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Journal of Mathematical Physics / Volume 28 / Issue 1

J. Math. Phys. 28, 15 (1987); http://dx.doi.org/10.1063/1.527799 (13 pages)

Approximate representations of SU(2) ordered exponentials in the adiabatic and stochastic limits

M. E. Brachet1 and H. M. Fried2

1Ecole Normale Supérieure, Paris, France
2Physique Théorique, Université de Nice, 06034 Nice Cedex, France

(Received 24 June 1986; accepted 10 September 1986)

Approximate representations for the SU(2) ordered exponential U(t‖E) =(exp[i∫t0dt′ σ⋅E(t′)])+, written as a functional of its input field E(t), are derived in the adiabatic ( ρ≪1) and stochastic ( ρ≫1) limits, where ρ≡‖dmath/dt‖/E, math=E/E, E=+(E2)1/2. An algorithm is set up for the adiabatic case, and fixed‐point equations are obtained for situations of possible convergence. In the stochastic regime, ‘‘averaged’’ functions describing U(t‖E) are derived which reproduce its slowly varying dependence of large magnitude while missing, or approximating, rapid oscillations of small magnitude. Several functional integrals, analytic and machine are carried out over these approximate forms, and their results compared with the same functional integrals over the exact U(t‖E).

KEYWORDS and PACS

Keywords

SU GROUPS, ADIABATIC PROCESSES, STOCHASTIC PROCESSES, DEGREES OF FREEDOM, FUNCTIONS, FUNCTIONALS, CONVERGENCE, OSCILLATIONS, FUNCTIONAL ANALYSIS, GROUP THEORY, DIFFERENTIAL EQUATIONS, ANALYTICAL SOLUTION, ALGORITHMS

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.527799

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

For access to fully linked references, you need to log in.
    H. M. Fried, Phys. Rev. D 16, 1916 (1977).

    H. M. Fried and J. Tessendorf, J. Math. Phys. 25, 1144 (1984JMAPAQ000025000004001144000001).


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