The Lagrangian Grassmannian Λ(d)

The Lagrangian Grassmannian Λ(d) is the space of all ways to split a 2d-dimensional phase space into a “position” half and a “momentum” half. Each Lagrangian subspace, a d-dimensional plane on which the symplectic form vanishes, is one point of it. For d = 1 it is the set of lines through the origin of the (x, p) plane, which is a circle; in general it is the compact manifold U(d)/O(d), of dimension 10 when d = 4. It contains a loop that cannot be shrunk to a point, the Maslov circle, so any angle measured around that loop is periodic. It also forms the distinguished (Shilov) boundary of the Siegel domain. In the program it is the space of possible vacuum polarizations, and its circle direction carries the dark-energy field. The mathematics is standard.

Status: Established — standard mathematics [T] (#1 §2 tags the definition, the dimension and π₁ = ℤ [T]; #11 §6 tags the Shilov-boundary statement [T]; calc note #5 marks it “[standard]”) — not machine-checked as such; its d = 1 case, Λ(1), is the compact Goldstone direction of the toy model checked in verify/toy_coset.py.

What it is#

Lagrangian subspaces#

Phase space ℝ^{2d} carries the symplectic form ω, which pairs each position with its momentum: with ω = [[0, 1], [−1, 0]] in the (x, p) block basis, ω((x, p), (x′, p′)) = x·p′ − p·x′ [T]. A Lagrangian subspace is a d-dimensional subspace on which ω vanishes identically; d is the largest dimension for which that is possible [T]. The position plane {p = 0} and the momentum plane {x = 0} are Lagrangian. So is the graph p = Ax of any real symmetric d × d matrix A, and these graphs are all the Lagrangian subspaces that meet the momentum plane only at the origin [T].

Choosing one Lagrangian subspace to call “position” is choosing a polarization: it fixes which coordinates a wavefunction is written in, as a waveform is written as a function of time and its spectrum as a function of frequency (the Fourier transform; Lagrangian subspaces and polarizations). The toy-model note states the d = 1 case: the null lines of a neutral metric “are Lagrangian. Choosing K is choosing which line is ‘position’” (§1).

The space of them: U(d)/O(d)#

Lagrangian subspaces form a manifold, the Lagrangian Grassmannian. #1 §2: “The space of Lagrangian polarizations of ℝ^{2d} is the Lagrangian Grassmannian Λ(d) ≅ U(d)/O(d), compact, of dimension d(d+1)/2 = 10 for d = 4. [T]”

The form U(d)/O(d) comes from a symmetry argument [T]. Phase space can be identified with ℂ^d, positions as real parts and momenta as imaginary parts. The unitary group U(d) then acts by maps that preserve both ω and the unit metric, and it carries the position plane ℝ^d onto every Lagrangian subspace. The unitary maps that carry the position plane onto itself are the real orthogonal matrices, O(d). Each Lagrangian subspace is therefore a unitary matrix taken up to an orthogonal factor on the right: Λ(d) = U(d)/O(d). Its dimension is dim U(d) − dim O(d) = d² − d(d − 1)/2 = d(d + 1)/2, the number of independent entries of a symmetric d × d matrix, as the graph picture suggests. It is compact because U(d) is.

For d = 1, U(1)/O(1) is the circle of unit complex numbers e^{iθ} with e^{iθ} and −e^{iθ} identified: the lines through the origin of the plane, a line at angle θ being the same line as at θ + π [T].

The notes write Λ(d) with d = 4 the physical case, and #11 writes Λ(4) when it means that case (§7: the second frame “confined to compact Λ(4)”). The two agree.

The Maslov class#

#1 §2: “Its fundamental group is π₁(Λ(d)) = ℤ, generated by the Maslov circle via the map det²: U(d)/O(d) → U(1). [T]”

What this says [T]. A unitary matrix U has a determinant det U of modulus 1. Multiplying U on the right by an orthogonal matrix O describes the same Lagrangian subspace but multiplies det U by det O = ±1. So det U is not a function on Λ(d), while its square det² U is. The map det² sends each Lagrangian subspace to a point of the unit circle. Along a closed loop of Lagrangian subspaces, the number of times det² winds around the circle is an integer, the Maslov index of the loop (Maslov 1965 †; Arnold 1967 †). Loops with different counts cannot be deformed into each other, loops with the same count can be, and every integer occurs, so π₁(Λ(d)) = ℤ. A loop that winds once is a Maslov circle.

For d = 1, the line at angle θ is U = e^{iθ} applied to the x-axis, and det² U = e^{2iθ}. Turning the line through a half-turn brings it back to itself and carries e^{2iθ} once around the circle: the half-turn of a line is the generating loop [T].

The remaining directions of Λ(d) leave det² unchanged. The points with a given value of det² form a copy of SU(d)/SO(d) [T], and moving within such a set changes the polarization without moving around the circle. At d = 4 these are the directions that #1 calls “the nine non-Maslov directions” (§8), beside the one circle direction.

An angle measured around the Maslov circle is periodic because of this topology, whatever dynamics act on it. The program uses exactly this for the dark-energy field: τ as the Maslov phase of the boundary. #1 §2 tags that periodicity “[D]”: “Periodicity is topology”.

The boundary of the Siegel domain#

The Siegel domain of complex symmetric matrices Z = X + iY with Y > 0 can be mapped by the Cayley transform W = (Z − i)(Z + i)⁻¹ onto a bounded domain, the “Siegel disk” of complex symmetric W with 1 − W̄W positive definite [T]. A bounded domain has a distinguished part of its boundary, its Shilov boundary: the smallest closed part of the boundary on which every function holomorphic inside and continuous up to the edge reaches its largest absolute value [T]. For the unit disk (d = 1) it is the whole circle. For a product of two disks it is the torus of pairs of boundary points, smaller than the full boundary. For the Siegel disk it is the set of symmetric unitary matrices [T] (Hua 1963 †).

That set is Λ(d) [T]. Every symmetric unitary matrix can be written as W = UUᵀ with U unitary, and two unitary matrices give the same W exactly when they differ by an orthogonal factor on the right, so the symmetric unitary matrices are U(d)/O(d). On them det W = det(U)², so the Maslov map det² of the previous section is the determinant of the boundary point. In the original half-space picture the real symmetric matrices, the limit Y → 0, form a dense part of this boundary: one chart of Λ(d) whose points, like the graphs of the first section, are labelled by real symmetric matrices. The remaining Lagrangian subspaces are limits at infinity [T].

The program’s statements are brief. Calc note #5: “Λ(d) is the Shilov boundary of the Siegel domain [standard]” (§C). #11 §6: “Λ(d) is the Shilov boundary of the Siegel domain [T]”. Neither document gives a reference. The program reads the boundary as the place where the dark-energy field lives (τ as the Maslov phase).

The space of vacuum polarizations#

Lost under strict P2, restored under P2′#

Under the original P2, which froze both ω and the neutral metric η, calc note #5 found: “The space of polarizations compatible with both structures (ω-Lagrangian and η-null) is discrete — the 2^d coordinate-type splittings, GL(d)-rigid. No Lagrangian Grassmannian, no Maslov circle, no τ field” (§B; on “discrete”, see flag 1 in review/wiki-build-flags.md). This is part of the nontriviality theorem that forced the repair.

Under P2′, only ω is rigid, and “the vacuum spontaneously selects a real structure J (an involution anticommuting with I), and η ≡ HJ. The space of H-compatible real structures on (ℝ^{2d}, I) is {J} ≅ U(d)/O(d) = Λ(d), the Lagrangian Grassmannian — π₁ = ℤ, Maslov circle intact, τ restored” (#5 §B; #11 §4 writes P2′ with the same η = HJ). Here I = ω⁻¹H is the complex structure of Born geometry. The identification is standard [T]: such a J is fixed by its +1 eigenspace, which is a Lagrangian subspace, and every Lagrangian subspace arises this way. Choosing J is choosing which directions count as position, and the possible choices are the points of Λ(d). In the language of spontaneous symmetry breaking, Λ(d) is the vacuum manifold of the polarization.

A note on letters. This page follows #5 and #11’s P2′ in calling the vacuum’s real structure J, with η = HJ. The toy-model note calls it K, with η = ωK (§1), and so does #11 §14 item 11. No document reconciles the two namings; see the order parameter and notation.

Ten compact moduli#

At d = 4 the polarization has ten directions in which to vary. Calc note #7 makes this the basis of constrained bimetric gravity: “J carries only the 10 compact moduli of U(d)/O(d) — not a full second metric. So the ‘second vielbein’ E₀[J] is confined to a compact 10-dimensional moduli space” (§4). Calc note #5 counts the same ten as “Λ(d) σ-modes (10: τ plus nine, of which inverse-Higgs constraints likely eliminate most [O])” (§D; inverse-Higgs constraints); whether they survive is open. The lab notebook assumes that τ, the circle direction, is the light one: “that τ is the lightest modulus, with the other nine directions heavy, is assumed [P]” (#1 §2). Calc note #6 adds that none of the ten is absorbed when X acquires its mass: “The Λ(d) moduli, τ included, are not the eaten modes — they carry the compact phase directions, which constant shifts do not generate” (§3, a section tagged [D — structural], which flags the check for τ as [O]; τ is not eaten).

The toy model’s Λ(1)#

In the d = 1 coset model of the toy-model note, the vacua at H = 1 are the real structures K(φ) = [[cos φ, sin φ], [sin φ, −cos φ]], a circle of them (§2). “The potential’s Hessian has rank 1 at the vacuum. That leaves three Goldstone directions: the two non-compact half-plane directions, and the compact angle φ, which parametrizes the Lagrangian Grassmannian Λ(1)” (§2). The gate prints PASS potential Hessian has rank 1: three flat directions (Goldstones), one massive [[0, 4, 0, 0]] (verify/toy_coset.py; the toy-model scripts). The worklog: “The compact Goldstone is the rotation of K at fixed H: the Lagrangian Grassmannian Λ(1), which is the checkpoint’s Maslov circle (the τ field of §10)” (‘Chunk 1 results’). #11 §10: “In the coset model that angle parametrizes Λ(1): it is τ’s circle.” See model I vacua and the polarization as a Goldstone angle. Because Λ(1) is a single circle, the toy has no counterpart of the nine other directions.

Standard references#

  • V. P. Maslov, Theory of Perturbations and Asymptotic Methods (1965) † — the index that bears his name.
  • V. I. Arnold, “Characteristic class entering in quantization conditions”, Functional Analysis and Its Applications (1967) † — the Lagrangian Grassmannian as U(d)/O(d), its fundamental group ℤ and the det² description of the Maslov class.
  • L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains (English translation 1963) † — the Shilov boundaries of the classical domains.

The program’s documents cite none of these; they tag the facts [T] or “[standard]”.

Used by#

Sources#

  • notes/broken-born-theory.md §2 — definition, dimension and π₁ = ℤ, tagged [T]; τ the lightest modulus [P]; periodicity “is topology [D]”.
  • papers/broken-born-checkpoint-v2.md §6 — Λ(d) as the Shilov boundary [T].
  • papers/broken-born-checkpoint-v2.md §4, §7, §10, §14 item 11 — P2′ with the real structure J; Λ(4) as the physical case; Λ(1) as τ’s circle in the coset model; η = ωK in the toy-model item.
  • notes/debt1a-geometric-core.md §B–§D — the discrete polarizations under strict P2; {J} ≅ U(d)/O(d) = Λ(d) under P2′; the boundary fact “[standard]”; the ten σ-modes.
  • notes/debt1a-iii-tuning-test.md §4 — the ten compact moduli carried by J.
  • notes/debt1a-ii-X-classification.md §3 — the Λ(d) moduli are not the eaten modes.
  • notes/broken-born-theory.md §8 — “the nine non-Maslov directions”.
  • papers/reciprocity-toy-models.md §1, §2 — Lagrangian null lines; Λ(1) in model I’s Goldstone spectrum.
  • notes/toy-lagrangian.md ‘Chunk 1 results’ — the compact Goldstone as the checkpoint’s Maslov circle.
  • verify/toy_coset.py — the rank-1 Hessian: three flat directions, one massive.