User:EverettYou/Notes on QFT

Some notes prepared for the improvement of the following articles.

Effective Action

Action Formalism

In statistical field theory, the partition function reads

$Z\equiv e^{-F}=\int\mathcal{D}[\psi]\,e^{-S[\psi]}$.

In the path integral formalism, the free energy (functional) $F$ is obtained from the action $S$ by integrating out the $\psi$ field, denoted as a transformation from the action into the free energy.

$S[\psi]\overset{\int\psi}{\longrightarrow} F.$

The factor $\beta$ (inverse temperature) has been absorbed into the (dimensionless) free energy $F$.

Quadratic Action

If the action is of quadratic form

$S[\psi]=\psi^\dagger\cdot K \cdot\psi,$

then Gaussian integral can be performed to obtain the free energy, and hence the correlations of the field. The operator $K$ is the kernel of the action, whose explicit form depends on the dynamics of the field. The following two types of dynamics are of interests.

Diffusive Dynamics

Equation of motion

$-\partial_\tau\psi=H\cdot\psi.$

In the frequency representation ($\partial_\tau=-\mathrm{i}\omega$),

$\mathrm{i}\omega\psi=H\cdot\psi.$

So the kernel of the action is

$K = -\mathrm{i}\omega + H.$

The convension is that $H$ is of the same sign as $K$ (or the action $S$), because the path integral is derived from $Z=\operatorname{Tr} e^{-H}$. As a consequence, every term lowering from the action (or raising to the action) will aquire a minus sign.

Wave Dynamics

Equation of motion in imaginary time ($\tau=\mathrm{i}t$),

$(-\partial_\tau^2 + \Omega^2)\cdot\psi=0.$

In the frequency representation ($\partial_\tau=-\mathrm{i}\omega$),

$(-(\mathrm{i}\omega)^2 + \Omega^2)\cdot\psi=0.$

So the kernel of the action is

$K = -(\mathrm{i}\omega)^2 + \Omega^2.$

Here $\pm\Omega$ plays the role of boson energy.

Free Energy

Free energy is obtained from the action by integrating out the field

$F=\operatorname{sTr}\ln K.$

sTr denotes the supertrace, which equals to Tr for bosonic fields and -Tr for fermionic fields.

The Matsubara frequency summation can be carried out given the specific form of the kernel $K$. For diffusive dynamics, the result is

$F=\operatorname{sTr}\ln(1-\eta e^{-\beta H}).$

For wave dynamics, the result is

$F=2\operatorname{Tr}\ln 2 \operatorname{sinh}\frac{\beta\Omega}{2}$ (bosonic),

$F=-2\operatorname{Tr}\ln 2\mathrm{i} \operatorname{cosh}\frac{\beta\Omega}{2}$ (fermionic).

Correlation of Fields

Connected Diagrams

To probe the field correlation, a source term coupled with the field is introduced. The quadratic action becomes

$S[\psi]=\psi^\dagger\cdot K \cdot\psi - J^\dagger\cdot\psi - \psi^\dagger\cdot J.$

Integrating over the field leads to the free energy with source

$F[J]=\operatorname{sTr}\ln K -J^\dagger\cdot K^{-1}\cdot J,$

where $\eta$ depends on the statistics of the field (bosonic: $\eta=+1$, fermionic: $\eta=-1$).

The negative free energy $\ln Z[J]=-F[J]$ serves as the generator of connected diagrams (i.e. the cumulants),

$\langle\psi(1)\psi^\dagger(2)\cdots\rangle_\text{con}= - \left.\delta_{J^\dagger(1)}\eta\delta_{J(2)}\cdots F[J]\right|_{J=0}.$

Note that the arrangement of the derivatives $\delta_{J^\dagger}$, $\eta\delta_J$ should follow the same ordering as that of the fields $\psi$, $\psi^\dagger$ in the vacuum expectation value (the ordering is particularly important for the Grassmann field). Note that each $\eta\delta_{J}$ operator must carry a statistical sign $\eta$, because the operator must commute through the field $\psi^\dagger$ to reach the field $J$, i.e. $\eta\delta_{J}\psi^\dagger\cdot J=\psi^\dagger\cdot\delta_{J}J=\psi^\dagger$, which will cosume the sign $\eta$. Intuitively, $-F$ can be considered as a kind of averaged $\langle -S\rangle\sim\langle J^\dagger\cdot\psi + \psi^\dagger\cdot J\rangle$, therefore applying the derivative operators on $-F$ yields the fields.

Bilinear Correlation

Define the bilinear correlation function (aka Green's function)

$G \equiv -\langle \psi\psi^\dagger \rangle_\text{con} = -\longleftarrow = - K^{-1}$.

It is diagrammatically represented as a line propagating from right to left (creation followed by annihilation, representing $\langle \psi\psi^\dagger\rangle_\text{con}$) with a minus sign in the front. The bilinear correlation function can be evaluated from

$\langle \psi\psi^\dagger \rangle_\text{con}=-\delta_{J^\dagger}\eta\delta_{J}F[J]=\delta_{J^\dagger}\eta\delta_{J}J^\dagger\cdot K^{-1}\cdot J=K^{-1}.$

This result is universal for both bosonic and fermionic fields.

Reversing the ordering leads to transpose of the correlation function and a statistical sign $\eta$ (+1 for bosons, -1 for fermions),

$\langle (\psi^\dagger)^\intercal \psi^\intercal \rangle_\text{con} = \eta (K^{-1})^\intercal = - \eta G^\intercal.$

So the advantage of defining the propagator as $-\langle\psi\psi^\dagger\rangle$ other than $\langle(\psi^\dagger)^\intercal\psi^\intercal\rangle$ is to avoid both the transpose field indices and the statistical sign dependancy.

Effective Action

Response to Perturbations

The response to perturbations is simply obtained by partial derivatives. By introducing the Green's function $G=-K^{-1}$, the results can be written in a compact from: to the first order

$\partial_\mu F= -\operatorname{sTr} G\cdot\partial_\mu K,$

and to the second order,

$\partial_\mu\partial_\nu F= -\operatorname{sTr} G\cdot\partial_\mu \partial_\nu K - \operatorname{sTr} G\cdot\partial_\mu K\cdot G\cdot\partial_\nu K.$

This is because by definition $G\cdot K=-1$, so $\partial (G\cdot K)=0$, from which we have $\partial G = G\cdot \partial K \cdot G$.

Beyond Bilinear Form

Consider trilinear vertex terms

S[\psi _{a},\psi _{b}]=\psi _{a}^{\dagger }\cdot K_{a}\cdot \psi _{a}+\psi _{b}^{\dagger }\cdot K_{b}\cdot \psi _{b}+(V^{\dagger }\vdots \psi _{a}^{\dagger }\psi _{a}\psi _{b}+h.c.)

.

Tree Diagram

Integrating out field ψ_b results in the effective action for ψ_a.

S[\psi _{a},\psi _{b}]\;{\overset {\int \psi _{b}}{\longrightarrow }}\;S[\psi _{a}]=\psi _{a}^{\dagger }\cdot K_{a}\cdot \psi _{a}-\psi _{a}^{\dagger }\psi _{a}:V^{\dagger }\cdot K_{b}^{-1}\cdot V:\psi _{a}^{\dagger }\psi _{a}+\operatorname {sTr} K_{b}

.

This correspond to a tree diagram, which leads to the effective interaction of the field ψ_a.

Loop Diagram

Integrating out field ψ_a results in the effective action for ψ_b.

S[\psi _{a},\psi _{b}]\;{\overset {\int \psi _{a}}{\longrightarrow }}\;S[\psi _{b}]=\operatorname {sTr} (K_{a}+V^{\dagger }\cdot \psi _{b}+\psi _{b}^{\dagger }\cdot V)+\psi _{b}^{\dagger }\cdot K_{b}\cdot \psi _{b}

,

which may be expand to the 2nd order of ψ_b

S[\psi _{b}]=\psi _{b}^{\dagger }\cdot (K_{b}-\operatorname {sTr} G_{a}\cdot V\cdot G_{a}\cdot V^{\dagger })\cdot \psi _{b}+\operatorname {sTr} K_{a}

.

This corresponds to a loop diagram, which gives the self-energy correction Σ_a to the action kernel K_a

\Sigma _{a}=-\operatorname {sTr} G_{a}\cdot V\cdot G_{a}\cdot V^{\dagger }

,

such that K_a → K_a+Σ_a.

Appendix: Gaussian Integral

If the field action is in a quadratic form of the field, Gaussian integral can be performed to obtained the effective action.

Real Field and Majorana Field

With source J:

{\frac {1}{2}}\psi ^{\text{T}}\cdot K\cdot \psi -J^{\text{T}}\cdot \psi \;{\overset {\int \psi }{\longrightarrow }}\;{\frac {1}{2}}\operatorname {sTr} \ln K-{\frac {1}{2}}J^{\text{T}}\cdot K^{-1}\cdot J

.

Complex Field and Grassmann Field