The renormalization group in curved space

As the main purpose of renormalization is not to remove divergences but to get essential information about the finite part of effective action, this chapter discusses some of the existing methods of solving this problem; such methods can be denoted the renormalization group. First, the minimal subtraction renormalization group in curved space is formulated. Next, the chapter shows how the overall μ‎-independence of the effective action enables one to interpret μ‎-dependence in some situations. As an example, the effective potential is restored from the renormalization group and compared with the expression calculated directly in chapter 13. In addition, the global conformal (scaling) anomaly is derived from the renormalization group.

Scaling Anomaly in the Mechanical Response in Microscale Reverse Extrusion of Copper

The continuing trend of device miniaturization brings increasing demand for small metal parts and, consequently, significant interest in microscale metal forming technologies. In this work, the influence of grain size on mechanical response in microscale axisymmetric reverse extrusion of Cu 110 alloy was investigated in detail. A characteristic plastic strain associated with material deformation in the extrusion process was, for the first time to the best of our knowledge, defined, measured, and used to evaluate the material's bulk flow stress at this corresponding strain. This flow stress was then used to scale measured mechanical response in reverse extrusion and help identify deviations from scaling behavior expected in continuum plasticity. A scaling anomaly was indeed observed, indicating a dependence of mechanical response on both the initial grain size and the characteristic dimension of microforming operations. Detailed microstructural examination of grains in extruded Cu parts was conducted, and points to directions for future study to better understand mechanisms behind the observed scaling anomaly.

DIRAC SPINORS IN SOLENOIDAL FIELD AND SELF-ADJOINT EXTENSIONS OF ITS HAMILTONIAN

We discuss Dirac equation and its solution in the presence of solenoid (infinitely long) field in (3+1) dimensions. Starting with a very restricted domain for the Hamiltonian, we show that a one-parameter family of self-adjoint extensions are necessary to make sure the correct evolution of the Dirac spinors. Within the extended domain bound state (BS) and scattering state (SS) solutions are obtained. We argue that the existence of bound state in such system is basically due to the breaking of classical scaling symmetry by the quantization procedure. A remarkable effect of the scaling anomaly is that it puts an open bound on both sides of the spectrum, i.e. E ∈ (-M, M) for ν2[0, 1)! We also study the issue of relationship between scattering state and bound state in the region ν2 ∈ [0, 1) and recovered the BS solution and eigenvalue from the SS solution.