\[ \newcommand{\j}{\jmath} \newcommand{\ep}{\mathfrak{e}} \newcommand{\em}{\bar{\mathfrak{e}}} \newcommand{\Sin}{\boldsymbol{\operatorname{sin}}} \newcommand{\Cos}{\boldsymbol{\operatorname{cos}}} \]

In this article, we explore a possible connection between the wave function and the square-root law for the number of independent random steps, using a common representation in terms of amplitude channels of splitting. We show how the way these channels are combined, together with a quadratic operation, yields intensities, probability weights, and statistical characteristics of observed results. First, however, we introduce simple examples involving sound waves to illustrate the roles of amplitudes, phases, and randomness in this construction.

Imagine two loudspeakers producing the same pure tone. At a particular point in a room, each speaker alone produces the same sound pressure amplitude. What happens when both are switched on? If the arriving waves are in phase, their amplitude doubles, and the corresponding intensity becomes four times that of a single source. With opposite phases and equal amplitudes, the sound at that point cancels under ideal conditions. If the relative phase varies uniformly over a full cycle, the average intensity equals the sum of the intensities of the individual sources.

The individual characteristics of the sources are identical, yet their combined result depends on their phase relationship. Its description therefore requires an amplitude with a phase, while the observed intensity is obtained through a quadratic transformation. The simple acoustic model below considers a locally traveling wave in a homogeneous medium: intensity is proportional to the square of the pressure amplitude. In a general standing-wave field, the energy flux also requires accounting for the particle velocity of the medium.

The article “Random Choice in Idempotent Splitting” develops a probabilistic description of a tree of histories. In “The Wave Function as a Representation of a Split Event,” the channels are described in terms of amplitudes and phases. Can these approaches be represented by a common mathematical object, with observable weights obtained by combining its components?

Let the complex amplitudes of the two arriving waves be \(z_1\) and \(z_2\). Their sum and the intensity of the resultant wave are

\[\tag{1} z=z_1+z_2,\qquad I=K|z|^2, \]

where \(K>0\) is determined by the medium and the chosen amplitude convention. If \(z_1=A_1\), \(z_2=A_2e^{i\theta}\), and \(A_1,A_2\geq0\), then

\[\tag{2} I=K\left(A_1^2+A_2^2+2A_1A_2\cos\theta\right). \]

For \(A_1=A_2=A\), matching phases give \(I=4KA^2\), while opposite phases give \(I=0\). This is a local result; it cannot automatically be extended to the total power of the entire setup.

Thus, combining the waves involves two stages: first the amplitudes are added, and then the squared modulus is taken. Adding intensities is valid only under conditions that eliminate the cross term. This term contains information about the relationship between the components that is absent from their individual squares.

Now suppose that the source volumes and their relative phase vary randomly. Denote the configuration by \(\omega\). Each realization has amplitudes \(A_1(\omega)\), \(A_2(\omega)\), and a phase \(\theta(\omega)\). We assume that the variations are slow enough to define the tone intensity for a given configuration and that the second moments of the amplitudes are finite.

\[\tag{3} Z(\omega)=A_1(\omega)+A_2(\omega)e^{i\theta(\omega)}, \qquad I(\omega)=K|Z(\omega)|^2. \]

The average intensity is

\[\tag{4} \boxed{\mathbb E[I]=K\left(\mathbb E[A_1^2]+\mathbb E[A_2^2] +2\mathbb E[A_1A_2\cos\theta]\right).} \]

If the phase is uniformly distributed over a full period and independent of the amplitudes, the final expectation is zero. With aligned phases, or when phases are related to amplitudes, it may remain nonzero. Randomness alone does not rule out interference.

It is important to distinguish the square of the mean amplitude from the mean square of the amplitude:

\[\tag{5} \mathbb E[|Z|^2]=|\mathbb E[Z]|^2 +\mathbb E\!\left[|Z-\mathbb E[Z]|^2\right]. \]

For example, if the resultant amplitude takes the values \(+A\) and \(-A\) with equal probabilities, its mean is zero, but the average intensity is \(KA^2\). Averaging amplitudes and averaging intensities therefore describe different procedures.

If the possible configurations form a finite set with probabilities \(w_\alpha\), then

\[\tag{6} \mathbb E[I]=K\sum_\alpha w_\alpha|Z_\alpha|^2, \qquad \sum_\alpha w_\alpha=1. \]

Here, \(w_\alpha\) is the probability of a configuration, while \(Z_\alpha\) is its sound amplitude. These are different data. They can, however, be combined into the coefficient \(\sqrt{w_\alpha}Z_\alpha\), leading to a common space of amplitude states.

Let the state be described in a complex inner-product space. Introduce orthogonal channel projectors:

\[\tag{7} P_\alpha^\dagger=P_\alpha,\qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_\alpha P_\alpha=\mathbf 1. \]

The idempotent structure identifies the components of a single whole:

\[\tag{8} \Phi=\sum_\alpha\Phi_\alpha,\qquad \Phi_\alpha=P_\alpha\Phi,\qquad \|\Phi\|^2=\sum_\alpha\|\Phi_\alpha\|^2. \]

For one-dimensional channels, choose orthonormal states \(\xi_\alpha\). Then

\[\tag{9} \boxed{\Phi=\sum_\alpha c_\alpha\xi_\alpha, \qquad \|\Phi\|^2=\sum_\alpha|c_\alpha|^2.} \]

We call this vector the amplitude state of splitting. Its coefficients contain magnitudes and relative phases. Splitting specifies the channels, but does not automatically determine the coefficient values, their evolution law, or their physical meaning.

A zero product of algebraic idempotents does not by itself specify an inner product either. In this construction, we additionally choose to represent them as self-adjoint orthogonal projectors. This choice makes it possible to use a positive quadratic norm.

To obtain an observable output, we must specify a registration map \(R\). For a finite set of outputs, denote its matrix elements by \(u_{r\alpha}\). Then

\[\tag{10} \Psi_{\mathrm{out}}=R\Phi,\qquad \psi_r=\sum_\alpha u_{r\alpha}c_\alpha. \]

If registration is complete and preserves the state weight, we assume \(R^\dagger R=\mathbf 1\), which gives

\[\tag{11} \sum_r|\psi_r|^2=\|\Phi\|^2. \]

A single output may receive contributions from several channels. However, the remaining outputs are also required for normalization: one cannot arbitrarily add all amplitudes into a single output and declare its square to be the total probability. Registration losses require including additional outputs or explicitly using probabilities conditional on successful registration.

For a nonzero state, the quadratic probability rule is

\[\tag{12} \boxed{p_r=\frac{|\psi_r|^2}{\|\Phi\|^2}.} \]

In a continuous representation, \(\psi_r\) is replaced by a function \(\psi(x)\), and the sum by an integral with respect to a specified measure \(d\nu(x)\). The quantity \(|\psi(x)|^2/\|\Phi\|^2\) is then a density with respect to that measure, and the probability of a region is obtained by integrating it.

Equation (12) is adopted here as a rule for the physical interpretation of a quadratic weight. In the acoustic example, the same type of expression gives an intensity with a dimensional coefficient \(K\); it cannot be converted into a probability without additionally defining the observation procedure.

Expanding the square of the output amplitude gives

\[\tag{13} \boxed{|\psi_r|^2= \sum_{\alpha,\beta} u_{r\alpha}c_\alpha\, u_{r\beta}^{*}c_\beta^{*}.} \]

Let \(C\) be the set of channels. The indices of this sum run over its Cartesian square:

\[\tag{14} C\times C=\{(\alpha,\beta):\alpha\in C\eta\in C\}. \]

The quadratic construction can therefore be understood as a transition from individual channels to pairs of channels. Diagonal pairs yield the individual weights, while off-diagonal pairs yield cross contributions:

\[\tag{15} |\psi_r|^2= \sum_\alpha|u_{r\alpha}c_\alpha|^2 +2\operatorname{Re}\sum_{\alpha<\beta} u_{r\alpha}c_\alpha\, u_{r\beta}^{*}c_\beta^{*}. \]

Cross contributions may be positive, negative, or zero. They are not separate probabilities. Nonnegativity applies to the full squared modulus.

The Cartesian product merely lists the pairs. A numerical result arises after assigning amplitude products to those pairs and applying the registration rule. For the coefficient vector \(c\), the same construction can be written using the matrix

\[\tag{16} D=cc^\dagger,\qquad D_{\alpha\beta}=c_\alpha c_\beta^*, \qquad |\psi_r|^2=(RDR^\dagger)_{rr}. \]

The matrix \(D\) stores quadratic relationships between channels. For a normalized pure state, it is the density matrix. For a random ensemble of states, its generalization is \(\overline D=\mathbb E[cc^\dagger]\), and the average output weight is \((R\overline D R^\dagger)_{rr}\), provided that the map \(R\) is fixed.

There is also a familiar geometric picture. If \(\psi_r=x_r+iy_r\), then

\[\tag{17} |\psi_r|^2=x_r^2+y_r^2. \]

This is the squared length of a vector in the amplitude plane. It agrees with the construction using pairs, but is not the same as the Cartesian product viewed as a set: a squared length requires a specified metric.

Consider \(n\) independent binary choices. A complete history is denoted by \(\boldsymbol\varepsilon=(\varepsilon_1,\ldots,\varepsilon_n)\), where \(\varepsilon_k=\pm1\). Under the symmetric measure, each history has weight \(2^{-n}\). Introduce the normalized amplitude representation

\[\tag{18} \Phi_n=2^{-n/2}\sum_{\boldsymbol\varepsilon} e^{i\theta_{\boldsymbol\varepsilon}}\xi_{\boldsymbol\varepsilon}. \]

The phases may be arbitrary here: measurements in the history basis still give equal probabilities. The classical measure itself does not determine the phases.

Assign to each history the result \(S_n(\boldsymbol\varepsilon)=\sum_k\varepsilon_k\). The operator for this observable and the projector for the result \(s\) are

\[\tag{19} \widehat S_n=\sum_{\boldsymbol\varepsilon} S_n(\boldsymbol\varepsilon)P_{\boldsymbol\varepsilon}, \qquad Q_s=\sum_{S_n(\boldsymbol\varepsilon)=s}P_{\boldsymbol\varepsilon}. \]

Combining histories with the same result gives

\[\tag{20} \Pr(S_n=s)=\|Q_s\Phi_n\|^2 =\sum_{S_n(\boldsymbol\varepsilon)=s}2^{-n}. \]

The histories remain orthogonal components, so there are no cross terms. For \(s=2m-n\), the number of matching histories is \(\binom nm\), giving the binomial distribution:

\[\tag{21} \Pr(S_n=2m-n)=2^{-n}\binom nm. \]

The mean square of the observable is also a quadratic characteristic of the state:

\[\tag{22} \langle S_n^2\rangle =\|\widehat S_n\Phi_n\|^2 =\mathbb E\!\left[\left(\sum_k\varepsilon_k\right)^2\right] =n. \]

The final equality uses zero means, unit squares, and independence of the choices. Hence \(S_{\mathrm{rms}}=\sqrt n\). For spatial steps of length \(\ell\), analogous conditions give \(R_{\mathrm{rms}}=\ell\sqrt n\). Correlations or a nonzero mean step change this conclusion.

Thus, the probability of a result and its mean square are obtained from the same state, but through different operators. The value of an individual result \(s\) is specified by the observable \(\widehat S_n\); it need not itself be a square.

For two channels, consider the state

\[\tag{23} \Phi=c_+\xi_++c_-\xi_-, \qquad |c_+|^2+|c_-|^2=1. \]

In this representation, the phase dynamics of the operator \(J\) used in the related article changes the coefficients to \(c_+e^{i\pi b}\) and \(c_-e^{i\pi a}\) for real \(a,b\). It preserves their magnitudes and does not redistribute the weights on its own.

Consider a specific norm-preserving combination into two outputs:

\[\tag{24} \psi_1=\frac{c_++c_-}{\sqrt2},\qquad \psi_2=\frac{c_+-c_-}{\sqrt2}. \]

If \(c_+=1/\sqrt2\) and \(c_-=e^{i\theta}/\sqrt2\), then

\[\tag{25} \boxed{p_1=\frac{1+\cos\theta}{2},\qquad p_2=\frac{1-\cos\theta}{2},\qquad p_1+p_2=1.} \]

For \(\theta=0\), all the weight appears in the first output; for \(\theta=\pi\), in the second. Uniform averaging over the relative phase gives an average weight of \(1/2\) for each output. In the latter case, the off-diagonal elements of the average matrix \(\overline D\) vanish.

This shows the connection with the loudspeaker example: the result depends on amplitudes and their relative phases. For a probabilistic description, however, the combination is normalized over all outputs. If the paths are marked by distinguishable orthogonal states of the apparatus or environment, their cross contribution also vanishes when these markers are ignored.

Classical addition of probabilities can be obtained by registering orthogonal alternatives and, for the outputs considered here, also by averaging that eliminates coherence. This does not mean that every random splitting is automatically a quantum process: the physical meaning of the channels and the registration law must be established separately.

The quadratic construction determines output weights and expectations of observables. It does not select a unique outcome in an individual experiment. To generate an outcome mathematically, one can introduce a uniform random variable \(r\in[0,1)\) and cumulative probabilities:

\[\tag{26} F_0=0,\qquad F_k=\sum_{j=1}^{k}p_j, \qquad \text{исход }k\, \Longleftrightarrow\, F_{k-1}\le r<F_k. \]

This rule reproduces the distribution exactly, but does not explain the origin of \(r\) or the physical selection mechanism. To develop such an interpretation, one can investigate a more general state of the system, apparatus, and environment.

Suppose that additional initial parameters have a measure \(\nu\), and that the registration dynamics partitions their space into regions \(B_k(\Phi)\) leading to different outcomes. The task of physical justification is then to obtain the equality

\[\tag{27} \nu\bigl(B_k(\Phi)\bigr) =\frac{|\psi_k|^2}{\|\Phi\|^2} \]

from an independently specified measure and dynamics. Assigning the required sizes to the regions in advance merely restates the probability rule. A full derivation must explain why precisely these sizes arise during the interaction.

We propose taking the amplitude state of a unified system of channels as the primary mathematical object. Splitting reveals its components, the registration map determines their combination, and a quadratic form specifies the observable weight.

\[\tag{28} \boxed{ \Phi\, \longrightarrow\, \{P_\alpha\Phi\} \, \xrightarrow{\, R\, }\, \Psi_{\mathrm{out}} \, \longrightarrow\, \text{quadratic output weights}. } \]

The Cartesian square of the channel set specifies the pair indices involved in the quadratic construction. Individual weights correspond to diagonal pairs, while phase relationships correspond to off-diagonal pairs. The metric and registration procedure determine which contributions survive in the observed result.

Probabilities of orthogonal outcomes, mean squares of random sums, and intensities of coherently combined waves acquire a common language. In the random-choice example, it reproduces the specified probability measure and accumulation statistics; in the wave example, it reproduces the dependence of output weights on phases.

The physical content of the hypothesis is the assumption that this state structure corresponds to the actual organization of the system being studied. Testing it requires specifying the channels, amplitude dynamics, registration map, and measurable consequences. The quadratic norm ensures consistency of the construction, but does not independently derive the Born rule, an equation of motion, or a mechanism for an individual realization.

The proposed construction suggests a possible connection between the wave function and the square-root law for the number of independent random steps. Both descriptions can be built on a common foundation: splitting reveals the channels of a unified amplitude state, while their combination and registration determine the observable quadratic quantity.

In the wave case, the channel components combine coherently into an amplitude \(\psi\), whose squared modulus gives an intensity or, under the adopted probabilistic interpretation, a probability density. For independent symmetric random steps, the quadratic characteristic of the sum accumulates differently: cross contributions vanish upon averaging, so the mean square of the result is proportional to the number of steps. For unit steps:

Thus, a possible common origin of these regularities lies in the amplitude structure of the split state and the quadratic procedure for obtaining observable characteristics. The wave function describes the amplitudes and phases involved in registration, while the law \(\sqrt n\) expresses the root-mean-square scale of accumulation of independent contributions. The difference between the results is determined by which relationships between channels survive their combination: coherent relationships produce interference terms, whereas independent random accumulation causes them to vanish on average.

This connection should be regarded as a general mathematical hypothesis. Splitting alone does not yet determine the quadratic norm, the probability rule, or the independence of steps. However, once these conditions are explicitly introduced, it allows the wave description and the square-root law to be represented as two applications of a single formalism. The next task is to establish whether the required quadratic structure and the nature of the relationships between channels can follow from the dynamics of splitting and registration.