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User:EverettYou/Notes on QFT

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Some notes prepared for the improvement of the following articles.

Action Formalism

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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

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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

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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

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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

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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

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Connected Diagrams

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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

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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

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Response to Perturbations

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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

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Consider trilinear vertex terms

.

Tree Diagram

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Integrating out field ψb results in the effective action for ψa.

.

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

Loop Diagram

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Standard loop diagram.

Integrating out field ψa results in the effective action for ψb.

,

which may be expand to the 2nd order of ψb

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

,

such that Ka → Kaa.

Appendix: Gaussian Integral

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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

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With source J:

.

Complex Field and Grassmann Field

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With source J:

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See Also

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