Quantum field theory
Feynman diagram
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Background Field theory Electromagnetism Weak force Strong force Quantum mechanics Special relativity General relativity Gauge theory Yang–Mills theory
Symmetries Symmetry in quantum mechanics C-symmetry P-symmetry T-symmetry Lorentz symmetry Poincaré symmetry Gauge symmetry Explicit symmetry breaking Spontaneous symmetry breaking Noether charge Topological charge
Tools Anomaly Background field method BRST quantization Correlation function Crossing Effective action Effective field theory Expectation value Feynman diagram Lattice field theory LSZ reduction formula Partition function Path Integral Formulation Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem Wightman axioms
Equations Dirac equation Klein–Gordon equation Proca equations Wheeler–DeWitt equation Bargmann–Wigner equations Schwinger-Dyson equation Renormalization group equation
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Inquantum field theory,partition functions aregenerating functionals forcorrelation functions, making them key objects of study in thepath integral formalism. They are theimaginary time versions ofstatistical mechanicspartition functions, giving rise to a close connection between these two areas of physics. Partition functions can rarely be solved for exactly, althoughfree theories do admit such solutions. Instead, aperturbative approach is usually implemented, this being equivalent to summing overFeynman diagrams.

In a${\displaystyle d}$-dimensional field theory with a realscalar field${\displaystyle \varphi }$ andaction${\displaystyle S[\varphi ]}$, the partition function is defined in the path integral formalism as thefunctional^[1]

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

where${\displaystyle J(x)}$ is a fictitioussource current. It acts as a generating functional for arbitrary n-point correlation functions

${\displaystyle G_n(x_1,\dots ,x_n)=(-1{)}^n\frac{1}{Z[0]}\frac{{\delta }^{n}Z[J]}{\delta J(x_1)\cdots \delta J(x_n)}{|}_{J=0}.}$

The derivatives used here arefunctional 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 apower series in source currents is given by^[2]

${\displaystyle Z[J]=\sum _{n\ge 0}\frac{1}{n!}\int \prod _{i=1}^n{d}^d{x}_{i}G(x_1,\dots ,x_n)J(x_1)\cdots J(x_n).}$

Incurved spacetimes there is an added subtlety that must be dealt with due to the fact that the initialvacuum state need not be the same as the final vacuum state.^[3] 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.^[4] For example, the partition function for a composite operator${\displaystyle \mathcal{O}(x)}$ is given by

${\displaystyle Z_{\mathcal{O}}[J]=\int \mathcal{D}\varphi e^{iS[\varphi ]+i\int d^{d}xJ(x)\mathcal{O}(x)}.}$

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 weakcoupling perturbatively, which amounts to regular perturbation theory using Feynman diagrams with${\displaystyle J}$ insertions on the external legs.^[5] The symmetry factors for these types of diagrams differ from those of correlation functions since all external legs have identical${\displaystyle J}$ insertions that can be interchanged, whereas the external legs of correlation functions are all fixed at specific coordinates and are therefore fixed.

By performing aWick transformation, the partition function can be expressed inEuclidean spacetime as^[6]

${\displaystyle Z[J]=\int \mathcal{D}\varphi e^{-(S_E[\varphi ]+\int d^d{x}_{E}J\varphi )},}$

where${\displaystyle S_E}$ is the Euclidean action and${\displaystyle x_E}$ are Euclidean coordinates. This form is closely connected to the partition function in statistical mechanics, especially since the EuclideanLagrangian is usually bounded from below in which case it can be interpreted as anenergy 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.

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, withantiparticle 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 withGrassmann valuedfermion fields, the sources are also Grassmann valued.^[7] For example, a theory with a singleDirac fermion${\displaystyle \psi (x)}$ requires the introduction of two Grassmann currents${\displaystyle \eta }$ and${\displaystyle \overline{\eta }}$ so that the partition function is

${\displaystyle Z[\overline{\eta },\eta ]=\int \mathcal{D}\overline{\psi }\mathcal{D}\psi e^{iS[\psi ,\overline{\psi }]+i\int d^{d}x(\overline{\eta }\psi +\overline{\psi }\eta )}.}$

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

Athermal field theory attemperature${\displaystyle T}$ is equivalent in Euclidean formalism to a theory with acompactified temporal direction of length${\displaystyle \beta =1/T}$. Partition functions must be modified appropriately by imposing periodicity conditions on the fields and the Euclidean spacetime integrals

${\displaystyle Z[\beta ,J]=\int \mathcal{D}\varphi e^{-S_{E,\beta }[\varphi ]+{\int }_{\beta }{d}^d{x}_{E}J\varphi }{|}_{\varphi (\mathit{x},0)=\varphi (\mathit{x},\beta )}.}$

This partition function can be taken as the definition of the thermal field theory in imaginary time formalism.^[8] Correlation functions are acquired from the partition function through the usual functional derivatives with respect to currents

${\displaystyle G_{n,\beta }(x_1,\dots ,x_n)=\frac{{\delta }^{n}Z[\beta ,J]}{\delta J(x_1)\cdots \delta J(x_n)}{|}_{J=0}.}$

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

${\displaystyle Z_0[J]=N\exp (-\frac{1}{2}\int d^{d}x{d}^{d}yJ(x){\mathrm{\Delta }}_F(x-y)J(y)),}$

where${\displaystyle {\mathrm{\Delta }}_F(x-y)}$ is the position space Feynmanpropagator

${\displaystyle {\mathrm{\Delta }}_F(x-y)=\int \frac{d^{d}p}{(2\pi {)}^d}\frac{i}{p^2-m^2+iϵ}{e}^{-ip\cdot (x-y)}.}$

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

${\displaystyle Z_0[\overline{\eta },\eta ]=N\exp (\int d^{d}x{d}^{d}y\overline{\eta }(y){\mathrm{\Delta }}_D(x-y)\eta (x)),}$

where${\displaystyle {\mathrm{\Delta }}_D(x-y)}$ is the position space Dirac propagator

${\displaystyle {\mathrm{\Delta }}_D(x-y)=\int \frac{d^{d}p}{(2\pi {)}^d}\frac{i(p/+m)}{p^2-m^2+iϵ}{e}^{-ip\cdot (x-y)}.}$

• Ashok Das,Field Theory: A Path Integral Approach, 2nd edition, World Scientific (Singapore, 2006); paperbackISBN 978-9812568489.
• Kleinert, Hagen,Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, World Scientific (Singapore, 2004); paperbackISBN 981-238-107-4 (also available online:PDF-filesArchived 2008-06-15 at theWayback Machine).
• Jean Zinn-Justin (2009),Scholarpedia,4(2): 8674.

• Quantum field theory

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