Variations and Tangent Vectors on Field Space

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A classical field is not one point in spacetime but an assignment over all of spacetime. Its configuration space is therefore generally infinite-dimensional. A variation is a tangent direction in that space: it tells us how an entire field changes inside a smooth one-parameter family. This geometric viewpoint separates three operations that are easy to confuse—spacetime differentiation, differentiation with respect to a family parameter, and exterior differentiation on field space—and supplies the language needed for variational equations and covariant phase space.

Required background. Exterior Algebra and Differential Forms supplies differential forms, while the Exterior Derivative and Poincaré Lemma supplies the base-manifold exterior derivative used for comparison with field-space differentiation. Tensor Fields and Natural Operations supplies smooth tensor fields, pullbacks, and local components.

Helpful background. Calculus and Linear Algebra supplies directional derivatives and dual spaces; Variational Mechanics and Symmetries supplies the finite-dimensional model of a varied trajectory.

Field configurations and their tangent directions

Fix a spacetime manifold $M$ and a bundle $E\to M$ . A configuration is a sufficiently regular section $\phi$ of $E$ satisfying whatever boundary or falloff conditions define the problem. Denote the resulting space by $\mathcal F$ . A tangent vector $\delta\phi\in T_\phi\mathcal F$ is represented by a curve $\phi_s\in\mathcal F$ with $\phi_0=\phi$ :

\delta\phi=\left.\frac{d\phi_s}{ds}\right|_{s=0}.

For a vector bundle, $\delta\phi$ is again a section of the same bundle. For nonlinear target spaces, it is instead a section of the pulled-back vertical tangent bundle along $\phi$ . The admissible tangent vectors must obey the linearization of the configuration-space conditions. Thus a fixed boundary value implies $\delta\phi|_{\partial M}=0$ , whereas allowing the boundary value to vary gives a larger tangent space.

A functional $F:\mathcal F\to\mathbb R$ has differential $\boldsymbol\delta F$ , a field-space one-form defined by

(\boldsymbol\delta F)_\phi[\delta\phi] =\left.\frac{d}{ds}F[\phi_s]\right|_{s=0}.

Bold $\boldsymbol\delta$ will denote the field-space exterior derivative; ordinary $d$ denotes the spacetime exterior derivative. This is a formal differential-geometric notation unless a topology, differentiability class, and functional-analytic model for $\mathcal F$ have been specified. The formal tangent and cotangent constructions used in covariant phase space, together with the hypotheses needed to make them precise, are organized in Khavkine 2014, §§3.2–3.3.

Spacetime d versus field-space δ

For a scalar field $\phi_s(x)$ on a fixed manifold, smoothness of the two-variable map gives

\boldsymbol\delta(d\phi)=d(\boldsymbol\delta\phi).

In coordinates this is simply $\partial_s\partial_\mu\phi_s=\partial_\mu\partial_s\phi_s$ . The equality extends to fixed background tensors and fixed differential operators. It is not automatic when the operation being varied itself depends on the field. For example, if the metric varies, then

\boldsymbol\delta(\star\alpha) =\star(\boldsymbol\delta\alpha)+(\boldsymbol\delta\star)\alpha,

and the second term cannot be discarded. Likewise, varying the pullback by a moving embedding produces a Lie-derivative contribution. Saying that $d$ and $\boldsymbol\delta$ commute therefore presupposes a fixed base manifold and a field-independent identification of nearby configurations.

The field-space exterior derivative satisfies $\boldsymbol\delta^2=0$ on differential forms on $\mathcal F$ , just as $d^2=0$ on spacetime forms. This does not say that any two chosen variation vector fields commute. For field-dependent variations $\delta_1$ and $\delta_2$ ,

(\boldsymbol\delta^2 F)(\delta_1,\delta_2) =\delta_1\delta_2F-\delta_2\delta_1F-[\delta_1,\delta_2]F=0.

Omitting the bracket term is justified only for commuting extensions of the tangent vectors. This qualification becomes important for field-dependent gauge parameters.

Local functionals and integration by parts

A local functional has the schematic form

F[\phi]=\int_M f(x,\phi,d\phi,\ldots),

where $f$ is a top-degree spacetime form. Its first variation is initially a linear expression in $\delta\phi$ and derivatives of $\delta\phi$ . Spacetime integration by parts transfers those derivatives to coefficients, separating a bulk linear functional from a boundary term. Field-space differentiation and spacetime integration play different roles: $\boldsymbol\delta$ measures displacement in $\mathcal F$ , while $d$ is what turns the local boundary contribution into an exact spacetime form.

The familiar finite-dimensional case is recovered by taking $M$ to be a time interval and $\phi=q(t)$ . Then $\mathcal F$ is a path space, $\delta q(t)$ is a tangent vector to that path space, and fixed endpoints select tangent vectors satisfying $\delta q(t_1)=\delta q(t_2)=0$ . The field-space language adds no new dynamics by itself; it makes the domain and the status of variations explicit.

Formal scope and failure modes

Several distinctions prevent overinterpretation. A tangent vector to the space of all configurations need not solve the linearized equations; it becomes tangent to the solution space only after that equation is imposed. A gauge variation may be a nonzero tangent vector on configuration space yet become a degeneracy after restriction and quotient. Distributional fields, low-regularity solutions, moving boundaries, and nonlocal functionals require additional analysis not supplied by the formal calculus.

The notation also does not choose boundary conditions. Those are part of the definition of $\mathcal F$ and determine which variations exist. The later decomposition of a varied Lagrangian uses this framework, but whether its boundary term vanishes is a separate question about the variational problem. Iyer and Wald formulate the spacetime-form and field-variation bookkeeping used in covariant Lagrangian theories in Iyer and Wald 1994, §3, pp. 6–10 of the preprint, PDF.

Exercise: commuting and field-dependent variations

Let $F[\phi]=\int_M \phi^2\varepsilon$ for a fixed volume form $\varepsilon$ . First take two field-independent variations $\delta_1\phi=f$ and $\delta_2\phi=g$ . Then let $\widehat\delta_1\phi=\phi$ and $\widehat\delta_2\phi=h\phi^2$ , where $f,g,h$ are fixed functions. Verify $\boldsymbol\delta^2F=0$ in both cases, including the variation bracket in the second.

Solution

For fixed $f$ and $g$ ,

\delta_1\delta_2F=2\int_M fg\,\varepsilon =\delta_2\delta_1F,

and $[\delta_1,\delta_2]\phi=0$ . For the field-dependent pair,

[\widehat\delta_1,\widehat\delta_2]\phi =\widehat\delta_1(h\phi^2)-\widehat\delta_2(\phi) =h\phi^2.

Direct differentiation gives

(\widehat\delta_1\widehat\delta_2-\widehat\delta_2\widehat\delta_1)F =2\int_M h\phi^3\varepsilon =[\widehat\delta_1,\widehat\delta_2]F.

Hence the three terms in the exterior-derivative formula cancel. Treating the variations as commuting would give the wrong conclusion.

References

Iyer, Vivek, and Robert M. Wald. “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy.” Physical Review D 50 (1994): 846–864. DOI.