As mentioned above, external and internal symmetries are usually distinguished. Internal symmetries are geometric and gauge symmetries of matter itself, reflecting the invariance (independence) of the properties of elementary particles and their interactions with respect to certain transformations. Most of them clearly manifest themselves only in the microcosm, being present at the macro- and mega-levels in a hidden form. External symmetries are symmetries of the space-time continuum, equally clearly manifested at all levels of organization of matter.
The following are distinguished: space-time symmetries :
1. Homogeneity of space . This is the shear symmetry of space. It lies in equivalence, equality of all points in space, that is absence of any selected points in space . Parallel transfer (shift) of the system as a whole in space does not lead to a change in its properties, that is physical laws are invariant with respect to shifts in space .
2. Isotropy of space . This is the rotational symmetry of space. It lies in the equality of all directions in space, that is, in absence of designated directions in space . Rotation of the system as a whole in space does not lead to a change in its properties, that is physical laws are invariant with respect to rotations in space.
3. Uniformity of time . Shift symmetry of time reflects the equality of all points in time, that is lack of dedicated time reference points . Transfer of the system as a whole in time does not lead to a change in its properties, that is physical laws do not change over time .
Concerning time isotropy , then the question of the presence of this symmetry remained open for a long time and in many respects remains debatable to this day. Thus, in classical mechanics, time is symmetrical: ideal mechanical processes are completely reversible, and a “turn in time” does not lead to a change in the laws of mechanics. In general relativity, where time, along with space, is considered as one of the geometric coordinates, the equivalence of its forward and reverse flow is also postulated. The vast majority of elementary processes occurring as a result of strong, electromagnetic and weak interactions are also symmetric with respect to this transformation (with the exception of the decays of K0L mesons). But at the same time, the development of thermodynamics (see topic 2.5) showed that in macroscopic processes associated with the transformation of energy, its irreversible dissipation occurs. Thus, all real processes occurring at the level of macro- and megascopic material systems are not invariant with respect to the direction of time. Changing it to the opposite would lead to a change in the laws of thermodynamics: the irreversible dissipation of energy would be replaced by its spontaneous concentration. Therefore, for these processes the time anisotropic , does not have rotational symmetry.
Relationship between conservation laws and symmetry (Noether's theorem)
The development of mathematical methods for describing symmetry, in particular the analytical mechanics of Lagrange and Hamilton, showed that both Newton's laws of classical mechanics and Maxwell's equations of electrodynamics can be derived mathematically from symmetry considerations. The methods of analytical mechanics can be extended to quantum mechanics, where classical theories lose their applicability.
The most important result in this area of theoretical physics is associated with the name of the outstanding female mathematician Amalia (Emmy) Noether (1882–1935). In 1918, Noether proved a theorem, later named after her, from which it follows that if a certain system is invariant (unchangeable) under some transformation, then there is a certain conserved quantity for it. In other words, the existence of any particular symmetry leads to a corresponding conservation law.
This theorem is valid for any symmetries - in space-time, degrees of freedom of elementary particles and physical fields - that is, it carries universal character . Noether's theorem became the most important tool of theoretical physics, establishing a special interdisciplinary role of symmetry principles in the construction of physical theory .
Continuous symmetries lead to the existence of conservation laws that manifest themselves at all levels of the organization of matter. Thus, according to Noether’s theorem, from the homogeneity (shift symmetry) of space it follows law of conservation of momentum (amount of motion), from the isotropy (rotational symmetry) of space – law of conservation of angular momentum (angular momentum), from the homogeneity of time it follows law of energy conservation . From the gauge symmetry of the dynamics of charged particles in electromagnetic fields it follows law of conservation of electric charge.
As for discrete symmetries, in classical mechanics they do not lead to any conservation laws. However, in quantum mechanics, in which the state of the system is described by a wave function, or for wave fields (for example, the electromagnetic field), where the principle of superposition is valid, the existence of discrete symmetries also implies conservation laws for some specific quantities that have no analogues in classical mechanics. Thus, mirror symmetry, or spatial inversion ( R), leads to the law of conservation of spatial parity; symmetry of replacing all particles with antiparticles, or charge conjugation ( WITH) – to the law of conservation of charge parity, etc.
Noether's theorem provides the simplest and most universal method for obtaining conservation laws. Noether's theorem is especially important in quantum field theory, where conservation laws resulting from the existence of a certain symmetry group are often the main source of information about the properties of the objects being studied.
Amalia (Emmy) Noether, Queen Without a Crown
According to the most prominent living mathematicians, Emmy Noether was the greatest creative mathematical genius to emerge from the world since higher education was opened to women.
Albert Einstein
Einstein was right and Emmy Noether (1882–1935) , with whom he never had the chance to work together at the Institute for Advanced Study at Princeton (although she deserved it more than anyone), was an amazing mathematician - perhaps the greatest female mathematician of all time. And Einstein was not the only one to hold this point of view: Norbert Wiener placed Noether on a par with the winner of two Nobel prizes, Marie Curie, who was also an excellent mathematician.
Also, Emmy Noether became the object of a number of bad jokes - let us recall at least the immortal phrase of the intemperate Edmund Landau: “I can believe in her mathematical genius, but I cannot swear that this is a woman.” Emmy was indeed distinguished by her masculine appearance, and besides this, she did not think at all about how she looked, especially during classes or scientific debates.
According to eyewitnesses, she forgot to style her hair, clean her dress, chew food thoroughly, and was distinguished by many other traits that made her not very feminine in the eyes of her decent fellow Germans. Emmy also suffered from severe myopia, which is why she wore ugly glasses with thick lenses and looked like an owl. Here we should also add the habit of wearing (for reasons of convenience) a man’s hat and a leather suitcase stuffed with papers, like an insurance agent’s. Hermann Weil himself, Emmy’s student and admirer of her mathematical talent, quite balancedly expressed his general opinion about her mentor with the words: “The Graces did not stand at her cradle.”
Portrait Emmy Noether in youth.
Transformation into a beautiful swan
Emmy Noether was born into a society where women, one might say, were shackled hand and foot. At that time, Germany was ruled by the all-powerful Kaiser Wilhelm II, a lover of ceremonies and receptions. He came to the city, decorously got off the train, and then the local mayor gave a speech. Iron Chancellor Bismarck did all the dirty work. He was the true head of state and society, the inspirer of its conservative structure, which prevented the education of women (universal education was considered a sign of hated socialism). The model of a woman was the Kaiser's wife, Empress Augusta Victoria. Her life credo was the four Ks: Kaiser, Kinder(children), Kirche(church), K"uche(kitchen) - an expanded version of the three Ks from the folk trilogy " Kinder, Kirche, K"uche" In such an environment, women were assigned a clearly defined role: on the social ladder they were lower than men and one step above domestic animals. Thus, women could not get an education. Actually, the education of women was not completely prohibited - for the homeland of Goethe and Beethoven this would have been too much. After overcoming many obstacles, women could study, but did not have the right to hold positions. The result was the same, but the game was more subtle. Some teachers, demonstrating particular ideological zeal, refused to start classes if at least one woman was present in the audience. The situation was completely different, for example, in France, where freedom and liberalism reigned.
Emmy was born in the small town of Erlangen, into an upper-middle class family of teachers. Erlangen occupied an unusual place in the history of mathematics - it was the small birthplace of the creator of the so-called synthetic geometry Christian von Staudt (1798–1867) Moreover, it was in Erlangen that the young genius Felix Klein (1849–1925) published his famous Erlangen program, in which he classified geometries from the point of view of group theory.
Emmy's father, Max Noether, taught mathematics at the University of Erlangen. His intellect was inherited by his son Fritz, who devoted his life to applied mathematics, and his daughter Emmy, who resembled the ugly duckling from Andersen’s fairy tale - no one could have imagined what scientific heights she would reach. In childhood and adolescence, Emmy was no different from her peers: she really liked to dance, so she willingly attended all the celebrations. At the same time, the girl did not show much interest in music, which distinguishes her from other mathematicians who often love music and even play different instruments. Emmy professed Judaism - at that time this circumstance was unimportant, but it affected her future fate. With the exception of rare flashes of genius, Emmy's education was no different from that of her peers: she knew how to cook and run a house, showed success in learning French and English, and was predicted to become a language teacher. To everyone's surprise, Emmy chose mathematics.
Facade of Kollegienhaus - one of the oldest buildings of the University of Erlangen.
Endless race
Emmy had everything she needed to devote herself to her chosen occupation: she knew mathematics, her family could provide her with funds for living (albeit very meager), and personal acquaintance with her father’s colleagues allowed her to count on the fact that studying at the university would not become unbearable . To continue her studies, Emmy had to become a student - she was prohibited from attending classes as a full student. She successfully completed her studies and passed the exam that gave her the right to receive a doctorate. Emmy chose algebraic invariants of ternary quadratic forms as her dissertation topic. The teacher of this discipline was Paul Gordan (1837–1912) , whom his contemporaries called the king of the theory of invariants; he was a longtime friend of Noether's father and a supporter of constructive mathematics. In search of algebraic invariants, Gordan turned into a real bulldog: he clung to an invariant and did not unclench his jaws until he singled it out among the intricacy of calculations, which sometimes seemed endless. It is not too difficult to explain what an algebraic invariant and form are, but these concepts are not of interest to modern algebra, so we will not dwell on them in more detail.
In his doctoral dissertation entitled “On the definition of formal systems of ternary biquadratic forms”, Emmy found 331 invariants of ternary biquadratic forms. The work earned her a doctorate and gave her the opportunity to practice mathematical gymnastics. Emmy herself later, in a fit of self-criticism, called this hard work nonsense. She became the second woman Doctor of Science in Germany after Sofia Kovalevskaya.
Emmy received a teaching position in Erlangen, where she worked for eight long years without receiving any salary. Sometimes she had the honor of replacing her own father - his health had weakened by that time. Paul Gordan retired and was replaced by Ernst Fischer, who had more modern views and got along well with Emmy. It was Fischer who introduced her to the works of Hilbert.
Fortunately, Noether’s insight, intelligence and knowledge were noticed by two luminaries of the University of Göttingen, “the most mathematical university in the world.” These luminaries were Felix Klein and David Gilbert (1862–1943) . The year was 1915, the First World War was in full swing. Both Klein and Gilbert were extremely liberal in matters of women's education (and their participation in research) and were specialists of the highest level. They convinced Emmy to leave Erlangen and move to Göttingen with them to work together. At that time, the revolutionary physical ideas of Albert Einstein were thundering, and Emmy was an expert on algebraic and other invariants, which made up the extremely useful mathematical apparatus of Einstein’s theory (we will return to the conversation about invariants a little later).
All this would be funny if it weren’t so sad - even the support of such authorities did not help Emmy overcome the resistance of the academic council of the University of Gottingen, from whose members one could hear statements in the spirit: “What will our heroic soldiers say when they return to their homeland, and in the classrooms will they have to sit in front of a woman who will address them from the pulpit? Gilbert, who was present at such a conversation, objected indignantly: “I don’t understand how the gender of the candidate prevents her from being elected as a private assistant professor. After all, this is a university, not a men’s bathhouse!”
But Emmy was never elected as a private assistant professor. The Academic Council declared real war on her. The conflict soon ended, the Weimar Republic was proclaimed, and the situation for women improved: they gained the right to vote, Emmy was able to take a professorship (but without a salary), but it was only in 1922, with great effort, that she finally began to receive money for her work. Emmy was annoyed that her time-consuming work as editor of the Annals of Mathematics was not appreciated.
In 1918, Noether's sensational theorem was published. Many called her that way, although Emmy proved many other theorems, including very important ones. Noether would have earned immortality even if she had died the day after the theorem was published in 1918, although she had actually found the proof three years earlier. This theorem does not relate to abstract algebra and is located at the junction between physics and mathematics, more precisely, it belongs to mechanics. Unfortunately, in order to explain it in a language understandable to the reader, even in a simplified form, we cannot do without higher mathematics and physics.
To put it simply, without symbols and equations, Noether’s theorem in its most general formulation states: “If a physical system has continuous symmetry, then it will contain corresponding quantities that retain their values over time.”
The concept of continuous symmetry in higher physics is explained using Lie groups. We will not go into details and say that in physics, symmetry is understood as any change in a physical system with respect to which the physical quantities in the system are invariant. This change, through a mathematically continuous transformation, must affect the coordinates of the system, and the value in question must remain unchanged before and after the transformation.
Where did the term “symmetry” come from? It belongs to a purely physical language and is used because its meaning is similar to the term “symmetry” in mathematics. Imagine the rotations of space forming a group of symmetry. If we apply one of these rotations to the coordinate system, we will get a different coordinate system. The change in coordinates will be described by continuous equations. According to Noether's theorem, if a system is invariant with respect to such continuous symmetry (in this case, rotation), then the law of conservation of one or another physical quantity automatically exists in it. In our case, after carrying out the necessary calculations, we can verify that this value will be the angular momentum.
We will not dwell on this topic and present some types of symmetry, symmetry groups and the corresponding physical quantities that will be conserved.
This theorem received much praise, including from Einstein, who wrote to Hilbert:
« Yesterday I received a very interesting article by Mrs. Noether on the construction of invariants. I am impressed that such things can be viewed from such a general point of view. It would not do any harm to the old guard at Göttingen if they were sent to study with Mrs. Noether. Looks like she knows her craft well».
The praise was deserved: Noether's theorem played a non-trivial role in solving problems in the general theory of relativity. This theorem, according to many experts, is fundamental, and some even put it on a par with the well-known Pythagorean theorem.
Let's move on to the simple and understandable world of experiments described Karl Popper (1902–1994) , and suppose that we have created a new theory that describes a certain physical phenomenon. According to Noether's theorem, if within the framework of our theory there is a certain kind of symmetry (it is quite reasonable to assume such a thing), then a certain quantity that can be measured will be conserved in the system. In this way we can determine whether our theory is correct or not.
THEOREM NOTER
A physical system in mechanics is defined in rather complex terms, including the concept of action, which can be considered as the product of released energy and the time spent absorbing it. The behavior of a physical system in the language of mathematics is described by its Lagrangian L, which is a functional (function of functions) of the form
Where q- position, q- speed (the dot at the top in Newton’s notation denotes the derivative of q), t- time. note that q- position in a general coordinate system, which is not necessarily Cartesian.
Action A in the language of mathematics it is expressed by an integral along the path chosen by the system:
Let us precisely formulate and prove Noether’s theorem.
Let us consider some system described by the Lagrange function
The form of the Lagrange-Euler equations obtained from the variational principle with such a Lagrange function is invariant under transformations of the form, as well as under more general transformations
involving replacement of the independent variable. However, the specific form for the new expression for the action, as a functional of new coordinates depending on the new time, can undergo any changes with such a change.
Noether's theorem is only interested in the case when such changes do not occur.
Using (4), we get:
Let the transformations be such that
those. forming a one-parameter group. Let us consider an infinitesimal transformation corresponding to the parameter.
Actually, the variations of generalized coordinates that occur during the transformation under consideration are the difference between the values of the new coordinates at some moment of the new time and the values of the old coordinates at the corresponding moment of the old time, i.e.
Along with them, it is convenient to introduce variations of the form into consideration
dependences of coordinates on time that are nonzero, even if our transformation affects only time and not coordinates.
For any function the following relation is valid:
Then there is a relationship between the two introduced types of variations, which can be obtained as follows: subtract equation (9) from (8), we obtain:
let's take into account that
then we have:
Variations without asterisks related to the same argument value are commutable with differentiation in time
while for variations with asterisks this is, generally speaking, not true.
The corresponding two types of variations can be introduced for any dynamic variable. For example, for the Lagrange function
where includes differentiation both by explicitly included time and by time included implicitly, through coordinates and velocities.
We now require that the integral of the action does not change under our transformation - this is the exceptional case required by the conditions of the theorem - i.e. that was
Where T"- the same domain of integration as T in the second integral, but expressed in terms of new variables. Then, substituting (11) into (13), we get
We express in (15) through (11) and taking into account the relation, proceeding to integration over t instead of t", we get:
Considering that
We get: (15)
Let's find the differential
Substituting (17) into (16), we get:
Under the sign of the first sum is the Lagrange equation, i.e.
Are common properties of space and time:
1. Space and time are objective and real, i.e. do not depend on the consciousness and will of people.
2. Space and time are universal, general forms of existence of matter. There are no phenomena, events of objects that exist outside of space or outside of time.
Basic properties of space:
1. Homogeneity - all points in space have the same properties, there are no selected points in space, parallel transfer does not change the form of the laws of nature.
2. Isotropy - all directions in space have the same properties, there are no preferred directions, and rotation by any angle keeps the laws of nature unchanged.
3. Continuity - between two different points in space, no matter how close they are, there is always a third.
4. Euclideanity is described by Euclidean geometry. A sign of a Euclidean space is the possibility of constructing Cartesian rectangular coordinates in it. But according to Einstein’s general relativity, in the presence of gravitating masses in space, space is curved and becomes non-Euclidean.
5. Three-dimensionality - each point in space is uniquely determined by a set of three real coordinate numbers. This position follows from the connection between the structure of space and the law of gravity. (P. Ehrenfest in 1917 investigated the question of why we are able to perceive only the space of three dimensions. He proved that the “inverse square law”, according to which point gravitational masses or electric charges act on each other, is due to the three-dimensionality of space. In space n dimensions, point particles would interact according to the inverse power law (n–1). Therefore, for n=3 the inverse square law is valid, since 3–1=2. He showed that, corresponding to the inverse cube law, the planets would move in spirals and would quickly fall to the Sun. In atoms with a number of dimensions greater than three, there would also be no stable orbits, i.e. there would be no chemical processes in life.
Basic properties of time:
1. Homogeneity - any phenomena occurring under the same conditions, but at different points in time, proceed in exactly the same way, according to the same laws.
2. Continuity is when between two moments of time, no matter how close they are located, a third can always be identified.
3. Unidirectionality or irreversibility is a property of time, which can be considered as a consequence of the second law of thermodynamics or the law of increasing entropy. All changes in the world occur from the past to the future.
The indicated properties of space and time are associated with the main laws of physics - the laws of conservation. If the properties of a system do not change due to the transformation of variables, then it corresponds to a certain conservation law. This is one of the essential expressions of symmetry in the world. According to E. Noether's theorem, each symmetry transformation, characterized by one continuously changing parameter, corresponds to a value that is conserved for a system that has this symmetry.
From the symmetry of physical laws regarding:
1) the displacement of a closed system in space (homogeneity of space) follows the law of conservation of momentum;
2) the rotation of a closed system in space (isotropy of space) follows the law of conservation of angular momentum;
3) changes in the origin of time (uniformity of time) follows the law of conservation of energy.
Questions for repetition and self-control
1. What were the ideas about space and time in the pre-Newtonian period?
2. How did I. Newton interpret space and time?
3. What ideas about space and time became decisive in A. Einstein’s theory of relativity?
4. What basic properties of space do you know?
5. What basic properties of time do you know?
6. State E. Noether’s theorem?