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  • 1
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 39 (1998), S. 5199-5230 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We consider quantum dynamical systems whose degrees of freedom are described by N×N matrices, in the planar limit N→∞. Examples are gauge theories and the M(atrix)-theory of strings. States invariant under U(N) are "closed strings," modeled by traces of products of matrices. We have discovered that the U(N)-invariant operators acting on both open and closed string states form a remarkable new Lie algebra which we will call the heterix algebra. (The simplest special case, with one degree of freedom, is an extension of the Virasoro algebra by the infinite-dimensional general linear algebra.) Furthermore, these operators acting on closed string states only form a quotient algebra of the heterix algebra. We will call this quotient algebra the cyclix algebra. We express the Hamiltonian of some gauge field theories (like those with adjoint matter fields and dimensionally reduced pure QCD models) as elements of this Lie algebra. Finally, we apply this cyclix algebra to establish an isomorphism between certain planar matrix models and quantum spin chain systems. Thus we obtain some matrix models solvable in the planar limit; e.g., matrix models associated with the Ising model, the XYZ model, models satisfying the Dolan–Grady condition and the chiral Potts model. Thus our cyclix Lie algebra describes the dynamical symmetries of quantum spin chain systems, large-N gauge field theories, and the M(atrix)-theory of strings. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 2259-2269 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The symplectic structure and Hamiltonian dynamics for a class of Grassmannian manifolds are described here. Using the two-dimensional sphere (S2) and disc (D2) as illustrative cases, we write their path integral representations using coherent state techniques. These path integrals can be evaluated exactly by semiclassical methods, thus providing examples of the localization formula. Along the way, we also give a local coordinate description for a class of Grassmannians.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 1870-1890 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We formulate Yang–Mills theory in terms of the large-N limit, viewed as a classical limit, of gauge-invariant dynamical variables, which are closely related to Wilson loops, via deformation quantization. We obtain a Poisson algebra of these dynamical variables corresponding to normal-ordered quantum (at a finite value of (h-dash-bar)) operators. Comparing with a Poisson algebra one of us introduced in the past for Weyl-ordered quantum operators, we find, using ideas closely related to topological graph theory, that these two Poisson algebras are, roughly speaking, the same. More precisely speaking, there exists an invertible Poisson morphism between them. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 3308-3319 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Using an approach based on the canonical formalism, the Yang–Mills theories on a cylinder are rigorously analyzed. In this way the moduli space A/G, can be explicitly described with A being the space of connections and G the group of gauge transformations. In particular A/G0, G0 being the group of the pointed gauge transformations, is diffeomorphic to the structure group of the theory G, whereas A/G is G modulo the group of inner automorphisms. It is also proven that A → G is a principal fiber bundle with structure group G0. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 39 (1998), S. 749-759 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We obtain for the attractive Dirac δ-function potential in two-dimensional quantum mechanics a renormalized formulation that avoids reference to a cutoff and running coupling constant. Dimensional transmutation is carried out before attempting to solve the system, and leads to an interesting eigenvalue problem in N−2 degrees of freedom (in the center of momentum frame) when there are N particles. The effective Hamiltonian for N−2 particles has a nonlocal attractive interaction, and the Schrodinger equation becomes an eigenvalue problem for the logarithm of this Hamiltonian. The three-body case is examined in detail, and in this case a variational estimate of the ground-state energy is given. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 637-649 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We describe a finite analog of the Poisson algebra of Wilson loops in Yang–Mills theory. It is shown that this algebra arises in an apparently completely different context: as a Lie algebra of vector fields on a noncommutative space. This suggests that noncommutative geometry plays a fundamental role in the manifestly gauge invariant formulation of Yang–Mills theory. We also construct the deformation of the algebra of loops induced by quantization, in the large-Nc limit. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 3845-3865 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A model of quantum Yang–Mills theory with a finite number of gauge invariant degrees of freedom is studied. The gauge field has only a finite number of degrees of freedom since it is assumed that space–time is a two-dimensional cylinder. The gauge field is coupled to matter, modeled by either one or two nonrelativistic point particles. These problems can be solved without any gauge fixing, by generalizing the canonical quantization methods of S. G. Rajeev [Phys. Lett. B 212, 203 (1988)] to the case including matter. For this, the geometry of the space of connections is used, which has the structure of a principal fiber bundle with an infinite-dimensional fiber. Both problems are reduced to finite-dimensional, exactly solvable, quantum mechanics problems. In the case of one particle, it is found that the ground state energy will diverge in the limit of infinite radius of space, consistent with confinement. In the case of two particles, this does not happen if they can form a color singlet bound state ("meson'').
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 2171-2180 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We obtain direct, finite, formulations of a renormalized quantum mechanical system with no reference to ultraviolet cutoffs and running coupling constants, in both the Hamiltonian and path integral pictures. The path integral description requires a modification to the Wiener measure on continuous paths that describes an unusual diffusion process wherein colliding particles occasionally stick together for a random interval of time before going their separate ways. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Communications in mathematical physics 116 (1988), S. 365-400 
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We extend the methods of Pressley and Segal for constructing cocycle representations of the restricted general linear group in infinite-dimensions to the case of a larger linear group modeled by Schatten classes of rank 1≦p〈∞. An essential ingredient is the generalization of the determinant line bundle over an infinite-dimensional Grassmannian to the case of an arbitrary Schatten rank,p≧1. The results are used to obtain highest weight representations of current algebras (with the operator Schwinger terms) ind+1-dimensions when the space dimensiond is any odd number.
    Type of Medium: Electronic Resource
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Communications in mathematical physics 135 (1991), S. 401-411 
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We point out that the coset space DiffS 1/S 1 is a dense complex submanifold of the Universal Teichmüller SpaceS of compact Riemann spaces of genus g≧1. A holomorphic map ofS into the inifinite dimensional Segal diskD 1 is constructed. This is the Universal analogue of the map of Teichmüller spaces into the Siegel disk provided by the period matrix. The Kähler potential for the general homogenous metric on DiffS 1/S 1 is computed explicitly using the map intoD 1. Some applications to string theory are discussed.
    Type of Medium: Electronic Resource
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