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Register a new userIntroduction to localization in quantum field theory
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This is the introductory chapter to the volume. We review the main idea of the localization technique and its brief history both in geometry and in QFT. We discuss localization in diverse dimensions and give an overview of the major applications of the localization calculations for supersymmetric theories. We explain the focus of the present volume.
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The expectation values of energy density and pressure of a quantum field inside a wedge-shaped region appear to violate the expected relationship between torque and total energy as a function of angle. In particular, this is true of the well-known Deutsch--Candelas stress tensor for the electromagnetic field, whose definition requires no regularization except possibly at the vertex. Unlike a similar anomaly in the pressure exerted by a reflecting boundary against a perpendicular wall, this problem cannot be dismissed as an artifact of an ad hoc regularization.
One of the most important mathematical tools necessary for Quantum Field Theory calculations is the field propagator. Applications are always done in terms of plane waves and although this has furnished many magnificent results, one may still be allowed to wonder what is the form of the most general propagator that can be written. In the present paper, by exploiting what is called polar form, we find the most general propagator in the case of spinors, whether regular or singular, and we give a general discussion in the case of vectors.
The study of the heat-trace expansion in noncommutative field theory has shown the existence of Moyal nonlocal Seeley-DeWitt coefficients which are related to the UV/IR mixing and manifest, in some cases, the non-renormalizability of the theory. We show that these models can be studied in a worldline approach implemented in phase space and arrive to a master formula for the $n$-point contribution to the heat-trace expansion. This formulation could be useful in understanding some open problems in this area, as the heat-trace expansion for the noncommutative torus or the introduction of renormalizing terms in the action, as well as for generalizations to other nonlocal operators.
After a brief introduction to Heun type functions we note that the actual solutions of the eigenvalue equation emerging in the calculation of the one loop contribution to QCD from the Belavin-Polyakov-Schwarz-Tyupkin instanton and the similar calculation for a Dirac particle coupled to a scalar $CP^1$ model in two dimensions can be given in terms of confluent Heun equation in their original forms. These equations were previously modified to be solved by more elementary functions. We also show that polynomial solutions with discrete eigenvalues are impossible to find in the unmodified equations.
In a previous paper it was shown how to calculate the ground-state energy density $E$ and the $p$-point Greens functions $G_p(x_1,x_2,...,x_p)$ for the $PT$-symmetric quantum field theory defined by the Hamiltonian density $H=frac{1}{2}( ablaphi)^2+frac{1}{2}phi^2(iphi)^varepsilon$ in $D$-dimensional Euclidean spacetime, where $phi$ is a pseudoscalar field. In this earlier paper $E$ and $G_p(x_1,x_2,...,x_p)$ were expressed as perturbation series in powers of $varepsilon$ and were calculated to first order in $varepsilon$. (The parameter $varepsilon$ is a measure of the nonlinearity of the interaction rather than a coupling constant.) This paper extends these perturbative calculations to the Euclidean Lagrangian $L= frac{1}{2}( ablaphi)^2+frac{1}{2}mu^2phi^2+frac{1}{2} gmu_0^2phi^2big(imu_0^{1-D/2}phibig)^varepsilon-ivphi$, which now includes renormalization counterterms that are linear and quadratic in the field $phi$. The parameter $g$ is a dimensionless coupling strength and $mu_0$ is a scaling factor having dimensions of mass. Expressions are given for the one-, two, and three-point Greens functions, and the renormalized mass, to higher-order in powers of $varepsilon$ in $D$ dimensions ($0leq Dleq2$). Renormalization is performed perturbatively to second order in $varepsilon$ and the structure of the Greens functions is analyzed in the limit $Dto 2$. A sum of the most divergent terms is performed to {it all} orders in $varepsilon$. Like the Cheng-Wu summation of leading logarithms in electrodynamics, it is found here that leading logarithmic divergences combine to become mildly algebraic in form. Future work that must be done to complete the perturbative renormalization procedure is discussed.
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