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Symmetry, Anti-Symmetry, and Symmetry Breaking IV


We asked: what other interpretations of physical phenomena are possible with the flat vectors and their operations?

Recall the important formula of calculating the elementary work that the force z performs on a body along the displacement w, given by:

z·w = |z|.|w| waistband The = aa + bB,

Where The is the smallest of the two angles between z = (The, B) and w = (THE, B). We call this calculation “scalar product of z per w”.

We have found that there is yet another permissible multiplication for complex plane vectors of important physical significance: it is the so-called vector product known also from our reader whose length is the area of ​​the parallelogram of sides |z| and |w| given by |z ´ w | = |z|.|w| sen The, Where z ´ w. It is interesting to remember that z ´ w It is a vector as well, but a vector in space outside the plane of complex vectors. This invention is not arbitrary, it has an important physical motivation. This product was invented as a vector perpendicular to the z and w, having as sense that given by the Right Hand Rule, or Corkscrew Rule, and as length the area of ​​the parallelogram generated by z and w.

But how to calculate it? Let us take the opportunity to note that physicists have given a generalized interpretation to the vector product of two vectors in space and not just on the complex plane. Two vectors z = (The, B, ç) and w = (THE, B, Ç) in space produce a torque, or a wide variety of other physical interpretations. Surprisingly, the mathematical operation that provides the product is one and is performed by a procedure of combining the coordinates of the two vectors. For mathematics, this combination has a motivation in the beauty of symmetry.

To understand the genesis of the vector product, we remember that a physicists need to have a product that behaves as follows when applied to unitary basic vectors. i, j and k, which generate the three spatial axes, respectively, axis xshaft y and shaft z:

i ´ j = k, j ´ i = - k, j ´ k = i, k ´ j = - i, i ´ k = - i, k ´ i = j

(Right Hand Rule or Corkscrew Rule)

i ´ i = 0, j ´ j = 0, k ´ k = 0.

These specifications meet the calculation needs, for example, of torque. For the mathematician, however, one need is to satisfy the curiosity to see what happens if multiplication is distributive in relation to addition. Then uniting the wishes of physicists and mathematicians, we obtain the vector product. So to get the vector product of z = (The, B, ç) and w = (THE, B, Ç) proceed mathematically as follows:

z ´ w =

(Thei + Bj + ck) ´ (THEi + Bj + Çk) = Thei ´ THEi + Thei ´Bj + Thei ´ Çk + Bj ´ THEi + Bj ´ Bj + Bj ´ Çk + çk ´ THEi + çk ´ Bj + çk ´ Çk = aa i´i + ab i´j + B.C i´k + BA j´i + bB j´j + bC j´ k + ca k´i + cb k´j + cc k´k = ab i´j + BA j´i + B.C i´k + ca k´i + bC j´k + cb k´j = ab i´j BA i´j + B.C i´k here i´k + bC j´k cB j´k = (ab BA) i´j + (B.C here) i´k + (bC cB) j´k = (ab BA) k + (B.C here) (─ j) + (bC cB) i = (bC cB) i ─ (B.C here) j + (ab BA) k.

Let's look at the "anti-symmetrical" combination of the coordinates. In the coefficient of i the coefficients of j it's from k antisymmetrically combined in the form bC cB. Similarly, the other two coefficients are obtained by the same anti-symmetrical philosophy. There is a minus sign in obtaining the coefficient of j That seems a bit mysterious. The reason for this signal is the need to satisfy the desire of physicists and the desire of mathematicians, as we have already noted, but it seemed very strange until the moment mathematicians discovered that it exists naturally in higher dimensions.

What happened was that by investigating the determinants of matrices of order greater than two, it was observed that it is possible to develop a naturally symmetrical theory that, in dimension three, appears this way, with this minus sign in this position. The symmetry of the determinant theory lies precisely in the antisymmetric combination bC cB, B.C here and ab BA of the coordinates of the two vectors. The term ─ (B.C here) is naturally explained by the structure of the theory of determinants in dimensions larger than two.

Taking stock, we ask how can one explain that the wishes of physicists do not conflict with the wishes of mathematicians? On the contrary, the union of the two kinds of desires seems to have a powerful force in unraveling Nature's behavior. How far can Homo sapiens sapiens go through this marriage?

To try to understand this story a little more, let's take a closer look at the geometric meaning, since we cannot do physical experiments here on the vector product. The physicists wanted that i ´ j = kbut realized that the product vector k has the length of the area of ​​the rectangle formed by i and j. Surprisingly, this pattern is generalized, that is, the vector product z ´ w = (bC cB) i ─ (B.C here) j + (ab BA) k is a vector whose length is the area of ​​the parallelogram formed by z and w in space. This is a remarkable fact: an area that measures a two-dimensional figure reappears in the third dimension by measuring a length. It is as if Nature were reproducing itself in new dimensions with the same measurements as it used in the lower dimensions. In other words, the product vector z ´ w measures the area formed by your factors z and w. What's more, the product is oriented, that is, it is a vector that points to a well-defined direction chosen between two possible by means of a rotation rule. If we calculate w ´ z the rotation is opposite now being “of w for z”And we get w ´ z = (cB bC) i ─ (hereB.C) j + (BAab) k. So we see that w ´ z = ─ (z ´ w) where we say the vector product is anti-commutative.

The area |z ´ w | = |z|.|w| sen The = |(bC cB) i ─ (B.C here) j + (ab BA) k| of the parallelogram formed by z and w has important physical interpretations. We conclude that while the product scales z . w = |z|.|w| waistband The = aa + bB + cC of two vectors z = (The, B, ç) and w = (THE, B, Ç) measures one type of energy (eg work performed by a force), the vector product also measures another type of energy, eg torque, energy that produces a rotation. Therefore, a vector can represent a nature phenomenon and operations between vectors continue to represent nature phenomena.

It is inevitable, then, to ask: what other operations between vectors represent nature's phenomena?

Which vector algebras are able to unravel the behavior of nature?

How far can Homo sapiens sapiens go in this investigation of nature?

In the next column we will look at how engineers took advantage of the usual multiplication of imaginary numbers.

zw = |z| andiwhat |w| andif = |z| |w| andi(what + f) = (The + Bi) (THE + Bi) = (aa - bB) + (ab + BA) i

when studying electrical circuits.

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