A quantum field theoretical representation of Euler‐Zagier sums

Müller, Uwe; Schubert, Christian

doi:10.1155/s0161171202106028

Cited by 4 publications

(4 citation statements)

References 35 publications

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“…[16]. Let us note that the Drinfeld's associator is also related to multiple zeta values in a linear way [17].…”

Section: Introductionmentioning

confidence: 99%

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Quantum correlations and number theory

Boos¹,

Korepin²,

Nishiyama³

et al. 2002

J. Phys. A: Math. Gen.

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We study spin-1/2 Heisenberg XXX antiferromagnet. The spectrum of the Hamiltonian was found by Hans Bethe in 1931, [1]. We study the probability of formation of ferromagnetic string in the antiferromagnetic ground state, which we call emptiness formation probability P (n). This is the most fundamental correlation function. We prove that for the short strings it can be expressed in terms of the Riemann zeta function with odd arguments, logarithm ln 2 and rational coefficients. This adds yet another link between statistical mechanics and number theory. We have obtained an analytical formula for P (5) for the first time. We have also calculated P (n) numerically by the Density Matrix Renormalization Group. The results agree quite well with the analytical ones. Furthermore we study asymptotic behavior of P (n) at finite temperature by Quantum Monte-Carlo simulation. It also agrees with our previous analytical results.

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“…[16]. Let us note that the Drinfeld's associator is also related to multiple zeta values in a linear way [17].…”

Section: Introductionmentioning

confidence: 99%

“…These identities can be derived using the reccurent relations (4.1), (4.5),..., (4.7) of ref. [16]. Let us note that the Drinfeld's associator is also related to multiple zeta values in a linear way [17].…”

Section: Introductionmentioning

confidence: 99%

Quantum correlations and number theory

Boos¹,

Korepin²,

Nishiyama³

et al. 2002

J. Phys. A: Math. Gen.

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“…Starting from the four-point case, the use of (39) does not immediately lead to integrals that can all be done just by applying the completeness relation (such integrals are called 'chain integrals'), but it can be shown that it is always possible to achieve a complete reduction to chain integrals using an integration-by-parts algorithm that was initially developed for a somewhat different purpose [12].…”

Section: Bernoulli Numbers and Polynomials In The Worldline Formalismmentioning

confidence: 99%

One-loop amplitudes in the worldline formalism

Edwards¹,

Mata²,

Schubert³

2022

Phys. Scr.

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We summarize recent progress in applying the worldline formalism to the analytic calculation of one-loop N-point amplitudes. This string-inspired approach is well-adapted to avoiding some of the calculational ineﬃciencies of the standard Feynman diagram approach, most notably by providing master formulas that sum over diagrams diﬀering only by the position of external legs and/or internal propagators. We illustrate the mathematical challenge involved with the low-energy limit of the N-photon amplitudes in scalar and spinor QED, and then present an algorithm that, in principle, solves this problem for the much more diﬃcult case of the N-point amplitudes at full momentum in φ3 theory. The method is based on the algebra of inverse derivatives in the Hilbert space of periodic functions orthogonal to the constant ones, in which the Bernoulli numbers and polynomials play a central role.

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“…In [69] two of the authors along these lines constructed a worldline QFT geared towards the derivation of identities between multiple zeta values, defined by…”

mentioning

confidence: 99%

New Techniques for Worldline Integration

Edwards

Mata²,

ller³

et al. 2021

SIGMA

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The worldline formalism provides an alternative to Feynman diagrams in the construction of amplitudes and effective actions that shares some of the superior properties of the organization of amplitudes in string theory. In particular, it allows one to write down integral representations combining the contributions of large classes of Feynman diagrams of different topologies. However, calculating these integrals analytically without splitting them into sectors corresponding to individual diagrams poses a formidable mathematical challenge. We summarize the history and state of the art of this problem, including some natural connections to the theory of Bernoulli numbers and polynomials and multiple zeta values.

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A quantum field theoretical representation of Euler‐Zagier sums

Cited by 4 publications

References 35 publications

Quantum correlations and number theory

Quantum correlations and number theory

One-loop amplitudes in the worldline formalism

New Techniques for Worldline Integration

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