In quantum field theory, partition functions are generating functionals for correlation functions, making them key objects of study in the path integral formalism. They are the imaginary time versions of statistical mechanics partition functions, giving rise to a close connection between these two areas of physics. Partition functions can rarely be solved for exactly, although free theories do admit such solutions. Instead, a perturbative approach is usually implemented, this being equivalent to summing over Feynman diagrams.
Generating functional
Scalar theories In a d {\displaystyle d} -dimensional field theory with a real scalar field ϕ {\displaystyle \phi } and action S [ ϕ ] {\displaystyle S[\phi ]} , the partition function is defined in the path integral formalism as the functional
Z [ J ] = ∫ D ϕ e i S [ ϕ ] + i ∫ d d x J ( x ) ϕ ( x ) {\displaystyle Z[J]=\int {\mathcal {D}}\phi \ e^{iS[\phi ]+i\int d^{d}xJ(x)\phi (x)}}
where J ( x ) {\displaystyle J(x)} is a fictitious source current. It acts as a generating functional for arbitrary n-point correlation functions
G n ( x 1 , … , x n ) = ( − i ) n 1 Z [ 0 ] δ n Z [ J ] δ J ( x 1 ) ⋯ δ J ( x n ) | J = 0 . {\displaystyle G_{n}(x_{1},\dots ,x_{n})=(-i)^{n}{\frac {1}{Z[0]}}{\frac {\delta ^{n}Z[J]}{\delta J(x_{1})\cdots \delta J(x_{n})}}{\bigg |}_{J=0}.}
The derivatives used here are functional derivatives rather than regular derivatives since they are acting on functionals rather than regular functions. From this it follows that an equivalent expression for the partition function reminiscent to a power series in source currents is given by
Z [ J ] = ∑ n ≥ 0 1 n ! ∫ ∏ i = 1 n d d x i G ( x 1 , … , x n ) J ( x 1 ) ⋯ J ( x n ) . {\displaystyle Z[J]=\sum _{n\geq 0}{\frac {1}{n!}}\int \prod _{i=1}^{n}d^{d}x_{i}G(x_{1},\dots ,x_{n})J(x_{1})\cdots J(x_{n}).}
In curved spacetimes there is an added subtlety that must be dealt with due to the fact that the initial vacuum state need not be the same as the final vacuum state. Partition functions can also be constructed for composite operators in the same way as they are for fundamental fields. Correlation functions of these operators can then be calculated as functional derivatives of these functionals. For example, the partition function for a composite operator O ( x ) {\displaystyle {\mathcal {O}}(x)} is given by
Z O [ J ] = ∫ D ϕ e i S [ ϕ ] + i ∫ d d x J ( x ) O ( x ) . {\displaystyle Z_{\mathcal {O}}[J]=\int {\mathcal {D}}\phi e^{iS[\phi ]+i\int d^{d}xJ(x){\mathcal {O}}(x)}.}
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