2026-08-20
Associated random phases
From particle uncertainty to two-particle interference
This work is a direct continuation of the article “Heisenberg uncertainty as a consequence of the uncertainty of the internal phase.” previous work considered a single particle whose initial position in the internal phase cycle is unknown in advance. We will now move from a single particle to a system of two or more particles and ask the following question: can the absolute phase of each component remain random, while a certain combination of phases of the entire system will be consistent and physically observable?
Such a transition is fundamentally important. The randomness of the phases of individual particles does not mean that all relationships between them are random. When a coupled system is formed, two independent uncertainties can turn into one common uncertainty, while the relative phase becomes fixed. In experiment, it does not appear as a directly observable angle, but through interference fringes, coincidence probabilities, atomic cloud density, or electric current noise.
The main idea of the article: communication does not have to eliminate the randomness of the general phase of the system. It can maintain a certain difference or a more complex combination of relative phases.
1. Random phase of a single particle
In Wave Electricity, the state of a particle is specified by a normalized operator
\[\tag{1} J(a,b)=\j^{a}(-\j)^{b}, \qquad J\bar J=1, \] where the parameter \(a\) describes the internal state of the particle, and the parameter \(b\) describes its external motion. For regular internal evolution
\[\tag{2} a(t)=\varpi t, \qquad \pi\varpi=\omega_{\mathrm{int}}. \] If the initial position of the internal wave is unknown, a random variable \(\phi\) is added to the phase parameter:
\[\tag{3} \boxed{ J_{\phi}(t)=J\!\left(a(t)+\phi,b(t)\right). } \] Here \(\phi\) is measured in the normalization of the parameter \(a\), so the corresponding physical angle is equal to \(\pi\phi\). If the phase is denoted directly by the angle \(\theta\) in radians, then we should write \(J(a+\theta/\pi,b)\).
Randomness \(\phi\) does not mean the absence of an internal wave. It only means that an external observer does not know at what point in the phase cycle the wave was at the moment the state was prepared. With a uniform phase distribution, the single-particle phase amplitude is averaged to zero:
\[\tag{4} \left\langle e^{i\pi\phi}\right\rangle=0. \] In previous work it was shown that the transition from the operator to the spatial display is carried out through the speed of the constant norm and its integration:
\[\tag{5} J(t) \longrightarrow V(t)=cJ(t) \longrightarrow L(t)=\int V(t)\,dt. \] The uncertainty of the initial phase thereby turns into the uncertainty of the position of the spatial representation of the wave. The wave, that is, Fourier conjugate, structure relates localization and spatial frequency, which leads to an uncertainty relationship. In this article we will be interested in another result of the same idea: what happens to random phases if particles are created or interact together?
2. Two independent particles
Consider two phase states:
\[\tag{6} J_{1}=J\!\left(a_{1}+\phi_{1},b_{1}\right), \qquad J_{2}=J\!\left(a_{2}+\phi_{2},b_{2}\right). \] If the particles are prepared independently and there is no connection between their phases, the joint distribution is expanded into the product:
\[\tag{7} \boxed{ P(\phi_{1},\phi_{2}) =P_{1}(\phi_{1})P_{2}(\phi_{2}). } \] In this case, knowledge of the phase of the first particle does not tell anything about the phase of the second. For independent uniform phases, the average phase correlation also disappears:
\[\tag{8} C_{12} = \left\langle e^{i\pi(\phi_{1}-\phi_{2})} \right\rangle =0. \] In one particular event, the difference \(\phi_{1}-\phi_{2}\) has some meaning. However, when the experiment is repeated many times, this value changes randomly, so the stable interference pattern disappears after averaging.
3. Common and relative phases of a coupled pair
For a bound or jointly prepared pair, it is more convenient to move from the two initial phases to the common and relative phases:
\[\tag{9} \phi_{1}=\Phi+\frac{\Delta}{2}, \qquad \phi_{2}=\Phi-\frac{\Delta}{2}. \] From here
\[\tag{10} \Phi=\frac{\phi_{1}+\phi_{2}}{2}, \qquad \boxed{\Delta=\phi_{1}-\phi_{2}}. \] The general phase \(\Phi\) can be completely unknown and uniformly distributed. But if the connection fixes \(\Delta\), then the mutual phase position of the two components remains determineddata:
\[\tag{11} \Delta=\Delta_{0}=\operatorname{const}\pmod 2. \] The period here is equal to two, since the physical angle is equal to \(\pi\Delta\). With a general shift of both phases
\[\tag{12} \phi_{1}\longrightarrow\phi_{1}+\Lambda, \qquad \phi_{2}\longrightarrow\phi_{2}+\Lambda \] their difference does not change:
\[\tag{13} (\phi_{1}+\Lambda)-(\phi_{2}+\Lambda) =\phi_{1}-\phi_{2}=\Delta. \] Thus, randomness does not disappear, but is transferred from two separate phases to one common phase of the entire system. The outside world may not know where the joint wave is located within the general cycle, but the internal connection preserves the relative positions of its components.
The absolute phase of each particle can be completely uncertain, while the relative phase of a pair can be determined with high accuracy.
4. Three degrees of phase consistency
The degree of connection can be conveniently characterized by a correlator
\[\tag{14} \boxed{ C_{12} = \left\langle e^{i\pi(\phi_{1}-\phi_{2})} \right\rangle. } \] Independent phases. With uniform independent distributions
\[\tag{15} C_{12}=0. \] Ideal phase coupling. If \(\phi_{1}-\phi_{2}=\Delta_{0}\), then
\[\tag{16} C_{12}=e^{i\pi\Delta_{0}}, \qquad |C_{12}|=1. \] Partial consistency. Let the relative phase contain additional noise:
\[\tag{17} \phi_{1}-\phi_{2}=\Delta_{0}+\eta. \] Then
\[\tag{18} C_{12} =e^{i\pi\Delta_{0}} \left\langle e^{i\pi\eta}\right\rangle, \qquad 0<|C_{12}|<1. \] For example, for Gaussian phase noise with dispersion \(\sigma_{\eta}^{2}\) it turns out
\[\tag{19} |C_{12}| = \exp\!\left(-\frac{\pi^{2}\sigma_{\eta}^{2}}{2}\right). \] The correlator module shows the degree of phase consistency, and its argument specifies the average relative shift. Therefore, phase coupling does not have to be complete or completely absent: it can gradually be destroyed under the influence of external noise and interaction with the environment.
5. Observed phase combinations
For a system of \(N\) components, a linear combination can be considered
\[\tag{20} \Xi=\sum_{k=1}^{N}n_{k}\phi_{k}. \] If \(\phi_k\) are phases of the particles themselves and the physical result should not depend on the general shift of all phases, the coefficients must satisfy the condition
\[\tag{21} \boxed{ \sum_{k=1}^{N}n_{k}=0. } \] For two particles, the simplest invariant combination is \(\Xi=\phi_{1}-\phi_{2}\). For three components, two independent relative phases are possible, as well as, for example, a collective combination
\[\tag{22} \Xi=\phi_{1}+\phi_{2}-2\phi_{3}. \] It can persist even when individual pairwise differences change. In this case, the agreed object is not each pair separately, but the entire three-component system.
It is necessary to distinguish between the internal phases of particles and the controlled phase delays of the measuring setup. In a two-photon interferometer, the observed quantity may depend on the sum \(\varphi_{A}+\varphi_{B}\). This is not the sum of two absolute phases of photons, but the relative phase between two joint alternatives for the passage of the entire pair. The total unobserved phase is still reduced when calculating the probability.
6. How a combination of phases becomes observable
Suppose that the same result can be obtained in two indistinguishable ways. Their amplitudes add up:
\[\tag{23} \mathcal A =\mathcal A_{1} +e^{i\pi\Xi}\mathcal A_{2}. \] The probability of the result contains the interference term:
\[\tag{24} \begin{aligned} P &=|\mathcal A|^{2}\ &=|\mathcal A_{1}|^{2} +|\mathcal A_{2}|^{2} +2\operatorname{Re} \!\left( \mathcal A_{1}^{*}\mathcal A_{2}e^{i\pi\Xi} \right). \end{aligned} \] If a controlled phase shift \(\delta\) is introduced between the alternatives, then after statistical averaging the observed signal can be written as
\[\tag{25} \boxed{ P(\delta) =P_{0} \left[ 1+V\cos\!\left(\delta+\arg C_{\Xi}\right) \right], } \] where
\[\tag{26} C_{\Xi}=\left\langle e^{i\pi\Xi}\right\rangle, \qquad V\le |C_{\Xi}|. \] In an ideal symmetrical installation, visibility \(V\) is directly determined by the module of the phase correlator. In practice, it is further reduced by losses, mismatch of wave packets, finite resolution of detectors and installation noise.
Consequently, the phase does not need to be measured as a separate mechanical coordinate. It manifests itself through a change in the number of events with a controlled change \(\delta\). This is exactly how relative phases are observed in interferometric experiments.
7. Geometric interpretation of a coupled system
The joint state of two particles can be preliminarily represented as
\[\tag{27} \mathcal J_{12} = J_{1}\!\left(a_{1}+\phi_{1},b_{1}\right) \otimes J_{2}\!\left(a_{2}+\phi_{2},b_{2}\right), \] and for a coupled system, not only the algebraic product of states is essential, but also the indecomposable joint distribution of their phases:
\[\tag{28} P(\phi_{1},\phi_{2}) \ne P_{1}(\phi_{1})P_{2}(\phi_{2}). \] Each operator generates its own one-dimensional motion
\[\tag{29} V_{k}(t)=cJ_{k}(t), \qquad L_{k}(t)=\int V_{k}(t)\,dt. \] If the relative phase is fixed, the spatial mappings of these processes receive a stable relative position. In geometric language, this suggests that the connection of particles does not correspond to the stopping of their internal waves, but to the coordination of phase cycles.
The condition for returning a relative configuration after a full period can be written as
\[\tag{30} \boxed{ \Xi(T)-\Xi(0)=2m, \qquad m\in\mathbb Z. } \] If \(\dot\Xi=0\), the relative configuration is saved. If the phase combination changes slowly, beats are observed:
\[\tag{31} \Omega_{\mathrm{beat}} =\pi|\dot\Xi|. \] They can manifest themselves as a periodic change in the interference signal or an exchange of energy between components. However, a specific coupling strength cannot yet be automatically derived from a single phase condition. To do this, it is necessary to separately specify the interaction dynamics and show how the phase mismatch changes the energy of the system.
8. Phase correlation does not mean entanglement
Model with general random phase
\[\tag{32} \phi_{1}=\Phi+\frac{\Delta}{2}, \qquad \phi_{2}=\Phi-\frac{\Delta}{2} \] already creates strong statistical correlations. But by itself it can remain a classical model with a common random variable \(\Phi\). Therefore, it is necessary to distinguish between three levels of description:
The first level is general randomness. Particles receive the same random addition to the phase.
The second level is phase correlation. A certain difference or combination of phases is preserved and appears in joint statistics.
The third level is quantum entanglement. A joint state cannot be represented as a statistical mixture of local states with predetermined parameters.
To obtain quantum entanglement from Wave Electricity, it is necessary to additionally derive the measurement rule, the probabilities of results for different detector settings and the impossibility of a local factorized description. In particular, the theory must reproduce the violation of suitable Bell inequalities, and not just the presence of a nonzero correlator \(C_{12}\).
Phase consistency is necessary for many interference effects, but the presence of phase consistency in itself is not evidence of quantum entanglement.
9. Franson interferometer: phase visible only to the pair
The closest experimental analogue of the idea under consideration is Franson's two-photon interferometer. The source creates a pair of energy-temporally coupled photons. Each photon is sent to a separate asymmetric interferometer having a short path \(S\) and a long path \(L\).
The path difference is chosen to be greater than the coherence length of an individual photon. Therefore, events in which one photon has traveled a short and a long path can be distinguished by time, and single-photon interference disappears. However, the two joint alternatives remain indistinguishable:
\[\tag{33} |S_{1}S_{2}\rangle, \qquad |L_{1}L_{2}\rangle. \] In the first case, both photons arrive earlier, in the second - both later. If the birth time of a pair is not known in advance within the coherence of the source, these alternatives cannot be distinguished by the central group of matches alone. Then the joint state has the form
\[\tag{34} |\Psi\rangle = \frac{1}{\sqrt2} \left( |S_{1}S_{2}\rangle +e^{i(\varphi_{1}+\varphi_{2})} |L_{1}L_{2}\rangle \right). \] The probability of joint registration depends on the sum of the controlled phases of two interferometers:
\[\tag{35} \boxed{ P_{12} =\frac12 \left[ 1+V\cos(\varphi_{1}+\varphi_{2}) \right]. } \] In this case, single detector counting rates do not contain a corresponding interference pattern. The phase is detected only after comparing the registration times of two photons and identifying joint events. In other words, individual parts of the pair appear phase blurred, butjoint statistics contain a stable combination of phases.
The original scheme was proposed by John Franson in Bell Inequality for Position and Time. In the experiment of Quiatt, Steinberg and Chiao High-Visibility Interference in a Bell-Inequality Experiment for Energy and Time a visibility of \(80.4\pm0.6\%\) was recorded. Further experiments showed two-photon interference and violation of Bell's inequalities on lines of many kilometers; one of these experiments is described in the work Violation of Bell Inequalities by Photons More Than 10 km Apart.
For Wave Electricity, Franson's experiment is important not as a direct proof of the operator \(J\), but as confirmation of a more general principle: the observed phase information can belong to the entire pair and be absent in the statistics of individual components.
10. Hong-Wu-Mandela Interference
In the Hong-Wu-Mandel experiment, two photons arrive at different inputs of a symmetrical beam splitter. If their spatial, temporal, frequency and polarization states are indistinguishable, two alternatives - “the first photon is reflected, the second is transmitted” and “the first is transmitted, the second is reflected” - interfere destructively in the separate registration channel.
The probability of detecting photons in different outputs is equal
\[\tag{36} \boxed{ P_{\mathrm c} =\frac12 \left[ 1-|\langle\psi_{1}|\psi_{2}\rangle|^{2} \right]. } \] If the wave packets completely coincide, \(|\langle\psi_{1}|\psi_{2}\rangle|=1\), therefore the ideal probability of coincidence becomes zero. As the relative delay increases, packets become distinguishable and \(P_{\mathrm c}\) returns to the classic value \(1/2\). The experiment is observed as a characteristic failure in the number of matches.
Original work: C. K. Hong, Z. Y. Ou, L. Mandel. Measurement of Subpicosecond Time Intervals between Two Photons by Interference.
This experiment proves the coherent addition of two-particle amplitudes and measures the overlap of complete states of photons. But it is not a direct measurement of a fixed difference in the absolute optical phases of two single-photon states. For states with a certain number of photons, the overall optical phase itself is not observed.
11. Induced coherence
An even more unusual manifestation of consistency was found in the experiments of Zou, Wang and Mandel. Two nonlinear crystals could create pairs of signal and idler photons. The idler photon of the first crystal was directed into the second crystal so that the paths of the idler photons became indistinguishable.
After erasing information about which crystal the pair was born in, the joint state is schematically written as
\[\tag{37} |\Psi\rangle \sim |s_{1},i\rangle +e^{i\Phi}|s_{2},i\rangle. \] Signal photons from two crystals begin to interfere:
\[\tag{38} I_{s}(\Phi) =I_{0}\left[1+V\cos\Phi\right]. \] Changing the phase in the idler channel shifts the interference pattern of signal photons, although the idler photon is not directly detected. The observed phase here refers not to a single localized photon, but to the coordination of two alternatives for the production of the entire pair.
Source: X. Y. Zou, L. J. Wang, L. Mandel. Induced Coherence and Indistinguishability in Optical Interference.
12. Relative phase of atomic condensates
Phase consistency is observed not only in photons. In experiments with Bose-Einstein condensates, two clouds of ultracold atoms are released from traps, expand and overlap. The density of atoms in the overlap region has the form
\[\tag{39} \boxed{ n(\mathbf r) =n_{1}(\mathbf r)+n_{2}(\mathbf r) +2\sqrt{n_{1}(\mathbf r)n_{2}(\mathbf r)} \cos\!\left( \Delta\mathbf k\cdot\mathbf r+\Delta\phi \right). } \] The position of the interference fringes is determined by the relative phase \(\Delta\phi\). In each individual run, a distinct pattern appears, but for two independently prepared condensates, its position may randomly change from run to run.
It is this result that is especially close to the original idea of this article. The absolute phases of the two clouds are not predetermined, but when they are compared in a separate implementation, an observable relative phase emerges. Theoretically, the result can be described by two equivalentin different ways: the measurement either detects a random relative phase or gradually shapes it through a sequence of registrations.
Experimental work: M. R. Andrews et al. Observation of Interference Between Two Bose Condensates. Theoretical analysis: Y. Castin, J. Dalibard. Relative Phase of Two Bose-Einstein Condensates.
13. Two-electron interference
An electronic analogue of the Hong-Wu-Mandel experiment was implemented with two independent synchronized sources of single electrons. They emitted electron wave packets at different inputs of the electronic beam splitter.
Photons are bosons and, with complete coincidence of states, are grouped in one output. Electrons are fermions, so they have the opposite effect: indistinguishable electrons preferentially exit through different channels. The observable quantity is the low-frequency noise of the electric current:
\[\tag{40} S =S_{0} \left[ 1-|\langle\psi_{1}|\psi_{2}\rangle|^{2} \right]. \] When electrons arrive synchronously, the noise decreases. With a temporary mismatch, the wave packets no longer overlap, and the classical separation noise is restored. This type of failure is called a Pauli failure.
The experiment showed two-particle interference of single electrons emitted by independent sources: E. Bocquillon et al. Coherence and Indistinguishability of Single Electrons Emitted by Independent Sources.
The result confirms that the consistency of joint wave states is a property not only of light. However, as in the Hong-W-Mandel photon experiment, the overlap of electron wave packets is directly measured, rather than the individual hidden internal phase of each electron.
14. What exactly has been established experimentally
The experiments considered differ in physical systems and recording methods, but they are united by a common principle. When one observable outcome can occur in several indistinguishable ways, the amplitudes of these ways are added. The result depends on their relative phase.
The following provisions have been established experimentally:
1. The global phase of the state is not directly observed, but the relative phases of the amplitudes change the probabilities of the results.
2. The joint statistics of a pair may contain a phase dependence that is not present in single signals. The most direct example is the Franson interferometer.
3. Consistency can be manifested through the sum of the controlled phases of two remote installations, through the overlap of wave packets or through the position of interference fringes.
4. Many-particle interference is observed both in photons and in material particles - atoms and electrons.
5. The destruction of indistinguishability or the growth of phase noise reduces the visibility of interference, that is, they experimentally reduce the modulus of phase coherence.
15. What remains the Wave Electricity hypothesis
Experiments confirm the physical significance of the relative phases of quantum amplitudes, but they do not in themselves prove that the observed phase is identical to the internal parameter \(a\) of the operator \(J(a,b)\). This identification is an interpretation that still needs to be turned into a verifiable dynamic rule.
Within the framework of Wave Electricity, the following working hypotheses can be put forward:
1. The random addition \(\phi\) characterizes the unknown initial position of the internal closed wave.
2. During the joint creation or binding of particles, not the absolute phase is fixed, but a certain combination \(\Xi\) of their internal phases.
3. Phase consistency of internal operators manifests itself as coherence of observed spatial amplitudes.
4. Bond destruction corresponds to an increase in the dispersion of the relative phase and a decrease in \(|C_{\Xi}|\).
5. A stable bound state can correspond to the condition of complete phase closure \(\Xi(T)-\Xi(0)=2m\).
For these provisions to become an independent physical theory, it is necessary to derive from the operator \(J\) a specific form of the joint state, a probability rule, the dependence of correlations on the settings of the measuring setup, and at least one quantitative prediction that differs from standard quantum mechanics.
16. Conclusions
The uncertainty of the internal phase of an individual particle does not prohibit the existence of a phase-consistent system. When a bond is formed, the two initial phases naturally separate into a common phase \(\Phi\) and a relative phase \(\Delta\):
\[\tag{41} (\phi_{1},\phi_{2}) \longrightarrow (\Phi,\Delta). \] The general phase may remain random, while the relative phase or a more complex combination \(\Xi\) is preserved. It is this combination that can enter into the observed probability:
\[\tag{42} \boxed{ P(\delta) =P_{0} \left[ 1+V\cos\!\left(\delta+\arg C_{\Xi}\right) \right]. } \] In experiments, consistency is manifested in various ways: through coincidences of two photons in the Franson interferometer, the Hong-W-Mandel dip, induced coherence, interference fringes of atomic condensates, and noise suppression in the collision of single electron packets.
Franson’s experience is most important for our concept. In it, single signals do not contain corresponding phase interference, while joint events depend on the combination of the phases of two distant interferometers. This clearly shows that physically observable phase information can belong to the system as a whole, and not to each of its components separately.
Nevertheless, phase correlation is not yet equal to quantum entanglement, and experimental coherence of amplitudes does not yet prove the existence of the internal phase of the operator \(J\). The next step is to construct a measurement rule for the joint state and test whether Wave Electricity can reproduce the angular correlations of entangled particles and the violation of Bell's inequalities.
The final principle: the randomness of the parts does not exclude the phase certainty of the whole.