Loop Dynamics for Gauge Theories: A Numerical Algorithm
Oct, 198326 pages
Published in:
- Nucl.Phys.B 239 (1984) 135-160
- Published: 1984
Report number:
- UFPR-0105
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Abstract: (Elsevier)
The loop equations defining the lattice U( N ) gauge theory are recalled and a formal solution is presented (for N = 1 and N → ∞). Wilson's expectation function W (C) for a loop C is expressed as a matrix element of a resolvent in the space of all the loops between C and 0 (no loop). It is shown that such a solution provides a new numerical algorithm to compute physical quantities. This is based on a Ulam-von Neumann type of method for computing inverse matrix elements by introducing importance sampling for the paths in the space of the matrix indices, which in this case are the loops. As a result W (C) is obtained by summing over important paths in the loop space connecting C to 0. A Monte Carlo program is presented, for the N → ∞ case, where a very simple form for the importance sampling is introduced so that the computer time for each step in the construction of the path is minimized. The rates for successful paths (i.e. path C → 0 within a given finite number of steps) are computed for D = 2 and D = 4. Both rates and computer time involved encourage us to attempt a large scale calculation. Here the numerical studies of the convergence and of the fluctuations are presented only for D = 2. Convergence is rather fast, but specially in the weak-coupling region rare and large fluctuations appear thus suggesting that a better tuning for the importance sampling is needed.- GAUGE FIELD THEORY: U(N)
- GAUGE FIELD THEORY: WILSON LOOP
- GAUGE FIELD THEORY: U(1)
- EXPANSION 1/N
- LATTICE FIELD THEORY: TWO-DIMENSIONAL
- LATTICE FIELD THEORY: FOUR-DIMENSIONAL
- FIELD EQUATIONS: WILSON LOOP
- FIELD EQUATIONS: SOLUTION
- LATTICE FIELD THEORY: CRITICAL PHENOMENA
- NUMERICAL CALCULATIONS: MONTE CARLO
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