Partition function (quantum field theory)

In quantum field theory, partition functions are the generating functionals for all correlation functions, making them key objects of study in the path integral formalism. They are closely related to the partition function in statistical mechanics, where the latter are imaginary time versions of the former. Partition functions can rarely be solved for exactly, although free theories do admit such solutions. Instead, a perturbative approach to solving for partition functions gives rise to the familiar method of calculating correlation functions using Feynman diagrams.

Generating functional

Scalar theories

In a $d$ -dimensional field theory with a real scalar field $\phi$ and action $S[\phi ]$ , the partition function is defined in the path integral formalism to be^[1]

Z[J]=\int {\mathcal {D}}\phi \ e^{iS[\phi ]+i\int d^{d}xJ(x)\phi (x)}

where $J(x)$ is a fictitious source current. The function acts as a generating functional (mathematics) for arbitrary n-point correlation functions

G_{n}(x_{1},\dots ,x_{n})=(-1)^{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^[2]

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 associated with the fact that the initial vacuum state need not be the same as the final vacuum state.^[3]

Knowing the partition function completely solves the theory since it allows for the direct calculation of all of its correlation functions. However, there are very few cases where the partition function can be calculated exactly. While free theories do admit exact solutions interacting theories generally do not. Instead the partition function can be evaluated at weak coupling which amounts to regular perturbation theory using Feynman diagrams.^[4]

By performing a Wick transformation, the partition function can be expressed in Euclidean spacetime as^[5]

Z[J]=\int {\mathcal {D}}\phi \ e^{-(S_{E}[\phi ]+\int d^{d}x_{E}J\phi )},

where $S_{E}$ is the Euclidean action and $x_{E}$ is the Euclidean coordinate. This form is closely connected to the partition function in statistical mechanics, especially since the Euclidean Lagrangian is generally bounded from below and thus can be interpreted as an energy. It also allows for the interpretation of the exponential factor as a statistical weight for the field configurations, with larger fluctuations with gradients or field values leading to greater suppression. This connection with statistical mechanics also lends additional intuition for how correlation functions should behave in a quantum field theory.

General theories

Most of the same principles of the scalar case hold for more general theories with additional fields. Each field requires introducing its own current as was done in the scalar case, with antiparticle fields requiring their own separate currents. Acting on the partition function with a derivative of a current brings down its associated field from the exponential, allowing for the construction of arbitrary correlation functions. Usually all currents are set to zero after the differentiation has been performed since they are merely fictitious currents that are not present in the theory.

For partition functions with Grassmann valued fermion fields, the sources are also Grassmann valued.^[6] For example, a theory with a single Dirac fermion $\psi (x)$ requires the introduction of two Grassmann currents $\eta$ and ${\bar {\eta }}$ so that the partition function is

Z[{\bar {\eta }},\eta ]=\int {\mathcal {D}}{\bar {\psi }}{\mathcal {D}}\psi \ e^{iS+\int d^{d}x{\bar {\eta }}\psi +{\bar {\psi }}\eta }.

Functional derivatives with respect to ${\bar {\eta }}$ give fermion fields while derivatives with respect to $\eta$ give anti-fermion fields in the correlation functions.

Thermal field theory

A thermal field theory at temperature $T$ is equivalent in Euclidean formalism to a theory with a compactified temporal direction of length $\beta =1/T$ . Partition functions must be modified appropriately by imposing periodicity conditions on the fields and the Euclidean spacetime integrals

Z[\beta ,J]=\int {\mathcal {D}}\phi e^{-S_{E,\beta }[\phi ]+\int _{\beta }d^{4}xJ\phi }{\bigg |}_{\phi ({\boldsymbol {x}},0)=\phi ({\boldsymbol {x}},\beta )}.

This partition function can be taken as a defining feature of thermal field theory in imaginary time formalism.^[7] The correlation functions are acquired from the partition function through the usual functional derivatives with respect to the currents

G_{\beta }(x_{1},\dots ,x_{n})={\frac {\delta ^{n}Z[\beta ,J]}{\delta J(x_{1})\cdots \delta J(x_{n})}}{\bigg |}_{J=0}.

Free theories

The partition function can be solved exactly in free theories by completing the square in terms of the scalar fields. Since a shift by a constant does not affect the path integral measure, this allows for separating the partition function into a constant of proportionality $N$ arising from the path integral, and a second term that only depends on the current. For the scalar theory this yields

Z_{0}[J]=N\exp {\bigg (}-{\frac {1}{2}}\int d^{d}xd^{d}y\ J(x)\Delta _{F}(x-y)J(y){\bigg )},

where $\Delta _{F}(x-y)$ is the position space Feynman propagator

\Delta _{F}(x-y)=\int {\frac {d^{d}p}{(2\pi )^{d}}}{\frac {i}{p^{2}-m^{2}+i\epsilon }}e^{-ip\cdot (x-y)}.

This partition function fully determines the free field theory.

A free Dirac fermion theory can similarly be solved exactly, with a partition function of the form

Z_{0}[{\bar {\eta }},\eta ]=N\exp {\bigg (}\int d^{d}xd^{d}y\ {\bar {\eta }}(y)\Delta _{D}(x-y)\eta (x){\bigg )},

where $\Delta _{D}(x-y)$ is the position space Dirac propagator

\Delta _{D}(x-y)=\int {\frac {d^{d}p}{(2\pi )^{d}}}{\frac {i({p\!\!\!/}+m)}{p^{2}-m^{2}+i\epsilon }}e^{-ip\cdot (x-y)}.

References

^ Rivers, R.J. (1988). "1". Path Integral Methods in Quantum Field Theory. Cambridge: Cambridge University Press. p. 14-16. ISBN 978-0521368704.
^ Năstase, H. (2019). "9". Introduction to Quantum Field Theory. Cambridge University Press. p. 78. ISBN 978-1108493994.
^ Birrell, N.C.; Davis, P.C.W. (1984). "6". Quantum Fields in Curved Spacetime. Cambridge University Press. p. 155-156. ISBN 978-0521278584.
^ Srednicki, M. (2007). "9". Quantum Field Theory. Cambridge: Cambridge University Press. p. 58-60. ISBN 978-0521864497.
^ Peskin, Michael E.; Schroeder, Daniel V. (1995). "9". An Introduction to Quantum Field Theory. Westview Press. p. 289-292. ISBN 9780201503975.
^ Schwartz, M. D. (2014). "34". Quantum Field Theory and the Standard Model (14 ed.). Cambridge University Press. p. 272. ISBN 9781107034730.
^ Le Bellac, M. (2008). "3". Thermal Field Theory. Cambridge University Press. p. 36-37. ISBN 978-0521654777.