Current Research Interests - Theodore J. Allen
Hybrid Mesons, QCD, Relativistic Boundstates and Strings
Hybrid Mesons from relativistic strings
Hybrid mesons, quark antiquark boundstates in which the gluonic degrees of
freedom are in an excited state, are currently a subject of intense
theoretical and experimental interest. I recently showed in collaboration
with M.G. Olsson and S. Veseli that the excitation states of the gluonic
degrees of freedom seen in lattice simulations of QCD with fixed quark and
antiquark sources are well described by string-like excitations of the QCD
field. This should also hold true when the quarks at the end of the string
are not fixed. I have developed a relativistic description of these
string-like excitations starting from the Nambu-Goto action. We are
working to predict the spectrum of the resulting hybrid mesons numerically.
We are also investigating quark mass corrections to the energies when the
quark masses are small.
Exactness of Born-Oppenheimer straight-string assumption to
first order
In my most recent work, we have shown that for two quarks connected by a
relativistic string, the Born-Oppenheimer assumption of a straight string
for the lowest energy state is a good approximation. In addition, we find
the shape of the string when the end has a small angular acceleration and
show that the string develops a small radial momentum, whose consequences
we calculate.
Lorentz vector potential model for quarkonium
Last year we published a QCD-inspired model that reproduces all the good
features of scalar potential models, such as linearly rising Regge
trajectories, but which is more realistic as it contains a Lorentz vector
potential. In the limit of small quark mass, the wave equation reduces to
that of the scalar potential. I have derived the Salpeter kernel for this
model and we are working to find its spectrum numerically.
QCD and strings
I am very interested in the dynamics associated with the formation of
chromoelectric ``flux tubes'' in QCD and am working to connect the monopole
currents and their fields that are seen in lattice simulations with simple
geometric constructions in first-quantized string theory.
BRST Quantization and Constrained Mechanical Systems
The quantum description of constrained theories such as Yang-Mills, general
relativity, and string theory, has been a continuing interest of mine. My
thesis work was on the use of constrained quantization in point particle
mechanics and string theory, especially the problem of quantizing the
manifestly supersymmetric string and particle theories. With my graduate
student Dennis Crossley, I extended the BRST quantization method to systems
with complex classical constraint functions and non-hermitian quantum
constraints. We published two papers, one describing a formalism in which
it is possible to quantize systems with complex first-class constraints,
for which the BRST charge is not hermitian and another paper in which we
solve the same problem when the complex conjugates of the constraints are
linearly dependent upon the constraints themselves.
BRST Quantization of Gravity in Self-dual Variables
Much of the work I have been doing on BRST quantization in the last few
years has been in the direction of applying these methods to quantum
general relativity in Ashtekar's new canonical variables. BRST methods can
be used to factor out the infinite gauge volume in an inner product and
generally yields a larger class of physical states than Dirac quantization.
It is also our hope to obtain a larger class of physical states than the
knot invariants that are the only physical states known at present, and to
try to connect the BRST formulation with the spin network description that
has recently been developed. For his Ph.D. thesis, my student Dennis Crossley
applied the methods we worked out together to Ashtekar gravity. We found
that we can construct a hermitian BRST charge for this theory that is
nearly polynomial.
The Fractional Quantum Hall Effect and Vortex Dynamics
I have continuing interest in the physics associated with the fractional
quantum Hall effect and, more generally, the physics of charged particles
in magnetic fields. With former UW-Madison graduate student Andy Bordner,
I developed an effective field theory description of the fractional quantum
Hall state. We determined the most general non-relativistic theory that
admits charged vortex states and we performed a collective coordinate
quantization of the vortices. Together with Dennis Crossley, we compared
the reduced phase space and Dirac-Gupta-Bleuler quantizations of charged
vortex mechanics and found that the equivalence of these two methods of
quantization depends upon the quantum statistics of the vortices. These
two methods produce Hamiltonians with different spectra when the vortices
have anyonic statistics.
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© 1997, 1998 Ted Allen