Lagrangian Submanifolds, Generating Functions, and Semiclassical Phases

A Lagrangian submanifold is a half-dimensional submanifold on which the symplectic form vanishes. That compact definition unifies three objects that can look unrelated in coordinates: a family of classical solutions, the canonical relation defined by time evolution, and the phase carried by a semiclassical wave. A scalar generating function is only a local coordinate description of that geometry. It can become multivalued or singular when a projection develops a caustic even though the Lagrangian submanifold itself remains smooth.

This page develops that distinction carefully. It fixes the signs in endpoint generating functions, explains why auxiliary phase variables survive caustics, and derives the Van Vleck and Maslov data of semiclassical propagation. The discussion is finite-dimensional. Extending it to a QFT functional integral requires separate choices of regulator, measure, gauge treatment, and integration cycle.

Required background. Symplectic Forms, Hamiltonian Vector Fields, and Poisson Brackets supplies symplectic complements, Hamiltonian flow, and the canonical two-form sign convention; Stationary Phase, Coalescing Saddles, and Stokes Geometry supplies the regulated Fresnel factor, the Airy fold, and the rule that only contour-accessible stationary points contribute.

Lagrangian data and sign conventions

Let $Q$ be an $n$-dimensional configuration manifold. In canonical coordinates on $T^*Q$, this page uses

$$\theta=p_a\,\mathrm dq^a, \qquad \omega=-\mathrm d\theta =\mathrm dq^a\wedge\mathrm dp_a.$$

The bar on a symplectic manifold reverses its symplectic form:

$$\overline{(M,\omega)}=(M,-\omega).$$

We take $\hbar>0$ and use $e^{+i\Phi/\hbar}$ for oscillatory phases. Reversing that exponential conjugates every Fresnel and Maslov phase below.

Lagrangian submanifolds

Let $\iota:L\hookrightarrow(M,\omega)$ be an embedded submanifold of a $2n$-dimensional symplectic manifold. It is Lagrangian when

$$\dim L=n, \qquad \iota^*\omega=0.$$

Equivalently, at every $x\in L$,

$$T_xL=(T_xL)^\omega,$$

where the right-hand side is the symplectic complement. The vanishing condition alone says that $L$ is isotropic; the dimension condition makes it maximally isotropic. A merely half-dimensional submanifold need not be Lagrangian.

Cotangent bundles provide the basic examples.

Graphs of one-forms

For a one-form $\alpha\in\Omega^1(Q)$, let

$$s_\alpha:Q\longrightarrow T^*Q, \qquad q\longmapsto(q,\alpha_q).$$

Because $s_\alpha^*\theta=\alpha$,

$$s_\alpha^*\omega=-\mathrm d\alpha.$$

The graph of $\alpha$ is therefore Lagrangian exactly when $\alpha$ is closed. On a contractible coordinate neighborhood, $\alpha=\mathrm dS$, so the graph has the familiar form

$$p_a=\frac{\partial S}{\partial q^a}.$$

The local qualifier matters. For example, on $Q=S^1$, the section $p\,\mathrm d\varphi=c\,\mathrm d\varphi$ is Lagrangian, but for $c\neq0$ it has no single-valued real generating function on the whole circle. Its period is $2\pi c$. Lagrangian implies locally exact in a cotangent chart; it does not imply a global scalar $S$.

These Lagrangian graph and local generating-function statements are treated in Cannas da Silva 2001, §§ 3.2–4.2, pp. 16–23, PDF.

Fibers and conormals

A cotangent fiber $T_q^*Q$ is Lagrangian but is not a graph over any open subset of $Q$. More generally, if $Y\subset Q$ is a submanifold, its conormal bundle

$$N^*Y = \left\{ (q,p)\in T^*Q: q\in Y,\ p(v)=0\ \text{for every }v\in T_qY \right\}$$

is Lagrangian. Indeed, $\theta$ vanishes on $N^*Y$, and

$$\dim N^*Y = \dim Y+\operatorname{codim}Y =n.$$

These examples already show why a single position-space function cannot be the definition: some perfectly regular Lagrangians are vertical, and others have nontrivial topology.

Canonical relations and endpoint signs

A canonical relation from $(M_i,\omega_i)$ to $(M_f,\omega_f)$ is a Lagrangian submanifold

$$C\subset\overline{M_i}\times M_f$$

for the product form

$$\Omega=-\omega_i+\omega_f.$$

A canonical transformation $F:M_i\to M_f$ gives the special case $C=\Gamma_F$, its graph. Pulling back the product form to that graph gives

$$\Omega|_{\Gamma_F} = -\omega_i+F^*\omega_f,$$

so $\Gamma_F$ is Lagrangian exactly when $F$ is symplectic. A general canonical relation need not be the graph of a single-valued map; this flexibility is essential at caustics and in composition.

For generating families, caustics, Maslov correction, and canonical relations, see Bates and Weinstein 1997, §§ 4.2–4.3, pp. 41–55, and § 5.2, pp. 76–78, PDF.

Suppose now that $M_i=T^*Q_i$ and $M_f=T^*Q_f$. Define

$$\beta=\theta_f-\theta_i = p_f\,\mathrm dq_f-p_i\,\mathrm dq_i.$$

Since

$$\Omega=-\mathrm d\beta,$$

the restriction $\beta|_C$ is closed. It is locally exact, so on $C$ there is a local function $S$ satisfying

$$\mathrm dS = p_f\,\mathrm dq_f-p_i\,\mathrm dq_i.$$

This does not yet make $S$ a function of the two endpoints. That step requires the projection

$$\pi_{if}:C\longrightarrow Q_i\times Q_f$$

to be a local diffeomorphism. In such a type-I chart,

$$S=S(q_f,q_i),$$

and coefficient matching fixes the signs:

$$\boxed{ p_f=\frac{\partial S}{\partial q_f}, \qquad p_i=-\frac{\partial S}{\partial q_i}. }$$

The minus sign on the initial momentum comes from the barred initial phase space. It is not optional notation.

When the endpoint projection fails, another polarization may still work. Here a polarization means a choice of which half of the canonical variables serve as independent coordinates. For example, if $q_i$ can be eliminated in favor of $p_i$, the $(q_f,p_i)$ choice is described by the partial Legendre transform

$$F_2(q_f,p_i) = S(q_f,q_i)+p_iq_i$$

obeys

$$\mathrm dF_2 = p_f\,\mathrm dq_f+q_i\,\mathrm dp_i.$$

Thus $p_f=\partial_{q_f}F_2$ and $q_i=\partial_{p_i}F_2$. Changing polarization changes the coordinate chart, not the underlying canonical relation.

Generating families with auxiliary variables

Ordinary generating functions are not flexible enough near a caustic. Let $\vartheta=(\vartheta^1,\ldots,\vartheta^k)$ be auxiliary variables and let

$$\Phi:Q\times\mathbb R^k\longrightarrow\mathbb R.$$

Its fiber-critical set is

$$C_\Phi = \left\{ (q,\vartheta): \partial_{\vartheta^\alpha}\Phi(q,\vartheta)=0 \right\}.$$

The phase is nondegenerate in the relevant sense when

$$\mathrm d(\partial_{\vartheta^1}\Phi),\ldots, \mathrm d(\partial_{\vartheta^k}\Phi)$$

are linearly independent on $C_\Phi$. This regular-value condition makes $C_\Phi$ a smooth $n$-dimensional manifold. The map

$$\iota_\Phi:C_\Phi\longrightarrow T^*Q, \qquad (q,\vartheta) \longmapsto \left(q,\partial_q\Phi(q,\vartheta)\right)$$

is then a Lagrangian immersion. The proof contains the main idea. On $C_\Phi$,

$$\begin{aligned} \iota_\Phi^*\theta &= \partial_q\Phi\,\mathrm dq\\ &= \mathrm d\!\left(\Phi|_{C_\Phi}\right), \end{aligned}$$

because $\partial_\vartheta\Phi=0$ there. Hence $\iota_\Phi^*\omega=0$, and the regularity condition supplies the correct dimension and immersion rank. After restricting to a neighborhood on which $\iota_\Phi$ is an embedding, its image is an exact Lagrangian.

The condition is not

$$\det\Phi_{\vartheta\vartheta}\neq0.$$

That stronger condition says that the stationary auxiliary variable can be solved smoothly as a function of $q$. It is precisely what can fail at a caustic while the full phase remains nondegenerate. Conversely, every smooth Lagrangian in a cotangent bundle has a local generating-family description after sufficiently many auxiliary variables are introduced. A global description may still require several charts and transition data. The local existence statement is proved in Guillemin and Sternberg 2010, §§ 5.1–5.2, pp. 133–138, and § 5.9, pp. 153–157, PDF.

The fold that stays smooth

Consider

$$\Phi(x,\vartheta) = x\vartheta-\frac{\vartheta^3}{3}.$$

The fiber-critical equation and its differential are

$$\partial_\vartheta\Phi=x-\vartheta^2=0, \qquad \mathrm d(\partial_\vartheta\Phi) = \mathrm dx-2\vartheta\,\mathrm d\vartheta.$$

The differential never vanishes, including at $\vartheta=0$. The generated Lagrangian is therefore the smooth parabola

$$\Lambda = \left\{ (x,p):x=p^2 \right\}, \qquad p=\partial_x\Phi=\vartheta.$$

Along its critical set,

$$\Phi|_{C_\Phi} = \frac23\vartheta^3, \qquad p\,\mathrm dx = \mathrm d\!\left(\frac23\vartheta^3\right).$$

The projection $\Lambda\to\mathbb R_x$ folds at $\vartheta=0$. For $x>0$ it has two graph charts,

$$p_\pm=\pm\sqrt{x}, \qquad S_\pm(x)=\pm\frac23x^{3/2},$$

but neither graph extends smoothly through $x=0$. The auxiliary phase does.

The associated Abel-prescribed oscillatory integral is

$$\begin{aligned} I_\hbar(x) &= \lim_{\epsilon\downarrow0} \int_{\mathbb R} e^{-\epsilon\vartheta^2} \exp\!\left[ \frac{i}{\hbar} \left( x\vartheta-\frac{\vartheta^3}{3} \right) \right]\mathrm d\vartheta\\ &= 2\pi\hbar^{1/3} \operatorname{Ai}\!\left( -\frac{x}{\hbar^{2/3}} \right). \end{aligned}$$

For fixed $x>0$, stationary phase at $\vartheta_\pm=\pm\sqrt{x}$ gives

$$I_\hbar(x) \sim 2\sqrt{\pi\hbar}\,x^{-1/4} \cos\!\left( \frac{2x^{3/2}}{3\hbar}-\frac{\pi}{4} \right).$$

The two graph amplitudes diverge as $x^{-1/4}$ when the fold is approached, whereas the Airy expression remains finite on the $x=O(\hbar^{2/3})$ scale. The divergence belongs to the projected graph chart, not to $\Lambda$ or to the uniform oscillatory integral. The Airy integral normalization and the asymptotic matching used here are recorded in NIST DLMF 2026, §§ 9.5 and 9.7.

Semiclassical phases and Maslov data

A local semiclassical state associated with a Lagrangian can be written as

$$u_\hbar(q) = (2\pi\hbar)^{-k/2} \int a(q,\vartheta;\hbar) e^{i\Phi(q,\vartheta)/\hbar} \,\mathrm d^k\vartheta,$$

with a specified contour or oscillatory prescription. Its fiber-critical points map to

$$\left(q,p=\partial_q\Phi\right)\in\Lambda.$$

If $\Phi_{\vartheta\vartheta}$ is invertible, stationary phase removes the auxiliary variables locally. For the convention used here, a real auxiliary Hessian $H$ contributes

$$e^{i\pi\operatorname{sig}H/4} \frac{1}{\sqrt{|\det H|}}$$

apart from powers of $2\pi\hbar$ and the transformed amplitude. Different phase functions can describe the same Lagrangian, so these signature factors must be included in their transition rules.

The resulting integer phase bookkeeping is the Maslov index. With the short-time propagator chosen as the baseline, this page denotes it by $\nu$. At a simple crossing where one Hessian eigenvalue changes from positive to negative,

$$\frac{e^{-i\pi/4}}{e^{+i\pi/4}} = e^{-i\pi/2}.$$

Thus that crossing contributes $-\pi/2$ and increases $\nu$ by one; the reverse crossing has the opposite sign. For ordinary positive-kinetic mechanical propagation forward in time, these crossings are conjugate points: losses of rank of the endpoint map from initial momentum to final position, counted with the rank loss as multiplicity. In a general Hamiltonian flow, the crossing form determines the sign instead. Reversing the oscillatory sign or the index orientation reverses the phase rule. An unqualified formula involving a symbol $\mu$ is therefore incomplete unless its baseline and crossing convention are stated.

Geometrically, the Maslov index records how the tangent Lagrangian planes meet the locus where the chosen projection becomes singular. It is what lets local phase charts glue into a global semiclassical object even when no global single-valued $S(q)$ exists.

Hamilton’s principal function and the propagator

Let a regular classical trajectory $\gamma$ join $q_i$ to $q_f$ in time $T$. Its on-shell action is

$$S_\gamma(q_f,q_i;T) = \int_0^T \left( p_a\dot q^a-H(q,p) \right)\mathrm dt.$$

Varying the endpoints at fixed $T$ gives

$$\mathrm dS_\gamma = p_f\,\mathrm dq_f-p_i\,\mathrm dq_i.$$

For an autonomous Hamiltonian, varying the final time also gives

$$\frac{\partial S_\gamma}{\partial T} + H\!\left( q_f,\frac{\partial S_\gamma}{\partial q_f} \right) =0.$$

Hamilton’s principal function is therefore a type-I generating function for the graph of Hamiltonian time evolution wherever the endpoint projection is regular.

Choose endpoint coordinate charts and refer the kernel to their coordinate measures. For isolated classical trajectories, define the Van Vleck matrix

$$D_\gamma = -\frac{\partial^2S_\gamma} {\partial q_f\,\partial q_i}.$$

The real-time quantum propagator is the position-space kernel

$$K(q_f,q_i;T) = \left\langle q_f \middle| e^{-i\widehat H T/\hbar} \middle| q_i \right\rangle.$$

Invariantly, its leading amplitude is a bi-half-density; the determinant below is that amplitude’s representative in the chosen coordinates.

With the square-root branch fixed by the positive short-time limit, the local semiclassical propagator is

$$\boxed{ \begin{aligned} K_{\mathrm{sc}}(q_f,q_i;T) ={}& \frac{1}{(2\pi i\hbar)^{n/2}} \sum_\gamma \left| \det D_\gamma \right|^{1/2}\\ &\times \exp\!\left[ \frac{i}{\hbar}S_\gamma(q_f,q_i;T) -\frac{i\pi}{2}\nu_\gamma \right]. \end{aligned} }$$

The sum is over trajectories compatible with the prescribed boundary conditions. It is not a sum over every formal solution after complexification.

The determinant has a useful geometric interpretation. Parameterize the flow graph by $(q_i,p_i)$ and put

$$B_\gamma = \left. \frac{\partial q_f}{\partial p_i} \right|_{q_i}.$$

Where the type-I chart exists,

$$\left|\det D_\gamma\right| = \left|\det B_\gamma\right|^{-1}.$$

A configuration-space caustic occurs when $\det B_\gamma=0$. Consequently the type-I Van Vleck matrix does not vanish there: it ceases to be a regular coordinate expression and typically diverges on approach. One must retain an auxiliary phase, change polarization, or use a uniform canonical integral. The relation among the Van Vleck determinant, Jacobi fields, caustics, and Morse-index phases is developed in Horváthy 2011, §§ 6–7, arXiv pp. 11–14.

Exact benchmark: the harmonic oscillator

For

$$H=\frac{p^2}{2m}+\frac12m\omega^2q^2, \qquad m,\omega>0,$$

and $\sin\omega T\neq0$, the unique endpoint trajectory is

$$q_{\mathrm{cl}}(t) = \frac{ q_i\sin\!\left(\omega(T-t)\right) + q_f\sin(\omega t) }{ \sin(\omega T) }.$$

Its action is

$$S_{\mathrm{cl}} = \frac{m\omega}{2\sin\omega T} \left[ (q_f^2+q_i^2)\cos\omega T -2q_fq_i \right].$$

Direct differentiation checks all signs:

$$\begin{aligned} p_f &= \frac{m\omega(q_f\cos\omega T-q_i)} {\sin\omega T},\\ p_i &= \frac{m\omega(q_f-q_i\cos\omega T)} {\sin\omega T},\\ D &= -\frac{\partial^2S_{\mathrm{cl}}} {\partial q_f\,\partial q_i} = \frac{m\omega}{\sin\omega T}. \end{aligned}$$

Meanwhile,

$$B = \left. \frac{\partial q_f}{\partial p_i} \right|_{q_i} = \frac{\sin\omega T}{m\omega}, \qquad D=B^{-1}.$$

This quadratic system is especially valuable because the semiclassical answer is exact. For $T>0$ away from a caustic, let

$$N=\left\lfloor\frac{\omega T}{\pi}\right\rfloor.$$

Then

$$\boxed{ K(q_f,q_i;T) = \sqrt{ \frac{m\omega} {2\pi i\hbar\,|\sin\omega T|} } \exp\!\left[ \frac{i}{\hbar}S_{\mathrm{cl}} -\frac{i\pi N}{2} \right]. }$$

At $T=N\pi/\omega$, with $N\geq1$, the coordinate formula must be replaced by its distributional limit:

$$\boxed{ K\!\left( q_f,q_i;\frac{N\pi}{\omega} \right) = e^{-i\pi N/2} \delta\!\left(q_f-(-1)^Nq_i\right). }$$

Thus the first caustic gives $-i\,\delta(q_f+q_i)$, while one full period gives $-\delta(q_f-q_i)$. The vanishing of $B$, the divergence of $D$, the $-\pi/2$ phase increment, and the delta-supported exact kernel are four views of the same projection failure. The exact propagator both between and at harmonic-oscillator caustics is checked independently in Funahashi 2010, pp. 2 and 6–8.

Composition is stationary elimination

Let

$$C_{12}\subset\overline{M_1}\times M_2, \qquad C_{23}\subset\overline{M_2}\times M_3.$$

Their set-theoretic composition matches the two copies of the intermediate phase space and then forgets them. It is Lagrangian only under suitable transversality or clean-intersection hypotheses, together with a well-behaved projection of the matched set. Without those hypotheses, the result can be immersed, multiply covered, or singular.

In compatible type-I charts, this geometric operation becomes

$$S_{13}(q_3,q_1) = \operatorname*{stat}_{q_2} \left[ S_{23}(q_3,q_2)+S_{12}(q_2,q_1) \right].$$

The stationary equation is

$$\partial_{q_2}S_{23} + \partial_{q_2}S_{12} =0.$$

It says that the initial momentum of the second relation equals the final momentum of the first. Several stationary values of $q_2$ produce several branches; a degenerate stationary value signals that the scalar composition formula needs a uniform phase description. This is the geometric reason that semiclassical propagators compose by integrating over an intermediate configuration and then applying stationary phase.

Scope of the QFT bridge

For a field theory, the classical action plays the role of a phase functional, and its second variation supplies the formal analogue of the finite-dimensional Hessian and Van Vleck data. That analogy is useful but incomplete. Functional determinants require regularization and renormalization; gauge directions and collective coordinates must be separated; boundary conditions select the operator domain; and negative modes do not by themselves determine an integration cycle.

Likewise, a middle-dimensional contour in a complexified field space is not automatically the same object as a real Lagrangian submanifold in a cotangent phase space. A symplectic or Kähler structure, a reality condition, and a contour prescription must first be specified. The finite-dimensional Airy model and Chern–Simons application in Witten 2010, §§ 2.1–2.3, pp. 8–11 show how thimble coefficients depend on the original integration cycle, but that is additional global data rather than a consequence of the local generating-function equations.

Negative Modes and Instability Indices develops the physical Morse-index, contour-deformation, and instability questions. The present page supplies its finite-dimensional phase geometry; it does not preselect the physical saddle sectors.

Common pitfalls

Treating half-dimensional as sufficient. A half-dimensional submanifold is Lagrangian only if the symplectic form also restricts to zero.

Assuming every Lagrangian is a global graph. Cotangent fibers, conormals, folded projections, and closed nonexact sections are counterexamples. Use local charts or generating families.

Losing the initial minus sign. For $C\subset\overline{M_i}\times M_f$, $\mathrm dS=p_f\,\mathrm dq_f-p_i\,\mathrm dq_i$. Reversing the product order or the symplectic convention changes the displayed formulas and must be stated.

Calling a phase degenerate when its fiber Hessian vanishes. The relevant condition is independence of the full differentials $\mathrm d(\partial_{\vartheta^\alpha}\Phi)$. The fold phase remains nondegenerate at its caustic.

Saying the Van Vleck determinant vanishes at a caustic. The endpoint Jacobian $\det(\partial q_f/\partial p_i)$ vanishes. Its inverse determinant in the type-I Van Vleck formula becomes singular.

Identifying a phase-space Lagrangian with a QFT integration cycle. The two can be related only after additional geometric and analytic structures are specified.

Check your understanding

1. A Lagrangian with no global scalar phase

On $T^*S^1$, consider the section $\alpha=c\,\mathrm d\varphi$. Show that its graph is Lagrangian and decide when it has a single-valued real generating function on all of $S^1$.

Solution

Although $\varphi$ itself is only a local real coordinate, the forms $\mathrm d\varphi$ on angular charts agree on overlaps and define a global one-form. Because

$$\mathrm d\alpha = c\,\mathrm d(\mathrm d\varphi) =0,$$

the graph is Lagrangian. If $\alpha=\mathrm dS$ globally, its integral around the circle must vanish. Instead,

$$\oint_{S^1}\alpha=2\pi c.$$

Therefore a single-valued real $S$ exists only for $c=0$. Every sufficiently small angular chart still has the local primitive $S=c\varphi$.

2. Recover the endpoint signs

For a free particle in one dimension,

$$S(q_f,q_i;T) = \frac{m(q_f-q_i)^2}{2T}, \qquad T>0.$$

Use the canonical-relation convention to recover the initial and final momenta.

Solution

Differentiate with respect to the final endpoint:

$$p_f = \frac{\partial S}{\partial q_f} = \frac{m(q_f-q_i)}{T}.$$

At the initial endpoint,

$$p_i = -\frac{\partial S}{\partial q_i} = \frac{m(q_f-q_i)}{T}.$$

The two momenta agree, as free motion requires. Omitting the initial minus sign would incorrectly reverse $p_i$.

3. Diagnose the fold

For

$$\Phi(x,\vartheta) = x\vartheta-\frac{\vartheta^3}{3},$$

find the critical set, its generated Lagrangian, and the point where the position projection fails. Why is the phase nevertheless nondegenerate?

Solution

The fiber-critical equation is $x-\vartheta^2=0$, so

$$C_\Phi=\{x=\vartheta^2\}.$$

Since $p=\partial_x\Phi=\vartheta$, the image is

$$\Lambda=\{x=p^2\}.$$

The derivative of $x=\vartheta^2$ vanishes at $\vartheta=0$, so the projection to $x$ folds there. However,

$$\mathrm d(\partial_\vartheta\Phi) = \mathrm dx-2\vartheta\,\mathrm d\vartheta$$

never vanishes. The fiber-critical set and the generated Lagrangian remain smooth even though $\Phi_{\vartheta\vartheta}=0$ at the fold.

4. Check the oscillator caustic

For the harmonic oscillator, compute

$$B=\left.\frac{\partial q_f}{\partial p_i}\right|_{q_i}$$

from the classical solution. What happens to $B$, the type-I Van Vleck matrix, and the exact kernel at $T=\pi/\omega$?

Solution

The initial-value solution is

$$q_f = q_i\cos\omega T + \frac{p_i}{m\omega}\sin\omega T,$$

so

$$B=\frac{\sin\omega T}{m\omega}.$$

At $T=\pi/\omega$, $B=0$: all initial momenta focus at $q_f=-q_i$. Away from the caustic,

$$D=B^{-1} = \frac{m\omega}{\sin\omega T},$$

so the type-I expression diverges rather than vanishes. Its correct distributional limit is

$$K\!\left(q_f,q_i;\frac{\pi}{\omega}\right) = -i\,\delta(q_f+q_i).$$

The factor $-i=e^{-i\pi/2}$ is the first Maslov increment in the convention of this page.

What has been established

A Lagrangian submanifold is the invariant object; ordinary generating functions, mixed polarizations, and auxiliary phases are local descriptions. Canonical evolution becomes a Lagrangian relation with fixed endpoint signs. When a position projection folds, graph phases and type-I amplitudes fail while a nondegenerate auxiliary phase can remain smooth. Stationary-phase transition factors then assemble into Maslov data, and the Van Vleck determinant measures the inverse endpoint Jacobian. The harmonic oscillator checks these statements against an exact kernel, including its distributional value at a caustic.

Where the method continues

WKB and Eikonal Methods and Turning-Point Matching applies the same Lagrangian projection and Airy-fold geometry to differential equations. Stationary Phase, Coalescing Saddles, and Stokes Geometry develops the local asymptotics behind the Fresnel and Airy formulas. The later Negative Modes and Instability Indices page supplies the field-theoretic stability and contour analysis that the finite-dimensional construction deliberately leaves open.

References

• Sean Bates and Alan Weinstein, Lectures on the Geometry of Quantization, PDF, Berkeley Mathematics Lecture Notes 8, American Mathematical Society (1997), §§4.2–4.3, pp. 41–55, and §5.2, pp. 76–78. Maslov correction, generating families, caustics, and canonical relations.

• Ana Cannas da Silva, Lectures on Symplectic Geometry, PDF, Springer Lecture Notes in Mathematics 1764 (2001), author-hosted revision (January 2006), §3.2, pp. 16–17, §§3.3–3.4, pp. 17–19, and §§4.1–4.2, pp. 22–23. Lagrangian submanifolds, graphs of closed one-forms, symplectic graphs, and generating functions.

• Kunio Funahashi, “Extended Feynman Formula for the Harmonic Oscillator by the Discrete Time Method”, Modern Physics Letters A 25 (2010), 179–188, pp. 2 and 6–8. The exact propagator between and at harmonic-oscillator caustics.

• Victor Guillemin and Shlomo Sternberg, Semi-Classical Analysis, PDF, author manuscript (2010), §§5.1–5.2, pp. 133–138, and §5.9, pp. 153–157. Generating functions relative to a fibration and local existence for Lagrangians and canonical relations.

• P. A. Horváthy, “The Maslov Correction in the Semiclassical Feynman Integral”, Central European Journal of Physics 9 (2011), 1–12, §§6–7, arXiv PDF pp. 11–14. Van Vleck determinants, Jacobi fields, caustics, and Morse-index phases.

• NIST Digital Library of Mathematical Functions, version 1.2.7, released June 15, 2026, National Institute of Standards and Technology, §9.5, “Integral Representations” and §9.7, “Asymptotic Expansions”. Airy normalization and asymptotic matching across the fold.

• Edward Witten, “Analytic Continuation of Chern–Simons Theory”, arXiv:1001.2933v4 (2010), §§2.1–2.3, pp. 8–11. Finite-dimensional Airy models, thimble decompositions, and the additional cycle data required in a gauge-theory path integral.