Effective action ($\Gamma$)

The quantum effective action is the Legendre transform of the generating functional of connected correlation functions. Its functional derivatives are the one-particle-irreducible vertex functions.

Write $S=\frac{1}{2}\varphi A\varphi$ and use the vacuum prescription $A\mapsto A+i0$. Completing the Gaussian path integral gives $Z[J]/Z[0]=\exp [-iJ{A}^{-1}J/2]$, hence $W=-J{A}^{-1}J/2$, $\phi =-A^{-1}J$ and $\Gamma =\phi A\phi /2=S[\phi ]$. The source-independent determinant cancels in vacuum normalization. Without that normalization, the equality holds up to a field-independent vacuum constant.

// Target: perturbative-quantum-field-theory.bigb

With $W=-i\log Z$, $\phi =\delta W/\delta J$ and $\Gamma =W-\int J\phi$, variation gives $\delta \Gamma /\delta \phi =-J$. Differentiating through the source-to-field map givesThe inverse is taken on the regulated nonsingular fluctuation space. Since $W^{(2)}=i{C}_2$, the full connected propagator is determined by the inverse effective-action quadratic kernel. Factors of $i$ and the minus sign depend on the explicitly chosen Minkowski Legendre transform convention.

// Target: perturbative-quantum-field-theory.bigb

Use the Minkowski generating functional convention $Z=e^W$, source insertion $iJ\varphi$, classical field $\phi =-i\delta W/\delta J$, and $i\Gamma =W-i\int J\phi$. With $G_r^c=i^{-r}{\delta }^{r}W/\delta J^r$, the response kernel is $\delta \phi /\delta J=i{G}_2^c$. Differentiating $\delta \Gamma /\delta \phi =-J$ once gives the displayed two-point vertex, where the vertex means a derivative of $i\Gamma$. Differentiating the inverse kernel gives the three-point vertex: the connected three-point function is amputated by one inverse two-point kernel on each leg. In the alternative convention $W_c=-i\log Z$, the factors of $i$ are redistributed.

The quantum effective potential is the derivative-free part of the quantum effective action evaluated on a spatially constant background. The displayed sign uses $\Gamma =-W+\int J\phi$ for $Z=e^{iW}$ and the source term $+J\varphi$; the opposite effective-action convention gives $-\int V_{\mathrm{eff}}$. Its derivatives give zero-momentum one-particle-irreducible vertices. The value depends on a renormalization scheme and scale; an arbitrary additive vacuum-energy constant may be fixed separately.

For one canonically normalized real scalar field and $V^{\prime \prime }(\phi )>0$, the modified minimal subtraction scheme gives the displayed one-loop term. The Gaussian fluctuation approximation produces $\frac{1}{2}\int d^d{k}_E(2\pi {)}^{-d}\log (k_E^2+V^{\prime \prime }(\phi ))$. Its derivative in $V^{\prime \prime }$ is half the Euclidean massive loop integral with $n=1$. Integrating and subtracting its pole yields the formula, up to a field-independent constant. Other finite counterterms alter the scheme-dependent nonlogarithmic part. Negative curvature requires an analytic continuation and can yield an imaginary part indicating instability of that background.

For $d=4-ϵ$, the scalar one-loop determinant has the displayed local divergent potential. If $V$ is a quartic polynomial, then $V^{\prime \prime }$ has degree at most two and $(V^{\prime \prime }{)}^2$ has degree at most four. Its subtraction therefore uses the original polynomial couplings and a vacuum-energy term. A polynomial potential of degree $r>4$ instead produces degree $2r-4>r$, requiring new couplings. This is a direct potential-level illustration of power counting in quantum field theory and renormalizability.

For real scalar multiplets with antisymmetric generators $t_a$, transforming a source $J$ in the same representation leaves $J\cdot \varphi$ invariant. With an invariant functional measure, local changes of integration variables imply the Ward identity for the quantum effective action. In the convention $Z=e^{iW}$, $\phi =W_J$, $\Gamma =-W+\int J\phi$, it isThe Legendre transform gives ${\Gamma }_{\phi }=J$ and ${\Gamma }_A{|}_{\phi }=-W_A{|}_J$. Its classical limit is $\Gamma =-S_0$, not $S_0$, for the source term $+J\varphi$. Differentiating twice in $\phi$ produces two contact terms and the momentum-space identity $p_{1\mu }{\tau }_{a,ij}^{\mu }=i(t_a{)}_{ik}{\tau }_{kj}-{\tau }_{ik}i(t_a{)}_{kj}$ with the appropriate shifted two-point momenta. Changing the overall convention for $\Gamma$ changes both two-point and mixed vertices together; changing just one violates the identity. A gauge anomaly would invalidate the assumed invariant-measure argument.

A functional derivative of the quantum effective action gives an amputated one-particle-irreducible correlation function. Its perturbative graphs stay connected after cutting any single internal propagator. A free quadratic effective action has no vertices of order greater than two; interactions generate higher vertices.

Use the Minkowski generating functional $Z=e^{iW}$, source $+J\varphi$, mean field $\phi =\delta W/\delta J$, and quantum effective action $\Gamma =W-\int J\phi$. Then $W^{(2)}=i{G}_2^c$, $W^{(3)}=-G_3^c$, and ${\Gamma }^{(2)}=-(W^{(2)}{)}^{-1}$. Differentiating this inverse gives ${\Gamma }^{(3)}=(W^{(2)}{)}^{-1\otimes 3}{W}^{(3)}=-i(G_2^c{)}^{-1\otimes 3}{G}_3^c$, with integration over the repeated spacetime arguments. Thus amputation of external propagators gives $i{\Gamma }^{(3)}$ in this convention. The full connected two-point functions, rather than bare propagators, perform the exact amputation.

A finite higher-point scalar vertex is a term in the quantum effective action with more external scalar fields than the original interaction, whose coefficient remains finite after renormalization. It can survive removal of an ultraviolet cutoff even when no corresponding independent counterterm is needed. Its coefficient is then predicted by the renormalized parameters and external momenta.

In Euclidean signature, for a canonically normalized real scalar field with potential $V$, the one-loop contribution to the quantum effective action is ${\Gamma }_1[\varphi ]=\frac{1}{2}\mathrm{Tr}\log [-{\partial }^2+V^{\prime \prime }(\varphi )]$, with a suitable regularization in quantum field theory. Expanding this functional determinant generates one-particle-irreducible Feynman diagrams with arbitrarily many external fields.

Expanding an action about a classical stationary field and retaining its second variation gives a Gaussian path integral. Its nonzero fluctuation modes determine a functional determinant and a one-loop soliton mass correction. Symmetry-generated zero modes in field theory must be removed from that determinant and integrated as collective coordinates.

Expanding the Hamiltonian about a static classical solution removes the linear term by its Euler-Lagrange field equation. For a canonical real scalar field with potential $U$, the quadratic fluctuation operator is $\mathcal{H}=-{\partial }_X^2+U^{\prime \prime }({\phi }_{\mathrm{cl}})$. Positive-frequency normal modes become quantum harmonic oscillators under canonical quantization, yielding the Gaussian contribution to a one-loop soliton mass correction. A negative eigenvalue signals an instability, while a physical zero mode in field theory must be replaced by a collective coordinate.

A Wilsonian effective action is obtained by integrating out field modes above a chosen momentum scale while retaining lower-momentum modes as backgrounds. Its local couplings change with the scale according to a renormalization-group flow.

The Polchinski equation changes an interaction action with its regulated Gaussian covariance so that the normalized functional integral stays unchanged. In operator notation, ${\dot{e}}^{-S_1}=-\dot{C}:{\delta }^2{e}^{-S_1}/2$. The resulting action flow has a product of first functional derivatives, joining two vertices, and a second derivative, closing two legs into a loop. It generates a Wilsonian effective action, which is not the same functional as the one-particle-irreducible correlation functions encoded by the quantum effective action.

After shell integration and $x^{\prime }=bx$, restore the kinetic term by ${\varphi }^{\prime }=b^{-(d-2)/2}\sqrt{1+\Delta Z}\varphi$. An operator with $n$ scalar fields and $m$ derivatives scales by $b^{m+n(d-2)/2-d}$. In four dimensions this gives the displayed factor with $m+n-4$; the general exponent is $m+n(d-2)/2-d$. The coupling shift and wavefunction renormalization arise before the rescaling.

The cumulant expansion of the shell integral keeps precisely connected Feynman diagrams built from shell-restricted scalar propagators. The external low fields are backgrounds, not propagators. Connected graphs with bridges can contribute, so the expansion is not limited to one-particle-irreducible Feynman diagrams.

If a bridge is the only line carrying momentum away from a low external field, tadpole pairs do not change its momentum and the shell-restricted scalar propagator has zero support there. A bridge attached to a product such as ${\varphi }_{<}^3$ can survive because the sum of three low momenta can enter the shell. This matters before a local derivative expansion.

Split Fourier transform modes into a low-momentum ball and its disjoint shell. The quadratic scalar action has no cross term between these supports. Integrating the shell at fixed low field defines the lower-cutoff Wilsonian effective action.