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After establishing the positivity constraint and spin content of the theory for gravitons interacting with a necessarily, and  a priori ,  non ‐conserved external energy‐momentum tensor, the expectation value formalism of the theory is developed at  finite  temperature in the functional  differential  treatment of quantum field theory. The necessity of having,  a priori , a non‐conserved external energy‐momentum tensor is an obvious technical requirement so that its respective ten components may be varied  independently  in order to generate expectation values and non‐linearities in the theory. The covariance of the  induced  Riemann curvature tensor, in the initial vacuum, is established even for the quantization in a gauge corresponding only to two physical states of the gravitons as established above. As an application, the  induced  correction to the metric and the underlying geometry is investigated due to a closed string arising from the Nambu action as a solution of a circularly oscillating string as, perhaps, the simplest generalization of a limiting point‐like object. Finally it is discussed on why the geometry of spacetime may, in general, depend on temperature due to radiative corrections and its physical significance is emphasized.

Gravitons, induced geometry and expectation value formalism at finite temperature