presented their algebraic approach in context with the work of Arthur Stanley Eddington. Hiley later pointed out that Eddington attributed to a particle not a metaphysical existence but a structural existence as an idempotent of an algebra, similarly as in process philosophy an object is a system which continuously transforms onto itself. With their approach based on algebraic idempotents, Bohm and Hiley "incorporate Niels Bohr's notion of 'wholeness' and d'Espagnat's concept of 'non-separability' in a very basic way".
In 1981, Bohm and Hiley introduced the "characteristic matrix", a non-Hermitian extension of the density matrix. The Wigner and Moyal transformation of the characteristic matrix yields a complex function, for which the dynamics can be described in terms of a (generalized) Liouville equation with the aid of a matrix operating in phase space, leading to eigenvalues that can be identified with stationary states of motion. From the characteristic matrix, they constructed a further matrix that has only non-negative eigenvalues which can thus be interpreted as a quantum "statistical matrix". Bohm and Hiley thus demonstrated a relation between the Wigner–Moyal approach and Bohm's theory of an implicate order that allows to avoid the problem of negative probabilities. They noted that this work stands in close connection with Ilya Prigogine's proposal of a Liouville space extension of quantum mechanics. They extended this approach further to relativistic phase space by applying the phase space interpretation of Mario Schönberg to the Dirac algebra. (1983). 9781468488326 ISBN 9781468488326 Their approach was subsequently applied by Peter R. Holland to and by Alves O. Bolivar to .A.O. Bolivar: Classical limit of bosons in phase space, Physica A: Statistical Mechanics and its Applications, vol. 315, no. 3–4, December 2002, pp. 601–615
In 1984, Hiley and Frescura discussed an algebraic approach to Bohm's notion of implicate and explicit orders: the implicate order is carried by an algebra, the explicate order is contained in the various representations of this algebra, and the geometry of space and time appear at a higher level of abstraction of the algebra.F. A. M. Frescura, B. J. Hiley: Algebras, quantum theory and pre-space, p. 3–4 (published in Revista Brasileira de Fisica, Volume Especial, Julho 1984, Os 70 anos de Mario Schonberg, pp. 49–86) Bohm and Hiley expanded on the concept that "relativistic quantum mechanics can be expressed completely through the interweaving of three basic algebras, the bosonic, the fermionic and the Clifford" and that in this manner "the whole of relativistic quantum mechanics can also be put into an implicate order" as suggested in earlier publications of David Bohm from 1973 and 1980.D. Bohm, B. J. Hiley: Generalisation of the twistor to Clifford algebras as a basis for geometry, published in Revista Brasileira de Fisica, Volume Especial, Os 70 anos de Mario Schönberg, pp. 1–26, 1984 ( PDF) On this basis, they expressed the twistor theory of Penrose as a Clifford algebra, thereby describing structure and forms of ordinary space as an explicit order that unfolds from an implicate order, the latter constituting a pre-space. The spinor is described mathematically as an ideal in the Pauli algebra, the twistor as an ideal in the conformal Clifford algebra.B. J. Hiley, F. David Peat: General Introduction: The development of Bohm's ideas from plasma to the implicate order, in: Basil . Hiley, F. David Peat (eds.): Quantum implications: essays in honour of David Bohm, Routledge, 1987, , pp. 1–32, therein: p. 25
The notion of another order underlying space was not new. Along similar lines, both Gerard 't Hooft and John Archibald Wheeler, questioning whether space-time was the appropriate starting-point for describing physics, had called for a deeper structure as starting point. In particular, Wheeler had proposed a notion of pre-space which he called pregeometry, from which spacetime geometry should emerge as a limiting case. Bohm and Hiley underlined Wheeler's view, yet pointed out that they did not build on the quantum foam proposed by Wheeler and by Stephen Hawking but rather worked towards a representation of the implicate order in form of an appropriate algebra or other pre-space, with spacetime itself considered part of an explicit order that is connected to pre-space as implicit order. The spacetime manifold and properties of locality and non-locality then arise from an order in such pre-space.
In the view of Bohm and Hiley, "things, such as particles, objects, and indeed subjects, are considered as semi-autonomous quasi-local features of this underlying activity". These features can be considered to be independent only up to a certain level of approximation in which certain criteria are fulfilled. In this picture, the classical limit for quantum phenomena, in terms of a condition that the action function is not much greater than the Planck constant, indicates one such criterion. Bohm and Hiley used the word holomovement for the underlying activity in the various orders together. This term is intended to extend beyond the movement of objects in space and beyond the notion of process, covering movement in a wide context such as for instance the "movement" of a symphony: "a total ordering which involves the whole movement, past and anticipated, at any one moment". This concept, which avowedly has similarities with the notion of organic mechanism of Alfred North Whitehead, underlies Bohm and Hiley's efforts to establish algebraic structures that relate to quantum physics and to find an ordering that describes thought processes and the mind.
They investigated non-locality of spacetime also in terms of the time dimension. In 1985, Bohm and Hiley showed that Wheeler's delayed-choice experiment does not require the existence of the past to be limited to its recording in the present. Hiley and R. E. Callaghan later confirmed this view, which stands in stark contrast to Wheeler's earlier statement that "the past has no existence except as it is recorded in the present",John Wheeler, cited after Huw Price: Time's Arrow & Archimedes' Point: New Directions for the Physics of Time, Oxford University Press, 1996, , p. 135 by a detailed trajectory analysis for delayed choice experiments and by an investigation into welcher Weg experiments. Hiley and Callaghan in fact showed that, an interpretation of Wheeler's delayed choice experiment based on Bohm's model, the past is an objective history that cannot be altered retroactively by delayed choice ( see also: Bohmian interpretation of Wheeler's delayed choice experiment).
Bohm and Hiley sketched also how Bohm's model could be treated under the point of view of statistical mechanics, and their joint work on this was published in their book (1993) and a subsequent publication (1996).
Hiley has pursued work on algebraic structures in quantum theory throughout his scientific career.B. J. Hiley: A note on the role of idempotents in the extended Heisenberg algebra, Implications, Scientific Aspects of ANPA 22, pp. 107–121, Cambridge, 2001Basil J. Hiley: Towards a Dynamics of Moments: The Role of Algebraic Deformation and Inequivalent Vacuum States, published in: Correlations ed. K. G. Bowden, Proc. ANPA 23, 104–134, 2001 ( PDF)B.J. Hiley: Non-Commutative Quantum Geometry: A Reappraisal of the Bohm Approach to Quantum Theory. In: Avshalom C. Elitzur, Shahar Dolev, Nancy Kolenda (eds.): Quo Vadis Quantum Mechanics? The Frontiers Collection, 2005, pp. 299–324, ( abstract, preprint) After Bohm's death in 1992, he published several papers on how different formulations of quantum physics, including Bohm's, can be brought in context. (2025). 9783540706229 ISBN 9783540706229 (2025). 9783642128202 ISBN 9783642128202 Hiley also pursued further work on the thought experiments set out by the EPR paradox of Albert Einstein Boris Podolsky and Nathan Rosen and by Hardy's paradox of Lucien Hardy, in particular considering the relation to special relativity. (2025). 9783540706229 ISBN 9783540706229
In the late 1990s, Hiley expanded further on the notion he had developed with Bohm on the description of quantum phenomena in terms of processes.Basil Hiley: Mind and matter: aspects of the implicate order described through algebra, published in: Karl H. Pribram, J. King (eds.): Learning as Self-Organization, pp. 569–586, Lawrence Erlbaum Associates, New Jersey, 1996, Basil J. Hiley, Marco Fernandes: Process and time, in: H. Atmanspacher, E. Ruhnau: Time, temporality, now: experiencing time and concepts of time in an interdisciplinary perspective, pp. 365–383, Springer, 1997, ( preprint) Hiley and his co-worker Marco Fernandes interpret time as an aspect of process that should be represented by a mathematically appropriate description in terms of an algebra of process. For Hiley and Fernandes, time should be considered in terms of "moments" rather than extensionless points in time, in conventional terms implying an integration over time, recalling also that from the "characteristic matrix" of Bohm and Hiley a positive definite probability can be obtained. They model the unfolding of implicate and explicate orders and the evolution of such orders by a mathematical formalism which Hiley has termed the Clifford algebra of process.
In 2000, Brown and Hiley showed that the Schrödinger equation can be written in a purely algebraic form that is independent of any representation in a Hilbert space. This algebraic description is formulated in terms of two operator equations. The first of these (formulated in terms of the commutator) represents an alternative form of the quantum Liouville equation, which is well known to describe the conservation of probability, the second (formulated in terms of the anticommutator), which they dubbed the "quantum phase equation", describes the conservation of energy. This algebraic description in turn gives rise to descriptions in terms of multiple vector spaces, which Brown and Hiley call "shadow phase spaces" (adopting the term "shadow" from Michał Heller. As cited by ). These shadow phase space descriptions include the descriptions in terms of the x-space of the Bohm trajectory description, of the quantum phase space, and of the p-space. In the classical limit, the shadow phase spaces converge to one unique phase space. In their algebraic formulation of quantum mechanics the equation of motion takes on the same form as in the Heisenberg picture, except that the bra and ket in the bra–ket notation each stand for an element of the algebra and that the Heisenberg time evolution is an inner automorphism in the algebra.
In 2001, Hiley proposed to extend the Heisenberg Lie algebra, which is defined by the pair () satisfying the commutator bracket and which is nilpotent, by additionally introducing an idempotent into the algebra to yield a symplectic Clifford algebra. This algebra makes it possible to discuss the Heisenberg equation and Schrödinger equation in a representation-free manner. He later noted that the idempotent can be the projection formed by the outer product of the standard ket and the standard bra, which had been presented by Paul Dirac in his work The Principles of Quantum Mechanics.B. J. Hiley: Non-commutative quantum geometry: A Reappraisal of the Bohm approach to Quantum Theory. In: (2006). 9783540266693, Springer Science & Business Media. ISBN 9783540266693 p. 316.
The set of two operator equations, first derived and published by Brown and Hiley in 2000, was re-derived and expanded upon in Hiley's later publications. Hiley also pointed out that the two operator equations are analogous to the two equations that involve the sine and cosine bracket, and that the quantum phase equation has apparently not been published prior to his work with Brown, except that such an equation was hinted at by P. Carruthers and F. Zachariasen.
Hiley has emphasized that quantum processes cannot be displayed in phase space for reason of lacking commutativity. As Israel Gelfand had shown, commutative algebras allow a unique manifold to be constructed as a sub-space which is dual to the algebra; non-commutative algebras in contrast cannot be associated with a unique underlying manifold. Instead, a non-commutative algebra requires a multiplicity of shadow manifolds. These shadow manifolds can be constructed from the algebra by means of projections into subspaces; however, the projections inevitably lead to distortions, in similar manner as Mercator projections inevitably result in distortions in geographical maps.
The algebraic structure of the quantum formalism can be interpreted as Bohm's implicate order, and shadow manifolds are its necessary consequence: "The order of process by its very essence cannot be displayed in one unique manifest (explicate) order. ... we can only display some aspects of the process at the expense of others. We are inside looking out."
Hiley demonstrated the equivalence between Moyal's characteristic function for the Wigner quasi-probability distribution F(x,p,t) and von Neumann's idempotent within the proof of the Stone–von Neumann theorem, concluding: "In consequence, F(x,p,t) is not a probability density function but a specific representation of the quantum mechanical density operator", thus the Wigner–Moyal formalism exactly reproduces the results of quantum mechanics. This confirmed an earlier result by George A. Baker that the quasi-probability distribution can be understood as the density matrix re-expressed in terms of a mean position and momentum of a "cell" in phase space, and furthermore revealed that the Bohm interpretation arises from the dynamics of these "cells" if the particle is considered to be at the center of the cell.B. Hiley: Moyal's characteristic function, the density matrix and von Neumann's idempotent ( preprint, 2006) Hiley pointed out that the equations defining the Bohm approach can be taken to be implicit in certain equations of the 1949 publication by José Enrique Moyal on the phase space formulation of quantum mechanics; he emphasized that this link between the two approaches could be of relevance for constructing a quantum geometry.
In 2005, building on his work with Brown, Hiley showed that the construction of subspaces allows the Bohm interpretation to be understood in terms of the choice of the x-representation as shadow phase space as one particular choice among an infinite number of possible shadow phase spaces. Hiley noted a conceptual parallel in the demonstration given by mathematician Maurice A. de Gosson that " the Schrödinger equation can be shown rigorously to exist in the of the symplectic group of classical physics and the quantum potential arises by projecting down onto the underlying group".Maurice A. de Gosson: " The Principles of Newtonian and Quantum Mechanics – The Need for Planck's Constant, h", Imperial College Press, World Scientific Publishing, 2001, More succinctly yet, Hiley and Gosson later stated: The classical world lives in a symplectic space, while the quantum world unfolds in the covering space. In mathematical terms, the covering group of the symplectic group is the metaplectic group, and De Gosson summarizes the mathematical reasons for the impossibility of constructing simultaneous position and momentum representations as follows: " Hiley's 'shadow phase space' approach is a reflection of the fact that we cannot construct a global chart for the metaplectic group, when it is viewed as a Lie group, that is, as a manifold equipped with a continuous algebraic structure".Maurice A. de Gosson: " The Principles of Newtonian and Quantum Mechanics – The Need for Planck's Constant, h", Imperial College Press, World Scientific Publishing, 2001, , p. 34 In Hiley's framework, the quantum potential arises as "a direct consequence of projecting the non-commutative algebraic structure onto a shadow manifold" and as a necessary feature which ensures that both energy and momentum are conserved.B.J. Hiley: Phase space description of quantum mechanics and non-commutative geometry: Wigner–Moyal and Bohm in a wider context, In: Theo M. Nieuwenhuizen et al. (eds.): Beyond the quantum, World Scientific Publishing, 2007, , pp. 203–211, therein p. 204 ( preprint) Similarly, the Bohm and the Wigner approach are shown to be two different shadow phase space representations.B. J. Hiley: Phase space descriptions of quantum phenomena, in: A. Khrennikov (ed.): Quantum Theory: Re-consideration of Foundations–2, pp. 267–286, Växjö University Press, Sweden, 2003 ( PDF)
With these results, Hiley gave evidence to the notion that the ontology of implicate and explicate orders could be understood as a process described in terms of an underlying non-commutative algebra, from which spacetime could be abstracted as one possible representation. The non-commutative algebraic structure is identified with an implicate order, and its shadow manifolds with the sets of explicate orders that are consistent with that implicate order. (2025). 9783642549427 ISBN 9783642549427
Here emerges, in Hiley's words, "a radically new way of looking at the way quantum processes enfold in time", built on the work of Bohm and Hiley in the 1980s: in this school of thought, processes of movement can be seen as automorphisms within and between inequivalent representations of the algebra. In the first case, the transformation is an inner automorphism, which is a way of expressing the enfolding and unfolding movement in terms of potentialities of the process; in the second case it is an outer automorphism, or transformation to a new Hilbert space, which is a way of expressing an actual change.
| Clifford algebras Clp,q and wave equations ! algebra !! Metric signature !! equation !! | |||
| Cl4,2 | +, +, +, +, −, − | Twistor space | Twistor theory |
| Cl1,3 | +, −, −, − | Dirac equation | relativistic spin-1/2 |
| Cl3,0 | +, +, + | Pauli equation | spin-1/2 |
| Cl0,1 | − | Schrödinger | spin-0 |
Using this approach to describe relativistic particle quantum mechanics, Hiley and R. E. Callaghan presented a complete relativistic version of the Bohm model for the Dirac particle in analogy to Bohm's approach to the non-relativistic Schrödinger equation, thereby refuting the long-standing misconception that the Bohm model could not be applied in the relativistic domain. Hiley pointed out that the Dirac particle has a 'quantum potential' which is the exact relativistic generalisation of the quantum potential found originally by de Broglie and Bohm. Within the same hierarchy, the twistor of Roger Penrose links to the conformal Clifford algebra Cl4,2 over the reals, and what Hiley calls the Bohm energy and the Bohm momentum arises directly from the standard energy-momentum tensor.Basil J. Hiley: Clifford algebras as a vehicle for quantum mechanics without wave functions: The Bohm model of the Dirac equation, Vienna Symposium on the Foundations of Modern Physics 2009 ( abstract) The technique developed by Hiley and his co-workers demonstrates
Hiley refers to Bohm's ink droplet analogy for a rather easily understandable analogy of the notion of implicate and explicate order. Regarding the algebraic formulation of the implicate order, he has stated: "An important new general feature that emerges from these considerations is the possibility that not everything can be made explicit at a given time" and adding: 'Within the Cartesian order, complementarity seems totally mysterious. There exists no structural reason as to why these incompatibilities exist. Within the notion of the implicate order, a structural reason emerges and provides a new way of searching for explanations."B.J. Hiley: Particles, fields, and observers, Volume I The Origins of Life, Part 1 Origin and Evolution of Life, Section II The Physical and Chemical Basis of Life, pp. 87–106 ( PDF)
Hiley has worked with Maurice A. de Gosson on the relation between classical and quantum physics, presenting a mathematical derivation of the Schrödinger equation from Hamiltonian mechanics. Together with mathematicians Ernst Binz and Maurice A. de Gosson, Hiley showed how "a characteristic Clifford algebra emerges from each ( 2n-dimensional) phase space" and discussed relations of quaternion algebra, symplectic geometry and quantum mechanics.
Later in 2011, for the first time experimental results were published that showed paths that display the properties expected for Bohm trajectories. More specifically, photon trajectories were observed by means of weak measurements in a double-slit interferometer, and these trajectories displayed the qualitative features that had been predicted ten years earlier by Partha Ghose for Bohm trajectories.Sacha Kocsis, Sylvain Ravets, Boris Braverman, Krister Shalm, Aephraim M. Steinberg: Observing the trajectories of a single photon using weak measurement, 19th Australian Instuturte of Physics (AIP) Congress, 2010 [42] The same year, Hiley showed that a description of weak processes – "weak" in the sense of weak measurements – can be included in his framework of an algebraic description of quantum processes by extending the framework to include not only (orthogonal) Clifford algebras but also the Moyal algebra, a symplectic Clifford algebra.
Glen Dennis, de Gosson and Hiley, expanding further on de Gosson's notion of quantum blobs, emphasized the relevance of a quantum particle's internal energy – in terms of its kinetic energy as well as its quantum potential – with regard to the particle's extension in phase space.
In 2018, Hiley showed that the Bohm trajectories are to be interpreted as the mean momentum flow of a set of individual quantum processes, not as the path of an individual particle, and related the Bohm trajectories to Richard Feynman's path integral formulation as an average of an ensemble of Feynman paths. Cited by
For his purely algebraic approach, Hiley takes reference to foundations in the work of Gérard Emch,Gérard Emch: Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, 1972 the work of Rudolf HaagRudolf Haag: Local quantum physics on local quantum field theory, and the work of Ola Bratteli and D.W. Robertson.Ola Bratteli, DW Robertson, Operator Algebras and Quantum Statistical Mechanics. I, Springer, New York, Heidelberg, Berlin, 1979. He points out that the algebraic mental representation allows to establish a connection to the thermo field dynamics of Hiroomi Umezawa, using a bialgebra constructed from a two-time quantum theory. Hiley has stated that his recent focus on noncommutative geometry appears to be very much in line with the work of Fred van Oystaeyen on noncommutative topology.Basil J. Hiley: The Bohm approach re-assessed, p. 9
Ignazio Licata cites Bohm and Hiley's approach as formulating "a quantum event as the expression of a deeper quantum process" that connects a description in terms of space-time with a description in non-local, quantum mechanical terms. Hiley is cited, together with Whitehead, Bohr and Bohm, for the "stance of elevating processes to a privileged role in theories of physics". His view of process as fundamental has been seen as similar to the approach taken by the physicist Lee Smolin. This stands quite in contrast to other approaches, in particular to the blockworld approach in which spacetime is static.
Philosopher Paavo Pylkkänen, Ilkka Pättiniemi and Hiley are of the view that Bohm's emphasis on notions such as "structural process", "order" and "movement" as fundamental in physics point to some form of scientific structuralism, and that Hiley's work on symplectic geometry, which is in line with the algebraic approach initiated by Bohm and Hiley, "can be seen as bringing Bohm's 1952 approach closer to scientific structuralism".
Referring to two fundamental notions of René Descartes, Hiley states that "if we can give up the assumption that space-time is absolutely necessary for describing physical processes, then it is possible to bring the two apparently separate domains of res extensa and res cogitans into one common domain", and he adds that "by using the notion of process and its description by an algebraic structure, we have the beginnings of a descriptive form that will enable us to understand quantum processes and will also enable us to explore the relation between mind and matter in new ways".
In Bohm and Hiley's work on implicate and explicate order, mind and matter are considered to be different aspects of the same process.
In this context, Hiley spoke of his aim of finding "an algebraic description of those aspects of this implicate order where mind and matter have their origins".Basil J. Hiley: Non-commutative geometry, the Bohm interpretation and the mind-matter relationship, CASYS 2000, Liège, Belgium, August 7–12, 2000, page 15
Hiley also worked with biologist Brian Goodwin on a process view of biological life, with an alternate view on Darwinism.
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