The symplectic form and Sp(2d, ℝ)

The symplectic form ω is the antisymmetric pairing of positions with momenta that underlies Hamiltonian mechanics and the quantum commutator; in the plane of one position and one momentum it measures signed area. The symplectic group Sp(2d, ℝ) is the set of linear maps of phase space (d positions, d momenta) that preserve it. The group contains rotations of phase space — the quarter-turn Born reciprocity is one — together with stretches that lengthen one direction and shorten another, called boosts and squeezes. In the program ω is the one structure held fixed: its corrected postulate reads the rigidity of ω as the constancy of Planck’s constant, and every other piece of phase-space geometry is allowed to vary.

Status: Established — standard mathematics [T]; its rigidity in the program is a postulate [P] (P2′, #11 §4) — the toy model’s invariance under a generic symplectic matrix is machine-checked [V] in verify/toy_coset.py (see the toy-model scripts).

Definition#

A symplectic form on phase space ℝ^{2d} is an antisymmetric, non-degenerate pairing ω(u, v) = uᵀωv of two phase-space vectors [T]. In the (x, p) block basis it takes the standard form

ω = [[0, 1], [−1, 0]]          (d × d blocks)
ω(u, v) = u_x · v_p − u_p · v_x

where u = (u_x, u_p) splits a vector into its position and momentum parts. The lab notebook writes this form with d = 4 (#1 §2); the toy-model note uses d = 1, where the blocks are single numbers (§1).

Three properties make ω the structure of mechanics [T]:

  • Antisymmetry. ω(u, u) = 0, so ω measures oriented area rather than length. For d = 1, ω(u, v) is the signed area of the parallelogram spanned by u and v.
  • Non-degeneracy. Only the zero vector pairs to zero with every vector. Each position direction is paired with its own momentum direction and with nothing else.
  • Dynamics. Hamilton’s equations use ω to turn the gradient of the energy into a velocity in phase space, and the Poisson bracket {x, p} = 1 is the same pairing written for functions.

Quantization turns the Poisson bracket into the commutator, so [x, p] = iℏ is the quantum form of ω (the canonical commutator).

The symplectic group#

A linear map S of phase space is symplectic when it preserves the pairing, ω(Su, Sv) = ω(u, v) for all u and v. In matrices:

Sᵀ ω S = ω

These matrices form the group Sp(2d, ℝ) [T]. Three facts about it matter here [T]:

  • Volume. Every symplectic matrix has det S = 1, so symplectic maps preserve phase-space volume. Hamiltonian time evolution is symplectic, so it preserves phase-space volume too: Liouville’s theorem.
  • One degree of freedom. For d = 1 the condition reduces to det S = 1, so Sp(2, ℝ) = SL(2, ℝ), the real 2 × 2 matrices of unit determinant. verify/toy_coset.py uses the generic element S = [[s₁, s₂], [s₃, (1 + s₂s₃)/s₁]], whose determinant is 1 by construction, and checks “generic S is symplectic”.
  • Quantum implementation. In quantum mechanics each symplectic map is carried out by a unitary operator, unique up to a phase (the metaplectic representation). For ρ that operator is the Fourier transform. Linear symplectic maps leave the commutator unchanged because they preserve ω, and the unitary operator that carries out each one is why they count, ordinarily, as changes of description; see unitary equivalence versus Noether symmetry.

Rotations, boosts and squeezes#

For d = 1 every element of Sp(2, ℝ) combines a rotation with a stretch [T]. Three one-parameter families appear in the toy-model note and its review:

rotation   R(θ) = [[cos θ, sin θ], [−sin θ, cos θ]]     compact: the circle SO(2)
squeeze    diag(a, 1/a)                                  stretches x by a, compresses p by 1/a
boost      S(t) = [[cosh t, sinh t], [sinh t, cosh t]]   the same stretch along the diagonals x ± p

The boost matrix is the one the referee of review round 4 used (“S(t)=[[cosh t, sinh t],[sinh t, cosh t]] (symplectic: True)”, audit notes Step 2(a)). It stretches the direction (1, 1) by eᵗ and compresses (1, −1) by e⁻ᵗ [T], so a boost is a squeeze along the diagonals. The referee’s notes call diag(a, 1/a) a boost as well (see flag 10 in review/wiki-build-flags.md); this page follows the toy note, which calls it a squeeze.

The quarter-turn ρ = [[0, 1], [−1, 0]] is the rotation R(π/2). It “is symplectic and squares to −1” (toy-model note §1), and ρ² = −1 is the half-turn, the parity (x, p) → (−x, −p) (#1 §2). The rotations form the compact subgroup SO(2), a circle. Squeezes and boosts are non-compact: their parameter runs to infinity [T].

Why compactness matters#

By the Mermin–Wagner–Coleman theorem, a continuous compact symmetry cannot break spontaneously in a two-dimensional system [T]. The toy-model note applies it to the rotations only: “In 1+1 the compact angle φ cannot order: Mermin–Wagner [13, 14] applies to the compact SO(2) containing ρ. The non-compact half-plane sector is a hyperbolic model, where symmetry breaking in 2D does occur [15]. Whether ρ’s SO(2) stays unbroken in the full coupled model is open.” (§4). See Open: does ρ’s SO(2) break in the coupled model?

The boosts are where the first ρ-preserving potential of model I, the toy note’s coset sigma model, failed. “Along H = SᵀS, K = S⁻¹K_ref S with S a boost, the compatibility term stays zero while the potential reaches −8.9 × 10⁶ μ at boost parameter 4” (toy-model note §4). That naive potential is dead as a stabilizing potential (Obituary: the naive ρ-preserving potential). Its replacement, the bounded potential V_b, penalizes boosts: “the boost family rises from −1 to −0.5” (round-4 adjudication, F2). Along the squeezes it keeps a flat direction: “The zero mode is the classical flat line H = diag(y, 1/y), K = K_ref. It is not symmetry-protected, because V_b breaks the squeeze away from K = ±K_ref. Boosts are penalized.” (toy-model note §4). The round-4 adjudication had called this zero mode an exact symmetry; the toy note’s v2, which ranks above the adjudication, says it is not symmetry-protected (see flag 12 in review/wiki-build-flags.md; conflicts between sources, entry C28).

The referee also worked out, with sympy, which maps of Sp(2, ℝ) the naive potential leaves unbroken. A conjugation S⁻¹K_ref S that returns +K_ref forces S = diag(a, 1/a); one that returns −K_ref forces S = [[0, −1/c], [c, 0]]. So the potential “leaves a non-compact boost subgroup unbroken plus {+-1, +-rho} (Z4) in SO(2)” (audit notes Step 2(c), capitals lowered here; its “boost subgroup” is the squeeze family diag(a, 1/a), see flag 10 in review/wiki-build-flags.md): among the rotations only ±1 and ±ρ survive.

How the toy models use the group#

Model I, the coset sigma model of a phase-space metric H and a real structure K (the order parameter), takes Sp(2, ℝ) as its symmetry group. Its metric field is itself a symplectic matrix: H(Z) = [[Y + X²/Y, X/Y], [X/Y, 1/Y]] for Z = X + iY in the upper half-plane (toy-model note §2; gate: “H(Z) is symplectic and I = w^-1 H squares to -1”). The group acts on both fields:

H → Sᵀ H S        K → S⁻¹ K S

“Sp(2,ℝ) acts by H → SᵀHS, K → S⁻¹KS, and the action is invariant (checked for generic symplectic S, so for ρ)” (§2) [V]. The gate checks each of the three terms of the Lagrangian separately (“H kinetic term invariant under generic symplectic S”, and likewise the K kinetic and potential terms), then ρ itself (“all three terms invariant under rho”). The vacua break the group: “the stabilizer of the vacuum (1, K0) in Sp(2,R) is {+1, -1} (so the orbit is 3-dimensional)”. See Model I: action and symmetries and the toy-model scripts.

Rigidity: the one fixed structure#

In the program’s postulates ω is the only structure that does not vary. “P2′ Only ω is rigid (symplectic rigidity ≡ constancy of ℏ). The neutral metric η is emergent — the order parameter” [P] (#11 §4; P2′; for η and the generalized metric H, see the structures ω, η, H). The calc note that forced the change put it as “Freeze ω only — that is genuinely ℏ” (#5 §B). The lab notebook’s gloss on the original postulate was “the symplectic structure is the constancy of ℏ — phase-space cells do not stretch” (#1 §1). The original P2 froze the neutral metric η as well; the compatibility theorem then left the program no content of its own (the nontriviality theorem).

What a rigid ω leaves free:

  • The Siegel domain. The positive generalized metrics H compatible with ω alone form the Siegel domain Sp(2d, ℝ)/U(d), coordinatized by a complex symmetric matrix Z = X + iY with Y > 0 (#5 §C; the Siegel domain). For d = 1 it is the upper half-plane of model I.
  • Möbius maps. Sp(2d, ℝ) acts on Z by fractional linear (Möbius) maps, and ρ acts as Z → −Z⁻¹ with fixed point Z = i [D] (#11 §6; reciprocity as modular inversion).
  • Shifts. “The rigid Sp(2d,ℝ) contains Z → Z + B (B symmetric)” (#11 §7.1, tagged [D + T]), the Möbius action of the block matrix [[1, B], [0, 1]] (#6 §1). It shifts the momentum graviton X = Re Z, so X “is shift-symmetric — axion-like, never auxiliary” (shift symmetry).
  • Unimodularity. With η also rigid, det H = 1 follows (#5 §A; the corollaries). #11 §5 adds, calling it machine-verified, that it “persists across the whole modular family”. Battery test T2 checks “det H_Z = 1” at one randomly sampled member of the family (d = 4), so the statement for the whole family is derived [D], not machine-verified (the verification battery).

A second symplectic space: charges#

#11 §11.3 finds the same structure on electric and magnetic charges. The Dirac–Schwinger–Zwanziger pairing ⟨q, q′⟩ = e₁g₂ − e₂g₁ ∈ 2πℤ “is an antisymmetric integer form: the charge lattice is a symplectic space [17†]” [D — structural; math T]. Duality frames are then Lagrangian polarizations of charge space, and S-duality is “generated by the same [[0,1],[−1,0]] rotation as Born’s ρ and our Z → −Z⁻¹: the third appearance of one matrix in this program”. See the symplectic charge lattice.

Standard references#

Added by the wiki from model memory (†); the program’s documents do not cite them.

  • V. I. Arnold, Mathematical Methods of Classical Mechanics (1978)†: symplectic forms, Hamiltonian flows, Liouville’s theorem.
  • G. B. Folland, Harmonic Analysis in Phase Space (1989)†: the symplectic group and the metaplectic representation.

The toy note’s own references for the two-dimensional theorems are Mermin–Wagner [13] and Coleman [14], and for the hyperbolic exception Niedermaier–Seiler and Duncan–Niedermaier–Seiler [15]; the round-4 referee found all of them on INSPIRE and supporting the statements they are cited for (audit notes, Step 1, under the note’s earlier numbering).

Used by#

Sources#

  • papers/broken-born-checkpoint-v2.md §4 — P2′: symplectic rigidity ≡ constancy of ℏ.
  • papers/broken-born-checkpoint-v2.md §5 — the compatibility theorem, with ω and η standard; unimodularity across the modular family.
  • papers/broken-born-checkpoint-v2.md §6, §7.1 — ρ as Z → −Z⁻¹; the rigid group contains Z → Z + B.
  • papers/broken-born-checkpoint-v2.md §11 — the charge lattice as a symplectic space.
  • papers/reciprocity-toy-models.md §1, §2, §4 — ρ symplectic; the Sp(2, ℝ) action of model I; boosts, squeezes and the compact SO(2).
  • review/round4-adjudication.md F1–F2 — the boost family and the bounded repair.
  • review/round4/referee/audit-notes.md Step 1, Step 2(a), Step 2(c) — the two-dimensional references checked; the boost matrix; the residual symmetry of the naive potential.
  • notes/debt1a-geometric-core.md §A–§C — det H = 1; “Freeze ω only”; the Siegel domain of ω-compatible metrics.
  • notes/debt1a-ii-X-classification.md §1 — the Möbius subgroup Z → Z + B of the rigid group.
  • notes/broken-born-theory.md §1, §2 — the original P2’s gloss; the standard block form of ω; ρ² as parity.
  • verify/toy_coset.py — invariance of model I under a generic symplectic S and under ρ; the vacuum’s stabilizer.
  • verify/battery.py T2 — det H_Z = 1 at one sampled member of the modular family.