Feynman integration over octonions with application to quantum mechanics

Lüdkovsky, S. V.

doi:10.1002/mma.1243

Cited by 7 publications

(6 citation statements)

References 12 publications

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“…since Tr(AA * ) = ∑ l < AA * e l , e l >= ∑ l,k < A * e l , e k >< e k , A * e l >. This implies Formula (31). From the Cauchy-Bunyakovskii-Schwarz inequality, Remark 4, Formulas (31) and (34), one obtains Inequality (32).…”

Section: Definitionmentioning

confidence: 78%

Noncommutative Integration of Generalized Diffusion PDE

Lüdkovsky

2022

Symmetry

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The article is devoted to the noncommutative integration of a diffusion partial differential equation (PDE). Its generalizations are also studied. This is motivated by the fact that many existing approaches for solutions of PDEs are based on evolutionary operators obtained as solutions of the corresponding stochastic PDEs. However, this is restricted to PDEs of an order not higher than 2 over the real or complex field. This article is aimed at extending such approaches to PDEs of an order higher than 2. For this purpose, measures and random functions having values in modules over complexified Cayley–Dickson algebras are investigated. Noncommutative integrals of hypercomplex random functions are investigated. By using them, the noncommutative integration of the generalized diffusion PDE is scrutinized. Possibilities are indicated for a subsequent solution of higher-order PDEs using their decompositions and noncommutative integration.

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Section: Definitionmentioning

confidence: 78%

Noncommutative Integration of Generalized Diffusion PDE

Lüdkovsky

2022

Symmetry

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“…Then use Formulas 33(1−16) to find fundamental solutions E Υ , E Υ 1 and E A or iterate this procedure (see also §35). A generalization of Feynman's formula over the Cayley-Dickson algebras for the second order partial differential operators with the first order addendum Q with variable coefficients from [20] also can be used.…”

Section: Terms Of the Formmentioning

confidence: 99%

“…)...)c c/2 n instead of exp(πc(n−k + )i/2) as above and putting |D| = 1. Thus(20) Ψ(z 1 , ..., z n ) = −Γ((n/2)−1)(P * (z 1 , ..., z n )−cl0) 1−(n/2) [(...(c )...)c c/2 n ] * /(4π n/2 ) for 3 ≤ n, while (21) Ψ(z 1 , z 2 ) = 4 −1 [c * Ln(P * (z 1 , z 2 ) − cl0) for n = 2, since c * j = c −1 j for |c j | = 1, y j q j = y j (c c/q 2 )...)dc c/2 n q n ] = dq 1 ...dq n [(...(c )...)c c/2 n | = 1. 36.…”

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confidence: 99%

Multidimensional Laplace transforms over Cayley-Dickson algebras and partial differential equations

Ludkovsky

2010

Preprint

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Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.

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“…Octonion has been widely studied by Baez [11]. Recently, Many experts and scholars are dedicated to octonionic analysis and obtain some results such as Cauchy integral formula for regular function, Hardy space, Bergman space [12][13][14][15]. H. Y. Wang and his collaborators studied the right inverse of Dirac in octonion space and generalized octonionic analysis to octonionic analysis of several variables [16][17].…”

Section: Introductionmentioning

confidence: 99%