The essence of the theory of quantum wave fields. quantum field theory

Producer: "Regular and chaotic dynamics"

In his monograph, the well-known theoretical physicist Anthony Zee introduces one of the most important and complex sections of theoretical physics, quantum field theory, into the subject. The book deals with a very wide range of issues: renormalization and gauge invariance, renormalization group and effective action, symmetries and their spontaneous breaking, elementary particle physics and the condensed state of matter. Unlike previously published books on this topic, E. Zee's work focuses on gravity, and also discusses the application of quantum field theory in the modern theory of the condensed state of matter. ISBN:978-5-93972-770-9

Publisher: "Regular and Chaotic Dynamics" (2009)

ISBN: 978-5-93972-770-9

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Author	Description	Year	Price	book type
Anthony Zee		2009	3330	paper book
Zee E.	In his monograph, the well-known theoretical physicist Anthony Zee introduces one of the most important and complex sections of theoretical physics, quantum field theory, into the subject. The book deals with a very wide ... - Regular and Chaotic Dynamics, Institute for Computer Research, (format: 60x84/16, 632 pages) -	2009	1506	paper book
Anthony Zee	In his monograph, the well-known theoretical physicist Anthony Zee introduces one of the most important and complex sections of theoretical physics, quantum field theory, into the subject. The book deals with a very wide ... - Regular and chaotic dynamics, (format: 60x84/16, 632 pages)	2009	1889	paper book

Fundamental discoveries in quantum physics

Isaac Newton, Nikola Tesla, Albert Einstein and many others are the great guides of mankind in the wonderful world of physics, who, like prophets, revealed to mankind the greatest secrets of the universe and the ability to control physical phenomena. Their bright heads cut through the darkness of ignorance of the unreasonable majority and, like a guiding star, showed the way to humanity in the darkness of the night. One of these conductors in the world of physics was Max Planck, the father of quantum physics.

Max Planck is not only the founder of quantum physics, but also the author of the world famous quantum theory. Quantum theory is the most important component of quantum physics. In simple terms, this theory describes the movement, behavior and interaction of microparticles. The founder of quantum physics also brought us many other scientific works that have become the cornerstones of modern physics:

theory of thermal radiation;
special theory of relativity;
research in the field of thermodynamics;
research in the field of optics.

The theory of quantum physics about the behavior and interaction of microparticles became the basis for condensed matter physics, elementary particle physics and high energy physics. Quantum theory explains to us the essence of many phenomena of our world - from the functioning of electronic computers to the structure and behavior of celestial bodies. Max Planck, the creator of this theory, thanks to his discovery allowed us to comprehend the true essence of many things at the level of elementary particles. But the creation of this theory is far from the only merit of the scientist. He was the first to discover the fundamental law of the universe - the law of conservation of energy. The contribution to science of Max Planck is difficult to overestimate. In short, his discoveries are priceless for physics, chemistry, history, methodology and philosophy.

quantum field theory

In a nutshell, quantum field theory is a theory of the description of microparticles, as well as their behavior in space, interaction with each other and mutual transformations. This theory studies the behavior of quantum systems within the so-called degrees of freedom. This beautiful and romantic name says nothing to many of us. For dummies, degrees of freedom are the number of independent coordinates that are needed to indicate the motion of a mechanical system. In simple terms, degrees of freedom are characteristics of motion. Interesting discoveries in the field of interaction of elementary particles were made by Steven Weinberg. He discovered the so-called neutral current - the principle of interaction between quarks and leptons, for which he received the Nobel Prize in 1979.

The Quantum Theory of Max Planck

In the nineties of the eighteenth century, the German physicist Max Planck took up the study of thermal radiation and eventually received a formula for the distribution of energy. The quantum hypothesis, which was born in the course of these studies, marked the beginning of quantum physics, as well as quantum field theory, discovered in the 1900th year. Planck's quantum theory is that during thermal radiation, the energy produced is emitted and absorbed not constantly, but episodically, quantumly. The year 1900, thanks to this discovery made by Max Planck, became the year of the birth of quantum mechanics. It is also worth mentioning Planck's formula. In short, its essence is as follows - it is based on the ratio of body temperature and its radiation.

Quantum-mechanical theory of the structure of the atom

The quantum mechanical theory of the structure of the atom is one of the basic theories of concepts in quantum physics, and indeed in physics in general. This theory allows us to understand the structure of everything material and opens the veil of secrecy over what things actually consist of. And the conclusions based on this theory are very unexpected. Consider the structure of the atom briefly. So what is an atom really made of? An atom consists of a nucleus and a cloud of electrons. The basis of the atom, its nucleus, contains almost the entire mass of the atom itself - more than 99 percent. The nucleus always has a positive charge, and it determines the chemical element of which the atom is a part. The most interesting thing about the nucleus of an atom is that it contains almost the entire mass of the atom, but at the same time it occupies only one ten-thousandth of its volume. What follows from this? And the conclusion is very unexpected. This means that the dense matter in the atom is only one ten-thousandth. And what about everything else? Everything else in the atom is an electron cloud.

The electron cloud is not a permanent and even, in fact, not a material substance. An electron cloud is just the probability of electrons appearing in an atom. That is, the nucleus occupies only one ten thousandth in the atom, and everything else is emptiness. And if we take into account that all the objects around us, from dust particles to celestial bodies, planets and stars, consist of atoms, it turns out that everything material is actually more than 99 percent of emptiness. This theory seems completely unbelievable, and its author, at least, a deluded person, because the things that exist around have a solid consistency, have weight and can be felt. How can it consist of emptiness? Has a mistake crept into this theory of the structure of matter? But there is no error here.

All material things appear dense only due to the interaction between atoms. Things have a solid and dense consistency only due to attraction or repulsion between atoms. This ensures the density and hardness of the crystal lattice of chemicals, of which everything material consists. But, an interesting point, when, for example, the temperature conditions of the environment change, the bonds between atoms, that is, their attraction and repulsion, can weaken, which leads to a weakening of the crystal lattice and even to its destruction. This explains the change in the physical properties of substances when heated. For example, when iron is heated, it becomes liquid and can be shaped into any shape. And when ice melts, the destruction of the crystal lattice leads to a change in the state of matter, and it turns from solid to liquid. These are clear examples of the weakening of bonds between atoms and, as a result, the weakening or destruction of the crystal lattice, and allow the substance to become amorphous. And the reason for such mysterious metamorphoses is precisely that substances consist of dense matter only by one ten-thousandth, and everything else is emptiness.

And substances seem to be solid only because of the strong bonds between atoms, with the weakening of which, the substance changes. Thus, the quantum theory of the structure of the atom allows us to take a completely different look at the world around us.

The founder of the theory of the atom, Niels Bohr, put forward an interesting concept that the electrons in the atom do not radiate energy constantly, but only at the moment of transition between the trajectories of their movement. Bohr's theory helped explain many intra-atomic processes, and also made a breakthrough in the science of chemistry, explaining the boundary of the table created by Mendeleev. According to , the last element that can exist in time and space has the serial number one hundred thirty-seven, and elements starting from one hundred and thirty-eighth cannot exist, since their existence contradicts the theory of relativity. Also, Bohr's theory explained the nature of such a physical phenomenon as atomic spectra.

These are the interaction spectra of free atoms that arise when energy is emitted between them. Such phenomena are typical for gaseous, vaporous substances and substances in the plasma state. Thus, quantum theory made a revolution in the world of physics and allowed scientists to advance not only in the field of this science, but also in the field of many related sciences: chemistry, thermodynamics, optics and philosophy. And also allowed humanity to penetrate the secrets of the nature of things.

There is still a lot to be done by humanity in its consciousness in order to realize the nature of atoms, to understand the principles of their behavior and interaction. Having understood this, we will be able to understand the nature of the world around us, because everything that surrounds us, starting with dust particles and ending with the sun itself, and we ourselves - everything consists of atoms, the nature of which is mysterious and amazing and fraught with a lot of secrets.

QUANTUM FIELD THEORY (QFT), a quantum theory of relativistic systems with an infinite number of degrees of freedom (relativistic fields), which is the theoretical basis for describing microparticles, their interactions and mutual transformations.

quantum fields. The quantum (quantized) field is a synthesis of the concepts of the classical electromagnetic field and the field of probabilities of quantum mechanics. According to modern concepts, the quantum field is the most fundamental and universal form of matter.

The idea of a classical electromagnetic field arose in the Faraday-Maxwell theory of electromagnetism and acquired a modern form in the special theory of relativity, which required the rejection of the ether as a material carrier of electromagnetic processes. In this case, the field is not a form of motion of any medium, but a specific form of matter. Unlike particles, a classical field is continuously created and destroyed (emitted and absorbed by charges), has an infinite number of degrees of freedom and is not localized at certain points in space-time, but can propagate in it, transmitting a signal (interaction) from one particle to another with finite speed not exceeding the speed of light c.

The emergence of ideas about quantization led to a revision of the classical ideas about the continuity of the mechanism of emission and absorption of light and to the conclusion that these processes occur discretely - by emission and absorption of electromagnetic field quanta - photons. The picture that arose contradictory from the point of view of classical physics, when photons were compared with an electromagnetic field and some phenomena could be interpreted only in terms of waves, while others - only with the help of the concept of quanta, was called corpuscular-wave dualism. This contradiction was resolved by the consistent application of the ideas of quantum mechanics to the field. The dynamic variables of the electromagnetic field - the potentials A, φ and the strength of the electric and magnetic fields E, H - have become quantum operators, subject to certain permutation relations and acting on the wave function (amplitude or state vector) of the system. Thus, a new physical object arose - a quantum field that satisfies the equations of classical electrodynamics, but has quantum mechanical operators as its values.

The introduction of the concept of a quantum field is also connected with the wave function of a particle ψ(x, t), which is not an independent physical quantity, but the amplitude of the state of the particle: the probabilities of any physical quantities related to the particle are determined by expressions bilinear in ψ. Thus, in quantum mechanics, a new field is associated with each material particle - the field of probability amplitudes. Generalization to the case of many particles that satisfy the principle of indistinguishability (identity to the principle) means that one field in four-dimensional space-time, which is an operator in quantum mechanics, is sufficient to describe all particles. This is achieved by passing to a new quantum mechanical representation - the representation of occupation numbers (or the second quantization representation).

The operator field introduced in this way is similar to the quantized electromagnetic field and differs from it only in the choice of the representation of the Lorentz group and, possibly, in the method of quantization. Like an electromagnetic field, one such field corresponds to the totality of identical particles of a given sort; for example, one Dirac operator field describes all the electrons (and positrons) of the Universe.

Thus, the fields and particles of classical physics have been replaced by single physical objects - quantum fields in four-dimensional space-time, one for each kind of particles or fields (classical). The elementary act of any interaction was the interaction of several fields at one point in space-time or - in corpuscular language - the local and instantaneous transformation of some particles into others. The classical interaction in the form of forces acting between particles turns out to be a secondary effect resulting from the exchange of quanta of the field that transfers the interaction.

Free fields and wave-particle duality. There are field and corpuscular representations of QFT. In the field approach, the theory of the corresponding classical field is considered, which is then quantized according to the model of electromagnetic field quantization proposed by W. Heisenberg and W. Pauli, and then its corpuscular interpretation is constructed. The initial concept here is the field u a (x) (the index a enumerates the components of the field), defined at each space-time point x = (ct, x) and carrying out some kind of representation of the Lorentz group. Further, the theory is constructed using the Lagrangian formalism: one chooses a local [i.e. i.e. depending only on the field components u a (x) and their first derivatives ∂ μ u a (x) = ∂u a (x) / ∂x μ = u μ a (x) (μ = 0,1,2 3) at one point x] the quadratic Poincaré-invariant Lagrangian L(x) = L(u a , ∂ μ u b) and from the principle of least action δS = δ∫d 4 xL(x) = 0 we obtain the equations of motion. For a quadratic Lagrangian, they are linear - free fields satisfy the superposition principle.

By virtue of Noether's theorem, the invariance of the action S with respect to each one-parameter group implies the conservation (time independence) of one integral function of u a and ∂ μ u b explicitly indicated by the theorem. Since the Poincaré group itself contains 10 parameters, 10 quantities (which are sometimes called fundamental dynamic quantities) are necessarily preserved in QFT: four components of the energy-momentum vector Р μ and six components of the angular momentum - three components of the three-dimensional angular momentum М i = (1/2) ε ijk M jk and three so-called. boost N i = c -1 M 0i (i,j,k= 1,2,3, ε ijk - unit completely antisymmetric tensor; summation is implied over repeated indices). From a mathematical point of view Р μ , M i , N i are generators of the Poincaré group.

Canonical quantization, according to the general principles of quantum mechanics, is that the generalized coordinates (i.e., the set of values of all field components u 1 ,..., u N at all points x of space at some time t) and the generalized momenta π b (x, t) = ∂L/∂u b (x, t) are declared as operators acting on the amplitude of the state (state vector) of the system, and commutation relations are imposed on them:

An alternative variant of quantization, covariant quantization, consists in establishing permutation relations on the field operators themselves at two arbitrary points x and y in a relativistically symmetric form:

where D m is the Pauli - Jordan permutation function that satisfies the Klein - Fock - Gordon equation (hereinafter, the system of units ħ = c = 1, ħ is Planck's constant is used).

In the corpuscular approach, the state vectors of free particles must form an irreducible representation of the Poincaré group, which is fixed by setting the values of the Casimir operators (operators commuting with all ten generators of the group P μ , M i and N i): the mass squared operator m 2 = Ρ μ Ρ μ and the square of the ordinary (three-dimensional) spin, and at zero mass - the helicity operator (the projection of the spin on the direction of motion). The spectrum m 2 is continuous, and the spin spectrum is discrete, it can have integer or half-integer values: 0.1/2.1,... in units of the Bohr magneton. In addition, it is necessary to specify the behavior of the state vector when reflecting an odd number of coordinate axes. If the particle has some other characteristics (electric charge, isospin, etc.), then new quantum numbers correspond to this; let us denote them by the letter τ.

In the representation of occupation numbers, the state of a set of identical particles is fixed by the occupation numbers n p,s,τ of all one-particle states. In turn, the state vector |n p,s,τ) is written as the result of the action on the vacuum state |0) (a state in which there are no particles at all) of the creation operators a + (p, s, τ):

(3)

The creation operators a + and the Hermitian conjugate annihilation operators a - satisfy the permutation relations

(4)

where the plus and minus signs correspond respectively to the Fermi - Dirac and Bose - Einstein quantization, and the occupation numbers are the eigenvalues of the particle number operators n р, s, τ = a + aˉ.

To take into account the local properties of the theory, it is necessary to translate the operators a ± into a coordinate representation and construct a superposition of the creation and annihilation operators. For neutral particles, this can be done directly by defining the local Lorentz-covariant field as

But for charged particles, this approach is unacceptable: the operators a τ + and a τ ˉ in (5) will increase one and decrease the charge on the other, and their linear combination will not have certain properties in this respect. Therefore, to form a local field, it is necessary to pair the creation operators a τ + with the annihilation operators a τ ˉ not of the same particles, but of new particles realizing the same representation of the Poincaré group, i.e., having exactly the same mass and spin, but differing from the initial sign of charge (signs of all charges τ).

It follows from the Pauli theorem that for fields of integer spin, whose field functions uniquely represent the Lorentz groups, when quantized according to Bose-Einstein, the commutators - or - are proportional to the function Dm(x - y) and disappear outside the light cone, while for realizing two-valued representation of fields of half-integer spin, the same is achieved for the anticommutators [u(x), u(y)] + or + with Fermi-Dirac quantization. The relationship between the field functions u or v, v* satisfying linear equations and the creation and annihilation operators a τ ± and a ~ τ ± of free particles in stationary quantum-mechanical states is an exact mathematical description of wave-particle duality. The new particles “born” by the operators a ~ τ ±, without which it was impossible to construct local fields, are called antiparticles in relation to the original ones. The inevitability of the existence of an antiparticle for each charged particle is one of the main conclusions of the quantum theory of free fields.

Field interaction. The solutions of the free field equations are proportional to the operators of creation and annihilation of particles in stationary states, i.e., they can only describe situations where nothing happens to the particles. To also consider cases where some particles affect the motion of others or turn into others, it is necessary to make the equations of motion nonlinear, that is, to include in the Lagrangian, in addition to terms quadratic in fields, also terms with higher degrees. The interaction Lagrangian L int (x) can be any function of the fields and their first derivatives that satisfies a number of conditions: point of space-time x; 2) relativistic invariance, for which L int (x) must be a scalar with respect to Lorentz transformations; 3) invariance under transformations from the groups of internal symmetries, if any, for the considered model. For theories with complex fields, there is also a requirement that the Lagrangian be Hermitian, which ensures that the probabilities of all processes are positive.

In addition, one can require the theory to be invariant under certain discrete transformations, such as spatial inversion P, time reversal T, and charge conjugation C (replacing particles with antiparticles). It is proved (the CPT theorem) that any interaction that satisfies conditions 1-3 must necessarily be invariant with respect to the simultaneous execution of these three discrete transformations.

The variety of interaction Lagrangians satisfying conditions 1-3 is as wide as the variety of Lagrange functions in classical mechanics. However, after quantization in theory, the problem of singularities arises when operators are multiplied at one point, which leads to the so-called problem of ultraviolet divergences (see Divergences in QFT). Their elimination by means of renormalizations in quantum electrodynamics (QED) singled out a class of renormalizable interactions. Condition 4 - the renormalizability condition - turns out to be very restrictive, and its addition to conditions 1-3 allows only interactions with L int , which have the form of polynomials of low degree in the fields under consideration, and fields of any high spins are generally excluded from consideration. Thus, the interaction in a renormalizable QFT does not allow (unlike classical and quantum mechanics) any arbitrary functions: as soon as a specific set of fields is chosen, the arbitrariness in L int is limited to a fixed number of interaction constants (coupling constants).

The complete system of QFT equations with interaction (in the Heisenberg representation) consists of the equations of motion obtained from the full Lagrangian and the canonical permutation relations (1). The exact solution of such a problem can be found only in a small number of cases (for example, for some models in two-dimensional space-time).

The method based on the transition to the representation of interaction, in which the fields u a (x) satisfy the linear equations of motion for free fields, and the entire influence of interaction and self-action is transferred to the temporal evolution of the amplitude of the state Ф, which is now not constant, but changes in accordance with an equation like the Schrödinger equation:

moreover, the interaction Hamiltonian H int (t) in this representation depends on time through the fields u a (x), obeying free equations and relativistic-covariant permutation relations (2); thus, the explicit use of canonical commutators (1) for interacting fields turns out to be unnecessary. For comparison with experience, the problem of particle scattering is solved, in the formulation of which it is assumed that asymptotically, as t → -∞ (+∞), the system was in a stationary state (will come to a stationary state) Ф -∞ (Ф +∞), and Ф ±∞ are such that the particles in them do not interact due to large mutual distances, so that all the mutual influence of particles occurs only at finite times near t = 0 and transforms Ф -∞ into Ф +∞ = SF -∞ . The operator S is called the scattering matrix (or S-matrix); through the squares of its matrix elements

(7)

the probabilities of transitions from a given initial state Ф i to some final state Ф f are expressed, that is, the effective sections of various processes. Thus, the S-matrix makes it possible to find the probabilities of physical processes without delving into the details of the time evolution described by the amplitude Ф(t). Nevertheless, the S-matrix is usually built on the basis of equation (6), which admits a formal solution in a compact form

(8)

using the chronological ordering operator T, which arranges all field operators in descending order of time t \u003d x 0. Expression (8) is a symbolic record of the procedure of successive integration of equation (6) from - ∞ to + ∞ over infinitely small time intervals (t, t + ∆t), and not a usable solution. To calculate the matrix elements (7), it is necessary to represent the scattering matrix in the form of a normal product, rather than a chronological one, in which all creation operators are to the left of the annihilation operators. The transformation of one work into another is the true difficulty of solving the problem.

Perturbation theory. For this reason, to constructively solve the problem, one has to resort to the assumption that the interaction is weak, i.e., that the interaction Lagrangian L int is small. Then it is possible to expand the chronological exponent in expression (8) into a perturbation series, and the matrix elements (7) will be expressed in each order of the perturbation theory through the matrix elements of simple chronological products of the corresponding number of interaction Lagrangians. This task is practically accomplished using the technique of Feynman diagrams and Feynman rules. Moreover, each field u a (x) is characterized by its causal Green's function (propagator, or distribution function) D c aa '(x - y), depicted on the diagrams by a line, and each interaction - by a coupling constant and a matrix factor from the corresponding term in L int , depicted on the diagram as a vertex. The Feynman diagram technique is easy to use and very visual. The diagrams make it possible to present the processes of propagation (lines) and mutual transformations (vertices) of particles - real in initial and final states and virtual in intermediate (on internal lines). Particularly simple expressions are obtained for the matrix elements of any process in the lowest order of perturbation theory, which correspond to the so-called tree diagrams that do not have closed loops - after the transition to the impulse representation, there are no integrations left in them. For the main QED processes, such expressions for matrix elements were obtained at the dawn of the emergence of QFT in the late 1920s and turned out to be in reasonable agreement with experience (the level of correspondence is 10ˉ 2 -10ˉ 3 , i.e., of the order of the fine structure constant α). However, attempts to calculate radiative corrections (related to higher approximations) to these expressions ran into specific difficulties. Such corrections correspond to diagrams with closed loops of lines of virtual particles whose momenta are not fixed by conservation laws, and the total correction is equal to the sum of contributions from all possible momenta. It turned out that in most cases the integrals over the momenta of virtual particles arising from the summation of these contributions diverge in the UV region, that is, the corrections themselves turn out to be not only not small, but infinite. According to the uncertainty relation, large impulses correspond to small distances. Therefore, we can assume that the physical origins of the divergences lie in the idea of the locality of the interaction.

Divergencies and renormalizations. Mathematically, the appearance of divergences is due to the fact that the propagators D c (x) are singular (more precisely, generalized) functions that, in the vicinity of the light cone at x 2 ≈ 0, have singularities like poles and delta functions in x 2 . Therefore, their products arising in matrix elements, which correspond to closed loops in diagrams, are poorly defined from a mathematical point of view. The momentum Fourier transforms of such products may not exist, but can be formally expressed in terms of divergent momentum integrals.

The problem of UV divergences was practically solved (i.e., finite expressions for most important physical quantities were obtained) in the second half of the 1940s on the basis of the idea of renormalizations (renormalizations). The essence of the latter is that the infinite effects of quantum fluctuations corresponding to closed loops of diagrams can be separated into factors that have the nature of corrections to the initial characteristics of the system. As a result, the masses and coupling constants g change due to the interaction, i.e., they are renormalized. In this case, due to the UV divergences, the renormalizing additions turn out to be infinitely large. Renormalization relations relating the initial, so-called bare, masses m 0 and bare charges (coupling constants) g 0 with physical m, g:

(9)

(where Z m , Z g are renormalization factors) turn out to be singular. To avoid singularity, an auxiliary regularization of divergences is introduced. Along with m 0 and g 0 , the arguments of the radiative corrections ∆m, ∆g and renormalization factors Z i , along with m 0 and g 0 , contain singular dependences on the auxiliary regularization parameters. The divergences are eliminated by identifying the renormalized masses and charges (coupling constants) with their physical values.

The class of QFT models for which all UV divergences without exception can be "removed" into the renormalization factors of masses and coupling constants is called the class of renormalizable theories. In these theories, all matrix elements and Green's functions, as a result, are expressed in a non-singular way in terms of physical masses, charges, and kinematic variables. The mathematical basis of this assertion is the Bogolyubov-Parasyuk renormalizability theorem, on the basis of which finite single-valued expressions for matrix elements are quite simply obtained.

In non-renormalizable models, it is not possible to "collect" all the divergences into renormalizations of masses and charges. In such theories, in each new order of the perturbation theory, new divergent structures arise, i.e., they contain an infinite number of parameters. This class of theories includes, for example, the quantum theory of gravity.

Renormalizable QFT models are characterized, as a rule, by dimensionless coupling constants, logarithmically divergent contributions to the renormalization of coupling constants and fermion masses, and quadratically divergent radiative corrections to the masses of scalar particles (if any). For such models, as a result of renormalization, a renormalized perturbation theory is obtained, which serves as the basis for practical calculations.

Transformations (9) connecting the bare and renormalizable interaction constants have a group character and form a continuous group called the renormalization group (renormalization group). When the scale changes, the Green's functions are multiplied by factors that depend nonlinearly on the interaction constants and are calculated by perturbation theory, while the interaction constants themselves change according to (9). Solving the differential equations of the renormalization group corresponding to such a scale transformation, one can obtain closed solutions as functions of the effective interaction constants depending on the scale, which correspond to the summation of an infinite series of perturbation theory. This allows, in particular, to find high-energy and low-energy asymptotics of the Green's functions.

Functional integral. An important role in QFT is played by the complete Green's functions, which include interaction effects. They can be represented by infinite sums of terms corresponding to increasingly complex Feynman diagrams with a fixed number and type of external lines. For such quantities, one can give formal definitions either through the vacuum averages of the chronological products of field operators in the interaction representation and the S-matrix (which is equivalent to the vacuum averages of the Γ-products of the complete, that is, Heisenberg operators), or through the functional derivatives of the generating functional presented in in the form of a functional integral depending on the auxiliary classical sources J a (x) of the fields u a (x). The formalism of generating functionals in QFT is analogous to the corresponding formalism of statistical physics. It allows one to obtain equations in functional derivatives for the complete Green's functions and vertex functions, from which, in turn, one can obtain an infinite chain of integro-differential equations similar to the chain of equations for the correlation function of statistical physics.

The functional integral method, which has received significant development since the 1970s, especially in the theory of non-Abelian gauge fields, is a generalization to QFT of the quantum mechanical method of path integrals. In QFT, such integrals can be considered as formulas for averaging the corresponding classical expressions (for example, the classical Green's function for a particle moving in a given external field) over quantum field fluctuations.

Initially, the idea of transferring the functional integral method to QFT was associated with the hope of obtaining compact closed expressions for the main quantum field quantities suitable for constructive calculations. However, it turned out that due to difficulties of a mathematical nature, a rigorous definition can only be given to integrals of the Gaussian type, which alone can be calculated exactly. Therefore, the functional integral representation was considered for a long time as a compact formalization of the quantum field perturbation theory. Later, a finite-time representation of the functional integral in Euclidean space began to be used to perform computer calculations on a spatial lattice (see Lattice field theories), which makes it possible to obtain results that are not based on perturbation theory. The representation of the functional integral also played an important role in the work on the quantization of Yang-Mills fields and the proof of their renormalizability.

Lit .: Akhiezer A. I., Berestetsky V. B. Quantum electrodynamics. 4th ed. M., 1981; Weisskopf VF How we grew up together with field theory // Uspekhi fizicheskikh nauk. 1982. T. 138. No. 11; Bogolyubov N. N., Shirkov D. V. Introduction to the theory of quantized fields. 4th ed. M., 1984; they are. quantum fields. 2nd ed. M., 1993; Itsikson K., Zuber J.-B. Quantum field theory. M., 1984. T. 1-2; Berestetsky V. B., Lifshits E. M., Pitaevsky L. P. Quantum electrodynamics. 4th ed. M., 2002; General principles of quantum field theory. M., 2006.

D. V. Shirkov, D. I. Kazakov.

Portal for the student. Self-training

The essence of the theory of quantum wave fields. quantum field theory

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Fundamental discoveries in quantum physics

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Portal for the student. Self-training

Other books on similar topics:

See also other dictionaries:

Fundamental discoveries in quantum physics

quantum field theory

The Quantum Theory of Max Planck

Quantum-mechanical theory of the structure of the atom

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