On the relations between ℝ^2, ℂ and Pauli matrices

An article about how each of the Pauli matrices is a simple geometric object – a unit vector of a regular coordinate system. Without stories about infinitesimal rotations and other quantum-mechanical and beautiful things.

Just like in the previous two articles, the structure of the article is as follows: first the recipe, then everything else. At the request of the comments of the previous articles, the font in the formulas was made larger.

I sincerely wish everyone a pleasant reading of interesting texts and daily simplicity of thinking, for example, as in pre-revolutionary textbook Cliffordwhich I recently read. (For those who haven't heard of old Clifford, he is the author of the algebras of the same name, part of which, as it turned out already in the 20th century, is the Pauli matrix algebra).

The textbook is called “Common Sense of the Exact Sciences”. It is interesting that the textbook under this title was published after Clifford's death and was renamed at his request shortly “before”, and the title he initially conceived was “The Elements of Mathematical Knowledge as Presented to Non-Mathematicians”.

The language there is absolutely simple and, if desired, understandable to any high school student. And this despite the fact that at that time this mathematics was one of the peaks of scientific thought, and Clifford himself was already one of the luminaries, not to mention his immortalized name now.

For some reason, 100 years ago they were able to explain complex things simply, but now some texts are scary to open, let alone understand. I hope that the presentation in the text below will be more enjoyable for non-mathematicians interested in high matters than what is in purely scientific articles

Recipe

Let us introduce matrix operators (Pauli matrices)

Decomposition of an arbitrary 2×2 real matrix into two such bases. For such matrices, when transposed, only the sign changes at t. This is the formula for dividing any such matrix into components.

The basis of the space is indeed defined by the Pauli matrices, as promised in the textbooks, but with one not so weak nuance.

z*σ1 and x*σ3 define the usual vectors (x,z), which in matrix form are written as column vectors, and to which everyone is accustomed. They define the usual coordinate system – the x-axis is horizontal, the z-axis is vertical.

zc*σ0 and xc*σ13 describe both ordinary complex numbers (a+i*b) and vectors in a basis reflected relative to the axis (1,1), that is, vectors of the form (zc, xc), which, although also written as column vectors, have the meaning of a second reflected coordinate system, in which the xc axis is vertical and the zc axis is horizontal.

The picture shows a description of such a vector in the form of a 2×2 matrix and two coordinate systems in which it operates.

Mathematics

Let us also define the concept of “operator” and “isomorphism“(this doesn’t sound very good, but it’s still necessary). Wikipedia says something. It is either not for people, or you can think about the dislike of Wikipedia authors for people. Whoever likes it.

In simple terms.

An operator is an object that performs an action on a vector, as a result of which the vector changes. The same operator and vector can have several types of recording (representation). For example, in this article there will be operators and vectors in the form of matrices and the same ones in the form of complex numbers. If we work with matrices, then the operator is called matrix, if with complex numbers, then complex.

Isomorphic mathematical objects are those that change in the same way when the same actions are performed on them. For example, the result of multiplying an operator by a vector in different isomorphic representations should give the same result. In this case, the operator and the vector themselves are the same, only their representations are different.

For those who don't know, get acquainted, these are Pauli matrix operators, isomorphic to the coordinates of the space ℝ^2 (Moreover, some textbooks even use the definition “these matrices can be identified with unit vectors”)

Identity operator and negative operator;

Two reflection operators: relative to the axes (1,1) and (1,0);

Two rotation operators: one isomorphic to the imaginary unit and one opposite in sign.

Please note that, in the Wikipedia article matrix representation of a complex numberthe sign of the matrix isomorphic to 1i is opposite to σ31.

Property σ1²=-1 is the same as 1i. The point is that 1i is isomorphic to σ13= -σ31.

Decomposition of an arbitrary 2×2 matrix operator in the Pauli basis of real matrices

You can check that any operator matrix can be decomposed into the following construction of Pauli matrices with rotation operator isomorphic to the imaginary unit.

You can see that when transposing, only the rotation operator is conjugated, that is, multiplication by the conjugate for complex numbers is the same as multiplication by the transpose operator for vectors in matrix form.

You can also see that in the original “A” the rotation operator has a minus sign.

Correspondence between vectors of the space ℝ^2, complex numbers ℂ and real matrices 2×2

We will write the vector in the 2*1 matrix representation and in the complex number representation in the following form (the letter “v” in the middle of rvc means a column vector, the letters “c” at the end of rvc, xc, zc mean the correspondence to complex numbers ℂ).

The minus sign for a complex number is adopted for the reason stated above.

Let's assume that they are isomorphic, which means that there must be two operators in two representations that act on these vectors in the same way. Let's take an operator that acts on a column vector with its basic rows, and in the column components we get the products “matrix row by column”.

Let's compare the elements for x and z in these two representations.

the operator matrix exists, its determinant is equal to the square of the modulus of the complex number, but since these are not all 2×2 matrices, we can conclude that

not all vectors ℝ^2 are isomorphic to complex numbers.

Also, then this all means that vectors of this type are isomorphic not only to complex numbers, but also to some 2×2 matrices, namely

Then the multiplication of such a transposed operator by a vector, in the elements of the vector-
column, gives the scalar and vector product.

About conjugates and inverses

To check this, we will compare the multiplication by the conjugate complex number to the operation on matrices in an identical way

Multiplication by the inverse of such an operator is isomorphic to multiplication by the inverse of a complex number.

And so there must be vectors that complete the space of vectors isomorphic to complex numbers to the full space. These vectors must be isomorphic to something else.

The construction obtained for the matrix representation of the vector Rvс corresponds to the components u,t.

Then vectors that are not isomorphic to complex numbers are described by matrices.

This matrix is obtained by acting on Rv with the reflection operator about the (1,1) axis.

Let's write it all down together:

Which means that in the usual two-dimensional space, in fact, there is not one, but two bases. Each has its own coordinate system, and one is obtained from the other by reflection relative to (1,1).

The analogue of a complex number for ordinary vectors corresponds to a proper ordinary vector from linear algebra.

Also, such a system of vectors must satisfy the condition that when R acts on a certain vector re, isomorphic re` and R` must be obtained. Let us take into account that we now know that the column vector corresponding to complex numbers must change sign when transposed.

Then all real 2×2 matrices:

And they are isomorphic to vectors:

Which means that in the usual two-dimensional space, in fact, there is not one, but two bases. Each has its own coordinate system, and one is obtained from the other by reflection relative to (1,1).

Also, such a system of vectors must satisfy the condition that when R acts on a certain vector re, isomorphic re` and R` must be obtained. Let's take into account that we now know that we need to multiply by the transposed operator (the column vector also becomes an operator after transposition)

Indeed, they change in the same way, but the fewer 2×2 matrices are involved, the more information is lost. Therefore, it is preferable to work with 2×2 matrices rather than column vectors.

And then comparing the coordinates in the matrix representation of the vector and the operator:

We obtained the standard scalar product in the zc` component, the scalar product of codirectional coordinates in the x` component, and the vector product of orthogonal coordinates in the z` component.

It is interesting that the standard scalar product of ordinary vectors is obtained in the very basis in which x, y are reflected relative to (1, 1) in the basis corresponding to complex numbers.

To put it more specifically, the scalar product of a vector with itself from the basis “r” gives a mapping to the zc axis of the basis “rvс”, without the need to invent any additional spaces.

I explained it as simply as I could.