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).

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.

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
 * $$S[\psi_a,\psi_b]=\psi^\dagger_a\cdot K_a\cdot\psi_a + \psi^\dagger_b\cdot K_b\cdot\psi_b + (V^\dagger \vdots \psi_a^\dagger\psi_a\psi_b +h.c.)$$.

Tree Diagram
Integrating out field ψb results in the effective action for ψa.
 * $$S[\psi_a,\psi_b]\;\overset{\int\psi_b}{\longrightarrow}\; S[\psi_a]=\psi^\dagger_a\cdot K_a\cdot\psi_a - \psi_a^\dagger\psi_a:V^\dagger\cdot K_b^{-1}\cdot V:\psi_a^\dagger \psi_a + \operatorname{sTr} K_b$$.

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

Loop Diagram
Integrating out field ψa results in the effective action for ψb.
 * $$S[\psi_a,\psi_b]\;\overset{\int\psi_a}{\longrightarrow}\;S[\psi_b]=\operatorname{sTr} (K_a+V^\dagger\cdot \psi_b + \psi_b^\dagger \cdot V)+\psi^\dagger_b\cdot K_b\cdot\psi_b $$,

which may be expand to the 2nd order of ψb
 * $$S[\psi_b]=\psi^\dagger_b\cdot (K_b - \operatorname{sTr}G_a\cdot V \cdot G_a\cdot V^\dagger)\cdot\psi_b +\operatorname{sTr} K_a$$.

This corresponds to a loop diagram, which gives the self-energy correction Σa to the action kernel Ka
 * $$\Sigma_a= - \operatorname{sTr}G_a\cdot V \cdot G_a\cdot V^\dagger$$,

such that Ka → Ka+Σa.

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
With source J:
 * $$\psi^\dagger\cdot K\cdot\psi - J^\dagger\cdot\psi - \psi^\dagger\cdot J \;\overset{\int\psi}{\longrightarrow}\; \operatorname{sTr}\ln K -J^\dagger\cdot K^{-1}\cdot J$$.