Conjugation

Conjugation is certain type of transformation of mathematical objects. There are several kinds of such conjugation.

In addition,there are more meanings of verb conjugation , that are not yet used in TORI.

Inner automorphism
By default, in TORI, Conjugation refers to the inner automorphism or Topological conjugacy . This conjugation should be specified with some function \(P\); it is assumed that the inverse function \(Q=P^{-1}\) exists. Then, for some function \(f\), its conjugate \(g\) is expressed with

\((1) ~ ~ ~ g(z)=P(f(Q(z)))=P\circ f\circ Q (z) \)

In general, no specific properties of the transform function \(P\) (except existing of the inverse funcion) are assumed. Wikipedia suggests the following illustration of the application of function \(P\) to the argument of the function and the inverse function \(Q\) to the result:"raise umbrella, walk through rain, lower umbrella" has a different result from just "walk through rain". However in some cases, the operations \(P\) and \(f\) commute, and \(P\) can be "drawn through" function \(f\), for example, "take off left glove, take off right glove, put on left glove" may have the same effect as "take off right glove only".

Other types of the conjugation specified below are just special cases of the expression (1) for some specific \(P\) and \(Q\).

Complex conjugation
Complex conjugation is function defined at the set of complex numbers. The result of operation is also complex number.

Often, the complex conjugation is denoted with asterisk * as the superscript; conjugation of a complex number \(z\) is written as \(z^*\).

If the order of operations is not indicated with parenthesis, the complex conjugation has highest priority, id est, performed before infix operation. For example, \(ab^*\) is interpreted as \(a\times (b^*)\) but not as \((ab)^*\).

For complex number \(z=\Re(z) + \mathrm i \Im(z)\), its complex conjugation is \(z^*=\Re(z) - \mathrm i \Im(z)\)

In programming languages, the special name for the operation of complex conjugation is required, while the symbol asterisk is used to denote multiplication (instead of symbol \(\times\), which is not ascii character and, therefore, not recommended at all for the programming code.

In C++, the complex conjugation is denoted with built-in finction conj.

In Mathematica, the complex conjugation is denoted with built-in finction Conjugate; the complex conjugation of number \(z\) is written as Conjugate[z]. (In Mathematica, the argument of a function is always written in rectangular parenthesis.)

With complex conjugation, operation square of modulus, id est, \(z\mapsto |z|^2\), can be expressed as \(z z^*\).

Similarly, the Real part and the Imaginary part of a number \(z\) can be expressed with

\(\Re(z)=\frac{1}{2}(z+z^*)\) ,

\(\Im(z)=\frac{1}{2}(z-z^*)\)

Complex conjugation is not a holomorphic function.

Linear conjugation
Linear conjugation is special kind of conjugation (1), characterized in that, that \(P\) is linear function; \(P(z)=a+bz\) for some numbers \(a\) and \(b\), then \(P^{-1}(z)=b^{-1}\times (z-a)=\frac{1}{b}(z-a)=(z-a)/b\)

Often, it is assumed, that \(a\) and \(b\) are just complex numbers, and \(b\ne 0\). Then, linear conjugation \(f\) of a function \(F\) can be written as follows:

\(f(z)=a+b~ F(b^{-1} \times (z-a))\)

Character \(\times\) is used above to avoid confusion with the case when \(b\) denotes some function and \(b^-1\) is the inverse function.

Complex conjugation of a function
For any function \(F\), its complex conjugation or complex conjugate is function \(G\) such that

\(G(z)=F^*(z)= \Big(F(z^*)\Big)^*\)

For a function \(F\), expression

\(F^*(z)\)

should not be confused with

\(F(z)^*\)

nor with

\(F(z^*)\)

At the use of the superscript after the name of a function, the special care is required:

if the character of is just prime, then, perhaps, it denotes derivative,

\(\displaystyle F'(z)=\lim_{t\rightarrow 0} \frac{F(z+t)-F(z)}{t}\)

If the character is number, it indicates the iteration number; for example, \(F^2(z)=F(F(z))\).

Use in superscripts of some object, that may mean prime, and may mean a number, and may mean asterisk, is not recommended, as it easy may lead to confusion. In particular, this refer to a case, when the number of derivative is some number, that is either too long (and it is difficult to write the corresponding amount of primes) or not yet specified. In this case, the parenthesis are used; so, if the whole superscript appear in parenthesis, then, perhaps, this supersctipt indicate derivative and not iteration.

Self-inverse
Complex conjugation is self-inverse, id est, \(f^{**}=f\).

This applies both to functions and to numbers.

Hermitian conjugation
\(|\psi\rangle^\dagger = \langle \psi |\)

Keywords
Holomorphic function , Complex number , Operator , Mathematical notations, Quantum mechanics