Basil Hiley#Projections into shadow manifolds

Basil James Hiley was born November 15, 1935 in Burma,{{Cite web |title=Wholeness and the Implicate Order Revisite |url=https://paricenter.com/basil-hiley/ |access-date=2025-02-07 |website=The Pari Center |language=en-US}} where his father, James Hiley, worked for the military of the British Raj. He moved to Hampshire, England, at the age of twelve, where he attended secondary school. His interest in science was stimulated by his teachers at secondary school and by books, in particular The Mysterious Universe by James Hopwood Jeans and Mr Tompkins in Wonderland by George Gamow.{{Cite web |last=Freire |first=Olival |date=2021-09-24 |title=Interview of Basil Hiley |url=https://www.aip.org/history-programs/niels-bohr-library/oral-histories/33822 |access-date=2023-10-27 |website=AIP |language=en}}

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 ferromagnets and long chain polymer models, under the supervision of Cyril Domb and Michael Fisher.{{cite journal |last1=Hiley |first1=B. J. |year=2010 |title=On the Relationship Between the Wigner-Moyal and Bohm Approaches to Quantum Mechanics: A Step to a More General Theory? |url=http://www.bbk.ac.uk/tpru/BasilHiley/W-MandBohm.pdf |journal=Foundations of Physics |volume=40 |issue=4 |pages=356–367 |bibcode=2010FoPh...40..356H |doi=10.1007/s10701-009-9320-y |s2cid=3169347}}{{Cite journal |date=2005 |title=About Authors |url=https://www.mindmatter.de/resources/pdf/authors3_2www.pdf |journal=Mind and Matter |volume=3 |issue=2 |pages=117–118}}

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.{{cite journal |quote=My own interests were very much directed towards trying to base physics on the general notion of process, an idea that attracted me to Bohm in the first place, as he had similar thoughts. |doi=10.1007/BF02058632|title=On the role of hidden variables in the fundamental structure of physics|year=1996|last1=Bohm|first1=David|journal=Foundations of Physics|volume=26|issue=6|pages=719–786|bibcode=1996FoPh...26..719B|s2cid=189834866}} He reports that during the seminars he held together with Roger Penrose he {{blockquote|text=was particularly fascinated by John Wheeler's "sum over three geometries" ideas that he was using to quantise gravity.|sign=Hiley|source=}}

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.{{Cite journal | last1 = Hiley | first1 = B. J. | author-link = Basil Hiley| doi = 10.1098/rsbm.1997.0007 | title = David Joseph Bohm. 20 December 1917--27 October 1992: Elected F.R.S. 1990 | journal = Biographical Memoirs of Fellows of the Royal Society | volume = 43 | pages = 107–131 | year = 1997 | title-link = David Bohm | s2cid = 70366771 }}

In 1995, Basil Hiley was appointed to the chair in physics at Birkbeck College at the University of London.[http://www.scimednet.org/assets/public/pdf-leaflets/bohmleafletnov20091.pdf#page=3 Basil Hiley] {{Webarchive|url=https://web.archive.org/web/20110728083712/http://www.scimednet.org/assets/public/pdf-leaflets/bohmleafletnov20091.pdf#page=3 |date=2011-07-28 }} (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".[http://www.ems.bbk.ac.uk/news/majorana Department Fellow wins Majorana Prize], Birkbeck College (downloaded 12 June 2013)[http://www.majoranaprize.com/ The Majorana Prize], www.majoranaprize.com (downloaded 12 June 2013)

Hiley died on 25 January 2025.{{Cite web |date=31 January 2025 |title=Basil Hiley (1935–2025) |url=https://www.hep.ucl.ac.uk/news/ |access-date=2025-02-07 |website=UCL HEP Group — News Archive}}{{Cite web |date=2025-02-09 |title=Basil J. Hiley (1935‑2025) |url=https://paricenter.com/ |access-date=2025-02-07 |website=The Pari Center Homepage |language=en-US}}{{Cite web |title=Radical Visions: Celebrating the Lives and Work of the Two Davids, David Bohm and David Peat. In memory of Basil J. Hiley (1935–2025) |url=https://paricenter.com/event/radical-visions/ |access-date=2025-02-07 |website=The Pari Center |language=en-US}}

{{Primary sources|section|date=February 2025}}

In the 1970s Bohm, Hiley and co-workers at Birkbeck College expanded further on the theory presented by David Bohm in 1952.Paavo Pylkkänen: Foreword by the Editor, in: David Bohm and Charles Biederman, and Paavo Pylkkänen (ed.): Bohm-Biederman Correspondence, {{ISBN|978-0-415-16225-8}}, [https://books.google.com/books?id=gh56bfGthoIC&pg=PR14 p. xiv] They suggested to re-express the field equations of physics in a way that is independent of their spacetime description. They interpreted Bell's theorem as a test of spontaneous localization, meaning a tendency of a many-body system to factorize into a product of localized states of its constituent particles, pointing out that such spontaneous localization removes the need for a fundamental role of the measuring apparatus in quantum theory.{{Cite journal|doi = 10.1007/BF02726670|title = On some new notions concerning locality and nonlocality in the quantum theory|year = 1975|last1 = Baracca|first1 = A.|last2 = Bohm|first2 = D. J.|last3 = Hiley|first3 = B. J.|last4 = Stuart|first4 = A. E. G.|journal = Il Nuovo Cimento B |series=Series 11|volume = 28|issue = 2|pages = 453–466|bibcode = 1975NCimB..28..453B|s2cid = 117001918}} They proposed that the fundamental new quality introduced by quantum physics is non-locality.{{Cite journal|doi = 10.1007/BF01100319|title = On the intuitive understanding of nonlocality as implied by quantum theory|year = 1975|last1 = Bohm|first1 = D. J.|last2 = Hiley|first2 = B. J.|journal = Foundations of Physics|volume = 5|issue = 1|pages = 93–109|bibcode = 1975FoPh....5...93B|s2cid = 122635316}}{{Cite journal|doi = 10.1007/BF02726290|title = Nonlocality and polarization correlations of annihilation quanta|year = 1976|last1 = Bohm|first1 = D. J.|last2 = Hiley|first2 = B. J.|journal = Il Nuovo Cimento B |series=Series 11|volume = 35|issue = 1|pages = 137–144|bibcode = 1976NCimB..35..137B|s2cid = 117932612}} In 1975, they presented how in the causal interpretation of the quantum theory introduced by Bohm in 1952 the concept of a quantum potential leads to the notion of an "unbroken wholeness of the entire universe", and they proposed possible routes to a generalization of the approach to relativity by means of a novel concept of time.

File:doppelspalt.svg. The resultant trajectories were first presented by Philippidis, Dewdney and Hiley in 1979.Statement on "first presented" quoted from 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., [https://books.google.com/books?id=jXRub48gzuIC&pg=PA p. 319], which takes reference to: {{cite journal |doi=10.1007/BF02743566|title=Quantum interference and the quantum potential|year=1979|last1=Philippidis|first1=C.|last2=Dewdney|first2=C.|last3=Hiley|first3=B. J.|journal=Il Nuovo Cimento B |series=Series 11|volume=52|issue=1|pages=15–28|bibcode=1979NCimB..52...15P|s2cid=53575967}}]]

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{{Cite journal|doi = 10.1007/BF02743566|title = Quantum interference and the quantum potential|year = 1979|last1 = Philippidis|first1 = C.|last2 = Dewdney|first2 = C.|last3 = Hiley|first3 = B. J.|journal = Il Nuovo Cimento B |series=Series 11|volume = 52|issue = 1|pages = 15–28|bibcode = 1979NCimB..52...15P|s2cid = 53575967}} and worked out descriptions of scattering processes.{{Cite journal|doi = 10.1007/BF00726873|title = A quantum potential description of one-dimensional time-dependent scattering from square barriers and square wells|year = 1982|last1 = Dewdney|first1 = C.|last2 = Hiley|first2 = B. J.|journal = Foundations of Physics|volume = 12|issue = 1|pages = 27–48|bibcode = 1982FoPh...12...27D|s2cid = 18771056}} 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, [http://www.controversia.fis.ufba.br/index_arquivos/freire-se.pdf#page=4 p. 576] {{Webarchive|url=https://web.archive.org/web/20140310040550/http://www.controversia.fis.ufba.br/index_arquivos/freire-se.pdf#page=4 |date=2014-03-10 }} In 1979, Bohm and Hiley discussed the Aharonov–Bohm effect which had recently found experimental confirmation.{{Cite journal|doi = 10.1007/BF02770900|title = On the Aharonov-Bohm effect|year = 1979|last1 = Bohm|first1 = D.|last2 = Hiley|first2 = B. J.|journal = Il Nuovo Cimento A|volume = 52|issue = 3|pages = 295–308|bibcode = 1979NCimA..52..295B|s2cid = 124958019}} They called attention to the importance of the early work of Louis de Broglie on pilot waves, 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, ([http://leopard.physics.ucdavis.edu/rts/p298/pilotwavetheory.pdf PDF] {{Webarchive|url=https://web.archive.org/web/20110819234038/http://leopard.physics.ucdavis.edu/rts/p298/pilotwavetheory.pdf |date=2011-08-19 }}) They offered ways of understanding quantum non-locality and the measurement process,David J. Bohm, Basil J. Hiley: Some Remarks on 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{{Cite journal|doi = 10.1007/BF00726935|title = Nonlocality in quantum theory understood in terms of Einstein's nonlinear field approach|year = 1981|last1 = Bohm|first1 = D.|last2 = Hiley|first2 = B. J.|journal = Foundations of Physics|volume = 11|issue = 7–8|pages = 529–546|bibcode = 1981FoPh...11..529B|s2cid = 121965108}} the limit of classicality,{{Cite journal|doi = 10.1103/PhysRevLett.55.2511|title = Unbroken Quantum Realism, from Microscopic to Macroscopic Levels|year = 1985|last1 = Bohm|first1 = D.|last2 = Hiley|first2 = B. J.|journal = Physical Review Letters|volume = 55|issue = 23|pages = 2511–2514|pmid = 10032166|bibcode = 1985PhRvL..55.2511B}} interference and quantum tunneling.See also the citation of Bohm and Hiley's article [http://prl.aps.org/abstract/PRL/v55/i23/p2511_1 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.{{Cite journal|doi = 10.1007/BF00730211|title = Measurement understood through the quantum potential approach|year = 1984|last1 = Bohm|first1 = D.|last2 = Hiley|first2 = B. J.|journal = Foundations of Physics|volume = 14|issue = 3|pages = 255–274|bibcode = 1984FoPh...14..255B|s2cid = 123155900}} 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 [https://books.google.com/books?hl=en&id=QN9HW1Oi7d4C&oi=fnd&pg=PA238 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[http://www.goertzel.org/dynapsyc/1997/interview.html 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 [https://books.google.com/books?hl=en&id=QN9HW1Oi7d4C&oi=fnd&pg=PA239 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, [http://www.bbk.ac.uk/tpru/BasilHiley/ActInfoTeleWein.pdf#page=7 p. 7]For 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, {{ISBN|0-8122-1002-6}} 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, {{ISBN|90-272-5194-0}}, pp. 197–214, see [https://books.google.com/books?id=Fvs3sw9ZVfsC&pg=PA207 p. 207] and [https://books.google.com/books?id=Fvs3sw9ZVfsC&pg=PA212 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 [https://books.google.com/books?id=jXRub48gzuIC&pg=PA326 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 qubits 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 ([http://www.bbk.ac.uk/tpru/BasilHiley/ActInfoTeleWein.pdf PDF]){{cite journal |last1=Maroney |first1=O. |last2=Hiley |first2=B. J. |title=Quantum State Teleportation Understood Through the Bohm Interpretation |journal=Foundations of Physics |date=1999 |volume=29 |issue=9 |pages=1403–1415 |doi=10.1023/A:1018861226606|s2cid=118271767 }}

With Pan N. Kaloyerou, Hiley extended the quantum potential approach to quantum field theory in Minkowski spacetime.P.N. Kaloyerou, Investigation of the Quantum Potential in the Relativistic Domain, PhD. Thesis, Birkbeck College, London (1985){{cite journal|doi=10.1016/0370-1573(87)90024-X|url=http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/bohm_hiley_kaloyerou_1986.pdf|title=An ontological basis for the quantum theory|year=1987|last1=Bohm|first1=D.|last2=Hiley|first2=B.J|last3=Kaloyerou|first3=P.N|journal=Physics Reports|volume=144|issue=6|pages=321–375|bibcode=1987PhR...144..321B|access-date=3 June 2011|archive-date=19 March 2012|archive-url=https://web.archive.org/web/20120319181802/http://www.tcm.phy.cam.ac.uk/~mdt26/local_papers/bohm_hiley_kaloyerou_1986.pdf}}, therein: D. Bohm, B. J. Hiley: I. Non-relativistic particle systems, pp. 321–348, and D. Bohm, B. J. Hiley, P. N. Kaloyerou: II. A causal interpretation of quantum fields, pp. 349–375{{cite journal |doi=10.1016/0370-1573(94)90155-4|title=The casual interpretation of the electromagnetic field|year=1994|last1=Kaloyerou|first1=P.N.|journal=Physics Reports|volume=244|issue=6|pages=287–358|bibcode=1994PhR...244..287K}}P.N. Kaloyerou, in "Bohmian Mechanics and Quantum Theory: An Appraisal", eds. J.T. Cushing, A. Fine and S. Goldstein, Kluwer, Dordrecht, 155 (1996). Bohm and Hiley proposed a new interpretation of the Lorentz transformation{{cite journal |doi=10.1119/1.14300|title=Active interpretation of the Lorentz boosts as a physical explanation of different time rates|year=1985|last1=Bohm|first1=D.|last2=Hiley|first2=B. J.|journal=American Journal of Physics|volume=53|issue=8|pages=720–723|bibcode=1985AmJPh..53..720B}} and considered the relativistic invariance of a quantum theory based on the notion of beables, a term coined by John BellJohn Bell, Speakable and Unspeakable in Quantum Mechanics to distinguish these variables from observables.{{Cite journal|doi = 10.1007/BF01889535|title = On the relativistic invariance of a quantum theory based on beables|year = 1991|last1 = Bohm|first1 = D.|last2 = Hiley|first2 = B. J.|journal = Foundations of Physics|volume = 21|issue = 2|pages = 243–250|bibcode = 1991FoPh...21..243B|s2cid = 121090344}} Hiley and a co-worker later extended the work further to curved spacetime.B. J. Hiley, A. H. Aziz Muft: The ontological interpretation of quantum field theory applied in a cosmological context. In: Miguel Ferrero, Alwyn Van der Merwe (eds.): Fundamental problems in quantum physics, Fundamental theories of physics, Kluwer Academic Publishers, 1995, {{ISBN|0-7923-3670-4}}, [https://books.google.com/books?hl=en&id=1wUFoP7HC38C&oi=fnd&pg=PA141 pages 141–156] Bohm and Hiley demonstrated that the non-locality of quantum theory can be understood as limit case of a purely local theory, provided the transmission of active information is allowed to be greater than the speed of light, and that this limit case yields approximations to both quantum theory and relativity.{{Cite journal|doi=10.1016/0370-1573(89)90160-9|title=Non-locality and locality in the stochastic interpretation of quantum mechanics|year=1989|last1=Bohm|first1=D.|last2=Hiley|first2=B.J.|journal=Physics Reports|volume=172|issue=3|pages=93–122|bibcode=1989PhR...172...93B}}

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:{{cite arXiv|eprint=1104.5594|last1=Miranda|first1=Abel|title=Particle Physics challenges to the Bohm Picture of Relativistic Quantum Field Theory|year=2011|class=hep-ph}}

{{blockquote|I emphasize that Bohm–Hiley ontological reformulation of RQFT always treats Bose fields as continuous distributions in spacetime – basically because these quantum fields have perfectly well-defined classical analogs. The textbook spin-0, spin-1 and spin-2 bosons, such as the Higgs, photons, gluons, electroweak bosons and gravitons [...] are, according to this viewpoint, not "particles" in any naive sense of the word, but just dynamical structural features of coupled continuous scalar, vector, and symmetric tensor fields that first become manifest when interactions with matter particles (elementary or otherwise) occur [...].}}

Much of Bohm and Hiley's work in the 1970s and 1980s has expanded on the notion of implicate, explicate and generative orders proposed by Bohm.{{Cite journal|doi = 10.1007/BF00671000|title = On a new mode of description in physics|year = 1970|last1 = Bohm|first1 = David|last2 = Hiley|first2 = Basil J.|last3 = Stuart|first3 = Allan E. G.|journal = International Journal of Theoretical Physics|volume = 3|issue = 3|pages = 171–183|bibcode = 1970IJTP....3..171B|s2cid = 121080682}}{{Cite journal|doi = 10.1007/BF00708436|title = Quantum theory as an indication of a new order in physics. B. Implicate and explicate order in physical law|year = 1973|last1 = Bohm|first1 = David|journal = Foundations of Physics|volume = 3|issue = 2|pages = 139–168|bibcode = 1973FoPh....3..139B|s2cid = 121061984}} This concept is described in the books Wholeness and the Implicate OrderDavid Bohm: Wholeness and the Implicate Order, 1980 by Bohm and Science, Order, and Creativity by Bohm and F. David Peat.David Bohm, F. David Peat: Science, Order, and Creativity, 1987 The theoretical framework underlying this approach has been developed by the Birkbeck group over the last decades. In 2013 the research group at Birkbeck summarized their over-all approach as follows:[http://www.bbk.ac.uk/tpru/ Relativity, Quantum Gravity and Space-time Structures], Birkbeck, University of London (downloaded 12 June 2013)

: "It is now quite clear that if gravity is to be quantised successfully, a radical change in our understanding of spacetime will be needed. We begin from a more fundamental level by taking the notion of process as our starting point. Rather than beginning with a spacetime continuum, we introduce a structure process which, in some suitable limit, approximates to the continuum. We are exploring the possibility of describing this process by some form of non-commutative algebra, an idea that fits into the general ideas of the implicate order. In such a structure, the non-locality of quantum theory can be understood as a specific feature of this more general a-local background and that locality, and indeed time, will emerge as a special feature of this deeper a-local structure."

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 spinors with minimal left ideals 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 ([http://www.birkbeck.ac.uk/tpru/BasilHiley/Algebraic%20Quantum%20Mechanic%205.pdf 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.{{Cite journal|doi = 10.1007/BF00709014|title = The implicate order, algebras, and the spinor|year = 1980|last1 = Frescura|first1 = F. A. M.|last2 = Hiley|first2 = B. J.|journal = Foundations of Physics|volume = 10|issue = 1–2|pages = 7–31|bibcode = 1980FoPh...10....7F|s2cid = 121251365}}{{Cite journal|doi = 10.1007/BF00708417|title = The algebraization of quantum mechanics and the implicate order|year = 1980|last1 = Frescura|first1 = F. A. M.|last2 = Hiley|first2 = B. J.|journal = Foundations of Physics|volume = 10|issue = 9–10|pages = 705–722|bibcode = 1980FoPh...10..705F|s2cid = 122045502}}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 ([http://ajp.aapt.org/resource/1/ajpias/v49/i2/p152_s1 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 quantum theory], namely the operators on the one hand and the state vectors on the other. Rather, one uses only a single type of object, the algebraic element".{{cite book |arxiv=quant-ph/0612002|doi=10.1063/1.2158735|chapter=Algebraic Quantum Mechanics and Pregeometry|title=AIP Conference Proceedings|year=2006|last1=Bohm|first1=D. J.|last2=Davies|first2=P. G.|last3=Hiley|first3=B. J.|volume=810|pages=314–324|s2cid=9836351}}, and its introductory note {{cite book |doi=10.1063/1.2158734|chapter=Quantum Space-Times: An Introduction to "Algebraic Quantum Mechanics and Pregeometry"|title=AIP Conference Proceedings|year=2006|last1=Hiley|first1=B. J.|volume=810|pages=312–313}} 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.{{cite journal |doi=10.1007/s10825-015-0728-7 |arxiv=1211.2098|title=On the relationship between the Wigner–Moyal approach and the quantum operator algebra of von Neumann|year=2015|last1=Hiley|first1=B. J.|journal=Journal of Computational Electronics|volume=14|issue=4|pages=869–878|s2cid=122761113}} With their approach based on algebraic idempotents, Bohm and Hiley "incorporate 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.{{Cite journal|doi = 10.1007/BF00726266|title = On a quantum algebraic approach to a generalized phase space|year = 1981|last1 = Bohm|first1 = D.|last2 = Hiley|first2 = B. J.|journal = Foundations of Physics|volume = 11|issue = 3–4|pages = 179–203|bibcode = 1981FoPh...11..179B|s2cid = 123422217}} They extended this approach further to relativistic phase space by applying the phase space interpretation of Mario Schönberg to the Dirac algebra.{{Cite book|doi = 10.1007/978-1-4684-8830-2_5|chapter = Relativistic Phase Space Arising out of the Dirac Algebra|title = Old and New Questions in Physics, Cosmology, Philosophy, and Theoretical Biology|year = 1983|last1 = Bohm|first1 = D.|last2 = Hiley|first2 = B. J.|pages = 67–76|isbn = 978-1-4684-8832-6}} Their approach was subsequently applied by Peter R. Holland to fermions and by Alves O. Bolivar to bosons.{{Cite journal|doi = 10.1007/BF00735377|title = Relativistic algebraic spinors and quantum motions in phase space|year = 1986|last1 = Holland|first1 = P. R.|journal = Foundations of Physics|volume = 16|issue = 8|pages = 701–719|bibcode = 1986FoPh...16..701H|s2cid = 122108364}}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: [http://www.bbk.ac.uk/tpru/BasilHiley/P12FrescandHiley3.pdf 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 ([http://www.birkbeck.ac.uk/tpru/BasilHiley/PS16Twistors.pdf 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 Clifford 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, {{ISBN|0-415-06960-2}}, pp. 1–32, therein: [https://books.google.com/books?id=Avvxbw-NONYC&pg=PA25 p. 25]

File:Antony Gormley Quantum Cloud 2000.jpg by Antony Gormley, influenced by an exchange of thoughts among Hiley and Gormley on algebra and pre-space."During our discussions the physicist Basil Hiley explained his notions of pre-space—a mathematical structure existing before space-time and matter—to the sculptor Gormley. This led Gormley to make a radical change to his work with the piece Quantum Cloud that is now mounted over the river Thames." F. David Peat: Pathways of Chance, Pari Publishing, 2007, {{ISBN|978-88-901960-1-0}}, [https://books.google.com/books?id=S3lze94Tm1kC&pg=PA127 p. 127]]]

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 foam-like structure 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".{{Cite web|author=Basil J. Hiley|title=Process and the Implicate Order: their relevance to Quantum Theory and Mind|url=http://www.ctr4process.org/publications/Articles/LSI05/Hiley%20paper.pdf|access-date=2006-10-14|archive-url=https://web.archive.org/web/20061014020948/http://ctr4process.org/publications/Articles/LSI05/Hiley%20paper.pdf|archive-date=2006-10-14}} 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,{{Cite web |author=B. J. Hiley |title=Process and the Implicate Order: their relevance to Quantum Theory and Mind |website=Center for Process Studies |url=http://www.ctr4process.org/publications/Articles/LSI05/Hiley%20paper.pdf |archive-url=https://web.archive.org/web/20110926212330/http://www.ctr4process.org/publications/Articles/LSI05/Hiley%20paper.pdf |archive-date=2011-09-26}} 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.{{Cite journal|bibcode = 1985Natur.315..294B|title = A quantum potential approach to the Wheeler delayed-choice experiment|last1 = Bohm|first1 = D. J.|last2 = Dewdney|first2 = C.|last3 = Hiley|first3 = B. H.|journal = Nature|year = 1985|volume = 315|issue = 6017|page = 294|doi = 10.1038/315294a0|s2cid = 43168123}} 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, {{ISBN|0-19-510095-6}}, [https://archive.org/details/timesarrowarchim00pric/page/135 p. 135] by a detailed trajectory analysis for delayed choice experiments{{cite journal |doi=10.1088/0031-8949/74/3/007 |arxiv=1602.06100|title=Delayed-choice experiments and the Bohm approach|year=2006|last1=Hiley|first1=B. J.|last2=Callaghan|first2=R. E.|journal=Physica Scripta|volume=74|issue=3|pages=336–348|bibcode=2006PhyS...74..336H|s2cid=12941256}} and by an investigation into welcher Weg experiments.{{Cite journal|doi=10.1007/s10701-006-9086-4|title=What is Erased in the Quantum Erasure?|year=2006|last1=Hiley|first1=B. J.|last2=Callaghan|first2=R. E.|journal=Foundations of Physics|volume=36|issue=12|pages=1869–1883|bibcode=2006FoPh...36.1869H|s2cid=18972152}} 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).{{Cite journal|doi = 10.1007/BF02058636|title = Statistical mechanics and the ontological interpretation|year = 1996|last1 = Bohm|first1 = D.|last2 = Hiley|first2 = B. J.|journal = Foundations of Physics|volume = 26|issue = 6|pages = 823–846|bibcode = 1996FoPh...26..823B|s2cid = 121500818}}

Hiley has pursued work on algebraic structures in quantum theory throughout his scientific career.{{Cite journal|doi = 10.1007/BF00674278|title = Phase space, fibre bundles and current algebras|year = 1971|last1 = Hiley|first1 = Basil J.|last2 = Stuart|first2 = Allan E. G.|journal = International Journal of Theoretical Physics|volume = 4|issue = 4|pages = 247–265|bibcode = 1971IJTP....4..247H|s2cid = 120247206}}{{Cite journal|doi=10.1142/S0217732393002361|title=Quantum Phase Space and the Discrete Weyl Algebra|year=1993|last1=Hiley|first1=Basil|last2=Monk|first2=Nick|journal=Modern Physics Letters A|volume=08|issue=38|pages=3625–3633|bibcode=1993MPLA....8.3625H}}{{cite journal |last1=Monk |first1=N. A. M. |last2=Hiley |first2=B. J. |title=A Unified Algebraic Approach to Quantum Theory |journal=Foundations of Physics Letters |date=1998 |volume=11 |issue=4 |pages=371–377 |doi=10.1023/A:1022181008699|s2cid=118169064 }}{{cite arXiv |eprint=quant-ph/0005026|last1=Brown|first1=M. R.|last2=Hiley|first2=B. J.|title=Schrodinger revisited: An algebraic approach|year=2000}}B. J. Hiley: [http://www.bbk.ac.uk/tpru/BasilHiley/IdpsinEEHA5.pdf 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 ([http://www.birkbeck.ac.uk/tpru/BasilHiley/14MomentsANPA2001W.pdf 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, [https://books.google.com/books?id=6HEaNL40C0MC&pg=PA299 pp. 299–324], {{doi|10.1007/3-540-26669-0_16}} ([https://doi.org/10.1007%2F3-540-26669-0_16 abstract], [http://www.bbk.ac.uk/tpru/BasilHiley/TemplePaper%28sys9%2902.pdf preprint]){{Cite arXiv|eprint=1011.4031|last1 = Hiley|first1 = B. J.|last2 = Callaghan|first2 = R. E.|title = The Clifford Algebra approach to Quantum Mechanics A: The Schroedinger and Pauli Particles|year = 2010|class = math-ph}}{{Cite arXiv|eprint=1011.4033|last1 = Hiley|first1 = B. J.|last2 = Callaghan|first2 = R. E.|title = The Clifford Algebra Approach to Quantum Mechanics B: The Dirac Particle and its relation to the Bohm Approach|year = 2010|class = math-ph}}{{cite journal |doi=10.1007/s10701-011-9558-z |url=http://www.bbk.ac.uk/tpru/BasilHiley/Bohm-Vienna.pdf|title=Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation|year=2012|last1=Hiley|first1=B. J.|last2=Callaghan|first2=R. E.|journal=Foundations of Physics|volume=42|issue=1|pages=192–208|bibcode=2012FoPh...42..192H|s2cid=8822308}}{{Too many citations inline|date=February 2025}} 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.{{Cite book|doi=10.1007/978-3-540-70626-7_15|chapter=Bohm Interpretation of Quantum Mechanics|title=Compendium of Quantum Physics|year=2009|last1=Hiley|first1=Basil J.|pages=43–47|isbn=978-3-540-70622-9}}{{cite book |doi=10.1007/978-3-642-12821-9_12 |chapter-url=http://www.bbk.ac.uk/tpru/BasilHiley/Distinction.pdf|chapter=Process, Distinction, Groupoids and Clifford Algebras: An Alternative View of the Quantum Formalism|title=New Structures for Physics|series=Lecture Notes in Physics|year=2010|last1=Hiley|first1=B.J.|volume=813|pages=705–752|arxiv=1211.2107|isbn=978-3-642-12820-2|s2cid=119318272}} 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.{{Cite journal|doi = 10.1007/BF02057882|title = Retrodiction in quantum mechanics, preferred Lorentz frames, and nonlocal measurements|year = 1995|last1 = Cohen|first1 = O.|last2 = Hiley|first2 = B. J.|journal = Foundations of Physics|volume = 25|issue = 12|pages = 1669–1698|bibcode = 1995FoPh...25.1669C|s2cid = 120911522}}{{Cite journal|doi=10.1103/PhysRevA.52.76|title=Reexamining the assumption that elements of reality can be Lorentz invariant|year=1995|last1=Cohen|first1=O.|last2=Hiley|first2=B. J.|journal=Physical Review A|volume=52|issue=1|pages=76–81|pmid=9912224|bibcode=1995PhRvA..52...76C}}{{Cite journal|doi = 10.1007/BF02058886|title = Elements of reality, Lorentz invariance, and the product rule|year = 1996|last1 = Cohen|first1 = O.|last2 = Hiley|first2 = B. J.|journal = Foundations of Physics|volume = 26|issue = 1|pages = 1–15|bibcode = 1996FoPh...26....1C|s2cid = 55850603}}{{cite book |pages=[https://books.google.com/books?id=ekyAV8VtfuYC&pg=PA55 55–58] |doi=10.1007/978-3-540-70626-7_17|chapter=Bohm's Approach to the EPR Paradox|title=Compendium of Quantum Physics|year=2009|last1=Hiley|first1=Basil J.|isbn=978-3-540-70622-9}}

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: [http://www.bbk.ac.uk/tpru/BasilHiley/PS20MindMatterAlgebra.pdf 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, {{ISBN|978-0-8058-2586-2}}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, {{ISBN|978-3-540-62486-8}} ([http://www.bbk.ac.uk/tpru/BasilHiley/PS25P&T3.pdf 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.

Around the same time, in 1997, Hiley's co-worker Melvin Brown[http://www.bbk.ac.uk/tpru/MelvinBrown.html Melin Brown], Birkbeck College showed that the Bohm interpretation of quantum physics need not rely on a formulation in terms of ordinary space ($x$-space), but can be formulated, alternatively, in terms of momentum space ($p$-space).{{Cite arXiv|eprint=quant-ph/9703007v1|last1 = Brown|first1 = M. R.|title = The quantum potential: The breakdown of classical symplectic symmetry and the energy of localisation and dispersion|year = 1997}}{{Cite arXiv|eprint = quant-ph/0005026|last1 = Brown|first1 = M. R.|last2 = Hiley|first2 = B. J.|title = Schrodinger revisited: An algebraic approach|year = 2000}}Ignazio Licata: Emergence and computation at the edge of classical and quantum systems, in: Ignazio Licata, Ammar Sakaji (eds.): Physics of Emergence and Organization, World Scientific, 2008, [https://books.google.com/books?id=BSO1KA-5CDIC&pg=PA1 pp. 1–26], {{ISBN|978-981-277-994-6}}, {{arxiv|0711.2973}}

{{quote box|align=right|width=14em|title=Operator equations|quote=$i\backslash hbar\backslash frac\{\backslash partial\backslash hat\{\backslash rho\}\}\{\backslash partial\; t\}\; +[\backslash hat\{\backslash rho\},\backslash hat\{H\}]\_\{-\}=0$
$\backslash hat\{\backslash rho\}\backslash frac\{\backslash partial\backslash hat\{S\}\}\{\backslash partial\; t\}\; +\backslash frac\{1\}\{2\}[\backslash hat\{\backslash rho\},\backslash hat\{H\}]\_\{+\}=0$|source=Brown and Hiley (2000)}}

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{{cite conference |arxiv=gr-qc/9806011|doi=10.1063/1.57128|title=Einstein-Podolski-Rosen experiment from noncommutative quantum gravity|series=AIP Conference Proceedings|year=1998|last1=Heller|first1=Michael|last2=Sasin|first2=Wiesław|conference=Particles|volume=453|pages=234–241|bibcode=1998AIPC..453..234H|s2cid=17410172}}. As cited by {{cite arXiv |eprint=quant-ph/9703007|last1=Brown|first1=M. R.|title=The quantum potential: The breakdown of classical symplectic symmetry and the energy of localisation and dispersion|year=1997}}). 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 ($\backslash hat\{Q\},\backslash hat\{P\}$) satisfying the commutator bracket $[\backslash hat\{Q\},\backslash hat\{P\}]=i\backslash hbar$ 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.{{Cite arXiv|eprint = 1901.01979|last1 = Hiley|first1 = B. J.|last2 = Dennis|first2 = G.|title = Dirac, Bohm and the Algebraic Approach|year = 2019|class = quant-ph}}B. J. Hiley: Non-commutative quantum geometry: A Reappraisal of the Bohm approach to Quantum Theory. In: {{cite book|author1=Avshalom C. Elitzur|author2=Shahar Dolev|author3=Nancy Kolenda|title=Quo Vadis Quantum Mechanics?|date=30 March 2006|publisher=Springer Science & Business Media|isbn=978-3-540-26669-3|pages=299–324}} [https://books.google.com/books?id=mvZ_1rhGCbgC&pg=PA316 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.{{cite arXiv |eprint=1303.6057|last1=Hiley|first1=B. J.|title=Bohmian Non-commutative Dynamics: History and New Developments|year=2013|class=quant-ph}}{{cite journal|doi=10.1103/RevModPhys.55.245|title=Quantum collision theory with phase-space distributions|year=1983|last1=Carruthers|first1=P.|last2=Zachariasen|first2=F.|journal=Reviews of Modern Physics|volume=55|issue=1|pages=245–285|bibcode=1983RvMP...55..245C|url=https://authors.library.caltech.edu/32343/|access-date=21 December 2020|archive-date=18 September 2020|archive-url=https://web.archive.org/web/20200918115244/https://authors.library.caltech.edu/32343/|url-status=dead}}

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."

In 2001, picking up on the "characteristic matrix" developed with Bohm in 1981 and the notion of a "moment" introduced with Fernandes in 1997, Hiley proposed to use a moment as "an extended structure in both space and time" as a basis for a quantum dynamics, to take the place of the notion of a point particle.

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{{cite journal | last1 = Baker Jr | first1 = George A. | year = 1958 | title = Formulation of Quantum Mechanics Based on the Quasi-probability Distribution Induced on Phase Space | journal = Physical Review | volume = 109 | issue = 6| pages = 2198–2206 | doi = 10.1103/PhysRev.109.2198 | bibcode = 1958PhRv..109.2198B }} 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 ([http://www.bbk.ac.uk/tpru/BasilHiley/Moyal&DensityMatrix.pdf 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 covering groups 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, {{ISBN|1-86094-274-1}} 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.{{cite journal|author1=Maurice A. de Gosson|author2=Basil J. Hiley|url=http://www.mindmatter.de/journal/abstracts/mmabstracts11_2.html#gos|title=Hamiltonian Flows and the Holomovement|journal=Mind and Matter|volume=11|number=2|year=2013}} In mathematical terms, the covering group of the symplectic group is the metaplectic group,{{cite journal |doi=10.1007/s10701-011-9544-5 |arxiv=1001.4632|title=Imprints of the Quantum World in Classical Mechanics|year=2011|last1=De Gosson|first1=Maurice A.|last2=Hiley|first2=Basil J.|journal=Foundations of Physics|volume=41|issue=9|pages=1415–1436|bibcode=2011FoPh...41.1415D|s2cid=18450830}} 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, {{ISBN|1-86094-274-1}}, [https://books.google.com/books?id=IT0FMVgvqp4C&pg=PA34 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, {{ISBN|978-981-277-117-9}}, pp. 203–211, therein p. 204 ([http://www.bbk.ac.uk/tpru/BasilHiley/LeidenPaper.pdf 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 ([http://www.birkbeck.ac.uk/tpru/BasilHiley/ShadowPhaseVajxo03.pdf 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.{{cite book|title=Quantum Interaction|series=Lecture Notes in Computer Science|volume=8369|year=2014|pages=6–21|doi=10.1007/978-3-642-54943-4_2|arxiv=1408.5697|last1=Hiley|first1=B. J.|chapter=Quantum Mechanics: Harbinger of a Non-Commutative Probability Theory?|isbn=978-3-642-54942-7|s2cid=7640980}}{{Cite web|author=B. J. Hiley|url=http://www.cs.ox.ac.uk/conferences/categorieslogicphysics/clap2/clap2-basilhiley.pdf|title=Towards a Quantum Geometry, Groupoids, Clifford algebras and Shadow Manifolds.}}

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.

Hiley expanded on the notion of a process algebra as proposed by Hermann Grassmann and the ideas of distinction of Louis H. Kauffman. He took reference to the vector operators introduced by Mário Schönberg in 1957{{Cite journal|doi = 10.1007/BF02724793|title = Quantum kinematics and geometry|year = 1957|last1 = Schönberg|first1 = M.|journal = Il Nuovo Cimento|volume = 6|issue = S1|pages = 356–380|bibcode = 1957NCim....6S.356S|s2cid = 122425051}} and by Marco Fernandes in his PhD thesis of 1995, who had constructed orthogonal Clifford algebras for certain pairs of dual Grassmann algebras. Adopting a similar approach, Hiley constructed algebraic spinors as minimal left ideals of a process algebra built on the Kauffman's notion of distinction. By nature of their construction, these algebraic spinors are both spinors and elements of that algebra. Whereas they can be mapped (projected) into an external Hilbert space of ordinary spinors of the quantum formalism in order to recover the conventional quantum dynamics, Hiley emphasizes that the dynamic algebraic structure can be exploited more fully with the algebraic spinors than with the ordinary spinors. In this aim, Hiley introduced a Clifford density element expressed in terms of left and right minimal ideals of a Clifford algebra, analogous to the density matrix expressed as an outer product in bra–ket notation in conventional quantum mechanics. On this basis Hiley showed how three Clifford algebras Cl_0,1, Cl_3,0, Cl_1,3 form a hierarchy of Clifford algebras over the real numbers that describe the dynamics of the Schrödinger, Pauli and Dirac particles, respectively.

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 Cl_4,2 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 ([http://www.quantum.at/fileadmin/quantum/documents/Vienna_Symposium_2009_-_Book_of_Abstracts.pdf#page=17 abstract]{{Dead link|date=October 2023 |bot=InternetArchiveBot |fix-attempted=yes }}) The technique developed by Hiley and his co-workers demonstrates

:"that quantum phenomena per se can be entirely described in terms of Clifford algebras taken over the reals without the need to appeal to specific representation in terms of wave functions in a Hilbert space. This removes the necessity of using Hilbert space and all the physical imagery that goes with the use of the wave function".

This result is in line with Hiley's striving for a purely algebraic approach to quantum mechanics that is not a priori defined on any external vector space. In this purely algebraic approach, the information normally contained in the wave function is encoded in an element of a minimal left ideal of the algebra.{{Cite journal | last1 = Hiley | first1 = Basil J. | last2 = Dennis | first2 = Glen | doi = 10.3390/sym16010067 | title = de Broglie, General Covariance and a Geometric Background to Quantum Mechanics | journal = Symmetry | volume = 16 | number = 67 | year = 2024 | page = 67 | doi-access = free }}

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 ([http://www.bbk.ac.uk/tpru/BasilHiley/Frontiersoflife.pdf 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.{{Cite journal |author=Ernst Binz |author2=Maurice A. de Gosson |author3=Basil J. Hiley |title=Clifford Algebras in Symplectic Geometry and Quantum Mechanics |journal=Foundations of Physics |year=2013 |volume=2013 |number=43 |pages=424–439| doi=10.1007/s10701-012-9634-z |arxiv=1112.2378 |bibcode=2013FoPh...43..424B |s2cid=6490101}}

In 2011, de Gosson and Hiley showed that when in Bohm's model a continuous observation of a trajectory is performed, the observed trajectory is identical to the classical particle trajectory. This finding puts the Bohm model in connection to the well-known quantum Zeno effect.Maurice A. de Gosson, Basil J. Hiley: Zeno paradox for Bohmian trajectories: the unfolding of the metatron, January 3, 2011 ([http://www.freewebs.com/cvdegosson/ZenoPaper.pdf PDF] – retrieved 7 June 2011) They confirmed this finding when they showed that the quantum potential enters into the approximation for the quantum propagator only on time scales of the order of $O(\backslash Delta\; t^2)$, which means that a continuously observed particle behaves classically and furthermore that the quantum trajectory converges to a classical trajectory if the quantum potential decreases with time.{{Cite journal|doi = 10.1016/j.physleta.2013.08.031|title = Short-time quantum propagator and Bohmian trajectories|year = 2013|last1 = De Gosson|first1 = Maurice|last2 = Hiley|first2 = Basil|journal = Physics Letters A|volume = 377|issue = 42|pages = 3005–3008|pmid = 24319313|pmc = 3820027|arxiv = 1304.4771|bibcode = 2013PhLA..377.3005D}}

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.{{Cite journal|arxiv = quant-ph/0102071|doi = 10.1016/S0375-9601(01)00677-6|title = Bohmian trajectories for photons|year = 2001|last1 = Ghose|first1 = Partha|last2 = Majumdar|first2 = A.S|last3 = Guha|first3 = S.|last4 = Sau|first4 = J.|journal = Physics Letters A|volume = 290|issue = 5–6|pages = 205–213|bibcode = 2001PhLA..290..205G|s2cid = 54650214}}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 [http://www.aip.org.au/Congress2010/Abstracts/Monday%206%20Dec%20-%20Orals/Session_3E/Kocsis_Observing_the_Trajectories.pdf] {{Webarchive|url=https://web.archive.org/web/20110626194505/http://www.aip.org.au/Congress2010/Abstracts/Monday%206%20Dec%20-%20Orals/Session_3E/Kocsis_Observing_the_Trajectories.pdf|date=2011-06-26}}{{Cite journal|doi = 10.1126/science.1202218|title = Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer|year = 2011|last1 = Kocsis|first1 = S.|last2 = Braverman|first2 = B.|last3 = Ravets|first3 = S.|last4 = Stevens|first4 = M. J.|last5 = Mirin|first5 = R. P.|last6 = Shalm|first6 = L. K.|last7 = Steinberg|first7 = A. M.|journal = Science|volume = 332|issue = 6034|pages = 1170–1173|pmid = 21636767|bibcode = 2011Sci...332.1170K|s2cid = 27351467}} 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.{{Cite journal|doi = 10.1088/1742-6596/361/1/012014|title = Weak Values: Approach through the Clifford and Moyal Algebras|year = 2012|last1 = Hiley|first1 = B. J.|journal = Journal of Physics: Conference Series|volume = 361|issue = 1|page = 012014|arxiv = 1111.6536|bibcode = 2012JPhCS.361a2014H}}

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.{{cite journal|last1=de Gosson|first1=Maurice A|title=Phase space quantization and the uncertainty principle|journal=Physics Letters A|volume=317|issue=5–6|year=2003|pages=365–369|issn=0375-9601|doi=10.1016/j.physleta.2003.09.008|bibcode = 2003PhLA..317..365D }}{{cite journal|author=Maurice A. de Gosson|title=Quantum blobs|journal=Foundations of Physics|date=April 2013|volume=43|issue=4|pages=440–457|arxiv = 1106.5468 |bibcode = 2013FoPh...43..440D |doi = 10.1007/s10701-012-9636-x |pmid=25530623|pmc=4267529}}{{cite journal|last1=Dennis|first1=Glen|last2=de Gosson|first2=Maurice A.|last3=Hiley|first3=Basil J.|title=Fermi's ansatz and Bohm's quantum potential|journal=Physics Letters A|volume=378|issue=32–33|year=2014|pages=2363–2366|issn=0375-9601|doi=10.1016/j.physleta.2014.05.020|bibcode = 2014PhLA..378.2363D }}{{cite journal|author1=Glen Dennis |author2=Maurice de Gosson |author3=Basil Hiley |title=Bohm's Quantum Potential as an Internal Energy|journal=Physics Letters A|volume=379|issue=18–19|date=26 June 2015|pages=1224–1227|doi=10.1016/j.physleta.2015.02.038|arxiv = 1412.5133 |bibcode = 2015PhLA..379.1224D |s2cid=118575562 }}

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{{cite journal|author=Robert Flack, Basil J. Hiley|title=Feynman Paths and Weak Values|journal=Entropy|volume=20|number=5|page=367|year=2018|doi=10.3390/e20050367|pmid=33265457|pmc=7512885|bibcode=2018Entrp..20..367F|doi-access=free}}{{Cite arXiv|title=Stapp, Bohm and the Algebra of Process|date=2018-09-17|eprint=1809.06078|last1=Hiley|first1=B. J.|class=quant-ph}} as an average of an ensemble of Feynman paths.{{cite journal|author=Robert Flack, Basil J. Hiley|title=Feynman Paths and Weak Values|journal=Entropy|volume=20|number=5|page=367|year=2018|doi=10.3390/e20050367|pmid=33265457|pmc=7512885|bibcode=2018Entrp..20..367F|doi-access=free}} Cited by {{Cite journal |author=Basil J. Hiley, Peter Van Reeth |title=Quantum Trajectories: Real or Surreal? |journal=Entropy |publisher=MDPI |volume=20 |number=5 |page=353 |doi=10.3390/e20050353 |date=2018-05-08 |pmid=33265443 |pmc=7512873 |bibcode=2018Entrp..20..353H |doi-access=free }}

Hiley has repeatedly discussed the reasons for which the Bohm interpretation has met resistance, these reasons relating for instance to the role of the quantum potential term and to assumptions on particle trajectories.B. J. Hiley: The conceptual structure of the Bohm interpretation in quantum mechanics. In {{Interlanguage link multi|Kalervo Vihtori Laurikainen|fi}}, C. Montonen, K. Sunnarborg (eds.): Symposium on the Foundations of Modern Physics 1994: 70 years of matter waves, {{ISBN|2-86332-169-2}}, Éditions Frontières, 1994, [https://books.google.com/books?hl=en&id=GeGf9-q3AJkC&pg=PA99 pages 99–118]B. J. Hiley: 'Welcher Weg' experiments from the Bohm perspective, PACS: 03.65.Bz, ([http://www.bbk.ac.uk/tpru/BasilHiley/WelcherWegBohmBJH2.pdf PDF]){{Cite arXiv|eprint = quant-ph/0010020|last1 = Hiley|first1 = B. J.|last2 = E Callaghan|first2 = R.|last3 = Maroney|first3 = O.|title = Quantum trajectories, real, surreal or an approximation to a deeper process?|year = 2000}}B. J. Hiley: Some remarks on the evolution of Bohm's proposals for an alternative to standard quantum mechanics, January 30, 2011, downloaded February 13, 2012 ([http://www.bbk.ac.uk/tpru/BasilHiley/History_of_Bohm_s_QT.pdf PDF]) He has shown how the energy–momentum-relations in the Bohm model can be obtained directly from the energy–momentum tensor of quantum field theory. He has referred to this as "a remarkable discovery, so obvious that I am surprised we didn't spot it sooner", pointing out that on this basis the quantum potential constitutes the missing energy term that is required for local energy–momentum conservation.B. J. Hiley: The Bohm approach re-assessed ([http://www.bbk.ac.uk/tpru/RecentPublications.html 2010 preprint]), [http://www.bbk.ac.uk/tpru/BasilHiley/BohmReassed1.pdf#page=6 p. 6] In Hiley's view the Bohm model and Bell's inequalities allowed a debate on the notion of non-locality in quantum physics or, in Niels Bohr's words, wholeness to surface.Basil Hiley: [http://www.bbk.ac.uk/tpru/BasilHiley/OxfordConference.pdf Quantum reality unveiled through process and the implicate order], 20 February 2008, downloaded 5 February 2012

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.O. Bratteli, DW Robertson, Operator Algebras and Quantum Statistical Mechanics. I, Springer, New York, Heidelberg, Berlin, 1979. He points out that the algebraic representation allows to establish a connection to the thermo field dynamics of Hiroomi Umezawa, using a bialgebra constructed from a two-time quantum theory.{{cite book|author=Basil J. Hiley|chapter=Time and the Algebraic Theory of Moments|title=Re-Thinking Time at the Interface of Physics and Philosophy|series=On Thinking|volume=4|pages=147–175|doi=10.1007/978-3-319-10446-1_7|arxiv=1302.2323|year=2015|isbn=978-3-319-10445-4|s2cid=118683643}} 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, [http://www.bbk.ac.uk/tpru/BasilHiley/BohmReassed1.pdf#page=9 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".{{Cite journal|arxiv = 1107.6019|doi = 10.1007/s10701-012-9646-8|title = Causal Categories: Relativistically Interacting Processes|year = 2013|last1 = Coecke|first1 = Bob|last2 = Lal|first2 = Raymond|journal = Foundations of Physics|volume = 43|issue = 4|pages = 458–501|bibcode = 2013FoPh...43..458C|s2cid = 119294268}} 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.{{Cite journal|arxiv = 1108.2261|doi = 10.1007/s10701-012-9653-9|title = Being, Becoming and the Undivided Universe: A Dialogue Between Relational Blockworld and the Implicate Order Concerning the Unification of Relativity and Quantum Theory|year = 2013|last1 = Silberstein|first1 = Michael|last2 = Stuckey|first2 = W. M.|last3 = McDevitt|first3 = Timothy|journal = Foundations of Physics|volume = 43|issue = 4|pages = 502–532|bibcode = 2013FoPh...43..502S|s2cid = 19920497}}

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".{{Cite arXiv|eprint=1405.4772v3|last1=Pylkkänen|first1=P.|last2=Hiley|first2=B. J.|last3=Pättiniemi|first3=I.|title=Bohm's approach and Individuality|year=2014|class=quant-ph}}

Hiley and Pylkkänen addressed the question of the relation between mind and matter by the hypothesis of an active information contributing to quantum potential.Basil J. Hiley, Paavo Pylkkänen: Active information and cognitive science – A reply to Kieseppä, Brain, Mind and Physics, P. Pylkkänen et al. (Eds.), IOS Press, 1997, {{ISBN|90-5199-254-8}}, [https://books.google.com/books?id=IdEpIXyXew4C&pg=PA64 p. 64 ff.]Basil J. Hiley, Paavo Pylkkänen: Naturalizing the mind in a quantum framework. In Paavo Pylkkänen and Tere Vadén (eds.): Dimensions of conscious experience, Advances in Consciousness Research, Volume 37, John Benjamins B.V., 2001, {{ISBN|90-272-5157-6}}, [https://books.google.com/books?hl=en&lr=&id=CCX1NhhpQNQC&pg=PA119 pages 119–144]Basil J. Hiley: From the Heisenberg picture to Bohm: a new perspective on active information and its relation to Shannon information, Proc. Conf. Quantum Theory: reconsideration of foundations, A. Khrennikov (ed.), pp. 141–162, Växjö University Press, Sweden, 2002, ([http://www.birkbeck.ac.uk/tpru/BasilHiley/Vexjo2001W.pdf PDF])Basil J. Hiley, Paavo Pylkkänen: [http://www.bbk.ac.uk/tpru/BasilHiley/CanMindEffMatterBJHPP05.pdf Can mind affect matter via active information], Mind & Matter vol. 3, no. 2, pp. 7–27, Imprint Academic, 2005 Recalling notions underlying Bohm's approach, Hiley emphasises that active information "informs" in the sense of a literal meaning of the word: it "induces a change of form from within", and "this active side of the notion of information [...] seems to be relevant both to material processes and to thought".Basil Hiley: [http://www.ctr4process.org/publications/Articles/LSI05/Hiley%20paper.pdf Process and the implicate order: their relevance to quantum theory and mind] {{Webarchive|url=https://web.archive.org/web/20110926212330/http://www.ctr4process.org/publications/Articles/LSI05/Hiley%20paper.pdf |date=26 September 2011 }}, p. 14 and p. 25 He emphasizes: "even though the quantum level may be analogous to the human mind only in a rather limited way, it does help to understand the interlevel relationships if there are some common features, such as the activity of information, shared by the different levels. The idea is not to reduce everything to the quantum level but rather to propose a hierarchy of levels, which makes room for a more subtle notion of determinism and chance".

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.

: "Our proposal is that in the brain there is a manifest (or physical) side and a subtle (or mental) side acting at various levels. At each level, we can regard one side the manifest or material side, while the other is regarded as subtle or mental side. The material side involves electrochemical processes of various kinds, it involves neuron activity and so on. The mental side involves the subtle or virtual activities that can be actualised by active information mediating between the two sides.

: These sides [...] are two aspects of the same process. [...] what is subtle at one level can become what is manifest at the next level and so on. In other words if we look at the mental side, this too can be divided into a relatively stable and manifest side and a yet more subtle side. Thus there is no real division between what is manifest and what is subtle and in consequence there is no real division between mind and matter".Basil Hiley: Quantum mechanics and the relationship between mind and matter, in: P. Pylkkanen, P. Pylkko und Antti Hautamaki (eds.): Brain, Mind and Physics (Frontiers in Artificial Intelligence and Applications), IOS Press, 1995, {{ISBN|978-90-5199-254-0}}, pp. 37–54, see [https://books.google.com/books?hl=de&lr=&id=IdEpIXyXew4C&oi=fnd&pg=PA51 pp. 51,52]

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: [http://www.bbk.ac.uk/tpru/BasilHiley/CASYS%202000FWhereP.pdf 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.{{YouTube|9gFCj5PPEyw|David Bohm Quantum theory versus Copenhagen Interpretation}}

Hiley received the Majorana Prize by the Electronic Journal of Theoretical Physics for "Best person in physics" in 2012.

; Overview articles :

• {{cite journal | last1 = Hiley | first1 = Basil J. | last2 = Dennis | first2 = Glen | doi = 10.3390/sym16010067 | title = de Broglie, General Covariance and a Geometric Background to Quantum Mechanics | journal = Symmetry | volume = 16 | number = 67 | year = 2024 | page = 67 | doi-access = free }}
• {{cite journal | last1 = Hiley | first1 = Basil J. | last2 = Dennis | first2 = Glen | last3 = de Gosson | first3 = Maurice A. | doi = 10.1016/j.aop.2022.168759 | title = The role of geometric and dynamical phases in the Dirac–Bohm picture | journal = Annals of Physics | volume = 438 | pages = 168759 | year = 2022 | doi-access = free }}
• {{cite book|author=B. J. Hiley|title=Beyond Peaceful Coexistence |chapter=The Algebraic Way |year=2016 |pages=1–25 |doi=10.1142/9781783268320_0002 |arxiv=1602.06071|isbn=978-1-78326-831-3 |s2cid=119284839 }}
• {{cite book |author=B. J. Hiley|chapter=Aspects of Algebraic Quantum Theory: a Tribute to Hans Primas |editor1=Harald Atmanspacher|editor2=Ulrich Müller-Herold|title=From Chemistry to Consciousness: The Legacy of Hans Primas|chapter-url=https://books.google.com/books?id=csocDQAAQBAJ&pg=PA111|date=20 September 2016|publisher=Springer|isbn=978-3-319-43573-2|pages=111–125 |doi=10.1007/978-3-319-43573-2_7|arxiv=1602.06077|s2cid=118548614}}
• {{cite arXiv |eprint=1303.6057|last1=Hiley|first1=B. J.|title=Bohmian Non-commutative Dynamics: History and New Developments|year=2013|class=quant-ph}}
• B. J. Hiley: Particles, fields, and observers. In: Baltimore, D., Dulbecco, R., Jacob, F., Levi-Montalcini, R. (eds.) Frontiers of Life, vol. 1, pp. 89–106. Academic Press, New York (2002)

; Books :

• David Bohm, Basil Hiley: The Undivided Universe: An Ontological Interpretation of Quantum Theory, Routledge, 1993, {{ISBN|0-415-06588-7}}
• F. David Peat (Editor) and Basil Hiley (Editor): Quantum Implications: Essays in Honour of David Bohm, Routledge & Kegan Paul Ltd, London & New York, 1987 (edition of 1991 {{ISBN|978-0-415-06960-1}})

; Other :

• Foreword to: "The Principles of Newtonian and Quantum Mechanics – The Need for Planck's Constant, h" by Maurice A. de Gosson, Imperial College Press, World Scientific Publishing, 2001, {{ISBN|1-86094-274-1}}
• Foreword to the 1996 edition of: "The Special Theory of Relativity" by David Bohm, Routledge, {{ISBN|0-203-20386-0}}
• {{cite journal | last1 = Hiley | first1 = B. J. | doi = 10.1098/rsbm.1997.0007 | title = David Joseph Bohm. 20 December 1917--27 October 1992: Elected F.R.S. 1990 | journal = Biographical Memoirs of Fellows of the Royal Society | volume = 43 | pages = 107–131 | year = 1997 | title-link = David Bohm |jstor=770328| s2cid = 70366771 }}

{{reflist|2}}

• William Seager, [https://link.springer.com/article/10.1007/s10701-012-9672-6 Classical Levels, Russellian Monism and the Implicate Order]. Foundations of Physics, April 2013, Volume 43, Issue 4, pp. 548–567.

• [http://www.bbk.ac.uk/tpru/BasilHiley.html Basil Hiley], Birkbeck College – [http://www.bbk.ac.uk/tpru/BasilHiley/ASQT.html Publications on algebraic structures in quantum theory] – [http://www.bbk.ac.uk/tpru/RecentPublications.html Recent publications]
• [http://inspirehep.net/search?p=find+a+hiley,+basil find a hiley, basil – Search Results], High-Energy Physics Literature Database (INSPIRE-HEP)
• Daniel M. Greenberger, Klaus Hentschel, Friedel Weinert (eds.): Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy, Springer, 2009, {{ISBN|978-3-540-70622-9}}:
• [https://books.google.com/books?id=ekyAV8VtfuYC&pg=PA884 Basil J. Hiley & authors bios], Google Books
• [https://doi.org/10.1007%2F978-3-540-70626-7_88 Hidden variables] {{doi|10.1007/978-3-540-70626-7_88}}
• [https://doi.org/10.1007%2F978-3-540-70626-7_145 Pilot waves] {{doi|10.1007/978-3-540-70626-7_145}}
• Interviews with Basil Hiley:
• [http://www.bbc.co.uk/programmes/b00hv1dp The measurement problem in physics], In Our Time, BBC Radio 4, a discussion with Melvyn Bragg and guests Basil Hiley, Simon Saunders and Roger Penrose, 5 March 2009
• [http://www.aip.org/history/ohilist/31624.html Interview with Basil Hiley] {{Webarchive|url=https://web.archive.org/web/20130126122524/http://www.aip.org/history/ohilist/31624.html |date=26 January 2013 }} conducted by Alexei Kojevnikov on December 5, 2000, Oral History Transcript, Niels Bohr Library & Archives, American Institute of Physics
• [http://www.aip.org/history/ohilist/33822.html Interview with Basil Hiley] {{Webarchive|url=https://web.archive.org/web/20121015004915/http://www.aip.org/history/ohilist/33822.html |date=15 October 2012 }} conducted by Olival Freire on January 11, 2008, Oral History Transcript, Niels Bohr Library & Archives, American Institute of Physics
• George Musser: [http://blogs.scientificamerican.com/critical-opalescence/2013/11/04/the-wholeness-of-quantum-reality-an-interview-with-physicist-basil-hiley/ The Wholeness of Quantum Reality: An Interview with Physicist Basil Hiley], Scientific American Blogs, November 4, 2013
• [http://www.goertzel.org/dynapsyc/1997/interview.html Interview with Basil Hiley] conducted by M. Perus
• {{YouTube|9gFCj5PPEyw|David Bohm Quantum theory versus Copenhagen Interpretation}}
• {{YouTube|wayQn0uVIvE|David Bohm, Wholistic Universe, quantum physics}}
• {{YouTube|oaNhL7NHOyU|Taher Gozel interview with Basil Hiley}} (part 1)
• {{YouTube|6eK6jxD8UwM|Basil Hiley & Taher Gozel}}, further interview (part 1)
• Lecture slides by Basil Hiley:
• [http://www.hep.ucl.ac.uk/~robflack/BasilHiley_seminar_UCL.pdf Weak measurements: A new type of quantum measurement and its experimental implications] (slides)
• Moyal and Clifford algebras in the Bohm approach ([http://www.tcm.phy.cam.ac.uk/~mdt26/tti_talks/deBB_10/hiley_tti2010.pdf slides] {{Webarchive|url=https://web.archive.org/web/20120606184638/http://www.tcm.phy.cam.ac.uk/~mdt26/tti_talks/deBB_10/hiley_tti2010.pdf |date=6 June 2012 }})
• [http://www.tcm.phy.cam.ac.uk/~mdt26/esdg/basil_abstract.pdf Weak measurements: Wigner–Moyal in a new light] {{Webarchive|url=https://web.archive.org/web/20150610205603/http://www.tcm.phy.cam.ac.uk/~mdt26/esdg/basil_abstract.pdf |date=10 June 2015 }} ([http://www.tcm.phy.cam.ac.uk/~mdt26/esdg_slides/hiley_weak_measurements.pdf slides] {{Webarchive|url=https://web.archive.org/web/20150610204010/http://www.tcm.phy.cam.ac.uk/~mdt26/esdg_slides/hiley_weak_measurements.pdf |date=10 June 2015 }}, {{YouTube|q_jHmoxuxsY|audio}})
• [http://categorieslogicphysics.wikidot.com/people#basilhiley Towards a quantum geometry: Groupoids, Clifford algebras and shadow manifolds], May 2008 ([http://www.cs.ox.ac.uk/quantum/talksarchive/clap2/clap2-basilhiley.pdf slides], {{YouTube|OQpGJvlvUYg|audio}})
• Lectures by Basil Hiley recorded at the [http://www.archmathsci.org/conferences-and-workshops/askloster/ Åskloster Symposia]:
• [https://web.archive.org/web/20130102103809/http://www.archmathsci.org/conferences-and-workshops/askloster/2004/part-1/ 7-7-2004], [https://web.archive.org/web/20150611070157/http://www.archmathsci.org/conferences-and-workshops/askloster/2004/part-8/ 10-7-2004], [https://web.archive.org/web/20150611070201/http://www.archmathsci.org/conferences-and-workshops/askloster/2005/videos-1/ 29-6-2005], [https://web.archive.org/web/20150611070208/http://www.archmathsci.org/conferences-and-workshops/askloster/2006/videos-1/ 9-7-2006], [https://web.archive.org/web/20150610200858/http://www.archmathsci.org/conferences-and-workshops/askloster/2007/videos-july-5th/ 5-7-2007], [https://archive.today/20130414085415/http://www.archmathsci.org/conferences-and-workshops/askloster/2008/videos-page-2/ 25-7-2008], [https://archive.today/20130225235920/http://www.archmathsci.org/conferences-and-workshops/askloster/2008/videos-july-27/ 27-7-2008], [https://web.archive.org/web/20150610202948/http://www.archmathsci.org/conferences-and-workshops/askloster/2009/videos/ 23-7-2009], [https://web.archive.org/web/20150610205119/http://www.archmathsci.org/conferences-and-workshops/askloster/2009/videos-4/ 26-7-2009]

{{Authority control}}

{{DEFAULTSORT:Hiley, Basil}}

Category:1935 births

Category:2025 deaths

Category:British physicists

Category:British theoretical physicists

Category:British quantum physicists

Category:Academics of Birkbeck, University of London

Category:Alumni of King's College London