“We wonder why a group of talented and dedicated people are ready to devote their lives to pursuing such tiny objects that you can't even see? In fact, in the studies of particle physicists, human curiosity and a desire to know how the world in which we live is manifested "Sean Carroll
If you are still afraid of the phrase quantum mechanics and still don’t know what the standard model is, welcome to cat. In my publication I will try to explain as simply and visually as possible the basics of the quantum world, as well as the physics of elementary particles. We will try to figure out what are the main differences between fermions and bosons, why quarks have such strange names, and finally, why everyone was so eager to find the Higgs Boson.
What are we made of?
Well, we will begin our journey into the microcosm with a simple question: what are the objects around us made of? Our world, like a home, consists of many small bricks, which, when combined in a special way, create something new, not only outward appearance, but also by its properties. In fact, if you look closely at them, you can find that there are not so many different types of blocks, just each time they connect with each other in different ways, forming new forms and phenomena. Each block is an indivisible elementary particle, which will be discussed in my story.
For example, let's take some substance, let it be the second element. periodic system Mendeleev, inert gas, helium... Like other substances in the Universe, helium is composed of molecules, which in turn are formed by bonds between atoms. But in this case, for us, helium is a little special, because it consists of only one atom.
What is an atom made of?
The helium atom, in turn, consists of two neutrons and two protons that make up an atomic nucleus around which two electrons revolve. The most interesting thing is that absolutely indivisible here is only electron.
Interesting moment of the quantum worldHow less the mass of an elementary particle, the more it takes place. It is for this reason that electrons, which are 2000 times lighter than a proton, occupy much more space compared to the nucleus of an atom.
Neutrons and protons belong to a group of so-called hadrons(particles subject to strong interactions), or to be more precise, baryons.
Hadrons can be divided into groups
- Baryons, which are composed of three quarks
- Mesons, which consist of a pair: particle-antiparticle
The neutron, as its name implies, is neutrally charged and can be divided into two down quarks and one up quark. A proton, a positively charged particle, splits into one down quark and two up quarks.
Yes, yes, I'm not kidding, they really are called top and bottom. It would seem that if we discovered the up and down quarks, and even the electron, we can use them to describe the entire Universe. But this statement would be very far from the truth.
The main problem is that the particles must somehow interact with each other. If the world consisted only of this trinity (neutron, proton and electron), then the particles would simply fly through the endless expanses of space and would never gather into larger formations, like hadrons.
Fermions and Bosons
Quite a long time ago, scientists came up with a convenient and laconic form of representation of elementary particles, called the standard model. It turns out that all elementary particles are divided by fermions, of which all matter consists, and bosons that carry different kinds of interactions between fermions.
The difference between these groups is very clear. The fact is that in order to survive according to the laws of the quantum world, fermions need some space, and for bosons it is almost unimportant to have free space.
Fermions
A group of fermions, as already mentioned, creates visible matter around us. Whatever we see and wherever, is created by fermions. Fermions are divided into quarks strongly interacting with each other and trapped inside more complex particles like hadrons, and leptons that freely exist in space independently of their fellows.
Quarks are divided into two groups.
- Upper type. The up-type quarks, with a charge of +2 \ 3, include: up, charmed and true quarks
- Lower type. The down-type quarks, with a charge of -1/3, include: down, strange and adorable quarks
Leptons are also divided into two groups.
- The first group, with a charge of "-1", it includes: an electron, a muon (heavier particle) and a tau particle (the most massive)
- The second group, with a neutral charge, contains: electron neutrino, muonic neutrino and tau neutrino
The question arises whether physicists will not find several more generations of particles that will be even more massive than the previous ones. It is difficult to answer it, but theorists believe that the generations of leptons and quarks are limited to three.
Don't you find any similarities? Both quarks and leptons are divided into two groups, which differ from each other in charge per unit? But more on that later ...
Bosons
Without them, fermions would fly in a continuous stream across the universe. But by exchanging bosons, fermions communicate to each other some kind of interaction. The bosons themselves practically do not interact with each other.
In fact, some bosons do interact with each other, but this will be discussed in more detail in the next articles on the problems of the microworld.
The interactions transmitted by bosons are:
- Electromagnetic, particles are photons. Light is transmitted through these massless particles.
- Strong nuclear, particles are gluons. With their help, quarks from the atomic nucleus do not decay into separate particles.
- Weak nuclear, particles are ± W and Z bosons. With their help, fermions are transferred by mass, energy, and can transform into each other.
- Gravitational , particles - gravitons... Power is extremely weak on the scale of the microcosm. Becomes visible only on supermassive bodies.
Gravitational clause.
The existence of gravitons has not yet been experimentally confirmed. They exist only as a theoretical version. In the standard model, in most cases they are not considered.
That's it, the standard model is assembled.
The problems have just begun
Despite the very nice representation of the particles in the diagram, two questions remain. Where do particles get their mass and what is it Higgs boson, which stands out from the rest of the bosons.
In order to understand the idea of using the Higgs boson, we need to refer to quantum theory fields. Speaking simple language, it can be argued that the whole world, the entire Universe, does not consist of the smallest particles, but of many different fields: gluon, quark, electronic, electromagnetic, etc. All these fields are constantly subject to slight fluctuations. But we perceive the strongest of them as elementary particles. And this thesis is very controversial. From the point of view of corpuscular-wave dualism, one and the same object of the microworld in different situations behaves either as a wave or as an elementary particle, it depends only on how it is more convenient for a physicist observing the process to simulate the situation.
Higgs field
It turns out that there is a so-called Higgs field, the average value of which does not want to go to zero. As a result, this field tries to take some constant non-zero value in the entire Universe. The field makes up an ubiquitous and constant background, as a result of strong fluctuations of which the Higgs Boson appears.And it is thanks to the Higgs field that the particles are endowed with mass.
The mass of an elementary particle depends on how strongly it interacts with the Higgs field constantly flying inside it.
And it is because of the Higgs Boson, or rather because of its field, that the Standard Model has so many similar groups of particles. The Higgs field forced many additional particles to be made, such as neutrinos.
Outcomes
What I've been talking about is the most superficial understanding of the nature of the Standard Model and why we need the Higgs Boson. Some scientists still deeply hope that the particle found in 2012 and similar to the Higgs Boson in the LHC was just a statistical error. After all, the Higgs field breaks many of the beautiful symmetries of nature, making the calculations of physicists more confusing.
Some even believe that the standard model is living out its last years because of its imperfection. But this has not been experimentally proven, and the standard model of elementary particles remains a working model of the genius of human thought.
Standard Model of Fundamental Interactions
in the physics of elementary particles.
Fundamental interactions.
According to modern concepts, all currently known processes are reduced to 4 types of interactions, which are called fundamental (table 1).
Table 1. Fundamental interactions.
|
interactions (field) |
Constant interactions |
interactions |
Characteristic |
Particles - carriers (field quanta) |
|||
|
Name | |||||||
|
Gravitational |
Graviton (?) | ||||||
|
10 -17 ... 10 -18 m |
W +, W - - bosons Z 0 - boson | ||||||
|
Electromagnetic | |||||||
|
10 -14 ... 10 -15 m | |||||||
In quantum physics, each elementary particle is a quantum of a certain field, and vice versa, each field has its own particle-quantum. The energy and momentum of each field is composed of many separate portions - quanta. The simplest and best studied example: an electromagnetic field and its quantum - a photon. The quanta of the field of strong interactions are gluons. The quanta of the field of weak interactions - gauge bosons W ± and Z 0 ... All these particles have been discovered experimentally and their properties have been well studied. The carrier of the gravitational interaction is the graviton: a hypothetical particle that has not yet been experimentally discovered. The quanta-carriers of the fields have an integer spin, i.e. are Bose particles (bosons), which is reflected in the name of some of them.
Modern accelerators. All modern accelerators are colliders (i.e., they use colliding beams).
Table 2. The largest accelerators.
|
Accelerator name |
Accelerated particles |
Maximum energies |
Year of commencement of work |
Accelerating chamber length | |
|
proton-antiproton | |||||
|
(linear) |
electron-positron | ||||
|
electron-positron |
100 + 100 GeV |
Switzerland |
|||
|
electron-proton |
30 GeV + 920 GeV |
Germany |
|||
|
electron-positron | |||||
|
proton - proton |
Switzerland |
||||
|
(linear) |
electron-positron |
500 + 500 GeV |
under construction |
Germany |
|
|
proton - proton |
under construction |
Due to the fact that quarks and gluons interact with each other more strongly than electrons and positrons, and also due to the fact that the energy of proton-proton accelerators is greater, much more events occur in collisions of protons with protons than in collisions of electrons. This has both pros and cons; the downside is that it is more difficult to isolate the desired reactions. Therefore, proton-proton colliders are called discovery machines, and electron-positron colliders are called precision measurement machines.
Standard model.
To date, a quantum description has been developed for three of the four fundamental interactions: strong, electromagnetic and weak, and it has also been shown that weak and electromagnetic interactions actually have a common origin (electroweak interaction). The agreement with experiment is observed up to distances of 10 -18 m, which is the limit for modern experimental technology. Therefore, the theory of three non-gravitational interactions, which includes 12 fundamental particles that participate in them (Table 2), is called standard model particle physics.
Table 3. Fundamental particles.
|
Mass, Mev |
Mass, Mev |
Mass, Mev |
|||
|
Electron | |||||
|
Electronic neutrino |
Muonic neutrino |
Taon neutrino | |||
Symmetry and invariance.
In the case when the state of the system does not change as a result of any transformation, the system is said to have symmetry with respect to this transformation. The concept of symmetry is very important in particle physics, because each type of symmetry has its own conservation law and vice versa: each conservation law of any physical quantity has its own symmetry. The well-known connection of the symmetry of time and space with respect to shifts (homogeneity) and rotations (isotropy) with the laws of conservation of energy, momentum and angular momentum. These laws are universal, i.e. are performed in all types of interactions.
In addition to these well-known types of symmetry, there are so-called "internal symmetries", which in elementary particle physics are called "gauge symmetries (or invariants)". In quantum physics, there is gauge invariance to the phase change of the wave function, since there is no way to determine the absolute value of the phase of this function. In other words, quantum mechanics is invariant with respect to an arbitrary change in the phase of the wave function by a constant value, i.e. replacements ψ on the ψ· exp(i ) provided = const... This is the so-called "global gauge symmetry" with respect to the change in the phase of the wave function by the same amount at once in all space and at all times. This invariance is obvious, since factor exp(i ) upon substitution of the modified wave function into the Schrödinger equation
can be shortened.
If the phase is not equal to a constant, but is an arbitrary function of coordinates and time, then such a transformation is called local. When replacing ψ on the ψ· exp(i (r, t)), the Schrödinger equation will, of course, change, but it can be kept unchanged if we introduce a compensating field into it: a four-dimensional vector ( φ (r, t), A (r, t)), which is a set of scalar and vector potentials of the electromagnetic field, the quanta of which are photons. This is the main idea of the quantum description of electromagnetic interaction (QED).
Higgs boson.
A similar idea is used to construct a theory of all interactions, and the corresponding form of symmetry is called "local gauge invariance". However, this raises a problem. An obligatory requirement for the equations for any physical field is invariance with respect to the Lorentz transformations. And this is done only if the mass of the field quantum is zero. Table 1 shows that the quanta of the electromagnetic, strong and gravitational fields are massless (i.e., they have zero rest mass), but the quanta-carriers of weak interactions have rather large masses. The same problem arises when explaining the values of masses for other elementary particles. We can say that internal symmetries forbid elementary particles to have nonzero rest masses, which, of course, contradicts experimental data. This question - about the explanation of the different values of the masses of elementary particles - remained until recently unresolved in the standard model.
To explain this contradiction in 1964, F. Englert and R. Brout and independently of them P. Higgs almost simultaneously suggested that there is another field, the interaction with which gives particles mass. P. Higgs, in addition, predicted the existence of a quantum in this field - a boson with a spin equal to zero, so the hypothetical quantum of this field was called the Higgs boson. The mass of this particle, according to the then estimates, should be in the range from 60 to 1000 GeV. Until recently, there were no accelerators on which a particle with such a mass could be detected, so the Higgs boson remained the only particle of the Standard Model not yet discovered experimentally.
At a seminar at CERN on July 4, 2012, the discovery of a new particle was announced, the properties of which, as the authors of the discovery cautiously declare, correspond to the expected properties of the theoretically predicted Higgs boson - the elementary boson of the Standard Model of elementary particle physics. This new particle (for which the designation H is accepted) has no electric charge. The boson mass according to the data of one group of experiments is (125.3 ± 0.9) GeV, according to the data of the other group (126.0 ± 0.8) GeV. BosonH is unstable, its lifetime is about 10 -24 s, and it can decay in different ways. Decays into two photons and into two pairs: electron-positron and (or) muon-anti-muon were observed at the LHC:
H→γ+γ,
H→e - + e + + e - + e + ,
H→ e - + e + + μ - + μ + ,
H → μ - + μ + + μ - + μ + .
The last three decays can be briefly written as follows
H→ 4l,
where l- one of the leptons (electron, positron, muon). All of these decays are consistent with the predicted properties of the Higgs boson.
All this makes it possible to assert with a high probability that the Higgs boson is open, and Standard model received a fundamentally important experimental confirmation.
Literature.
Physical encyclopedia, vol. 5 / Ch. ed. A.M. Prokhorov. - M .: Great Russian Encyclopedia, 1998. - p. 596-608.
Kapitonov I.M. An introduction to the physics of the nucleus and particles. - M .: URSS, 2002.
Rubakov V.A. To the discovery at the Large Hadron Collider of a new particle with the properties of the Higgs boson. - UFN, 2012, v. 182, no. 10. - p. 1017-1025.
Rubakov V.A. The long-awaited discovery of the Higgs boson. - Science and Life, 2012, no. 10. - p. 2-17.
Physical encyclopedia, vol. 4 / Ch. ed. A.M. Prokhorov. - M .: Great Russian Encyclopedia, 1994. - p. 505-520.
Physics of the microworld: Little encyclopedia / Ch. ed. D. V. Shirkov. - M .: "Soviet Encyclopedia", 1980.
Green B. Elegant Universe. / Per. from English ed. V.O. Malyshenko. - Ed. 2nd. - M .: Editorial URSS, 2005 .-- 288 p.
Arinstein E.A. Elements of Theoretical Physics: Textbook. - Tyumen, Tyumen State University Publishing House, 2011. - p.103-105.
The Standard Model of particle physics, or simply the Standard Model, is a theoretical framework in physics that most accurately and successfully describes the current position of elementary particles, their meanings and behavior. The Standard Model is not and does not claim to be a "theory of everything" because it does not explain dark matter, dark energy, and does not include gravity. Constant confirmations of the Standard Model, for evil alternative model supersymmetries appear at the Large Hadron Collider. However, not all physicists love the Standard Model and wish it a speedy demise, because this could potentially lead to the development of more general theory everything, the explanation of black holes and dark matter, the unification of gravity, quantum mechanics and general relativity.
If particle physicists get their way, new accelerators may one day scrutinize the most curious subatomic particle in physics, the Higgs boson. Six years after the discovery of this particle at the Large Hadron Collider, physicists are planning new huge machines that will stretch tens of kilometers in Europe, Japan or China.
Scientists recently started talking about a new cosmological model known as Higgsogenesis. A document describing the new model has been published in Physical Review Lettres. The term "Higgsogenesis" refers to the first appearance of Higgs particles in the early universe, just as baryogenesis refers to the appearance of baryons (protons and neutrons) in the early moments after the Big Bang. And although baryogenesis is a fairly well-studied process, higgsogenesis remains purely hypothetical.
Dirac's equation for the electron was a turning point for physics in many ways. In 1928, when Dirac proposed his equation, of all the elementary particles, only electrons, protons and photons were known to science. Maxwell's free equations describe the photons predicted by Einstein in 1905. This early work was gradually developed by Einstein, Bose and others, and in 1927 Jordan and Pauli created a complete mathematical scheme for describing free photons by introducing quantization into Maxwell's free field theory. It also seemed that the proton, like the electron, is fairly well described by the Dirac equation. Dirac's theory fit perfectly with the electromagnetic interaction, which describes how photons act on electrons and protons, thanks to the idea of gauge (introduced by Weil in 1918). The beginning of the formulation of a complete theory of electrons (or protons) interacting with photons (i.e., quantum electrodynamics) was laid by Dirac himself in 1927. Thus, it seemed that there are at hand all more or less basic means for describing all the particles that exist in Nature, as well as the most obvious interactions between them.
The origins of modern particle physics
And yet the physicists of that time were, for the most part, not stupid enough to assume that all this was about to lead them to a "theory of everything." They realized that neither the forces keeping the nucleus from decay (now called strong interaction) nor the mechanism responsible for radioactive decay (now called weak interaction) could be explained without further progress. If the only ones constituent parts atoms, including atomic nuclei, were Dirac protons and electrons, interacting only through an electromagnetic field, then all ordinary nuclei (with the exception of a single proton that makes up the nucleus of a hydrogen atom) had to instantly decay due to electrostatic repulsion due to the predominance of positive charges. There must have been something hitherto unknown, creating strong attraction between the particles inside the nucleus!
In 1932, Chadwick discovered the neutron, and this eventually led to the replacement of the previously popular proton-electron model of the nucleus new model, according to which the nucleus contains protons and neutrons, the strong interaction between which keeps the nucleus from decay. But even this strong interaction was not all that eluded understanding at the time. The radioactivity of uranium, known since the observation of Henri Becquerel in 1896, turned out to be the result of another - weak - interaction, different from both the strong and the electromagnetic interaction. Even the neutron itself, being left to itself, decays in about 15 minutes.
One of the mysterious products of radioactive decay turned out to be the elusive neutrino, a tentative hypothesis of the existence of which was put forward by Pauli in 1929, but which was not directly detected until 1956. It was the study of radioactivity that ultimately brought physicists unexpected popularity and influence towards the end of World War II and after it ...
Much has changed since the initial penetration into the physics of elementary particles in the first third of the 20th century. Now, at the beginning of the 21st century, we have a much more complete picture, known as the standard model of particle physics. This model describes almost all the observed behavior of a wide class of currently known elementary particles. The photon, electron, proton, positron, neutron and neutrino were later joined by various other types of neutrinos, muons, pions (effectively predicted by Yukawa in 1934), kaons, lambda and sigma particles, as well as the omega minus particle, the famous thanks to the history of her predictions. In 1955, the antiproton was experimentally discovered, in 1956 - the antineutron. There are objects of a new type - quarks, gluons and W- and Z-bosons, as well as a whole flock of particles, whose existence is so fleeting that they have never been observed directly, they are referred to as "resonances". Formalism modern theory It also requires the existence of non-stationary objects called "virtual particles", as well as quantities called "spirits", in relation to which the possibility of direct observation is excluded.
There is also a confusing abundance of hypothetical (and not yet discovered) particles predicted by some theoretical models, but not yet fit into the generally accepted scheme of elementary particles - "X-bosons", "axions", "photino", "squarks", "gluino "," Magnetic monopoles "," dilatons ", etc. There is also the ghostly Higgs particle, not discovered at the time of this writing, the existence of which in one form or another (perhaps not as a single particle) is essential for today's physics of elementary particles, in which the Higgs field associated with this particle determines the mass of each elementary particle.
Dirac equation
$$ \ left (i \ hbar c \, \ gamma ^ \ mu \, \ partial_ \ mu - mc ^ 2 \ right) \ psi = 0 $$ It follows from the Dirac equation that the electron has its own mechanical moment of momentum - spin equal to ħ / 2, as well as the intrinsic magnetic moment equal to the Bohr magneton $ e \ hbar / 2Мc $, which were discovered experimentally (1925) (e and m are the charge and mass of an electron, c is the speed of light, $ \ hbar $ - Dirac's constant (reduced Planck's constant)). Using the Dirac equation, a more accurate formula for the energy levels of the hydrogen atom (and hydrogen-like atoms) was obtained, including the fine structure of the levels, and the Zeeman effect was also explained. On the basis of the Dirac equation, formulas were found for the probabilities of scattering of photons by free electrons (Compton effect) and the emission of an electron during its deceleration (bremsstrahlung), which received experimental confirmation. However, a consistent relativistic description of the motion of an electron is given by quantum electrodynamics.
Salient feature Dirac's equations - the presence among its solutions of those that correspond to states with negative values of energy for the free motion of the particle (which corresponds to the negative mass of the particle). This presented a difficulty for the theory, since all the mechanical laws for a particle in such states would be incorrect, but transitions to these states are possible in quantum theory. The actual physical meaning of transitions to levels with negative energy became clear later, when the possibility of interconversion of particles was proved. It followed from the Dirac equation that there should be a new particle (antiparticle with respect to the electron) with the mass of the electron and electric charge opposite sign; such a particle was actually discovered in 1932 by K. Anderson and called a positron. This was a huge success for Dirac's theory of the electron. The transition of an electron from a state with negative energy to a state with positive energy and the reverse transition are interpreted as the process of formation of an electron-positron pair and annihilation of such a pair.
The Dirac equation is also valid for other particles with spin 1/2 (in units of $ \ hbar $) - fermions, for example, muons, neutrinos, while good agreement with experiment is obtained by directly applying the Dirac equation to simple (not composite) particles, as the ones just mentioned. For a proton and a neutron (composite particles consisting of quarks bound by a gluon field, but also having spin 1/2), when applied directly (as to simple particles), it leads to incorrect values of the magnetic moments: the magnetic moment of the “Dirac” proton “must be »Is equal to the nuclear magneton $ e \ hbar / 2Mc $ (M is the proton mass), and the neutron (since it is not charged) - to zero. Experience shows that the magnetic moment of the proton is about 2.8 times greater than the nuclear magneton, and the magnetic moment of the neutron is negative and in absolute value is about 2/3 of the magnetic moment of the proton. The anomalous magnetic moments of these particles are due to their composite nature and strong interactions.
In fact, this equation is applicable to quarks, which are also elementary particles with spin 1/2. The modified Dirac equation can be used to describe protons and neutrons, which are not elementary particles (they are composed of quarks). Another modification of the Dirac equation, the Majorana equation, is used in some extensions of the Standard Model to describe neutrinos.
Zigzag representation of an electron
This article and a number of subsequent articles provide a concise guide to the standard model of modern particle physics.
Let's start in a somewhat non-standard way, reformulating the Dirac equation in the “2-spin representation. The Pauli spinor describing a particle with spin - is a two-component quantity $ \ psi_a $ - (The components are $ \ psi_0 $ - and $ \ psi_1 $.) Taking into account the requirements of the theory of relativity, we also need quantities with shaded indices $ A ", B ", C '$, ..., which appear when complex conjugation is applied to unshaded indices. It turns out that the Dirac spinor $ \ psi $ described above with its four complex components can be represented as a pair of 2-spinors, $ \ alpha_a $ and $ \ beta_ (a ') $, one of which has an unshaded index and the other a shaded one :
$$ \ psi = (\ alpha_a, \ beta_ (a ’)) $$
Then the Dirac equation can be written in the form of an equation connecting these two 2-spinors, with each of them playing in relation to the other the role of a "source" with a "coupling constant" $ 2 ^ (- 1/2) M $, which determines the "force of interaction" between them:
$$ \ nabla ^ (A) _ (B ') \ alpha_a = 2 ^ (- 1/2) M \ beta_ (B'), ~~ \ nabla ^ (B ') _ (A) \ beta_ (B' ) = 2 ^ (- 1/2) M, \ alpha _ (A '), $$
The operators $ \ nabla ^ (A) _ (B ’) $, and $ \ nabla ^ (B) _ (A’) $ are 2-spin translations of the usual gradient operator $ \ nabla $. Should not be attached of great importance all these indices, the factors $ 2 ^ (- 1/2) $ and the exact form of these equations - I am presenting them here only to show how the Dirac equation can be introduced into the general framework of 2-spin analysis and how it can help, once this has been done, in gaining some new insight into the nature of the Dirac equation.
The form of these equations shows that the Dirac electron can be considered to consist of two ingredients - $ \ alpha_A $ and $ \ beta_ (A ') $. They can be given some physical meaning.
You can imagine a picture in which there are two "particles", one of which is described by the value a $ \ alpha_A $ and the other - $ \ beta_ (A ') $, and both of them have no mass and each of them is continuously transformed into another. Let's give these particles the names zig and zag, so $ \ alpha_A $ will describe the zig particle, and $ \ beta_ (A ') $ will describe the zig particle. Being massless, they must travel at the speed of light, but instead, they can be considered to "swing" back and forth, with the forward motion of the zig particle continuously turning into the backward motion of the zag particle and vice versa. In fact, this is a realization of a phenomenon called "zitterbewegung" ("tremor"), which consists in the fact that the instantaneous motion of an electron due to participation in such oscillations always occurs at the speed of light, although the total average motion of an electron is characterized by a speed lower than the speed of light. Each of these ingredients has a spin of $ \ frac (1) (2) \ hbar $ in the direction of motion, corresponding to left rotation for a zig particle and right rotation for a zig particle. (This is due to the fact that the zig particle $ \ alpha_A $ has an unshaded index corresponding to negative helicity, and the zig particle $ \ beta_ (A ') $ has a shaded index corresponding to positive helicity.
Note that although the speed changes all the time, the direction of the spin in the electron rest frame remains constant (Fig. 1). With this interpretation, the zig particle acts as a source for the zig particle, and the zig particle as a source for the zig particle, the bond strength between them is determined by the value of $ M $.
Rice. 1. Zigzag representation of an electron, a) An electron (or another massive particle with spin $ \ frac (1) (2) \ hbar $) can be considered as oscillating in space-time between a massless zig particle with left helicity (helicity $ - \ frac (1) (2) $ is described by the unshaded 2-spinor $ \ alpha_A $ or, in notation more familiar to physicists, by the part projected by the operator - $ \ frac (1) (2) (1- \ gamma_5) $ )) and a massless zag particle with right-handed helicity (the helicity $ + \ frac (1) (2) $ is described by the shaded 2-spinor $ \ beta_ (A ") $ or the part projected by the operator $ \ frac (1) (2 ) (1+ \ gamma_5) $. Each of the particles serves as a source for the other with the rest mass as a coupling constant, b) From the point of view of 3-space, in the electron rest frame, there is a continuous change in speed (always equal to the speed of light) however, the direction of the spin remains constant. (For greater clarity, the picture is not fully depicted in the rest frame of the electron - instead, the electron is slowly moves to the right.)
In fig. Figure 2 shows a diagrammatic representation of the contribution of this process to the complete Feynman propagator. Each individual zigzag process has a finite length, but their entire set, including zigzags of increasing length, contributes to complete process electron propagation in accordance with the $ 2 \ times2 $ matrix shown in Fig. 2. In this case, the "zig" particle becomes a "zig" particle, then the "zig" turns into a "zig", that again into a "zig", and so on in a certain finite segment.
Considering the process as a whole, we will find that the average frequency with which this occurs is related inversely to the coupling parameter - the mass M; in fact it is the "de Broglie frequency" of the electron.
A comment must be made, however, as to how the Feynman diagrams should be interpreted. The displayed process can be legal grounds considered as a spatio-temporal description of what is happening, however, when considering at the quantum level, it must be borne in mind that even in the case of one particle, many such processes occur simultaneously. Each of them should be considered as one of the contributions to some quantum superposition huge number various processes. The real quantum state is determined by the entire superposition as a whole. Each individual Feynman diagram is just one of its components.
A comment must be made, however, as to how the Feynman diagrams should be interpreted. The depicted process can legitimately be considered as a spatio-temporal description of what is happening, however, when considering at the quantum level, it must be borne in mind that even in the case of one particle, many such processes occur simultaneously. Each of them should be considered as one of the contributions to some quantum superposition of a huge number of different processes. The real quantum state is determined by the entire superposition as a whole. Each individual Feynman diagram is just one of its components.
In the same spirit, the above description of the motion of the electron as swinging back and forth, in which the "zig" continuously turns into a "zag" and vice versa, should be understood. Real motion consists of a large (in fact, infinitely large) number of such separate processes, so that the observed motion of an electron can be considered as a result of some of their "averaging" (although, strictly speaking, there is a quantum superposition here). This is the case with just a free electron. In reality, the electron continuously interacts with other particles (for example, with photons - quanta of the electromagnetic field). All such interaction processes should also be included in the general superposition.
With all this in mind, let us ask ourselves the question: how “real” are the zig and zag particles? Or are they just artifacts of some mathematical formalism that I used here when describing the Dirac equation for the electron? The question arises more general: How justified from a physical point of view is to be guided by considerations of elegance of some mathematical description, and then try to pass it off as a description of "reality"? In this case, one should start by posing the question of the importance (as well as grace) of the 2-spinor formalism itself as a mathematical method. I must warn the reader that this formalism is not widely used by physicists who are concerned with the Dirac equation and its applications, in particular, quantum electrodynamics (QED), the most successful branch of quantum field theory.
Fig. 2. Each zigzag process individually contributes, as part of an infinite quantum superposition, to a complete "propagator" like Feynman diagrams. The straight line standard Feynman propagator on the left represents the entire matrix of infinite sums of finite zigzags shown on the right.
The reader who is already somewhat familiar with Feynman diagrams may be confused by the vertical time ordering used here. In quantum field theory, diagrams are usually drawn in which the time variable increases from left to right. This choice, in which time flows from bottom to top, is consistent with that adopted in the theory of relativity, since this direction of time is chosen for most space-time diagrams.
Most physicists use the "Dirac spinors" (or 4-spinors) formalism, in which the spinor indices are omitted. Instead of the 2-spinor $ \ alpha_A $, they use the 4-spinor $ (1- \ gamma_5) \ psi $ (calling it the "left-handed part of the Dirac electron" or
something like that, instead of my zig particle) LINK8. The value $ \ gamma_5 $ is the product
$$ \ gamma_5 = -i \ gamma_0 \ gamma_1 \ gamma_2 \ gamma_3 $$
and has the property of anticommuting with each of the elements of the Clifford algebra, with $ \ gamma_5 ^ 2 = 1 $. Similarly, instead of $ \ beta_ (A ') $, $ (1+ \ gamma_5) \ psi $ (right-handed part) is used.
One might notice that this is just a matter of notation, and it is indeed possible to go from 2-spin formalism to 4-spin formalism and vice versa. The zigzag representation is definitely applicable (although not always applied) to any formalism, but it is closer to the 2-spin formalism than to the 4-spin one. So are the zig and zag particles real? They can be said to be real to the extent that the "Dirac electron" itself is real — as an eminently useful idealized mathematical description of one of the most fundamental elements of the universe. But is this real "reality"?