Quantum field theory of partitions

Bender, Carl M.; Brody, Dorje C.; Meister, Bernhard K.

doi:10.1063/1.532883

Cited by 25 publications

(60 citation statements)

References 4 publications

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“…(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16), as well as with eqn. (4-9) [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] which can be solved recursively for the w n . One obtains These polynomials play a major role in combinatorics of necklaces, and in the study of the Burnside ring [22,57].…”

Section: [[T]]mentioning

confidence: 99%

“…(t − n + 1) , the falling factorial. It seems to be known since the seventies that the normal ordering process of creation and annihilation operators provides another way to obtain these numbers, for example see [58], the results of Bender et al [10], and also [35]. Identifying t =: a † a : one has…”

Section: Normal Ordering and Rota-baxter Operatorsmentioning

confidence: 99%

“…W R is called Witt ring. In fact there are universal polynomials F i , G j with integer coefficients such that (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) and the variables in the F i , G i run in the set of variables w d , v l where d, l are divisors of i . The functor which relates these rings is called the Witt functor.…”

Section: [[T]]mentioning

confidence: 99%

“…Note that the Möbius series is not a homomorphism since µ(2)µ(2) = −1 · −1 = 1 but µ(2 · 2) = µ(4) = 0 . Let ∂ 2 be the coboundary operator as introduced in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20), see [59], and employed in [12,29]. It maps 1-cochains into 2-coboundaries.…”

Section: Proofmentioning

confidence: 99%

“…Finally it should be noted that in eqn. (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) no explicit normalization factors are included.…”

Section: Algebraic Setupmentioning

confidence: 99%

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The Dirichlet Hopf Algebra of Arithmetics

Fauser

Jarvis

2007

J. Knot Theory Ramifications

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Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the "Dirichlet Hopf algebra of arithmetics" by dualizing the addition and multiplication maps. Then we study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, but form only a weaker Hopf gebra obeying a weakened homomorphism axiom. A careful identification of the algebraic structures involved is done featuring subtraction, division and derivations derived from coproducts and chochains using branching operators. The consequences of the weakened structure of a Hopf gebra on cohomology are explored, showing this has major impact on number theory. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an 'unrenormalized' coproduct and an 'unrenormalized' pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf gebra (literally: antipodal convolution). This can be modelled alternatively by employing Rota-Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures. The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.

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Section: [[T]]mentioning

confidence: 99%

Section: Normal Ordering and Rota-baxter Operatorsmentioning

confidence: 99%

Section: [[T]]mentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

“…Finally it should be noted that in eqn. (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18) no explicit normalization factors are included.…”

Section: Algebraic Setupmentioning

confidence: 99%

See 3 more Smart Citations

The Dirichlet Hopf Algebra of Arithmetics

Fauser

Jarvis

2007

J. Knot Theory Ramifications

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Combinatorial Physics, Normal Order and Model Feynman Graphs

Solomon

Błasiak

Duchamp

et al.

Symmetries in Science XI

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The general normal ordering problem for boson strings is a combinatorial problem. In this talk we restrict ourselves to single-mode boson monomials. This problem leads to elegant generalisations of well-known combinatorial numbers, such as Bell and Stirling numbers. We explicitly give the generating functions for some classes of these numbers. Finally we show that a graphical representation of these combinatorial numbers leads to sets of model field theories, for which the graphs may be interpreted as Feynman diagrams corresponding to the bosons of the theory. The generating functions are the generators of the classes of Feynman diagrams.

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