The Generating Functional and the Effective Action in Biquaternionic Form

Introduction

The generating functional $Z[\tilde J]$ of the previous article is a bookkeeping device: its functional derivatives with respect to the source $\tilde J$ produce the correlation functions. Two derived objects carry almost all of the quantum content of a field theory. The first is its logarithm, $$ W[\tilde J] = \log Z[\tilde J], $$ the connected generating functional, whose derivatives give the connected correlation functions. The second is its Legendre transform, $$ \Gamma[\tilde\phi] = \sup_{\tilde J}\Big(\langle \tilde J,\tilde\phi\rangle - W[\tilde J]\Big), \qquad \tilde\phi = \frac{\delta W}{\delta \tilde J}, $$ the effective action, whose derivatives give the one-particle-irreducible (1PI) correlation functions, and whose expansion in powers of $\hbar$ is the loop expansion. This article asks what the biquaternion algebra $\mathbb{B}$ contributes to that pair of objects.

The answer is a factorization statement and a convexity statement, both inherited from the trace pairing, and both sharpened by the sector split.

• Established, and recomputed below. For a free theory with central kinetic operator $\tilde K$ the connected and effective functionals are the quadratic forms $$ W[\tilde J] = \tfrac12\big\langle \tilde J,\tilde K^{-1}\tilde J\big\rangle , \qquad \Gamma[\tilde\phi] = \tfrac12\big\langle \tilde\phi,\tilde K\tilde\phi\big\rangle , $$ built from the real bilinear form $\langle \tilde{Q},\tilde Y\rangle=\mathrm{Re}\,\mathrm{Tr}(\tilde{Q}^{*}\tilde Y)$ of the previous article, with the inverse-Hessian relation $\Gamma^{(2)}W^{(2)}=I$ and the Legendre identity $W[\tilde J]+\Gamma[\tilde\phi]=\langle \tilde J,\tilde\phi\rangle$ at $\tilde\phi=\delta W/\delta\tilde J$. These were checked in finite dimension with explicit matrices.
• Established (algebra). Because the source $\tilde J$, the classical field $\tilde\phi=\delta W/\delta\tilde J$, and the fluctuations all live in the same module, the effective action is a functional on the module, and its convexity is convexity with respect to the real form of the algebra — a real form that is positive definite on $\mathbb{M}_+$ and negative definite on $\mathbb{M}_-$. For a central kinetic operator the effective action factorizes over the sectors, $$ \Gamma[\tilde\phi] = \Gamma_-[{\tilde\phi}_-]+\Gamma_+[{\tilde\phi}_+] , $$ since the Legendre transform of a sum of independent functionals is the sum of the Legendre transforms.
• The one-loop term. The first quantum correction to the effective action is the functional determinant, $$ \Gamma[\tilde\phi] = S[\tilde\phi] + \frac{\hbar}{2}\,\mathrm{Tr}\log S''[\tilde\phi] + O(\hbar^2), $$ where $S''[\tilde\phi]$ is the second variation of the action about the classical field. In the biquaternion framework $S''[\tilde\phi]$ is an operator on the module, and the trace is the trace pairing; the determinant is the subject of The Functional Determinant in Biquaternionic Form.
• The vacuum condition. The classical field that extremizes $\Gamma$ is the theory's vacuum expectation value, and in the framework the tree-level vacuum is a minimal idempotent of $\mathbb{M}_+$, selected by the sign of the quadratic term, as The Biquaternion Vacuum as a Minimal Idempotent establishes. The effective-action version of that statement is that the vacuum minimizes the effective potential on the manifold of normalizable states.

The article proceeds as follows. The next section defines the connected functional and its derivatives. A section defines the classical field and the Legendre transform, with the convexity in the real form. A section gives the effective action and the loop expansion, and a section the sector factorization. A section treats the effective potential and its relation to the vacuum, and a section the one-loop term and the determinant. A section separates what is established from what is interpretation, and the article closes with open questions.

Conventions. We use those of the companion articles, in particular The Functional Integral in Biquaternionic Form. The algebra is $\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$ with basis $e_0=1,e_1,e_2,e_3$, $e_k^2=-e_0$. The sectors are $\mathbb{M}_-$ (anti-Hermitian, material) and $\mathbb{M}_+$ (Hermitian, informational). The real bilinear form is $$ \langle \tilde{Q},\tilde Y\rangle = \mathrm{Re}\,\mathrm{Tr}\big(\tilde{Q}^{*}\tilde Y\big), $$ positive definite on $\mathbb{M}_+$ and negative definite on $\mathbb{M}_-$. The quadratic action is $S[\tilde\Phi]=\int d^4x\,\langle\tilde\Phi,\tilde K\tilde\Phi\rangle$ with $\tilde K=\Box-m^2$ and $\Box=\tilde\nabla\tilde\nabla^{\natural}=\partial_{ict}^2+\Delta$; the $ict$ metric is $\eta=\mathrm{diag}(-1,+1,+1,+1)$. The companion articles The Biquaternion Vacuum as a Minimal Idempotent and The GNS Construction in the Biquaternion Framework supply the vacuum state; the trace pairing and the module are those of The Feynman Propagator in Biquaternionic Form and The S-Matrix in Biquaternionic Form.

The Connected Generating Functional

Write the functional integral with a source as in the previous article, $$ Z[\tilde J] = \int\mathcal{D}\tilde\Phi\; e^{\,iS[\tilde\Phi]/\hbar + \langle \tilde J,\tilde\Phi\rangle}, $$ with $\tilde J$ valued in the module and the pairing as above. The connected functional is $$ W[\tilde J] = -i\hbar\,\log Z[\tilde J]. $$

Derivatives give correlators. The n-point connected correlation functions are the functional derivatives $$ \big\langle \tilde\Phi(x_1)\cdots\tilde\Phi(x_n)\big\rangle_{\!c} = \frac{\delta^n W}{\delta\tilde J(x_1)\cdots\delta\tilde J(x_n)}\Big|_{\tilde J=0} , $$ and the full correlators are the derivatives of $Z$ itself. The formalism is the standard one; the biquaternion content is that the source is a module-valued function and the derivative is taken with respect to that function, so the correlation functions take values in tensor powers of the module.

The free case. For the Gaussian integral of The Functional Integral in Biquaternionic Form the connected functional is exact: $$ W[\tilde J] = \tfrac12\big\langle \tilde J,\tilde K^{-1}\tilde J\big\rangle , \qquad\text{equivalently}\qquad W[\tilde J] = \tfrac12\int\frac{d^4k}{(2\pi)^4}\,\frac{\big|\tilde J(k)\big|^2}{\tilde k\tilde k^{\natural}+m^2}, $$ using the wave biquaternion $\tilde k=iEe_0+\mathbf{p}$ and $\tilde k\tilde k^{\natural}=-p^2$. The kernel displayed is that of the kinetic operator $-\Box+m^2$, whose symbol is the central mass-shell operator $\mathcal{M}(\tilde k)=\tilde k\tilde k^{\natural}+m^2$; the action's operator is $\tilde K=\Box-m^2$, whose symbol is $-\mathcal{M}$, so the kernel of $\tilde K^{-1}$ is its negative, $-1/\mathcal{M}$. Recording this once fixes the relative sign of the source kernel and of the classical field below. Its second derivative is the propagator $D_F$, the object of The Feynman Propagator in Biquaternionic Form; the framework's contribution is the central scalar kinetic operator $\mathcal{M}(\tilde k)=\tilde k\tilde k^{\natural}+m^2$, as before.

A remark on the trace. The derivative $\delta W/\delta\tilde J$ is defined through the pairing, and because the pairing is a real bilinear form on the module, it takes a module-valued source to a module-valued classical field. No privileged matrix basis is needed; the derivative is the module's own gradient. This will matter in the next section, where the Legendre transform is taken on the module and not on a coordinate vector space.

The Classical Field and the Legendre Transform

The classical field is the source-derivative of the connected functional, $$ \tilde\phi(x) = \frac{\delta W}{\delta\tilde J(x)} , $$ a module-valued function. It is the one-point function in the presence of the source, $\tilde\phi=\langle\tilde\Phi\rangle_{\tilde J}$.

The effective action is the Legendre transform $$ \Gamma[\tilde\phi] = \Big\langle \tilde J,\tilde\phi\Big\rangle - W[\tilde J]\Big|_{\tilde J=\tilde J[\tilde\phi]} , \qquad \frac{\delta\Gamma}{\delta\tilde\phi} = \tilde J . $$ The two functionals $\Gamma$ and $W$ differ by the product of the conjugate variables, and the map $\tilde J\leftrightarrow\tilde\phi$ is inverted by the second relation. The Legendre transform is defined by a supremum over $\tilde J$, and it is finite exactly when $W$ is convex.

Convexity is convexity in the real form. In this framework the appropriate notion of convexity is with respect to the real bilinear form $\langle\cdot,\cdot\rangle$: a functional $F[\tilde\phi]$ is convex if, on the module, $$ F[\lambda\tilde\phi_1+(1-\lambda)\tilde\phi_2] \le \lambda F[\tilde\phi_1]+(1-\lambda)F[\tilde\phi_2], \qquad \lambda\in[0,1], $$ with the comparison of the two module elements understood through the positive-definite part of the form. Because the form is positive definite on $\mathbb{M}_+$ and negative definite on $\mathbb{M}_-$, the convexity is a relative convexity on each sector separately, and the Legendre transform factorizes accordingly, as the next-but-one section shows. This is the first place where the sector signature enters the effective-action formalism rather than the measure.

The inverse-Hessian relation. The second derivatives satisfy $$ \Gamma^{(2)}[\tilde\phi]\;W^{(2)}[\tilde J]\Big|_{\tilde\phi=\delta W/\delta\tilde J} = \tilde I , $$ the module identity, and the Legendre identity $$ W[\tilde J]+\Gamma[\tilde\phi] = \langle \tilde J,\tilde\phi\rangle \qquad\text{at}\qquad \tilde\phi=\frac{\delta W}{\delta\tilde J} $$ holds exactly. These are the two standard identities of the Legendre transform; they are recorded here because they are what the finite-dimensional verification checks.

Finite-dimensional verification. With a Hermitian positive-definite $$ \tilde K \to K = \begin{pmatrix} 2 & 0.3 & 0 \\ 0.3 & 1.4 & 0.2 \\ 0 & 0.2 & 1.1 \end{pmatrix}, $$ and source $J=(0.5,-0.4,0.9)$, the classical field is $\phi=K^{-1}J=(0.322647,-0.484316,0.906239)$, and $$ W[J]=\tfrac12 J^T K^{-1}J = 0.58533264 = \tfrac12\phi^T K\phi = \Gamma[\phi], \qquad W+\Gamma = 1.17066529 = J^T\phi , $$ with $K\cdot K^{-1}=I$ to machine precision. The quadratic case has $W=\Gamma$ and $W+\Gamma=\langle J,\phi\rangle$, exactly as the identities require. The convexity is the positivity of $K$, whose leading principal minors are all positive.

The Effective Action and the Loop Expansion

The effective action is the generating functional of the 1PI correlation functions, $$ \Gamma^{(n)}[\tilde\phi=0] = \big\langle \tilde\Phi\cdots\tilde\Phi\big\rangle_{\!1\text{PI}} , $$ and it has a loop expansion in powers of $\hbar$, $$ \Gamma[\tilde\phi] = S[\tilde\phi] + \hbar\,\Gamma_1[\tilde\phi] + \hbar^2\,\Gamma_2[\tilde\phi] + \cdots . $$ The tree term is the classical action. The one-loop term is the functional determinant of the second variation, $$ \Gamma_1[\tilde\phi] = \frac12\,\mathrm{Tr}\log S''[\tilde\phi] , $$ evaluated on the module, with $S''[\tilde\phi]$ the operator of the quadratic fluctuations about the classical field. In the biquaternion framework $S''[\tilde\phi]$ is an operator on the module and the trace is the trace pairing of the algebra, so the one-loop effective action is the logarithm of a determinant of $\mathbb{B}$-module operators. This is the object that the next article computes.

Standard, and transcribed. The loop expansion, the identification of $\Gamma$ as the 1PI generator, and the form of $\Gamma_1$ are standard field theory (Weinberg, Peskin–Schroeder, Zinn-Justin) and are not rebuilt here. The biquaternion content is the module on which $S''$ acts and the trace pairing used to define its logarithm.

A structural consequence. Because $S''$ is obtained by differentiating the central action, it is central whenever the classical field multiplies a central vertex; for a scalar biquaternion field with central mass the fluctuation operator is central, so it does not mix sectors and $\Gamma_1$ factorizes. For a fermionic field the mass term is a right multiplication and $S''$ is not central; $\Gamma_1$ then does mix the chiralities, and this is where the anomaly and index structure of articles 11 and 12 enters.

Sector Factorization of the Effective Action

The sector factorization of the functional integral implies the same factorization of the effective action, and the implication is a property of the Legendre transform rather than a new computation.

Suppose the classical field splits as $\tilde\phi=\tilde\phi_-+\tilde\phi_+$ and the action as $S[\tilde\phi]=S_-[\tilde\phi_-]+S_+[\tilde\phi_+]$; then the source splits as $\tilde J=\tilde J_-+\tilde J_+$ with $\tilde J_\pm=\delta S_\pm/\delta\tilde\phi_\pm$, and $$ W[\tilde J] = W_-[J_-]+W_+[J_+] , $$ because $\log$ of a product is a sum. The Legendre transform of a sum of independent functionals is the sum of the Legendre transforms, so $$ \Gamma[\tilde\phi] = \Gamma_-[{\tilde\phi}_-]+\Gamma_+[{\tilde\phi}_+] . $$ At one loop the same statement holds with $\Gamma_{1,\pm}=\tfrac12\mathrm{Tr}\log S_\pm''$, and the determinant factorizes, $\det S''=\det S_-''\det S_+''$.

What breaks the factorization. Any term that is not sector-diagonal. The three candidates, in order of importance, are: an interaction that couples the sectors, such as a term built from $\mathrm{Sc}(\tilde\Phi\tilde\Phi^{\natural}\tilde\Phi\tilde\Phi^{\natural})$ with a non-central contraction; a kinetic operator that is not central; and the chirality-off-diagonal Dirac mass. A central scalar theory factorizes at all loops.

The factorized one-loop structure. For the free biquaternion scalar the one-loop effective action is the sum $$ \Gamma_1 = \frac12\mathrm{Tr}\log\big(-\Box+m^2\big)\Big|_{\mathbb{M}_-} + \frac12\mathrm{Tr}\log\big(-\Box+m^2\big)\Big|_{\mathbb{M}_+} , $$ and, because the two sectors are isomorphic as real vector spaces and the operator is the same central scalar, the two terms are equal up to the sign of the sector form. The two sectors are not independent field copies — they are related by multiplication by the central $i$, $\mathbb{M}_-=i\mathbb{M}_+$ — so this is a component count and not a doubling of the field content; the same caution applies as in The Functional Integral in Biquaternionic Form and The Harmonic Oscillator in Biquaternionic Form.

The Effective Potential and the Vacuum

For a translation-invariant classical field, $\Gamma$ reduces to an effective potential, and the potential decides the vacuum.

The effective potential. For a constant classical field $\tilde\phi$ the effective action per unit spacetime volume defines $$ V_{\mathrm{eff}}(\tilde\phi) = -\frac{\Gamma[\tilde\phi]}{\text{vol}}, \qquad \frac{\delta V_{\mathrm{eff}}}{\delta\tilde\phi} = 0 \;\Longrightarrow\; \text{vacuum}, \qquad \frac{\delta^2 V_{\mathrm{eff}}}{\delta\tilde\phi^2} \ge 0 \;\Longrightarrow\; \text{stability}. $$ The minimizer is the vacuum expectation value of the field, and the curvature at the minimum is the mass of the fluctuation. In the framework the derivatives are taken with respect to the module-valued $\tilde\phi$, so the vacuum condition is the vanishing of a module-valued gradient and the stability condition is the positive-semidefiniteness of the Hessian in the real form.

Tree-level vacuum. At tree level $V_{\mathrm{eff}}=V$, and for the biquaternion scalar $$ V(\tilde\phi) = \tfrac12 m^2\,\mathrm{Re}\,\mathrm{Tr}\big(\tilde\phi^{*}\tilde\phi\big) = \tfrac12 m^2\,\big\langle \tilde\phi,\tilde\phi\big\rangle , $$ whose minimum is $\tilde\phi=0$ for $m^2>0$, the symmetric vacuum. A one-mode truncation of this field is exactly the two-level structure of The Biquaternion Vacuum as a Minimal Idempotent: the constant mode's energy is quadratic in the norm, the number operator $\tilde N$ is a projector, and the ground state is a minimal idempotent of $\mathbb{M}_+$ selected by the sign of the quadratic term. The effective potential's minimum is the field-theoretic statement of the vacuum-selection rule that the earlier article states algebraically.

The one-loop potential and convexity. The one-loop correction $$ V_{\mathrm{eff}}(\tilde\phi) = V(\tilde\phi) + \frac{\hbar}{2}\int\frac{d^4k}{(2\pi)^4}\log\big(k^2+V''(\tilde\phi)\big) + \cdots $$ is the standard Coleman–Weinberg form, and it is manifestly convex in the region where $V''>0$; where the tree potential is non-convex the one-loop potential is amended by the Maxwell construction, which is exactly the statement that the Legendre transform is the convex envelope of $W$. The framework's contribution is again the real form with respect to which the convexity is taken, and the sectorwise sign. The standard results are transcribed and cited.

Finite-dimensional convexity check. For the one-dimensional model $V(\phi)=\tfrac12 m^2\phi^2+\tfrac{\lambda}{4}\phi^4$ with $m^2=1.3$, $\lambda=0.7$, the curvature $V''(\phi)=m^2+3\lambda\phi^2$ takes the values $1.3$, $1.825$, $4.324$ at $\phi=0,0.5,1.2$: positive at all three points, so the tree effective potential is convex on that range and the Legendre transform is unmodified there.

The One-Loop Term and the Determinant

The one-loop effective action is the entry point of the determinant, and this section states the reduction that the next article completes.

The quadratic expansion. Expand the action about the classical field, $$ S[\tilde\phi+\tilde\eta] = S[\tilde\phi] + \tfrac12\big\langle \tilde\eta, S''[\tilde\phi]\,\tilde\eta\big\rangle + O(\tilde\eta^3) , $$ where the linear term vanishes by the classical equation of motion, $\delta S/\delta\tilde\phi=0$. The Gaussian integral over the fluctuation $\tilde\eta$ gives $$ W[\tilde J] = S[\tilde\phi] + \tfrac12\big\langle \tilde J,\tilde\phi\big\rangle - \frac{i\hbar}{2}\,\mathrm{Tr}\log S''[\tilde\phi] + O(\hbar^2) , $$ so that $$ \Gamma[\tilde\phi] = S[\tilde\phi] - \frac{i\hbar}{2}\,\mathrm{Tr}\log S''[\tilde\phi] + O(\hbar^2) \;\;\xrightarrow[\text{Euclidean}]{}\;\; S_E[\tilde\phi] + \frac{\hbar}{2}\,\mathrm{Tr}\log S_E''[\tilde\phi] + O(\hbar^2), $$ the last form under the Wick rotation that turns $iS$ into $-S_E$. The logarithm of the fluctuation operator's determinant is thus the whole of the one-loop effective action, and its regularization is the source of the trace anomaly of The Trace Anomaly in Biquaternionic Form.

Biquaternion content. Two features are the algebra's. First, $S''[\tilde\phi]$ acts on the module, so its determinant is a determinant on a two-complex-dimensional fibre per mode, and the trace in $\mathrm{Tr}\log$ is the trace pairing $\mathrm{Tr}(\tilde{Q})=2\mathrm{Sc}(\tilde{Q})$. Second, when $S''$ is central the logarithm factorizes over the sectors, giving the factorized one-loop expressions above. These are the only two places the algebra enters; the rest — the loop integral, the regularization, the renormalization — is standard.

The Proper-Time Form

The determinant is often traded for an integral over a proper time $s$, by $$ \frac12\,\mathrm{Tr}\log S''[\tilde\phi] = -\frac12\int_0^\infty\frac{ds}{s}\;\mathrm{Tr}\,e^{-s\,S''[\tilde\phi]}, $$ which is exact when the integral converges and is defined by regularization otherwise. In the biquaternion framework $S''[\tilde\phi]$ is an operator on the module and the trace is the pairing, so the integrand is a one-loop heat kernel on the module, $$ K(s) = \mathrm{Tr}\,e^{-s\,S''[\tilde\phi]} , $$ whose small-$s$ expansion $$ K(s) \sim \frac{1}{(4\pi s)^{d/2}}\sum_{n\ge0} a_n\,s^n $$ has coefficients $a_n$ built from the curvature of $S''$ and the geometry of the module. The coefficient $a_{d/2}$ controls the ultraviolet divergence of the one-loop effective action, and it is the object whose trace-part is anomalous in The Trace Anomaly in Biquaternionic Form. The framework's contribution is that the heat kernel is taken on the module and that the small-$s$ coefficients carry the module's dimension; the heat-kernel method itself is standard (Seeley, DeWitt, Gilkey) and is cited rather than rebuilt. The proper-time form is also the reason the one-loop effective action is the natural home of the trace anomaly: the divergence and its regularization are visible in the $s\to0$ end of the integral.

What Is Established and What Is Interpretation

Established (algebra and functional analysis of the module). - The connected and effective functionals are the quadratic forms $W=\tfrac12\langle\tilde J,\tilde K^{-1}\tilde J\rangle$ and $\Gamma=\tfrac12\langle\tilde\phi,\tilde K\tilde\phi\rangle$ for a free theory, with the inverse-Hessian relation and the Legendre identity; verified in finite dimension. - The Legendre transform is taken on the module, and its convexity is convexity in the real form $\langle\tilde{Q},\tilde Y\rangle=\mathrm{Re}\,\mathrm{Tr}(\tilde{Q}^{*}\tilde Y)$, which is positive definite on $\mathbb{M}_+$ and negative definite on $\mathbb{M}_-$. - For a sector-diagonal action the effective action factorizes, $\Gamma=\Gamma_-+\Gamma_+$, at tree level and at one loop, with the determinant factorizing. - The one-loop effective action is $\Gamma_1=\tfrac12\mathrm{Tr}\log S''[\tilde\phi]$, the logarithm of a module determinant.

Standard, and transcribed. - The loop expansion, the 1PI interpretation, the Coleman–Weinberg potential, the Maxwell construction as convexification, and the free-energy Legendre structure.

Interpretation. - Reading the tree-level vacuum minimum as the field-theoretic counterpart of the minimal-idempotent vacuum is the interpretive link to The Biquaternion Vacuum as a Minimal Idempotent; the minimum itself is a computed fact of the quadratic potential.

Open. - Whether a non-central one-loop operator produces a framework-specific anomaly or index statement is the question of The Trace Anomaly in Biquaternionic Form and The Theta Vacuum in Biquaternionic Form; this article only identifies where it would enter ($S''[\tilde\phi]$). - The equal sector contributions to the one-loop effective action are a component count; whether a multiplicity is observable depends on the normalization conventions of the physical fields, which the spin-0 articles fix.

Summary

The connected functional and the effective action of a biquaternion field are functionals on the module, built from the real form $\langle\tilde{Q},\tilde Y\rangle=\mathrm{Re}\,\mathrm{Tr}(\tilde{Q}^{*}\tilde Y)$. For a free theory they are the quadratics $W[\tilde J]=\tfrac12\langle\tilde J,\tilde K^{-1}\tilde J\rangle$ and $\Gamma[\tilde\phi]=\tfrac12\langle\tilde\phi,\tilde K\tilde\phi\rangle$, with $\tilde K=\Box-m^2$, the inverse-Hessian relation $\Gamma^{(2)}W^{(2)}=I$, and the Legendre identity $W+\Gamma=\langle\tilde J,\tilde\phi\rangle$ at $\tilde\phi=\delta W/\delta\tilde J$ — all verified in finite dimension. The one-loop effective action is the logarithm of the module determinant of the second variation, $\Gamma_1=\tfrac12\mathrm{Tr}\log S''[\tilde\phi]$, which the next article computes and regularizes.

Because a central kinetic operator does not mix the material and informational sectors, the effective action factorizes, $\Gamma=\Gamma_-+\Gamma_+$, at tree level and at one loop, with the determinant factorizing with it; any non-central term — an interaction, a non-central kinetic operator, or the chirality-off-diagonal Dirac mass — breaks the factorization. The tree-level effective potential $V=\tfrac12 m^2\langle\tilde\phi,\tilde\phi\rangle$ has its minimum at the symmetric point and, in a one-mode truncation, at a minimal idempotent of $\mathbb{M}_+$: the field-theoretic statement of the vacuum-selection rule of The Biquaternion Vacuum as a Minimal Idempotent.

Summary of Notation

Further Reading

• J. Schwinger, "On the Green's functions of quantized fields. I, II," Proceedings of the National Academy of Sciences 37 (1951) 452–455, 455–459, for the effective action as the generator of 1PI functions.
• G. Jona-Lasinio, "Relativistic field theories with symmetry-breaking solutions," Il Nuovo Cimento 34 (1964) 1790–1795, for the effective potential and symmetry breaking.
• S. Coleman and E. Weinberg, "Radiative corrections as the origin of spontaneous symmetry breaking," Physical Review D 7 (1973) 1888–1910, for the one-loop effective potential.
• R. Jackiw, "Functional evaluation of the effective potential," Physical Review D 9 (1974) 1686–1701, for the functional derivation and the convexification.
• K. Symanzik, "Renormalizable models with broken symmetries," in Renormalization of Yang–Mills Fields and Applications to Particle Physics (CERN, 1972), for the effective action and the convexity of the Legendre transform.
• S. Weinberg, The Quantum Theory of Fields, Vol. 2 (Cambridge, 1996), for the effective action, the loop expansion, and the background-field method.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, 1995), for the generating functionals and the effective potential.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford, 2002), for the Legendre structure, convexity, and the loop expansion in the Euclidean formulation.
• E. S. Fradkin and A. A. Tseytlin, "Conformal supergravity," Physics Reports 119 (1985) 233–362, for the background-field effective action and its determinant.
• B. S. DeWitt, Dynamical Theory of Groups and Fields (Gordon and Breach, 1965), for the proper-time representation and the heat-kernel coefficients.
• P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem (Publish or Perish, 1984), for the Seeley–DeWitt coefficients and their geometric content.
• J. S. Schwinger, "On gauge invariance and vacuum polarization," Physical Review 82 (1951) 664–679, for the proper-time representation of the one-loop determinant.
• P. Lounesto, Clifford Algebras and Spinors (Cambridge, 2001), for the module on which the fluctuation operator and its determinant are taken.
• Companion articles: The Functional Integral in Biquaternionic Form, for $Z[\tilde J]$, the real form, and the sector factorization; The Functional Determinant in Biquaternionic Form, for the computation and regularization of $\mathrm{Tr}\log S''$; The Trace Anomaly in Biquaternionic Form, for the anomalous part of the one-loop effective action; The Biquaternion Vacuum as a Minimal Idempotent, for the vacuum as a minimal idempotent and the one-mode two-level structure; The GNS Construction in the Biquaternion Framework, for the state space on which the vacuum expectation is taken; The Feynman Propagator in Biquaternionic Form and The S-Matrix in Biquaternionic Form, for the propagator and the trace conventions.