A multi-world interpretation of quantum mechanics

    Surely most of you no, no, and even in the literature of popular science there have been references to the "many-world interpretation" of quantum mechanics (MMI). They like to remember her in the comments on Habré, but often in the wrong way or with serious inaccuracies.


    Let’s try to figure out what’s happening in MMI.

    Part 1: why do you need to "interpret" quantum physics?

    Quantum physics has firmly entered our lives: flash drives use the tunnel effect , lasers record and transmit information, and LED lamps illuminate our homes. We are perfectly able to describe all these phenomena using the mathematical apparatus of quantum physics, and the most accurate experiments do not find deviations from the effects predicted by the theory. On the other hand, the physical meaning of all these equations sometimes eludes us. Interpretations of quantum mechanics try to fill the equations with some physical (and philosophical) content.

    Important : all interpretations are reduced to the same equations of the standard QM and do not predict new physics!

    The main problem that interpretations are trying to solve is the measurement problem. In classical physics, everything is simple: there is space and time, there is matter in this space, there are system parameters (like momentum or position), and there are laws of physics that describe the change in these parameters. If you know exactly the initial state of the system, you can predict its future behavior with absolute accuracy. In quantum physics, this is not so ... The system is described by a wave function. It determines the probability of measuring the system in a certain state (for example, a certain coordinate or momentum). Before measuring the measurement process, it cannot be said that the system has a certain moment, it has only a wave function.

    It is important that the probability is given by the squared modulus of the wave function, and not by the wave function itself. In this case, the WF itself can take both positive and negative values. Moreover, two WFs (or parts of WFs) can interfere with each other.
    Probability calculation rule (Bourne rule). The squares of the coefficients in the wave function determine the probability of a specific outcome in the measurement. For example, the Schrödinger cat is described by the WF:

    $ \ Psi = \ alpha_1 | alive> + \ alpha_2 | dead>, \ alpha_1 = \ alpha_1 = \ frac {1} {\ sqrt {2}} $

    the probability of him being alive when you open the box is considered as $ P (alive) = | \ alpha_1 | ^ 2 = 0.5 $, i.e. fifty%. The same for the probability of him being dead:$ P (dead) = | \ alpha_2 | ^ 2 = 0.5 $again 50%.

    Small illustration


    Your friend - Vasya Pupkin - spends his days either at the computer, programming, or on the couch, playing playstation. You are standing in front of a closed door to his apartment. From the classical point of view, Vasya is either at the computer or on the couch, you just don’t know where exactly. But quantum Vasya is simultaneously in two places until you open the door and look (measure its condition). His condition before measurement:

    $ \ Psi = \ frac {1} {\ sqrt {2}} (| game> + | work>) $

    And after measuring with a probability of 50% he is at the game or at work.

    Let's continue the illustration. Suppose, before doing business, Vasya can either go to the refrigerator for a beer or smoke on the balcony. At the same time, if you caught him during these activities (watched by the refrigerator or on the balcony), he then with equal probability goes to play on the sofa or work. But it may be that when you are not looking, he is 100% of cases with a joystick in his hands. The reason for this is interference. Vasya’s state is described by a wave function, which can be negative, but at the same time correspond to the same probability as a positive WF.

    Let's take a closer look. First step: if we don’t look, Vasya is in a superposition of a refrigerator / balcony:

    $ \ Psi = \ frac {1} {\ sqrt {2}} (| refrigerator> + | balcony>) $

    Second step: let's say if Vasya comes from the refrigerator, his WF

    $ | refrigerator> = \ frac {1} {\ sqrt {2}} (| game> - | work>), $

    and if it comes from the balcony:

    $ | balcony> = \ frac {1} {\ sqrt {2}} (| game> + | work>) $

    If we observe it in its original state, we will reduce its state to either | fridge> or | balcony>, which will give a 50/50 probability at the output: he will go to play or work. But if we do not observe his movements, his WF:

    $ \ Psi = \ frac {1} {\ sqrt {2}} (| refrigerator> + | balcony>) = \ frac {1} {2} (| game> - | work> + | game> + | work> ) = | game> $

    That is, he always ends up on the couch! And all because of interference.

    So, we see that the fact of observing us for Vasya changes his final state. Why does measurement play such a significant role? Interpretations of CM are trying to answer this question.

    The classical (Copenhagen) interpretation postulates that the observation process is the process of the collapse of the wave function into one of the states. The collapse leads to the fact that the WF continues to evolve only as one part of the original WF, the object is no longer in a superposition state and cannot interfere. As a result, all sorts of effects such as quantum entanglement disappear. She does not explain how the collapse occurs, as well as why some interactions cause collapse, while others do not. The presence of such postulates is not liked by everyone, and scientists are trying to find alternative interpretations. One of the simplest and most developed is the multiworld.

    Part 2: A World-Wide Interpretation


    To begin with, recall what quantum entanglement is. By definition, two states are confused when it is not possible to separate them into two independent parts. Let's go back to the illustration from the first part, and imagine that Vasya has a girlfriend, Anya. Anya either reads a book in an armchair or walks in a park. Until they started dating, their choice was random:

    $ | Vasya, Anya> = 0.5 | game, book> +0.5 | game, park> +0.5 | work, book> +0.5 | work, park> $

    And the outcome of your measurement gave a probability of 25% to each specific set (and the probability of finding Vasya on the couch in total was 50%).

    Now they are in a confused state:

    $ | Vasya, Anya> = \ frac {1} {\ sqrt {2}} (| game, book> + | work, park>) $

    If we watch Vasya, then the probability of finding him on the couch is again 50%. However, if he is on the couch, then Anya is absolutely right behind the book, you don’t even need to check.

    This is how the absolute correlation between measurements appears when the system is in an entangled state.

    Next step: Vasya can either go to the balcony or to the refrigerator before sitting down to work or play, but we are not watching him. Let's say Anya and Vasya find themselves in a confused state:

    $ | Vasya, Anya> = \ frac {1} {\ sqrt {2}} (| balcony, book> + | refrigerator, park>) $

    Then the two parts of Vasya’s VF no longer interfere with each other, and we don’t always observe Vasya on the sofa, as it was in the first part:

    $ | Vasya, Anya> = \ frac {1} {2} (| game, book> + | work, book> + | game, park> - | work, park>) $

    Entanglement prevents WF from interfering. In principle, we can perform some operations on the Ani and Vasya system and unravel them, then interference will again be possible. However, for this we need to have access to both systems. In reality, we do not always have access to all parts of an entangled state. For example, when Vasya is confused not only with Anya, but also with two thousand anonymous names on the Internet, and all his neighbors (in other words, the system gets confused with her surroundings), we have no way to return the ability to interfere.

    This effect is called decoherence . The surroundingscalled the degrees of freedom with which the system is in contact, usually there are a lot of them. If the system turns out to be confused with the whole world around us, the different parts of the wave function are completely isolated from each other, although no “collapse” has occurred. As if they were in different worlds.

    This is the main idea of ​​a multi-world interpretation. Its only postulate is that the entire Universe is described by one wave function. There is no "classical" world, no observers, no collapse - all this is a unitary evolution of one WF under the influence of the Schrödinger equation. What we observe as collapse is exclusively a process of decoherence, our inability to “untie” the object and the environment with which it is entangled.

    Different "worlds" in this case arise every time a "collapse" occurs - the interaction of the system with the environment. In this case, one world is divided into several, in accordance with the branches of the WF, and these worlds no longer interact.

    Example with a Schrödinger cat: in a famous thought experiment, the cat is in a box with poison, which at a random moment poisons the cat. At the same time, according to KM, while the box is closed, the cat is in superposition$ | cat> = \ frac {1} {2} (| alive> + | dead>) $. According to the Copenhagen interpretation, when Schrödinger opens the box, he collapses the cat into a state of either “alive” or “dead”. According to the MMI, Schrödinger is confused:$ | cat, W> = \ frac {1} {2} (| alive, sees + | dead, sees "dead">) $ "data-tex =" inline ">. To this you need to add the environment: $ | cat, W> | o> = \ frac {1} {2} (| alive, sees + | dead, sees "dead">) | exists> $ "data-tex =" inline ">, which, as a result of the decoherence process, gets confused with both of them:
    $ | cat, W, o> = \ frac {1} {2} (| alive, sees + | dead, sees "dead", okr "dead">) | exists> $ "data-tex =" inline ">. In this option, Schrödinger no longer has the ability to" cancel "the measurement or do something to" unravel "Two states. Two worlds were divided: in one Schrödinger found a dead cat, in the other - a living cat. At the same time, no collapse occurred, all this is still just a unitary evolution of a large wave function.

    A bit more formal:

    Part 3: Details


    1. The problem of the existence of the classical world. From the point of view of MMI, everything in the world is quantum. Moreover, from the point of view of mathematics, we can choose an infinite number of ways to divide (choose a basis) WF into different "worlds" (orthogonal states). Question: why do we observe the world classic? How does the Universe “choose” one decomposition method that we observe? This is the so-called preferred basis problem. Answer: because the properties of physical interactions are such that they are all local. The values ​​of the fundamental constants and the Hamiltonian of the Universe are such that localized objects are stable. Macroscopic states can remain so for a long time; the wave function of the Universe does not branch continuously. As a result: we manage to observe macroscopic objects in their places. In another variant of decomposition into a basis, branching occurs so quickly that we could not have time to perceive it. This is the other side of the decoherence process: the decoherence speed is faster the more massive the object.

      More details can be found here: [1] , [2] , [3] , [4]
    2. What exactly is a dimension? How to distinguish measurement from simple interaction? Measurement in the MMI is simply the process of entanglement of the observer and the object as a result of the interaction. Sometimes the interaction can be “rewound” by unraveling the two systems, then this is not a measurement. Usually, a certain amplification process is involved in the measurement process. For example, you detect a photon on a photomultiplier; it knocks out one electron, which, as a result of the avalanche process, is converted into a current at the output from the detector. In MMI, the whole process is the process of entangling one photon with electrons (and other parts of the detector). But such a measurement cannot be rewound - most of the degrees of freedom in entanglement are inaccessible. Of course, for the measurement process it is not necessary that the observer is reasonable, the process is irreversible enough.
    3. When does the division of the worlds occur? Separation occurs when in the process of interaction many degrees of freedom are involved, and the measurement becomes irreversible. Those. after the interaction of the photon with the detector, but before the appearance of the current at the output. As an example, the Schrödinger cat again: the environment there can be considered the process of radioactive decay. At the moment when the core decays and the poison is released, the cat splits into two versions. And from the point of view of the cat, he can no longer interact with his copy. From Schrödinger's point of view, the cat is still alive and dead. Only when he opens the box does he get confused with the cat and the source of radiation. Because radioactive decay is irreversible, Schrödinger is also irreversibly split into two versions of itself.
    4. Is MMI a local theory? Because in MMI, the WF obeys the Schrödinger equation, which in turn obeys the special theory of relativity, all interactions in it are local, and the whole theory is local in the same way. The splitting of worlds extends from a measuring point no faster than the speed of light
    5. How many worlds? We do not know, it can be either a finite quantity or infinite. Based on the finiteness of the entropy of the Universe, it can be assumed that the number of worlds is finite.
    6. A multiworld theory is completely deterministic at the level of the WF of the Universe. WF evolves according to the Schrödinger equation. We only observe the world randomly due to the process of measurement and decoherence.
    7. What to do with energy conservation? Energy is saved in the process of dividing the worlds: each world receives “weight” in accordance with the probability associated with this world. The energy of the entire universe remains unchanged.
    8. If the MMI is correct, then can anything happen? No, firstly, the laws of physics act in exactly the same way, and what is not allowed by “ordinary” physics will not happen in the MMI either. Secondly, if the number of worlds is finite, some events may have too little probability to occur.
    9. How to determine the probabilities in MMI? The Bourne rule is not postulated in the MMI, but is derived from general provisions. See e.g. Here or here .
    10. Is it possible to test MMI? MMI is a “pure” version of quantum mechanics, so every time we test QM, we test MMI. To prove that it is the MMI that is the correct theory, and not some other one, is difficult, although different ideas were proposed, you can find it here .

    Bottom line: MMI is a minimalist interpretation of QM, requiring nothing but the mathematical apparatus of quantum mechanics itself. The best interpretation for Occam's razor.


    1. https://plato.stanford.edu/entries/qm-manyworlds/
    2. https://www.hedweb.com/everett/everett.htm
    3. Mad-Dog Everettianism: Quantum Mechanics at Its Most Minimal
    4. http://www.preposterousuniverse.com/blog/2014/06/30/why-the-many-worlds-formulation-of-quantum-mechanics-is-probably-correct/
    5. Making Sense of the Many Worlds Interpretation

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