John V. Corbett, BSc PhD *Adelaide Univ*.
john.corbett@mq.edu.au is
an *Associate Professor* of Mathematical Physics and a
*Senior Research Fellow* in the Department of Mathematics at Macquarie
University-Sydney, Australia. **Areas of Research:** Mathematical
aspects of quantum theory including quantum entanglement and intuisionist
logic. Group representations in quantum scattering theory, theory of
measurement. Dynamical systems associated with quadratic systems.

Visit the school's website: http://www.science.mq.edu.au/

**Mathematical Physics Research:**

Physics is one of the great traditional sources of mathematical problems and these take many forms. Some of the topics of mathematical physics in which Prof. John Corbett has contributed are as follows:

If we project *N* particles at each other and the number of particles
Is conserved, it is physically obvious that any clustering of the particles
which is possible under conservation of energy will occur with positive
probability. However, the proof of this within the usual mathematical
formulation of quantum mechanics has been very difficult to obtain for
an acceptable class of interactions. John Corbett has shown that
this problem is equivalent to an existence problem for certain holomorphic
representations of the group *SL*_2(**R**) on the subspace of the
continuous spectrum of the total Hamiltonian. This unifies earlier
work on this problem.

There is still considerable debate about the correct philosophical interpretation
of quantum mechanics, despite its successful applications in physics.
In this debate the status of the measurement problem is a central issue.
It seems that what is required is a mathematical framework that embraces
the standard formulation of quantum and classical mechanics, at least as
limiting cases. Prof. John Corbett and Dr. Murray Adelman
have analysed some simple physical systems with this end in mind.
The idea is to construct a topos for quantum systems which can be used
to measure the extent to which a physical quantity takes on various values.
This gives a useful mathematical description of a test of non-locality
and non-causality in quantum mechanics proposed recently by Dr. Dipankar
Home dhom@boseinst.ernet.in
(a frequent visitor to Macquarie Univ. from the Bose Institute in Calcutta).
The inspiration for this new topos for quantum systems is based on an
*Intuitionistic logic* which traces its geneology to Heyting algebras.

Recently, in collaboration with Thomas Durt, research has continued on intuitionistic logic and on the mathematical theory of what Prof. Corbett refers to as quantum real numbers (qr-numbers) in a quantum space with its own metric and topology, which is different from classical space. Quoting from his recent presentation, a preprint of which is in the publication list below, "The quantum real number (qr-number) interpretation of quantum mechanics is based on the claim that the ontological properties of microscopic entities differ from those of classical objects in that their attributes, which are represented by operators, always have values not as standard real numbers but as topos real numbers called qr-numbers. In place of the standard quantum states it uses conditions that are open subsets of quantum states [Heyting algebra tie in]. The qr-numbers, having extents as well as values, behave like functions whose domains are the extents."

John V. Corbett, "*Entanglement and the quantum spatial continuum*,"
at "75 Years of Quantum Entanglement**:** Foundations and Information Theoretic
Applications**:** S.N. Bose National Centre for Basic Sciences Silver Jubilee
Symposium," 6-10 Jan. 2011, Kolkata, India, AIP Conf. Proc. 1384, pp. 34-41
(2011).

John Corbett, "*Quantum particles are localized in quantum
space*." Preprint of presentation given at 16th UK and European
Meeting on the Foundations of Physics (Aberdeen, 5-7 July 2010).

pdf copy of above presentation: Corbett_Localization_in_Quantum_Mechanics.pdf 206.8 kB, pages 1-9.

J.V. Corbett and T. Durt, "*Spatial localization in quantum
theory based on qr-numbers*," Foundations of Physics, **40**,
607-628 (2010).

J.V. Corbett and T. Durt, "*An Intuitionistic Model of Single
Electron Interference*," Studia Logica - An International Journal
for Symbolic Logic, **95**, no.1, 81-100 (2010).

J.V. Corbett and T. Durt, "*Collimation processes in quantum
mechanics interpreted in quantum real numbers*," Studies in History
and Philosophy of Modern Physics, **40**, pp. 68-83 (2009).

J.V. Corbett, "*The mathematical structure of quantum real numbers*,"
arXiv:math-ph/0905.0944,
7 May (2009).

J.V. Corbett and D. Home, "*Bell's Inequality, Quantum Measurement
and Einstein Realism: A Unified Perspective*,"
arXiv:quant-ph/0802.2443,
18 Feb (2008).

J.V. Corbett, "*The Pauli Problem, state reconstruction and quantum
real numbers*," Reports on Mathematical Physics, **57**,
53-68 (2006).

J. Corbett and T. Durt, "*Quantum mechanics as a space-time theory*,"
arXiv:quant-ph/0512220,
23 Dec (2005).

Samuel Colin, John Corbett, Thomas Durt, and David Gross, "*About SIC
POVMs amd discrete Wigner distributions*," Journal of Optics B-Quantum
and Semiclassical Optics, **7**, no.12, S778-S785 (2005).

J.V. Corbett and D. Home, "*Information transfer and non-locality for
a tripartite entanglement using dynamics*." Phys. Lett. A
**333**, no.5, 382-388 (2004).

J. Corbett and T. Durt, "*Quantum mechanics interpreted in Quantum
Real Numbers*,"
arXiv:quant-ph/0211180,
27 Nov (2002).

John V. Corbett and Dipankar Home, "*Ipso-information-transfer*,"
arXiv:quant-ph/0103146,
27 Mar (2001).

John V. Corbett and Dipankar Home, "*Quantum effects involving interplay
between unitary dynamics and kinematic entanglement*." Phys. Rev. A
**62**, 062103 (2000).

M. Adelman and J.V. Corbett, "*A Sheaf Model for Intuitionistic Quantum
Mechanics*." Applied Categorical Structures **3** (1), 79-104 (1995).

M. Adelman and J.V. Corbett, "*Quantum Numbers Viewed Intuitionistically*."
in *Confronting the Infintite*, edited by Alan Carey, Wm. J. Ellis,
and Paul A. Pearce, World Scientific Press (1995).

Murray Adelman and John V. Corbett, "*A sheaf model for Intuitionistic
quantum mechanics*." ARC report, p. 1 (1993).

Murray Adelman, John V. Corbett, and C. A. Hurst, "*The geometry of
state space*." Found. Phys. **23** (2), 211-223 (1993).

Murray Adelman and John V. Corbett, "*Intuitionistic logic and Bell's
inequalities*." p. 45 (1991).

J. V. C. Corbett and C. A. Hurst, "*What is needed to determine a
state?*" Asia-Pacific Physics News, **4** (1990).

John V. Corbett, "*An Intuitionistic logic for quantum mechanics*."
in Proc., International Conf. "*In Search of Quantum Reality*" (New
Delhi, India, Jan. 1990).

John V. Corbett, "*On the state space structure of ideal incomplete
measurements.*" (1989).

J. V. Corbett, "*Quantum mechanical measurement of non-orthogonal states
and a test of nonlocality.*" Phys. Lett. A **130**, 419-25 (1988).

J. V. Corbett, "*Scattering theory for the dilation group. I. Simple
quantum mechanical scattering.*" J. Math. Phys. **24**, 1797-805 (1983).

J.V. Corbett and C.A. Hurst, "*Are wave functions uniquely
determined by their position and momentum distributions?*," Journal
of the Australian Mathematical Society Series B-Applied Mathematics,
vol.20, pt.2, pp. 182-201, Dec. 1977.

J.V. Corbett, "*Unbound motion and scattering*," Nuovo Cimento
della Societa Italiana di Fisica B-General Physics Relativity Astronomy
& Mathematical Physics Methods, vol.25B, ser.2, no.1, pp. 103-124,
11 Jan. 1975.

J.V. Corbett, "*Particles and simple scattering theory*," J.
Math.Phys. **16**, no.2, pp. 271-274, (1975).

J.V. Corbett, "*Galilean symmetry, measurement, and scattering
as an isomorphism between two subalgebras of observables*," Phys.
Rev. D **1**, no.12, pp. 3331-3344 (1970).

J.V. Corbett, "*Convergence of the Born series*," J.Math.Phys.
**9**, no.6, pp. 891-898 (1968).

**Supervised PhD Theses:**

Dr. Dale Alan Woodside
dale.woodside@mq.edu.au
was awarded his PhD in 1999 for a thesis entitled "*Investigation of
the Uniqueness Properties of Classical Four-Vector Fields in Euclidean and Minkowski
Spaces*", (1998). A copy of this thesis can be obtained through
Macquarie Univ. Library by following the link from Dale's web site
at URL:
http://www.physics.mq.edu.au/~dalew/ In his thesis Dale
Woodside develops Euclidean and Minkowski four-space extensions of
Helmholtz's uniqueness theorem for three-vector fields. He then goes on
to develop a new class of four-vector fields which rely on a new gauge
which he calls the *relativistic longitudinal gauge*, where the
four-curl (the Maxwell field tensor itself) is set to zero. An
article, which is based on his thesis, has been published in the
Journal of
Mathematical Physics.
The reference is: D. A. Woodside, "*Uniqueness theorems for classical
four-vector fields in Euclidean and Minkowski spaces*." J. Math. Phys.
**40**, 4911 (1999). A second article, which completes the essential
material from his thesis, has also been published
in J. Math. Phys. The reference is: D. A. Woodside, "*Classical
four-vector fields in the relativistic longitudinal gauge.*"
J. Math. Phys. **41**, 4622 (2000).
Preprints of these articles and others are available for examination
(in *.ps and *.pdf format) from his web site.

Dr. David L. Tilbrook was awarded his PhD in 1997 for a thesis entitled
"*The Quantisation of Fields in Flat and Curved Space-times*",
(1996). In his thesis David Tilbrook investigated the Fulling
generalisation of the standard canonical quantisation in Minkowski
space-time after first developing the generally covariant theory.
Previous authors have concluded that the spectrum of particles
associated with the Davies-Unruh effect is given by a Bose-Einstein
distribution, leading to what has become known as the
*thermalisation theorem*. In this work it is shown that
if the space-time under consideration is restricted to the right-Rindler
wedge then the spectrum is not thermal. It is demonstrated, however,
that the spectrum is well approximated by a Bose-Einstein distribution
in the limit of very large acceleration. In order to be able
to conclude that the Fulling quantisation leads to fundamentally different
Fock spaces with physically different vacuum states it is essential that
these vacuum states are not coordinate dependent, and this is
demonstrated with a flat space metric approach.

Dr. Greg Taylor was awarded his PhD in 1986 for a thesis entitled "*Superunification
in a model resembling Kaluza-Klein*". The problem of superunification
is that of finding a theory which describes the four fundamental forces
of nature while respecting the laws of quantum mechanics and relativity.
Greg Taylor's model assumes a universe of dimension greater than four in
which the usual four dimensional space-time is embedded. The gravitational
force is described by the geometry of four dimensional space-time and the
weak, strong and electromagnetic forces are represented geometrically in
the higher dimensions in the manner of the Kaluza-Klein theory of electromagnetism.
These forces leave their trace on the space-time manifold as potentials
that perturb the geodesics in this space. The model does not solve
all the problems, but it does give an interesting framework.

Dr. Keiko Yasukawa obtained her PhD for a thesis entitled "*Study
of a Nonlinear Control Algorithm Using Dynamical Systems Theory*",
(1990). In her thesis Keiko Yasukawa developed a new control
algorithm for nonlinear input-output systems which generalizes a linear
algorithm known as model-algorithmic control (MAC) by incorporating
some of the ideas from her previous study of Volterra and
Wiener functional expansions. In studying the stability properties
of the equations arising in the new algorithm, it was found that these
equations took a form which necessitated the development of a mathematical
framework within which the stability analysis could be
achieved. Application and generalization of results from discrete
dynamical systems theory proved to provide a suitable framework.

Last Modified: November 29, 2012 by:

Dale Alan Woodside, PhD ( dale.woodside@mq.edu.au)