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
- Finite spin matrices
- Tensor spin composition
- Ordered tensor registers, their inner product, and wire permutations
- The direct sum of an indexed family of modules
- Linear map between vector spaces over the same field
- The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥
- Real and complex inner product spaces, with the inner product linear in the first argument
- The polar surface set function on the unit sphere
- Sphere integrals in angle coordinates
- The Axiom of Countable Choice (AC_ω)
- The second fundamental theorem: if G is differentiable on [a,b] with G' = f and f is integrable, then ∫ₐᵇ f = G(b)-G(a)
- The derivatives of sine and cosine are cosine and minus sine
- The addition formulas for sine and cosine
- Parity and the Pythagorean identity for sine and cosine
- Fubini's theorem for L¹ functions on a sigma-finite product
- A function differentiable at c is continuous at c
- A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
- A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral
- Quarter-turn values and shifts by pi/2 and pi
- Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree
- A continuous map has Borel preimages of Borel sets
- Integrable real and complex functions, and their integrals
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- The nonnegative integral agrees with the simple integral on simple functions
- The integral of a nonnegative simple function
- The Lebesgue integral is linear on L¹(μ)
- Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases
- On Borel subsets of ℝᵐ⁺ⁿ, the product λₘ × λₙ agrees with λₘ₊ₙ
- The Borel product of ℝᵐ and ℝⁿ is the Borel sigma-algebra of ℝᵐ⁺ⁿ
- The radiation sector of the framework: modes and the number, energy and momentum assignments
- Cutoff dipole relative bound
- The Axiom of Choice
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Single-mode thermal trace
- Hilbert tensor construction: inner product, completion and product bases
- Diagonal adjoint and graph-core lemma
- Density operators and scalar spectral probability
- Kato-Rellich theorem
- Transverse continuum free Fock Hamiltonian
- Quantum and Atom Optics · Section 5.1, discussion of the long-wavelength approximation and equation (5.5)
- Quantum Optics · Table 1.2, entry (3), thermal state, PDF page 39
- Schwartz/distribution spaces and Hilbert Fock conventions
- Bosonic and fermionic Fock operators with exact domains
- One-mode factorization of symmetric Fock space
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- Self-adjoint trace-Gibbs normalization and form-energy optimization
- Trace class operator
- Trace of a trace class operator
- Nuclear series characterizes trace norm
- For |r| < 1, ∑_k ≥ 0 rᵏ = 1/(1-r), and for |r| ≥ 1 the series diverges
- Projection valued measure
- Regular Borel measure on an LCH space
- Bounded borel pvm integral
- Integral of a measurable function against a projection-valued measure
- Trace is absolutely convergent and basis independent
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
Definition
Finite spin blocks, spherical tensors and Clebsch–Gordan phases
Lemma
Transverse projector and isotropic angular contraction
Postulate
Regularized transverse photon dipole bath model
Lemma