And the dual relation is preserved in more general projective transformations. The notion of parallelism, which is preserved in affine geometry, has no meaning in projective geometry. Thus, by separating symmetry groups from geometries, relationships between symmetries can be established at the group level. Since the group of affine geometry is a subgroup of projective geometry, any notion of an invariant in projective geometry a priori makes sense in affine geometry, which is not true in the opposite direction. If you add the required symmetries, you get a stronger theory, but fewer concepts and theorems (which will be deeper and more general).
Thurston's point of view
Odd functions
ƒ (x) = x 3 is an example of an odd function.
Again let f(x) is a function of a real variable with real values. f is an odd, if in the domain of definition f
− f (x) = f (− x) , (\displaystyle -f(x)=f(-x)\,) f(x) + f(−x) = 0 . (\displaystyle f(x)+f(-x)=0\,.)Geometrically, the graph of an odd function has rotational symmetry about the origin, in the sense that the graph of the function does not change if it is rotated 180 degrees about the origin.
The odd functions are x, x 3 , sin ( x), sinh ( x) and erf ( x).
Integrals
Galois theory
Given a polynomial, it is possible that some roots are related by different algebraic equations. For example, it may turn out that for two roots, say, A and B, A 2 + 5 B 3 = 7 (\displaystyle A^(2)+5B^(3)=7). The central idea of Galois theory is the fact that when the roots are rearranged, they continue to satisfy all these equations. It is important that in doing so we restrict ourselves to algebraic equations whose coefficients are rational numbers. Thus, Galois theory studies symmetries inherited from algebraic equations.
Automorphisms of algebraic objects
In the case when the events represent an interval of real numbers, the symmetry that takes into account permutations of subintervals of equal length corresponds to a continuous uniform distribution.
In other cases, such as "choosing a random integer" or "choosing a random real", there is no symmetry in the probability distribution, allowing for permutations of numbers or intervals of equal length. Other acceptable symmetries do not lead to a particular distribution, or in other words, there is no unique probability distribution that provides maximum symmetry.
There is one type one-dimensional isometry, which can keep the probability distribution unchanged, is a reflection about a point, for example, zero.
A possible symmetry for random values with positive probability is that which applies to logarithms, i.e. when an event and its reciprocal have the same distribution. However, this symmetry does not lead to a definite probability distribution.
For a "random point" in a plane or in space, one can choose a center and consider the symmetry of the probability distribution with respect to a circle or sphere.
The concept of movement
Let us first consider such a concept as movement.
Definition 1
A plane mapping is called a plane motion if the mapping preserves distances.
There are several theorems related to this concept.
Theorem 2
The triangle, when moving, passes into an equal triangle.
Theorem 3
Any figure, when moving, passes into a figure equal to it.
Axial and central symmetry are examples of movement. Let's consider them in more detail.
Axial symmetry
Definition 2
Points $A$ and $A_1$ are said to be symmetric with respect to the line $a$ if this line is perpendicular to the segment $(AA)_1$ and passes through its center (Fig. 1).
Picture 1.
Consider axial symmetry using the problem as an example.
Example 1
Construct a symmetrical triangle for the given triangle with respect to any of its sides.
Decision.
Let us be given a triangle $ABC$. We will construct its symmetry with respect to the side $BC$. The side $BC$ in case of axial symmetry will go into itself (follows from the definition). The point $A$ will go to the point $A_1$ as follows: $(AA)_1\bot BC$, $(AH=HA)_1$. Triangle $ABC$ will turn into triangle $A_1BC$ (Fig. 2).
Figure 2.
Definition 3
A figure is called symmetric with respect to the line $a$ if each symmetric point of this figure is contained on the same figure (Fig. 3).
Figure 3
Figure $3$ shows a rectangle. It has axial symmetry with respect to each of its diameters, as well as with respect to two straight lines that pass through the centers of opposite sides of the given rectangle.
Central symmetry
Definition 4
Points $X$ and $X_1$ are said to be symmetric with respect to the point $O$ if the point $O$ is the center of the segment $(XX)_1$ (Fig. 4).
Figure 4
Let's consider the central symmetry on the example of the problem.
Example 2
Construct a symmetrical triangle for the given triangle at any of its vertices.
Decision.
Let us be given a triangle $ABC$. We will construct its symmetry with respect to the vertex $A$. The vertex $A$ under central symmetry will go into itself (follows from the definition). The point $B$ will go to the point $B_1$ as follows $(BA=AB)_1$, and the point $C$ will go to the point $C_1$ as follows: $(CA=AC)_1$. Triangle $ABC$ goes into triangle $(AB)_1C_1$ (Fig. 5).
Figure 5
Definition 5
A figure is symmetric with respect to the point $O$ if each symmetric point of this figure is contained on the same figure (Fig. 6).
Figure 6
Figure $6$ shows a parallelogram. It has central symmetry about the point of intersection of its diagonals.
Task example.
Example 3
Let us be given a segment $AB$. Construct its symmetry with respect to the line $l$, which does not intersect the given segment, and with respect to the point $C$ lying on the line $l$.
Decision.
Let us schematically depict the condition of the problem.
Figure 7
Let us first depict the axial symmetry with respect to the straight line $l$. Since axial symmetry is a movement, then by Theorem $1$, the segment $AB$ will be mapped onto the segment $A"B"$ equal to it. To construct it, we do the following: through the points $A\ and\ B$, draw the lines $m\ and\ n$, perpendicular to the line $l$. Let $m\cap l=X,\ n\cap l=Y$. Next, draw the segments $A"X=AX$ and $B"Y=BY$.
Figure 8
Let us now depict the central symmetry with respect to the point $C$. Since the central symmetry is a motion, then by Theorem $1$, the segment $AB$ will be mapped onto the segment $A""B""$ equal to it. To construct it, we will do the following: draw the lines $AC\ and\ BC$. Next, draw the segments $A^("")C=AC$ and $B^("")C=BC$.
Figure 9
The concept of matter as the indestructible and uncreatable basis of all that exists was formed back in antiquity. On the other hand, the observation of constant changes in nature led to the idea of the perpetual motion of matter as its most important property. The idea of "preservation" appeared in science as a purely philosophical conjecture about the presence of something stable in an ever-changing world. The unity of change and preservation finds expression in the concept of "symmetry". Symmetry - invariance (immutability) of an object with respect to the transformations imposed on it. Transformations that give a symmetrical object are called symmetrical. The level of symmetry is determined by the number (spectrum) of possible symmetrical transformations. The more homogeneous, more balanced the system, i.e. the more proportionate to its part, the greater the number of possible symmetric transformations for it, i.e. the more symmetrical it is. Therefore, the concept of symmetry is associated with the balance and proportionality of the parts of the system. The symmetry of physical systems manifests itself in the existence of conservation laws. At first, the conservation laws, like the principle of relativity, were established empirically, by generalizing a huge number of experimental facts. Much later came the understanding of the deep relationship between these laws and the symmetry properties of physical systems, which made it possible to comprehend their universality. In this case, symmetry is understood as the invariance of the laws, the quantities included in them, and the properties of natural objects described by them with respect to a certain group of transformations in the transition from one frame of reference to another. For example, in the special theory of relativity, for all inertial frames of reference moving at different speeds, the speed of light in vacuum, electric charge, and the laws of nature are invariant.
The presence of symmetry leads to the fact that for a given system there is a conserved quantity. Thus, if the symmetry properties of a system are known, it is possible to determine the conservation laws for it and vice versa.
The connection between the symmetry of space-time and the fundamental laws of conservation was established at the beginning of the 20th century. E. Noether (1882 - 1935). Space and time are homogeneous and, therefore, symmetrical with respect to arbitrary shifts of the origin. The isotropy of space makes it symmetrical with respect to the rotation of the coordinate axes.
The most important symmetry of nature was revealed in the relativistic theory: all natural phenomena are invariant under shifts, rotations and reflections in a single four-dimensional space-time. These symmetries are inherently "global", covering the entire space-time. The conservation laws due to global symmetry are the most fundamental laws of nature. These include:
law of conservation of momentum, connected with homogeneity of space;
law of conservation of angular momentum, connected with isotropy of space;
law of energy conservation, connected with uniformity of time.
Thus, each transformation of the global space-time symmetry corresponds to the law of conservation of a certain value. These laws are fulfilled for closed systems, the bodies of which interact with each other, and external influences are compensated.
In classical physics, many quantities (such as momentum, energy, and angular momentum) are conserved. Conservation theorems for the corresponding quantities also exist in quantum mechanics. The most beautiful thing about quantum mechanics is that, in a certain sense, conservation theorems can be deduced from something else; in classical mechanics, however, they themselves are practically the starting points for other laws. (It is possible, however, in classical mechanics to act in the same way as in quantum mechanics, but this is possible only at a very high level.) In quantum mechanics, however, conservation laws are very closely related to the principle of superposition of amplitudes and to the symmetry of physical systems with respect to various changes . This is the topic of this lecture. Although we will apply these ideas mainly to the conservation of angular momentum, it is essential here that all theorems on the conservation of any quantities are always connected - in quantum mechanics - with the symmetries of the system.
Let us therefore begin by studying the question of the symmetries of systems. A very simple example is provided by molecular hydrogen ions (however, ammonia molecules would be equally suitable), which have two states each. For the molecular hydrogen ion, we took for one basic state such a state when the electron is located near proton No. 1, and for another basic state, the one in which the electron was located near proton No. 2. These two states (we called them and ) we again show in Fig. 15.1, a. And so, since both nuclei are exactly the same, there is a certain symmetry in this physical system. In other words, if we had to reflect the system in a plane placed in the middle between two protons (meaning, if everything on one side of the plane symmetrically moved to the other side), then the picture presented in Fig. 15.1b. And since the protons are identical, the reflection operation translates into , and into . Let us denote this reflection operation and write
. (15.1)
So ours is an operator, in the sense that it “does something” with the state so that a new state comes out. What is interesting here is that, acting on any state, creates some other state of the system.
Fig. 15.1. If the states and are reflected in the plane , they go over to the states and , respectively.
are the matrix elements that are obtained if and are multiplied on the left by . According to equation (15.1), they are equal
(15.2)
In the same way, you can get and , and . The matrix with respect to the basic system is
We again see that the words operator and matrix in quantum mechanics are practically interchangeable. There are, of course, slight technical differences, as between the words "numeral" and "number", but we are not such pedants as to bother ourselves with this. So we will call either an operator or a matrix, regardless of whether it defines an operation or is actually used to obtain a numerical matrix.
Now we would like to draw your attention to something. Let us assume that the physics of the entire system of the molecular hydrogen ion is itself symmetrical. This might not be - it depends, for example, on what is next to her. But if the system is symmetric, then the following idea must necessarily be true. Suppose that at first, at , the system is in the state , and after a period of time we find that the system is in a more complex position - in some linear combination of both basic states. Remember that in ch. 6 (Issue 8), we used to represent "evolution in time" by multiplying by the operator . This means that the system in a moment (say, for definiteness, in 15 seconds) will be in some other state. For example, this state on can consist of the state and on of the state , and we would write
Now we ask: what happens if we first start the system in a symmetrical state and wait 15 seconds under the same conditions? It is clear that if the world is symmetric (which is what we assume), then we will definitely get a state symmetric with (15.4):
The same ideas are schematically depicted in Fig. 15.2. So, if the physics of the system is symmetric with respect to some plane and we have calculated the behavior of one state or another, then we also know the behavior of the state that would result after the reflection of the initial state in the plane of symmetry.
Fig. 15.2. If in a symmetric system the pure state develops in time as shown in part (a), then the pure state will develop in time as shown in part (b).
The same can be said a little more generally, that is, a little more abstractly. Let - any of the many operations that you can perform on the system without changing the physics. For example, for we can take the operation of reflection in a plane located in the middle between two atoms of the hydrogen molecule. Or in a system with two electrons one could mean the operation of exchanging two electrons. The third possibility would be, in a spherically symmetric system, the operation of rotating the entire system by a finite angle about some axis; this does not change the physics. Of course, in each individual case, we would designate in our own way. In particular, through we will usually denote the operation "rotate the system around the axis by an angle". By we simply mean one of the named operators or any other that leaves the whole physical situation unchanged. We will call the operator the symmetry operator for the system.
Here are some more examples of symmetry operators. If we have an atom, and there is no external magnetic or external electric field, then after rotating the coordinate system around any axis, the physical system remains the same. Again, the ammonia molecule is symmetrical with respect to reflection in a plane parallel to the one in which the three hydrogen atoms lie (as long as there is no electric field). If there is an electric field, then the field would also have to be reversed during reflection, and this changes the entire physical problem. But as long as there is no external field, the molecule is symmetrical.
Now consider the general case. Suppose we started with the state , and after some time or under the influence of other physical conditions, it turned into the state . Let's write
[Look at formula (15.4).] Now imagine that we are operating on the entire system. The state will be transformed into the state, which is also written as . And the state becomes . And now, if physics is relatively symmetric (do not forget about this, if this is by no means a general property of the system), then, after waiting the same time under the same conditions, we should get
[As in (45.5).] But one can write instead of , and instead write , so (15.7) is rewritten in the form, holds for matrices and .]
By the way, since for an infinitesimal time we have , where is the usual Hamiltonian [see. ch. 6 (issue 8)], it is easy to see that when (15.10) is satisfied, then
So (15.11) is a mathematical formulation of the conditions for the symmetry of the physical situation with respect to the operator . It defines symmetry.
Symmetrical (asymmetric) multi-phase electric current system according to GOST R 52002-2003
In which they are equal (not equal) in amplitude and (or) shifted relative to each other in equal (unequal) angles. Notes:
- In a symmetrical multi-phase system of electric currents, the shift of electric currents relative to each other in phase is an angle equal to 2 p / m, where m - number of phases.
- Similarly, symmetrical (asymmetrical) multi-phase systems are defined, etc.
[from clause 162 GOST R 52002-2003]
Symmetrical negative sequence system (currents) according to GOST R 52002-2003
The order of which is reversed to the main one. Notes:
- With the reverse order of the phases, the phase shifts of each of the phases of a symmetrical multi-phase system of electric currents relative to the phase taken as the first one decrease or increase by the same amount equal to 2 p (1-k) / m, where m - number of phases; k = 1, 2, ..., m - phase number.
- Symmetric systems of reverse sequences are defined similarly, and so on.
[from clause 165 GOST R 52002-2003]
Symmetric positive sequence system (currents) according to GOST R 52002-2003
The order of which is taken as the main one. Notes:
- With the main phase order, the phase shifts of each of the phases of a symmetrical multi-phase system of electric currents relative to the phase taken as the first one increase or decrease by the same amount equal to 2 p (1-k) / m, where m - number of phases; k = 1, 2, ..., m - phase number.
- Symmetric positive sequence systems are defined similarly, and so on.
[from clause 164 GOST R 52002-2003]
Symmetrical components (asymmetric -phase system of electric currents) according to GOST R 52002-2003
Symmetric m-phase sequences into which this asymmetric m-phase system of electric currents can be decomposed, namely, sequences with indices n=0, 1, ..., m-1, phase shifts in each of which relative to the first phase are 2 p ( 1-k)n/m, where k = 1, 2, ... , m - phase number. Notes:
- For the designations of phases A, B and C, the values k=1, 2 and 3 correspond, and the names of the sequences as zero, direct and reverse correspond to the values n = 0, 1 and 2.
- Similarly, the symmetrical components of asymmetric m-phase systems are determined, etc.
[from clause 166 GOST R 52002-2003]
