# Consistent Histories and Many Worlds

## Abstract

**:**

## 1. Introduction

## 2. Basic Formalism of Consistent Histories

_{0}, property ${F}_{1}^{\alpha}$ at t

_{1}, and so on. It is also possible to consider histories that are nontrivial superpositions of the projectors of the above kind, even though their physical interpretation is somewhat intricate. A family of histories $\left\{{Y}^{\alpha}\right\}$ contains orthogonal projectors of the above kind that sum up to unity, exactly as explained in the previous paragraph. Histories Y

^{α}are called “elementary”, and they constitute a sample space. In what follows, we will drop the distinction between families of histories and families of elementary histories (sample spaces) if this does not lead to confusion. Thus, such a family constitutes a framework, for which it is possible to define a classical probability function.

_{a}to t

_{b}is assumed to be governed by a unitary operator T(t

_{b}, t

_{a}), which depends on the Hamiltonian of the system. Next, we define for any history ${Y}^{\alpha}$ its corresponding chain operator $K\left({Y}^{\alpha}\right)$, as follows:

_{0}, t

_{1}and limit ourselves to histories with a fixed initial state $|{\psi}_{0}\u27e9$. That is, the selected family of histories consists of the following:

_{1}. When we apply Formula (2), we can quickly calculate that

_{1}will generally not be the result of a unitary evolution applied to the initial state $T\left({t}_{1},{t}_{0}\right)|{\psi}_{0}\u27e9$. We will return to the question of the proper ontological interpretation of histories later, but for now we can use a conceptual crutch, in the form of a Copenhagen-style explanation, with its irreducible use of measurements. A history of the form (7) may be provisionally interpreted as resulting from a series of measurements, each of which is associated with a particular projector ${F}_{i}^{\alpha}$. In other words, at every moment t

_{i}where i > 0, we ask the experimental question whether the system is in a state corresponding to ${F}_{i}^{\alpha}$. If the answer each time is “yes”, we have physically selected the history (7) out of many alternative possibilities. The probability associated with a particular history ${Y}^{\alpha}$ is precisely the probability that appropriate measurements will reveal a string of yes-answers to questions ${F}_{i}^{\alpha}$. However, we have to stress that the CH approach does not admit the concept of measurement understood as a special physical process different from the standard unitary evolution prescribed by the Schrödinger equation. We will discuss this issue shortly.

## 3. Example of a Consistent Family

_{0}and t

_{1}before entering the magnet, and t

_{3}after leaving the magnet (see Figure 1). The initial state at t

_{0}is assumed to be $|{\psi}_{0}\u27e9=$ $|{x}^{+}\u27e9|\omega \u27e9$, where $|\omega \u27e9$ is the spatial wave function associated with the particle at t

_{0}. We assume the standard dynamics in the following form:

_{1}and t

_{2}. That way we will arrive at the following unitary history:

_{0}). The probability assigned to this history is obviously 1. On the other hand, if we wanted to calculate the probabilities of obtaining definite values of z-spin at time t

_{2}, we would have to use a different family $\mathcal{F}$

_{1}, consisting of the following two histories (plus their complement, which I will ignore):

_{2}equals ½, as expected. However, we may be interested in asking a similar question regarding the z-spin at t

_{1}, before the particle enters the magnet. In order to answer this question, we have to select yet another family (let us call it $\mathcal{F}$

_{2}):

_{1}, conditional on the initial state being $|{\psi}_{0}\u27e9$, is one-half. However, if we conditionalize on the later values of spins at t

_{2}, the result will be different: the probability of the z-spin being “up” at t

_{1}, given that at t

_{2}it was “up”, equals one.

## 4. Frameworks and Worlds

“[…] if a single framework, a single consistent family, of histories is in view, the sample space, represented mathematically by an appropriate PD of the history identity, is a collection of mutually-exclusive possibilities, one and only one of which actually occurs.”[italics mine]

#### The Many-Worlds Variant of CH

_{0}includes just one possible world (let us call it ${w}_{0}$) with the unitary history (10), since this is the only history in this family that receives a non-zero probability. However, another group of worlds contains equally probable histories from family $\mathcal{F}$

_{1}. In one of these worlds (${w}_{1}^{+}$), the electron has a well-defined x-spin before entering the magnet and then acquires the value “up” of the z-spin, while simultaneously travelling along the upper trajectory. The alternative world (${w}_{1}^{-}$) differs, in that the electron leaves the magnet following the lower trajectory and possesses the “down” value of spin in the z direction. The third considered family $\mathcal{F}$

_{2}, which—it has to be stressed—is equally acceptable, also separates into two worlds. One world ${w}_{2}^{+}$ contains an electron that already exhibits the “up” value of its z-spin before entering the magnet, and consequently follows the upper trajectory, while in the other world ${w}_{2}^{-}$ the electron consistently possesses z-spin “down” from the moment t

_{1}. All in all, in our simple example we have five distinct worlds ${w}_{0}$, ${w}_{1}^{+}$, ${w}_{1}^{-}$, ${w}_{2}^{+}$, and ${w}_{2}^{-}$ grouped into three families $\mathcal{F}$

_{0}, $\mathcal{F}$

_{1}, and $\mathcal{F}$

_{2}.

_{2}the electron is in a superposition with no well-defined z-spin, while in ${w}_{1}^{+}$ it possesses a definite value z

^{+}. On the other hand, worlds ${w}_{1}^{-}$, ${w}_{2}^{+}$ diverge with respect to the definite values of z-spin at t

_{2}, as well as regarding the state of the electron at t

_{1}. In world ${w}_{1}^{-}$, the electron has a definite value of x-spin at t

_{1}, whereas in ${w}_{2}^{+}$ the electron is characterized by a definite z-spin at the same moment t

_{1}. In worlds ${w}_{1}^{+}$ and ${w}_{2}^{+}$ there are no differences regarding the possessed values of the same parameter, but nevertheless the worlds are different due to their incompatible characterizations of the electron’s state at t

_{1}.

_{0}. No other histories are admissible; they do not represent any real physical processes. However, MWI interprets the superposition $\frac{1}{\sqrt{2}}\left(|{z}^{+}\u27e9|{\omega}^{+}\u27e9+|{z}^{-}\u27e9|{\omega}^{-}\u27e9\right)$ characterizing the system at time t

_{2}as describing two independent realities: one in which the electron has spin “up” in the z direction, and the other in which the z-spin of the electron is “down”. Thus, MWI admits the existence of two worlds ${w}_{1}^{+}$ and ${w}_{1}^{-}$, even though no history corresponding to these worlds represents a genuine quantum-mechanical process, since these histories clearly violate the universal law of quantum mechanics, i.e., the Schrödinger equation.

_{2}, which eliminates the worlds ${w}_{2}^{+}$ and ${w}_{2}^{-}$, since they seem to move the moment of splitting back in time to point t

_{1}.

## 5. The Measurement Problem and Quasi-Classicality

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Bigaj, T.
Consistent Histories and Many Worlds. *Quantum Rep.* **2023**, *5*, 186-197.
https://doi.org/10.3390/quantum5010012

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Bigaj T.
Consistent Histories and Many Worlds. *Quantum Reports*. 2023; 5(1):186-197.
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Bigaj, Tomasz.
2023. "Consistent Histories and Many Worlds" *Quantum Reports* 5, no. 1: 186-197.
https://doi.org/10.3390/quantum5010012