Editing Partition function (quantum field theory)

{{short description|Generating function for quantum correlation functions}}
{{Quantum field theory|cTopic=Tools}}

In [[quantum field theory]], '''partition functions''' are [[generating function]]als for [[correlation function (quantum field theory)|correlation functions]], making them key objects of study in the [[path integral formulation|path integral formalism]]. They are the [[imaginary time]] versions of [[statistical mechanics]] [[partition function (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 field|free theories]] do admit such solutions. Instead, a [[perturbation theory (quantum mechanics)|perturbative]] approach is usually implemented, this being equivalent to summing over [[Feynman diagram]]s.

== Generating functional ==

=== Scalar theories ===

In a <math>d</math>-dimensional field theory with a real [[scalar field theory|scalar field]] <math>\phi</math> and [[action (physics)|action]] <math>S[\phi]</math>, the partition function is defined in the path integral formalism as the [[functional (mathematics)|functional]]<ref>{{cite book|last=Rivers|first=R.J.|author-link=|date=1988|title=Path Integral Methods in Quantum Field Theory|url=|doi=|location=Cambridge|publisher=Cambridge University Press|chapter=1|pages=14–16|isbn=978-0521368704}}</ref>

:<math>
Z[J] = \int \mathcal D\phi \ e^{iS[\phi] + i \int d^dx J(x)\phi(x)}
</math>

where <math>J(x)</math> is a fictitious [[source field|source current]]. It acts as a generating functional for arbitrary n-point correlation functions

:<math>
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}.
</math>

The derivatives used here are [[functional derivative]]s 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<ref>{{cite book|last=Năstase|first=H.|author-link=Horațiu Năstase|date=2019|title=Introduction to Quantum Field Theory|url=|doi=|location=|publisher=Cambridge University Press|chapter=9|page=78|isbn=978-1108493994}}</ref>

:<math>
Z[J] = \sum_{n\geq 0}\frac{1}{n!}\int \prod^n_{i=1} d^dx_i G(x_1, \dots, x_n) J(x_1)\cdots J(x_n).
</math>

In [[curved space]]times there is an added subtlety that must be dealt with due to the fact that the initial [[quantum vacuum state|vacuum state]] need not be the same as the final vacuum state.<ref>{{cite book|last1=Birrell|first1=N.C.|author-link1=|last2=Davis|first2=P.C.W.|author-link2=Paul Davies|date=1984|title=Quantum Fields in Curved Spacetime|url=|doi=|location=|publisher=Cambridge University Press|chapter=6|pages=155–156|isbn=978-0521278584}}</ref> 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.<ref>{{cite book|last=Năstase|first=H.|author-link=|date=2015|title=Introduction to the AdS/CFT Correspondance|url=|doi=|location=Cambridge|publisher=Cambridge University Press|chapter=1|pages=9–10|isbn=978-1107085855}}</ref> For example, the partition function for a composite operator <math>\mathcal O(x)</math> is given by

:<math>
Z_{\mathcal O}[J] = \int \mathcal D \phi e^{iS[\phi]+i\int d^d x J(x) \mathcal O(x)}.
</math>

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 constant|coupling]] perturbatively, which amounts to regular perturbation theory using Feynman diagrams with <math>J</math> insertions on the external legs.<ref>{{cite book|last=Srednicki|first=M.|author-link=|date=2007|title=Quantum Field Theory|url=|doi=|location=Cambridge|publisher=Cambridge University Press|chapter=9|pages=58–60|isbn=978-0521864497}}</ref> The symmetry factors for these types of diagrams differ from those of correlation functions since all external legs have identical <math>J</math> insertions that can be interchanged, whereas the external legs of correlation functions are all fixed at specific coordinates and are therefore fixed.

By performing a [[Wick rotation|Wick transformation]], the partition function can be expressed in [[Euclidean space|Euclidean]] spacetime as<ref>{{cite book|last1=Peskin|first1=Michael E.|author1-link=Michael Peskin|last2=Schroeder|first2=Daniel V.|date=1995|title=An Introduction to Quantum Field Theory|publisher=Westview Press|chapter=9|pages=289–292|isbn=9780201503975}}</ref>

:<math>
Z[J] = \int \mathcal D\phi \ e^{-(S_E[\phi] + \int d^d x_E J\phi)},
</math>

where <math>S_E</math> is the Euclidean action and <math>x_E</math> are Euclidean coordinates. This form is closely connected to the partition function in statistical mechanics, especially since the Euclidean [[Lagrangian (field theory)|Lagrangian]] is usually bounded from below in which case it can be interpreted as an [[energy]] density. It also allows for the interpretation of the exponential factor as a statistical weight for the field configurations, with larger fluctuations in the gradient 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 the introduction of its own fictitious current, 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. After differentiation, the currents are set to zero when correlation functions in a vacuum state are desired, but the currents can also be set to take on particular values to yield correlation functions in non-vanishing background fields.

For partition functions with [[Grassmann number|Grassmann]] valued [[fermionic field|fermion fields]], the sources are also Grassmann valued.<ref>{{cite book|first=M. D.|last=Schwartz|title=Quantum Field Theory and the Standard Model|publisher=Cambridge University Press|date=2014|chapter=34|page=272|isbn=9781107034730}}</ref> For example, a theory with a single [[Dirac fermion]] <math>\psi(x)</math> requires the introduction of two Grassmann currents <math>\eta</math> and <math>\bar \eta</math> so that the partition function is

:<math>
Z[\bar \eta, \eta] = \int \mathcal D \bar \psi \mathcal D \psi \ e^{iS[\psi, \bar \psi] + i\int d^d x (\bar \eta \psi + \bar \psi \eta)}.
</math>

Functional derivatives with respect to <math>\bar \eta</math> give fermion fields while derivatives with respect to <math>\eta</math> give anti-fermion fields in the correlation functions.

=== Thermal field theories ===

A [[thermal quantum field theory|thermal field theory]] at [[temperature]] <math>T</math> is equivalent in Euclidean formalism to a theory with a [[compactification (physics)|compactified]] temporal direction of length <math>\beta = 1/T</math>. Partition functions must be modified appropriately by imposing periodicity conditions on the fields and the Euclidean spacetime integrals

:<math>
Z[\beta,J] = \int \mathcal D\phi e^{-S_{E,\beta}[\phi]+\int_\beta d^d x_E J \phi}\bigg|_{\phi(\boldsymbol x, 0) = \phi(\boldsymbol x, \beta)}.
</math>

This partition function can be taken as the definition of the thermal field theory in imaginary time formalism.<ref>{{cite book|last=Le Bellac|first=M.|author-link=|date=2008|title=Thermal Field Theory|url=|doi=|location=|publisher=Cambridge University Press|chapter=3|pages=36–37|isbn=978-0521654777}}</ref> Correlation functions are acquired from the partition function through the usual functional derivatives with respect to currents

:<math>
G_{n,\beta}(x_1, \dots, x_n) = \frac{\delta^n Z[\beta, J]}{\delta J(x_1)\cdots \delta J(x_n)}\bigg|_{J=0}.
</math>

== Free theories ==

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

:<math>
Z_0[J] = N \exp\bigg(-\frac{1}{2}\int d^d x d^d y \ J(x)\Delta_F(x-y)J(y)\bigg),
</math>

where <math>\Delta_F(x-y)</math> is the position space Feynman [[propagator]]

:<math>
\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)}.
</math>

This partition function fully determines the free field theory.

In the case of a theory with a single free Dirac fermion, completing the square yields a partition function of the form

:<math>
Z_0[\bar \eta, \eta] = N \exp\bigg(\int d^d x d^d y \ \bar \eta(y) \Delta_D(x-y) \eta(x)\bigg),
</math>

where <math>\Delta_D(x-y)</math> is the position space Dirac propagator

:<math>
\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)}.
</math>

==References==
{{Reflist|2}}

==Further reading==
* Ashok Das, ''Field Theory: A Path Integral Approach'', 2nd edition, World Scientific (Singapore, 2006); paperback {{ISBN|978-9812568489}}.
* [[Hagen Kleinert|Kleinert, Hagen]], ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 4th edition, World Scientific (Singapore, 2004); paperback {{ISBN|981-238-107-4}} '' (also available online: [http://www.physik.fu-berlin.de/~kleinert/b5 PDF-files])''.
*  [http://www.scholarpedia.org/Path_integral   Jean Zinn-Justin (2009), ''Scholarpedia'', '''4'''(2): 8674].

[[Category:Quantum field theory]]