1804 Russian Physics Journal, Vol. 59, No. 11, March, 2017 (Russian Original No. 11, November, 2016) CURVED SPACE QUANTUM FIELD THEORY OF THE 1970S ELUCIDATES BOUNDARY CASIMIR ENERGY TODAY S. A. Fulling UDC 539 Results of investigations of the divergent vacuum energy at reflecting boundaries in quantum field theory are summarized. The boundary is modeled by a soft rapidly increasing potential barrier such as a power wall. In the model without pressure anomaly and the principle of virtual work is fulfilled. Keywords: quantum theory of a field with a boundary. This report summarizes an ongoing program [1–6] at Texas A&M University and the University of Oklahoma to understand better the divergent vacuum energy at reflecting boundaries [7–11] in quantum field theory. The main points are these: 1. Understanding local energy density and pressure is essential for general relativity. It also clarifies the physics of global energy and force calculations [12, 13]. Therefore, we devote primary attention to calculating the expectation values of the stress-energy-momentum tensor T    . We treat the model of a scalar field. 2. For hard (Dirichlet) walls, an ultraviolet cutoff yields physically inconsistent results for energy and pressure [1]. It is universally agreed that boundary divergences in the theory without cutoff represent a failure of the idealized model: No real surface is perfectly reflecting to arbitrarily high frequencies. One might have thought that an exponential frequency cutoff would provide a qualitatively acceptable model of a real boundary, but, instead, it violates the principle of virtual work: The pressure on a partition from one side is not equal to the negative of the derivative of the energy on that side with respect to the position of the partition. 3. The exponential ultraviolet cutoff can be generalized to a point-splitting regularization, and one then realizes that the pressure anomaly is, at root, an instance of the dependence of the regularized stress on the direction of point separation, thoroughly investigated by Christensen [14] in the context of an external gravitational field. The ultraviolet cutoff corresponds to time separation and gives the wrong energy. Point splitting in the spatial direction perpendicular to the partition gives the wrong pressure in that direction, by the same factor ( 1 2  ). Separating the points in a neutral direction (parallel to both the partition and the boundary) yields physically plausible results [4]. Logical justification for this procedure is lacking, and it can hardly be considered a permanent solution. 4. Therefore, we have sought [2] to model a boundary by a soft but rapidly increasing potential barrier, such as the power wall, = () V z z   (  – Heaviside function). As   , this scalar potential (or space-dependent Klein– Gordon mass) increasingly approximates a hard wall at =1 z . 5. The most efficient way to do calculations in this model is to Fourier-transform in the three trivial directions and study the resulting reduced Green’s function in the z direction [3]. The calculations of energy density and pressure hinge on reflection coefficients, ()    , in the Green’s function. (Here  is the Wick-rotated wave number.) Texas A&M University, College Station, USA, e-mail: fulling@math.tamu.edu. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 68–70, November, 2016. Original article submitted July 4, 2016. 1064-8887/17/5911-1804 2017 Springer Science+Business Media New York DOI 10.1007/s11182-017-0979-9