In quantum mechanics, each dynamic variable - coordinate, momentum, angular momentum, energy - is associated with a linear self-adjoint (Hermitian) operator.

All functional relations between quantities known from classical mechanics are replaced in quantum theory by analogous relations between operators. The correspondence between dynamic variables (physical quantities) and quantum mechanical operators is postulated in quantum mechanics and is a generalization of a huge amount of experimental material.

1.3.1. Coordinate operator:

As is known, in classical mechanics, the position of a particle (system N- particles) in space at a given time is determined by a set of coordinates - vector or scalar quantities. Vector mechanics is based on Newton's laws, the main ones here are vector quantities - speed, momentum, force, angular momentum (angular momentum), moment of force, etc. Here, the position of a material point is given by the radius vector, which determines its position in space relative to the selected reference body and the coordinate system associated with it, i.e.

If all vectors of forces acting on a particle are determined, then it is possible to solve the equations of motion and construct a trajectory. If movement is considered N- particles, then it is more expedient (regardless of whether the motion of bound particles is considered or the particles are free in their movements from any kind of constraints) to operate not with vector, but with scalar quantities - the so-called generalized coordinates, velocities, impulses and forces. This analytical approach is based on the principle of least action, which in analytical mechanics plays the role of Newton's second law. A characteristic feature of the analytical approach is the absence of a rigid connection with any particular coordinate system. In quantum mechanics, each observed dynamic variable (physical quantity) is associated with a linear self-adjoint operator. Then, obviously, the classical set of coordinates will correspond to a set of operators of the form: , whose action on a function (vector) will be reduced to multiplying it by the corresponding coordinates, i.e.

whence it follows that:

1.3.2. Momentum operator:

The classical expression for momentum by definition is:

given that:

we will have, respectively:

Since any dynamic variable in quantum mechanics is associated with a linear self-adjoint operator:

then, accordingly, the expression for the momentum, expressed through its projections on three non-equivalent directions in space, is transformed to the form:

The value of the momentum operator and its components can be obtained by solving the problem for the eigenvalues ​​of the operator:

To do this, we use the analytical expression for a de Broglie plane wave, which we have already obtained earlier:

considering also that:

we have thus:

Using the de Broglie plane wave equation, we now solve the problem for the eigenvalues ​​of the momentum operator (its components):

insofar as:

and the function is on both sides of the operator equation:

then the magnitudes of the wave amplitude will decrease, therefore:

thus we have:

since the momentum component operator (similarly to and ) is a differential operator, then its action on the wave function (vector) will obviously be reduced to calculating the partial derivative of the function of the form:

Solving the problem for the eigenvalues ​​of the operator, we arrive at the expression:

Thus, in the course of the above calculations, we came to an expression of the form:

then respectively:

given that:

after substitution, we get an expression of the form:

Similarly, one can obtain expressions for other components of the momentum operator , i.e. we have:

Given the expression for the total momentum operator:

and its component:

we have, respectively:

Thus, the total momentum operator is a vector operator and the result of its action on a function (vector) will be an expression of the form:

1.3.3. Angular momentum (angular momentum) operator:

Consider the classical case of an absolutely rigid body rotating about a fixed axis OO passing through it. Let's break this body into small volumes with elementary masses: located at distances: from the axis of rotation of OO. When a rigid body rotates about the fixed axis OO, its separate elementary volumes with masses , obviously, will describe circles of different radii and will have different linear velocities: . From the kinematics of rotational motion it is known that:

If a material point makes a rotational motion, describing a circle with radius , then after a short period of time it will turn by an angle from its original position.

The linear velocity of a material point, in this case, will be equal, respectively:

insofar as:

Obviously, the angular velocity of the elementary volumes of a solid body rotating around a fixed axis OO at distances from it will be equal, respectively:

When studying the rotation of a rigid body, they use the concept of the moment of inertia, which is a physical quantity equal to the sum of the products of the masses - material points of the system and the squares of their distances to the considered axis of rotation of the OO, relative to which the rotational movement is performed:

then we find the kinetic energy of a rotating body as the sum of the kinetic energies of its elementary volumes:

insofar as:

then respectively:

Comparison of formulas for the kinetic energy of translational and rotational motions:

shows that the moment of inertia of the body (system), characterizes the measure of inertia of this body. Obviously, the greater the moment of inertia, the greater the energy must be expended to achieve a given speed of rotation of the considered body (system) around the fixed axis of rotation of the OO. An equally important concept in solid mechanics is the momentum vector, so by definition, the work done to move a body over a distance is equal to:

because, as already mentioned above, with rotational motion:

then, respectively, we will have:

considering the fact that:

then the expression for the work of rotational motion, expressed in terms of the moment of forces, can be rewritten as:

because in general:

then, therefore:

Differentiating the right and left parts of the resulting expression with respect to , we will have, respectively:

given that:

we get:

The moment of force (rotational moment) acting on the body is equal to the product of its moment of inertia and angular acceleration. The resulting equation is an equation for the dynamics of rotational motion, similar to the equation of Newton's second law:

here, instead of force, the moment of force, the role of mass, plays the moment of inertia. Based on the above analogy between the equations for translational and rotational motions, the analogue of the momentum (momentum) will be the angular momentum of the body (angular momentum). The angular momentum of a material point by mass is the vector product of the distance from the axis of rotation to this point, by its momentum (momentum); we then have:

Considering that the vector is determined not only by the triple of components:

but also by an explicit expansion in the unit vectors of the coordinate axes:

we will have, respectively:

The components of the total angular momentum can be represented as algebraic complements of the determinant, in which the first row is unit vectors (orts), the second row is Cartesian coordinates and the third row is the momentum components, then, respectively, we will have an expression of the form:

whence it follows that:

From the formula of the angular momentum as a vector product, an expression of the form also follows:

or for a particle system:

taking into account the relations of the form:

we obtain an expression for the angular momentum of the system of material points:

Thus, the angular momentum of a rigid body relative to a fixed axis of rotation is equal to the product of the moment of inertia of the body and the angular velocity. The angular momentum is a vector directed along the axis of rotation in such a way that from its end one can see the rotation occurring clockwise. Differentiating the resulting expression with respect to time gives another expression for the dynamics of rotational motion, equivalent to the equation of Newton's second law:

analogous to Newton's second law equation:

"The product of the angular momentum of a rigid body with respect to the axis of rotation OO is equal to the moment of force with respect to the same axis of rotation." If we are dealing with a closed system, then the moment of external forces is zero, then, therefore:

The equation obtained above for a closed system is an analytical expression of the momentum conservation law. “The angular momentum of a closed system is a constant value, i.e. does not change over time." So, in the course of the above calculations, we came to the expressions we need in further reasoning:

and thus we have, respectively:

Since in quantum mechanics any physical quantity (dynamic variable) is associated with a linear self-adjoint operator:

then, respectively, the expressions:

are converted to the form:

because by definition:

and also given that:

Then, respectively, for each of the components of the angular momentum we will have an expression of the form:

based on an expression like:

1.3.4. The angular momentum square operator:

In classical mechanics, the square of the angular momentum is determined by an expression of the form:

Therefore, the corresponding operator will look like:

whence it follows, respectively, that:

1.3.5. Kinetic energy operator:

The classical expression for kinetic energy is:

given that the expression for momentum is:

we have, respectively:

expressing momentum in terms of its components:

we will have, respectively:

Since each dynamic variable (physical quantity) in quantum mechanics corresponds to a linear self-adjoint operator, i.e.

then, therefore:

considering expressions like:

and thus, we arrive at an expression for the kinetic energy operator of the form:

1.3.6. Potential energy operator:

The potential energy operator in describing the Coulomb interaction of particles with charges and has the form:

It coincides with a similar expression for the corresponding dynamic variable (physical quantity) - potential energy.

1.3.7. The total energy operator of the system:

The classical expression for the Hamiltonian, known from the analytical mechanics of Hamilton, is:

based on the correspondence between quantum mechanical operators and dynamical variables:

we arrive at the expression for the operator of the total energy of the system, the Hamilton operator:

taking into account the expressions for the potential and kinetic energy operators:

we arrive at an expression of the form:

Operators of physical quantities (dynamic variables) - coordinates, momentum, angular momentum, energy are linear self-adjoint (Hermitian) operators, therefore, based on the corresponding theorem, their eigenvalues ​​are real (real) numbers. It is this circumstance that served as the basis for the use of operators in quantum mechanics, since as a result of a physical experiment we obtain precisely real quantities. In this case, the operator eigenfunctions corresponding to different eigenvalues ​​are orthogonal. If we have two different operators, then their own functions will be different. However, if the operators commute with each other, then the eigenfunctions of one operator will also be the eigenfunctions of another operator, i.e. the systems of eigenfunctions of operators commuting with each other will coincide.

Using a well-known quantum mechanical approach in which units of information are the basic building blocks, Lloyd spent several years studying the evolution of particles in terms of shuffling ones (1) and zeros (0). He found that as particles become more and more entangled with each other, the information that described them (1 for spin clockwise and 0 for counterclockwise, for example) will transfer to the description of the system of entangled particles as a whole. As if the particles gradually lost their individual autonomy and became pawns of a collective state. At this point, as Lloyd discovered, the particles go into a state of equilibrium, their states stop changing, like a cup of coffee cools down to room temperature.

“What is really going on? Things become more interconnected. The arrow of time is the arrow of rising correlations.”

The idea presented in the 1988 doctoral dissertation was not heard. When the scientist sent it to the journal, he was told that "there is no physics in this work." Quantum information theory "was deeply unpopular" at the time, Lloyd says, and questions about the arrow of time "were left to nutters and retired Nobel laureates."

"I was pretty damn close to being a taxi driver," Lloyd said.

Since then, advances in quantum computing have turned quantum information theory into one of the most active areas of physics. Today, Lloyd remains a professor at MIT, recognized as one of the founders of the discipline, and his forgotten ideas resurface in a more confident form in the minds of Bristol physicists. The new evidence is more general, the scientists say, and applies to any quantum system.

“When Lloyd came up with the idea in his dissertation, the world was not ready,” says Renato Renner, head of the Institute for Theoretical Physics at ETH Zurich. - Nobody understood him. Sometimes you need ideas to come at the right time.”

In 2009, a proof by a group of Bristol physicists resonated with quantum information theorists, opening up new ways to apply their methods. It showed that as objects interact with their environment - like particles in a cup of coffee interact with air, for example - information about their properties "leaks and smears with the environment," explains Popescu. This local loss of information causes the state of the coffee to stagnate, even as the pure state of the entire room continues to evolve. With the exception of rare random fluctuations, the scientist says, "its state ceases to change with time."

It turns out that a cold cup of coffee cannot spontaneously heat up. In principle, as the clean state of the room evolves, the coffee can suddenly "become unmixed" with the air and enter the clean state. But there are so many more mixed states available than pure coffee that this will almost never happen - the universe will end sooner than we can witness it. This statistical improbability makes the arrow of time irreversible.

“Essentially, entanglement opens up a huge space for you,” Popescu comments. - Imagine that you are in a park with a gate in front of you. As soon as you enter them, you will fall into a huge space and get lost in it. You will never return to the gate either.

In the new story of the arrow of time, information is lost in the process of quantum entanglement, not due to the subjective lack of human knowledge, which leads to the balancing of a cup of coffee and a room. The room eventually balances with the outside environment, and the environment—even more slowly—drifts toward equilibrium with the rest of the universe. The thermodynamic giants of the 19th century viewed this process as a gradual dissipation of energy that increases the overall entropy, or chaos, of the universe. Today, Lloyd, Popescu and others in the field see the arrow of time differently. In their opinion, information becomes more and more diffuse, but never completely disappears. Although entropy grows locally, the total entropy of the universe remains constant and zero.

“On the whole, the universe is in a pure state,” says Lloyd. “But its individual parts, being entangled with the rest of the universe, remain mixed.”

One aspect of the arrow of time remains unresolved.

“There is nothing in these works that explains why you start with a gate,” Popescu says, returning to the park analogy. "In other words, they don't explain why the original state of the universe was far from equilibrium." The scientist hints that this question applies.

Despite recent progress in calculating equilibration times, the new approach still cannot be used as a tool for calculating the thermodynamic properties of specific things like coffee, glass, or exotic states of matter.

“The point is to find criteria under which things behave like window glass or a cup of tea,” says Renner. “I think I will see new work in this direction, but there is still a lot of work ahead.”

Some researchers have expressed doubt that this abstract approach to thermodynamics will ever be able to accurately explain how particular observable objects behave. But conceptual advances and new mathematical formalism are already helping researchers ask theoretical questions from the field of thermodynamics, such as the fundamental limits of quantum computers and even the ultimate fate of the universe.

“We are thinking more and more about what can be done with quantum machines,” says Paul Skrzypczyk of the Institute of Photon Sciences in Barcelona. - Suppose the system is not yet in equilibrium and we want to make it work. How much useful work can we extract? How can I step in to do something interesting?"

Sean Carroll, a theoretical cosmologist at the California Institute of Technology, applies the new formalism in his latest work on the arrow of time in cosmology. “I am interested in the most that neither is the long-term fate of cosmological space-time. In this situation, we still do not know all the necessary laws of physics, so it makes sense to turn to the abstract level, and here, I think, this quantum mechanical approach will help me.”

Twenty-six years after the grand failure of Lloyd's idea of ​​the arrow of time, he is happy to witness its rise and is trying to apply the ideas of the latest work to the paradox of information falling into a black hole.

“I think now they will still talk about the fact that there is physics in this idea.”

And philosophy - and even more so.

According to scientists, our ability to remember the past but not the future, another manifestation of the arrow of time, can also be seen as an increase in correlations between interacting particles. When you read something from a piece of paper, the brain correlates with the information through photons that reach the eyes. Only from now on will you be able to remember what is written on paper. As Lloyd notes:

"The present can be defined as the process of associating (or establishing correlations) with our environment."

The background for the steady growth of entanglements throughout the universe is, of course, time itself. Physicists emphasize that despite great advances in understanding how time changes occur, they are not one iota closer to understanding the nature of time itself or why it differs from the other three dimensions of space. Popescu calls this puzzle "one of the greatest misunderstandings in physics."

“We can discuss the fact that an hour ago our brain was in a state that correlated with fewer things,” he says. “But our perception that time is ticking is another matter entirely. Most likely, we will need a revolution in physics that will reveal this secret to us.”

A.Yu. Sevalnikov
Quantum and time in modern physical paradigm

The year 2000 marked the 100th anniversary of the birth of quantum mechanics. The transition through the turn of centuries and centuries is an occasion to talk about time, and in this case, just in connection with the anniversary of the quantum.

Linking the concept of time to the ideas of quantum mechanics might seem artificial and far-fetched, if not for one circumstance. We still do not understand the meaning of this theory. "It's safe to say that no one understands the meaning of quantum mechanics," said Richard Feynman. Faced with micro-phenomena, we are faced with a mystery that we have been trying to unravel for a century. How not to remember the words of the great Heraclitus, that "nature loves to hide."

Quantum mechanics is full of paradoxes. Do they reflect the very essence of this theory? We have a perfect mathematical apparatus, a beautiful mathematical theory, the conclusions of which are invariably confirmed by experience, and at the same time there are no “clear and distinct” ideas about the essence of quantum phenomena. The theory here is rather a symbol behind which another reality is hidden, manifested in irremovable quantum paradoxes. “The oracle does not open or hide, it hints,” as the same Heraclitus said. So what does quantum mechanics hint at?

M. Planck and A. Einstein stood at the origins of its creation. The focus was on the problem of emission and absorption of light, i.e. the problem of becoming in a broad philosophical sense, and, consequently, of movement. This problem as such has not yet become the focus of attention. During discussions around quantum mechanics, the problems of probability and causality, wave-particle duality, problems of measurement, nonlocality, participation of consciousness, and a number of others closely related directly to the philosophy of physics were considered. However, we dare to assert that it is the problem of formation, the oldest philosophical problem, that is the main problem of quantum mechanics.

This problem has always been closely related to quantum theory, from the problem of light emission and absorption in the works of Planck and Einstein to the latest experiments and interpretations of quantum mechanics, but always implicitly, implicitly, as some kind of hidden subtext. In fact, almost all of its debatable issues are closely related to the problem of becoming.

So the so-called is currently being actively discussed. "problem of measurement", which plays a key role in the interpretation of quantum mechanics. The measurement dramatically changes the state of the quantum system, the shape of the wave function Ψ(r,t). For example, if, when measuring the position of a particle, we obtain a more or less accurate value of its coordinate, then the wave packet, which was the function Ψ before the measurement, is “reduced” into a less extended wave packet, which can even be point, if the measurement is carried out very accurately. This is the reason for the introduction by W. Heisenberg of the concept of “reduction of a package of probabilities”, which characterizes such a sharp change in the wave function Ψ(r,t).

Reduction always leads to a new state, which cannot be foreseen in advance, since before the measurement we can only predict the probabilities of various possible options.

Quite a different situation in the classics. Here, if the measurement is carried out accurately enough, then this is only a statement of the “existing state”. We get the true value of the quantity, which objectively exists at the moment of measurement.

The difference between classical mechanics and quantum mechanics is the difference between their objects. In the classics, this is an existing state; in the quantum case, it is an object that arises, becomes, an object that fundamentally changes its state. Moreover, the use of the concept of "object" is not entirely legitimate, we have rather the actualization of potential being, and this act itself is not fundamentally described by the apparatus of quantum mechanics. The reduction of the wave function is always a discontinuity, a jump in the state.

Heisenberg was one of the first to argue that quantum mechanics brings us back to the Aristotelian notion of being in possibility. Such a point of view in quantum theory brings us back to the two-mode ontological picture, where there is a mode of being in possibility and a mode of being of the real, i.e. the world of the realized.

Heisenberg did not develop these ideas in a consistent way. This was carried out a little later by V.A. Fok. The concepts of "potential possibility" and "realized" introduced by him are very close to the Aristotelian concepts of "being in possibility" and "being in the stage of completion."

According to Fock, the state of the system described by the wave function is objective in the sense that it represents an objective (independent of the observer) characteristic of the potential possibilities of one or another act of interaction between a microobject and a device. Such an “objective state is not yet real, in the sense that for an object in a given state the indicated potential possibilities have not yet been realized, the transition from potential possibilities to the realized one occurs at the final stage of the experiment.” The statistical distribution of probabilities that arises during the measurement and reflects the potential opportunities objectively existing under given conditions. Actualization, "implementation" according to Fock is nothing more than "becoming", "change", or "movement" in a broad philosophical sense. Actualization of the potential introduces irreversibility, which is closely related to the existence of the “arrow of time”.

It is interesting that Aristotle directly connects time with movement (see, for example, his "Physics" - "time does not exist without change", 222b 30ff, book IV especially, as well as treatises - "On the sky", "On the emergence and destruction"). Without considering the Aristotelian understanding of time in detail, we note that for him it is, first of all, a measure of movement, and speaking more broadly, a measure of the formation of being.

In this understanding, time acquires a special, distinguished status, and if quantum mechanics really points to the existence of a potential being and its actualization, then this special character of time should be explicit in it.

It is precisely this special status of time in quantum mechanics that is well known and has been repeatedly noted by various authors. For example, de Broglie, in his book Heisenberg's Uncertainty Relations and the Wave Interpretation of Quantum Mechanics, writes that QM "does not establish a true symmetry between the space and time variables. The coordinates x, y, z of the particle are considered observable corresponding to certain operators and having in any state (described by the wave function Ψ) some probability distribution of values, while the time t is still considered to be a completely deterministic quantity.

This can be specified as follows. Imagine a Galilean observer making measurements. It uses x, y, z, t coordinates, observing events in its macroscopic frame of reference. The variables x, y, z, t are numerical parameters, and it is these numbers that enter the wave equation and wave function. But each particle of atomic physics corresponds to "observable quantities", which are the coordinates of the particle. The relationship between the observed quantities x, y, z and the spatial coordinates x, y, z of a Galilean observer is of a statistical nature; each of the observed values ​​x, y, z in the general case can correspond to a whole set of values ​​with a certain probability distribution. As for time, in modern wave mechanics there is no observable quantity t associated with a particle. There is only the variable t, one of the space-time variables of the observer, determined by the clock (essentially macroscopic) that this observer has.

Erwin Schrödinger claims the same. “In CM, time is allocated in comparison with coordinates. Unlike all other physical quantities, it does not correspond to an operator, not statistics, but only a value that is accurately read, as in good old classical mechanics, by the usual reliable clock. The distinguished nature of time makes quantum mechanics in its modern interpretation from beginning to end a non-relativistic theory. This feature of QM is not eliminated when a purely external "equality" of time and coordinates is established, i.e. formal invariance under Lorentz transformations, with the help of appropriate changes in the mathematical apparatus.

All CM statements have the following form: if now, at time t, a certain measurement is made, then with probability p its result will be equal to a. Quantum mechanics describes all statistics as functions of one exact time parameter... I can always choose the time of measurement at my own discretion.

There are other arguments showing the distinguished nature of time, they are known and I will not dwell on this here. There are also attempts to overcome such a distinction, up to the point where Dirac, Fock and Podolsky proposed the so-called covariance of the equations to ensure the covariance of the equations. "multi-time" theory, when each particle is assigned not only its own coordinate, but also its own time.

In the book mentioned above, de Broglie shows that such a theory cannot escape the special status of time, and it is quite characteristic that he ends the book with the following phrase: “Thus, it seems to me impossible to eliminate the special role that such a variable plays in quantum theory of time” .

On the basis of such reasoning, it can be confidently asserted that quantum mechanics makes us talk about the allocation of time, about its special status.

There is one more aspect of quantum mechanics, which has not yet been considered by anyone.

In my opinion, it is legitimate to speak of two "times". One of them is our usual time - finite, unidirectional, it is closely connected with actualization and belongs to the world of the realized. The other is that which exists for the mode of being in possibility. It is difficult to characterize it in our usual terms, since at this level there are no concepts of "later" or "earlier". The principle of superposition just shows that in potency all possibilities exist simultaneously. At this level of being, it is impossible to introduce the spatial concepts of “here”, “there”, since they appear only after the “unfolding” of the world, in the process of which time plays a key role.

It is easy to illustrate such a statement with the famous double-slit thought experiment, which, according to Richard Feynman, contains the whole mystery of quantum mechanics.

Let us direct a beam of light onto a plate with two narrow slits. Through them, light enters the screen placed behind the plate. If the light consisted of ordinary "classical" particles, then we would get two light bands on the screen. Instead, as is known, a series of lines is observed - an interference pattern. Interference is explained by the fact that light propagates not just as a stream of photon particles, but in the form of waves.

If we try to trace the path of photons and place detectors near the slits, then the photons begin to pass through only one slit and the interference pattern disappears. “It seems that photons behave like waves as long as they are “allowed” to behave like waves, i.e. spread through space without occupying any particular position. However, the moment one "asks" exactly where the photons are - either by identifying the slit they went through, or by making them hit the screen through only one slit - they instantly become particles...

In experiments with a double-slit plate, the physicist's choice of measuring instrument forces the photon to "choose" between passing through both slits simultaneously, like a wave, or only passing through one slit, like a particle. However, what would happen, Wheeler asked, if the experimenter could somehow wait until the light had passed through the slits before choosing the mode of observation?

Such an experiment with a "delayed choice" can be demonstrated more clearly in the radiation of quasars. Instead of a plate with two slits, “in such an experiment, a gravitational lens should be used - a galaxy or other massive object that can split the quasar radiation and then focus it in the direction of a distant observer, creating two or more images of the quasar ...

The astronomer's choice of how to observe photons from a quasar today is determined by whether each photon traveled both paths or only one path near the gravitational lens billions of years ago. At the moment when the photons reached the "galactic beam splitter", they should have had some kind of premonition, telling them how to behave in order to respond to the choice that will be made by unborn beings on a planet that does not yet exist.

As Wheeler rightly points out, such speculations arise from the erroneous assumption that photons have some shape before the measurement is made. In fact, “quantum phenomena in themselves have neither a corpuscular nor a wave character; their nature is not determined until the moment when they are measured.

Experiments carried out in the 1990s confirm such "strange" conclusions from quantum theory. A quantum object really "does not exist" until the moment of measurement, when it receives actual existence.

One of the aspects of such experiments has so far been practically not discussed by researchers, namely, the time aspect. After all, quantum objects get their existence not only in the sense of their spatial localization, but also begin to "be" in time. Having admitted the existence of potential being, it is necessary to draw a conclusion about a qualitatively different nature of existence at this level of being, including the temporal one.

As follows from the principle of superposition, different quantum states exist "simultaneously", i.e. a quantum object initially, before the actualization of its state, exists immediately in all admissible states. When the wave function is reduced from the "superposed" state, only one of them remains. Our usual time is closely connected with such "events", with the process of actualization of the potential. The essence of the “arrow of time” in this sense lies in the fact that objects come to being, “in-exist”, and it is with this process that the unidirectionality of time and its irreversibility are connected. Quantum mechanics, the Schrödinger equation describes the line between the level of being possible and being real, more precisely, it gives dynamics, the probability of the potential being realized. The potential itself is not given to us, quantum mechanics only points to it. Our knowledge is still fundamentally incomplete. We have an apparatus that describes the classical world, that is, the actual, manifest world - this is the apparatus of classical physics, including the theory of relativity. And we have the mathematical formalism of quantum mechanics that describes becoming. The formalism itself is “guessed” (here it is worth recalling how the Schrödinger equation was discovered), it is not deduced from anywhere, which gives rise to the question of a more complete theory. In our opinion, quantum mechanics only brings us to the brink of being manifest, makes it possible to reveal the secret of being and time, without revealing and not having such an opportunity to reveal it completely. We can only draw a conclusion about the more complex structure of time, about its special status.

An appeal to the philosophical tradition will also help substantiate this point of view. As you know, even Plato gives a distinction between two times - time itself and eternity. Time and eternity are incommensurable with him, time is only a moving likeness of eternity. When the demiurge created the Universe, as the Timaeus tells about it, the demiurge “planned to create some kind of moving likeness of eternity; arranging the sky, he together with it creates for eternity, which is in one, the eternal image, moving from number to number, which we called time.

Plato's concept is the first attempt to overcome, to synthesize two approaches to time and the world. One of them is the Parmenidean line, the spirit of the Eleatic school, where any movement, change was denied, where only eternal being was recognized as truly existing, the other is associated with the philosophy of Heraclitus, who claimed that the world is a continuous process, a kind of burning or unceasing flow.

Another attempt to overcome this duality was the philosophy of Aristotle. By introducing the concept of potential being, he succeeded for the first time in describing movement, the doctrine of which he expounds in close connection with the doctrine of nature.

Based on the Platonic dualistic scheme "being-non-being", it turns out to be impossible to describe the movement, it is necessary "to find the "underlying" third, which would be an intermediary between opposites."

The introduction by Aristotle of the concept of dynamis - "being in possibility" is caused by his rejection of the Platonic method, which proceeded from the opposites "existing-bearing". As a result of this approach, writes Aristotle, Plato cut off his path to the comprehension of change, which is the main feature of natural phenomena. “... If we take those who ascribe being-non-being together to things, it turns out from their words that all things are at rest, and not in motion: in fact, there is nothing to change into, because all properties are present<уже>all things." [Metaphysics, IV,5].

“So, the opposition of being-non-being, says Aristotle, must be mediated by something third: in Aristotle, the concept of “being in possibility” acts as such an intermediary between them. Aristotle introduces the concept of possibility in such a way that it would be possible to explain the change, the emergence and death of everything natural and thereby avoid the situation that has developed in the system of Platonic thinking: the emergence from the non-existent is an accidental occurrence. Indeed, everything in the world of transient things is unknowable for Plato, because it is random. Such a reproach against the great dialectician of antiquity may seem strange: after all, as you know, it is dialectics that considers objects from the point of view of change and development, which cannot be said about the formal-logical method, the creator of which is rightly considered Aristotle.

However, this reproach of Aristotle is fully justified. Indeed, in a paradoxical way, the change that occurs with sensible things does not fall into Plato's field of vision. His dialectic considers the subject in its change, but this, as P.P. Gaidenko rightly notes, is a special subject - a logical one. In Aristotle, the subject of change moved from the logical sphere to the realm of being, and the logical forms themselves ceased to be the subject of change. What is in Stagirite has a twofold character: what is in reality and what is in possibility, and since it has “a twofold character, then everything changes from what exists in possibility to what exists in reality ... Therefore, emergence can take place not only - in an incidental way - from non-existent , but also<можно сказать, что>everything arises from what exists, precisely from what exists in possibility, but does not exist in reality” (Metaphysics, XII, 2). The concept of dynamis has several different meanings, which Aristotle reveals in Book V of the Metaphysics. Two main meanings subsequently received a terminological distinction in Latin - potentia and possibilitas, which are often translated as “ability” and “possibility” (cf. German ability - Vermögen, and opportunity - Möglichkeit). “The name of possibility (dynamis) first of all designates the beginning of movement or change, which is in another or insofar as it is other, as, for example, the art of building is a capacity that is not in what is being built; and medical art, being a certain ability, can be in the one who is being treated, but not insofar as he is being treated ”(Metaphysics, V, 12).

Time for Aristotle is closely related to movement (in the broadest sense). "It is impossible for time to exist without movement." According to Aristotle, this is obvious, since "if there is time, it is obvious that there must also be movement, since time is a certain property of movement." This means that there is no movement in itself, but only a changing, becoming being, and “time is a measure of movement and being [of a body] in a state of movement.” From here it becomes clear that time with this becomes the measure of being, because "and for everything else, being in time means measuring its being by time."

There is a significant difference between the approaches of Plato and Aristotle in understanding time. In Plato, time and eternity are incommensurable, they are qualitatively different. For him, time is only a moving likeness of eternity (Timaeus, 38a), for everything that has arisen does not participate in eternity, having a beginning, and therefore an end, i.e. it was and will be, while eternity only is.

Aristotle denies the eternal existence of things, and although he introduces the concept of eternity, this concept is for him rather an infinite duration, the eternal existence of the world. His logical analysis, however ingenious it may be, is incapable of grasping the existence of a qualitatively different one. The Platonic approach, although it does not describe movement in the sensible world, turns out to be more far-sighted in relation to time. In the future, the concepts of time were developed within the framework of the Neoplatonic school and Christian metaphysics. Without being able to enter into an analysis of these teachings, we note only the common thing that unites them. All of them speak of the existence of two times - ordinary time associated with our world, and eternity, an eon (αιων), associated with being supersensible.

Returning to the analysis of quantum mechanics, we note that the wave function is defined on the configuration space of the system, and the function Ψ itself is a vector of an infinite-dimensional Hilbert space. If the wave function is not just an abstract mathematical construct, but has some referent in being, then it is necessary to draw a conclusion about its “otherness”, not belonging to the actual four-dimensional space-time. The same thesis demonstrates both the well-known "unobservability" of the wave function and its quite tangible reality, for example, in the Aharonov-Bohm effect.

Simultaneously with the Aristotelian conclusion that time is a measure of being, one can conclude that quantum mechanics allows at least to raise the question of the plurality of time. Here, modern science, according to the figurative expression of V.P. Vizgin, “enters into a fruitful“ ideological roll call ”with the ancient heritage.” Indeed, already “Einstein’s theory of relativity is closer to the ideas of the ancients about space and time as properties of being, inseparable from the order of things and the order of their movements, than to Newton’s ideas about absolute space and time, conceived as completely indifferent to things and their movements, if not dependent on them."

Time is closely related to the "event". “In a world where there is one “reality”, where “opportunity” does not exist, there is no time either, time is a difficultly predictable creation and disappearance, a re-formulation of the “opportunity package” of this or that existence.” But the “package of opportunities” itself exists, as we wanted to show, in the conditions of a different time. This statement is a kind of "metaphysical hypothesis", however, if we take into account that quantum mechanics has recently become "experimental metaphysics", then we can raise the question of the experimental detection of such "above-time" structures associated with the wave function of the system. The presence of such extratemporal structures is already indirectly indicated by the “delayed choice” experiments and Wheeler’s thought experiment with the “galactic lens”, which demonstrates the possible “delay” of the experiment in time. To what extent such a hypothesis is true, time itself will show.

Notes

Fok V.A. On the interpretation of quantum mechanics. M., 1957. S. 12.

L. de Broglie. Heisenberg uncertainty relations and wave interpretation of quantum mechanics. M., 1986. S. 141-142.

Schroedinger E. Special Theory of Relativity and Quantum Mechanics // Einstein's collection. 1982-1983. M., 1983. S. 265.

L. de Broglie. Decree. work. S. 324.

Horgan J. Quantum philosophy // In the world of science. 1992. No. 9-10. S. 73.

Horgan J. There. S. 73.

There. S. 74.

Plato. Timaeus, 38a.

There. 37 p.

Gaidenko P.P. The evolution of the concept of science. M., 1980. S. 280.

There. S. 282.

Aristotle. On Creation and Destruction, 337 a 23f.

Aristotle. Physics, 251b 27ff.

Ibid, 221a.

Ibid., 221a 9f.

For a description of the Neoplatonic concept, see, for example: Losev A.F. Being. Name. Space. M., 1993. S. 414-436; on the understanding of time in Christian theology: Lossky V.N. Essay on the mystical theology of the Eastern Church. M., 1991. Ch. v.

Vizgin V.P. Etude of time // Philos. research M., 1999. No. 3. S. 149.

There. S. 149.

There. S. 157.

Horgan, John. Quanten-Philosophie // Quantenphilosophie. Heidelberg, 1996. S. 130-139.

The obvious inapplicability of classical physics, mechanics and electrodynamics, to describe micro-objects, atoms, molecules, electrons and radiation. The problem of equilibrium thermal radiation. The problem of substance stability. Discreteness in the microcosm. Spectral lines. Experiments by Frank and Hertz.

Discreteness in classical physics. Analogy with eigenvalue problems. String vibrations, wave equation, boundary conditions. Necessity of wave description of microparticles. Experimental indications on the wave properties of micro-objects. Electron diffraction. Experiments by Davisson and Germer.

Wave and geometric optics. Description of wave fields in the limit of small wavelengths as particle flows. De Broglie's idea of ​​building quantum or wave mechanics.

Elements of classical mechanics: principle of least action, Lagrange function, action as a function of coordinates, notation of the principle of least action in terms of Hamilton's function. The equation Hamilton-Jacobi. Shortened action. The action of a freely moving particle

Wave equation in classical physics. monochromatic waves. Helmholtz equation.

Reconstruction of the wave equation for a free particle from the dispersion relation. Schrödinger equation for a free non-relativistic particle.

2. Physical quantities in classical and quantum mechanics.

The need to introduce physical quantities as operators, on the example of momentum and Hamilton operators. Interpretation of the wave function. Probability amplitude. The principle of superposition. Addition of amplitudes.

Thought experiment with two slits. transition amplitude. Transition amplitude as Green's function of the Schrödinger equation. Amplitude interference. Analogy with the principle Huygens-Fresnel. Composition of amplitudes.

Probability distribution for coordinate and momentum. Go to k- performance. Fourier transform as an expansion in terms of eigenfunctions of the momentum operator. Interpretation of eigenvalues ​​of operators as observable physical quantities.

Delta function as the core of the identity operator. Various views

delta functions. Calculation of Gaussian integrals. A bit of math. Memories of mathematical physics and a new look.

3. General theory of operators of physical quantities.

Problems for own values. quantum numbers. What does "a physical quantity has a certain value" mean? Discrete and continuous spectra.

Hermitian-definition. Validity of mean and eigenvalues. Orthogonality and normalization. Wave functions as vectors. Scalar product of functions.

Decomposition of functions in terms of the operator's own functions. Basis functions and expansions. Calculation of coefficients. Operators as matrices. Continuous and discrete indices. Representations of multiplication and differentiation operators as matrices.

Dirac notation. Abstract vectors and abstract operators. Representations and transition to different bases.

4. Measurement in quantum mechanics.

Macroscopic and classic measuring instrument. Measurement - "decomposition" in terms of the instrument's own functions.

5. Schrödinger equation for a free non-relativistic particle.

Solution by the Fourier method. wave package. The principle of uncertainty. Noncommutativity of momentum and coordinate operators. What variables does the wave function depend on? The concept of a complete set. No trajectory.

Commutability of operators and the existence of common eigenfunctions.

Necessity and sufficiency. Once again about the transition to different bases.

Transformations of operators and state vectors. Unitary operators are operators that preserve orthonormality.

Non-stationary Schrödinger equation. evolution operator. Green's function. Functions from operators. Construction of an evolution operator by expanding in eigenfunctions of a stationary equation. Operator of the derivative of a physical quantity with respect to time.

6. Heisenberg representation.

Heisenberg equations. Schrödinger equation for coupled and asymptotically free systems.

7. Entangled and independent states.

The condition for the existence of the wave function of the subsystem. Pure and mixed states of a subsystem. Description of mixed states using the density matrix. The rule for calculating averages. The evolution of the density matrix. The von Neumann equation.

8. One-dimensional movement.

One-dimensional Schrödinger equation. General theorems. Continuous and discrete spectra. Solving problems with piecewise constant potentials. Boundary conditions on potential jumps. Search for discrete levels and eigenfunctions in rectangular potentials. Oscillation theorem. variational principle. An example of a shallow hole. Existence of a bound state in a well of any depth in dimensions 1 and 2. One-dimensional scattering problem. Even potentials. The parity operator. The law of conservation of parity is fundamentally a quantum ZS that has no analogue in the classics.

9. Exactly solvable potentials.

Constant strength. Harmonic oscillator. Morse potential. Epstein potential. reflective potentials. Mention of the inverse problem of scattering theory. Laplace method. Hypergeometric and degenerate hypergeometric functions. Finding a solution in the form of a series. Analytical continuation. Analytical theory of differential equations. Three-dimensional Schrödinger equation. Centrally symmetrical potential. Isotropy.

10. Harmonic oscillator.

Approach of operators of birth and annihilation. A la Feinman, "Statistical Physics". Calculation of eigenfunctions, normalizations and matrix elements. Hermite equation. Laplace method. Finding a solution in the form of a series. Finding eigenvalues ​​from the series termination condition.

11. Orbital momentum operator.

Rotation transformation. Definition. Switching ratios. Own functions and numbers. Explicit Expressions for Orbital Momentum Operators in Spherical Coordinates. Derivation of eigenvalues ​​and operator functions. Matrix elements of orbital momentum operators. Symmetry with respect to the inversion transformation. True and pseudo scalars, vectors and tensors. Parity of various spherical harmonics. Recursive expression for moment eigenfunctions.

12. Movement in the central field.

General properties. centrifugal energy. Normalization and orthogonality. Free motion in spherical coordinates.

Spherical Bessel functions and their expressions in terms of elementary functions.

The problem of a three-dimensional rectangular well. Critical depth for the existence of a bound state. Spherical harmonic oscillator. Solution in Cartesian and spherical coordinate systems. own functions. Degenerate hypergeometric function. The equation. Solution in the form of a power series. Quantization is a consequence of the finiteness of the series.

13. Coulomb field.

Dimensionless variables, Coulomb system of units. Solution in a spherical coordinate system. discrete spectrum. Expression for energy eigenvalues. Relation between the principal and radial quantum numbers. Calculation of the degree of degeneracy. The presence of additional degeneracy.

14. Perturbation theory.

Stationary perturbation theory. General theory. Operator geometric progression. Stationary perturbation theory. Frequency corrections for a weakly anharmonic oscillator. Stationary perturbation theory in the case of degeneracy. secular equation. The problem of an electron in the field of two identical nuclei. Proper zero approximation functions. Overlap integrals. Non-stationary perturbation theory. General theory. resonance case. Fermi's golden rule.

15. semiclassical approximation.

Basic solutions. local accuracy. line layer. Airy function. VKB solution. Zwan's method. The problem of a potential well. Quantization rules Bora Sommerfeld. VKB approximation. The problem of under-barrier passage. The problem of over-barrier reflection.

16. Spin.

Multicomponent wave function. An analogue of the polarization of electromagnetic waves. The Stern-Gerlach Experience. spin variable. The infinitesimal transformation of rotation and the spin operator.

Switching ratios. Eigenvalues ​​and eigenfunctions of spin operators. matrix elements. Spin 1/2. Pauli matrices. Commutation and anticommutation relations. Pauli Matrix Algebra. Calculation of an arbitrary function from a spin scalar. Finite rotation operator. Derivation using a matrix differential equation. Linear Conversion s form. matrices U x,y,z . Determination of beam intensities in the Stern-Gerlach experiments with analyzer rotation.

17. Movement of an electron in a magnetic field.

Pauli equation. gyromagnetic ratio. The role of potentials in quantum mechanics. Gauge invariance. Bohm-Aronov effect. Switching ratios for speeds. The motion of an electron in a uniform magnetic field. Landau calibration. Equation solution. Landau levels. Lead center coordinate operator. Commutation relations for him.

1. L.D. Landau, E.M. Lifshits, Quantum mechanics, vol. 3, Moscow, Nauka, 1989
2. L. Schiff, Quantum mechanics, Moscow, IL, 1967
3. A. Messiah, Quantum mechanics, v.1,2, M. Nauka, 1978
4. A. S. Davydov, Quantum mechanics, M. Nauka, 1973
5. D.I. Blokhintsev, Fundamentals of Quantum Mechanics, Moscow, Nauka, 1976.
6. V.G. Levich, Yu. A. Vdovin, V. A. Myamlin, Theoretical Physics Course, v.2
7. L.I. Mandelstam, Lectures on optics, relativity theory and quantum mechanics.

additional literature

1. R. Feynman, Leighton, Sands, Feynman Lectures in Physics (FLP), vols. 3,8,9
2. E. Fermi, Quantum mechanics, M. Mir, 1968
3. G. Bethe, Quantum mechanics, M. Mir, 1965
4. P. Dirac, Principles of Quantum Mechanics, M. Nauka, 1979
5. V. Balashov, V. Dolinov, Course of Quantum Mechanics, ed. Moscow State University, Moscow

problem books

1. A.M. Galitsky, B. M. Karnakov, V. I. Kogan, Problems in quantum mechanics. Moscow, "Nauka", 1981.
2. M.Sh. Goldman, V. L. Krivchenkov, M. Nauka, 1968
3. Z. Flygge, Problems in quantum mechanics, vol. 1,2 M. Mir, 1974

Questions to control

1. Prove that the Schrödinger equation preserves the probability density.
2. Prove that the eigenfunctions of SL of an infinite motion are doubly degenerate.
3. Prove that the eigenfunctions of the SE of free motion corresponding to different impulses are orthogonal.
4. Prove that the eigenfunctions of the discrete spectrum are non-degenerate.
5. Prove that the eigenfunctions of the discrete spectrum of the SE with even well are either even or odd.
6. Find an eigenfunction of SL with a linear potential.
7. Determine the energy levels in a symmetrical rectangular well of finite depth.
8. Derive the boundary conditions and determine the reflection coefficient from delta potential.
9. Write an equation for the eigenfunctions of a harmonic oscillator and bring it to a dimensionless form.
10. Find the ground state eigenfunction of the harmonic oscillator. Normalize it.
11. Define birth and death operators. Write the Hamiltonian of the harmonic oscillator. Describe their properties.
12. Solving the equation in coordinate representation, find the ground state eigenfunction.
13. Using operators a, a+ calculate the matrix elements of the operators x 2 , p 2 in the basis of the eigenfunctions of the harmonic oscillator.
14. How coordinates are transformed during an infinitesimal (infinitely small) rotation.
15. Relationship between the torque and rotation operator. Definition of the moment operator. Derive commutation relations between torque components Derive commutation relations between torque projections and coordinates Derive commutation relations between torque projections and momentum l 2 ,l_z representation.
16. Momentum eigenfunctions in spherical coordinates. Write the equation and its solution using the separation of variables method. Expression in terms of associated Legendre polynomials.
17. State parity, inversion operator. Scalars and pseudoscalars, polar and axial vectors. Examples.
18. Inversion transformation in spherical coordinates. Relationship between parity and orbital momentum.
19. Reduce the problem of two bodies to the problem of the motion of one particle in a central field.
20. Divide the VN variables for the central field and write the overall solution.
21. Write a condition for orthonormality. How many quantum numbers and which form a complete set.
22. Determine particle energy levels with momentum l, equal to 0, moving in a spherical rectangular well of finite depth. Determine the minimum depth of the well required for the bound state to exist.
23. Determine the energy levels and wave functions of the spherical harmonic oscillator by separating the variables in Cartesian coordinates. What are quantum numbers. Determine the degree of degeneracy of the levels.
24. Write the SE for motion in the Coulomb field and reduce it to a dimensionless form. Atomic system of units.
25. Determine the asymptotics of the radial function of motion in the Coulomb field near the center.
26. What is the degree of degeneracy of the levels when moving in the Coulomb field.
27. Derive the formula for the first correction to the wave function corresponding to the non-degenerate energy
28. Derive the formula for the first and second energy corrections.
29. Using perturbation theory, find the first correction to the frequency of a weakly anharmonic oscillator due to the perturbation. Use the birth and death operators
30. Derive a formula for the energy correction in the case of m-fold degeneracy of this level. secular equation.
31. Derive a formula for the energy correction in the case of 2-fold degeneracy of this level. Determine the correct zero-approximation wavefunctions.
32. Get the non-stationary Schrödinger equation in the representation of the eigenfunctions of the unperturbed Hamiltonian.
33. Derive a formula for the first correction to the wave function of the system for an arbitrary non-stationary perturbation
34. Derive a formula for the first correction to the wave function of the system under a harmonic nonresonant perturbation.
35. Derive a formula for the transition probability under resonant action.
36. Fermi's golden rule.
37. Derive the formula for the leading term of the semiclassical asymptotic expansion.
38. Write local conditions for the applicability of the semiclassical approximation.
39. Write a semiclassical solution for SE that describes motion in a uniform field.
40. Write a semiclassical solution for SE that describes motion in a uniform field to the left and right of the turning point.
41. Use Zwan's method to derive boundary conditions for the transition from a semi-infinite classically forbidden region to a classically allowed one. What is the phase shift in reflection?
42. In the semiclassical approximation, determine the energy levels in the potential well. Quantization rule Bora Sommerfeld.
43. Using the quantization rule Bora Sommerfeld determine the energy levels of the harmonic oscillator. Compare with exact solution.
44. Use Zwan's method to derive boundary conditions for the transition from a semi-infinite classically allowed region to a classically forbidden one.
45. The concept of spin. spin variable. An analogue of the polarization of electromagnetic waves. The Stern-Gerlach Experience.
46. The infinitesimal transformation of rotation and the spin operator. What variables does the spin operator act on.
47. Write commutation relations for spin operators
48. Prove that the operator s 2 commutes with spin projection operators.
49. What s 2 , sz performance.
50. Write the Pauli matrices.
51. Write matrix s 2 .
52. Write eigenfunctions of operators s x , y , z for s=1/2 in s 2 , s z representation.
53. Prove the anticommutativity of Pauli matrices by direct calculation.
54. Write finite rotation matrices U x , y , z
55. A beam polarized along x is incident on the Stern-Gerlach device with its own z axis. What's the output?
56. A beam polarized along z is incident on the Stern-Gerlach device along the x axis. What is the output if the instrument axis z" is rotated relative to the x axis by an angle j?
57. Write the SE of a spinless charged particle in a magnetic field
58. Write the SE of a charged particle with spin 1/2 in a magnetic field.
59. Describe the relationship between spin and magnetic moment of a particle. What is the gyromagnetic ratio, Bohr magneton, nuclear magneton. What is the gyromagnetic ratio of an electron.
60. The role of potentials in quantum mechanics. Gauge invariance.
61. extended derivatives.
62. Write expressions for the operators of the velocity components and obtain commutation relations for them at a finite magnetic field.
63. Write the equations of motion of an electron in a uniform magnetic field in the Landau gauge.
64. Bring the SE of an electron in a magnetic field to a dimensionless form. Magnetic length.
65. Output the wave functions and energy values ​​of an electron in a magnetic field.
66. What quantum numbers characterize the state. Landau levels.

Coffee cools, buildings collapse, eggs shatter, and stars go out in a universe that seems doomed to transition into a gray monotony known as thermal equilibrium. The astronomer and philosopher Sir Arthur Eddington stated in 1927 that the gradual dissipation of energy was proof of the irreversibility of the "arrow of time".

But to the bewilderment of entire generations of physicists, the concept of the arrow of time does not correspond to the basic laws of physics, which act both in the forward direction and in the opposite direction in time. According to these laws, if someone knew the paths of all the particles in the universe and reversed them, energy would begin to accumulate, not dissipate: cold coffee would begin to heat up, buildings would rise from the ruins, and sunlight would go back to the sun.

“In classical physics, we had difficulties,” says Professor Sandu Popescu, who teaches physics at the British University of Bristol. “If I knew more, could I reverse the course of events and put together all the molecules of a broken egg?”

Of course, he says, the arrow of time is not controlled by human ignorance. And yet, since the dawn of thermodynamics in the 1850s, the only known way to calculate the propagation of energy has been to formulate the statistical distribution of unknown particle trajectories and demonstrate that over time, ignorance blurs the picture of things.

Now physicists are unearthing a more fundamental source of the arrow of time. Energy dissipates and objects come into balance, they say, because elementary particles get entangled when interacting. This strange effect they called "quantum mixing", or entanglement.

“We can finally understand why a cup of coffee in a room comes into equilibrium with it,” says Bristol-based quantum physicist Tony Short. “There is a confusion between the state of the coffee cup and the state of the room.”

Popescu, Short and their colleagues Noah Linden and Andreas Winter reported their discovery in the journal Physical Review E in 2009, stating that objects come into equilibrium, or a state of even distribution of energy, over an indefinite period of time. long time due to quantum mechanical mixing with the environment. A similar discovery was made a few months earlier by Peter Reimann of the University of Bielefeld in Germany, publishing his findings in Physical Review Letters. Short and colleagues backed up their argument in 2012 by showing that entanglement produces equilibrium in a finite time. And in a paper published in February on arXiv. org, two separate groups have taken the next step by calculating that most physical systems quickly equilibrate in a time directly proportional to their size. “To show that this applies to our real physical world, the processes must occur within a reasonable time frame,” says Short.

The tendency for coffee (and everything else) to balance is "very intuitive," says Nicolas Brunner, a quantum physicist at the University of Geneva. "But in explaining the reasons for this, for the first time, we have solid grounds in view of the microscopic theory."

© RIA Novosti, Vladimir Rodionov

If the new line of research is correct, then the story of the arrow of time begins with the quantum mechanical idea that, at its core, nature is inherently uncertain. An elementary particle is devoid of specific physical properties, and it is determined only by the probabilities of being in certain states. For example, at a certain moment, a particle can rotate clockwise with a 50 percent probability and counterclockwise with a 50 percent probability. Northern Irish physicist John Bell's experimentally verified theorem states that there is no "true" state of particles; probabilities are the only thing that can be used to describe it.

Quantum uncertainty inevitably leads to confusion, the supposed source of the arrow of time.

When two particles interact, they can no longer be described by separate, independently evolving probabilities called "pure states." Instead, they become intertwined components of a more complex probability distribution that describe the two particles together. They can, for example, indicate that the particles are spinning in opposite directions. The system as a whole is in a pure state, but the state of each particle is "mixed" with the state of the other particle. Both particles may be moving several light-years apart, but the rotation of one particle will correlate with the other. Albert Einstein well described it as "spooky action at a distance".

“Entanglement is, in a sense, the essence of quantum mechanics,” or the laws that govern interactions on a subatomic scale, says Brunner. This phenomenon underlies quantum computing, quantum cryptography and quantum teleportation.

The idea that confusion could explain the arrow of time first occurred to Seth Lloyd 30 years ago when he was a 23-year-old Cambridge University philosophy graduate with a Harvard degree in physics. Lloyd realized that quantum uncertainty, and its spread as particles become more entangled, could replace human uncertainty (or ignorance) of the old classical evidence and become the true source of the arrow of time.

Using a little-known quantum mechanical approach in which units of information are the basic building blocks, Lloyd spent several years studying the evolution of particles in terms of shuffling ones and zeros. He found that as the particles get more and more mixed with each other, the information that described them (for example, 1 for clockwise rotation and 0 for counterclockwise) will transfer to the description of the system of entangled particles as a whole. The particles seemed to gradually lose their independence and become pawns of the collective state. Over time, all information passes into these collective clusters, and individual particles do not have it at all. At this point, as Lloyd discovered, the particles enter a state of equilibrium, and their states stop changing, like a cup of coffee cools to room temperature.

“What is really going on? Things become more interconnected. The arrow of time is the arrow of rising correlations.”

This idea, set out in Lloyd's 1988 doctoral dissertation, fell on deaf ears. When the scientist sent an article about this to the editors of the journal, he was told that "there is no physics in this work." Quantum information theory "was deeply unpopular" at the time, Lloyd says, and questions about the arrow of time "were the domain of lunatics and wacky Nobel laureates."

“I was pretty damn close to being a taxi driver,” he said.

Since then, advances in quantum computing have turned quantum information theory into one of the most active areas of physics. Lloyd is currently a professor at the Massachusetts Institute of Technology, recognized as one of the founders of the discipline, and his forgotten ideas are being revived by the efforts of Bristol physicists. The new evidence is more general, the scientists say, and applies to any quantum system.

“When Lloyd came up with the idea in his dissertation, the world was not ready for it,” says Renato Renner, head of the Institute for Theoretical Physics at ETH Zurich. Nobody understood him. Sometimes you need ideas to come at the right time.”

In 2009, evidence from a team of Bristol physicists resonated with quantum information theorists, who discovered new ways to apply their methods. They showed that as objects interact with their environment—like particles in a cup of coffee interact with air—information about their properties “leaks and spreads through that environment,” Popescu explains. This local loss of information causes the state of the coffee to remain the same even as the net state of the entire room continues to change. With the exception of rare random fluctuations, the scientist says, "his state ceases to change in time."

It turns out that a cold cup of coffee cannot spontaneously warm up. In principle, as the clean state of the room evolves, coffee can suddenly escape from the air of the room and return to the clean state. But there are many more mixed states than pure ones, and in practice, coffee can never return to a pure state. To see this, we will have to live longer than the universe. This statistical improbability makes the arrow of time irreversible. “Essentially, blending opens up a huge space for us,” says Popescu. - Imagine that you are in a park, there is a gate in front of you. As soon as you enter them, you get out of balance, fall into a huge space and get lost in it. You will never return to the gate."

In the new story of the arrow of time, information is lost in the process of quantum entanglement, not because of human subjective lack of knowledge about what balances a cup of coffee and a room. The room eventually balances with the environment, and the environment moves even more slowly towards equilibrium with the rest of the universe. The thermodynamic giants of the 19th century viewed this process as a gradual dissipation of energy that increases the overall entropy, or chaos, of the universe. Today, Lloyd, Popescu, and others in the field view the arrow of time differently. In their opinion, information becomes more and more diffuse, but never completely disappears. Although entropy grows locally, the total entropy of the universe remains constant and zero.

“On the whole, the universe is in a pure state,” says Lloyd. “But its individual parts, intertwined with the rest of the universe, come into a mixed state.”

But one riddle of the arrow of time remains unsolved. “There is nothing in these works that explains why you start with a gate,” Popescu says, returning to the park analogy. “In other words, they don’t explain why the original state of the universe was far from equilibrium.” The scientist hints that this question refers to the nature of the Big Bang.

Despite recent advances in equilibration time calculations, the new approach still cannot be used as a tool for calculating the thermodynamic properties of specific things like coffee, glass, or unusual states of matter. (Some conventional thermodynamicists say they know very little about the new approach.) “The point is that you need to find criteria for what things behave like window glass and what things behave like a cup of tea,” says Renner. “I think I will see new work in this direction, but there is still a lot to be done.”

Some researchers have expressed doubt that this abstract approach to thermodynamics will ever be able to accurately explain how particular observable objects behave. But conceptual advances and a new set of mathematical formulas are already helping researchers ask theoretical questions from the field of thermodynamics, such as the fundamental limitations of quantum computers and even the ultimate fate of the universe.

“We are thinking more and more about what can be done with quantum machines,” says Paul Skrzypczyk of the Institute of Photon Sciences in Barcelona. Let's say the system is not yet in equilibrium and we want to make it work. How much useful work can we extract? How can I intervene to do something interesting?”

Context

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Nautilus 01/25/2016
Caltech cosmology theorist Sean Carroll applies new formulas in his latest work on the arrow of time in cosmology. “I’m most interested in the long-term fate of cosmological spacetime,” says Carroll, who wrote From Eternity to Here: The Quest for the Ultimate Theory of Time. “In this situation, we still do not know all the necessary laws of physics, so it makes sense to turn to the abstract level, and here, it seems to me, this quantum mechanical approach will help us.”

Twenty-six years after the failure of Lloyd's grandiose idea of ​​the arrow of time, he enjoys watching its revival and trying to apply the ideas of the latest work to the paradox of information falling into a black hole. “I think now they will still talk about the fact that there is physics in this idea,” he says.

And philosophy even more so.

According to scientists, our ability to remember the past but not the future, which is a confusing manifestation of the arrow of time, can also be seen as an increase in correlations between interacting particles. When you read a note on a piece of paper, the brain correlates with the information through photons that hit your eyes. Only from this moment you can remember what is written on paper. As Lloyd notes, "the present can be characterized as the process of establishing correlations with our environment."

The backdrop for the steady growth of weaves throughout the universe is, of course, time itself. Physicists point out that despite great advances in understanding how time changes occur, they are no closer to understanding the nature of time itself or why it differs from the other three dimensions of space (in conceptual terms and in the equations of quantum mechanics) . Popescu calls this mystery "one of the greatest unknowns in physics."

“We can discuss that an hour ago our brain was in a state that correlated with fewer things,” he says. “But our perception that time is ticking is another matter entirely. Most likely, we will need a new revolution in physics that will tell about it.”

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