{ "schema_version": "qft.bogoliubov-density-volume-scaling.v1", "artifact_id": "qft.artifact.curved-spacetime.states-hadamard-microlocal-control.bogoliubov-density-volume-scaling", "owner_page_id": "qft.topic.curved-states.bogoliubov-inequivalent-representations", "title": "Finite density does not imply a Hilbert-Schmidt homogeneous mixing map", "generated_by": "figures-src/curved-spacetime/bogoliubov-density-volume-scaling.mjs", "source_revision": 1, "generated_on": "2026-09-04", "reader_question": "Why can a spatially homogeneous Bogoliubov transformation have finite particle-number density but still fail the global Hilbert-Schmidt implementability criterion?", "takeaway": "The momentum integral controls density. In a periodic box the corresponding homogeneous Hilbert-Schmidt trace grows as L cubed, whereas a genuinely rank-one squeezed packet has the volume-independent trace b squared.", "conventions": { "spatial_dimension": "n=3 for the plotted benchmark; threshold statements are written for general n", "periodic_box": "side L, volume V=L^3, momenta k_m=2*pi*m/L", "dimensionless_size": "x=kappa*L", "gaussian_mixing": "beta(k)=b*exp[-k^2/(2*kappa^2)] with real b=1/2", "exact_bosonic_normalization": "alpha(k)=sqrt(1+|beta(k)|^2)", "pointwise_identity": "|alpha(k)|^2-|beta(k)|^2=1", "density_measure": "d^3k/(2*pi)^3", "logarithmic_shell_weight": "d rho/d ln k is proportional to k^n |beta(k)|^2" }, "integrability_thresholds": { "infrared_assumption": "|beta(k)| is asymptotic to k^(-s) as k approaches 0", "infrared_condition": "2*sn", "ultraviolet_equality": "2*r=n gives a logarithmic divergence", "derivation": "rho is proportional to integral d(ln k) k^n |beta(k)|^2", "marginal_cutoff_check": { "epsilon": 0.000001, "infrared_integral_domain": "[epsilon,1]", "ultraviolet_integral_domain": "[1,1/epsilon]", "infrared_exact_integral": 12.247366377401327, "ultraviolet_exact_integral": 12.24736637740133, "equality_residual": 1.7763568394002505e-15, "interpretation": "The reciprocal cutoffs give equal marginal integrals, each growing as log(1/epsilon)." } }, "shell_profiles": [ { "key": "admissible", "beta_over_b": "q^(-1/2)/(1+q)^(3/2)", "s": 0.5, "r": 2, "shell_weight_over_b_squared": "q^2/(1+q)^3", "exact_log_shell_integral": 0.5, "interpretation": "integrable at both endpoints" }, { "key": "ir_marginal", "beta_over_b": "q^(-3/2)/(1+q)^(3/2)", "s": 1.5, "r": 3, "shell_weight_over_b_squared": "1/(1+q)^3", "interpretation": "constant at the infrared endpoint; logarithmically divergent" }, { "key": "uv_marginal", "beta_over_b": "1/(1+q)^(3/2)", "s": 0, "r": 1.5, "shell_weight_over_b_squared": "q^3/(1+q)^3", "interpretation": "constant at the ultraviolet endpoint; logarithmically divergent" } ], "homogeneous_gaussian_benchmark": { "beta": "b*exp[-k^2/(2*kappa^2)]", "alpha": "sqrt(1+beta^2)", "exact_density_over_kappa3": 0.005612097566411455, "exact_density_formula": "b^2/(8*pi^(3/2))=1/(32*pi^(3/2))", "finite_box_direct": "N_beta(x)=b^2*sum_{m in Z^3} exp[-(2*pi*|m|/x)^2]", "poisson_dual": "N_beta(x)=b^2*x^3/(8*pi^(3/2))*sum_{ell in Z^3} exp[-x^2*|ell|^2/4]", "extensive_asymptote": "N_beta(x) is asymptotic to x^3/(32*pi^(3/2))=rho*L^3", "global_conclusion": "For nonzero homogeneous beta on nonatomic L2(R^3), finite density does not make the multiplication-reflection map Hilbert-Schmidt.", "noncompactness_witness": { "region": "|k|<=kappa", "lower_bound_on_abs_beta": 0.3032653298563167, "construction": "Partition the positive-measure ball into arbitrarily many disjoint positive-measure sets and use their normalized indicators.", "conclusion": "Their multiplication-reflection images are mutually orthogonal with norms at least b*exp(-1/2), so no image subsequence converges; the nonzero homogeneous beta map is not compact and hence not Hilbert-Schmidt." } }, "rank_one_packet_comparison": { "beta_map": "B_rank1=b*|f>