Second Variation, Hessians, and Jacobi Operators

At a stationary field configuration, the first variation vanishes on every admissible direction. The next term is the Hessian,

$$H_{\bar\phi}(\xi,\eta) =D^2S_{\bar\phi}[\xi,\eta],$$

a symmetric bilinear form on the tangent space of admissible fields. After an integration by parts, the same form is represented in the bulk by the Jacobi differential expression $\mathcal J_{\bar\phi}=D\mathcal E|_{\bar\phi}$, together with a boundary form. Its admissible kernel contains the linearized fluctuations that cost no action at quadratic order. Negative directions rule out a local minimum in a Euclidean or energy variational problem, while a zero mode makes the second-order test inconclusive unless it comes from a symmetry or a known family of solutions.

Those conclusions depend on the variation space, boundary conditions, normalizability, and—in spectral language—the operator domain. In particular, the indefinite Lorentzian action does not diagnose dynamical instability by the sign of its spacetime Hessian alone. This page derives the local machinery and its scalar-field check. Regulated determinants, integration contours, collective coordinates, and loop counting are left to the physical semiclassical treatment.

Required background. Field Variations and Boundary Terms supplies the bulk–boundary first-variation identity and admissible-variation language used below.

Hessian setting and admissible variations

We retain its fixed oriented coordinate domain $U\subset\mathbb R^d$, directed surface element $\mathrm d\Sigma_\mu$, and definitions

$$\begin{aligned} \pi_a^\mu &=\frac{\partial\mathcal L} {\partial(\partial_\mu\phi^a)},\\ \mathcal E_a(\mathcal L) &=\frac{\partial\mathcal L}{\partial\phi^a} -\partial_\mu\pi_a^\mu. \end{aligned}$$

For a classical pointwise derivation, take a $C^3$ first-order Lagrangian density, a background $\bar\phi\in C^2(\overline U)$, and $C^2$ variations. We work in an affine configuration space cut out by linear essential boundary conditions. Genuinely nonlinear constraints on the allowed boundary values require a constrained second variation and lie outside this treatment. Natural conditions generated by a boundary action are handled separately below. We also require $\bar\phi$ to be stationary for the full variational problem. A solution of the bulk Euler–Lagrange equations with a surviving boundary variation is not yet a saddle in that space.

We first use real fields and real variations, so the Hessian is real bilinear. Its complexification is complex bilinear; the Hilbert-space inner product used to normalize and compare complex modes is instead sesquilinear. We do not assume that a displayed differential expression already has a self-adjoint realization or a discrete spectrum.

From first variation to the Hessian

Choose two fixed admissible directions $\xi^a$ and $\eta^a$. The mixed second variation is

$$D^2S_{\bar\phi}[\xi,\eta] \equiv \left. \frac{\partial^2}{\partial s\,\partial t} S[\bar\phi+s\xi+t\eta] \right|_{s=t=0}.$$

Continuous second derivatives allow the $s$ and $t$ derivatives to be interchanged, so this Hessian is symmetric. It has no factor of $1/2$. That factor enters the Taylor expansion:

$$\begin{aligned} S[\bar\phi+\epsilon\eta] ={}&S[\bar\phi] +\epsilon\,DS_{\bar\phi}[\eta]\\ &+\frac{\epsilon^2}{2} D^2S_{\bar\phi}[\eta,\eta] +o(\epsilon^2). \end{aligned}$$

At a stationary background the linear term vanishes for every admissible $\eta$, and the quadratic fluctuation action is

$$S^{(2)}[\eta] \equiv\frac12D^2S_{\bar\phi}[\eta,\eta].$$

To calculate the Hessian, evaluate the following coefficient fields at $(x,\bar\phi,\partial\bar\phi)$:

$$\begin{aligned} A_{ab} &\equiv \frac{\partial^2\mathcal L} {\partial\phi^a\partial\phi^b},\\ B_{ab}^{\mu} &\equiv \frac{\partial^2\mathcal L} {\partial\phi^a\partial(\partial_\mu\phi^b)},\\ C_{ab}^{\mu\nu} &\equiv \frac{\partial^2\mathcal L} {\partial(\partial_\mu\phi^a) \partial(\partial_\nu\phi^b)}. \end{aligned}$$

Two applications of the chain rule give the unintegrated second variation:

$$\begin{aligned} D^2S_{\bar\phi}[\xi,\eta] =\int_U\mathrm d^d x\,\bigl[ &A_{ab}\xi^a\eta^b +B_{ab}^{\nu}\xi^a\partial_\nu\eta^b\\ &+B_{ba}^{\mu}(\partial_\mu\xi^a)\eta^b\\ &+C_{ab}^{\mu\nu} (\partial_\mu\xi^a)(\partial_\nu\eta^b) \bigr]. \end{aligned}$$

The equalities $A_{ab}=A_{ba}$ and $C_{ab}^{\mu\nu}=C_{ba}^{\nu\mu}$ make the symmetry under $\xi\leftrightarrow\eta$ explicit. This is the field-theory version of the ordinary Hessian matrix. Cristoferi’s Cristoferi 2016, §§ 5.1 and 8.4–8.5, PDF give the corresponding variational formula and identify the Jacobi expression as the linearized Euler–Lagrange expression.

The Jacobi expression and its boundary form

Define the momentum linearized along $\eta$ by

$$\mathcal P_a^\mu[\eta] \equiv D\pi_a^\mu|_{\bar\phi}[\eta] =B_{ba}^{\mu}\eta^b +C_{ab}^{\mu\nu}\partial_\nu\eta^b.$$

Linearizing the Euler–Lagrange expression defines our sign convention for the Jacobi expression:

$$(\mathcal J_{\bar\phi}\eta)_a \equiv D\mathcal E_a|_{\bar\phi}[\eta].$$

Substitution yields

$$\boxed{ \begin{aligned} (\mathcal J_{\bar\phi}\eta)_a ={}&A_{ab}\eta^b +B_{ab}^{\nu}\partial_\nu\eta^b\\ &-\partial_\mu\!\left( B_{ba}^{\mu}\eta^b +C_{ab}^{\mu\nu}\partial_\nu\eta^b \right). \end{aligned} }$$

Integrating the derivatives of $\xi$ by parts now gives the complete bulk–boundary split:

$$\boxed{ \begin{aligned} D^2S_{\bar\phi}[\xi,\eta] ={}&\int_U\mathrm d^d x\, \xi^a(\mathcal J_{\bar\phi}\eta)_a\\ &+\int_{\partial U}\mathrm d\Sigma_\mu\, \xi^a\mathcal P_a^\mu[\eta]. \end{aligned} }$$

The boundary term is part of the Hessian, not an optional correction. For fixed boundary values it vanishes because both variations have zero boundary trace. Periodic conditions make opposite faces cancel. When the boundary trace is free, however, arbitrary directions in the Hessian’s form domain are not required to obey the linearized natural equation.

For example, if the action includes the derivative-free boundary density $b(x,\phi)$ from the preceding page, define

$$S_{\mathrm{tot}} =S+\int_{\partial U}\mathrm d\sigma\,b(x,\phi).$$

Its Hessian adds

$$\int_{\partial U}\mathrm d\sigma\, b_{,ab}\xi^a\eta^b,$$

so the integrated form is

$$\begin{aligned} D^2S_{\mathrm{tot},\bar\phi}[\xi,\eta] ={}&\int_U\mathrm d^d x\, \xi^a(\mathcal J_{\bar\phi}\eta)_a\\ &+\int_{\partial U}\mathrm d\sigma\, \xi^a\left( s_\mu\mathcal P_a^\mu[\eta] +b_{,ab}\eta^b \right). \end{aligned}$$

The stationary background obeys the natural condition $s_\mu\pi_a^\mu+b_{,a}=0$ on a smooth free face. Its linearization is

$$s_\mu\mathcal P_a^\mu[\eta] +b_{,ab}\eta^b=0.$$

Here $\mathrm d\Sigma_\mu=s_\mu\mathrm d\sigma$. Together with the required regularity and essential data, this linearized condition places $\eta$ in the Jacobi operator domain and removes the last line of the integrated form; it is not an extra restriction on every direction used in the second-variation minimum test. Thus a boundary action can leave the bulk Jacobi expression unchanged while changing both the form’s boundary term and the operator realization.

The symmetric $b_{,ab}$ contribution cancels upon antisymmetrization. Subtracting the two orders of the Hessian therefore gives the Green identity

$$\begin{aligned} \int_U\mathrm d^d x\, \bigl[ \xi^a(\mathcal J\eta)_a -\eta^a(\mathcal J\xi)_a \bigr] =\int_{\partial U}\mathrm d\Sigma_\mu\, \bigl[ \eta^a\mathcal P_a^\mu[\xi] -\xi^a\mathcal P_a^\mu[\eta] \bigr]. \end{aligned}$$

The identity shows that the local differential expression agrees with its formal adjoint. When the right-hand side vanishes for all pairs in a proposed dense operator domain, the resulting realization is symmetric there. That does not yet make it self-adjoint: one must prove equality with the adjoint domain. The spectrum and kernel can change when the boundary conditions change even though the local differential formula does not. Teschl’s Teschl 2014, §§ 2.2 and 9.1–9.2 exhibit this distinction and derive the boundary form for second-order operators.

Jacobi fields and zero modes

A Jacobi field is a solution of the linearized Euler–Lagrange equation

$$\mathcal J_{\bar\phi}\eta=0.$$

For it to be an admissible fluctuation, it must also obey the linearized boundary conditions. The terminology is justified by a direct construction. If $\bar\phi(\lambda)$ is a differentiable one-parameter family of exact stationary configurations satisfying the same boundary data, then

$$\eta =\left. \frac{\partial\bar\phi(\lambda)}{\partial\lambda} \right|_{\lambda=\lambda_0}$$

obeys

$$\mathcal J_{\bar\phi}\eta =\left. \frac{\partial}{\partial\lambda} \mathcal E(\bar\phi(\lambda)) \right|_{\lambda_0} =0.$$

Differentiating the common boundary condition proves admissibility as well. The converse need not hold: a solution of the linearized equation may be obstructed at higher order and fail to integrate to a family of exact solutions.

A zero mode is more restrictive. On this page, the term means a nonzero, admissible, normalizable element of the kernel of a chosen operator realization. A local solution of $\mathcal J\eta=0$ that violates the boundary conditions or is not normalizable is not a zero mode of that spectral problem. Nor is a vector with merely $H(\eta,\eta)=0$ necessarily in the kernel: for $M=\operatorname{diag}(1,-1)$ and $v=(1,1)^{\mathsf T}$, one has $v^{\mathsf T}Mv=0$ but $Mv=(1,-1)^{\mathsf T}\ne0$.

Boundary conditions provide the simplest diagnostic. On $[0,L]$, the differential expression $-\mathrm d^2/\mathrm d x^2$ has a constant zero mode with Neumann or periodic data. With Dirichlet data its eigenvalues are $(n\pi/L)^2$, $n=1,2,\ldots$, so the constant function is excluded and the kernel is trivial.

Symmetry families often generate zero modes, but only when the symmetry preserves the action and the boundary domain. Translating a localized solution in infinite space may produce $\partial_\mu\bar\phi$; a fixed finite boundary can remove that direction. Devoto and collaborators’ Devoto et al. 2022, § 3.5.1 gives both normalizable translation modes and a non-normalizable scale-mode counterexample. Some physics sources still call the latter a zero mode of the formal fluctuation equation; under the operator convention used here, it is a non-normalizable Jacobi solution rather than a spectral zero mode.

Gauge directions require a separate interpretation. For the unreduced Maxwell Hessian,

$$\mathcal J^{\nu\rho} =\eta^{\nu\rho}\Box -\partial^\nu\partial^\rho, \qquad \mathcal J^{\nu\rho}\partial_\rho\lambda=0.$$

If $\lambda$ preserves the boundary data and belongs to the subgroup declared to be proper gauge, this kernel direction is redundancy rather than a physical flat fluctuation. Boundary-preserving transformations can instead act nontrivially on boundary degrees of freedom, so classifying them requires more than the local Hessian. Gauge fixing, quotienting, and the associated determinants lie beyond this page.

What the Hessian says about stability

For a finite-dimensional critical point, a negative Hessian direction rules out a local minimum, while a positive-definite Hessian gives a strict local minimum. The infinite-dimensional statement needs a topology and a uniform estimate. In a Euclidean or energy problem, a local minimum must satisfy

$$D^2S_{\bar\phi}[\eta,\eta]\ge0$$

for every admissible $\eta$. If $X$ is the chosen normed space of admissible variations, a bound of the form

$$D^2S_{\bar\phi}[\eta,\eta] \ge c\lVert\eta\rVert_X^2, \qquad c>0,$$

together with control of the Taylor remainder in the same norm, supplies the coercivity needed for a strict local result. Positivity on each nonzero direction without a uniform lower bound is generally weaker. Cristoferi’s Cristoferi 2016, §§ 5.1 and 8.1, PDF separate the necessary nonnegativity condition from a coercive sufficient condition.

If the Hessian is only semidefinite, second order is inconclusive along its kernel. The elementary functions $f_+(x)=x^4$ and $f_-(x)=-x^4$ are both stationary at the origin and have the same zero Hessian there, yet one has a minimum and the other a maximum. Higher orders, an exact symmetry quotient, or a moduli-space description must decide what happens along a zero mode.

The Lorentzian spacetime action is a different object. Its kinetic term makes its Hessian indefinite even for the stable free scalar. One should instead study the linearized evolution, a conserved energy when available, or a well-defined Euclidean variational problem. Calling every negative direction of a Lorentzian spacetime Hessian an instability is therefore incorrect.

Quadratic fluctuation operators around a saddle

Use the site’s $(+,-,\ldots,-)$ Lorentzian convention and the scalar density

$$\mathcal L =\frac12\partial_\mu\phi\,\partial^\mu\phi -V(\phi).$$

Let $\bar\phi$ solve

$$\Box\bar\phi+V'(\bar\phi)=0$$

and satisfy the boundary conditions that make the full first variation vanish. Expanding $\phi=\bar\phi+\epsilon\eta$ gives

$$D^2S_{\bar\phi}[\xi,\eta] =\int_U\mathrm d^d x\, \left[ \partial_\mu\xi\,\partial^\mu\eta -V''(\bar\phi)\xi\eta \right].$$

One integration by parts identifies

$$\begin{aligned} D^2S_{\bar\phi}[\xi,\eta] ={}&\int_U\mathrm d^d x\, \xi\bigl[-\Box-V''(\bar\phi)\bigr]\eta\\ &+\int_{\partial U}\mathrm d\Sigma_\mu\, \xi\partial^\mu\eta. \end{aligned}$$

Thus, with the sign convention fixed above,

$$\boxed{ \mathcal J_L=-\Box-V''(\bar\phi) }$$

and the Jacobi equation is equivalently

$$\bigl[\Box+V''(\bar\phi)\bigr]\eta=0.$$

For a homogeneous saddle $\bar\phi=v$, set $M^2=V''(v)$. The site’s Fourier rule $\partial_\mu\mapsto-ip_\mu$ gives

$$\mathcal J_L(p)=p^2-M^2.$$

The Jacobi equation therefore reproduces the mass shell $p^2=M^2$. This is an independent sign check against the scalar propagator denominator. The causal $+i0$ prescription is additional state and contour data; it is not determined by the classical Hessian.

For a static background, introduce the spatial fluctuation operator

$$K=-\boldsymbol\nabla^2+V''(\bar\phi).$$

Then

$$\mathcal J_L=-(\partial_t^2+K), \qquad \ddot\eta+K\eta=0.$$

After boundary conditions make $K$ self-adjoint, an eigenmode $Ku_n=\lambda_nu_n$ has time-dependent amplitude

$$\ddot q_n+\lambda_nq_n=0.$$

Positive $\lambda_n$ gives oscillation, $\lambda_n=0$ gives $q_n=a_n+b_nt$, and $\lambda_n=-\alpha_n^2<0$ gives exponential growth or decay. This is a dynamical statement obtained from the evolution equation, not from the signature of the Lorentzian spacetime action.

For a Euclidean variational problem, start instead from

$$S_E[\phi] =\int\mathrm d^d x_E\, \left[ \frac12(\partial_E\phi)^2+V(\phi) \right],$$

and let $\bar\phi$ be stationary for this Euclidean action, including its boundary terms and data. The corresponding elliptic expression is

$$\boxed{ \mathcal J_E=-\partial_E^2+V''(\bar\phi). }$$

If a self-adjoint realization has a normalized eigenmode $\mathcal J_Eu=\lambda u$, then

$$S_E[\bar\phi+\epsilon u]-S_E[\bar\phi] =\frac{\epsilon^2}{2}\lambda +o(\epsilon^2).$$

Now $\lambda<0$ directly exhibits a descending direction of the Euclidean action. A zero eigenvalue marks a degenerate quadratic approximation, while a strict positive lower bound supports local minimality. Devoto and collaborators’ Devoto et al. 2022, §§ 2.2 and 2.3 show how this Euclidean fluctuation operator enters a semiclassical expansion and why zero and negative modes invalidate a naive Gaussian product. The required contour, regularization, and determinant treatment is deliberately deferred.

The dimensions close the calculation. For a canonically normalized scalar,

$$[\eta]=\frac{d-2}{2}, \qquad [V'']=2, \qquad [\mathcal J]=2.$$

Hence $[\xi\mathcal J\eta]=d$ in the bulk and $[\xi\partial^\mu\eta]=d-1$ on the boundary, as required for a dimensionless second variation.

Checks, limits, and common pitfalls

Hessian versus quadratic action. The Hessian is $D^2S$; the quadratic term in the Taylor expansion is $S^{(2)}=D^2S/2$. Moving the factor of $1/2$ between these definitions corrupts Gaussian normalizations later.

Formal symmetry versus self-adjointness. Integration by parts establishes a boundary identity. It does not prove that the operator domain equals the adjoint domain, nor that the spectrum is discrete or complete.

Jacobi solution versus zero mode. A local solution of the linearized field equation need not satisfy the boundary conditions or be normalizable. A zero mode is a kernel vector in the specified spectral problem.

Potential curvature versus an eigenvalue. Pointwise $V''(\bar\phi)<0$ does not by itself guarantee a negative mode on a finite domain: the positive gradient term and the allowed wavelengths also matter. For a homogeneous background the spatial eigenvalues are $\mathbf k^2+V''(v)$, with $\mathbf k$ fixed by the boundary conditions.

Gauge kernel versus physical modulus. A gauge direction makes the unreduced Hessian degenerate but does not describe a new physical configuration. Its removal requires a controlled quotient or gauge-fixing procedure.

Exercises

1. Locate the factor of one half. If $D^2S_{\bar\phi}[\eta,\eta]=6$, what is the order-$\epsilon^2$ contribution to $S[\bar\phi+\epsilon\eta]$ at a stationary point?

Solution

Taylor’s formula gives

$$S[\bar\phi+\epsilon\eta]-S[\bar\phi] =\frac{\epsilon^2}{2}\,6+o(\epsilon^2) =3\epsilon^2+o(\epsilon^2).$$

The Hessian itself is $6$; the quadratic action along this direction is $3$.

2. Let the domain decide the zero mode. Find the kernel of $-\mathrm d^2/\mathrm d x^2$ on $[0,L]$ with (a) Dirichlet data and (b) Neumann data.

Solution

A kernel function has $u(x)=a+bx$. Dirichlet conditions $u(0)=u(L)=0$ force $a=b=0$, so the kernel is trivial. Neumann conditions $u'(0)=u'(L)=0$ force only $b=0$; every constant function remains, giving a one-dimensional kernel. The differential expression is identical, but the operator domains and kernels differ.

3. Recover the mechanical Jacobi expression. For

$$L(q,\dot q)=\frac m2\dot q^2-V(q),$$

derive the bilinear second variation and integrate it by parts.

Solution

At a stationary path $\bar q$,

$$D^2S_{\bar q}[\xi,\eta] =\int_{t_i}^{t_f}\mathrm dt\, \left[m\dot\xi\dot\eta -V''(\bar q)\xi\eta\right].$$

Integrating the first term by parts gives

$$\begin{aligned} D^2S_{\bar q}[\xi,\eta] ={}&\int_{t_i}^{t_f}\mathrm dt\, \xi\left[-m\frac{\mathrm d^2}{\mathrm dt^2} -V''(\bar q)\right]\eta\\ &+\left[m\xi\dot\eta\right]_{t_i}^{t_f}. \end{aligned}$$

Thus $\mathcal J=-m\mathrm d^2/\mathrm dt^2-V''(\bar q)$ in the same Euler–Lagrange sign convention used on this page. Fixed endpoints remove the displayed boundary term.

4. Check the scalar Fourier sign. For a homogeneous saddle with $V''(v)=M^2$, apply $\mathcal J_L$ to $e^{-ip\cdot x}$. What condition follows from the Jacobi equation, and what does it not determine?

Solution

Since $\Box e^{-ip\cdot x}=-p^2e^{-ip\cdot x}$,

$$\mathcal J_Le^{-ip\cdot x} =(p^2-M^2)e^{-ip\cdot x}.$$

The Jacobi equation gives $p^2=M^2$, the correct mass shell in the mostly-minus convention. It does not choose a Feynman, retarded, advanced, or other causal prescription; those require state and contour data beyond the classical second variation.

Synthesis and continuation

The second variation diagnoses a stationary configuration through three linked objects: a symmetric bilinear Hessian, its bulk Jacobi expression, and the boundary form that fixes the admissible domain. Negative directions rule out Euclidean or energy minima; admissible kernel vectors are zero modes; and coercivity, not pointwise positivity alone, controls strict local minimality. Lorentzian stability instead comes from the linearized evolution or an appropriate energy problem.

For the regulated QFT use of the quadratic fluctuation operator—including Gaussian integration, zero-mode treatment, determinant phases, and contour choices—continue to Saddles and the Semiclassical Expansion.

References

• Riccardo Cristoferi, Calculus of Variations: Lecture Notes — Open PDF, Carnegie Mellon University, 2016, §§ 5.1 and 8.1–8.5, especially pp. 57, 77–79, and 84–86. These notes supply the teaching route from second variation to the Jacobi linearization and distinguish necessary nonnegativity from a coercive sufficient condition. Their multidimensional spectral sketch is used only for orientation, not as operator-domain authority.
• Federica Devoto, Simone Devoto, Luca Di Luzio, and Giovanni Ridolfi, “False Vacuum Decay: An Introductory Review”, Journal of Physics G 49 (2022) 103001, §§ 2.2, 2.3, and 3.5.1, especially pp. 9–10, 23–24, and 39–41. This open-access QFT source supplies the Euclidean fluctuation-operator, negative-mode, symmetry-zero-mode, and normalizability checks. Its determinant and contour analysis belongs to the downstream semiclassical treatment rather than to the local derivation here.
• Gerald Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, second edition, American Mathematical Society, 2014, doi:10.1090/gsm/157, §§ 2.2 and 9.1–9.2, especially pp. 58–63 and 182–188. This source supports the distinction among a formal differential expression, a symmetric operator, and a self-adjoint realization with boundary conditions.