In this paper we discuss local averages of the energy density for the non-minimally coupled scalar quantum field, extending a previous investigation of the classical field. By an explicit example, we show that such averages are unbounded from below on the class of Hadamard states. This contrasts with the minimally coupled field, which obeys a state-independent lower bound known as a Quantum Energy Inequality (QEI). Nonetheless, we derive a generalised QEI for the non-minimally coupled scalar field, in which the lower bound is permitted to be statedependent. This result applies to general globally hyperbolic curved spacetimes for coupling constants in the range 0 < ξ ≤ 1/4. We analyse the state-dependence of our QEI in fourdimensional Minkowski space and show that it is a non-trivial restriction on the averaged energy density in the sense that the lower bound is of lower order, in energetic terms, than the averaged energy density itself.

The stress energy tensor for the classical non-minimally coupled scalar field is known not to satisfy the point-wise energy conditions of general relativity. In this paper we show, however, that local averages of the classical stress energy tensor satisfy certain inequalities. We give bounds for averages along causal geodesics and show, e.g., that in Ricci-flat background spacetimes, ANEC and AWEC are satisfied. Furthermore we use our result to show that in the classical situation we have an analogue to the phenomenon of quantum interest. These results lay the foundations for analogous energy inequalities for the quantised non-minimally coupled fields, which will be discussed elsewhere.

The stress-energy tensor for the non-minimally coupled scalar field is known not to satisfy the pointwise energy conditions, even on the classical level. We show, however, that local averages of the classical stress-energy tensor satisfy certain inequalities and give bounds for averages along causal geodesics. It is shown that in vacuum background spacetimes, ANEC and AWEC are satisfied. Furthermore we use our result to show that in the classical situation we have an analogue to the so called quantum interest conjecture. These results lay the foundations for averaged energy inequalities for the quantised non-minimally coupled fields.