1. Entrance
Phase–duality algebra is a unique structure that combines the geometric, algebraic and physical properties of trigonometric functions (sin, cos, sec, csc, tan, cot) and covers both circular and hyperbolic rotations. This algebra is reinterpreted within the framework of Clifford algebra and Lie groups, providing a strong basis for both mathematical consistency and physical modelling.
2. State Vector and Operators
2.1 State Vector
Composite vector containing primal and dual components: [ \mathbf{S}(x)=\begin{bmatrix} \cos x \ \sin x \ \sec x \ \csc x \end{bmatrix} ]
2.2 Operators
- Phase operator (quadrature rotation): [ J = \begin{bmatrix} 0 & -1 & 0 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & 0 & -1 \ 0 & 0 & 1 & 0 \end{bmatrix},\quad J^2 = -\mathbf{I}_4 ]
- Duality operator (primal ↔ extension): [ D(x) = \mathrm{diag}(\sec x,\ \csc x,\ \cos x,\ \sin x) ]
- Rate–balance operator: [ \Lambda(x) = \mathrm{diag}(1,\ 1,\ \tan x,\ \cot x) ]
3. Clifford Algebra Perspective
3.1 Primal Channel (SO(2))
- Bivector: ( J = e_1 e_2 ), ( J^2 = -1 )
- Rotor rotation: ( R(\theta) = \cos\theta + J\sin\theta )
3.2 Dual Channel (SO(1,1))
- Bivector: ( K = e_3 e_4 ), ( K^2 = +1 )
- Boost rotation: ( B(\eta) = \cosh\eta + K\sinh\eta )
3.3 Combined Clifford Module
[ \Psi(x) = \begin{bmatrix} \mathbf{p}(x) \ \mathbf{d}(x) \end{bmatrix},\quad \mathcal{U}(\theta,\eta) = \begin{bmatrix} R(\theta) & 0 \ 0 & B(\eta) \end{bmatrix} ]
4. Lie Group Perspective
4.1 Phase Rotation (SO(2))
[ R(\theta) = \exp(\theta J_2),\quad J_2 = \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} ]
4.2 Duality/Scale (SO(1,1))
[ B(\eta) = \exp(\eta H),\quad H = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}]
4.3 United Group
[ \mathcal{G} \cong SO(2) \times SO(1,1),\quad \mathfrak{g} = \mathfrak{so}(2) \oplus \mathfrak{so}(1,1) ]
5. Commutator Table
| Operator Pair | Commutator ([A,B]) | Comment |
|---|---|---|
| ([J, J]) | (0) | Phase operator is abelian with itself |
| ([D(x), D(x)]) | (0) | Duality operator diagonal |
| ([J, D(x)]) | x-dependent diagonal shift | Phase–dual interaction |
| ([J, Λ(x)]) | x-dependent diagonal shift | Phase–scale interaction |
| ([D(x), Λ(x)]) | (0) | Commutative |
6. Energy Function and Invariants
6.1 Definition
[ \mathcal{E}(x) = \alpha(\cos^2 x + \sin^2 x) + \beta(\tan^2 x + \cot^2 x) + \gamma(\sec^2 x + \csc^2 x) ]
6.2 Simplifying with Identities
[ \mathcal{E}(x) = \alpha + (\beta + \gamma)(\tan^2 x + \cot^2 x) + 2\gamma ]
6.3 Invariants
- Primal norm invariance: ( \cos^2 x + \sin^2 x = 1 )
- Dual channel invariance: ( \sec^2 x – \tan^2 x = 1 ), ( \csc^2 x – \cot^2 x = 1 )
7. Physical Modeling
7.1 Quantum Circuit Analogy
- Phase gate: ( J \rightarrow Z ) rotation
- Duality: Hadamard-like primal–dual transition
- Ratio channel: spin–phase stabilizer
7.2 Wave–Particle Duality
- Sin–cos: wave amplitude
- Sec–csc: energy inverses
- Tan–cot: orientation and feedback
7.3 Circuit and Signal Systems
- Phase shifter: ( J ) operator
- Dual filter: inverse amplitude with ( D(x) )
- Stability: check with ( \mathcal{E}(x) )
8. Conclusion
Phase–duality algebra combines trigonometric functions with Clifford algebra and Lie groups, providing a structure that is both mathematically and physically consistent.
Commutator relations, energy invariance and group structure show the usability of this algebra in both theoretical and applied systems. This structure is a unique yet compatible synthesis with classical algebras in the literature.