For a classical orbit of a system in oscillating motion, this quantity is equal to the area enclosed by said orbit in phase space.
In quantum field theory, the partition function is equal to an integral of e to the power of quantity (an integral plus this quantity). In QED, the invariance of this quantity, along with the electric and magnetic fields, under gauge transformation is an easy way to derive charge conservation. In a scalar field theory, considering an extremum of this quantity leads to an equation identical to the Klein-Gordon equation. One version of this quantity is equal to a constant times the integral of the Ricci curvature tensor times the square root of the negative determinant of the metric tensor. Another version of this quantity is defined in terms of derivatives with respect to both the spacelike and timelike worldsheet (*) coordinates. For a classical orbit of a system in oscillating motion, this quantity is equal to the area enclosed by said orbit in phase space. Besides the Einstein-Hilbert, Nambu-Goto, and Polyakov versions of this quantity, this quantity times i over h-bar, all exponentiated, gives the probability amplitude of a given path in the Feynman formulation. Hamilton's principal function is equal to this quantity, which is stationary in the path followed by a physical system. For 10 points, name this quantity equal to the integral of the Lagrangian.
In quantum field theory, the partition function is equal to an integral of e to the power of quantity (an integral plus this quantity).
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