Alphabeta MathRadiation, spin and code interfaces
Supply the regulated photon model, thermal moments, spectral kernels and conditional ULE contracts together with angular-momentum and dipole identities. Use the same spin conventions for absorption-emission code criteria, and specify the bosonic, inner/outer encoding and continuous-monitoring interfaces needed for platform comparisons.
1 · Prerequisites
- Mass and first moment of the ULE scalar envelope
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- The operator norm is a norm on the space of bounded linear operators
- Every finite-dimensional normed space is Banach
- Strongly measurable Banach-valued function
- Bochner-integrable function
- Bochner integrability criterion
- Bochner integral norm inequality
- Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
- Fubini's theorem for L¹ functions on a sigma-finite product
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
- Lebesgue outer measure on ℝⁿ
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- Every section of a product-measurable function is measurable
- The Axiom of Countable Choice (AC_ω)
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- The nonnegative Lebesgue integral
- The integral of a nonnegative simple function
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Additivity of the nonnegative Lebesgue integral
- The Lebesgue integral is linear on L¹(μ)
- The modulus of an integral is bounded by the integral of the modulus
- The induced length is a norm
- The product sigma-algebra and its finite iterates
- The Borel sigma-algebra of a topological space
- A measurable function between measurable spaces
- The product measure of two sigma-finite measure spaces
- The Borel product of ℝᵐ and ℝⁿ is the Borel sigma-algebra of ℝᵐ⁺ⁿ
- Arithmetic and lattice operations preserve measurability whenever they are defined
- Stationary correlation spectra and ULE kernel regularity
- ULE kernel adjoint symmetry and rate identities
- Convergent jump integrals and Hermitian Lamb operators
- Static Bohr-frequency evaluation of the jump integral
- A first-moment bound for the bath correlation convolution
- Density operator
- Universal Lindblad equation for open quantum systems · PDF pp. 4–5, Sec. IIA1; pp. 9–10, Sec. IIIC–IVA, Eqs. (24)–(29), (32)–(33); pp. 16–18, Appendix A opening and A1–A2, Eqs. (A2)–(A3), signs following (A6), and (A9); p. 20, Appendix A4, defining expression in (A25); pp. 21–22, Appendix A6, (A38)–(A41); p. 22, Appendix B, (B2), unnumbered convolution identity immediately before (B4), and norm inequality (B4).
- Erratum: Universal Lindblad equation for open quantum systems · PDF p. 1, Corrected expressions (1), (2), (4), (5), (6)
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- The nonnegative integral agrees with the simple integral on simple functions
- A nonnegative integral over a null set vanishes
- A C¹ diffeomorphism satisfies the change-of-variables formula for L¹ functions
- Finite-preparation residual budget and its remote-boundary limit
- Non-negative and positive operators
- The basis-independent trace of an endomorphism of a finite-dimensional vector space
- Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely
- The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral
- The vector space M_m × n(F) := F^ m × n of m by n matrices over a field, with entrywise operations
- Finite dimensional norm equivalence over a complete valued field
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Lemma
Bounded quadrant modification and generator norms
Physics theorem
ULE jump construction under independent bath hypotheses
Lemma
Derivative of the quadrant modification
Lemma