page_title: Basil Hiley
section_title: Implicate orders, pre-space and algebraic structures
hierarchical_section_title: Basil Hiley / Work / Implicate orders, pre-space and algebraic structures
caption_attribution_description: A less well known sculpture by a famous artist, Antony Gormley. Added to 'cream of the crop' as best of 2004Millenium Man
context_page_description: Basil J. Hiley, is a British quantum physicist and professor emeritus of the University of London.
  Long-time colleague of David Bohm, Hiley is known for his work with Bohm on implicate orders and for his work on algebraic descriptions of quantum physics in terms of underlying symplectic and orthogonal Clifford algebras. Hiley co-authored the book The Undivided Universe with David Bohm, which is considered the main reference for Bohm's interpretation of quantum theory.
  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.
context_section_description: 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. This concept is described in the books Wholeness and the Implicate Order by Bohm and Science, Order, and Creativity by Bohm and F. David Peat. 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:
  "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 spinor 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 Grassmann, Hamilton, and Clifford. 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". 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 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 M
caption: Quantum Cloud by Antony Gormley, influenced by an exchange of thoughts among Hiley and Gormley on algebra and pre-space.[69]
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