<i>Techniques and Applications of Path Integration</i>

Schulman, L. S.; DeWitt‐Morette, Cécile

doi:10.1063/1.2914703

Cited by 1,549 publications

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“…If the system is also isotropic, the two-particle correlation function is equal to its translational and orientational average: (27) Equation (27) defines an operator which averages over both translation and rotation of the system, where Ω j,i defines the orientation of the vector r⃑ j,i from the i to the j CG site. For an isotropic homogeneous system the gradient term in eqs (26) may then be simplified as Under the additional assumption that the pair interaction between CG sites is central such that, (28) eq (26) may be re-expressed to read (29) Projecting this equation onto the vector u⃑ i,j and shifting the integration variable, one obtains the following result: (30) Thus the dot product factor arises naturally in an integral equation theory, just as it did in the MS-CG equations. For a system described by a central pair potential without an external field, the average effect of a third particle on two-particle correlations must lie along the two-particle vector.…”

Section: Yvon-born-green Equationmentioning

confidence: 99%

Multiscale Coarse-Graining and Structural Correlations: Connections to Liquid-State Theory

et al. 2007

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(2005)]. If no approximations are made, the MS-CG method yields a many-body multi-dimensional potential of mean force describing the interactions between CG sites. However, numerical applications of the MS-CG method typically employ a set of pair potentials to describe non-bonded interactions. The analogy between coarse-graining and the inverse problem of liquid state theory clarifies the general significance of three-particle correlations for the development of such CG pair potentials. It is demonstrated that the MS-CG methodology incorporates critical three-body correlation effects and that, for isotropic homogeneous systems evolving under a central pair potential, the MS-CG equations are a discretized representation of the well-known Yvon-Born-Green equation. Numerical calculations validate the theory and illustrate the role of these structural correlations in the MS-CG method.

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Section: Yvon-born-green Equationmentioning

confidence: 99%

Multiscale Coarse-Graining and Structural Correlations: Connections to Liquid-State Theory

et al. 2007

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“…Furthermore, this way of thinking inspired his later work on quantum electrodynamics [Fey49b,Fey49a]. Since then the theory of path integrals has been placed on a firm mathematical footing [GJ81,Sch81,Sim05] …”

Section: Functional Integrals In Quantum Field Theorymentioning

confidence: 99%

Proof Nets as Formal Feynman Diagrams

Blute

Panangaden

2010

New Structures for Physics

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Summary. The introduction of linear logic and its associated proof theory has revolutionized many semantical investigations, for example, the search for fullyabstract models of PCF and the analysis of optimal reduction strategies for lambda calculi. In the present paper we show how proof nets, a graph-theoretic syntax for linear logic proofs, can be interpreted as operators in a simple calculus. This calculus was inspired by Feynman diagrams in quantum field theory and is accordingly called the φ-calculus. The ingredients are formal integrals, formal power series, a derivative-like construct and analogues of the Dirac delta function.Many of the manipulations of proof nets can be understood as manipulations of formulas reminiscent of a beginning calculus course. In particular, the "box" construct behaves like an exponential and the nesting of boxes phenomenon is the analogue of an exponentiated derivative formula. We show that the equations for the multiplicative-exponential fragment of linear logic hold.

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“…This can be seen in the formula of the transmission amplitude w D (E, V 0 ) with dissipation Eq. (18). It can be rewritten as…”

Section: A Traversal Time and Transmission Probabilitymentioning

confidence: 99%