Quantum theory. Quantum field theory What is field theory

This apparently measurement-induced collapse of the wave function has been the source of many conceptual difficulties in quantum mechanics. Before the collapse, there is no way to know for sure where the photon will end up; it can be anywhere with non-zero probability. There is no way to trace the path of a photon from source to detector. The photon is unreal in the same sense that a plane flying from San Francisco to New York is real.

Werner Heisenberg, among others, interpreted this mathematics to mean that reality does not exist until it is observed. “The idea of an objective real world, the smallest particles of which exist objectively in the same sense in which stones or trees exist, whether we observe them or not, is impossible,” he wrote. John Wheeler also used a variant of the double slit experiment to state that “no elementary quantum phenomenon is a phenomenon until it becomes a registered (“observable”, “reliably recorded”) phenomenon.”

But quantum theory gives absolutely no clues as to what constitutes a “measurement.” It simply postulates that the measuring device must be classical, without defining where the line between classical and quantum lies, and leaving the door open for those who believe that collapse is caused by human consciousness. Last May, Henry Stapp and his colleagues argued that the double slit experiment and its modern variants suggest that "a conscious observer may be necessary" to make sense of the quantum realm, and that transpersonal intelligence underlies the material world.

But these experiments do not provide empirical evidence for such claims. The double slit experiment, performed with single photons, can only test the probabilistic predictions of mathematics. If probabilities emerge from the process of sending tens of thousands of identical photons through a double slit, the theory states that each photon's wave function has collapsed—through a vaguely defined process called measurement. That's all.

Additionally, there are other interpretations of the double slit experiment. Take, for example, the de Broglie-Bohm theory, which states that reality is both a wave and a particle. The photon is directed towards the double slit with a certain position at any moment and passes through one slit or the other; therefore, each photon has a trajectory. It passes through a pilot wave, which penetrates both slits, interferes, and then directs the photon to the site of constructive interference.

In 1979, Chris Dewdney and his colleagues at Brickbeck College in London modeled this theory's prediction of the trajectories of particles that would pass through the double slit. Over the past ten years, experimenters have confirmed that such trajectories exist, although they used the controversial technique of so-called weak measurements. Although controversial, experiments have shown that the de Broglie-Bohm theory is still able to explain the behavior of the quantum world.

More importantly, this theory does not require observers, or measurements, or immaterial consciousness.

Just as the so-called collapse theories are not needed, from which it follows that wave functions collapse randomly: the greater the number of particles in a quantum system, the more likely the collapse. Observers simply record the result. Markus Arndt's team at the University of Vienna in Austria tested these theories by sending larger and larger molecules through a double slit. Collapse theories predict that when particles of matter become more massive than a certain threshold, they can no longer remain in quantum superposition and pass through both slits simultaneously, and this destroys the interference pattern. Arndt's team sent a molecule of 800 atoms through a double slit and still saw interference. The search for the threshold continues.

Roger Penrose had his own version of collapse theory, in which the higher the mass of an object in superposition, the faster it collapses into one state or another due to gravitational instabilities. Again, this theory does not require an observer or any consciousness. Dirk Bouwmeester of the University of California, Santa Barbara, is testing Penrose's idea using a version of the double-slit experiment.

Conceptually, the idea is not just to put a photon in a superposition of passing through two slits at the same time, but to put one of the slits in a superposition and force it to be in two places at the same time. According to Penrose, the replaced gap will either remain in superposition or collapse with the photon in flight, leading to different interference patterns. This collapse will depend on the mass of the cracks. Bouwmeester has been working on this experiment for ten years and may soon confirm or refute Penrose's claims.

In any case, these experiments show that we cannot yet make any statements about the nature of reality, even if these statements are well supported mathematically or philosophically. And given that neuroscientists and philosophers of mind cannot agree on the nature of consciousness, the claim that it leads to the collapse of wave functions is premature at best and misleading at worst.

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Basic principles of quantum field theory: 1). Vacuum state. Nonrelativistic quantum mechanics allows us to study the behavior of a constant number of elementary particles. Quantum field theory takes into account the birth and absorption or destruction of elementary particles. Therefore, quantum field theory contains two operators: the creation operator and the annihilation operator of elementary particles. According to quantum field theory, a state where there is neither field nor particles is impossible. Vacuum is a field in its lowest energy state. Vacuum is characterized not by independent, observable particles, but by virtual particles that appear and then disappear after a while. 2.) Virtual mechanism of interaction of elementary particles. Elementary particles interact with each other as a result of fields, but if a particle does not change its parameters, it cannot emit or absorb a real quantum of interaction, such energy and momentum and for such a time and distance, which are determined by the relations ∆E∙∆t≥ħ, ∆рх∙∆х≥ħ( quantum constant) uncertainty relation. The nature of virtual particles is such that they appear after some time, disappear or are absorbed. Amer. Physicist Feynman developed a graphical way to depict the interaction of elementary particles with virtual quanta:

Emission and absorption of a virtual quantum of a free particle

Interaction of two elements. particles by means of one virtual quantum.

Interaction of two elements. particles by means of two virtual quantums.

On the data in Fig. Graphic image of particles, but not their trajectories.

3.) Spin is the most important characteristic of quantum objects. This is the particle’s own angular momentum, and if the angular momentum of the top coincides with the direction of the rotation axis, then the spin does not determine any specific preferred direction. Spin sets direction, but in a probabilistic manner. Spin exists in a form that cannot be visualized. The spin is denoted s=I∙ħ, and I takes both integer values I=0,1,2,..., and half-numeric values I = ½, 3/2, 5/2,... In classical physics, identical particles are not spatially different, because occupy the same region of space, the probability of finding a particle in any region of space is determined by the square of the modulus of the wave function. The wave function ψ is a characteristic of all particles. ‌‌. corresponds to the symmetry of wave functions, when particles 1 and 2 are identical and their states are the same. the case of antisymmetry of wave functions, when particles 1 and 2 are identical to each other, but differ in one of the quantum parameters. For example: spin. According to the Paul exclusion principle, particles with half-integer spin cannot be in the same state. This principle allows us to describe the structure of the electronic shells of atoms and molecules. Those particles that have integer spin are called bosons. I =0 for Pi mesons; I =1 for photons; I = 2 for gravitons. Particles with half-numerical spin are called fermions. For an electron, positron, neutron, proton, I = ½. 4) Isotopic spin. The mass of a neutron is only 0.1% greater than the mass of a proton; if we abstract (ignore) the electric charge, then these two particles can be considered two states of the same particle, the nucleon. Similarly, there are mesons, but these are not three independent particles, but three states of the same particle, which are simply called Pi - meson. To take into account the complexity or multiplicity of particles, a parameter called isotopic spin is introduced. It is determined from the formula n = 2I+1, where n is the number of particle states, for example for a nucleon n=2, I=1/2. The isospin projection is designated Iз = -1/2; Iз = ½, i.e. a proton and a neutron form an isotopic doublet. For Pi mesons, the number of states = 3, i.e. n=3, I =1, Iз=-1, Iз=0, Iз=1. 5) Classification of particles: the most important characteristic of elementary particles is rest mass; on this basis, particles are divided into baryons (trans. heavy), mesons (from Greek: medium), leptons (from Greek: light). According to the principle of interaction, baryons and mesons also belong to the class of hadrons (from the Greek strong), since these particles participate in strong interaction. Baryons include: protons, neutrons, hyperons, of these particles, only the proton is stable, all baryons are fermions, mesons are bosons, are unstable particles, participate in all types of interactions, just like baryons, leptons include: electron, neutron , these particles are fermions and do not participate in strong interactions. The photon stands out especially, which does not belong to leptons, and also does not belong to the class of hadrons. Its spin = 1, and rest mass = 0. Sometimes interaction quanta are classified into a special class, the meson is a weak interaction quantum, and the gluon is a gravitational interaction quantum. Sometimes quarks are classified into a special class, having a fractional electric charge equal to 1/3 or 2/3 of the electric charge. 6) Types of interaction. In 1865, the theory of the electromagnetic field (Maxwell) was created. In 1915, the theory of the gravitational field was created by Einstein. The discovery of strong and weak interactions dates back to the first third of the 20th century. Nucleons are tightly bound together in the nucleus by strong interactions, which are called strong. In 1934, Fermet created the first theory of weak interactions that was sufficiently adequate to experimental research. This theory arose after the discovery of radioactivity, it was necessary to assume that minor interactions arise in the nuclei of an atom, which lead to the spontaneous decay of heavy chemical elements such as uranium, and rays are emitted. A striking example of weak interactions is the penetration of neutron particles through the ground, while neutrons have a much more modest penetrating ability; they are retained by a lead sheet several centimeters thick. Strong: electromagnetic. Weak: gravitational = 1: 10-2: 10-10: 10-38. The difference between electromagnetic and gravitational The interactions are that they smoothly decrease with increasing distance. Strong and weak interactions are limited to very small distances: 10-16 cm for weak ones, 10-13 cm for strong ones. But at a distance< 10-16 см слабые взаимодействия уже не являются малоинтенсивными, на расстоянии 10-8 см господствуют электромагнитные силы. Адроны взаимодействуют с помощью кварков. Переносчиками взаимодействия между кварками являются глюоны. Сильные взаимодействия появляются на расстояниях 10-13 см, т. Е. глюоны являются короткодействующими и способны долететь такие расстояния. Слабые взаимодействия осуществляются с помощью полей Хиггса, когда взаимодействие переносится с помощью квантов, которые называются W+,W- - бозоны, а также нейтральные Z0 – бозоны(1983 год). 7) Fission and synthesis of atomic nuclei. The nuclei of atoms consist of protons, which are denoted by Z and neutrons by N, the total number of nucleons is denoted by the letter - A. A = Z + N. To remove a nucleon from a nucleus, it is necessary to expend energy, therefore the total mass and energy of the nucleus is less than the sum of the ass and energies of all its components. The energy difference is called the binding energy: Eb=(Zmp+Nmn-M)c2 binding energy of nucleons in the nucleus – Eb. The binding energy passing per nucleon is called specific binding energy (Eb/A). The specific binding energy takes on a maximum value for the nuclei of iron atoms. In the elements following iron, an increase in nucleons occurs, and each nucleon acquires more and more neighbors. Strong interactions are short-range, this leads to the fact that with the growth of nucleons and with a significant growth of nucleons, chemical the element tends to decay (natural radioactivity). Let's write down the reactions in which energy is released: 1. During the fission of nuclei with a large number of nucleons: n+U235→ U236→139La+95Mo+2n a slowly moving neutron is absorbed by U235 (uranium) resulting in the formation of U236, which is divided into 2 nuclei La(laptam) and Mo(molybdenum), which fly away at high speeds and 2 neutrons are formed, which can cause 2 such reactions. The reaction takes on a chain character so that the mass of the initial fuel reaches a critical mass.2. Fusion reaction of light nuclei.d2+d=3H+n, if people were able to ensure stable fusion of nuclei, they would save themselves from energy problems. Deuterium contained in ocean water is an inexhaustible source of cheap nuclear fuel, and the synthesis of light elements is not accompanied by intense radioactive phenomena, as is the case with the fission of uranium nuclei.

Fock space, describing all possible excitations of the quantum field. An analogue of the quantum mechanical wave function in QFT is a field operator (more precisely, a “field” is an operator-valued generalized function, from which only after convolution with the main function we obtain an operator acting in the Hilbert state space), capable of acting on the vacuum vector of the Fock space (see vacuum ) and generate single-particle excitations of the quantum field. The physical observables here also correspond to operators composed of field operators [ style!] .

It is on quantum field theory that all elementary particle physics is based.

When constructing quantum field theory, the key point was understanding the essence of the phenomenon of renormalization.

History of origin

The basic equation of quantum mechanics - the Schrödinger equation - is relativistically non-invariant, as can be seen from the asymmetric inclusion of time and spatial coordinates in the equation. In 1926, a relativistically invariant equation for a free (spinless or zero spin) particle (the Klein-Gordon-Fock equation) was proposed. As is known, in classical mechanics (including non-relativistic quantum mechanics) energy (kinetic, since potential is assumed to be zero) and momentum of a free particle are related by the relation . The relativistic relationship between energy and momentum has the form . Assuming that the momentum operator in the relativistic case is the same as in the non-relativistic region, and using this formula to construct the relativistic Hamiltonian by analogy, we obtain the equation Klein-Gordon equation:

or, briefly, using natural units in addition:

, where is the D'Alembert operator.

However, the problem with this equation is that the wave function here is difficult to interpret as a probability amplitude, if only because - as can be shown - the probability density will not be a positive definite quantity.

The Dirac equation, proposed by him in 1928, has a slightly different justification. Dirac tried to obtain a first-order differential equation in which the equality of the time coordinate and spatial coordinates was ensured. Since the momentum operator is proportional to the first derivative with respect to the coordinates, the Dirac Hamiltonian must be linear in the momentum operator.

and taking into account the formula for the connection between energy and momentum, restrictions are imposed on the square of this operator, and therefore on the “coefficients” - their squares must be equal to one and they must be mutually anticommutative. So these definitely can't be numeric odds. However, they can be matrices, with dimensions of at least 4, and the “wave function” is a four-component object, called a bispinor. In this case, the Dirac equation formally has a form identical to the Schrödinger equation (with the Dirac Hamiltonian).

However, this equation, like the Klein-Gordon equation, has solutions with negative energies. This circumstance was the reason for the prediction of antiparticles, which was later confirmed experimentally (the discovery of the positron). The presence of antiparticles is a consequence of the relativistic relationship between energy and momentum.

At the same time, by the end of the 20s, a formalism for the quantum description of many-particle systems (including systems with a variable number of particles), based on the operators of creation and annihilation of particles, was developed. Quantum field theory also turns out to be based on these operators (expressed through them).

The Klein-Gordon and Dirac equations should be considered as equations for field operator functions acting on the state vector of a system of quantum fields satisfying the Schrödinger equation.

The essence of quantum field theory

Lagrangian formalism

In classical mechanics, many-particle systems can be described using the Lagrangian formalism. The Lagrangian of a many-particle system is equal to the sum of the Lagrangians of individual particles. In field theory, a similar role can be played by the Lagrangian density (Lagrangian density) at a given point in space. Accordingly, the Lagrangian of the system (field) will be equal to the integral of the Lagrangian density over three-dimensional space. The action, as in classical mechanics, is assumed to be equal to the integral of the Lagrangian over time. Consequently, the action in field theory can be considered as an integral of the Lagrangian density over four-dimensional space-time. Accordingly, one can apply the principle of least (stationary) action to this four-dimensional integral and obtain the equations of motion for the field - the Euler-Lagrange equations. The minimum requirement for the Lagrangian (Lagrangian density) is relativistic invariance. The second requirement is that the Lagrangian must not contain derivatives of the field function higher than the first degree in order for the equations of motion to be “correct” (corresponding to classical mechanics). There are also other requirements (locality, unitarity, etc.). According to Noether's theorem, the invariance of the action under k-parametric transformations leads to k dynamic field invariants, that is, to conservation laws. In particular, the invariance of the action with respect to translations (shifts) leads to the conservation of the 4-momentum.

Example: Scalar field with Lagrangian

The equations of motion for a given field lead to the Klein–Gordon equation. To solve this equation, it is useful to move to the momentum representation through the Fourier transform. From the Klein-Gordon equation it is easy to see that the Fourier coefficients will satisfy the condition

Where is an arbitrary function

The delta function establishes a connection between frequency (energy), wave vector (momentum vector) and parameter (mass): . Accordingly, for two possible signs we have two independent solutions in momentum representation (Fourier integral)

It can be shown that the momentum vector will be equal to

Therefore, the function can be interpreted as the average density of particles with mass, momentum and energy. After quantization, these products turn into operators having integer eigenvalues.

Field quantization. Operators of creation and destruction of quanta

Quantization means the transition from fields to operators acting on the vector (amplitude) of the state Φ . By analogy with conventional quantum mechanics, the state vector completely characterizes the physical state of a system of quantized wave fields. A state vector is a vector in some linear space.

The main postulate of quantization of wave fields is that the operators of dynamic variables are expressed in terms of field operators in the same way as for classical fields (taking into account the order of multiplication)

For a quantum harmonic oscillator, a well-known energy quantization formula has been obtained. The eigenfunctions corresponding to the indicated eigenvalues of the Hamiltonian turn out to be related to each other by certain operators - an increasing operator, - a decreasing operator. It should be noted that these operators are non-commutative (their commutator is equal to one). The use of an increasing or decreasing operator increases the quantum number n by one and leads to an equal increase in the energy of the oscillator (spectrum equidistance), which can be interpreted as the birth of a new one or the destruction of a field quantum with energy . It is this interpretation that allows the above operators to be used, like creation and destruction operators quanta of a given field. The Hamiltonian of the harmonic oscillator is expressed through the indicated operators as follows, where - quanta number operator fields. It is easy to show - that is, the eigenvalues of this operator - the number of quanta. Any n-particle field state can be obtained by the action of creation operators on the vacuum

For a vacuum state, the result of applying the annihilation operator is equal to zero (this can be taken as a formal definition of a vacuum state).

In the case of N oscillators, the Hamiltonian of the system is equal to the sum of the Hamiltonians of the individual oscillators. For each such oscillator, one can define its own creation operators. Therefore, an arbitrary quantum state of such a system can be described using fill numbers- the number of operators of a given type k acting on the vacuum:

This representation is called representation of filling numbers. The essence of this representation is that instead of specifying a function of a function of coordinates (coordinate representation) or as a function of impulses (pulse representation), the state of the system is characterized by the number of the excited state - the filling number.

It can be shown that, for example, the Klein-Gordon scalar field can be represented as a collection of oscillators. Expanding the field function into an infinite Fourier series in a three-dimensional momentum vector, it can be shown that from the Klein-Gordon equation it follows that the expansion amplitudes satisfy the classical second-order differential equation for an oscillator with a parameter (frequency). Let us consider a limited cube and impose a periodicity condition on each coordinate with a period. The periodicity condition leads to the quantization of the permissible impulses and energy of the oscillator:

Field operators, dynamic variable operators

Fock's representation

Bose-Einstein and Fermi-Dirac quantization. Connection with spin.

The Bose-Einstein commutation relations are based on an ordinary commutator (the difference between the “direct” and “inverse” product of operators), and the Fermi-Dirac commutation relations are based on an anticommutator (the sum of the “direct” and “inverse” product of operators). Quanta of the first fields obey Bose-Einstein statistics and are called bosons, and quanta of the second fields obey Fermi-Dirac statistics and are called fermions. Bose-Einstein quantization of fields turns out to be consistent for particles with integer spin, and for particles with half-integer spin, Fermi-Dirac quantization turns out to be consistent. Thus, fermions are particles with half-integer spin, and bosons are particles with integer spin.

S-matrix formalism. Feynman diagrams

The problem of divergences and ways to solve them

Axiomatic quantum field theory

Literature

Quantum field theory - Physical encyclopedia (editor-in-chief A. M. Prokhorov).
Richard Feynman, “The nature of physical laws” - M., Nauka, 1987, 160 p.
Richard Feynman, “QED - a strange theory of light and matter” - M., Nauka, 1988, 144 p.
Bogolyubov N. N., Shirkov D. V. Introduction to Quantized Field Theory. - M.: Nauka, 1984. - 600 p.
Wentzel G. Introduction to quantum theory of wave fields. - M.: GITTL, 1947. - 292 p.
Itsikson K., Zuber J.-B. Quantum field theory. - M.: Mir, 1984. - T. 1. - 448 p.
Ryder L. Quantum field theory. - M.: Mir, 1987. - 512 p.

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Quantum field theory

This doctrine allows us to determine the process of introducing quantum systems into a framework called degrees of freedom in science, which assume a certain number of independent coordinates, which are extremely important for indicating the overall movement of a mechanical concept.

In simple words, these indicators are the main characteristics of movement. It is worth noting that interesting discoveries in the field of harmonious interaction of elementary particles were made by researcher Steven Weinberg, who discovered the neutral current, namely the principle of the relationship between leptons and quarks. For his discovery in 1979, the physicist became a Nobel Prize laureate.

In quantum theory, an atom consists of a nucleus and a specific cloud of electrons. The basis of this element includes almost the entire mass of the atom itself - more than 95 percent. The nucleus has an exclusively positive charge, defining the chemical element of which the atom itself is a part. The most unusual thing about the structure of the atom is that, although the nucleus makes up almost all of its mass, it contains only one ten-thousandth of its volume. It follows from this that there is indeed very little dense matter in an atom, and the rest of the space is occupied by an electron cloud.

Interpretations of quantum theory - the principle of complementarity

The rapid development of quantum theory has led to a radical change in classical ideas about such elements:

structure of matter;
movement of elementary particles;
causality;
space;
time;
the nature of cognition.

Such changes in people's consciousness contributed to a radical transformation of the picture of the world into a clearer concept. The classical interpretation of a material particle was characterized by a sudden release from the environment, the presence of its own movement and a specific location in space.

In quantum theory, an elementary particle began to be represented as the most important part of the system in which it was included, but at the same time it did not have its own coordinates and momentum. In the classical cognition of movement, the transfer of elements that remained identical to themselves along a pre-planned trajectory was proposed.

The ambiguous nature of particle division necessitated the abandonment of such a vision of motion. Classical determinism gave way to the leading position to the statistical direction. If previously the entire whole in an element was perceived as the total number of component parts, then quantum theory determined the dependence of the individual properties of the atom on the system.

The classical understanding of the intellectual process was directly related to the understanding of a material object as fully existing in itself.

Quantum theory has demonstrated:

dependence of knowledge about the object;
independence of research procedures;
completeness of actions on a number of hypotheses.

Note 2

The meaning of these concepts was initially far from clear, and therefore the main provisions of quantum theory have always received different interpretations, as well as various interpretations.

Quantum statistics

In parallel with the development of quantum and wave mechanics, other components of quantum theory were rapidly developing - statistics and statistical physics of quantum systems, which included a huge number of particles. On the basis of classical methods of movement of specific elements, a theory of the behavior of their integrity was created - classical statistics.

In quantum statistics there is absolutely no possibility of distinguishing between two particles of the same nature, since the two states of this unstable concept differ from each other only by the rearrangement of particles of identical power of influence on the principle of identity itself. This is how quantum systems mainly differ from classical scientific systems.

An important result in the discovery of quantum statistics is the proposition that each particle that is part of any system is not identical to the same element. This implies the importance of the task of determining the specifics of a material object in a specific segment of systems.

The difference between quantum physics and classical

So, the gradual departure of quantum physics from classical physics consists in the refusal to explain individual events occurring in time and space, and the use of the statistical method with its probability waves.

Note 3

The goal of classical physics is to describe individual objects in a certain sphere and formulate laws governing the change of these objects over time.

Quantum physics occupies a special place in science in the global understanding of physical ideas. Among the most memorable creations of the human mind is the theory of relativity - general and special, which is a completely new concept of directions that combines electrodynamics, mechanics and the theory of gravity.

Quantum theory was able to finally break ties with classical traditions, creating a new, universal language and an unusual style of thinking, allowing scientists to penetrate the microworld with its energetic components and give its complete description by introducing specifics that were absent in classical physics. All these methods ultimately made it possible to understand in more detail the essence of all atomic processes, and at the same time, it was this theory that introduced an element of randomness and unpredictability into science.

Are our attempts to describe reality nothing more than playing dice and trying to predict the desired outcome? James Owen Weatherall, professor of logic and philosophy of science at the University of Irvine, reflected on the pages of Nautil.us about the mysteries of quantum physics, the problem of the quantum state and how much it depends on our actions, knowledge and subjective perception of reality, and why, predicting different probabilities, we all turn out to be right.

Physicists are well aware of how to apply quantum theory - your phone and computer are proof of this. But knowing how to use something is a far cry from fully understanding the world described by the theory, or even what the various mathematical tools scientists use mean. One such mathematical tool, the status of which physicists have long debated, is the “quantum state.” ⓘ A quantum state is any possible state that a quantum system can be in. In this case, the “quantum state” should also be understood as all the potential probabilities of getting one or another value when playing “dice”. — Approx. ed..

One of the most striking features of quantum theory is that its predictions are probabilistic. If you conduct an experiment in a laboratory and use quantum theory to predict the results of various measurements, at best the theory can only predict the probability of the result: for example, 50% for the predicted result and 50% for it to be different. The role of a quantum state is to determine the probability of outcomes. If the quantum state is known, you can calculate the probability of obtaining any possible result for any possible experiment.

Does a quantum state represent an objective aspect of reality or is it just a way of characterizing us, that is, what one knows about reality? This issue was actively discussed at the very beginning of the study of quantum theory and has recently become relevant again, inspiring new theoretical calculations and subsequent experimental tests.

“If you just change your knowledge, things will no longer seem strange.”

To understand why a quantum state illustrates someone's knowledge, imagine a case in which you are calculating a probability. Before your friend rolls the dice, you guess which way they will land. If your friend rolls a regular six-sided die, your guess will have about a 17% (one-sixth) chance of being correct no matter what you guess. In this case, probability says something about you, namely, what you know about the die. Suppose you turn your back while throwing, and your friend sees the result - let it be six, but this result is unknown to you. And until you turn around, the outcome of the throw remains uncertain, even though your friend knows it. Probability, which represents human uncertainty even though reality is certain, is called epistemic, from the Greek word for knowledge.

This means that you and your friend could determine different probabilities without either of you being wrong. You will say that the probability of getting a six on the die is 17%, and your friend, who is already familiar with the result, will call it 100%. This is because you and your friend know different things, and the probabilities you name represent different degrees of your knowledge. The only incorrect prediction would be one that rules out the possibility of rolling a six altogether.

For the last fifteen years, physicists have been wondering whether a quantum state might turn out to be epistemic in the same way. Suppose some state of matter, such as the distribution of particles in space or the outcome of a game of dice, is certain but unknown to you. The quantum state, according to this approach, is just a way of describing the incompleteness of your knowledge about the structure of the world. In different physical situations, there may be more than one way to determine the quantum state depending on the known information.

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It's tempting to think of a quantum state in this way because it becomes different when the parameters of a physical system are measured. Taking measurements changes this state from one where every possible outcome has a non-zero probability to one where only one outcome is possible. This is similar to what happens in a game of dice when you find out the result you get. It may seem strange that the world can change simply because you take measurements. But if it's just a change in your knowledge, it's no longer surprising.

Another reason to believe that a quantum state is epistemic is that it is impossible to determine from a single experiment what the quantum state was like before it was performed. This is also reminiscent of playing dice. Let's say your friend offers to play and claims that the probability of rolling a six is only 10%, while you insist on 17%. Can one single experiment show which of you is right? No. The fact is that the resulting result is comparable to both probability estimates. There is no way to know which of you two is right in any given case. According to the epistemic approach to quantum theory, the reason why most quantum states cannot be experimentally determined is like a game of dice: for each physical situation there are multiple probabilities consistent with the multiplicity of quantum states.

Rob Spekkens, a physicist at the Institute for Theoretical Physics in Waterloo, Ontario, published a paper in 2007 presenting a “toy theory” designed to simulate quantum theory. This theory is not entirely analogous to quantum theory, since it is simplified to an extremely simple system. The system has only two options for each of its parameters: for example, “red” and “blue” for color and “up” and “down” for position in space. But, like quantum theory, it included states that could be used to calculate probability. And the predictions made with its help coincide with the predictions of quantum theory.

Spekkens's "toy theory" was exciting because, like quantum theory, its states were "undeterminable"—and this indeterminacy was entirely explained by the fact that epistemic theory actually had to do with real physical situations. In other words, toy theory was like quantum theory, and its states were uniquely epistemic. Since, if the epistemic view is abandoned, the uncertainty of quantum states has no clear explanation, Spekkens and his colleagues considered this a sufficient reason to consider quantum states also epistemic, but in this case the “toy theory” must be extended to more complex systems ( i.e. on physical systems explained by quantum theory). Since then, it has entailed a series of studies in which some physicists tried to explain all quantum phenomena with its help, while others tried to show its fallacy.

“These assumptions are consistent, but that does not mean they are true.”

Thus, opponents of the theory raise their hands higher. For example, one widely discussed 2012 result published in Nature Physics showed that if one physical experiment can be performed independently of another, then there can be no uncertainty about the “correct” quantum state describing that experiment. That. All quantum states are “regular” and “true” except those that are completely “unreal,” namely “wrong” states such as those in which the probability of rolling a six is zero.

Another study, published in Physical Review Letters in 2014 by Joanna Barrett and others, showed that the Spekkens model cannot be applied to a system in which each parameter has three or more degrees of freedom - for example, "red", "blue" and "green" for colors, not just “red” and “blue”—without violating the predictions of quantum theory. Proponents of the epistemic approach propose experiments that could show the difference between the predictions of quantum theory and the predictions made by any epistemic approach. Thus, all experiments carried out within the framework of the epistemic approach could be consistent to some extent with standard quantum theory. In this regard, it is impossible to interpret all quantum states as epistemic, since there are more quantum states, and epistemic theories cover only part of the quantum theory, because they give results different from quantum ones.

Do these results rule out the idea that a quantum state indicates characteristics of our minds? Yes and no. Arguments against the epistemic approach are mathematical theorems proven from a special structure used for physical theories. Developed by Spekkens as a way to explain the epistemic approach, this framework contains several fundamental assumptions. One of them is that the world is always in an objective physical state, independent of our knowledge about it, which may or may not coincide with the quantum state. Another is that physical theories make predictions that can be represented using standard probability theory. These assumptions are consistent, but that does not mean they are true. The results show that in such a system there cannot be results that are epistemic in the same sense as Spekkens's "toy theory" as long as it is consistent with quantum theory.

Whether this can be put to rest depends on your view of the system. Here opinions differ.

For example, Ouee Maroney, a physicist and philosopher at the University of Oxford and one of the authors of a 2014 paper published in Physical Review Letters, said in an email that "the most plausible psi-epistemic models" (i.e. those that can be fitted to a system Speckens) are excluded. Also, Matt Leifer, a physicist at the University of Champagne who has written many papers on the epistemic approach to quantum states, said that the question was closed back in 2012 - if, of course, you agree to accept the independence of the initial states (which Leifer is inclined to do).

Speckens is more vigilant. He agrees that these results severely limit the application of the epistemic approach to quantum states. But he emphasizes that these results were obtained within his system, and as the creator of the system, he points out its limitations, such as assumptions about probability. Thus, an epistemic approach to quantum states remains appropriate, but if this is the case, then we need to reconsider the basic assumptions of physical theories that many physicists accept without question.

Nevertheless, it is clear that significant progress has been made in the fundamental questions of quantum theory. Many physicists tend to call the question of the meaning of the quantum state merely interpretive or, worse, philosophical, but only until they have to develop a new particle accelerator or improve the laser. By calling a problem “philosophical,” we seem to be taking it beyond the boundaries of mathematics and experimental physics.

But work on the epistemic approach shows that this is not true. Spekkens and his colleagues took the interpretation of quantum states and turned it into a precise hypothesis, which was then filled with mathematical and experimental results. This does not mean that the epistemic approach itself (without mathematics and experiments) is dead, it means that its defenders need to put forward new hypotheses. And this is undeniable progress - both for scientists and philosophers.

James Owen Weatherall is Professor of Logic and Philosophy of Science at the University of Irvine, California. His latest book, The Strange Physics of Empty Space, examines the history of the study of the structure of empty space in physics from the 17th century to the present day.