Dokument

Summary:
In this dissertation I work out a supersymmetric formulation of conformal geometric quantum mechanics, which based on ideas I started to develop in my Master’s thesis [77]. In this approach, supersymmetry provides a fundamental connection between conformal geometric quantum mechanics, the spectral geometry of Schrödinger operators and topology. I use these links to give a physics proof of the famous Yang-Yau estimate for the first eigenvalue of the Laplacian on compact Riemann surfaces [80] and to generalize this physics-based proof to Schrödinger operators. Furthermore, I apply the derived eigenvalue estimate to the Coulomb problem and the harmonic oscillator. Moreover, I motivate the application of supersymmetry to spin chain models by describing some properties of the 1D nearest neighbor Ising model in terms of supercharges [74]. By doing so, some important concepts are explained, which are necessary for the further work. Motivated by Ref. [53], I introduce the Witten index [74] for spin chains, which is an object on dual configuration spaces corresponding to Boltzmann weights. I establish a connection between Witten indices and n-point correlation functions. Thus, the spin-spin interactions can be interpreted by considering the Witten index of spin chains. Finally, by transferring the results to subspaces I obtain a rigorous expression of the vacuum expectation value for the density matrix of an arbitrary spin chain model in terms of correlation functions. Moreover, the special case of supersymmetric theories is analyzed and it is shown that no phase transitions can occur in spin chain models with supersymmetry. Furthermore, it is shown that my results are invariant under unitary transformations. There exist numerous approaches to the Riemann zeta function and the Riemann hypothesis using different concepts from physics, see, e.g., Ref. [69]. A promising and well-known approach is the primon gas, also called Riemann gas, which is a toy model combining concepts of number theory, quantum field theory and statistical physics, introduced by Julia [49] and Spector [70] . More precisely, the primon gas describes a canonical ensemble with the Riemann zeta function �(�) as partition function, where � = T−1 is the inverse temperature. Since the Riemann zeta function has a singularity at � = 1, see, e.g., Ref. [6], the primon gas reaches its Hagedorn temperature [36–40] at this point, see Refs. [49, 70]. The behavior of the primon gas beyond the Hagdorn temperature is still not clear, but there are investigations concerning this point [23, 50]. Generally, it is well-known in condensed matter physics that hadronic matter becomes unstable at the Hagedorn temperature [36–40]. A similar situation exists in string theory [7] and there are observations in this direction in the context of number-theoretical gases [50]. Here, I use Spector’s theory of the supersymmetric primon gas [70, 71] to analyze the behavior of a canonical ensemble, which is closely related to the primon gas. By doing so, I interpret the transition at the Hagedorn temperature as a coupling of the fermions of the supersymmetric primon gas and the fermions of an ensemble of harmonic oscillator states to boson-like pairs comparable with the formation of Cooper pairs in the BCS theory [9, 10, 21]. Based on this, I work out a novel link to the Riemann hypothesis.

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