2026-11-03
The Two-Phase Operator J
From Euler’s Formula to an Extended Normalized State
The hand of a clock rotates in a single plane, and its position can be specified by a single angle. How can we describe two independent rotations, each with its own phase, while retaining the simplicity of Euler's formula? In this article, we construct a two-phase operator \(J\) in a space with four real coordinates and derive its properties and trigonometric identities. We then introduce normalized amplitude weights and show how the extended construction relates to quaternion rotations and can be used to describe probabilistic phenomena.
Part I
The Original Operator J and Two-Phase Geometry
The Original Operator J and Two-Phase Geometry
1. From One Phase to Two
Euler's formula combines two real projections of a single complex rotation:
\[\tag{1} e^{i\alpha}=\cos\alpha+i\sin\alpha,\qquad \alpha\in\mathbb R. \] The complex plane has two real coordinates, but a point on the unit circle is specified by a single phase. Here, cosine and sine are complementary projections of the same state.
Let us pose a more general problem: to combine two independent complex rotations, each with its own phase, in a single object. This requires a space with two complex coordinates, or four real coordinates. At the same time, we want to retain the exponential form, straightforward composition rules, and trigonometric identities.
The algebraic foundations of this construction are described in “From Euler's Formula to Split Geometry” and “Split Sine and Split Cosine in Two-Phase Geometry”. Below, we bring the definitions together in a coherent sequence and develop the properties needed to introduce the extended operator.
2. The Hyperbolic Unit and Idempotents
We work over the real numbers in a commutative algebra generated by two units:
\[\tag{2} i^2=-1,\qquad \j^2=1,\qquad i\j=\j i,\qquad \j\ne\pm1. \] The unit \(i\) is the usual complex unit, while \(\j\) is a hyperbolic unit. Define
\[\tag{3} \ep=\frac{1+\j}{2},\qquad \em=\frac{1-\j}{2}. \] The condition \(\j^2=1\) immediately gives:
\[\tag{4} \ep^2=\ep,\qquad \em^{\,2}=\em,\qquad \ep\em=0,\qquad \ep+\em=1. \] The idempotents act as complementary projectors. For example, multiplication by \(\ep\) preserves the first component and annihilates the second. The bar in \(\em\) identifies the second idempotent; complex conjugation will be denoted separately.
In particular,
\[\tag{5} 1=\ep+\em,\qquad \j=\ep-\em. \] This decomposition is a property of the chosen extended algebra. It expresses the identity as the sum of two projectors onto independent components.
3. Two Complex Planes and Four Real Coordinates
A general element of the algebra has two equivalent representations:
\[\tag{6} Z=\ep z_++\em z_- =x_0+i x_1+\j x_2+i\j x_3, \qquad z_+,z_-\in\mathbb C. \] The coordinates are related by
\[\tag{7} z_+=(x_0+x_2)+i(x_1+x_3),\qquad z_-=(x_0-x_2)+i(x_1-x_3). \] Adding and subtracting these expressions gives the inverse transformation. Thus, the algebra is four-dimensional over \(\mathbb R\). The two distinguished complex planes have bases \((\ep,i\ep)\) and \((\em,i\em)\).
Ordinary multiplication is componentwise in idempotent coordinates:
\[\tag{8} (\ep z_++\em z_-) (\ep w_++\em w_-) =\ep z_+w_++\em z_-w_-. \] This formula expresses the independence of the components. The algebra contains zero divisors: the nonzero elements \(\ep\) and \(\em\) have a zero product. Consequently, not every nonzero element is invertible: \(Z^{-1}\) exists only when both \(z_+\ne0\) and \(z_-\ne0\) hold.
4. Continuous Powers and the Construction of J
Since \(-1=e^{i\pi}\), the hyperbolic unit can be written as
\[\tag{9} \j=\ep+\em e^{i\pi},\qquad -\j=\ep e^{i\pi}+\em. \] For real exponents, we choose the following continuous phase extension:
\[\tag{10} \j^a:=\ep+\em e^{i\pi a},\qquad (-\j)^b:=\ep e^{i\pi b}+\em. \] This definition corresponds to the chosen logarithms \(\log\j=i\pi\em\) and \(\log(-\j)=i\pi\ep\). It reproduces the usual integer powers and defines one-parameter groups. Specifying this choice is essential: the condition \(\j^2=1\) alone does not determine fractional powers.
The operator \(\j^a\) rotates the second component through \(\pi a\), while \((-\j)^b\) rotates the first through \(\pi b\). Their product is
\[\tag{11} \boxed{\begin{aligned} J(a,b)&:=\j^a(-\j)^b\\ &=(\ep+\em e^{i\pi a}) (\ep e^{i\pi b}+\em)\\ &=\ep e^{i\pi b}+\em e^{i\pi a}. \end{aligned}} \] \[\large{ J(a,b)=\ep e^{i\pi b}+\em e^{i\pi a}. } \]
The cross terms vanish because \(\ep\em=0\). The resulting operator combines two independent phase rotations.
5. A Two-Phase Extension of Euler's Formula
The idempotent decomposition allows the exponential to be evaluated componentwise:
\[\tag{12} J(a,b)= \exp\!\left[i\pi\left(\ep b+\em a\right)\right]. \] Substituting the definitions of the idempotents gives another useful form:
\[\tag{13} J(a,b)= \exp\!\left[i\pi\frac{a+b}{2}+i\j\pi\frac{b-a}{2}\right]. \] The two generators commute, and \(i^2=(i\j)^2=-1\). The exponential can therefore be expressed as the product of two rotations with the usual circular structure but different generators.
The diagonal and antidiagonal of the phase plane give simple special cases:
\[\tag{14} J(a,a)=e^{i\pi a},\qquad J(a,-a)=e^{-i\j\pi a} =\cos(\pi a)-i\j\sin(\pi a). \] Thus, the classical Euler formula is included in the two-phase construction as the case of coincident phases. When \(a\) and \(b\) are independent, a single ordinary complex phase parameter is no longer sufficient.
6. Representative Values
A few values show how the real, complex, and hyperbolic units arise within the same family.
| \(a\) | \(b\) | \(J(a,b)\) | \(J(a,b)^2\) |
|---|---|---|---|
| \(0\) | \(0\) | \(1\) | \(1\) |
| \(1\) | \(0\) | \(\j\) | \(1\) |
| \(0\) | \(1\) | \(-\j\) | \(1\) |
| \(1\) | \(1\) | \(-1\) | \(1\) |
| \(\tfrac12\) | \(\tfrac12\) | \(i\) | \(-1\) |
| \(\tfrac12\) | \(-\tfrac12\) | \(-i\j\) | \(-1\) |
| \(-\tfrac12\) | \(\tfrac12\) | \(i\j\) | \(-1\) |
| \(-\tfrac12\) | \(-\tfrac12\) | \(-i\) | \(-1\) |
For example,
\[\tag{15} J\!\left(\frac12,-\frac12\right) =-i\ep+i\em =-i\j. \] Rotating just one component gives roots of the hyperbolic units:
\[\tag{16} \begin{aligned} J\!\left(\frac12,0\right)&=\ep+i\em, &J\!\left(\frac12,0\right)^2&=\j,\\ J\!\left(0,\frac12\right)&=i\ep+\em, &J\!\left(0,\frac12\right)^2&=-\j. \end{aligned} \] 7. Composition, Conjugation, and Invertibility
Componentwise multiplication immediately gives the phase-addition rule:
\[\tag{17} J(a,b)J(c,d)=J(a+c,b+d). \] This product is commutative and associative. The identity state is \(J(0,0)=1\), and
\[\tag{18} J^{-1}(a,b)=J(-a,-b),\qquad J(a,b)^n=J(na,nb),\quad n\in\mathbb Z. \] Define complex conjugation of the coefficients while leaving the idempotents unchanged:
\[\tag{19} Z^\dagger:=\ep\overline{z_+} +\em\overline{z_-}. \] Then
\[\tag{20} J^\dagger(a,b)=J(-a,-b)=J^{-1}(a,b), \qquad J^\dagger J=1. \] The final equality is an algebraic normalization condition: each complex coefficient has modulus one. This should be distinguished from the scalar norm of an arbitrary pair of coefficients, which we will need in Part II.
8. Periodicity, Roots, and the Geometry of the Family J
Both phase coordinates are periodic with period \(2\):
\[\tag{21} J(a+2k,b+2l)=J(a,b),\qquad k,l\in\mathbb Z. \] For a positive integer \(n\), all solutions of \(X^n=1\) in this algebra have the form
\[\tag{22} X=J\!\left(\frac{2k}{n},\frac{2l}{n}\right), \qquad k,l=0,\ldots,n-1. \] Indeed, each complex coefficient must independently be an \(n\)th root of unity. There are therefore \(n^2\) distinct solutions. In particular,
\[\tag{23} X^2=1\, \Longleftrightarrow\, X\in\{1,-1,\j,-\j\}, \qquad X^2=-1\, \Longleftrightarrow\, X\in\{i,-i,i\j,-i\j\}. \] The pair \((e^{i\pi b},e^{i\pi a})\) ranges over the product of two circles. The family \(J\) therefore has the geometry of a torus \(S^1\times S^1\): a two-parameter surface in the four-dimensional space of the algebra.
Multiplication of an arbitrary element \(Z\) by \(J\) rotates the two distinguished planes independently. In coordinates \((\operatorname{Re}z_+,\operatorname{Im}z_+,\operatorname{Re}z_-,\operatorname{Im}z_-)\), the corresponding matrix is
\[\tag{24} L_J=\begin{pmatrix} R(\pi b)&0\\ 0&R(\pi a) \end{pmatrix},\qquad R(\phi)=\begin{pmatrix}\cos\phi&-\sin\phi\\ \sin\phi&\cos\phi\end{pmatrix}. \] Thus, \(J\) acts in a four-dimensional space but describes a particular commutative subgroup of rotations in two fixed planes, isomorphic to \(SO(2)\times SO(2)\). Arbitrary rotations in four dimensions require a more general framework.
9. Split Cosine and Split Sine
Expanding both exponentials in the definition of \(J\) and collecting the real and imaginary parts gives a two-phase Euler formula:
\[\tag{25} \boxed{J(a,b)=\Cos(a,b)+i\Sin(a,b),} \] where
\[\tag{26} \Cos(a,b)= \ep\cos(\pi b)+\em\cos(\pi a), \] \[\tag{27} \Sin(a,b)= \ep\sin(\pi b)+\em\sin(\pi a). \] Bold type distinguishes the split functions from ordinary scalar functions. Their values lie in the real subalgebra generated by \(1\) and \(\j\).
The expressions for \(J\) and \(J^{-1}\) give symmetric and antisymmetric decompositions:
\[\tag{28} \boxed{\Cos(a,b) =\frac{J(a,b)+J(-a,-b)}{2} =\frac{J+J^{-1}}{2} ,}\] \[\tag{29} \boxed{\Sin(a,b) =\frac{J(a,b)-J(-a,-b)}{2i} =\frac{J-J^{-1}}{2i} .} \] Split cosine extracts the part that is symmetric under simultaneous reversal of both phases. Split sine extracts the antisymmetric part and divides it by \(i\).
When the phases coincide,
\[\tag{30} \Cos(a,a)=\cos(\pi a),\qquad \Sin(a,a)=\sin(\pi a). \] Further details on the properties of split sine and split cosine, their trigonometric identities, and addition formulas can be found in “Split Sine and Split Cosine in Two-Phase Geometry”.
10. Derivatives and Phase Independence
Differentiating the original exponential representation gives
\[\tag{31} \frac{\partial J}{\partial a}=i\pi\em J, \qquad \frac{\partial J}{\partial b}=i\pi\ep J. \] Each derivative selects only its own component. In particular,
\[\tag{32} \frac{\partial^2J}{\partial a\,\partial b}=0, \qquad \left(\frac{\partial^2}{\partial a^2} +\frac{\partial^2}{\partial b^2}\right)J=-\pi^2J. \] For the split functions, we obtain
\[\tag{33} \begin{aligned} \partial_a\Cos&=-\pi\em\Sin, &\partial_a\Sin&=\pi\em\Cos,\\ \partial_b\Cos&=-\pi\ep\Sin, &\partial_b\Sin&=\pi\ep\Cos. \end{aligned} \] If the phases depend on a parameter \(t\), then
\[\tag{34} \frac{dJ}{dt} =i\pi\bigl(\ep\dot b+\em\dot a\bigr)J. \] This last formula describes two independent phase motions at different rates within a single operator representation.
11. Repeated Idempotent Splitting
The two-phase construction admits a further generalization: each original component can be split into two independent channels. This requires an extension of the algebra, because \(\ep\) and \(\em\) are already minimal idempotents in the original algebra. Introduce a new independent hyperbolic unit \(\j_2\), satisfying \(\j_2^2=1\) and commuting with \(i\) and \(\j\).
It generates an additional pair of idempotents:
\[\tag{35} \begin{gathered} \pmp=\frac{1+\j_2}{2},\qquad \pme=\frac{1-\j_2}{2},\\ \pmp^2=\pmp,\quad \pme^2=\pme,\quad \pmp\pme=0,\quad \pmp+\pme=1. \end{gathered} \] Multiplying each original component by this sum gives repeated splitting:
\[\tag{36} \ep=\ep\pmp+\ep\pme,\qquad \em=\em\pmp+\em\pme. \] The four resulting products are mutually annihilating idempotents whose sum is the identity. Assigning an independent phase to each one defines a four-phase extension of the operator:
\[\tag{37} \boxed{\begin{aligned} J_2(a_+,a_-;b_+,b_-)={}& \ep\pmp\,e^{i\pi b_+} +\ep\pme\,e^{i\pi b_-}\\ &+\em\pmp\,e^{i\pi a_+} +\em\pme\,e^{i\pi a_-}. \end{aligned}} \] When the daughter phases coincide in pairs, the original operator is recovered:
\[\tag{38} J_2(a,a;b,b)=\ep e^{i\pi b}+\em e^{i\pi a}=J(a,b). \] The original two-phase geometry is therefore a special case of the four-phase geometry. The extended algebra has four complex components, or eight real coordinates. The detailed construction and further levels of splitting are discussed in “Multilevel Idempotent Splitting of Phase Planes”.
Repeated splitting increases the number of independent components. In Part II, we consider a different generalization: we retain two components but introduce variable amplitude weights. The two approaches can subsequently be combined.
Part II
The Extended Operator and Normalized Amplitudes
The Extended Operator and Normalized Amplitudes
The original operator \(J\) describes two independent phase rotations, but both complex components retain unit modulus. To describe states with different amplitude ratios, we need to allow the weights to be redistributed between the components while keeping the sum of their squared moduli equal to one. We therefore introduce the extended operator \(\widehat J\), which supplements the independent phases with an amplitude parameter. This construction can also be applied to probabilistic phenomena if the squared moduli of the components are interpreted as the probabilities of two alternative outcomes.
12. Introducing an Amplitude Parameter
In the original \(J\), both complex coefficients have modulus \(1\). We now allow their relative magnitudes to vary. Introduce a real parameter \(\varsigma\) and define
\[\tag{39} \boxed{\widehat J(a,b,\varsigma)= \ep\cos\varsigma\,e^{i\pi b} +\em\sin\varsigma\,e^{i\pi a}.} \] The relationship to the original operator is expressed by the ordinary idempotent product:
\[\tag{40} \widehat J=W(\varsigma)J(a,b),\qquad W(\varsigma)=\ep\cos\varsigma +\em\sin\varsigma. \] Denote the complex coefficients by
\[\tag{41} u=\cos\varsigma\,e^{i\pi b},\qquad v=\sin\varsigma\,e^{i\pi a},\qquad \widehat J=\ep u+\em v. \] Two parameters now specify the phases, while a third distributes the magnitude between the components. For equal weights,
\[\tag{42} \widehat J\!\left(\frac\pi4,a,b\right) =\frac{J(a,b)}{\sqrt2}. \] 13. Normalization: Which Equality Is Preserved?
For a pair of complex coefficients, define the scalar squared norm:
\[\tag{43} N(\ep u+\em v):=|u|^2+|v|^2. \] Then, for any real \(\varsigma\),
\[\tag{44} \boxed{N(\widehat J)=\cos^2\varsigma+\sin^2\varsigma=1.} \] This scalar norm and the algebraic product with the conjugate are different constructions. For comparison:
\[\tag{45} \begin{aligned} J^\dagger J&=1,&N(J)&=2,\\ \widehat J^\dagger\widehat J &=\ep\cos^2\varsigma+\em\sin^2\varsigma, &N(\widehat J)&=1. \end{aligned} \] In coordinates \((x_0,x_1,x_2,x_3)\) relative to the basis \((1,i,\j,i\j)\), the sum of the coordinate squares is \(N(Z)/2\). Thus, the original \(J\) has unit Euclidean norm in this basis, while \(\widehat J\) has unit norm in the coordinates of the pair \((u,v)\). Specifying the coordinates explicitly removes the ambiguity in the term “unit norm.”
In coordinates \((\operatorname{Re}u,\operatorname{Im}u,\operatorname{Re}v,\operatorname{Im}v)\), the extended operator ranges over the sphere \(S^3\) in \(\mathbb R^4\). It is sufficient to take \(0\le\varsigma\le\pi/2\), with both phases defined modulo \(2\pi\). When \(u=0\) or \(v=0\), the phase of the zero component is arbitrary.
Thus, four-dimensionality refers to the ambient real space; the normalized family \(\widehat J\) has three independent parameters.
14. Correspondence with Unit Quaternions
Introduce the quaternion units \(\mathbf i,\mathbf j,\mathbf k\), satisfying \(\mathbf i^2=\mathbf j^2=\mathbf k^2=-1\) and \(\mathbf i\mathbf j=\mathbf k=-\mathbf j\mathbf i\). They differ from the commuting units of the original algebra.
Define the real-linear correspondence
\[\tag{46} \Phi(\ep u+\em v) =q_0+q_1\mathbf i+q_2\mathbf j+q_3\mathbf k, \] where
\[\tag{47} \begin{aligned} q_0&=\cos\varsigma\cos(\pi b), &q_1&=\cos\varsigma\sin(\pi b),\\ q_2&=\sin\varsigma\cos(\pi a), &q_3&=\sin\varsigma\sin(\pi a). \end{aligned} \] It follows that
\[\tag{48} |q|^2=q_0^2+q_1^2+q_2^2+q_3^2=|u|^2+|v|^2=N(\widehat J)=1. \] The correspondence covers all unit quaternions. For a given \(q\), we recover
\[\tag{49} u=q_0+i q_1,\qquad v=q_2+i q_3, \qquad \varsigma=\operatorname{atan2}(|v|,|u|),\quad b=\frac{\arg u}{\pi},\quad a=\frac{\arg v}{\pi}. \] The arguments are defined modulo \(2\pi\); the phase of a zero coefficient is unspecified. This is a feature of the phase coordinates, rather than a loss of the state itself.
15. A Separate Composition Operation
The linear correspondence \(\Phi\) does not, by itself, turn the original commutative product into quaternion multiplication. To transfer the composition of rotations, define a new operation:
\[\tag{50} X_2\star X_1:=\Phi^{-1}\!\left(\Phi(X_2)\Phi(X_1)\right). \] Writing a quaternion as \(u+v\mathbf j\), identifying the complex unit with \(\mathbf i\), and using \(\mathbf j z=\bar z\mathbf j\), we obtain
\[\tag{51} \boxed{(u_2,v_2)\star(u_1,v_1)= \left(u_2u_1-v_2\overline{v_1},\quad u_2v_1+v_2\overline{u_1}\right).} \] This operation is associative and generally noncommutative. Its identity is the pair \((1,0)\), that is, \(\ep\). In particular,
\[\tag{52} \em\star\em=-\ep, \qquad \ep\star\em =\em\star\ep=\em. \] The idempotency relation \(\em^{\,2}=\em\) therefore applies to the original product. Under \(\star\), the same symbols serve as a basis for a different algebraic structure. The nonzero zero divisors of the original algebra likewise do not carry over to quaternion multiplication.
The inverse of a normalized pair under \(\star\) is \((\bar u,-v)\). It differs from componentwise conjugation, \((\bar u,\bar v)\).
Direct expansion of the squared moduli gives
\[\tag{53} \begin{aligned} N(X_2\star X_1) &=|u_2u_1-v_2\overline{v_1}|^2 +|u_2v_1+v_2\overline{u_1}|^2\\ &=(|u_2|^2+|v_2|^2)(|u_1|^2+|v_1|^2). \end{aligned} \] The cross terms cancel. The composition of any two normalized operators is therefore normalized:
\[\tag{54} N(\widehat J_2\star\widehat J_1)=1. \] 16. An Arbitrary Spatial Rotation
Let the unit vector \(\mathbf n=(n_x,n_y,n_z)\) specify the axis, and let \(\theta\) be the angle of an active rotation in a right-handed coordinate system. The corresponding quaternion is
\[\tag{55} q=\cos\frac\theta2 +(n_x\mathbf i+n_y\mathbf j+n_z\mathbf k)\sin\frac\theta2. \] The coefficients of the extended operator are then
\[\tag{56} u=\cos\frac\theta2+i n_x\sin\frac\theta2, \qquad v=(n_y+i n_z)\sin\frac\theta2. \] They automatically satisfy \(|u|^2+|v|^2=1\); the phase parameters are recovered using the formulas above. This constructs any spatial rotation directly from its axis and angle.
For a vector \(\mathbf r\), represented as a purely imaginary quaternion,
\[\tag{57} \mathbf r'=q\mathbf r q^{-1}. \] If \(q=q_0+\mathbf s\), where \(\mathbf s=(q_1,q_2,q_3)\), the calculation can be performed without explicit quaternion multiplication:
\[\tag{58} \mathbf r'=(q_0^2-|\mathbf s|^2)\mathbf r +2(\mathbf s\cdot\mathbf r)\mathbf s +2q_0(\mathbf s\times\mathbf r). \] Applying \(q_1\) first and \(q_2\) second gives \(q_{\mathrm{total}}=q_2q_1\). Hence
\[\tag{59} \boxed{\widehat J_{\mathrm{total}} =\widehat J_2\star\widehat J_1.} \] The rightmost factor acts first. The quaternions \(q\) and \(-q\) represent the same spatial rotation; correspondingly, \(\widehat J\) and \(-\widehat J\) have the same action on vectors through this correspondence.
For example, rotations through \(\pi/2\) about the \(x\) and \(y\) axes are represented by the pairs
\[\tag{60} X=\left(\frac{1+i}{\sqrt2},0\right),\qquad Y=\left(\frac1{\sqrt2},\frac1{\sqrt2}\right). \] Their compositions differ:
\[\tag{61} Y\star X=\left(\frac{1+i}{2},\frac{1-i}{2}\right), \qquad X\star Y=\left(\frac{1+i}{2},\frac{1+i}{2}\right). \] The first order maps the initial vector \((0,0,1)\) to \((0,-1,0)\), while the second maps it to \((1,0,0)\). Both resulting pairs have unit norm.
17. Computational Implications
The usual quaternion representation stores four real components subject to one normalization constraint. The form \(\widehat J\) uses three parameters and explicitly separates the relative component magnitudes from their phases.
For repeated composition, it is convenient to store \((u,v)\) directly, apply \(\star\), and recover \((\varsigma,a,b)\) only for analysis or display. This avoids evaluating arguments and trigonometric functions after every step.
The quaternion and complex-pair representations express the same arithmetic in different coordinates. The new notation therefore does not imply a universal reduction in the number of operations. The advantage of the phase form lies in its parameter structure and explicit normalization; computational speed depends on the algorithm and implementation.
In exact arithmetic, normalization is preserved identically. Floating-point calculations can accumulate norm drift, just as in ordinary quaternion calculations. If necessary, the pair is divided by \(\sqrt{|u|^2+|v|^2}\). The three-parameter representation ensures normalization analytically, but does not eliminate rounding errors.
18. A Probabilistic Interpretation of the Weights
The normalized pair \((u,v)\) can serve as an amplitude description of two alternatives. Adopting the squared-modulus rule, define
\[\tag{62} \boxed{p_1=|u|^2=\cos^2\varsigma,\qquad p_2=|v|^2=\sin^2\varsigma,\qquad p_1+p_2=1.} \] Both numbers are nonnegative. The value \(\varsigma=0\) gives \((p_1,p_2)=(1,0)\), the value \(\pi/2\) gives \((0,1)\), and \(\pi/4\) gives equal probabilities. Any pair \((p,1-p)\), with \(0\le p\le1\), can be obtained by choosing \(\varsigma=\arccos\sqrt p\).
The phases \(a,b\) do not directly change these two weights. However, they can affect the probabilities after a transformation that adds amplitudes. For example, the unitary mixing
\[\tag{63} u'=\frac{u+v}{\sqrt2},\qquad v'=\frac{u-v}{\sqrt2} \] gives
\[\tag{64} \begin{aligned} p'_1&=\frac12\left[1+\sin(2\varsigma)\cos\bigl(\pi(b-a)\bigr)\right],\\ p'_2&=\frac12\left[1-\sin(2\varsigma)\cos\bigl(\pi(b-a)\bigr)\right]. \end{aligned} \] The sum remains one, while the distribution depends on the relative phase. Here, amplitude addition and the subsequent probability calculation take place in the specified common channel; the idempotent decomposition alone does not define such mixing.
If the parameters depend on \(x\), then \(p_1(x)+p_2(x)=1\) at every point. This is useful for two complementary probability functions. However, pointwise normalization does not replace the integral normalization of a density over \(x\). For example, given a normalized density \(\rho(x)\), we can set
\[\tag{65} \psi_1(x)=\sqrt{\rho(x)}\,u(x),\qquad \psi_2(x)=\sqrt{\rho(x)}\,v(x),\qquad \int\rho(x)\,dx=1. \] Then
\[\tag{66} \int\left(|\psi_1(x)|^2+|\psi_2(x)|^2\right)dx=1. \] More alternatives require more normalized components, for example by distributing weights successively through additional angles. A single pair \((u,v)\) directly describes two alternatives.
The probabilistic interpretation here applies specifically to the extended \(\widehat J\), because its coefficients have the required scalar normalization. The original \(J\) specifies a phase structure in which each coefficient has unit modulus.
19. Possible Physical Meaning
The mathematical framework supports several applications: independent phase dynamics of the components, spatial rotations, and normalized amplitudes of alternatives. In physical models, the two components can be associated with states or channels, and the parameters with quantities that determine their phases and relative weights.
A specific model must separately specify what is measured, how the state evolves, and how its coefficients relate to observed outcomes. A sum of squares equal to one ensures consistent normalization under the adopted amplitude interpretation, but does not by itself derive a physical probability law or an equation of motion.
The four real coordinates of the algebra describe a mathematical state space. Relating them to spatial coordinates, time, or internal degrees of freedom is a separate task of physical interpretation.
Conclusion
The operator \(J(a,b)\) extends the Euler exponential from one phase to two independent phases in a four-dimensional real algebra. The idempotent decomposition retains simple composition rules, allows split sine and split cosine to be defined, and leads directly to their identities.
The extended operator \(\widehat J\) introduces adjustable amplitude weights whose squares sum to one. Its correspondence with unit quaternions, together with the separately defined product \(\star\), provides the composition of arbitrary spatial rotations. The same normalized pair can describe two probabilistic alternatives when the model adopts a rule relating amplitudes to probabilities.