Maslov index¶

Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis.

Version: v1 · 2026-09-28 · History
Domain-specific #: 10580
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Symplectic Geometry, Microlocal Analysis → Mathematics

Core Idea¶

Maslov index is treated here as the recurring mathematics_logic_statistics identity summarized by this source-grounded definition: Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis.

Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. It is associated most classically with a loop or path of Lagrangian subspaces in a symplectic vector space, and measures how that path meets a distinguished singular hypersurface known as the Maslov cycle. Informally, it records the winding of a family of Lagrangian subspaces relative to the locus of subspaces that fail to be transverse to a chosen reference Lagrangian.

The Maslov index was introduced in the context of asymptotic methods in mathematical physics and later reformulated in topological and geometric terms. It plays an important role in the theory of Fourier integral operators, geometric quantization, Hamiltonian systems, spectral theory, and Floer homology. For a sufficiently generic path \gamma\colon [a,b]\to \Lambda(V) whose endpoints are transverse to L_0 , the Maslov index \mu_{L_0}(\gamma) is defined as the signed intersection number of \gamma with the Maslov cycle \Sigma(L_0).

For Maslov index, the abstraction is narrower than the article's general subject matter: a positive case must preserve Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — Fixing the horizontal axis as reference, the Maslov cycle consists of the single point represented by that axis.
Constitutive relation — When an oscillatory solution passes through a caustic, the phase jumps by a multiple of \pi/2.
Operating condition — At a transverse crossing time t_* for which \gamma(t_)\cap L_0\neq 0 , the sign is determined by the crossing form, a quadratic form on the intersection \gamma(t_)\cap L_0.
Recognition evidence — Then the edge A_kA_{k+1} in the polygon is parameterized by u\in[0,1] , {x + uSx\mid x\in A_k}.
Admissible variation — Homotopy invariance: it is unchanged under homotopies through paths with fixed admissible endpoints.
Characteristic consequence — A path of Lagrangian lines is therefore a path of unoriented lines through the origin.
Failure boundary — The Maslov index of a generic loop counts how many times, with sign and appropriate normalization, the moving line passes through the horizontal direction.

What It Is Not¶

Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis.
Not an over-broad reading. Different authors adopt slightly different normalizations, especially when comparing the Maslov index of Lagrangian paths, the Maslov class, and the Conley-Zehnder index.
Not an over-broad reading. If an oscillatory integral is written using two different phase functions that.
Not an over-broad reading. These constructions agree in overlapping settings but are adapted to different geometric and analytic problems.
Not automatically Hosoya Index. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Maslov index applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Fourier integral operators. If an oscillatory integral is written using two different phase functions that.
Microlocal analysis. In the theory of Fourier integral operators, phase functions may be replaced under changes of coordinates or generating functions.
Infinite-dimensional generalizations. intersections in real Hilbert spaces, and a widely cited functional-analytic definition for continuous curves in the.
Fourier integral operators. parametrize the same Lagrangian submanifold, the method of stationary phase.
Documented setting. The Maslov index was introduced in the context of asymptotic methods in mathematical physics and later reformulated in topological and geometric terms.
Overview. Let (V,\omega) be a finite-dimensional symplectic vector space, and let \Lambda(V) denote its Lagrangian Grassmannian, the manifold of all Lagrangian subspaces of V.

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Measurement or should be marked as analogy.

Clarity¶

A clear use of Maslov index names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. The strongest recognition evidence in the frozen account is: Then the edge A_kA_{k+1} in the polygon is parameterized by u\in[0,1] , {x + uSx\mid x\in A_k}. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification Different authors adopt slightly different normalizations, especially when comparing the Maslov index of Lagrangian paths, the Maslov class, and the Conley-Zehnder index. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Maslov index compresses multiple mathematics_logic_statistics details into a stable diagnostic relation. The source shows both the central mechanism—when an oscillatory solution passes through a caustic, the phase jumps by a multiple of \pi/2 .—and the practical consequence—a path of Lagrangian lines is therefore a path of unoriented lines through the origin. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics_logic_statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis.
Check operation and conditions. At a transverse crossing time t_* for which \gamma(t_)\cap L_0\neq 0 , the sign is determined by the crossing form, a quadratic form on the intersection \gamma(t_)\cap L_0.
Demand recognition evidence. Then the edge A_kA_{k+1} in the polygon is parameterized by u\in[0,1] , {x + uSx\mid x\in A_k}.
Test variation. Change an implementation or setting while preserving homotopy invariance: it is unchanged under homotopies through paths with fixed admissible endpoints.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Measurement.

Knowledge Transfer¶

Within the home domain. Knowledge about Maslov index transfers literally when a new case preserves the same carrier type, relation, and recognition test. If an oscillatory integral is written using two different phase functions that. In the theory of Fourier integral operators, phase functions may be replaced under changes of coordinates or generating functions.

Beyond the home domain. No canonical parent is asserted for Maslov index. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

Let (V,\omega) be a finite-dimensional symplectic vector space, and let \Lambda(V) denote its Lagrangian Grassmannian, the manifold of all Lagrangian subspaces of V. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis; recognition evidence → Then the edge A_kA_{k+1} in the polygon is parameterized by u\in[0,1] , {x + uSx\mid x\in A_k}

Applied / In Practice¶

The topology of \Lambda(V) is nontrivial: its fundamental group is isomorphic to \mathbf{Z}. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Overview; invariant → Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis; boundary → the case exits the class when different authors adopt slightly different normalizations, especially when comparing the Maslov index of Lagrangian paths, the Maslov class, and the Conley-Zehnder index

Structural Tensions¶

T1 — Stable identity versus admissible variation. Different authors adopt slightly different normalizations, especially when comparing the Maslov index of Lagrangian paths, the Maslov class, and the Conley-Zehnder index. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. If an oscillatory integral is written using two different phase functions that. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. These constructions agree in overlapping settings but are adapted to different geometric and analytic problems. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. The index is named after the Soviet mathematician and physicist Viktor Pavlovich Maslov, who introduced it in the study of asymptotic solutions of differential equations. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. Fixing the horizontal axis as reference, the Maslov cycle consists of the single point represented by that axis. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Maslov index literally, co-instantiate Measurement, or only resemble it?

T6 — Autonomy versus reduction. When an oscillatory solution passes through a caustic, the phase jumps by a multiple of \pi/2. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Maslov index distinguish that the broader parent Measurement leaves together?

Terminal boundary synthesis. For Maslov index, the terminal identity test begins with the definition Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis.. A reviewer must then establish the carrier and operation described by Fixing the horizontal axis as reference, the Maslov cycle consists of the single point represented by that axis. and When an oscillatory solution passes through a caustic, the phase jumps by a multiple of \pi/2.. Recognition is constrained by At a transverse crossing time t for which \gamma(t)\cap L0\neq 0 , the sign is determined by the crossing form, a quadratic form on the intersection \gamma(t)\cap L0., while admissible variation is limited by Then the edge AkA{k+1} in the polygon is parameterized by u\in[0,1] , \{x + uSx\mid x\in Ak\}. and the collapse boundary Homotopy invariance: it is unchanged under homotopies through paths with fixed admissible endpoints.. The source-domain setting in mathematics logic statistics matters because If an oscillatory integral is written using two different phase functions that. and In the theory of Fourier integral operators, phase functions may be replaced under changes of coordinates or generating functions. specify where those roles have literal occupants. The strongest negative controls are The node requires the specific identity stated by Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. and Different authors adopt slightly different normalizations, especially when comparing the Maslov index of Lagrangian paths, the Maslov class, and the Conley-Zehnder index.; a case satisfying either exclusion should not be rescued merely because its label or examples look familiar.

Terminal adjudication sequence. First, bind the claimed instance to a concrete carrier and state the criterion by which Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. is recognized. Second, vary implementation, scale, notation, and example while holding When an oscillatory solution passes through a caustic, the phase jumps by a multiple of \pi/2. fixed; persistence supports one identity rather than several topic fragments. Third, remove At a transverse crossing time t for which \gamma(t)\cap L0\neq 0 , the sign is determined by the crossing form, a quadratic form on the intersection \gamma(t)\cap L0. or trigger Homotopy invariance: it is unchanged under homotopies through paths with fixed admissible endpoints. and verify that the classification fails. Fourth, compare the result with the two negative controls instead of relying on name similarity. Fifth, check scope against If an oscillatory integral is written using two different phase functions that. and record any qualification supplied by mathematics logic statistics. Finally, audit the graph claim. The approved unparented placement prevents a weak lexical resemblance from becoming a false ontological claim; a later edge must preserve every constitutive role stated here. This sequence makes the entry rejectable, keeps analogy separate from literal transfer, and exposes which fact would require revision.

Structural–Framed Character¶

Maslov index is structural-leaning. Its structural side is the repeatable organization summarized by Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: At a transverse crossing time t_* for which \gamma(t_)\cap L_0\neq 0 , the sign is determined by the crossing form, a quadratic form on the intersection \gamma(t_)\cap L_0. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Measurement. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: Fixing the horizontal axis as reference, the Maslov cycle consists of the single point represented by that axis. When an oscillatory solution passes through a caustic, the phase jumps by a multiple of \pi/2. It further constrains recognition and variation through: At a transverse crossing time t for which \gamma(t)\cap L0\neq 0 , the sign is determined by the crossing form, a quadratic form on the intersection \gamma(t)\cap L0. Then the edge AkA{k+1} in the polygon is parameterized by u\in[0,1] , {x + uSx\mid x\in Ak}.

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Maslov index literal. Its documented scope includes the condition that If an oscillatory integral is written using two different phase functions that. Another bounded application condition is that In the theory of Fourier integral operators, phase functions may be replaced under changes of coordinates or generating functions. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—Homotopy invariance: it is unchanged under homotopies through paths with fixed admissible endpoints.—and future graph densification may discover a defensible relation only if it preserves that boundary.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Maslov index. The reviewed identity is: Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Neighborhood in Abstraction Space¶

Maslov index sits in a moderately populated region (54^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Continuum Mechanics & Field Models (42 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Measurement. The parent omits the specialist differentia. Tell: Can the case establish Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis?
Hosoya Index. Count every matching of a graph, including the empty matching, to obtain a graph invariant used in matching theory and as a molecular topological descriptor. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Einstein–Brillouin–Keller method. A semiclassical quantization method assigning action-integral conditions, including Maslov phase corrections, to invariant tori of integrable classical systems. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Hyper-Wiener Index. Compress a connected molecular graph's shortest-path distance distribution into a scalar by summing both distance and squared distance over unordered vertex pairs. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Maslov index remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Measurement?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Maslov_index (revision 1351136987).
Preserved source candidate: https://dlmf.nist.gov/9.7

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.