{"id":"4c24b448-d397-410a-85e7-b26b7b242799","arxiv_id":"2509.08082","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The complex Weyl correspondence is shown to be a Stratonovich-Weyl correspondence for the generic representations of the generalized diamond group, with explicit Berezin and Weyl symbols.","lead":"This paper builds explicit quantization maps, called Stratonovich-Weyl correspondences, for a family of solvable groups known as generalized diamond groups. It gives closed formulas for the symbols of the representation operators and recovers known Moyal product identities, offering a concrete testbed for geometric quantization.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (4.3) omits complex conjugates, so the anti-holomorphic/holomorphic dichotomy in Proposition 4.1 is false as printed; the classification is likely repairable but the proof needs correction before the covariance and Stratonovich-Weyl results rest on it.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: Proposition 4.1's printed equation (4.3) is missing complex conjugates, and the anti-holomorphic conclusion in its proof is false as written. This matters because Proposition 4.1 is the only place where the form of σ(t) is derived; Proposition 5.3's covariance and Proposition 7.2's Stratonovich-Weyl conclusion both depend on it. I checked that with the conjugates restored the functional equation has the stated holomorphic/anti-holomorphic dichotomy and yields the claimed extension, so the theorem statement is correct but the proof as printed is incomplete. I also noted the printed group law in Section 4 writes the second slot of ω as t·z' instead of \\overline{t·z'}; this is another fixable typo that does not affect the intended semidirect product used in the rest of the paper. The Corollary 9.3 solvability issue mentioned in the reader's rationale is a condition on an application, not on the central Stratonovich-Weyl claim. Since the central claim appears mathematically sound and independently derivable, the appropriate disposition is to require the corrections rather than reject; I therefore leave the reader's CONDITIONAL verdict unchanged.","tokens_in":18499,"tokens_out":18363,"duration_ms":155113,"concrete_test":"Independently recompute equation (4.3) from the formula for ρ_λ in Section 2: the left factor must be exp(λ/2 \\overline{t·z0} z). Substitute w=0 and then z0→w to obtain b_t(z,w) exp(−λ/2 \\overline{t·w} z) = b_t(z−t·w,0). Verify that the left side is anti-holomorphic in w and the right side holomorphic in w; the equality then forces b_t(z,w)=χ(t) exp(λ/2 \\overline{t·w} z), and inserting this into (4.2) gives exactly σ(t)f(z)=χ(t) f(t^{-1}·z). Alternatively, derive the same classification from Stone-von Neumann by noting that Hom(ρ_λ, ρ_λ∘t^{-1}) is one-dimensional and contains U_t f(z)=f(t^{-1}z); imposing σ(t+s)=σ(t)σ(s) forces χ to be a unitary character.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing spot is Proposition 4.1's derivation of the intertwining equation (4.3). From the definition of ρ_λ, the left factor in (4.3) should be exp(λ/2 \\overline{t·z0} z), not exp(λ/2 (t·z0) z). With the printed version, after w=0 and z0→w the equation reads b_t(z,w) exp(−λ/2 (t·w)z) = b_t(z−t·w,0); the exponential factor is holomorphic in w, so the left-hand side is a product of an anti-holomorphic kernel and a holomorphic exponential and is not anti-holomorphic. The claimed dichotomy with the holomorphic right-hand side therefore does not force the kernel's form. With the conjugate restored, both factors on the left are anti-holomorphic, the right side is holomorphic, and the equality forces b_t(z,w)=χ(t) exp(λ/2 \\overline{t·w} z), which yields σ(t)f(z)=χ(t) f(t^{-1}·z). This classification underpins the covariance propositions in Section 5 and the Stratonovich-Weyl conclusion in Proposition 7.2, so as printed the proof chain is incomplete. The defect is a repairable typo, not an error in the statement: the same classification follows from Stone-von Neumann, since Hom(ρ_λ, ρ_λ∘t^{-1}) is one-dimensional and contains U_t f(z)=f(t^{-1}z). A second typo of the same kind appears in the printed multiplication law for G, where the second slot of ω should be \\overline{t·z'}, not t·z'; the later formulas use the correct semidirect product.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the generalized diamond group G = R^m ⋉ H_n, where R^m acts on C^n by diagonal phase rotations. It constructs generic unitary representations π of G on the Fock space F_λ by extending the Bargmann-Fock representations of the Heisenberg group, and it computes the Berezin and complex Weyl symbols of the representation operators π(g) and dπ(X). The main structural claim is that the complex Weyl correspondence W_0 is a Stratonovich-Weyl correspondence for the triple (G,π,C^n), and that via a moment map it becomes a Stratonovich-Weyl correspondence on the coadjoint orbit O(ξ_0). The paper also develops a Schrödinger model, derives Mehler-type kernel formulas, and applies the results to recover the Moyal product of Gaussians and to compute star exponentials of quadratic polynomials.","tokens_in":18788,"tokens_out":27680,"duration_ms":211820,"significance":"If the central claims hold, the paper gives explicit, covariant quantization maps on coadjoint orbits for a family of solvable groups, with closed formulas for the symbols of representation operators. This is a natural extension of the author's earlier work on the diamond group and the complex Weyl calculus, and the paper is valuable for bringing together the Berezin calculus, the complex Weyl correspondence, and the orbit method in a concrete setting. The final symbol formulas, once the conjugate typos are corrected, are consistent with known identities: W_0(I)=1, the covariance identities in Section 5 are coherent, and the Gaussian star-product computation in Section 9.2 recovers the classical Moyal formula. The paper does not ship machine-checked proofs, but it does provide explicit integral formulas that are independently verifiable. The main weakness is not the mathematical strategy but the number of missing complex conjugates in load-bearing formulas, which currently make the proof of the classification of extensions and the displayed symbol formula invalid as printed.","major_comments":[{"comment":"Equation (4.3) is missing a complex conjugate: the first exponential on the left should be exp(λ/2 z \\overline{t·z0}), not exp(λ/2 (t·z0) z). As printed, after setting w=0 and replacing z0 by w, the left-hand side is a product of an anti-holomorphic kernel and a holomorphic exponential, so it is not anti-holomorphic; the claimed dichotomy with the holomorphic right-hand side does not force the kernel to have the stated form. With the conjugate restored, the anti-holomorphic/holomorphic argument works, but the proof must also justify that the resulting factor C(z) is independent of z; this can be done by substituting the solution back into (4.3) or by invoking the one-dimensionality of Hom(ρ_λ, ρ_λ∘t^{-1}) from the Stone-von Neumann theorem. This correction is load-bearing: Proposition 4.1 underpins the covariance results in Section 5 and the Stratonovich-Weyl conclusion in Proposition 7.2.","section":"§4, Eq. (4.3)"},{"comment":"The displayed multiplication law for G is printed with the second slot of ω as (t·z', t·z'), but it should be (t·z', \\overline{t·z'}). As written, the third component c+c'+1/2 ω((z,\\bar z),(t·z',t·z')) is not real-valued in general, so the formula does not define a real Lie group law. The later formulas in the paper implicitly use the corrected semidirect product, so this is a typo, but it appears at the definition of the central object of the paper and must be fixed.","section":"§4, multiplication law"},{"comment":"The displayed formula for W_0(π(g))(z) is missing conjugates in the first factor of the quadratic exponential: it should read exp( λ/2 ( \\overline{t^{-1}·z0} + 2\\bar z ) (I_n + A(t))^{-1} ( t^{-1}·z0 + 2z ) ). As printed, taking t=0 and z0=0 gives W_0(I)(z) = exp(-λ|z|^2 + λ z^2), which contradicts the Stratonovich-Weyl normalization W_0(I)=1. The same missing conjugate appears in the definition of I(t,z0,z) and of u in the proof. The correction is also needed for consistency with the Section 9.2 formula for W_0(σ(t)), which contains |z_k|^2 rather than z_k^2; this is a load-bearing computation for the later applications.","section":"§6, Proposition 6.3"}],"minor_comments":[{"comment":"The kernel of σ'(t) is denoted b_t(x,y) instead of b'_t(x,y) in the proof, and the displayed double integral defining it contains a spurious φ(y) factor in the integrand; both are typos that make the derivation harder to follow.","section":"§8, proof of Proposition 8.1"},{"comment":"The formula \"α_k(t) = − 2/λ b_k\" is ambiguous; it should read α_k(t) = −(2/λ)b_k to be consistent with the subsequent choice of c = c_0/λ + (1/λ^2)∑ b_k.","section":"§9.3, proof of Corollary 9.3"},{"comment":"The notation L^2(F_λ) for the Hilbert space of Hilbert-Schmidt operators on F_λ is unusual and collides with the L^2 notation for function spaces; consider using HS(F_λ) or S_2(F_λ).","section":"§2, after Eq. (2.1)"},{"comment":"The word \"Melher\" should be \"Mehler\" in the sentence introducing the Mehler-type formula.","section":"§8, final paragraph before Proposition 8.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is a direct continuation of the author's prior work on the diamond group and the complex Weyl calculus, and the extension to G = R^m ⋉ H_n is natural. The three missing-conjugate errors are localized and repairable, but they occur in load-bearing positions: the classification of extensions, the definition of the group law, and the main symbol formula. I recommend asking the author to correct these formulas and to add the missing step in Proposition 4.1 showing that the factor C(z) is constant; with those changes the central claims appear sound. The paper fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what it advertises: it constructs generic representations of the generalized diamond group G = R^m ⋉ H_n by extending the Heisenberg representations, proves covariance of the Berezin and complex Weyl correspondences, computes explicit symbols for the representation operators, and shows the complex Weyl map is a Stratonovich–Weyl correspondence with respect to coadjoint orbits. The genuinely new part is the family R^m ⋉ H_n with independent rotation angles α_k, which forces the matrix-valued Gaussian integrals; those computations look right. The formulas for W_0(π(g)) and the Mehler-type kernel in Section 8 are real additions, and the sanity checks check out: W_0(I)=1 and the recovered Moyal product of Gaussians are the known results. The paper leans heavily on the author’s earlier work, but that is legitimate here because the earlier results are used as black boxes for the Heisenberg group; the new claims are derived, not assumed.\n\nThe soft spots are real but repairable. The stress-test note is correct about equation (4.3): as printed, the first exponential should contain \\overline{t·z_0} z, not (t·z_0) z. Without the conjugate, the left-hand side is not anti-holomorphic in w, so the dichotomy with the holomorphic right-hand side does not force the kernel’s form. This matters because Proposition 4.1 underpins covariance and therefore the whole Stratonovich–Weyl conclusion. The classification itself is true and can be salvaged cleanly by Stone–von Neumann: Hom(ρ_λ, ρ_λ∘t^{-1}) is one-dimensional and contains the shift operator, so σ(t) is that shift up to a character. The paper does not supply that argument, so a referee should ask for it or for a corrected derivation. There is also a matching missing conjugate in the printed multiplication law in Section 4, though the subsequent formulas use the correct semidirect product.\n\nA smaller issue: in Corollary 9.3, the proof silently assumes one can choose t ∈ R^m with α_k(t) = -2/(λ b_k) for all k. Since the α_k are fixed linear forms, for arbitrary b_k there may be no such t. This is an existence gap in an application, not in the main correspondence results, and it is fixable by stating the result for the image of the map R^m → R^n given by t ↦ (α_1(t), …, α_n(t)) or by adjusting the setup.\n\nWho is this for? People working on quantization of solvable groups, orbit-method test cases, and Berezin–Weyl calculus. It deserves a serious referee: the central argument is correct in substance, the typos are minor in the sense that the statements are true and repairable, and the applications are concrete. My recommendation: send it to peer review, and ask the referee to verify the corrected equation (4.3) and to check the solvability assumption in Corollary 9.3. Conditional acceptance after those fixes would be appropriate.","headline":"Extends the complex Weyl correspondence to the generalized diamond group with explicit symbol formulas and a Stratonovich–Weyl correspondence; the main results hold up, but a conjugate-sign typo in the key extension proof and a small existence gap in one corollary need fixing before the paper is fully rigorous.","tokens_in":19396,"tokens_out":1847,"would_cite":true,"duration_ms":18709,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E45","22E70","81R05","81S10","81R30"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every generic representation of the generalized diamond group $G=\\mathbb{R}^m\\ltimes H_n$ on the Fock space, the complex Weyl correspondence $W_0$ is a Stratonovich-Weyl correspondence: a $G$-covariant, tracial isomorphism from…","keywords":["complex Weyl correspondence","Stratonovich-Weyl correspondence","generalized diamond group","Fock space","Bargmann-Fock representation","Berezin quantization","coadjoint orbits","Moyal product"],"falsifier":"Take the exact intertwining relation (4.1) with the correct kernels: for $h=(z_0,c_0)$ the left hand side contains $\\exp\\big(\\tfrac{\\lambda}{2}\\overline{t\\cdot z_0}\\, z\\big)$ and the right hand side contains $\\exp\\big(-\\tfrac{\\lambda}{2}\\overline{w}\\, z_0\\big)$; substitute $w=0$ and solve for $b_t(z,w)$. The classification claim holds exactly if the unique solution is $b_t(z,w)=\\chi(t)\\exp\\big(\\tfrac{\\lambda}{2}\\overline{t\\cdot w}\\, z\\big)$; any other solution disproves it.","tokens_in":18208,"feed_emoji":"🧮","tokens_out":10036,"duration_ms":82850,"temperature":0.7,"pith_summary":"This paper extends the Stratonovich–Weyl quantization program from the Heisenberg group and the classic diamond group to the whole family of generalized diamond groups $G=\\mathbb{R}^m\\ltimes H_n$. Its main claim is that, for each generic representation $\\pi$ of $G$ on the Fock space $F_\\lambda$, the complex Weyl correspondence $W_0$ satisfies the four defining axioms: it maps the identity to 1, respects conjugation, is covariant with respect to $\\pi$, and is isometric/tracial. The paper also gives explicit formulas for the Berezin and complex Weyl symbols of all representation operators and their Lie-algebra counterparts, and uses them to recover known Moyal-product formulas and to compute star exponentials of quadratic polynomials. If the main claim is right, the generalized diamond groups become one of the few families of solvable groups for which quantization on coadjoint orbits is realized by closed, computable maps.","feed_headline":"Complex Weyl map quantizes all generic diamond-group representations","feed_subtitle":"One covariance map sends Fock-space operators to functions on coadjoint orbits, with closed symbol formulas.","key_machinery":"The machinery has three components. The first is the Fock-space model: $F_\\lambda$ consists of holomorphic functions on $\\mathbb{C}^n$ square-integrable against $e^{-\\lambda|z|^2/2}d\\mu_\\lambda(z)$, with coherent states $e_z(w)=\\exp(\\lambda\\bar z w/2)$. The second is the integral formula $W_0(A)(z)=2^n\\int k_A(z+w,z-w)\\exp\\big(\\tfrac\\lambda2(-z\\bar z-w\\bar w+z\\bar w-\\bar z w)\\big)d\\mu_\\lambda(w)$ for the complex Weyl symbol, which extends $W_0$ beyond trace-class operators and is the unitary part of the polar decomposition of the Berezin map. The third is the equivariant moment map $\\psi$ above, whose covariance under the coadjoint action lifts the $G$-action on $\\mathbb{C}^n$ to the orbit $O(\\xi_0)$; this is what turns the covariance of $W_0$ into the orbit picture that defines a Stratonovich–Weyl correspondence.","core_discovery":"The central discovery is that the map $W_0$ constructed from the Stratonovich–Weyl quantizer $\\Omega_0(z)=\\rho_\\lambda(z,0)R_0\\rho_\\lambda(z,0)^{-1}$ is a Stratonovich–Weyl correspondence for the triple $(G,\\pi,\\mathbb{C}^n)$, and that, after composing with the moment map $\\psi:\\mathbb{C}^n\\to \\mathfrak{g}^*$, it becomes one for the triple $(G,\\pi,O(\\xi_0))$ on a genuine coadjoint orbit. The proof combines: (a) a classification of all unitarizable extensions of $\\rho_\\lambda$ to $G$, each of the form $\\sigma(t)f(z)=\\chi(t)f(t^{-1}\\cdot z)$; (b) covariance identities for both the Berezin symbol $S_\\lambda$ and the complex Weyl symbol $W_0$ under the $G$-action $g\\cdot z=t\\cdot z+z_0$; and (c) the identification $\\psi(z)=(-i\\,d\\chi+\\tfrac12\\sum_k(1-\\lambda|z_k|^2)\\alpha_k,\\ -\\lambda z,\\ \\lambda)$, a $G$-equivariant bijection onto $O(\\xi_0)$. On the way the paper computes the kernels and symbols of $\\pi(g)$ and $d\\pi(X)$ in closed form.","pith_inferences":["Editorial: the same scheme should work when the action of $\\mathbb{R}^m$ on $\\mathbb{C}^n$ is any unitary action diagonalized by characters, not just the phase action $e^{i\\alpha_k(t)}$; the Gaussian-integral technology would then produce analogous closed formulas for a wider class of solvable semi-direct products.","Editorial: because $\\psi$ exhibits $\\mathbb{C}^n$ as the coadjoint orbit $O(\\xi_0)$, the construction suggests the complex Weyl correspondence is the quantization map attached to the complex polarization defining the Fock model; the formulas could be recast as a comparison of two polarizations, a reading the paper does not spell out.","Editorial: the star-exponential formulas for quadratic Hamiltonians should reproduce unitary evolution generated by the corresponding Weyl operators; checking the $\\tan$/$\\cos$ expressions against direct time-evolution calculations would give a clean independent test of the whole chain."],"forward_implications":["For each generic $\\pi$, the operator-to-function map $W_0$ gives a concrete quantization of the coadjoint orbit $O(\\xi_0)$, with the constant function 1 corresponding to the identity and the trace pairing matched to the Hilbert–Schmidt inner product.","The covariance identity means quantizing first and translating on $\\mathbb{C}^n$ is the same as conjugating by the representation first; this is exactly what makes the correspondence geometric rather than an arbitrary symbol choice.","In the Schrödinger model, the corresponding map $W_1$ is a Stratonovich–Weyl correspondence on $\\mathbb{R}^{2n}$, and the identity $W_1(W(f))(x,y)=f(x,\\lambda y)$ ties the construction to ordinary Weyl calculus and to the Moyal product.","The group law itself yields star-product identities: the relation $\\sigma(t+t')=\\sigma(t)\\sigma(t')$ becomes a Moyal/Gaussian identity for exponentials of quadratic phase functions, reproducing the classical formula for the Moyal product of two Gaussians.","Star exponentials of quadratic polynomials are obtained in closed form from $\\exp_{*_0}(W_0(d\\pi(X)))=W_0(\\pi(\\exp X))$, giving the $\\cos$-$\\tan$ formulas of Corollaries 9.3 and 9.4."],"supporting_citations":[{"why":"Establishes the diamond-group case that this paper generalizes and supplies the result that W0 is the unitary part of the polar decomposition of Sλ.","marker":"[14]"},{"why":"Provides the extension-by-kernel method used in Proposition 4.1 and the Gaussian-integral lemma (Lemma 6.2) used for closed symbol formulas.","marker":"[15]"},{"why":"Gives the integral formula for W0 and the complex Weyl correspondence on the Fock space for Heisenberg and Heisenberg motion groups.","marker":"[12]"},{"why":"Supplies the definition of Stratonovich-Weyl correspondence and the axioms the paper verifies.","marker":"[23]"},{"why":"Introduces the notion of distribution on representation space that underlies Stratonovich-Weyl correspondences.","marker":"[35]"},{"why":"Provides the Berezin quantization calculus and covariant symbols that the paper extends and compares with W0.","marker":"[6]"},{"why":"Develops quantization in complex symmetric domains, the background for the Berezin correspondence on Fock space.","marker":"[7]"},{"why":"Supplies standard facts on the Heisenberg group, the Bargmann transform, Weyl calculus, and Gaussian integrals used throughout.","marker":"[22]"},{"why":"Provides the orbit-method context and the Stone-von Neumann theorem underlying the generic Heisenberg representations.","marker":"[28]"}],"fun_headline_variants":["Stratonovich–Weyl correspondence covers all generic diamond reps","Weyl map on orbits quantizes diamond-group reps","Closed symbol formulas for all diamond-group reps","Stratonovich–Weyl for every diamond-group rep","Explicit Weyl symbols via coadjoint orbit map"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification of extensions in Proposition 4.1 carries everything: the proof assumes every unitary extension of the Heisenberg representation to $G$ is given by a unitary character times the shift $f(t^{-1}\\cdot z)$, and the printed derivation of that classification solves an intertwining equation whose displayed form omits complex conjugates, so the classification needs a corrected computation to stand.","fun_headline_variants_meta":{"raw":{"variants":["Stratonovich–Weyl correspondence covers all generic diamond reps","Weyl map on orbits quantizes diamond-group reps","Closed symbol formulas for all diamond-group reps","Stratonovich–Weyl for every diamond-group rep","Explicit Weyl symbols via coadjoint orbit map"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000734,"raw_usage":{"total_tokens":3312,"prompt_tokens":1004,"completion_tokens":2308,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":620,"completion_tokens_details":{"reasoning_tokens":2225}},"tokens_in":620,"tokens_out":2308,"duration_ms":14885,"temperature":1.0,"reasoning_tokens":2225,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:13:06.212915+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the exact intertwining relation (4.1) with the correct kernels: for $h=(z_0,c_0)$ the left hand side contains $\\exp\\big(\\tfrac{\\lambda}{2}\\overline{t\\cdot z_0}\\, z\\big)$ and the right hand side contains $\\exp\\big(-\\tfrac{\\lambda}{2}\\overline{w}\\, z_0\\big)$; substitute $w=0$ and solve for $b_t(z,w)$. The classification claim holds exactly if the unique solution is $b_t(z,w)=\\chi(t)\\exp\\big(\\tfrac{\\lambda}{2}\\overline{t\\cdot w}\\, z\\big)$; any other solution disproves it.","supporting_citations":[{"cited_title":"A., Quantization in complex symmetric domains","cited_arxiv_id":null,"evidence_quote":"Develops quantization in complex symmetric domains, the background for the Berezin correspondence on Fock space."},{"cited_title":"Tsukuba J","cited_arxiv_id":null,"evidence_quote":"Establishes the diamond-group case that this paper generalizes and supplies the result that W0 is the unitary part of the polar decomposition of Sλ."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the extension-by-kernel method used in Proposition 4.1 and the Gaussian-integral lemma (Lemma 6.2) used for closed symbol formulas."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the integral formula for W0 and the complex Weyl correspondence on the Fock space for Heisenberg and Heisenberg motion groups."},{"cited_title":"M., Generalized Moyal quantization on homogeneous symplectic spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of Stratonovich-Weyl correspondence and the axioms the paper verifies."},{"cited_title":"L., On distributions in representation space","cited_arxiv_id":null,"evidence_quote":"Introduces the notion of distribution on representation space that underlies Stratonovich-Weyl correspondences."},{"cited_title":"A., Quantization","cited_arxiv_id":null,"evidence_quote":"Provides the Berezin quantization calculus and covariant symbols that the paper extends and compares with W0."},{"cited_title":"Princeton Univ","cited_arxiv_id":null,"evidence_quote":"Supplies standard facts on the Heisenberg group, the Bargmann transform, Weyl calculus, and Gaussian integrals used throughout."},{"cited_title":"A., Lectures on the orbit method","cited_arxiv_id":null,"evidence_quote":"Provides the orbit-method context and the Stone-von Neumann theorem underlying the generic Heisenberg representations."}],"review_version":2}