Papers by Thomas Curtright
Systems built out of N-body interactions, beyond 2-body interactions, are formulated on the plane... more Systems built out of N-body interactions, beyond 2-body interactions, are formulated on the plane, and investigated classically and quantum mechanically (in phase space). Their Wigner Functions—the density matrices in phase-space quantization—are given and analyzed. In this brief note, we consider systems of N particles on a plane, interacting through N-body potentials, over and above more familiar 2-body potentials. Below, we provide the compact generating function, G(a,a ∗,b,b ∗;z,z ∗,p,p ∗ ) = 1 1 |a|
Physics Letters A, 2017
The number of times spin s appears in the Kronecker product of n spin j representations is comput... more The number of times spin s appears in the Kronecker product of n spin j representations is computed, and the large n asymptotic behavior of the result is obtained. Applications are briefly sketched.
AIP Conference Proceedings, 2003
We relate classical and quantum Dirac and Nambu brackets. At the classical level, we use the rela... more We relate classical and quantum Dirac and Nambu brackets. At the classical level, we use the relations between the two brackets to gain some insight into the Jacobi identity for Dirac brackets, among other things. At the quantum level, we suggest that the Nambu bracket is the preferred method for introducing constraints, although at the expense of some unorthodox behavior, which we describe in detail.
Journal of Physics A: Mathematical and Theoretical, 2014
Recent results for rotations expressed as polynomials of spin matrices are derived here by elemen... more Recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods. Structural features of the results are then examined in the framework of biorthogonal systems, to obtain an alternate derivation. The central factorial numbers play key roles in both derivations.
Branes, Strings, and Odd Quantum Nambu Brackets
Quantum Theory and Symmetries - Proceedings of the 3rd International Symposium, 2004
Lecture Notes in Physics
This is a pedagogical digest of results reported in [Curtright, Fairlie, & Zachos 1997], and an e... more This is a pedagogical digest of results reported in [Curtright, Fairlie, & Zachos 1997], and an explicit implementation of Euler's construction for the solution of the Poisson Bracket dual Nahm equation. But it does not cover 9 and 10-dimensional systems, and subsequent progress on them [Fairlie 1997]. Cubic interactions are considered in 3 and 7 space dimensions, respectively, for bosonic membranes in Poisson Bracket form. Their symmetries and vacuum configurations are explored. Their associated first order equations are transformed to Nahm's equations, and are hence seen to be integrable, for the 3-dimensional case, by virtue of the explicit Lax pair provided. Most constructions introduced also apply to matrix commutator or Moyal Bracket analogs.
Physical Review D, 1986
%'e explore the classical dynamics of strings with rigidity, i.e. , with terms added to the strin... more %'e explore the classical dynamics of strings with rigidity, i.e. , with terms added to the string action which depend on the extrinsic curvature of the world sheet. %e study classical solutions of the string equations of motion using both analytical and numerical methods, and we determine the leading Regge trajectories J(E2) for a set of classical rotating string configurations. %e observe that for open strings the dominant classical solutions are identical to those for the conventional Nambu string, and correspondingly give linear trajectories. However, for closed strings we find new solutions that include finite-energy, static configurations, and that give trajectories which are nonlinear as the lowest-energy solution is approached, but become linear asymptotically as J(E')~ao.
Progress of Theoretical Physics Supplement, 1999
In this lecture, a limited introduction of gauge invariance in phase-space is provided, predicate... more In this lecture, a limited introduction of gauge invariance in phase-space is provided, predicated on canonical transformations in quantum phase-space. Exact characteristic trajectories are also specified for the time-propagating Wigner phase-space distribution function: they are especially simple-indeed, classical-for the quantized simple harmonic oscillator. This serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase-space. This is a pedagogical selection from work published in Refs. 1) and 2).
Physics Letters B, 1997
Cubic interactions are considered in 3 and 7 space dimensions, respectively, for bosonic membrane... more Cubic interactions are considered in 3 and 7 space dimensions, respectively, for bosonic membranes in Poisson Bracket form. Their symmetries and vacuum configurations are discussed. Their associated first order equations are transformed to Nahm's equations, and are hence seen to be integrable, for the 3-dimensional case, by virtue of the explicit Lax pair provided. The constructions introduced also apply to commutator or Moyal Bracket analogues.
Physics Letters B, 2009
We consider several ternary algebras relevant to physics. We compare and contrast the quantal ver... more We consider several ternary algebras relevant to physics. We compare and contrast the quantal versions of the algebras, as realized through associative products of operators, with their classical counterparts, as realized through classical Nambu brackets. In some cases involving infinite algebras, we show the classical limit may be obtained by a contraction of the quantal algebra, and then explicitly realized through classical brackets. We illustrate this classical-contraction method by the Virasoro-Witt example.
Physics Letters A, 2002
Systems built out of N-body interactions, beyond 2-body interactions, are formulated on the plane... more Systems built out of N-body interactions, beyond 2-body interactions, are formulated on the plane, and investigated classically and quantum mechanically (in phase space). Their Wigner Functions-the density matrices in phase-space quantization-are given and analyzed.
Trajectories of Strings with Rigidity
Physical Review Letters, 1986
Physical Review D, 1995
A generating functional F is found for a canonical nonabelian dual transformation which maps the ... more A generating functional F is found for a canonical nonabelian dual transformation which maps the supersymmetric chiral O(4) σ-model to an equivalent supersymmetric extension of the dual σmodel. This F produces a mapping between the classical phase spaces of the two theories in which the bosonic (coordinate) fields transform nonlocally, the fermions undergo a local tangent space chiral rotation, and all currents (fermionic and bosonic) mix locally. Purely bosonic curvature-free currents of the chiral model become a symphysis of purely bosonic and fermion bilinear currents of the dual theory. The corresponding transformation functional T which relates wavefunctions in the two quantum theories is argued to be exactly given by T = exp(iF).
Features of time-independent Wigner functions
Physical Review D, 1998
Currents, charges, and canonical structure of pseudodual chiral models
Physical Review D, 1994
Physical Review D, 2003
The classical and quantum features of Nambu mechanics are analyzed and fundamental issues are res... more The classical and quantum features of Nambu mechanics are analyzed and fundamental issues are resolved. The classical theory is reviewed and developed utilizing varied examples. The quantum theory is discussed in a parallel presentation, and illustrated with detailed specific cases. Quantization is carried out with standard Hilbert space methods. With the proper physical interpretation, obtained by allowing for different time scales on different invariant sectors of a theory, the resulting non-Abelian approach to quantum Nambu mechanics is shown to be fully consistent.
Nuclear Physics B, 1996
The Supersymmetric Dual Sigma Model (SDSM) is a local field theory introduced to be nonlocally eq... more The Supersymmetric Dual Sigma Model (SDSM) is a local field theory introduced to be nonlocally equivalent to the Supersymmetric Chiral nonlinear σ-Model (SCM), this dual equivalence being proven by explicit canonical transformation in tangent space. This model is here reconstructed in superspace and identified as a chiral-entwined supersymmetrization of the Dual Sigma Model (DSM). This analysis sheds light on the boson-fermion symphysis of the dual transition, and on the new geometry of the DSM. 1 i.e. preserve canonical commutation relations. Canonical transformations in quantum mechanics underlie Dirac's path integral formulation [4], and have been discussed extensively in field theory, e.g. [5, 6, 7].
Nuclear Physics B, 1986
We discuss some general effects produced by adding Wess-Zumino terms to the actions of nonlinear ... more We discuss some general effects produced by adding Wess-Zumino terms to the actions of nonlinear sigma models, an addition which may be made if the underlying field manifold has appropriate homological properties. We emphasize the geometrical aspects of such models, especially the role played by torsion on the field manifold. For general chiral models, we show explicitly that the torsion is simply the structure constant of the underlying Lie group, converted by vielbeine into an antisymmetric rank-three tensor acting on the field manifold. We also investigate in two dimensions the supersymmetric extensions of nonlinear sigma models with torsion, showing how the purely bosonic results carry over completely. We consider in some detail the renormafization effects produced by the Wess-Zumino terms using the background field method. In particular, we demonstrate to two-loop order the existence of geometrostasis, i.e. fixed points in the renormalized geometry of the field manifold due to parallelism.
Deformation quantization of superintegrable systems and Nambu mechanics
New Journal of Physics, 2002
Negative Probability and Uncertainty Relations
Modern Physics Letters A, 2001
A concise derivation of all uncertainty relations is given entirely within the context of phase-s... more A concise derivation of all uncertainty relations is given entirely within the context of phase-space quantization, without recourse to operator methods, to the direct use of Weyl's correspondence, or to marginal distributions of x and p.
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Papers by Thomas Curtright