What are symmetries? What does the mathematical concept of a group have to do with them?? What are group views? Why physics pays so much attention to symmetries?

First, let's talk about household, close to every idea of ​​symmetry. As always, in order to better understand what it is about, I will try to reveal the meaning of this word itself. As with many other concepts, the meaning of this term is by no means the only one. Therefore, you need to go through all the meanings (Well, or at least, for their main part). The literal meaning of the word “symmetry” in his, so to speak, Russian version (although this word itself has long been an integral part of the Russian language), this co-dimension. Ie. the original meaning of the word implies a certain connection, compatibility in size, by the size of something with something. Although the first, what usually comes to mind, when we hear this word, this is an idea of ​​something beautiful, perfect. Symmetrical face — this face, in which everything is balanced, right matches left, top and bottom do not create a feeling of contradiction between themselves. A symmetrical geometric figure does not consist of many chaotically connected elements, and vice versa, is a collection of elements, possibly equal, showing order. for instance, circle. All points are located at an equal distance from the center and in this sense are identical, indistinguishable from each other. Or an equilateral triangle. All sides are equal. All angles are equal. You can rotate like this, that when the vertices match, then the triangle is essentially unchanged, remains the same. Two parts of a triangle can be flipped, obtained by dividing it by any height, on each other and they will match, and the new triangle will be indistinguishable from the old.

So what do we need, so that you can say about the subject of conversation, that it has some symmetry?

The first. The subject of conversation about symmetry, generally speaking, should have some complexity, to consist of “details”, be a collection of other items, and not to be something unique. The simple existence of a single element does not allow talking about symmetry. There is nothing to compare it with, proportion. I will note, what here it is already appropriate to use the mathematical concept “lots of”. Many elements.

Second. For the concept of symmetry to arise, it is necessary that the elements of a set be involved in some kind of relationship with each other, so that the concept of a set was added to the concept of an operation  over the elements of this set. Only judgment about the result of some operation allows us to say, are there any elements in this set (figure, volumetric body, a phrase or something else) signs of the presence or absence of symmetry, proportions. Because of, what are we talking about proportionality, it could seem, that such an operation must necessarily be somehow related to the measurement, at least in its rudimentary form of comparison. The comparison is really always done, but, usually, at the very last stage, after operation, which is here. Compared then, what happened before the operation, with that, what happened after applying it. But the operation itself, which this paragraph is about, not necessarily limited to measurement operations. As an example, I can offer the operation of permutation of elements of some set.

Third. Symmetry implies, what the use of a certain operation leaves some property (or multiple properties) multitudes (the subject under discussion as a whole) unchanged. Something is invariant of the operation. Remains an unchanged property of the set as a whole (or those elements, to which the operation was applied). Perhaps, because of, that the first two properties are kind of self-evident for us, available by default, this property prevails in our ideas about the presence or absence of symmetry.

It is accepted to consider, which is the basis for describing symmetries using as accurate, mathematical concepts is the concept of a group. It's right, but not really. First let's say, what group — is a set of elements with an operation defined in it, the application of which leaves the elements of a set in this set itself. This part of the definition of the group corresponds exactly to our ideas about the properties of symmetry. But in the definition of a group there are also some important conditions on the elements of the set and the properties of the operation. We will clarify them later.. These conditions are important for the concept of a group, but often excessive to describe symmetries. Slightly broader is another mathematical concept — group presentation. It provides some additional properties, which we expect to find in the formal description of the intuitive concept of various kinds of symmetries. From a purely formal point of view, it is more convenient to discuss the concept of group representation on the basis of the already described concept, groups. Generally speaking, some of our basic intuitions about symmetry need to be described in a mapping language, and by no means groups and their representations. but, then the idea of ​​symmetries, which formed in physics, is most directly related to the concept of a group. Therefore I, primarily, I will follow this line and below I will try to explain, what is it connected with.

Definition (mathematical) groups. Non-empty set with a given in it binary  (ie. applied to two elements of the set) operation (and the result of the operation is also an element of the same set, and not some other) called a group, if the following conditions are met:

1. The set has neutral element, often called a unit, such, that its participation as one of the operands (elements of the set involved in the operation) anywhere (the result of an operation in the general case may depend on the place of the operand in a binary operation) does not change the second operand. Ie. operation, that includes this element results in the second element involved in the operation. Name “unit” historically related to, that basic ideas about the properties of groups are often derived from the study of a group of numbers with the operation of multiplication. By the way, therefore very often, the group operation itself is called “multiplication”. what, naturally, is nothing more than jargon, with limited scope. These names (“unit”, “multiplication”) very conditional. Numbers are a group and in relation to the addition operation (and in a stricter sense, due to the strict implementation of one more, below condition 2, than with respect to multiplication). In this case, the neutral element is the number zero, not a unit.
2. Each element of the set can be associated with back element, such, that the operation with these two elements (direct and reverse), regardless of their place in the operation, results in a neutral element. These two conditions can be defined as one condition for the existence of the reverse operation. (Addition — subtraction, multiplication — division).
3. Associativity condition. The operation can be applied sequentially. Since the result of a binary operation is again an element of the set, then three or more elements from it can be involved in it. In this case the result of the operation should not depend on, how pairs are selected in this chain (because the operation is basically binary!). It doesn't matter, that first the operation of the first is performed with the second operands, and then the result becomes the first operand in the operation with the third. Or first the second with the third, and then the result becomes the second operand in the operation with the first. But the place of all elements in the chain should not change..

Let's see how it looks on specific, all clear examples, and that in the concept of a group it may be superfluous to describe the symmetry.

Symmetrical face. Here the set consists of two halves of the face, left and right. Each of the halves includes a subset of very different elements. — eyes, cheeks, eyebrows, halves of nose and mouth, forehead chin. You can further refine the description of the face, but that's enough, to see the essence of the idea. And the idea of ​​a symmetrical face is, that each element of the left half of the face corresponds to exactly the same element of the right half (well and vice versa, of course). Moreover, this correspondence is provided by the operation of mirror reflection along the lines, perpendicular to the axis of the face — straight lines, passing through the center of the forehead, tip of nose and tip of chin. Lots of, describing this symmetry in a mathematical sense, consists of at least three elements (an axis of symmetry has been added to the halves of the face, more precisely, points, or very small areas, located on it) and operations of mirroring elements into each other. Left to right and vice versa, and the axis with this display remains unchanged (can also be said, what is displayed in itself).  But there is a question — and does the described set with the operation satisfy all of the above conditions to be called a group in the mathematical sense? Not. And that's why. The reflection operation is actually not binary, but unary. It applies to a single element of the set. Reflection maps any element to another element. Or the same, if the set contains an element neutral with respect to the operation (coincidence of the result with the operand itself is in this case the definition of neutrality).  The presence of a neutral element is also not necessary in this case.. After all, you can truncate the description of a face to two halves, which are reflected into each other. And the idea of ​​symmetry still remains explicit in this description.. To describe this type of symmetry, a mathematical concept is sufficient “display”.

Circle. This example is much richer. It already allows you to see how the concept of a mathematical group appears.. That symmetry, which appeared when discussing facial properties, obvious immediately — it appears at the same moment, how we complement (at least imaginary) circle of any diameter, ie. straight line, passing through the center of the circle. Ie. a circle is symmetrical about any of its diameters. But! If there is only one symmetry line for the face, then for a circle of such lines, there are already infinitely many diameters and it is possible to turn the circle (and any one selected diameter) around the center to any angle. And at the same time on the one hand, left / right symmetry with respect to this diameter is preserved, and, Besides, the circle itself coincides with itself (more precisely, any point, lying on a circle on it and remains). A new operation appears, more precisely, infinitely many operations, their continuous collection,  turns through some angle. In everyday language we will say, that a circle is like a geometric figure, more symmetrical, than a face. Let's try to describe this new wealth in the language of mathematics.

Let's stop at the bends. Turns can be carried out at different angles. This means that this operation has a qualifying parameter — angle of rotation. Further, turns can be done two, one by one and the result will also be a rotation by some angle (as we know, corners add up; this property is essentially a definition as a rotation parameter, and the pivot operation itself). Where are the two turns, there are three and more. And the overall result will always be a turn. Now let's take a step forward in our constructions. Not even forward, but you can say up, build one more floor above our structure. Ground floor — set of circle points and rotation operations, preserving this circle as a whole. The second floor will be a lot of operations — turns. Every turn, corresponding to some parameter value, angle will be considered an element of the new set, many twists and turns. And already in this set we introduce the operation, combining any two consecutive turns. Let's call it “multiplication”. Although they could well have called “addition” (if only on the basis, that the angles of turns in our case, by definition, add up). Take a close look at this new floor. After all, this construction exactly corresponds to the mathematical definition of the group. There are many. Many operations on something else? so what? Operation is. Operation on operations? And what's wrong? Is there a neutral element, “unit”? there is. Zero angle rotation (no turning). A little weird? No stranger than zero for empty set. In fact, this is it.. The same face in profile. Inverse elements? Turns back and forth. Allow positive rotation angles, and negative. Well, there is also associativity in the sequence of turns.. The group of turns turns out to be identical to the group of real numbers with the group operation “addition”. With one significant difference. Not all numbers, and numbers in the interval from 0 to 360 (if the angles are measured in degrees), or from 0 up to 2π (measured in fractions of the length of the radius, ie. in radians).

We saw, as an attempt to describe a certain type of symmetry of a particular figure, circles, led us to the concept of a group. Transformation groups, in this case turns. And you can rotate other figures? Can, of course. And they will all coincide with themselves at the same time? Not, of course. And they, which will match, even if not for every turn, and for some — eg, shapes such as a line segment, equilateral triangle, square, regular pentagon and all other regular ones “squares”.  We call them symmetrical. Ie, turns out, if the figure does not change after rotating by some angle (or corners), then it has a certain symmetry. The turn group helped us see this symmetry. For different shapes. And the group is one (well, or different samples from it, subgroups). This is how our building turns over, second floor (transformation group, turns) becomes the main, foundation. And on top you can attach different specific floors (figures).

What conclusions can be drawn from the above examples?

Regarding the description of symmetries, we can observe, that although the concept of a group does not correspond to the concept of symmetry in full, but transformation groups allow you to describe quite fully (and identify the presence) if not all symmetries without exception, then, probably, very many. At least all those, with which we can associate the invariants available in this group (ie. constructions, left by all transformations from this group unchanged). Different groups — different invariants — different types of symmetries.

With regard to the groups themselves, our focus has been on the special role of groups., elements of which are transformations, moreover, active transformations. Respectively, they reveal symmetries in those objects, on which these transformations act. However, it's clear enough, that the group properties themselves are not strictly tied to the transformations of certain specific objects. The same properties, fully or partially, can have transformation groups, acting on many completely different objects. for instance, on geometric shapes (turns) and numbers (folding group). Etc. This is where the idea of ​​introducing a group comes in. Group, as abstract, perfect concept, can be implemented by a variety of transformation groups. Exactly or with certain, well described deviations. Any such implementation is called a group representation.. About the same, how a set of five elements can be realized with five stones or five fingers.

The most convenient representation of the representation of transformation groups is their representation by matrices. The reason, that matrix groups are central in group theory is sufficiently simple and at the same time fundamental. It also makes the use of group theory in physics inevitable and all-encompassing., penetrating into different corners of physics. And mathematicians too. And together with groups, the concept of symmetries is naturally introduced into physics. Truth has to be said, what exactly because, that for physics these two concepts become almost synonymous, the concept of symmetry in physics is somewhat different from that everyday concept, which we discussed above.   What is the reason for doing transformation groups and, in particular, groups of matrices so dedicated to physics?

Several times in my articles I wrote, that any description of the world, using as language mathematics, relies on the measurement procedure. Numbers by themselves are just meaningless symbols. They acquire their values ​​only then, when these characters are associated with the results of counting or comparing something with something, taken as base, scale. And this is the measurement procedure. Score — its simplest form. After all, there is a scale here — need to indicate, what exactly do we think. And the measurement procedure is by no means the only one. A lot of them. And they differ, first of all, their sets of scales, their bases. The measurement procedures themselves, used to describe the world are set, which can be described, specifying the properties of its elements, similarities and differences between them. AND the scope of some procedures can be measured by others. And vice versa too. And so these cross-measurements give rise to the idea of ​​transformations and are described by it. Number conversions, measurement results, in other numbers, also measurement results, but obtained in a different way. Scale transformations, coordinate transformations, conversions of all and all kinds of measurement results. In this way, transformations are an integral part of any of our descriptions of the world, claiming to be. The very many transformations, at least, some parts of it, can also be described by some idea. Namely, group idea (different groups, for different subsets of transformations).

I want to note, what these transformations are by nature passive, describing changing the point of view of the selected object, changing the way it is described. And when discussing symmetries, we came to active transformations, manipulating an object or parts of it. But is there a difference? And that, and no.

From the point of view of the application of group theory, from the point of view of an abstract mathematical idea, there is no difference. Both these transformations are representations, implementations of the same abstract groups, regarded as purely mathematical, ideal concepts. Everything, what can be described with the help of a passive point of view can be fully discovered and in the transition to active transformations. And vice versa, of course.

Physically, there is a difference and is fundamental. In the case of passive transformations, changes take place in the way of describing, roughly speaking in the properties of an observer, not in the described object. But with active transformations, changes, if they are, these are changes in the described object (parts of the world).

For this reason, when applying the concept of symmetry in physics, two ideas about possible symmetries arise.

One of them, based on active transformations, has the same genesis, like our everyday idea and, therefore it is quite easy to perceive. We have something. Turning, reflect, move — matches (remains invariant under transformations) — means symmetrical. The more invariants the group has, the more symmetries. I call such symmetry symmetries first kind.

 But when applying passive transformations, changing points of view, a completely different understanding of symmetries comes to the fore. It would seem, here too, the more the group of invariants has, the more symmetrical the object. In this sense, everything is correct. But let's pay attention to this.:  When the transformation group (ie. points of view taken into account) getting wider and wider (the number of its invariants decreases and decreases) then object descriptions, with limited points of view (groups of admissible transformations) which seemed completely different, incompatible, become evidently different descriptions of the same object. Ponder! The object is the same (symmetry!), the whole reason for that, that we considered a certain set of objects to be different, lies in that, that this object is visible to us (we describe) from different points of view. And the unification of these different points of view into a single group is difficult for one reason or another., subjective or objective. This kind of symmetry I call symmetry of the second kind. For physics, their search and the presence of symmetries of the second kind are much more important., than the usual first kind. The presence of symmetries of the first kind breaks symmetries of the second kind. Symmetries of the second kind are found only then, when the symmetries of the first die out.

Examples of.

If we consider only orthogonal rotations on the plane (keeping the length, calculated by the Pythagorean theorem), then only circles of the same radius with a single center at the origin go into each other. Ie. just one single circle. Add offsets by one coordinate — all circles with centers on this coordinate axis become images of one single. Add shifts in two coordinates — on a plane, all circles of the same radius are images of one. Add general scaling, simultaneous change in the same proportion of scales in both coordinates (ie. change in length) — in general, any circles turn out to be images of one single. And if we resolve different scales in two coordinates, then we can realize, what's the difference between all kinds of ellipses (including circles) occurs due to, or can be compensated for by just a change in point of view, measurement procedures.

Here's an example from the cutting edge of science. Quite a long time ago in physics the understanding of, that elementary particles are representations of the Poincaré group. Let's ask a question, how wide is this group, does it include all admissible transformations of measurement procedures without exception?, acceptable point of view changes? Clear, what no. At the very least, we think of a transition to transformation from much wider groups, included one into the other, up to the most general group of locally non-singular transformations, preserving only the number of dimensions of space-time. However, such an expansion of the group must be analyzed from the point of view of not only the admissibility of a conceivable, but also from the point of view of our real possibilities it (expansion) to realize. On this path you can get (understand) reasons for combining elementary particles into families, as well as the reasons for the difference in the characteristics of individual particles, united in these families. Then, what is now called in physics broken internal symmetries.

And now about the reason for finding matrix groups at the center of group theory. She is transparent — passive transformations locally generate transition matrices from one coordinate to another (from one measurement procedure to another, from one description of the world to another). In the case of a single scale in the measurement procedure, the matrix is ​​reduced to a single number, the ratio of the scale of different procedures. If there are two scales, such numbers become four and they are combined into a table, square matrix 2x2. Square because, that the number of scales in each measurement procedure should be necessary and sufficient for a complete description of the measured object, which means the same. More scales are needed to describe the object — more numbers in the column and row of the matrix. 3x3, 4x4, etc.. Respectively, numerically, transformation groups are matrix groups. This is the most important implementation of any group., allowing you to study its properties directly.

In mathematics and physics, in addition to the symmetry concepts discussed above, there is one more, seemingly formal, but in fact, also a very important concept of symmetry (and antisymmetry) mathematical and physical objects depicted by them. It goes back to the concept of immutability (or the presence of changes of a given type) any composite object when rearranging its parts. A simple example — object is a set of two (or more) exactly the same items. How not to swap these items, the object itself (whole set, as a whole) does not change.

Geometric objects, except for the simplest, scalar, are this kind of sets, sets of measurement results. For, to distinguish them from each other, each component is assigned indices. At the same time, the object itself is marked with a letter, label. And to this letter on the right, at the top and (or) at the bottom add an index, which can have a scale number value, with which the measurement of this particular property is associated, this object component. There are two kinds of measurements. Direct, telling in what relation a given property is (component) object to the specified scale, roughly speaking, how many times the scale fits in this object. In this case, the index, scale number, put on top and the component acquires the dimension of the given scale. for instance, meter, centimeter, second. This index is called contravariant.. There are also conjugate, specific measurements, talking about, how much of a given component, of this property of the object falls on the corresponding unit scale. Dimension, appropriately, 1 divided by meter, per centimeter or second. Quite understandable, that direct and conjugate measurements are conjugate by multiplication. for instance, if we are talking about a single scale, then the simplest objects will have one component each and the product of conjugate components of the same object, measured differently, will give one (dimensionless! one subject, no matter what) at any choice of scale. To distinguish between direct and conjugate measurements, the index of the latter is always written below and is called covariant.

If a geometric object has two or more indices of the same sort (contravariant or covariant, but not mixed), then it becomes possible to construct two new geometric objects from the components of this object. These two objects are called the symmetric and antisymmetric parts of the original object.. for instance, take an object Bⁱ_jk. Two objects can be formed from its components,  Sⁱ _jk = 1/2 ( Bⁱ_jk + Bⁱ_kj) and Aⁱ_jk = 1/2 ( Bⁱ_jk – Bⁱ_kj),  so the original object is their sum:   Bⁱ_jk = Sⁱ_jk + Aⁱ_jk. All these formulas must be understood as follows, what if instead of indexes j and k put their specific values, then for all such values ​​the written out equalities will hold. for instance, Sⁱ₁₂ = 1/2 ( Bⁱ₁₂ + Bⁱ₂₁)  for any value i. ObjectsSⁱ_jk andAⁱ_jk have special names — symmetrical part   and antisymmetric part object Bⁱ_jk. These names are equivalent to the relationsSⁱ_jk=Sⁱ _kj      andAⁱ_jk= –Aⁱ_kj    , which are obviously fulfilled for them due to their structure. The operations of selecting symmetric and antisymmetric parts are usually called symmetrization and antisymmetrization. Since these operations are invariant with respect to coordinate transformations (choice of measurement procedure), ie. relationships are preserved in all coordinate systems, if they are correct in any one, then the equality to zero of any of the parts is an absolute fact, independent of the choice of the coordinate system. Ie. if any object is symmetrical (the antisymmetric part is zero), then this is true for any of its descriptions. The same goes for antisymmetry. As a matter of fact, these two parts do not depend on each other. Symmetrization (selection of a symmetrical part) geometric objects can be carried out only by an even number of indices of the same type. But antisymmetrization can be performed using an arbitrary number of such indices (if any,of course), adding to the formula defining this operation the terms with cyclically permuted indices and a plus sign for an even total number of permutations and a minus sign for an odd. Besides, instead of a multiplier 1/2 the number of possible unique permutations should be used as a weighting factor, ie. you need to divide the resulting sum by the factorial of the number of indices of the same type participating in the operation. For example, there are three contravariant (or covariant) index. Then the sum will include terms with indices of the form: +ijk -ikj +kij -kji +jki -jik, Total 6. And the weighting factor will be 1/3!=1/6. This is not an accidental difference between these two operations.. Somewhat, antisymmetrization is a more fundamental operation, it even deals with a special branch of mathematics, theory of forms. There are very important facts in geometry proper, concerning completely antisymmetric objects. In particular, antisymmetrization by the number of indices, exceeding the dimension of space automatically gives zero. And with equality — or volume (for contravariant indices), or its conjugate quantity, bulk density.   Symmetrization is also a very important operation., but it is in a way more private, has a slightly more specific, not universal meaning. for instance, when metric spaces. Although these remarks are quite relative. After all, one of the directions of classification of structures that are most important for geometry, affine connection and curvature tensor is based precisely on the separation of their symmetrized and antisymmetrized parts.

These seemingly purely formal operations with indices actually allow us to describe some very fundamental properties of real world objects.. Using these concepts, it becomes possible to classify, list these properties.

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