Supersymmetry in Conformal Geometric and Number-Theoretical Quantum Mechanics
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
ζ(β) as partition function, where β = T−1 is the inverse temperature. Since the Riemann zeta
function has a singularity at β = 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.
Philipps-Universität Marburg
2018-02-28
Marburg
Universitätsbibliothek Marburg
UB Marburg
http://archiv.ub.uni-marburg.de/diss/z2018/0092
https://doi.org/10.17192/z2018.0092
urn:nbn:de:hebis:04-z2018-00926
Supersymmetry in Conformal Geometric and Number-Theoretical Quantum Mechanics
Yam, Kevin
Universitätsbibliothek Marburg
mailto:auskunft@ub.uni-marburg.de
https://archiv.ub.uni-marburg.de/adm/img/unilogo-dfg.gif
https://www.uni-marburg.de/bis
https://archiv.ub.uni-marburg.de/diss/z2018/0092