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I am reading the following notes https://munsal.files.wordpress.com/2014/10/marino-lectures2014.pdf. On section 4.3 the euclidean Yang-Mills theory is considered. It is said that renormalizability and gauge invariance allow the following Lagrangian form
where we have the usual YM lagrangian and the theta term. In equation (4.54) he defines what he calls the partition function in the presence of the $\theta$ angle as
he then says that he will write this expression above as an exponential
where $V$ is the volume of the space. I need more motivation dor this step. Why can we write this as an exponential? He then says that the ground state energy density is given by
it is completely far from obvious why this should be the ground state energy. What is behind this?
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