Academia.eduAcademia.edu

Outline

Title
Abstract
Introduction
References
All Topics
Mathematics
Algebra

Prepotential formulation ofSU(3) lattice gauge theory

Profile image of manu mathurmanu mathur

2009, Journal of Physics A: Mathematical and Theoretical

https://doi.org/10.1088/1751-8113/43/3/035403
visibility

…

description

25 pages

link

1 file

Abstract

The SU(3) lattice gauge theory is reformulated in terms of SU(3) prepotential harmonic oscillators. This reformulation has enlarged SU (3) ⊗ U (1) ⊗ U (1) gauge invariance under which the prepotential operators transform like matter fields. The Hilbert space of SU(3) lattice gauge theory is shown to be equivalent to the Hilbert space of the prepotential formulation satisfying certain color invariant Sp(2,R) constraints. The SU(3) irreducible prepotential operators which solve these Sp(2,R) constraints are used to construct SU(3) gauge invariant Hilbert spaces at every lattice site in terms of SU(3) gauge invariant vertex operators. The electric fields and the link operators are reconstructed in terms of these SU(3) irreducible prepotential operators. We show that all the SU(3) Mandelstam constraints become local and take very simple form within this approach. We also discuss the construction of all possible linearly independent SU(3) loop states which solve the Mandelstam constraints. The techniques can be easily generalized to SU(N).

Related papers

Toward a Many-Body Treatment of Hamiltonian Lattice SU(N) Gauge Theory
Norbert Ligterink, Raymond Bishop

Annals of Physics, 2000

We develop a consistent approach to Hamiltonian lattice gauge theory, using the maximaltree gauge. The various constraints are discussed and implemented. An independent and complete set of variables for the colourless sector is determined. A general scheme to construct the eigenstates of the electric energy operator using a symbolic method is described. It is shown how the one-plaquette problem can be mapped onto an N-fermion problem. Explicit solutions for U(1), SU(2), SU , SU(4), and SU(5) lattice gauge theory are shown.

View PDFchevron_right
Loop approach to lattice gauge theories
manu mathur

Nuclear Physics B, 2007

We solve the Gauss law and the corresponding Mandelstam constraints in the loop Hilbert space H L using the prepotential formulation of (d + 1) dimensional SU(2) lattice gauge theory. The resulting orthonormal and complete loop basis, explicitly constructed in terms of the d(2d − 1) prepotential intertwining operators, is used to transcribe the gauge dynamics directly in H L without any redundant gauge and loop degrees of freedom. Using generalized Wigner-Eckart theorem and Biedenharn-Elliot identity in H L , we show that the above loop dynamics for pure SU(2) lattice gauge theory in arbitrary dimension, is given by real and symmetric 3nj coefficients of the second kind (e.g., n=6, 10 for d=2, 3 respectively). The corresponding "ribbon diagrams" representing SU(2) loop dynamics are constructed. The prepotential techniques are trivially extended to include fundamental matter fields leading to a description in terms of loops and strings. The SU(N) gauge group is briefly discussed.

View PDFchevron_right
Towards a Many-Body Treatment of Hamiltonian Lattice SU(N) Gauge Theory
Niels Walet

2000

We develop a consistent approach to Hamiltonian lattice gauge theory, using the maximaltree gauge. The various constraints are discussed and implemented. An independent and complete set of variables for the colourless sector is determined. A general scheme to construct the eigenstates of the electric energy operator using a symbolic method is described. It is shown how the one-plaquette problem can be mapped onto an N-fermion problem. Explicit solutions for U(1), SU(2), SU , SU(4), and SU(5) lattice gauge theory are shown.

View PDFchevron_right
Analytic variational investigation of euclidean SU (3) gauge theory
N D Hari Dass

Nuclear physics, 1993

Analytic variational techniques for lattice gauge theories based on the Rayleigh-Ritz(RR) method were previously developed for euclidean SU(2) gauge theories in 3 and 4 dimensions. Their extensions to SU(3) gauge theory including applications to correlation functions and mass gaps are presented here. * This work was carried out under the project on Lattice Gauge Theories (SP/S2/K22/87) sponsored by DST

View PDFchevron_right
Coulomb gauge studies of SU(3) Yang-Mills theory on the lattice
Ernst-michael Ilgenfritz

2007

We study the infrared behaviour of lattice SU(3) Yang-Mills theory in Coulomb gauge in terms of the ghost propagator, the Coulomb potential and the transversal and the time-time component of the equal-time gluon propagator. In particular, we focus on the Gribov problem and its impact on the observables. We observe that the simulated annealing method is advantageous for fixing the

View PDFchevron_right
Simulations in SO(3) × Z(2) Lattice Gauge Theory
Richard Haymaker

2000

Lattice QCD continues to maintain an important role in the search for the physics of color confinement. The lattice regulator maintains gauge invariance at all costs. Dynamical variables are group elements rather than elements of a Lie algebra. As a consequence many of the topological features that are prominent candidates for elucidating the physics of confinement have natural lattice definitions. These include U(1), Z(N), SU(N)/Z(N) monopole loops, Dirac sheets, Z(N) and SU(N) vortex sheets etc. These objects are often abundant in U(1) and SU(N) lattice gauge theories. They become singular only as one approaches the continuum limit. Yaffe , Tomboulis 2 and Kovacs and Tomboulis 3 have developed a formulation of SU(N) gauge theory that is manifestly SU(N)/Z(N) invariant. In this formulation, center elements, Z(N), multiplying each link leave the action and measure invariant. New Z(N) variables, defined on plaquettes, σ(p), carry the Z(N) degrees of freedom. This formalism, equivalen...

View PDFchevron_right
Loop states in lattice gauge theory
manu mathur

Physics Letters B, 2006

We solve the Gauss law as well as the corresponding Mandelstam constraints of (d+1) dimensional SU(2) lattice gauge theory in terms of harmonic oscillator prepotentials. This enables us to explicitly construct a complete orthonormal and manifestly gauge invariant basis in the physical Hilbert space. Further, we show that this gauge invariant description represents networks of unoriented loops carrying certain non-negative abelian fluxes created by the harmonic oscillator prepotentials. The loop network is characterized by 3(d − 1) gauge invariant integers at every lattice site which is the number of physical degrees of freedom. Time evolution involves local fluctuations of these loops. The loop Hamiltonian is derived. The generalization to SU(N) gauge group is discussed.

View PDFchevron_right
On the influence of colour magnetic currents on the confining properties of SU(3) lattice gauge theory
Martin Zach

Physics Letters B, 1998

We modify the standard Wilson action of SU (3) lattice gauge theory by adding an extra term which suppresses colour magnetic currents. We present numerical results of simulations at zero and finite temperature and show that colour magnetic currents strongly influence the confining properties of SU (3) lattice gauge theory. * Supported in part by Fonds zur Förderung der wissenschaftlichen Forschung, P11387-PHY

View PDFchevron_right
Discriminating between two reformulations of SU(3) Yang-Mills theory on a lattice
Kei-ichi Kondo

AIP Conference Proceedings, 2016

In oder to investigate quark confinement, we give a new reformulation of the SU(N) Yang-Mills theory on a lattice and present the results of the numerical simulations of the SU(3) Yang-Mills theory on a lattice. The numerical simulations include the derivation of the linear potential for static interquark potential, i.e., non-vanishing string tension, in which the "Abelian" dominance and magnetic monopole dominance are established, confirmation of the dual Meissner effect by measuring the chromoelectric flux tube between quark-antiquark pair, the induced magnetic-monopole current, and the type of dual superconductivity, etc.

View PDFchevron_right
Phase structure of lattice gauge theories for non-abelian subgroups of SU(3)
Harald Grosse

Physics Letters B, 1981

View PDFchevron_right

References (32)

  1. S. Mandelstam, Ann. Phys. (N.Y.) 19 (1962) 1;
  2. S. Mandelstam, Phys. Rev. 175 (1968) 1580;
  3. S. Mandelstam, Phys. Rev. D 19 (1979) 2391.
  4. K. G. Wilson, Phys. Rev. D 10 (1974) 2445.
  5. A. M. Polyakov, Phys. Lett. B 82 (1979) 247; (1983) 199;
  6. A. M. Polyakov, Nucl. Phys. B164 (1979) 171; N. Nambu, Phys. Lett. B 80 (1979) 372; J. L. Gervais, A. Neveu, Phys. Lett. B 80 (1979) 255; Yu. M. Makeenko and A. A. Migdal, Phys. Lett. B88 (1979) 135; Y. M. Makeenko, A. A. Migdal, Nucl. Phys. B 188 (1981) 269; F. Gliozzi, T. Reggeand M. A. Virasaro, Phys. Lett. B81 (1979) 178;
  7. M. Virasaro, Phys. Lett. B 82 (1979) 436; A. Jevicki and B. Sakita, Phys. Rev. D 22 (1974) 467; A. M. Polyakov, Gauge Fields and Strings (Harwood, New York, 1987).
  8. A. A. Migdal, Phys. Rep. 102 199-290 (1983).
  9. J. Kogut, L Susskind, Phys Rev. D 10 (1974) 3468, D 11, 395 (1975).
  10. R. Gambini, Jorge Pullin, Loops, Knots, Gauge Theories and Quantum Gravity (Cambridge University Press, 2000).
  11. As emphasized in [6]: "The proliferation of loops when one considers larger lattices and higher dimensions completely washes out the advantages provided by the (loop) formalism.", Chapter 12, page 303-304;
  12. Gambini R, Leal L, Trias A, Phys. Rev. D 39 (1989) 3127; Bartolo C, Gambini R, Leal L, Phys. Rev. D 39 (1989) 1756.
  13. Manu Mathur, Nucl. Phys. B 779, 32 (2007), Phys. Letts. B 640 (2006) 292-296.
  14. Manu Mathur, J. Phys. A: Math. Gen. 38 (2005) 10015.
  15. H. S. Sharatchandra, Nucl. Phys. B 196 (1982) 62.
  16. Ramesh Anishetty, H. S. Sharatchandra, Phys. Rev. Letts. 65 (1990) 81; B. Gnanapragasam, H. S. Sharatchandra, Phys. Rev. D 45 (1992) R1010;
  17. D. Robson, D. M. Weber, Z. Phys. C15 (1982) 199.
  18. W. Furmanski, A. Kolawa, Nucl. Phys. B 291 (1987) 594.
  19. G. Burgio, R. De. Pietri, H. A. Morales-Tecotl, L. F. Urrutia, J. D. Vergara, Nucl. Phys. B 566 (2000), 547.
  20. Brügmann B, Phys. Rev. D 43 (1991) 566.
  21. Loll R. Nucl. Phys B 368 (1992) 121, Nucl. Phys. B 400 (1993) 126.
  22. Watson N. J., Phys. Letts. B 323 (1994) 385; Nucl. Phys. Proc. Suppl. (1995) 39B, C 224, hep-th/9408174.
  23. C. Rovelli, Quantum Gravity, Cambridge University Press (2004).
  24. C. Rovelli, L. Smolin, Phys. Rev. D 52 (1995) 5743, [arXiv:gr-qc/9505006].
  25. N. D. H. Dass and Manu Mathur, Class. Quant. Grav. 24, 2179 (2007) [arXiv:gr-qc/0611156].
  26. S. Chaturvedi and n. Mukunda, J. Math. Phys. 43 (2002) 5262
  27. Ramesh Anishetty, Manu Mathur and Indrakshi Raychowdhury, J. Math. Phys. 50 (2009) 053503.
  28. P. Jordan, Z. Physik 94 (1935) 531, J. Schwinger U.S Atomic Energy Commission Report NYO- 3071, 1952 or D. Mattis, The Theory of Magnetism (Harper and Row, 1982).
  29. H. Georgi, Lie Algebras in Particle Physics (Benjamin/Cummings, Reading, 1982).
  30. Manu Mathur and Diptiman Sen, J. Math. Phys. 42 (2001) 4181.
  31. Manu Mathur and H. S. Mani, J. Math. Phys. 43 (2002) 5351.
  32. R. Anishetty, H. Gopalkrishna Gadiyar, Manu Mathur and H. S. Sharatchandra, Phys. Lett. B 271, (1991) 391; J. S. Prakash and H. S. Sharatchandra, J. Math. Phys. 37, (1996) 6530.

Related papers

3-dimensionalSU(2) lattice gauge theory in terms of gauge invariant variables
Annette Holtkamp

Zeitschrift für Physik C Particles and Fields, 1982

As a first step towards a duality transformation for the SU(2) lattice gauge theory in 3 dimensions, the integration over all gauge variant variables is performed explicitly after introducing gauge invariant auxiliary variables. The resulting new Hamiltonian is complex and involves a sum over closed loops. Each of these loops is confined to an elementary cube of a dual lattice. Like in a previous investigation for the 0(4) symmetric Heisenberg ferromagnet Riihl's boson representation is used to derive the result.

View PDFchevron_right
The Physical Hilbert Space of SU(2) Lattice Gauge Theory
manu mathur

2004

We solve the Gauss law of SU(2) lattice gauge theory using the harmonic oscillator prepotential formulation. We construct a generating function of a manifestly gauge invariant and orthonormal basis in the physical Hilbert space of (d+1) dimensional SU(2) lattice gauge theory. The resulting orthonormal physical states are given in closed form. The generalization to SU(N) gauge group is discussed.

View PDFchevron_right
Harmonic oscillator pre-potentials inSU(2) lattice gauge theory
manu mathur

Journal of Physics A: Mathematical and General, 2005

We write the SU(2) lattice gauge theory Hamiltonian in (d+1) dimensions in terms of prepotentials which are the SU(2) fundamental doublets of harmonic oscillators. The Hamiltonian in terms of prepotentials has SU(2) ⊗ U(1) local gauge invariance. In the strong coupling limit, the color confinement in this formulation is due to the U(1) gauge group. We further solve the SU(2) ⊗ U(1) Gauss law to characterize the physical Hilbert space in terms of a set of gauge invariant integers. We also obtain certain novel gauge invariant operators in terms of the above oscillators. The corresponding prepotential formulation of SU(N) lattice gauge theory is also simple and discussed.

View PDFchevron_right
Canonical transformations and loop formulation of SU(N) lattice gauge theories
manu mathur

Physical Review D, 2015

We construct canonical transformations to reformulate SU(N) Kogut-Susskind lattice gauge theory in terms of a set of fundamental loop & string flux operators along with their canonically conjugate loop & string electric fields. The canonical relations between the initial SU(N) link operators and the final SU(N) loop & string operators, consistent with SU(N) gauge transformations, are explicitly constructed over the entire lattice. We show that as a consequence of SU(N) Gauss laws all SU(N) string degrees of freedom become cyclic and decouple from the physical Hilbert space H p. The Kogut-Susskind Hamiltonian rewritten in terms of the fundamental physical loop operators has global SU(N) invariance. There are no gauge fields. We further show that the (1/g 2) magnetic field terms on plaquettes create and annihilate the fundamental plaquette loop fluxes while the (g 2) electric field terms describe all their interactions. In the weak coupling (g 2 → 0) continuum limit the SU(N) loop dynamics is described by SU(N) spin Hamiltonian with nearest neighbour interactions. In the simplest SU(2) case, where the canonical transformations map the SU(2) loop Hilbert space into the Hilbert spaces of hydrogen atoms, we analyze the special role of the hydrogen atom dynamical symmetry group SO(4, 2) in the loop dynamics and the spectrum. A simple tensor network ansatz in the SU(2) gauge invariant hydrogen atom loop basis is discussed.

View PDFchevron_right
A novel improved action for SU (3) lattice gauge theory
Kurt Langfeld

SU(3) lattice gauge theory is studied by means of an improved action where a 2 × 2 Wilson loop is supplemented to the standard plaquette term. By contrast to earlier studies using a tree level improvement, the prefactor of the 2 × 2 Wilson term is determined by minimizing the breaking of rotational symmetry detected from the static quark-antiquark potential. On coarse lattices, the novel action is superior to the Iwasaki action and comparable with DBW2 action. The scaling behavior of the novel action is studied by using the static quark potential and the ratio of the deconfinement temperature and the string tension.

View PDFchevron_right

Related topics

  • Mathematics
  • Physics
  • Lattice Theory
  • Lattice gauge theory
  • Gauge theory
  • Gauge invariance

    • Cited by

    Simulating lattice gauge theories within quantum technologies
    Luca Tagliacozzo

    The European Physical Journal D

    Lattice gauge theories, which originated from particle physics in the context of Quantum Chromodynamics (QCD), provide an important intellectual stimulus to further develop quantum information technologies. While one long-term goal is the reliable quantum simulation of currently intractable aspects of QCD itself, lattice gauge theories also play an important role in condensed matter physics and in quantum information science. In this way, lattice gauge theories provide both motivation and a framework for interdisciplinary research towards the development of special purpose digital and analog quantum simulators, and ultimately of scalable universal quantum computers. In this manuscript, recent results and new tools from a quantum science approach to study lattice gauge theories are reviewed. Two new complementary approaches are discussed: first, tensor network methods are presented – a classical simulation approach – applied to the study of lattice gauge theories together with some r...

    View PDFchevron_right
    SU(N) irreducible Schwinger bosons
    manu mathur

    Journal of Mathematical Physics, 2010

    We construct SU(N) irreducible Schwinger bosons satisfying certain U(N-1) constraints which implement the symmetries of SU(N) Young tableaues. As a result all SU(N) irreducible representations are simple monomials of (N − 1) types of SU(N) irreducible Schwinger bosons. Further, we show that these representations are free of multiplicity problems. Thus all SU(N) representations are made as simple as SU(2).

    View PDFchevron_right
    Canonical transformations and loop formulation of SU(N) lattice gauge theories
    manu mathur

    Physical Review D, 2015

    We construct canonical transformations to reformulate SU(N) Kogut-Susskind lattice gauge theory in terms of a set of fundamental loop & string flux operators along with their canonically conjugate loop & string electric fields. The canonical relations between the initial SU(N) link operators and the final SU(N) loop & string operators, consistent with SU(N) gauge transformations, are explicitly constructed over the entire lattice. We show that as a consequence of SU(N) Gauss laws all SU(N) string degrees of freedom become cyclic and decouple from the physical Hilbert space H p. The Kogut-Susskind Hamiltonian rewritten in terms of the fundamental physical loop operators has global SU(N) invariance. There are no gauge fields. We further show that the (1/g 2) magnetic field terms on plaquettes create and annihilate the fundamental plaquette loop fluxes while the (g 2) electric field terms describe all their interactions. In the weak coupling (g 2 → 0) continuum limit the SU(N) loop dynamics is described by SU(N) spin Hamiltonian with nearest neighbour interactions. In the simplest SU(2) case, where the canonical transformations map the SU(2) loop Hilbert space into the Hilbert spaces of hydrogen atoms, we analyze the special role of the hydrogen atom dynamical symmetry group SO(4, 2) in the loop dynamics and the spectrum. A simple tensor network ansatz in the SU(2) gauge invariant hydrogen atom loop basis is discussed.

    View PDFchevron_right
    580 California St., Suite 400
    San Francisco, CA, 94104
    © 2025 Academia. All rights reserved