**48
Commutator Symmetry**

We make here a mathematical bond in quantum mechanics. The term of the commutator comes from a household context. Among the commonly used in everyday life numbers are numbers that are subject to some simple group symmetries. One such group is the commutativity properties, or rewrite it as. This symmetrie can be written abbreviated as

(48.1).

The brackets [] is called the *"commutator"*, and the right side of the equation *"commutator-residue"*. This residue must be but not
zero in the general case. If the commutator-residue
is zero, it is called Abel's numbers, if it is not equal to zero, whereas
called non-Abelian numbers. Elementary particles are
represented in particular by such non-Abelian
numbers. The mathematical "trick" here is that those numbers are
represented by so-called "operators". Operators are, as the name
suggests, some mathematical gadgets that do anything when applied to other
numbers, say, in the case of differential operators that make a differenciation. The commutator-residue
if it is not zero.

Let, therefore, look us again on the derivation of the previous one. When considering the quantity equation in the SMF, we have the relationship

to derive the average monetary velocity. This structure enables us to sort something:

or

or

(48.2).

`We can now write in the so called `*operator *notation:

(48.3)

with the operators

and (48.4).

`From quantum mechanics we know the simple coherences of this elementary symmetries`

(48.5)

known about the properties of atoms and their
constituents do. This meant that the KV-operator defined here is applied to *Y*
and so that *KV* product is produced. Thus we get:

(48.6)

The commutator-residue should we expect equal zero as in the economy we use always finite abelian numbers. So either, which almost never happens, or specify:

or (48.7)

Now isprecisely the nominal return on capital (and derived from that also namelyor):

(48.8)

what gives the desired results in relation to the *average*
return on all assets of an economy. The interest rate we can now compare with
the supply-demand estimate at the beginning of the second part of the book.

In the next graph we see the result. The real change in the total nominal
capital stock, is roughly equivalent to the nominal
interest rate on all assets. The curve (37.1) with) comes very close. This corresponds to the fact that in the underlying function,
also for the product money, there is an effective supply-demand ratio.

The curve (:48.8) on the other hand, goes faster to zero than the real numbers.
The reason for this lies in our simplified derivation in
which onlyoccurs but not. The investment business is therefore *not
included*. Therefore (48.8), follows the curve of the real numbers only up
to the region of the phase change in 1967, and then falls off more quickly, *since
without the bank's own business can not be achieved then actually higher
interest rates. Before phase change the interest rates would have been even
higher, due to the same effect in the other direction, as then loans to the
real economy would have been the better choice.*