Purpose: mathematics path toward QFT-readiness.
Role: curriculum map.
Scaffold: Xianxia cultivation realms.
Method: symbol → meaning → example → code → physics use.
Status: personal learning curriculum. Not a textbook. Not a claim of mastery.
Orthodox Mathematics Cultivation Path Toward Quantum Field Theory
| Cultivation Realm | Chinese | Core Techniques | Role |
|---|---|---|---|
| Mortal Condensation | 凡人凝期 | Single-variable calculus, limits, derivatives, integrals, basic algebra, elementary vectors and matrices | computational survival |
| Qi Condensation | 炼气期 | Multivariable calculus, gradients, divergence, curl, multiple integrals, complex numbers, basic ODEs | change, motion, local behaviour |
| Foundation Establishment | 筑基期 | Linear algebra, vector spaces, linear maps, eigenvalues, diagonalisation, abstract algebra, complex analysis | structure, symmetry, transformation |
| Golden Core | 金丹期 | Functional analysis, Hilbert spaces, operators, Fourier series, Fourier transforms, ODEs, PDEs | waves, operators, spaces of functions |
| Nascent Soul | 元婴期 | Differential geometry, manifolds, tangent spaces, tensors, differential forms, topology, calculus of variations | geometry becomes physics |
| Soul Formation | 化神期 | Lie groups, Lie algebras, Lorentz group, Poincaré group, SU(n), representations, probability, statistics | symmetry, particles, inference |
| Tribulation Transcendence | 渡劫期 | Measure theory, Lebesgue integration, Banach spaces, advanced PDEs, path integrals, canonical quantisation | mathematical machinery behind QFT |
| Mahayana | 大乘期 | QED, Yang-Mills, gauge theory, renormalisation group, symmetry breaking, functional methods, statistical mechanics | serious QFT territory |
| Ascension | 飞升期 | CFT, string theory, supersymmetry, AdS/CFT, algebraic geometry, K-theory, modular forms, vertex algebras | frontier research; not current target |
凡人凝期: calculate before proving.
炼气期: understand change in more than one dimension.
筑基期: learn spaces, maps, symmetry, and complex structure.
金丹期: functions become objects.
元婴期: geometry enters physics.
化神期: symmetry becomes law.
渡劫期: analysis becomes brutal; hand-waving dies.
大乘期: fields, particles, symmetries, and renormalisation meet.
飞升期: research frontier. Not the current battle.
The current mathematics code lives in:
math_library/
Current modules:
constants.py
calculus.py
linear_algebra.py
complex_analysis.py
fourier.py
ode_pde.py
functional.py
differential_geometry.py
lie_groups.py
probability.py
qft_math.py
These modules are anchors.
They do not mean the full subject is mastered.
They mean there is now a place to convert abstract ideas into code.
Minimal spine:
calculus
linear algebra
complex analysis
ODE / PDE
Fourier analysis
functional analysis
differential geometry
Lie groups
probability
quantum mechanics
QFT mathematics
More specific QFT chain:
vector spaces
→ Hilbert spaces
→ operators
→ Fourier modes
→ fields
→ symmetries
→ Lie groups
→ action principles
→ path integrals
→ renormalisation
Each important mathematical idea should eventually have:
- definition
- notation
- plain English meaning
- minimal example
- Python implementation if useful
- test
- physics connection
A concept is not owned just because the name is familiar.
| Chinese | English |
|---|---|
| 数理合一，道法自然 | Mathematics and physics must meet; otherwise both float. |
| 层层递进，不可逾越 | Progress layer by layer; skipped foundations return as confusion. |
| 知其名，不如验其式 | Knowing the name is weaker than testing the formula. |
| 会其式，不如明其义 | Using the formula is weaker than understanding its meaning. |
数学为骨，物理为血。
Mathematics is the skeleton; physics is the blood.
符号不明，则大道不显。
If the symbols are unclear, the path remains hidden.
逐式而修，终向场论。
Cultivate formula by formula; the road points toward field theory.