Hamiltonian Group Actions and Moment Maps

A moment map packages all infinitesimal symmetry generators into one Lie-algebra-dual-valued function. If a Lie group $G$ acts on a symplectic manifold $(M,\omega)$, a moment map

$$\mu:M\longrightarrow\mathfrak g^*$$

assigns to every $\xi\in\mathfrak g$ a component $\mu_\xi=\langle\mu,\xi\rangle$. The defining equation says that $\mu_\xi$ generates the infinitesimal transformation associated with $\xi$. Equivariance says that these generators reproduce the Lie bracket under the Poisson bracket. Invariance of the Hamiltonian then makes them conserved.

Those are three distinct statements. A symplectic action can fail to admit global generators; global generators can fail to form the intended Lie algebra; and a valid generator need not be conserved by a Hamiltonian that breaks the symmetry. This page separates the three issues, fixes their signs, and ends with one classical scalar-field example. Constraint reduction, quantum charge operators, Ward identities, anomalies, and boundary charges require additional input and are not inferred here.

Required background. Symplectic Forms, Hamiltonian Flows, and Poisson Brackets supplies symplectic contraction, Hamiltonian vector fields, Poisson brackets, and the local-versus-global distinction for symplectic vector fields; Lie Groups, Lie Algebras, and Exponential and Adjoint Maps supplies one-parameter subgroups, Lie brackets, and the adjoint action needed to turn a family of Hamiltonians into a $\mathfrak g^*$-valued map.

Hamiltonian group-action setting and sign convention

We first work with finite-dimensional smooth manifolds. The inherited symplectic convention is

$$\boxed{ \omega=\sum_i\mathrm dq^i\wedge\mathrm dp_i, \qquad \iota_{X_f}\omega=\mathrm df, \qquad \{f,g\}=\omega(X_f,X_g). }$$

It implies

$$X_f[u]=-\{f,u\}=\{u,f\}, \qquad [X_f,X_g]=-X_{\{f,g\}}, \qquad \dot f=\partial_t f+\{f,H\}.$$

Let $G$ act on $M$ from the left, and define the direct fundamental field

$$\xi_M(x) = \left.\frac{\mathrm d}{\mathrm dt}\right|_{t=0} \exp(t\xi)\mathbin{\cdot}x.$$

With this $+t$ definition, a left action gives an anti-homomorphism:

$$[\xi_M,\eta_M]=-[\xi,\eta]_M.$$

This sign is not an anomaly. For the left action of $G$ on itself, the direct fundamental fields are right-invariant, and right-invariant vector fields have precisely this bracket sign.

Some references instead differentiate $\exp(-t\xi)$, define $\iota_{X_f}\omega=-\mathrm df$, reverse $\omega$, or reverse the Poisson bracket. Each convention can be consistent. A translation between sources must move the entire package together.

Symplectic actions and infinitesimal generators

Write $\Phi_g(x)=g\mathbin{\cdot}x$. The action is symplectic when

$$\Phi_g^*\omega=\omega \qquad\text{for every }g\in G.$$

Differentiating this identity along a one-parameter subgroup gives $\mathcal L_{\xi_M}\omega=0$. Cartan’s formula and $\mathrm d\omega=0$ then yield

$$\mathrm d(\iota_{\xi_M}\omega) = \mathcal L_{\xi_M}\omega -\iota_{\xi_M}\mathrm d\omega =0.$$

Thus $\iota_{\xi_M}\omega$ is closed. It is locally exact, so every infinitesimal symplectic symmetry is locally generated by a Hamiltonian function.

On this page, a moment map means a smooth map $\mu:M\to\mathfrak g^*$ satisfying the differential condition

$$\begin{aligned} \mu_\xi(x)&=\langle\mu(x),\xi\rangle,\\ \mathrm d\mu_\xi&=\iota_{\xi_M}\omega \qquad\text{for every }\xi\in\mathfrak g. \end{aligned}$$

This usage is sometimes called a weak moment map: equivariance is not yet part of the definition. Cannas da Silva 2006, §§ 22.1 and 22.4, pp. 133–138, PDF develops the same differential condition and its Hamiltonian generators. Nondegeneracy of $\omega$ immediately gives

$$X_{\mu_\xi}=\xi_M.$$

Consequently the direct infinitesimal change of an observable $F$ is

$$\delta_\xi F =\xi_M[F] =\{F,\mu_\xi\}.$$

This is the first sense in which $\mu_\xi$ is a generator.

There is a nearby sign that should not be conflated with this one. The induced left action on functions uses inverse pullback,

$$(g\mathbin{\cdot}F)(x)=F(g^{-1}\mathbin{\cdot}x).$$

Its infinitesimal operator is therefore

$$D_\xi F =-\xi_M[F] =\{\mu_\xi,F\}.$$

The direct motion of points and the representation on functions carry opposite signs because the latter contains $g^{-1}$.

Equivariance and the generator algebra

Define the coadjoint action explicitly by

$$\left\langle\operatorname{Ad}_g^*\nu,\xi\right\rangle = \left\langle\nu,\operatorname{Ad}_{g^{-1}}\xi\right\rangle.$$

A moment map is equivariant when

$$\mu(g\mathbin{\cdot}x)=\operatorname{Ad}_g^*\mu(x).$$

We call a symplectic action equipped with an equivariant moment map a Hamiltonian action. Terminology varies in the literature, so this distinction between a moment map and a Hamiltonian action is part of the page’s convention.

Pairing equivariance with $\xi$ gives

$$\mu_\xi(g\mathbin{\cdot}x) = \mu_{\operatorname{Ad}_{g^{-1}}\xi}(x).$$

Set $g=\exp(t\eta)$ and differentiate at $t=0$. Since

$$\left.\frac{\mathrm d}{\mathrm dt}\right|_{0} \operatorname{Ad}_{\exp(-t\eta)}\xi =-[\eta,\xi] =[\xi,\eta],$$

one obtains

$$\eta_M[\mu_\xi]=\mu_{[\xi,\eta]}.$$

But $X_{\mu_\eta}=\eta_M$, so the left-hand side is

$$X_{\mu_\eta}[\mu_\xi] =\{\mu_\xi,\mu_\eta\}.$$

Therefore an equivariant moment map obeys

$$\boxed{ \{\mu_\xi,\mu_\eta\}=\mu_{[\xi,\eta]}. }$$

This is the second sense in which the components are symmetry generators: their Poisson brackets reproduce the Lie algebra. There is a useful independent sign check. Both maps

$$f\longmapsto X_f, \qquad \xi\longmapsto\xi_M$$

are anti-homomorphisms with the present conventions. Their two minus signs cancel, leaving the positive generator algebra above.

For connected $G$, the Poisson-bracket relation is also sufficient for equivariance: the infinitesimal condition integrates throughout the identity component. If $G$ is disconnected, it does not test the other components; the finite equivariance equation must still be checked there. Meinrenken 2024, §§ 7.3–7.4, pp. 82–88, PDF gives the equivariance and component-bracket equivalence, with its sign conventions translated above.

Global existence, ambiguity, and equivariance defects

Symplecticity only made $\iota_{\xi_M}\omega$ closed. A global component $\mu_\xi$ exists exactly when

$$\left[\iota_{\xi_M}\omega\right]=0 \qquad\text{in }H^1_{\mathrm{dR}}(M).$$

If this holds for every $\xi$, one may choose Hamiltonians on a basis of $\mathfrak g$ and extend linearly to obtain a moment map. In particular, $H^1_{\mathrm{dR}}(M)=0$ is sufficient for all component Hamiltonians to exist. It says nothing by itself about equivariance.

Assume now that $M$ is connected. Two moment maps satisfying the differential condition differ by a constant element $\lambda\in\mathfrak g^*$:

$$\mu'=\mu+\lambda.$$

If both maps are equivariant, this constant must be fixed by the coadjoint action. For an abelian group the coadjoint action is trivial, so every constant shift remains allowed.

The failure of a chosen moment map to reproduce the Lie algebra is measured by

$$c(\xi,\eta) = \{\mu_\xi,\mu_\eta\} -\mu_{[\xi,\eta]}.$$

Its Hamiltonian vector field vanishes. Indeed,

$$\begin{aligned} X_{\{\mu_\xi,\mu_\eta\}} &=-[X_{\mu_\xi},X_{\mu_\eta}]\\ &=-[\xi_M,\eta_M]\\ &=[\xi,\eta]_M =X_{\mu_{[\xi,\eta]}}. \end{aligned}$$

Connectedness of $M$ therefore makes $c(\xi,\eta)$ a real constant. The Jacobi identity implies

$$\begin{aligned} 0={}&c([\xi,\eta],\zeta) +c([\eta,\zeta],\xi)\\ &+c([\zeta,\xi],\eta), \end{aligned}$$

so $c$ is a Lie-algebra two-cocycle. A constant shift changes it by

$$c'(\xi,\eta) =c(\xi,\eta)-\lambda([\xi,\eta]).$$

Thus the cohomology class of $c$, not a particular normalization of $\mu$, is the infinitesimal obstruction to an equivariant choice. This brief statement is all that is needed here; central extensions are not developed on this page.

Two examples keep the two obstructions separate.

Closed need not be exact. Let $q,p$ be angular coordinates on $T^2$, let $\omega=\mathrm dq\wedge\mathrm dp$, and let $S^1$ translate $q$. Its fundamental field is $\partial_q$, and

$$\iota_{\partial_q}\omega=\mathrm dp.$$

The one-form $\mathrm dp$ is closed but has nonzero integral around the $p$-circle, so it is not exact. This symplectic action has no real-valued global moment map.

Exact generators need not be equivariant. Let the additive group $\mathbb R^2$ translate phase space $(\mathbb R^2,\mathrm dq\wedge\mathrm dp)$. For $\xi=(u,v)$,

$$\xi_M=u\,\partial_q+v\,\partial_p, \qquad \mu_\xi=up-vq.$$

For $\eta=(u',v')$,

$$\{\mu_\xi,\mu_\eta\}=uv'-vu'.$$

The Lie algebra is abelian, so $\mu_{[\xi,\eta]}=0$. The nonzero constant is the cocycle $c$. Because a constant shift contributes $-\lambda([\xi,\eta])=0$, no shift removes it. Contractibility of the phase space removed the de Rham obstruction but did not remove the equivariance obstruction.

Conservation from an invariant Hamiltonian

Let the moment map be time independent. Along evolution generated by $H$,

$$\dot\mu_\xi=\{\mu_\xi,H\}.$$

The generator identity gives

$$\xi_M[H]=\{H,\mu_\xi\},$$

and antisymmetry therefore yields the precise Noether relation

$$\boxed{ \dot\mu_\xi=-\xi_M[H]. }$$

If $H$ is $G$-invariant, then $\xi_M[H]=0$ and every component $\mu_\xi$ is conserved. This is the third role of the moment map. Conversely, if all components are conserved as functions on phase space, then $H$ is invariant under the identity component of $G$. A disconnected symmetry again requires a separate finite check.

The statement presupposes that $H$ and $\mu_\xi$ are differentiable functions on the chosen phase space and that their Hamiltonian vector fields exist. In field theory, those qualifications include domain and boundary conditions; an uncancelled surface variation can invalidate the displayed Poisson bracket before any conservation argument begins.

Cotangent lifts, translations, and rotations

Moment maps arise canonically on cotangent bundles. Suppose $G$ acts on a configuration manifold $Q$. Its cotangent lift is

$$\begin{aligned} g\mathbin{\cdot}\alpha_q&\in T^*_{g\cdot q}Q,\\ (g\mathbin{\cdot}\alpha_q)(v_{g\cdot q}) &= \alpha_q\!\left( T_{g\cdot q}\Phi_{g^{-1}}\,v_{g\cdot q} \right). \end{aligned}$$

Let $\theta$ be the tautological one-form and $\omega=-\mathrm d\theta$. The cotangent lift preserves $\theta$. Writing $\xi_{T^*Q}$ for its direct fundamental field, Cartan’s formula gives

$$\begin{aligned} 0 &=\mathcal L_{\xi_{T^*Q}}\theta\\ &=-\iota_{\xi_{T^*Q}}\omega +\mathrm d\!\left( \iota_{\xi_{T^*Q}}\theta \right). \end{aligned}$$

Hence the canonical moment-map component is

$$\mu_\xi(\alpha_q) = \iota_{\xi_{T^*Q}}\theta = \alpha_q(\xi_Q(q)).$$

If $\xi_Q=\xi^i(q)\partial_{q^i}$, then

$$\mu_\xi=p_i\xi^i(q),$$

and its Hamiltonian vector field is

$$X_{\mu_\xi} = \xi^i\frac{\partial}{\partial q^i} -p_j\frac{\partial\xi^j}{\partial q^i} \frac{\partial}{\partial p_i},$$

which is exactly the infinitesimal cotangent lift. The construction is equivariant.

For translations on $Q=\mathbb R^n$, the basis generators are simply

$$\mu_i=p_i.$$

For rotations in three dimensions, identify $\mathfrak{so}(3)\simeq\mathbb R^3$ by

$$\xi_Q(q)=\boldsymbol\xi\times\mathbf q, \qquad [\boldsymbol\xi,\boldsymbol\eta] =\boldsymbol\xi\times\boldsymbol\eta.$$

The cotangent-lift formula becomes

$$\begin{aligned} \mu_\xi(\mathbf q,\mathbf p) &=\mathbf p\mathbin{\cdot} (\boldsymbol\xi\times\mathbf q)\\ &=\boldsymbol\xi\mathbin{\cdot} (\mathbf q\times\mathbf p). \end{aligned}$$

Thus $\mathbf L=\mathbf q\times\mathbf p$, and equivariance gives

$$\{L_i,L_j\}=\epsilon_{ijk}L_k.$$

For

$$H=\frac{\mathbf p^2}{2m}+V(|\mathbf q|),$$

rotational invariance implies $\dot L_i=0$. Linear and angular momentum are therefore not merely examples that inspired the terminology: they are the cotangent-lift moment maps for translations and rotations.

Noether generators in Hamiltonian form

The same structure appears on a classical field phase space. Consider two real scalar fields $\phi^a(\mathbf x)$, $a=1,2$, with conjugate momenta $\pi_a(\mathbf x)$ on a fixed spatial slice $\Sigma$. Formally,

$$\Omega = \int_\Sigma\mathrm d^{d-1}x\, \boldsymbol\delta\phi^a(\mathbf x) \wedge \boldsymbol\delta\pi_a(\mathbf x).$$

Assume that $\Sigma$ is compact without boundary, periodic, or equipped with falloff and variation conditions that make the functionals below differentiable. Let $SO(2)$ act on both $\phi$ and $\pi$, with

$$T= \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix}, \qquad T\phi=(-\phi^2,\phi^1).$$

The moment-map component for this one-dimensional Lie algebra is

$$\begin{aligned} Q &= \int_\Sigma\mathrm d^{d-1}x\, \pi_aT^a{}_b\phi^b\\ &= \int_\Sigma\mathrm d^{d-1}x\, \left(\pi_2\phi^1-\pi_1\phi^2\right). \end{aligned}$$

Its functional derivatives are

$$\frac{\delta Q}{\delta\pi}=T\phi, \qquad \frac{\delta Q}{\delta\phi}=-T\pi.$$

Using the equal-time symplectic convention inherited from the prerequisite,

$$X_Q = \left( \frac{\delta Q}{\delta\pi}, -\frac{\delta Q}{\delta\phi} \right) =(T\phi,T\pi).$$

Thus $Q$ generates the intended internal rotation:

$$\begin{array}{ll} \{\phi^1,Q\}=-\phi^2, & \{\phi^2,Q\}=\phi^1, \\[3pt] \{\pi_1,Q\}=-\pi_2, & \{\pi_2,Q\}=\pi_1. \end{array}$$

Equivalently, the complex field $\psi=(\phi^1+i\phi^2)/\sqrt2$ transforms as $\psi\mapsto e^{i\alpha}\psi$. The sign of $Q$ is tied to this choice of phase orientation. Tong 2006–2007, §§ 1.3–1.4, pp. 13–20, PDF supplies the boundary-qualified Noether and scalar Hamiltonian comparison.

For the invariant Hamiltonian

$$\begin{aligned} H =\frac12\int_\Sigma\mathrm d^{d-1}x\, \bigl[ &\pi_a\pi_a +\boldsymbol\nabla\phi^a \mathbin{\cdot}\boldsymbol\nabla\phi^a\\ &+m^2\phi_a\phi_a \bigr], \end{aligned}$$

the antisymmetry of $T$ gives $\xi_M[H]=0$. Hence

$$\dot Q=\{Q,H\}=0.$$

An $O(2)$-invariant interaction $V(\phi_a\phi_a)$ preserves the same conclusion. Dimensional analysis provides a quick check:

$$[Q] =-(d-1)+[\pi]+[\phi] =0.$$

This calculation is classical and formal. It identifies a differentiable Hamiltonian functional that generates the global internal rotation under the stated boundary assumptions. It does not by itself construct a local current, prove hypersurface independence, define a quantum charge operator, exclude anomalies, diagnose spontaneous symmetry breaking, or treat boundary and asymptotic charges. Those physical developments belong to Continuous Symmetries, Generators, and Charges.

Checks, limits, and common pitfalls

Symplectic does not imply Hamiltonian. Symplecticity makes $\iota_{\xi_M}\omega$ closed, not globally exact. The torus translation is the minimal counterexample.

A moment map need not be equivariant. Solving $\mathrm d\mu_\xi=\iota_{\xi_M}\omega$ supplies generators. It does not guarantee that their Poisson brackets reproduce $\mathfrak g$; the phase-space translation cocycle demonstrates the gap.

Constants are dynamically invisible but algebraically relevant. Adding $\lambda(\xi)$ does not change $X_{\mu_\xi}$, but it can change the equivariance defect by a coboundary. A normalization should not be discarded before checking the generator algebra.

Direct transformations and inverse pullbacks have opposite signs. With the convention here, $\xi_M[F]=\{F,\mu_\xi\}$, whereas the induced representation on functions has $D_\xi F=\{\mu_\xi,F\}$.

A generator is not automatically conserved. The exact relation is $\dot\mu_\xi=-\xi_M[H]$. Conservation follows only when the Hamiltonian has the symmetry, with the required differentiability and boundary conditions in place.

A moment-map level set is not yet a reduced phase space. Sets such as $\mu^{-1}(\nu)$ are natural inputs to reduction. For an equivariant moment map, this level is preserved by the coadjoint stabilizer $G_\nu$; all of $G$ preserves it only when $\nu$ is coadjoint-fixed. Even then, the level set need not be smooth and $\mu^{-1}(\nu)/G_\nu$ need not be a manifold. Regularity, stabilizers, properness, and the reduced symplectic form require a separate treatment.

Exercises

1. Retrieve the sign of the generator algebra. Starting from $X_{\mu_\xi}=\xi_M$, derive the Poisson bracket of two components without differentiating equivariance.

Solution

Use the two anti-homomorphism identities:

$$\begin{aligned} [X_{\mu_\xi},X_{\mu_\eta}] &=-X_{\{\mu_\xi,\mu_\eta\}},\\ [\xi_M,\eta_M] &=-[\xi,\eta]_M =-X_{\mu_{[\xi,\eta]}}. \end{aligned}$$

The left-hand sides are equal, so

$$X_{\{\mu_\xi,\mu_\eta\}} =X_{\mu_{[\xi,\eta]}}.$$

The two Hamiltonians can still differ by a constant. Equivariance fixes that constant to zero, giving

$$\{\mu_\xi,\mu_\eta\}=\mu_{[\xi,\eta]}.$$

2. Prove the torus obstruction. Why can no single-valued function $\mu$ on $T^2$ satisfy $\mathrm d\mu=\mathrm dp$?

Solution

The integral of an exact form around any closed loop is zero. Around the $p$-circle,

$$\oint_{S^1_p}\mathrm dp=2\pi$$

when the angular period is $2\pi$. Hence $\mathrm dp$ is not exact. The translation field is symplectic and locally Hamiltonian, but it has no global real-valued generator.

3. Compute an equivariance defect. For phase-space translations, take $\xi=(1,0)$ and $\eta=(0,1)$. Evaluate the two generators and their bracket. Can a constant shift remove the result?

Solution

The components are

$$\mu_\xi=p, \qquad \mu_\eta=-q.$$

Therefore

$$\{\mu_\xi,\mu_\eta\}=\{p,-q\}=1.$$

Since $\mathbb R^2$ is abelian, $\mu_{[\xi,\eta]}=0$, so $c(\xi,\eta)=1$. A shift changes $c$ by $-\lambda([\xi,\eta])=0$; it cannot remove the defect.

4. Check angular momentum directly. Let $L_z=q_xp_y-q_yp_x$. Show that $X_{L_z}$ rotates both $\mathbf q$ and $\mathbf p$, and verify $\{L_x,L_y\}=L_z$.

Solution

Hamilton’s coordinate formula gives

$$\begin{aligned} X_{L_z} ={}&-q_y\frac{\partial}{\partial q_x} +q_x\frac{\partial}{\partial q_y}\\ &-p_y\frac{\partial}{\partial p_x} +p_x\frac{\partial}{\partial p_y}. \end{aligned}$$

This is the direct infinitesimal rotation $(\delta\mathbf q,\delta\mathbf p) =(\mathbf e_z\times\mathbf q,\mathbf e_z\times\mathbf p)$. A coordinate calculation using $L_x=q_yp_z-q_zp_y$ and $L_y=q_zp_x-q_xp_z$ gives

$$\{L_x,L_y\}=q_xp_y-q_yp_x=L_z.$$

5. Verify the scalar-field generator. Differentiate $Q$ with respect to $\phi^a$ and $\pi_a$, recover the four infinitesimal transformations, and explain why the free $H$ is invariant.

Solution

Varying

$$Q=\int_\Sigma\mathrm d^{d-1}x\, (\pi_2\phi^1-\pi_1\phi^2)$$

gives

$$\frac{\delta Q}{\delta\pi_1}=-\phi^2, \quad \frac{\delta Q}{\delta\pi_2}=\phi^1, \quad \frac{\delta Q}{\delta\phi^1}=\pi_2, \quad \frac{\delta Q}{\delta\phi^2}=-\pi_1.$$

Thus

$$\{\phi^1,Q\}=-\phi^2, \quad \{\phi^2,Q\}=\phi^1, \quad \{\pi_1,Q\}=-\pi_2, \quad \{\pi_2,Q\}=\pi_1.$$

Each term in $H$ contracts two internal vectors with $\delta_{ab}$. Its variation contains a contraction of a vector with $T$ times itself, which vanishes because $T^\mathsf T=-T$. Therefore $\xi_M[H]=0$, and the Noether relation gives $\dot Q=0$.

Synthesis and continuation

A moment map answers the page’s question component by component:

$$\mathrm d\mu_\xi=\iota_{\xi_M}\omega \quad\Longrightarrow\quad X_{\mu_\xi}=\xi_M.$$

Equivariance upgrades these Hamiltonians into a representation of the Lie algebra under Poisson brackets, and invariance of $H$ upgrades them into conserved quantities. De Rham cohomology controls whether global generators exist, while the constant two-cocycle controls whether a chosen family can be made equivariant. The cotangent-lift and scalar-field examples show the same geometry behind linear momentum, angular momentum, and an internal Noether charge.

This page has deliberately stopped before quotienting a level set of $\mu$. That next operation needs hypotheses ensuring that the constraint surface and quotient carry the intended geometry; the moment-map equation alone does not supply them.

References

• Ana Cannas da Silva, Lectures on Symplectic Geometry — Open PDF, revised 2006, Lecture Notes in Mathematics 1764, Springer, doi:10.1007/978-3-540-45330-7, § 22.1, pp. 133–134; § 22.4, pp. 137–138; § 24.1, p. 147; and §§ 26.1–26.4, pp. 164–167. These sections support the moment-map and comoment-map definitions, linear and angular momentum examples, Hamiltonian Noether conservation, the de Rham obstruction, the Lie-algebra cocycle, constant ambiguity, and the torus counterexample.
• Eckhard Meinrenken, Symplectic Geometry — Open PDF, University of Toronto lecture notes, Fall 2024, §§ 3.2 and 7.3–7.4, pp. 33 and 82–88. These notes support weak versus equivariant moment maps, coadjoint equivariance, the component-generator algebra, central extensions, cotangent lifts, and momentum examples. They define infinitesimal action fields using $\exp(-t\xi)$ and Hamiltonian fields by $\iota_{X_H}\omega=-\mathrm dH$; both choices have been translated to this site’s direct $\exp(+t\xi)$ convention and $\iota_{X_f}\omega=\mathrm df$ convention.
• David Tong, Lectures on Quantum Field Theory — Open PDF, Cambridge Part III lecture notes, 2006–2007, §§ 1.3–1.4, pp. 13–20. These sections support the boundary-qualified relation between continuous symmetries and conserved charges, the complex-scalar internal rotation, and the scalar Hamiltonian formalism used in the controlled QFT example.