2026-09-04
One Reality and Many Realities
The Unifying Geometry of Quantum Interpretations
Quantum mechanics accurately predicts the probabilities of outcomes, but it does not provide a single, generally accepted answer to the question of what happens to reality during a measurement. Is one outcome chosen, are all possible outcomes preserved, is there a single hidden trajectory, or does the state itself only make sense relative to another physical object?
This article proposes a common geometric language for mapping such answers. It is based on an idempotent splitting of a unified state into mutually orthogonal channels. We will distinguish between reality as a complete global state and realities as its internally consistent orthogonal realizations. This construction allows us to represent the main interpretations of quantum mechanics as different rules for the physical interpretation of a single branching geometry.
The proposed approach is a unifying geometric model of quantum interpretations, but not yet a new, complete physical theory. It reveals the common mathematical core of interpretations without declaring their mutually exclusive assertions simultaneously true.
1. The Problem of Quantum Measurement
Let a quantum system admit two measurement outcomes. Before interacting with the device, we write its state in standard form.
\[\tag{1} |\psi\rangle =c_+|q_+\rangle+c_-|q_-\rangle, \qquad |c_+|^2+|c_-|^2=1. \] The device and observer are initially in states \(|A_0\rangle\) and \(|O_0\rangle\). Therefore, the complete initial state is as follows
\[\tag{2} |\Psi_0\rangle =\left(c_+|q_+\rangle+c_-|q_-\rangle\right) |A_0\rangle|O_0\rangle. \] The unitary interaction links each result to the corresponding states of the apparatus and observer:
\[\tag{3} |\Psi\rangle =c_+|q_+\rangle|A_+\rangle|O_+\rangle +c_-|q_-\rangle|A_-\rangle|O_-\rangle. \] Formula (3) creates a correlation, but by itself does not indicate whether one of its terms has disappeared, whether both remain, or whether one of them contains the only actual configuration. This question is answered by interpretations of quantum mechanics.
2. Reality and Reality
The words "reality" and "reality" are often used synonymously. Here, we will specifically separate them and use them as model terms.
Reality \(\mathcal D\) is the complete global state, including all admissible orthogonal continuations.
Reality \(\mathcal R_h\) is a single internally consistent projection of reality, corresponding to a specific history \(h\):
The index \(h\) denotes a specific development of events, or history, and \(P_h\) is the projector that identifies the corresponding component of the complete state. At this stage, such a selection does not yet mean that history \(h\) necessarily exists as an independent physical reality: the formula only defines a possible decomposition of reality.
\[\tag{4} \boxed{ \mathcal R_h=P_h\mathcal D. } \] If the projectors form a complete system, then
\[\tag{5} P_hP_{h'}=\delta_{hh'}P_h, \qquad \sum_hP_h=1. \] Therefore, a single reality is represented by the sum of relative realities:
\[\tag{6} \boxed{ \underbrace{\mathcal D}_{\text{single reality}} =\sum_h \underbrace{\mathcal R_h}_{\text{relative reality }h}. } \] No single \(\mathcal R_h\) is equal to the entire \(\mathcal D\). It contains a specific result, the corresponding state of the device, the environment, and the observer. Other results belong to other orthogonal channels.
Reality in the proposed terminology is single. There may be multiple realities, but they are not independent copies of the original whole: each represents its orthogonal projection.
3. Three Different Meanings of Splitting
To avoid slipping unnoticed from a mathematical formula to a statement about multiple worlds, it is necessary to distinguish three levels.
\[\tag{7} \boxed{ \begin{aligned} \text{algebraic splitting} &:\quad Q=\sum_hP_hQ,\\ \text{dynamic splitting} &:\quad \text{emergence of physical correlations},\\ \text{ontological splitting} &:\quad \text{recognition of channels as separate realities}. \end{aligned} } \] The first is a mathematical decomposition. The second occurs when the system is entangled with the device and the environment. The third already represents an interpretative postulate. From the existence of orthogonal termsIt does not yet follow that each of them is an independent physical reality.
4. Idempotent Basis of Splitting
In the unified concept of Wave Electricity, the initial normalized state allows for decomposition into two mutually complementary idempotents:
\[\tag{8} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] For any state \(Q\)
\[\tag{9} \boxed{ Q=\ep Q+\em Q. } \] Equality (9) does not create two copies of \(Q\). It reveals two orthogonal channels in the original whole. Their sum restores the state, and the product is zero.
This splitting yields the geometric form of two mutually exclusive but jointly complete alternatives. However, their physical meaning particle states, measurement outcomes, spatial directions, or entire histories must be specified by an additional mapping.
At the same time, the algebraic idempotents \(\ep,\em\) and the quantum outcome projectors \(P_h\) should not be considered the same object a priori. They share the properties of idempotency, orthogonality, and completeness. It is further assumed that each stable algebraic channel can be associated with a corresponding projection of the space of physical states. Constructing an exact mapping between these two levels remains a separate task of the model.
5. Independent Binary Events
A primary pair of idempotents describes a single splitting of the whole into two mutually exclusive channels. However, a sequence of measurements or other quantum interactions contains several distinct events. Therefore, for each event with number \(k\), we associate its own pair of projectors \(p_k^+\) and \(p_k^-\):
\[\tag{10} p_k^++p_k^-=1, \qquad (p_k^\pm)^2=p_k^\pm, \qquad p_k^+p_k^-=0. \] The first equality expresses the completeness of the splitting: the two channels together preserve the original whole. The second demonstrates the stability of each channel: repeated application of the same projector does not create a new splitting and does not change the already allocated state. The third equality denotes mutual exclusivity of outcomes: a state belonging to channel \(p_k^+\) cannot simultaneously belong to channel \(p_k^-\) of the same event.
The signs \(+\) and \(-\) here denote not positive and negative values, but two possible outcomes. These could be, for example, two spin directions, two positions of the device pointer, or two alternative continuations of the same history. At this stage, the projectors describe only the structure of possibilities; the question of whether both channels become physical realities will be decided later by the chosen interpretation.
The next event receives a different pair: \(p_l^+\) and \(p_l^-\). Separate indices prevent the outcome information from being mixed: the projector \(p_k^{s_k}\) indicates the outcome of event \(k\), and \(p_l^{s_l}\) indicates the outcome of event \(l\). In the simplest algebraic model, independent splittings are considered commutative:
\[\tag{11} p_k^{s_k}p_l^{s_l} =p_l^{s_l}p_k^{s_k}, \qquad k\ne l, \qquad s_k,s_l\in\{+,-\}. \] Commutativity means that the same common channel can be identified regardless of the order in which the projectors are written. For example, selecting the outcome of event \(k\) and then the outcome of event \(l\) results in the same composite channel as the reverse order of algebraic projection. This preserves information about both outcomes: the projectors do not merge into one and do not replace each other.
Condition (11) is an assumption of the simplest model of independent or already decohered events. Projectors of incompatible quantum observables may not commute in general. Therefore, we are not talking about arbitrary measurements here, but about channels that can be jointly included in a single consistent history.
Thus, each new binary event does not destroy previous outcomes, but adds a new independent difference to them. Successive application of such pairs forms a branched system of composite channels, from which a tree of possible histories will be constructed.
6. Tree of Possible Histories
Let's denote the sequential history of results by
\[\tag{12} h=(s_1,s_2,\ldots,s_N), \qquad s_k\in\{+,-\}. \] It corresponds to a joint idempotent
\[\tag{13} \boxed{ P_h=\prod_{k=1}^{N}p_k^{s_k}. } \] For different histories, the orthogonality and completeness conditions are satisfied:
\[\tag{14} P_hP_{h'}=\delta_{hh'}P_h, \qquad \boxed{\sum_hP_h=1.} \] After \(N\) binary events, \(2^N\) possible histories arise:
\[\tag{15} 1 =\prod_{k=1}^{N}(p_k^++p_k^-) =\sum_{h\in\{+,-\}^N}P_h. \] Orthogonality of different histories follows from the fact that any two distinct sequences \(h\) and \(h'\) differ in at least one result. Therefore, the product \(P_hP_{h'}\) necessarily contains a factor of the form \(p_k^+p_k^-\), equal to zero. Completeness, on the contrary, is achieved by expanding all the brackets \(p_k^++p_k^-\): every admissible sequence of results appears in the sum.
At this level, the tree describes all algebraically admissible continuations. Which of these become physical realities and whether they persist after measurement is determined not by formula (15), but by the chosen quantum interpretation.
The number of channels increases, but the whole is not multiplied. The total sum remains equal to one. This is why geometric branching should not be understood as a mechanical copying of matter or energy.
7. An example of two consecutive measurements
For two binary events, we have
\[\tag{16} 1=(p_1^++p_1^-)(p_2^++p_2^-). \] Expanding the product, we obtain four orthogonal histories:
\[\tag{17} \boxed{ 1 =p_1^+p_2^+ +p_1^+p_2^- +p_1^-p_2^+ +p_1^-p_2^-. } \] They correspond to the sequences \((++),(+-),(-+),(--)\). Each history contains not only two results, but also all the physical records that arose during the process of their registration.
8. Global state of the tree
We write the generalized state of reality as
\[\tag{18} \boxed{ \mathcal D=\sum_hc_hP_hJ_h. } \] Here \(P_h\) defines the history channel, \(J_h\) describes the state of physical objects within it, and \(c_h\) is the complex amplitude. Relative reality takes the form
\[\tag{19} \mathcal R_h=P_h\mathcal D=c_hP_hJ_h. \] The three factors in \(c_hP_hJ_h\) perform different functions. The projector \(P_h\) determines which history the state belongs to; The operator \(J_h\) contains its physical internal content; the coefficient \(c_h\) defines the amplitude, the square of whose modulus subsequently determines the weight of this history. Therefore, the projector and the amplitude do not duplicate each other: the former distinguishes channels qualitatively, the latter characterizes them quantitatively.
The operator \(J_h\) can contain final states of the form \(J(a,b)=\j^a(-\j)^b\), where \(a\) describes the internal state, and \(b\) the external movement. Projectors \(P_h\) belong to a higher level: they distinguish not two components of a single operator \(J(a,b)\), but entire alternative histories of the global state.
9. Measurement as Establishing Correlation
Within each story, the measured system, the instrument, the environment, and the observer must be consistent. Symbolically, this can be written as
\[\tag{20} J_h =J_h^{(\mathrm S)} \otimes J_h^{(\mathrm A)} \otimes J_h^{(\mathrm E)} \otimes J_h^{(\mathrm O)}. \] Then after the measurement
\[\tag{21} \mathcal D =\sum_hc_hP_h \left( J_h^{(\mathrm S)} \otimes J_h^{(\mathrm A)} \otimes J_h^{(\mathrm E)} \otimes J_h^{(\mathrm O)} \right). \] The tensor product sign indicates that the system, instrument, environment, and observer remain distinct subsystems but form a single joint state. After the measurement, their components can no longer be selected independently: the system's result, the instrument's reading, and the observer's record must belong to the same history \(h\).
A measurement in such a record is not a simple reading of a predetermined value. It creates a correlation: a certain output of the system corresponds to a certain pointer position, a certain state of the environment, and a certain entry in the observer's memory.
10. Decoherence and Branch Autonomy
When interacting with the environment, the state takes the form
\[\tag{22} |\Psi\rangle =\sum_hc_h|\psi_h\rangle|E_h\rangle. \] If the states of the environment associated with different outputs become practically orthogonal, then
\[\tag{23} \langle E_h|E_{h'}\rangle\approx0, \qquad h\ne h'. \] In the reduced density matrix, the interference terms are suppressed:
\[\tag{24} \boxed{ P_h\rho P_{h'}\approx0, \qquad h\ne h'. } \] Here \(\rho\) denotes the reduced density matrix of the observed system, obtained after eliminating inaccessible degrees of freedom of the environment. The sign \(\approx0\) is important: the mixed components usually do not disappear completely, but become so small that it is practically impossible to detect interference between the macroscopic branches.
Thus, decoherence explains why the branches behave like autonomous classical realsreality, but does not by itself select one of them. The question of whether all branches are preserved or only one physically remains remains a matter of interpretation.
Not every formal decomposition is a splitting of reality into realities. It is reasonable to call reality only a stable channel in which a consistent system of physical correlations and records has formed.
11. An Observer within Reality
An observer belonging to branch \(h\) has access to the projection.
\[\tag{25} \mathcal D_h=P_h\mathcal D. \] For the other branch
\[\tag{26} P_hP_{h'}\mathcal D=0, \qquad h\ne h'. \] Therefore, the observer does not perceive the sum of incompatible macroscopic results. Within their reality, there exists one continuous sequence of memories and one local trajectory:
\[\tag{27} \mathbf r_h(t)=\mathbf r_h\!\left(s_h(t)\right). \] This is consistent with the postulate of local one-dimensionality of realization: the multiplicity of possible continuations of a global state does not mean that the same local observer simultaneously moves in several directions. In the many-worlds interpretation, each branch corresponds to the observer's own relative state.
12. Norm Preservation
Orthogonality allows us to decompose the norm of the global state into branches:
\[\tag{28} \|\mathcal D\|^2 =\sum_h|c_h|^2\|P_hJ_h\|^2. \] If each state of a branch is normalized, then
\[\tag{29} \boxed{ \sum_h|c_h|^2=1. } \] Branching does not create an additional norm. It distributes the components of a single state into orthogonal channels. This continues the general principle of Wave Electricity: splitting reveals the structure of the whole without destroying or doubling it.
13. Is energy created during branching?
A common objection to the literal many-worlds picture is that each dimension should create new copies of matter and energy. In the geometry under consideration, such addition is incorrect. The energy of a global state is determined by taking into account the weights of the orthogonal components:
\[\tag{30} \langle E\rangle =\sum_h|c_h|^2E_h. \] This decomposition is valid if, after decoherence, the Hamiltonian practically does not mix different branches, and the interbranch matrix elements can be considered negligible. Then \(E_h\) has the meaning of a conditional average energy value within the history \(h\), and the global energy is determined by the weighted sum of these values.
Therefore, the conditional energies of different branches cannot be summed as if they were all simultaneously in a single observable spatial channel.
\[\tag{31} \boxed{ \text{increase in the number of branches} \ne \text{increase in the total energy}. } \] A rigorous proof of energy conservation requires specifying the Hamiltonian of the full state and its dynamics. Therefore, formula (30) expresses the principle of constructing the model, rather than replacing such a derivation.
14. Reality Weights and the Born Rule
The natural measure of a branch is
\[\tag{32} w_h=|c_h|^2, \qquad \sum_hw_h=1. \] The weight \(w_h\) should not be understood as a fraction of matter or a simple number of identical worlds. In particular, probability cannot be obtained by simply counting branches: a single interaction can create a huge or even continuous set of components. The weight is determined by the amplitude norm and characterizes the measure of history in the complete state, not the number of its geometric copies.
Before measurement, \(w_h\) is interpreted as the probability of the outcome. In the many-worlds picture, after measurement, it can be viewed as a measure of the relative branch. However, the relations of idempotency and orthogonality do not yet derive the Born rule.
The rule \(w_h=|c_h|^2\) is currently accepted from quantum mechanics. Its possible derivation from the norm of the deep state space should be the subject of a separate work.
15. The Unified Core of Quantum Interpretations
We introduce the actualization rule \(\mathcal A\), which determines the physical status of the branches:
\[\tag{33} \mathcal A: \{\mathcal R_h\} \longrightarrow \{\text{physically realized states}\}. \] Then the difference in interpretations can be expressed by a general scheme:
\[\tag{34} \boxed{ \text{quantum interpretation} = \text{splitting geometry} + \text{actualization rule}. } \] The word "actualization" is used here broadly. It can mean collapse, preservation of all branches, selection of the actual configuration, selection of a consistent family, or definition of a state relative to another object.
16. Copenhagen IntReduction
In the schematic Copenhagen interpretation, measurement identifies a single result \(h_*\):
\[\tag{35} \mathcal D \longrightarrow \frac{P_{h_*}\mathcal D} {\|P_{h_*}\mathcal D\|}. \] The remaining branches cease to be included in the description of the obtained result. In some versions, reduction is understood as a physical change, in others, as an update of knowledge after the registration of an event. Therefore, the "Copenhagen interpretation" is not a single strictly fixed model, but a family of related approaches.
17. Many-Worlds Interpretation
In the relative states formulation, global dynamics does not require collapse. After decoherence, all branches with non-zero weights are preserved:
\[\tag{36} \mathcal A_{\mathrm E} =\{\mathcal R_h:\,w_h>0\}. \] The condition \(w_h>0\) alone is not sufficient to call a component a separate reality. The global state may contain small interfering components that do not form a stable history. Only decohered channels, within which consistent and sufficiently long-lasting physical records have formed, attain the status of relative reality.
It is here that the distinction between reality and realities acquires a direct physical meaning:
\[\tag{37} \boxed{ \mathcal D=\sum_h\mathcal R_h, \qquad \mathcal R_h=\text{physically realized relative reality}. } \] The observer is not transported to one of the pre-prepared universes. During the interaction, correlated relative states of the observer arise. In each reality, its observer sees its own specific outcome.
In this sense, idempotent geometry offers a more cautious image than the literal "birth of new universes"=> global reality is preserved, while its internal structure unfolds into virtually autonomous orthogonal realizations.
18. The Bohmian Interpretation
In Bohmian mechanics, the global wave function preserves all components, but the actual configuration \(Q(t)\) is located in one branch:
\[\tag{38} Q(t)\in\operatorname{supp}\!\left(\mathcal R_{h_*}\right). \] Other branches may persist as "empty waves" that do not contain the actual configuration. This is particularly close to the combination of a deep possibility space and a single locally realized trajectory. However, a fully-fledged Bohmian model requires a separate guiding equation defining the motion \(Q(t)\). Idempotent splitting by itself does not provide such a law.
19. Objective Collapse
In objective reduction models, branches initially emerge, but special stochastic dynamics enhance one of them and suppress the others:
\[\tag{39} |c_{h_*}|^2\longrightarrow1, \qquad |c_{h\ne h_*}|^2\longrightarrow0. \] Here, collapse is independent of the observer's consciousness and must be an objective physical process. To obtain such an interpretation within Wave Electricity, it would be necessary to supplement the normalized dynamics of \(J\) with a new law for the variation of branch amplitudes.
20. Consistent Histories
The consistent histories approach is particularly close to the idea of an event tree. In the standard quantum formalism, history is specified by a class operator.
\[\tag{40} C_h =P_{s_N}(t_N)\cdots P_{s_2}(t_2)P_{s_1}(t_1). \] The projector \(P_{s_k}(t_k)\) expresses the assertion that at time \(t_k\), the system belongs to the state or range of states \(s_k\). The successive product of such projectors no longer defines a single result, but an entire time-ordered history. The order of the factors is significant here, since quantum projectors related to different moments or incompatible observables may not commute in general.
For two histories, a decoherence functional is introduced.
\[\tag{41} D(h,h') =\operatorname{Tr}\!\left(C_h\rho C_{h'}^\dagger\right). \] The family is considered consistent if
\[\tag{42} \boxed{ D(h,h')\approx0, \qquad h\ne h'. } \] Then the diagonal elements of \(D(h,h)\) admit a probabilistic interpretation. Moreover, the product of projections at different points in time need not itself be idempotent. Therefore, equality between \(C_h\) and the algebraic \(P_h\) cannot be postulated without an additional mapping. The common denominator is the structure of orthogonal histories, not the literal coincidence of all operators.
21. Relational Quantum Mechanics
In the relational approach, the state of an object is defined relative to another physical system. To do this, we introduce a relational index:
\[\tag{43} \mathcal R_h^{(A|B)} =P_h^{(A|B)}\mathcal D. \] Here, the outcome of system \(A\) is defined relative to the interacting system \(B\). One object can have a certain state relative to one observer and not yet have such a state relative to another. It is the structure of physical relations, not absolute space, that splits primarily.
22. Informational Reading
In QBism and related informational approaches, the quantum state expresses the agent's expectations regarding its future experience. Then \(P_h\) describes a possible event in the system of expectations, not an objectively existing branch of the Universe.
\[\tag{44} P_h \longleftrightarrow \text{possible experience of the agent}. \] This limiting case shows that the same algebraic form allows for both ontological and informational interpretations. Consequently, the physical meaning of splitting cannot be derived solely from identities for projectors.
23. Comparison of Interpretations
The interpretations considered are based on the same quantum formalism, but they answer the question of the physical status of its orthogonal components differently. For some, they remain only possible outcomes, for others, they form a set of realized realities, and in some approaches, all components are preserved mathematically, but only one configuration or one relative history is recognized as relevant. Let's compare these approaches based on three main features: what happens to the branches after measurement, whether physical collapse is assumed, and what exactly is considered a realized state.
| Interpretation | Branch status | Collapse | What is relevant |
|---|---|---|---|
| Copenhagen | Possible results | Postulated or understood as an update of the description | Result obtained |
| Many-worlds | All stable branches are physically preserved | No | Each relative reality within itself |
| Bohmian | All components are preserved in wave | No | One configuration and its trajectory |
| Objective collapse | Unrealized branches are dynamically suppressed | Physical | One stable branch |
| Consistent histories | A consistent family is admissible | No | History within a selected family |
| Relational | States are relative to interacting systems | No | Outcome relative to the system |
| QBism | Possible outcomes in the agent's expectations | No Physical Collapse | New Agent Experience |
The table shows that interpretations cannot be simply lumped together into a single ontology. They provide incompatible answers about the existence of unchosen outcomes. The unifying factor is a deeper level the structure of possible orthogonal continuations.
24. What Place Does Wave Electricity Occupy?
The modern concept of Wave Electricity does not yet fully coincide with any single interpretation. Its different provisions point to different, closely related approaches:
The listed correspondences should be understood as structural similarities, not as proof of mathematical equivalence. The global state of Wave Electricity has not yet been strictly identified with the universal wave function, nor its deep idempotent channels with Hilbert space projectors. Nevertheless, the coincidence of the fundamental relations allows us to determine which interpretations the individual propositions of the model gravitate toward.
A single normalized global state brings the model closer to the Everett picture without fundamental collapse.
Orthogonal channels of deep splitting are naturally associated with consistent quantum histories.
The local one-dimensionality of each realization is reminiscent of the idea of a single actual trajectory, characteristic of Bohmian mechanics.
The dependence of the observed manifestation on projection and interaction is close to the relational understanding of state.
Therefore, the most natural scheme for the model is view
\[\tag{45} \boxed{ \begin{aligned} &\text{single global state}\\ &\quad+\, \text{orthogonal histories}\\ &\quad+\, \text{relative local realization}. \end{aligned} } \] 25. The Principle of Relative Realities
The following principle can be proposed for the many-worlds extension of the model.
The Principle of Relative Realities. Global reality represents a complete normalized state. With stable decoherence, its orthogonal channels form relative realities. Each such reality has its own consistencyThe history of objects, devices, and observers is a continuous process, while the totality of all realities preserves the unity of global reality.
Formally:
\[\tag{46} \boxed{ \mathcal D=\sum_h\mathcal R_h, \qquad \mathcal R_h=P_h\mathcal D, \qquad P_hP_{h'}=0\quad(h\ne h'). } \] This principle does not follow automatically from the original idempotent algebra. It is an additional interpretative hypothesis about the physical status of stable branches.
26. A Unifying Model and the Boundaries of Unification
The proposed construction can be called a unifying geometric model of quantum interpretations. It shows that many interpretive disagreements arise after the construction of a common tree of alternatives at the moment when branches are assigned a specific physical status.
\[\tag{47} \boxed{ \text{single branching geometry} + \text{different update rules} = \text{different interpretations}. } \] Unification does not mean that all interpretations are simultaneously correct. It means that they can be placed within a single formal framework and the precise additional postulate where they diverge can be shown.
27. Unresolved Questions
To transform the geometric scheme into an independent physical theory, the following questions must be answered:
1. How does the physical tree of projections \(P_h\) emerge from the dynamics \(J(a,b)\)?
2. What determines the preferred set of branches and the stable basis for decoherence?
3. Is it possible to derive the Born rule rather than introduce it additionally?
4. Are all nonzero branches physically realized?
5. Is there a single global actual trajectory or a separate trajectory within each reality?
6. Is backward interference of already formed realities possible?
7. Does the model yield a testable prediction that distinguishes it from standard quantum mechanics?
Until these questions are resolved, the correct status of the construction is expressed by the formula
\[\tag{48} \boxed{ \text{idempotent splitting} = \text{unifying geometric model of interpretations}. } \] Conclusion
This article proposes distinguishing between a unified reality and the relative realities that comprise it. Reality is described by a complete normalized state, while reality is described by a stable orthogonal projection within which the measurement result, instrument readings, the state of the environment, and the observer's memory are consistent.
Idempotent algebra naturally describes the completeness and mutual exclusivity of channels. A sequence of independent splits forms a tree of histories, but does not propagate the original whole: the sum of all projectors remains equal to one.
The main quantum interpretations can be compared on a common geometric basis. The Copenhagen approach singles out a single outcome, the many-worlds approach preserves all stable realities, the Bohmian approach places the current configuration on a single branch, objective collapse suppresses the remaining components, consistent histories constrain admissible families, and the relational and informational approaches alter the very meaning of state.
The strongest claim that can be made at this stage is not a final solution to the measurement problem, but rather the discovery of a general geometry of its possible solutions:
\[\tag{49} \boxed{ \underbrace{\mathcal D}_{\text{unified reality}} =\sum_h \underbrace{P_h\mathcal D}_{\text{relative reality }h}, \qquad \sum_hP_h=1. } \] Perhaps quantum interpretations describe not different mathematical worlds, but different ways of answering the same question: how do orthogonal possibilities of a single reality become reality for an observer within it.
Materials used
- H. Everett III. “Relative State” Formulation of Quantum Mechanics, Reviews of Modern Physics, 29, 454–462 (1957).
- D. Bohm. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. I, Physical Review, 85, 166–179 (1952).
- R. B. Griffiths. Consistent Histories and the Interpretation of Quantum Mechanics, Journal of Statistical Physics, 36, 219–272 (1984).
- G. C. Ghirardi, A. Rimini, T. Weber. Unified Dynamics for Microscopic and Macroscopic Systems, Physical Review D, 34, 470–491 (1986).
- C. Rovelli. Relational Quantum Mechanics, International Journal of Theoretical Physics, 35, 1637–1678 (1996).
- W. H. Zurek. Decoherence, Einselection, and the Quantum Origins of the Classical, Reviews of Modern Physics, 75, 715–775 (2003).