Ginsparg-Wilson Relation and Admissibility Condition in Noncommutative Geometry

Ginsparg-Wilson relation and admissibility condition have the key role to construct lattice chiral gauge theories. They are also useful to define the chiral structure in finite noncommutative geometries or matrix models. We discuss their usefulness briefly.Comment: Latex 4 pages, uses ptptex.cls. Talk given at Nishinomiya-Yukawa Memorial Symposium on Theoretical Physics ``Noncommutative Geometry and Quantum Spacetime in Physics", Japan, Nov.11-15, 2006. (To be published in the Proceedings

Chiral Anomaly and Ginsparg-Wilson Relation on the Noncommutative Torus

We evaluate chiral anomaly on the noncommutative torus with the overlap Dirac operator satisfying the Ginsparg-Wilson relation in arbitrary even dimensions. Utilizing a topological argument we show that the chiral anomaly is combined into a form of the Chern character with star products.Comment: 19 pages, uses ptptex.cls and ptp-prep.clo, references added, typo corrected, the final version to appear in Prog.Theor.Phy

Formulation of Complex Action Theory

We formulate a complex action theory which includes operators of coordinate and momentum $\hat{q}$ and $\hat{p}$ being replaced with non-hermitian operators $\hat{q}_{new}$ and $\hat{p}_{new}$, and their eigenstates ${}_m <_{new} q |$ and ${}_m <_{new} p |$ with complex eigenvalues $q$ and $p$. Introducing a philosophy of keeping the analyticity in path integration variables, we define a modified set of complex conjugate, real and imaginary parts, hermitian conjugates and bras, and explicitly construct $\hat{q}_{new}$, $\hat{p}_{new}$, ${}_m <_{new} q |$ and ${}_m <_{new} p |$ by formally squeezing coherent states. We also pose a theorem on the relation between functions on the phase space and the corresponding operators. Only in our formalism can we describe a complex action theory or a real action theory with complex saddle points in the tunneling effect etc. in terms of bras and kets in the functional integral. Furthermore, in a system with a non-hermitian diagonalizable bounded Hamiltonian, we show that the mechanism to obtain a hermitian Hamiltonian after a long time development proposed in our letter works also in the complex coordinate formalism. If the hermitian Hamiltonian is given in a local form, a conserved probability current density can be constructed with two kinds of wave functions.Comment: 29 pages, 2 figures, references added, presentation improved, typos corrected. (v5)The definition of $\hat{q}_{new}$ and $\hat{p}_{new}$ are corrected by replacing them with their hermitian conjugates. The errors and typos mentioned in the errata of PTP are corrected. arXiv admin note: substantial text overlap with arXiv:1009.044