Renormalization for a Scalar Field in an External Scalar Potential

The Pauli--Villars regularization procedure confirms and sharpens the conclusions reached previously by covariant point splitting. The divergences in the stress tensor of a quantized scalar field interacting with a static scalar potential are isolated into a three-parameter local, covariant functional of the background potential. These divergences can be naturally absorbed into coupling constants of the potential, regarded as a dynamical object in its own right; here this is demonstrated in detail for two different models of the field-potential coupling. here is a residual dependence on the logarithm of the potential, reminiscent of the renormalization group in fully interacting quantum field theories; these terms are finite but numerically dependent on an arbitrary mass or length parameter, which is purely a matter of convention. This work is one step in a program to elucidate boundary divergences by replacing a sharp boundary by a steeply rising smooth potential.

[1]  R. Wald.  The back reaction effect in particle creation in curved spacetime , 1977 .

[2]  U. Leonhardt,et al.  Casimir stress inside planar materials , 2017, 1703.02211.

[3]  K. Milton.  Hard and soft walls , 2011, 1107.4589.

[4]  D. Deutsch,et al.  Boundary effects in quantum field theory , 1979 .

[5]  Kimball A. Milton.  The Casimir Effect: Physical Manifestations of Zero-Point Energy , 1999 .

[6]  Conservation of the Stress Tensor in Perturbative Interacting Quantum Field Theory in Curved Spacetimes , 2004, gr-qc/0404074.

[7]  M. Krech.  The Casimir Effect , 1994 .

[8]  F. Mazzitelli,et al.  Ultraviolet cutoffs for quantum fields in cosmological spacetimes , 2015, 1504.02993.

[9]  J. Nery,et al.  Boundary divergences in vacuum self-energies and quantum field theory in curved spacetime , 2011, 1110.3554.

[10]  S. Christensen.  Vacuum Expectation Value of the Stress Tensor in an Arbitrary Curved Background: The Covariant Point Separation Method , 1976 .

[11]  S. Fulling,et al.  Vacuum energy density and pressure of a massive scalar field , 2014, 1412.8381.

[12]  S. Fulling,et al.  Investigating the Spectral Geometry of a Soft Wall , 2011, 1106.1162.

[13]  J. Serbenta.  Quantum Field Theory in Curved Spacetime , 2016, Spacetime and Geometry.

[14]  K. Milton,et al.  Quantum electromagnetic stress tensor in an inhomogeneous medium , 2018, Physical Review D.

[15]  Comments on the Stress-Energy Tensor Operator in Curved Spacetime , 2001, gr-qc/0109048.

[16]  J. Wagner,et al.  Vacuum energy density and pressure near a soft wall , 2015, 1512.09121.

[17]  R. Wald.  Trace anomaly of a conformally invariant quantum field in curved spacetime , 1978 .

[18]  U. Leonhardt,et al.  Casimir stress in materials: Hard divergency at soft walls , 2017, 1704.03078.

[19]  Umar Mohideen,et al.  Advances in the Casimir Effect , 2009 .

[20]  K. Milton,et al.  Stress tensor for a scalar field in a spatially varying background potential: Divergences, "renormalization", anomalies, and Casimir forces , 2016, 1602.00916.

[21]  Yvonne Herz,et al.  A First Course In General Relativity , 2016 .