John V. Corbett, BSc PhD Adelaide Univ. jvc@ics.mq.edu.au is Associate Professor of Mathematical Physics in the Department of Mathematics at Macquarie University-Sydney, Australia. Areas of Research: Mathematical aspects of quantum theory. Group representations in quantum scattering theory, theory of measurement. Dynamical systems associated with quadratic systems.
Visit the school's website: http://www.ics.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.
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), 211-23 (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).
Preprints:
You will find some preprints by myself and Murray Adelman at ftp://ftp.ics.mq.edu.au/maths/murray.
At the moment there are three preprints by Murray Adelman (retired) and myself:
Supervised PhD Theses:
Dr. Dale Alan Woodside dalew@physics.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 are available for examination (in *.ps 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: August 18, 2004 by:
Dale Alan Woodside, PhD ( dalew@physics.mq.edu.au)