Conservation Law and Trace Anomaly for the Stress-Energy Tensor of a Self-interacting Scalar Field

Abstract We consider a self-interacting, massive, real scalar field on a four-dimensional globally hyperbolic spacetime and the associated stress-energy tensor. Using techniques proper of the algebraic approach to perturbative quantum field theory, we study the associated, Wick-ordered, quantum observable. In particular, we generalize a construction, first developed in the free field theory scenario by Moretti in (Commun Math Phys 232: 189–221, 2003. arXiv:gr-qc/0109048 [gr-qc]), aimed at exploiting the existing freedoms in the definition of the classical stress-energy tensor, in order to define a quantum counterpart which is divergence free. We focus on Minkowski spacetime proving that this procedure can be adapted also to cubic or quartic self-interactions, at least up to order $${\mathcal {O}}(\lambda ^3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>λ</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in perturbation theory. We remark that this result can be extended to arbitrary globally hyperbolic spacetimes, although in this case one needs to exploit the existing regularization freedom in the construction of the Wick ordered stress-energy tensor.