Multi-world interpretation of quantum mechanics
Surely most of you no-no and even found in the popular science literature references to the "many-world interpretation" of quantum mechanics (MMI). She is loved to be remembered in the comments on Habré, but often in the wrong key or with serious inaccuracies.
Let's try to figure out what is what in the MMI.
Quantum physics has firmly entered our lives: flash drives use a tunnel effect , lasers record and transmit information, and LED lamps illuminate our homes. We are perfectly able to describe all these phenomena with the help of 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 are trying to fill the equations with some physical (and philosophical) content.
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Important : all interpretations are reduced to the same equations of standard CM and do not predict a new physics!
The main problem that interpretation is trying to solve is the problem of measurement. In classical physics, everything is simple: there is space and time, there is matter in this space, there are parameters of the system (like an impulse or position), and there are laws of physics that describe the change of these parameters. If you know exactly the initial state of the system, you can predict its behavior in the future with absolute accuracy. In quantum physics, everything is wrong ... 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 only has 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 HFs (or parts of HFs) can interfere with each other.
Your friend - Vasya Pupkin - spends his days either at the computer, programming, or on the couch, playing in the playstation. You are standing in front of a closed door to his apartment. From the classical point of view, Vasya, either at the computer or on the couch, you just do not know exactly where. But the quantum Vasya is in two places at the same time, until you open the door and look at it (measure its state). Its state before measurement:
And after measuring with a probability of 50%, he is at play or at work.
Continue the illustration. Suppose, before you go about your business, Vasya can either go to the refrigerator for a beer or smoke on the balcony. At the same time, if you caught him doing these exercises (watched from the refrigerator or on the balcony), he then goes with equal probability to play on the sofa or work. But it may be that when you do not look, it is 100% of cases with a joystick in hand. 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 see more in detail. First step: if we do not look, Vasya is in the superposition state of a refrigerator / balcony:
That is, he always turns on the couch! And all because of the interference.
So, we see that the fact that we observe Vasya changes his final state. Why does measurement play such a significant role? Interpretations of the CM are trying to answer this question.
The classical (Copenhagen) interpretation postulates that the process of observation is the process of the collapse of the wave function into one of the states. Collapse leads to the fact that the VF continues evolution only as one part of the original VF, the object is no longer in a state of superposition and can not interfere. As a result, any effects like quantum entanglement disappear. How does the collapse occur, it does not explain, as well as why some interactions cause collapse, while others do not. Not everyone likes these postulates, and scientists are trying to find alternative interpretations. One of the simplest and most developed is multiworld.
First, let us recall what quantum entanglement is. By definition, two states are tangled 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 likes reading a book in her chair, or walks in the park. Until they started dating, their choice was random:
And the outcome of your measurement gave a probability of 25% to each particular set (and the probability of finding Vasya on the sofa in the amount was 50%).
Now they are in a confused state:
If we watch Vasya, he is likely to find him on the sofa again 50%. However, if he is on the couch, then Anya is absolutely exactly behind the book, you don’t even need to check.
This is the absolute correlation between measurements 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. Suppose Anya and Vasya at the same time find themselves in a confused state:
Then the two parts of the WF Vasya no longer interfere with each other, and we do not always see Vasya on the sofa, as it was in the first part:
Entanglement does not allow the VF to interfere. In principle, we can perform some operations on the system of Ani and Vasya 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 the entangled state. For example, when Vasya is confused not only with Anya, but with two thousand anonymous Internet users, and all his neighbors (in other words, the system is confused with her surroundings), we have no way to regain the ability to interfere.
This effect is called decoherence . Environment is the degree of freedom with which the system is in contact, usually a lot of them. If the system is entangled with the whole world, 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 multi-world interpretation. Its only postulate is that the whole Universe is described by a single wave function. There is no “classical” world, no observers, no collapse — all this is a unitary evolution of a single WF under the influence of the Schrödinger equation. What we see as a collapse is an exclusively decoherence process, our impossibility to “untie” the object and the environment with which it is confused.
At the same time, different “worlds” 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.
An example of a Schrödinger cat: in a well-known thought experiment, a cat is in a box of poison, which poisons a cat at random times. At the same time, according to the CM, while the box is closed, the cat is in superposition |cat>= frac12(|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 MMI, Schrödinger finds itself in a tangled state: |cat,W>= frac12(|alive,sees"alive">+|dead,sees"dead">) . Add to this the environment: |cat,W>|o>= frac12(|alive,sees"alive">+|dead,sees"dead">)|exists> which as a result of the decoherence process is confused with both of them:
|cat,W,o>= frac12(|alive,sees"alive",env"alive">+|dead,sees"dead",okr"dead">)|exists> . In this version, Schrödinger no longer has the ability to “cancel” a measurement or do something to “unravel” two states. The two worlds were divided: in one, Schrödinger found a dead cat, in the other a living one. In this case, no collapse has occurred, all of this is still just a unitary evolution of a large wave function.
Bottom line: MMI is a minimalist interpretation of CM, which does not require anything other than the mathematical apparatus of quantum mechanics itself. The best interpretation for Occam's razor.
Literature:
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/
Let's try to figure out what is what in the MMI.
Part 1: why do we need to "interpret" quantum physics?
Quantum physics has firmly entered our lives: flash drives use a tunnel effect , lasers record and transmit information, and LED lamps illuminate our homes. We are perfectly able to describe all these phenomena with the help of 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 are trying to fill the equations with some physical (and philosophical) content.
')
Important : all interpretations are reduced to the same equations of standard CM and do not predict a new physics!
The main problem that interpretation is trying to solve is the problem of measurement. In classical physics, everything is simple: there is space and time, there is matter in this space, there are parameters of the system (like an impulse or position), and there are laws of physics that describe the change of these parameters. If you know exactly the initial state of the system, you can predict its behavior in the future with absolute accuracy. In quantum physics, everything is wrong ... 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 only has 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 HFs (or parts of HFs) can interfere with each other.
Probability calculation rule (Born rule). The squares of the coefficients in the wave function set the probability of a particular outcome in the measurement. For example, Schrödinger's cat is described by the VF:Psi= alpha1|live>+ alpha2|dead>, alpha1= alpha1= frac1 sqrt2
at the same time the probability of it being alive when opening a box is considered as P(live)=| alpha1|2=0.5 i.e. 50%. The same for the likelihood of him being dead: P(dead)=| alpha2|2=0.5 , again 50%.
Small illustration
Your friend - Vasya Pupkin - spends his days either at the computer, programming, or on the couch, playing in the playstation. You are standing in front of a closed door to his apartment. From the classical point of view, Vasya, either at the computer or on the couch, you just do not know exactly where. But the quantum Vasya is in two places at the same time, until you open the door and look at it (measure its state). Its state before measurement:
Psi= frac1 sqrt2(|game>+|work>)
And after measuring with a probability of 50%, he is at play or at work.
Continue the illustration. Suppose, before you go about your business, Vasya can either go to the refrigerator for a beer or smoke on the balcony. At the same time, if you caught him doing these exercises (watched from the refrigerator or on the balcony), he then goes with equal probability to play on the sofa or work. But it may be that when you do not look, it is 100% of cases with a joystick in hand. 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 see more in detail. First step: if we do not look, Vasya is in the superposition state of a refrigerator / balcony:
Psi= frac1 sqrt2(|fridge>+|balcony>)
The second step: let's say if Vasya goes from the refrigerator, his VF
|fridge>= frac1 sqrt2(|game>−|work>),
and if it comes from the balcony:
|balcony>= frac1 sqrt2(|game>+|work>)
If we observe it in its original state, we reduce its state to either | refrigerator> or | balcony>, which will give a 50/50 probability at the output: it will go to play or work. But if we do not observe his movements, his WF:
Psi= frac1 sqrt2(|fridge>+|balcony>)= frac12(|game>−|work>+|game>+|work>)=|game>
That is, he always turns on the couch! And all because of the interference.
So, we see that the fact that we observe Vasya changes his final state. Why does measurement play such a significant role? Interpretations of the CM are trying to answer this question.
The classical (Copenhagen) interpretation postulates that the process of observation is the process of the collapse of the wave function into one of the states. Collapse leads to the fact that the VF continues evolution only as one part of the original VF, the object is no longer in a state of superposition and can not interfere. As a result, any effects like quantum entanglement disappear. How does the collapse occur, it does not explain, as well as why some interactions cause collapse, while others do not. Not everyone likes these postulates, and scientists are trying to find alternative interpretations. One of the simplest and most developed is multiworld.
Part 2: Multi-world interpretation
First, let us recall what quantum entanglement is. By definition, two states are tangled 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 likes reading a book in her chair, or walks in the 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 particular set (and the probability of finding Vasya on the sofa in the amount was 50%).
Now they are in a confused state:
|Vasya,Anya>= frac1 sqrt2(|game,book>+|work,park>)
If we watch Vasya, he is likely to find him on the sofa again 50%. However, if he is on the couch, then Anya is absolutely exactly behind the book, you don’t even need to check.
This is the absolute correlation between measurements 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. Suppose Anya and Vasya at the same time find themselves in a confused state:
|Vasya,Anya>= frac1 sqrt2(|balcony,book>+|refrigerator,park>)
Then the two parts of the WF Vasya no longer interfere with each other, and we do not always see Vasya on the sofa, as it was in the first part:
|Vasya,Anya>= frac12(|game,book>+|work,book>+|game,park>−|work,park>)
Entanglement does not allow the VF to interfere. In principle, we can perform some operations on the system of Ani and Vasya 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 the entangled state. For example, when Vasya is confused not only with Anya, but with two thousand anonymous Internet users, and all his neighbors (in other words, the system is confused with her surroundings), we have no way to regain the ability to interfere.
This effect is called decoherence . Environment is the degree of freedom with which the system is in contact, usually a lot of them. If the system is entangled with the whole world, 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 multi-world interpretation. Its only postulate is that the whole Universe is described by a single wave function. There is no “classical” world, no observers, no collapse — all this is a unitary evolution of a single WF under the influence of the Schrödinger equation. What we see as a collapse is an exclusively decoherence process, our impossibility to “untie” the object and the environment with which it is confused.
At the same time, different “worlds” 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.
An example of a Schrödinger cat: in a well-known thought experiment, a cat is in a box of poison, which poisons a cat at random times. At the same time, according to the CM, while the box is closed, the cat is in superposition |cat>= frac12(|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 MMI, Schrödinger finds itself in a tangled state: |cat,W>= frac12(|alive,sees"alive">+|dead,sees"dead">) . Add to this the environment: |cat,W>|o>= frac12(|alive,sees"alive">+|dead,sees"dead">)|exists> which as a result of the decoherence process is confused with both of them:
|cat,W,o>= frac12(|alive,sees"alive",env"alive">+|dead,sees"dead",okr"dead">)|exists> . In this version, Schrödinger no longer has the ability to “cancel” a measurement or do something to “unravel” two states. The two worlds were divided: in one, Schrödinger found a dead cat, in the other a living one. In this case, no collapse has occurred, all of this is still just a unitary evolution of a large wave function.
A bit more formal:
Part 3: Details
- The problem of the existence of the classical world. From the point of view of the 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 (select a basis) WF into different "worlds" (orthogonal states). Question: why do we see the world classic? How does the universe "choose" one mode of decomposition 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 turn out to be stable. Macroscopic states can remain so for a long time, the wave function of the Universe does not branch constantly. 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 another side of the decoherence process: decoherence is faster, the more massive the object.
You can read more here: [1] , [2] , [3] , [4] - What exactly is the measurement? How to distinguish measurement from simple interaction? Measurement in an MWI is simply the process of entangling an observer and an object as a result of an interaction. Sometimes interaction can be “rewound” back by disentangling two systems, then it is not a measurement. Usually, some amplification process is involved in the measurement process. For example, you detect a photon on a photomultiplier, it knocks out one electron, which is converted into a current at the output from the detector as a result of the avalanche process. In MMI, the whole process is the process of entangling one photon with electrons (and other parts of the detector). But to rewind such a measurement does not work out - most of the degrees of freedom in entanglement are not available. Of course, for the measurement process it is not necessary for the observer to be reasonable, the process’s irreversibility is sufficient.
- When is the separation of worlds? Separation occurs when many degrees of freedom are involved in the interaction process, and the measurement becomes irreversible. Those. after the interaction of the photon with the detector, but before the appearance of current at the output. As an example, again Schrödinger's cat: the environment there can be considered the process of radioactive decay. At the moment when the nucleus disintegrates, 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 the point of view of Schrödinger, the cat is still alive and dead. Only when he opens the box does he become entangled in the cat and the source of the radiation. Since radioactive decay is irreversible; Schrödinger is also irreversibly split into two versions of itself.
- Is MMI a local theory? Since in MWI, the VF 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 as well. The splitting of worlds spreads from the point of measurement not faster than the speed of light.
- How many worlds are there? We do not know, there may be both a finite quantity and an infinite one. Based on the finiteness of the entropy of the Universe, it can be assumed that the number of worlds is finite.
- The multiworld theory is completely deterministic at the level of the Universe's WF. The WF evolves according to the Schrödinger equation. We are only observing the world at random due to the process of measurement and decoherence.
- How to be with energy conservation? Energy is conserved in the process of dividing worlds: each world gains "weight" in accordance with the probability associated with this world. The energy of the entire universe remains unchanged.
- If MMI is correct, then anything can 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 be too small a probability to occur.
- How to determine the probability in the MMI? The Born rule is not postulated in the MMI, but is derived from general provisions. See ex. Here or here .
- Is it possible to test MMI? MMI is a “pure” version of quantum mechanics, so every time we test a CM, we test the MMI. It is difficult to prove that MMI is the correct theory, and not some other one, although different ideas were suggested, you can find it here .
Bottom line: MMI is a minimalist interpretation of CM, which does not require anything other than the mathematical apparatus of quantum mechanics itself. The best interpretation for Occam's razor.
Literature:
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/
Source: https://habr.com/ru/post/444880/
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