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Hilbert space, a linear infinite-dimensional space in which a scalar product is given and the completeness condition is fulfilled with respect to the norm generated by this scalar product. It is named after D. Hilbert, who used these spaces to solve equations of mathematical physics. The Hilbert space is a natural generalization of a finite-dimensional vector (Euclidean) space. It is usually assumed that the linear structure of the Hilbert space is defined over the field of complex numbers C In other words, multiplication of elements by complex numbers is introduced in Hilbert space, their sum is introduced for each pair of elements and these operations are subject to the axioms of vector space. If the linear structure is defined over the field of real numbers R , then we speak of a real Hilbert space. A scalar product in a Hilbert space H is a function, which is usually denoted (x,y) , with values in the field C (for a real Hilbert space – in the field R ), defined for an arbitrary pair of elements x,y∈H and having the following properties: (x,x)⩾0 , and (x,x)=0 if and only if x=0 ; (x+y,z)=(x,z)+(y,z) for all x,y,z∈H ; (αx,y)=α(x,y) for all α∈C and x,y∈H ; (x,y)= (x,y) , where the dash means complex conjugation. In a Hilbert space, a norm can be introduced ‖x‖ of the element x∈H , putting ‖x‖= (x,x) ​ . In this embodiment, all the properties of norm: ‖x‖⩾0, ‖αx‖=∣α∣⋅‖x‖, ‖x+y‖we‖x‖+‖y‖. The Cauchy–Bunyakovsky inequality is valid for the scalar product and the norm ∣(x,y)∣⩽‖x‖⋅‖y‖. Completeness of the Hilbert space H means that for any sequence of elements x 1 ,x 2 ,...∈H , for which ‖x k −x n ‖→0 when k,n→∞ , there exists an element x∈H such that ‖x k −x‖→0 for k→∞ . A linear infinite-dimensional space with a scalar product in which the completeness condition is not fulfilled is called a pre-Hilbert space. If H 0 is a pre–Hilbert space, then there exists a unique (up to isomorphism) Hilbert space H such that H 0 ⊂H and H 0 are dense in H . Such a Hilbert space H it is called the replenishment of the H 0 space . Every Hilbert space is a Banach space. The parallelogram equality is fulfilled in the Hilbert space ‖x+y‖ 2 +‖x−y‖ 2 =2(‖x‖ 2 +‖y‖ 2 ). Conversely, if the parallelogram equality is satisfied in a Banach space, then this space can be considered as a Hilbert space, since in this case the function (x,y)= 4 1 (∣∣x+y∣∣ 2 −∣∣x−y∣∣ 2 +i∣∣x+iy∣∣ 2 −i∣∣x-iy∣∣ 2 ), where i – an imaginary unit has all the properties of a scalar product (in the case of a real Hilbert space, the last two terms are omitted). Important examples of Hilbert spaces are the spaces l2 and L2 (a,b) . The space l 2 consists of sequences of complex numbers x={x 1 ​ ,x 2 ,...} satisfying the condition ∑ k=1 ∞ ​ ∣x k ​ ∣ 2 <∞ . If x={x 1 ​ ,x 2 ,...} and y={y 1 ​ ,y 2 ,...} are two such sequences, then the series ∑ k=1 ∞ x k y k ​ converges and defines the scalar product (x,y) in l2 . The space L 2 (a,b) is introduced as a completion of the space of continuous functions C[a,b] according to the norm defined by the scalar product (f,g)=∫ a b f(x) g (x)dx . The description of this replenishment can be given in terms of the Lebesgue integral. The space L 2 (a,b) consists of Lebesgue-measurable functions f for which there is a Lebesgue integral ∫ a b ∣f(x)∣ 2 dx In this case, the functions are identified if they differ only on the set of Lebesgue measure zero. Similarly, Hilbert spaces L 2 (Ω) are defined , where Ω is a region in n -dimensional space R n . The spaces L2 and L2 (Ω) (like most spaces found in applications) are separable. The separability of a space means that there exists a countable set of elements in it such that any element of the space can be approximated arbitrarily precisely by elements from this set. In a separable Hilbert space H there are countable orthonormal bases, i.e. systems of elements {e k } k=1 ∞ , having the properties: ‖e k ‖=1,k=1,2,...,(e k ,e j )=0 when k  =j any item x∈H we will present in the form of a series x=∑ k=1 ∞ x k e k where x k – the number and the series converges in norm, i.e., ku x−∑ k=1 n x k e k Ku→0 when n→∞ . This representation is unambiguous, the numbers x k are equal (x,e k ) and are called the Fourier coefficients of the element x according to the system {e k ​ } k=1 ∞ ​ . Equality is fair k=1 ∑ ∞ ​ ∣x k ​ ∣ 2 =∣∣x∣∣ 2 , called Parseval's equality; it is an infinite-dimensional analogue of the Pythagorean theorem. An example of an orthonormal basis in l 2 is the set {e n } n=1 ∞ , where e n is a sequence whose nth coordinate is 1 , and the other coordinates are zeros. An example of an orthonormal basis in L2 (0,1) is the system of functions { 2 sinπnx} n=1 ∞ . All separable Hilbert spaces are isomorphic to each other. In the theory of Banach spaces , together with spaces B , conjugate spaces B * consisting of linear continuous functionals on B are considered . The space conjugate to the Hilbert space H is arranged simply, namely: for every linear continuous functional f on H , there exists an element x * ∈H such that f(x)=(x,x * ) and ‖f‖=‖x * ‖ , i.e. the conjugate space H * turns out to be isomorphic to the original Hilbert space H . This fact makes Hilbert space convenient for constructing the theory of linear operators and makes it possible to introduce the concepts of self-adjoint and unitary operators on Hilbert space, which play an important role in functional analysis and mathematical physics. The abstract definition of Hilbert space and the foundations of the general theory of linear self-adjoint and unitary operators were given in the works of J. von Neumann, F. Riess and the American mathematician M. Stone. Hilbert space methods began to play a special role after the basic principles of quantum mechanics were formulated in the mid-1920s, according to which the states of a quantum mechanical system are interpreted as vectors of Hilbert space H , and observables (energy, momentum, etc.) as self–adjoint operators in H . An important role in the formation of the theory of operators in Hilbert space was played by the works of P. L. Chebyshev, A. A. Markov (Sr.) and T. Stieltjes on the problem of moments, Jacobi matrices, the theory of orthogonal polynomials and continuous fractions. Many World Interpretation Part 1: Why is it necessary to "interpret" quantum physics? Quantum physics has become firmly embedded in our lives: flash drives use the tunneling 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 equations with some physical (and philosophical) content. Important: all interpretations are reduced to the same equations of standard QM and do not predict new physics! The main problem that interpretations are 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 (such as momentum 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. This is not the case in quantum physics… The system is described by a wave function. It determines the probability of measuring a system in a certain state (for example, a certain coordinate or momentum). Before the measurement, 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 square of the modulus of the wave function, and not by the wave function itself. At the same time, the VF itself can take both positive and negative values. Moreover, two VFS (or parts of the VF) can interfere with each other. The probability calculation rule (Born's rule). The squares of the coefficients in the wave function set the probability of a particular outcome during measurement. For example, Schrodinger's cat is described in: $ \Psi = \alpha_1 |\text{alive}\rangle + \alpha_2 |\text{dead}\rangle, \alpha_1 = \alpha_1 =\frac{1}{\sqrt{2}}$ at the same time, the probability of him being alive when opening the box is considered as $P(alive) = |\alpha_1|^2 = 0.5$, i.e. 50%. The same is true for the probability of him being dead: $P(dead) = |\alpha_2|^2 = 0.5$, again 50%. A small illustration Your friend, Jane Peaks, spends her days either at the computer, programming, or on the couch, playing playstation. You are standing in front of the closed door to his apartment. From a classical point of view, Jane is either at the computer or on the couch, you just don't know where exactly. But the quantum Jane is in two places at the same time, until you open the door and take a look (measure its condition). His condition before the measurement: And after the measurement, with a 50% probability, he is at play or at work. Let's continue the illustration. Let's say that 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 doing these activities (watching at the refrigerator or on the balcony), he is equally likely to go to play on the sofa or work afterwards. But it may be that when you are not looking, he turns out to be holding a joystick 100% of the time. The reason for this is interference. The Vasya state is described by a wave function, which can be negative, but at the same time correspond to the same probability as a positive VF. Let's look into it in more detail. The first step: if we are not looking, Vasya is in a state of superposition refrigerator/balcony:The second step: let's say if Vasya goes from the refrigerator, his VFA if he goes from the balcony:If we observe him in his original state, we reduce his state to either | refrigerator or |balcony, which will give a 50/50 probability at the exit: he will go to play or work. But if we don't watch his movements, his: That is, he always ends up on the couch! And all because of interference. So, we see that the fact that we are observing Vasya changes his final state. Why does measurement play such a significant role? This is the question that KM interpretations are trying to answer. The classical (Copenhagen) interpretation postulates that the observation process is the process of collapse of the wave function into one of the states. The collapse leads to the fact that the VF continues to evolve only as one part of the original VF, the object is no longer in a state of superposition and cannot interfere. As a result, all sorts of effects such as quantum entanglement disappear. It does not explain how collapse occurs, nor does it explain why some interactions cause collapse and others do not. Not everyone likes the presence of such postulates, and scientists are trying to find alternative interpretations. One of the simplest and most developed is the multi—world one. Part 2: A Multi-world interpretation To begin with, let's remember what quantum entanglement is. By definition, two states are entangled when there is no way to separate them into two independent parts. Let's go back to the illustration from the first part, and imagine that Jane has a boyfriend, Mark.Mark is either reading a book in an armchair or walking in the park. Until they started dating, their choice was random: And the outcome of your measurement gave a 25% probability of each specific set (and the probability of finding Jane on the couch was 50% in total). Now they are in a confused state: If we watch Jane, there's a 50% chance of finding him on the couch again. However, if he is on the couch, then Anya is definitely at the book, you don't even need to check. This is how the absolute correlation between measurements manifests itself when the system is in an entangled state. The next step: Jane 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 Mark and Jane find themselves in a confused state at the same time: Then the two parts of Jane's VF no longer interfere with each other, and we don't always see Jane on the couch, as it was in the first part.: Entanglement prevents the VF from interfering. In principle, we can perform some operations on the Mark and Jane system and untangle them, then interference will be possible again. However, to do 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 finds himself entangled not only with Mark, but also with two thousand anonymous people on the Internet, and all his neighbors (in other words, the system gets entangled with its environment), we have no way to regain the ability to interfere. This effect is called decoherence. The environment is called the degrees of freedom with which the system comes into contact, usually there are a lot of them. If the system turns out to be entangled with the entire surrounding world, 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 the multi-world interpretation. Its only postulate is that the entire universe is described by a single wave function. There is no "classical" world, there are no observers, there is no collapse — all this is a unitary evolution of one VF under the action of the Schrodinger equation. What we observe as collapse is solely a process of decoherence, our inability to "untie" the object and the environment with which it is entangled. At the same time, different "worlds" arise every time there is a "collapse" — the interaction of the system with the environment. At the same time, one world is divided into several, according to the branches of the VF, and these worlds no longer interact. The example of Schrodinger's 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. According to the Copenhagen interpretation, when Schrodinger opens the box, he collapses the cat into either a "alive" or "dead" state. According to the MMI, Schrodinger finds himself in a confused state: . You need to add an environment to this:  which, as a result of the decoherence process, gets entangled with both of them:. In this case, Schrodinger no longer has the opportunity to "cancel" the measurement or do something to "untangle" the two states. Two worlds separated: in one Schrodinger found a dead cat, in the other — a living one. At the same time, no collapse occurred, all this is still just a unitary evolution of a large wave function. Part 3: Details 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) VF to different "worlds" (orthogonal states). Question: why do we observe the classical world? How does the universe "choose" one way 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 are 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 version of the decomposition into a basis, branching occurs so quickly that we would not be able to perceive it in time. This is the other side of the decoherence process: The faster the decoherence rate, the more massive the object.You can read more here: [1], [2], [3], [4] What exactly is a dimension? How do you distinguish measurement from simple interaction? Measurement in MMI is simply the process of entangling an observer and an object as a result of interaction. Sometimes the interaction can be "rewound" by untangling 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, as a result of an avalanche process, is converted into a current at the detector output. In MMI, the whole process is the process of entangling a single photon with electrons (and other parts of the detector). But it will not work to rewind such a measurement — most of the degrees of freedom in entanglement are inaccessible. Of course, for the measurement process, it is not necessary that the observer be reasonable, the irreversibility of the process is enough. When does the separation of the worlds take place? Separation occurs when many degrees of freedom are involved in the interaction process, and the measurement becomes irreversible. I.e. after the photon interacts with the detector, but before the output current appears. As an example, Schrodinger's cat again: the environment there can be considered the process of radioactive decay. At the moment when the core 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 Schrodinger's point of view, the cat is still in a live-dead state. It is only when he opens the box that he finds himself entangled with the cat and the source of the radiation. Since radioactive decay is irreversible, Schrodinger also irreversibly splits into two versions of himself. Is MMI a local theory? Since in MMI, the VF obeys the Schrodinger 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 no faster than the speed of light How many worlds are there in total? We don't know, there can be either a finite number or an infinite number. Based on the finiteness of the entropy of the universe, it can be assumed that the number of worlds is finite. The multidimensional theory is completely deterministic at the level of the VF universe. The VF evolves according to the Schrodinger equation. We only observe the world randomly due to the process of measurement and decoherence. What about energy conservation? Energy is conserved in the process of dividing worlds: each world receives a "weight" according to the probability associated with this world. The energy of the entire universe remains unchanged. If the MMI is correct, does it mean that anything can happen? No, first of all, the laws of physics work exactly the same way, and what is not allowed by "ordinary" physics will not happen in MMI either. Secondly, if the number of worlds is finite, some events may have too low a probability to occur. How to determine the probabilities in MMI? The Born rule is not postulated in the MMI, but is derived from general provisions. See e.g. here or here. Is it possible to test the MMI? MMI is a "pure" version of quantum mechanics, so every time we test KM, we test MMI. It is difficult to prove that MMI is the correct theory, and not some other one, although different ideas have been proposed, you can find here. Result: MMI is a minimalistic interpretation of QM that requires nothing but the mathematical apparatus of quantum mechanics itself. The best interpretation for Occam's razor.