Supersymmetry and the Möbius inversion function

Spector, Donald

doi:10.1007/bf02096755

Cited by 58 publications

(87 citation statements)

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“…Spector [31] has shown why the Mobius inversion function of number theory can be interpreted as the operator (−1) F ( Witten index where F is the fermion number ) in QFT. Physical interpretations of various properties of the Mobius function are provided due to the central role played by Supersymmetry and the Witten index.…”

Section: The Operator That Yields the Riemann Zerosmentioning

confidence: 99%

On Strategies Towards the Riemann Hypothesis: Fractal Supersymmetric Qm and a Trace Formula

Castro

2007

Int. J. Geom. Methods Mod. Phys.

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The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn = 1/2+iλn. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert-Polya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu-Sprung potential ( that capture the average level density of zeros ) by recurring to a weighted superposition of Weierstrass functions p W (x, p, D) and where the summation has to be performed over all primes p in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schroedinger equation whose fluctuating potential (relative to the smooth Wu-Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss-Jacobi theta series. An explicit trace formula related to a superposition of eigenfunctions of these scaling-like operators is defined. If the trace relation is satisfied this could be another test of the Riemann Hypothesis

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Section: The Operator That Yields the Riemann Zerosmentioning

confidence: 99%

On Strategies Towards the Riemann Hypothesis: Fractal Supersymmetric Qm and a Trace Formula

Castro

2007

Int. J. Geom. Methods Mod. Phys.

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“…First, I will state briefly what is meant by partial supersymmetry. Then I will use partial supersymmetry to study a theory with a logarithmic spectrum, a spectrum which is of interest for its role connecting quantum mechanics both to number theory [5] and to string theory [6]. Partial supersymmetry will reveal the underlying structure of this theory.…”

Section: Introductionmentioning

confidence: 99%

Applications of partial supersymmetry

Spector

2004

J. Phys. A: Math. Gen.

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I examine quantum mechanical Hamiltonians with partial supersymmetry, and explore two main applications. First, I analyze a theory with a logarithmic spectrum, and show how to use partial supersymmetry to reveal the underlying structure of this theory. This method reveals an intriguing equivalence between two formulations of this theory, one of which is one-dimensional, and the other of which is infinite-dimensional. Second, I demonstrate the use of partial supersymmetry as a tool to obtain the asymptotic energy levels in non-relativistic quantum mechanics in an exceptionally easy way. In the end, I discuss possible extensions of this work, including the possible connections between partial supersymmetry and renormalization group arguments.

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“…Supersymmetry provides some of the richest insights into the connections between physics and mathematics, with the Witten index [5] serving as one of the central tools in forging such connections. Perhaps what is most striking is the range of the applications of supersymmetry to mathematics; supersymmetry has been used to prove the Atiyah-Singer index theorem [2], to compute the topological invariants of manifolds [5] [6], and to derive a variety of results in arithmetic number theory [3]. The central role of the Witten index in these and in many other physical and mathematical applications stems from the invariance of the index under deformations of the parameters of a theory.…”

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confidence: 99%

Supersymmetry, vacuum statistics, and the fundamental theorem of algebra

Spector

1996

Commun.Math. Phys.

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I give an interpretation of the fundamental theorem of algebra based on supersymmetry and the Witten index. The argument gives a physical explanation of why a real polynomial of degree n need not have n real zeroes, while a complex polynomial of degree n must have n complex zeroes. This paper also addresses in a general and model-independent way the statistics of the perturbative ground states (the states which correspond to classical vacua) in supersymmetric theories with complex and with real superfields. 8/94; rev. 6/95 † spector@hws.edu

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Supersymmetry and the Möbius inversion function

Cited by 58 publications

References 20 publications

On Strategies Towards the Riemann Hypothesis: Fractal Supersymmetric Qm and a Trace Formula

On Strategies Towards the Riemann Hypothesis: Fractal Supersymmetric Qm and a Trace Formula

Applications of partial supersymmetry

Supersymmetry, vacuum statistics, and the fundamental theorem of algebra

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