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Gravitons, induced geometry and expectation value formalism at finite temperature

After establishing the positivity constraint and spin content of the theory for gravitons interacting with a necessarily, and \textit{a priori}, \textit{non}-conserved external energy-momentum tensor, the expectation value formalism of the theory is developed at \textit{finite} temperature in the functional \textit{differential} treatment of quantum field theory. The necessity of having, \textit{a priori}, a non-conserved external energy-momentum tensor is an obvious technical requirement so that its respective ten components may be varied \textit{independently} in order to generate expectation values and non-linearities in the theory. The covariance of the \textit{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 \textit{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.