What is the physical meaning of W [ J ] = ℏ i ln ⁡ Z [ J ]

Question

What is the physical meaning of   W  [  J  ]  =      ℏ    i    ln  ⁡  Z  [  J  ]

vibrerentb · Accepted Answer

Recall that, by definition,  Z  [  J  ]  =      e          −      i      W      [      J      ]        =  ⟨  Ω      |        e          −      i      H      T            |    Ω  ⟩where H is the Hamiltonian of your system, and   Ω is the vacuum state. Therefore,   W  [  J  ] can be interpreted as the energy of the vacuum       E    Ω   in the presence of a source J, where the origin is chosen at E=0 for J=0. In other words, W[J] is how much more energy does the vacuum have when we turn on an external source J.More precisely, and as per   Z  [  J  ]  =      e          −      i      W      [      J      ]      , you can think of W as the Helmholtz free energy of the system.

What is the physical meaning of W[J]=ℏ/i lnZ[J]?

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