Algebra of generalized functions (Shirokov)

An associative algebra $$\mathcal{A}$$ equipped with involution and differentiation, is constructed for generalized functions of one variable that at one fixed point can have singularities like the delta function and its derivatives and also finite discontinuities for the function and all its derivatives. The elements of $$\mathcal{A}$$ together with the differentiation operator form the algebra of local observables for a quantum theory with indefinite metric and state vectors that are also generalized functions. By going over to a smaller space, one can obtain quantum models with positive metric and with strongly singular concentrated potentials.

Introduction
One of the key mathematical problems of relativistic (and, quite generally, local) field theory is the construction of a satisfactory operation of multiplication for a certain class of generalized functions (see, for example, [1]). The idea behind the present work is that to obtain the renormalized equations of local (including relativistic) quantum field theory one needs appropriate associative algebras of functions, including the necessary generalized functions, and these algebras must be equipped with involution and differentiation. In other words, it is the operation of multiplication of functions that must be the object of renormalization. Hitherto, only individual classes of generalized functions have, as a rule, been studied for the purpose of constructing products of generalized functions (see, for example, [1-5]), the existence of an algebra of generalized functions not being required for the construction of renormalized perturbation theory. The simplest model for which perturbative calculations lead to divergences of the same kind as in relativistic quantum field theory is provided by the motion of a single particle in the field of a concentrated $$\delta(x)$$ potential (see [6], and also [7]). The simplest variant of this model is one-dimensional motion of a nonrelativistic particle in the field of such a potential. It is true that for a one-dimensional potential proportional to $$\delta(x)$$, the model is superrenormalizable, but for potentials contacting $$\delta'(x)$$, $$\delta''(x)$$, etc., one can obtain various nonsuperrenormalizable models, Including nonrenormallzable models. In the present paper, we therefore construct an algebra of one-dimensionalfunctions that include the delta function and its derivatives.

Notations
We introduce the notation and definitions needed to formulate the basic assumptions. Let

$$\{ {\rm A} \}$$ be the space of complex-valued functions, including the necessary generalized functions, $$\{ \Psi \}$$ be the dual space of test functions, and $$\langle...,...\rangle $$ denote a bilinear functional on $$\{ {\rm A} \}$$ and $$\{ \Psi \}$$. We shall denote elements in $$\{ A \}$$ by $$A$$, $$B$$, ...; elements in $$\{ \Psi \}$$ by $$~\psi~$$, $$~\chi~$$, ...

Assumptions
Our basic assumptions are as follows.

A. The set $$\{ \rm A \} $$ is a topological vector space.

B. On $$\{  {\rm  A}   \}$$, there exists a renormalized operation of multiplication, this being associative, defined for any ordered pair $$A$$, $$B$$ from $$ \{ {\rm A} \}$$, does not carry the product outside $$\{ {\rm A} \}$$, and coincides with ordinary multiplication of functions for ordinary (not generalized) functions. Thus, the elements in $$\{ {\rm A} \}$$ form an associative algebra, which we shall denote by $$\mathcal{A}$$.

C. The set $$\{ \Psi \}$$ is also a topological vector space and forms a dual pair with $$\{\rm A\}$$ (see, for example,[8]). D. Each element in $$\{ {\rm A}\}$$ can act on each element in $$\{ \Psi \}$$ as an operator. This means that there is defined the bilinear product $$A \psi \in \{\Psi\}$$. Such multiplication satisfies the condition of associativity

$$(1) \langle AB, \psi \rangle = \langle A, B\psi\rangle $$

for all $$ A$$, $$B$$ belonging to $$ \{ \rm A \} $$ and $$ \psi \in \{\Psi\} $$. The relation (1) shows that, in general, the space $$\{ \Psi \}$$ also contains generalized

functions. On products of the type $$A \psi$$, we

assume that they coincide with the ordinary products of functions for nongeneralized functions $$ A $$ and $$ \psi $$. The same we assume with respect to multiplication by any C-number. E. For all functions in $$\{ \rm A \} $$ and $$\{ \Psi \}$$, there is defined an operation of differentiation that coincides with ordinary differentiation for all functions in open domainson where these functions are differentiable. F. For all functions in $$\{ \rm A \} $$ and $$\{ \Psi \}$$, there is defined an operation of involution that coincides with complex conjugation for ordinary functions and commutes with differentiation. Algebras containing $$\delta (x)$$ were studied in [9,10] under different systems of basic assumptions (which did not allow to include their derivatives).

Construction of Algebra $$\mathcal{A}$$
We begin the construction of the required algebra with the spaces $$ \{ {\rm A} \} $$ and $$\{ \Psi \}$$ and the functional $$\langle ...,...\rangle$$. We first introduce the smaller spaces $$ \{ {\rm   A_f} \}$$ and $$ \{ {\rm \Psi_f} \}$$. Both spaces consist of complex functions of a single variable $$x \in \mathbb{R}_1 $$ that are finite, with all their derivatives, everywhere except at the point $$x = 0$$, and at $$x = 0$$ have finite limits from the left and the right for the functions themselves and for all derivatives.

Slow rise and fast decay
At infinity, functions in $$ \{ {\rm A_f} \}$$, together with all their derivatives, can have slow (i.e., not faster than polynomial) growth, while functions in $$ \{ {\rm \Psi_f} \}$$ decrease rapidly with all their derivatives (i.e., faster than any polynomial). The expression for a bilinear functional has the standard form

$$(2)~\left\langle A_{\rm f}, \psi_{\rm f}\right\rangle=\int {\rm d} x~ A_{\rm f}(x) \psi_{\rm f}(x)$$.

The spaces $$ \{ {\rm A} \} $$ and $$\{ \Psi \}$$ obviously satisfy the duality condition. Topologies can be introduced on them in the same way as in [11] for analogous functions without discontinuity. The only difference will be that in the norms of the corresponding auxiliary spaces it will be necessary to introduce everywhere in the suprema left and right derivatives instead of the derivatives at the point$$ x=0 $$.

Construct Functionals with compact support
We now begin construct functionals over $$ \{ {\rm \Psi_f} \}$$ with support at the point $$x=0$$. We first introduce the notations

$$\psi_{\pm}=({\rm d}/{{\rm d}x})^n \psi(\pm 0)$$

$$ \psi_{\rm s}^n=\frac{1}{2} \psi^{(n)}_- + \frac{1}{2}\psi^{(n)}_+ $$

$$ \psi_{\rm a}^n=\frac{1}{2} \psi^{(n)}_- - \frac{1}{2}\psi^{(n)}_+ $$

Note that for all real $$ y \ne 0$$ the functionals $$\delta(x-y)$$ are defined. This allows to define the functionals $$\delta^{(n)}_{\rm s}(x)$$ and $$\delta^{(n)}_{\rm a}(x)$$ as follows:

$$(3)\left\langle \delta^{(n)}_{\rm s}(x) ,\psi(x)_{\rm f}(x) \right\rangle = \frac{1}{2} \left\langle \left(\delta^{(n)}(x-0)+\delta^{(n)}(x+0)\right) , \psi_{\rm f}(x) \right\rangle =(-1)^n\psi^{(n)}_{\rm s} $$

$$(4)\left\langle \delta^{(n)}_{\rm s}(x) ,\psi(x)_{\rm f}(x) \right\rangle = \frac{1}{2} \left\langle \left(\delta^{(n)}(x-0)-\delta^{(n)}(x+0)\right) , \psi_{\rm f}(x) \right\rangle =(-1)^n\psi^{(n)}_{\rm a} $$

Eash of these functionals has supports at $$x=0$$.

We complete the space $$\{ {\rm A_f}\}$$ with the generalized functions $$\delta^{(n)}_{\rm s}(x)$$ and $$\delta^{(n)}_{\rm a}(x)$$. We denote the completed space by $$ \{ {\rm A} \}$$. It contains all functions of the form

$$(5)~A(x)=A_{\rm f}(x)+ \sum_{m=0}^M A_{m}\delta^{(m)}_{\rm s}(x)+ \sum_{n=0}^N A^{n}\delta^{(n)}_{\rm a}(x) $$

Here, $$ M$$ and $$N$$ are non-negative integers and $$A_m$$ and $$A^n$$ are complex numbers. (Here, the superscript $$^n$$  not  an exponentiation.)

The functional $$\langle ...,...\rangle$$ is now defined for all $$A(x)$$ in $$\{ {\rm A} \}$$ and for all $$\psi(x)$$ in $$\{ \Psi \}$$. The topology is first introduced separately for each space with fixed $$M$$ and $$N$$, after which the topology on $$\{ {\rm A} \}$$ is obtained through the rigorous inductive limit with respect to $$M$$ and $$N$$.

Functions with compact support
Our next task is to complete the space $$\{ {\rm A_f} \}$$ with generalized functions with support at $$x=0$$. Note that on functions of the space {A} for all real $$ y \ne 0$$, there is defined the functional $$~ \langle ~A(x)~, ~ \theta(x \!+\! y)-\theta(x\!-\! y)~ \rangle~ $$, which corresponds to the generalized function $$ \theta(x+y)-\theta(x-y) $$ (where $$~\theta(x)=0~$$ for $$~x<0~$$ and $$~\theta(x)=1~$$ for $$~ x>0 ~$$.

Regilarizing functionals
Functions $$\theta$$ allow to introduce new functionals on functions of the space $$\{ {\rm A} \}$$ (which here play the part of test functions). The functionals of a new type have support concentrated at the same point but they differ from $$\delta^{(n)}(x)$$. We denote these functlonals by $$\eta^{(n)+}_{\rm s}(x)$$ and $$\eta^{(n)+}_{\rm a}(x)$$; we define them as follows: $$(6)~ \left\langle~A(x)~,~ \eta_{\rm s}^{(n)+}(x)~\right\rangle= \lim_{x\rightarrow +0}~ \left\langle ~A(x)~,~ \frac{(-x)^n}{n!} \left( \theta(x\!+\!y)-\theta(x\!-\!y) \right) ~\right\rangle $$ $$(7)~ \left\langle~A(x)~,~ \eta_{\rm a}^{(n)+}(x)~\right\rangle= \lim_{x\rightarrow +0}~ \left\langle ~A(x)~,~ \frac{(-x)^n}{n!} \varepsilon(x) \left( \theta(x\!+\!y)-\theta(x\!-\!y) \right) ~\right\rangle $$, where $$ \varepsilon(x)= \begin{cases} 1, & x>0 \\ -1,& x<0 \end{cases} $$

We emphasize that the limits in (6) and (7) are made after the limits in (3) and (4); it is the trick which allows to make the algebra assotiative. For example,

$$(8) ~ \left\langle \delta^{(m)} _{\rm s}(x), \eta^{(n)+} _{\rm s}(x) \right\rangle =\frac{1}{2}~ \lim_{z\rightarrow 0}~ \lim_{y\rightarrow 0}~ \left\langle \Big( \delta^m(x\!-\!y)+\delta^m(x\!+\!y) \Big) \frac{(-x)^n}{n!} \Big( \theta(x\!+\!y\!+\!z)-\theta(x\!-\!y\!-\!z) \Big) \right\rangle =\delta_{m,n} $$

where $$\delta_{m,n}$$ is the Kronecker diagonal tensor. Similarly, $$(9) ~ \left\langle \delta^{(m)} _{\rm a}(x), \eta^{(n)+} _{\rm a}(x) \right\rangle \delta_{m,n} $$

$$(10) ~ \left\langle \delta^{(m)} _{\rm s}(x), \eta^{(n)+} _{\rm a}(x) \right\rangle = \left\langle \delta^{(m)} _{\rm a}(x), \eta^{(n)+} _{\rm s}(x) \right\rangle =0 $$

It also follows from (6) and (7) that

$$(11) ~ \left\langle A_{\rm f}(x), \eta^{(n)+} _{\rm s}(x) \right\rangle = \left\langle A_{\rm f}(x), \eta^{(n)+} _{\rm a}(x) \right\rangle =0 $$

Thus, the functionals $$ \eta_{\rm s}^{(n)+} $$ and $$ \eta_{\rm a}^{(n)+} $$ vanish on all functions in the usual spaces of test functions.

Note that the order of the limits in formulas of the type (8) will not change, so that we do not require uniformity in the tending to the limits.

Regularization
Using the functionals $$\eta_{\rm a}^{(n)+}$$ and $$\eta_{\rm a}^{(n)+}$$, we can construct a regularizing operator, which we denote by $$ p_{\rm f}^{+}(x) $$. This operator is applied to the left and maps from $$~\{{\rm A }\}~$$ to $$~\{ {\rm A_f }\}~$$; define it with $$ (12) \left\langle A(x) p_{\rm f}^+ \right\rangle =~ A(x)~-~ \sum_{n=0}^\infty \left\langle ~A(x)~ \eta_{\rm s}^{(n)+}~\right\rangle~ \delta_{\rm s}^{(n)}(x) ~-~ \sum_{n=0}^\infty \left\langle ~A(x)~ \eta_{\rm a}^{(n)+}~\right\rangle~ \delta_{\rm s}^{(n)}(x) $$ Both summations in fact always terminate, since the number of singular terms in $$A(x)$$ is finite. It follows from (8)-(11) that

$$(13) \left\langle A(x)\eta_{\rm s}^{(n)+}(x) \right\rangle = A_n ,~ \left\langle A(x)\eta_{\rm a}^{(n)+}(x) \right\rangle = A^n $$ so that $$ (14) \left\langle ~A(x)~ p_{\rm f}^+(x)~ \right\rangle=A_{\rm f}(x) $$

Now it is seen, why we call $$p_{\rm f}^+(x)$$ "reguladising operator": after its application, the only regular part of a function survives.

Using (14), we can introduce generalized functions of the type (3) and (4) already as functlonals on the space $$\{{\rm A}\}$$. Namely, on $$\{{\rm A}\}$$ we define the functlonals $$\delta_{\rm s}^{(n)+}(x) $$ and $$\delta_{\rm a}^{(n)+}(x) $$ by

$$(15) \left\langle A(x), \delta_{\rm s}^{(n)+}(x) \right\rangle = \frac{1}{2} \left\langle~ A(x) p_{\rm f}^+(x) ~,~\delta^{(n)}(x\!-\!0)+\delta^{(n)}(x\!+\!0)~\right\rangle~=~(-1)^n\!~A_{\rm s}(x) $$

$$(16) \left\langle A(x), \delta_{\rm a}^{(n)+}(x) \right\rangle = \frac{1}{2} \left\langle~ A(x) p_{\rm f}^+(x) ~,~\delta^{(n)}(x\!-\!0)-\delta^{(n)}(x\!+\!0)~\right\rangle~=~(-1)^n\!~A_{\rm a}(x) $$

Accordingly, we complete the space $$\{ {\rm \Psi_f } \}$$ by the functions $$\delta^{(n)+}_{\rm s}(x)$$ and $$\delta^{(n)+}_{\rm a}(x)$$. We denote the resulting space by $$\{\Psi \}$$. It contains all functions of the form

$$(17) \Psi(x)=\Psi_{\rm f}(x) + \sum_{m=0}^{M} \delta_{\rm s}^{(m)+}\psi_m + \sum_{n=0}^{N} \delta_{\rm a}^{(n)+}\psi^n $$,

where, in analogy with (5), $$~M~$$ and $$~N~$$ are non-negative integers, and $$\psi_m$$ amd $$\psi_n$$ are complex numbers.

Concentrated functionals
On functions in $$~\{\Psi \}~$$, we now introduce concentrated functionals $$\eta_{\rm s}^{(n)}(x) $$ and $$\eta_{\rm a}^{(n)}(x) $$, defining them in analogy to (6) and (7):

$$(18) \left\langle \eta_{\rm s}^{(n)}(x), \psi \right\rangle ~=~\lim_{y \rightarrow +0} \left\langle \Big(\theta(x\!+\!y)-\theta(x\!-\!y)\Big)\frac{(-x)^n}{n!}, \psi(x) \right\rangle$$,

$$(19) \left\langle \eta_{\rm a}^{(n)}(x), \psi \right\rangle ~=~\lim_{y \rightarrow +0} \left\langle \Big(\theta(x\!+\!y)-\theta(x\!-\!y)\Big)\varepsilon(x)\frac{(-x)^n}{n!}, \psi(x) \right\rangle~ $$.

All the comments made after the dual formulas (6) and (7) apply to (18) and (19). In analogy with (13), $$(20) \left\langle \eta_{\rm s}^{(n)}(x)\!~ \psi(x) \right\rangle = \psi_n , \left\langle \eta_{\rm a}^{(n)}(x)\!~ \psi(x) \right\rangle = \psi^n $$.

As the dual to $$~p_{\rm f}^+(x)~$$, we have the operator $$~P_{\rm f} (x)~$$, which acts to the right and maps from $$~ \{ {\rm \Psi }\}~ $$ to $$~\{ {\rm \Psi_f }\}~$$:

$$(21) P_{\rm f}(x)\psi(x)~ =~\psi(x) ~-~ \sum_{n=0} ^{\infty} \delta_{\rm s}^{(n)+}(x) \left\langle \eta_{\rm s}^{(n)}(x), \psi(x) \right\rangle ~-~ \sum_{n=0} ^{\infty} \delta_{\rm a}^{(n)+}(x) \left\langle \eta_{\rm a}^{(n)}(x), \psi(x) \right\rangle $$. As in (12), the summations here are also in fact always finite. In accordance with (20),

$$(22)P_{\rm f}(x) \psi(x) = \psi_{\rm f}(x) $$.

With expressions (21), we finish the tools necessary fot correct definition of bilinear functional at the extendedspaces.

Generalized bilinear form
Only now that we have the possibility of defining the definition of the required renormalized bilinear functional of the type $$ \left\langle A(x), \psi(x) \right\rangle $$, suitable for all functions in the spaces $$ \{ {\rm A} \}$$ and $$~ \{ \Psi \} ~$$, respectively:

$$(23) \langle A(x), \psi(x) \rangle = \langle A(x) p_{\rm f}^{+}(x),              \psi(x) \rangle + \langle A(x)                       , P_{\rm f}\psi(x) \rangle - \langle A(x) p_{\rm f}^{+}(x), P_{\rm f}\psi(x) \rangle~ $$.

Assume, the functions $$A(x)$$ and $$\psi(x)$$ are decomposed into regular and singular parts:

$$ A(x)=A_{\rm f} (x)+A_{\rm sing}(x),~ \psi(x)=\psi_{\rm f} (x)+\psi_{\rm sing}(x)$$;

then the definition (22) can be rewritten in the more perspicuous form $$(24) \langle A(x), \psi(x) \rangle = \langle A_{\rm f},\psi_{\rm f}(x) \rangle + \langle A_{\rm f},\psi          (x)  \rangle + \langle A           ,\psi_{\rm f}(x) \rangle $$.

We emphasize that the associativity condition(l) is not satisfied for the regularizing operators $$p_{\rm f}^+(x)$$ and $$~P_{\rm f}(x)~$$. Fortunately, these operators are not included in the set of elements of the renormalized algebra.

Associative multiplication
This section, constructs a renormalized associative operation of multiplication for functions in $$\left\{ {\rm A}\right\}$$. Usually, such operation is believed to not exist, to, we begin with the most important part.

First note, that the operators $$~\{ {\rm A} \} ~$$ and $$~\{\Psi\}~$$ form a dual pair; it follows from $$ \left\langle A(x), \psi(x)\right\rangle = 0 $$

for all $$~A(x)~$$ (all $$~\psi(x)~$$ ) such that that $$~\psi(x) = 0~$$ (respectively, $$~A(x) = 0$$). It therefore follows from the associativity condition (1) that all renormalized products of the type $$ A(x)!~B(x)$$ are defined.

Products
The operator renormalized products of the type $$ A(x)\psi(x) $$ that do not take one out of $$~\{\Psi\}~$$ are defined. We begin with the construction of these last. We give first two definitions of two new operators $$ g(x) $$ and  $$ g^+(x) $$ such that for them there exist the products $$ A(x)g(x)\psi_{\rm f}(x) \in \{\Psi\}$$, $$ A_{\rm f}(x)g^{+}(x)\psi_{\rm f}(x) \in \{\Psi\} $$:

$$(25)~ \begin{align} A(x)g(x)\psi_{\rm f}(x) &= \sum_{m=0}^{\infty} \sum_{n=0}^{m} \left\langle A(x),\eta_{\rm s}^{(m)+}(x) \right\rangle {C_m}^n \left( \delta_{\rm s}^{(m-n)+}(x) \langle \delta_{\rm s}^{(n)} (x),\psi_{\rm f}(x) \rangle - \delta_{\rm a}^{(m-n)+}(x) \langle \delta_{\rm a}^{(n)} (x),\psi_{\rm f}(x) \rangle \right) \\ &+ \sum_{m=0}^{\infty} \sum_{n=0}^{m} \left\langle A(x),\eta_{\rm a}^{(m)+}(x) \right\rangle {C_m}^n \left( - \delta_{\rm a}^{(m-n)+}(x) \langle \delta_{\rm s}^{(n)} (x),\psi_{\rm f}(x) \rangle + \delta_{\rm s}^{(m-n)+}(x) \langle \delta_{\rm a}^{(n)} (x),\psi_{\rm f}(x) \rangle \right) \\ \end{align} $$

$$(26)~ \begin{align} A_{\rm f}(x)g^+(x)\psi(x) &= \sum_{m=0}^{\infty} \sum_{n=0}^{m} \left( \left\langle A_{\rm f}(x),\delta_{\rm s}^{(n)+}(x) \right\rangle \delta_{\rm s}^{(m-n)+}(x)  +\left\langle A_{\rm f}(x),\delta_{\rm a}^{(n)+}(x) \right\rangle \delta_{\rm a}^{(m-n)+}(x)  \right) {C_m}^n \langle \eta_{\rm s}^{(m)} (x),\psi(x) \rangle \\ &+ \sum_{m=0}^{\infty} \sum_{n=0}^{m} \left( \left\langle A_{\rm f}(x),\delta_{\rm s}^{(n)+}(x) \right\rangle \delta_{\rm a}^{(m-n)+}(x)  +\left\langle A_{\rm f}(x),\delta_{\rm a}^{(n)+}(x) \right\rangle \delta_{\rm s}^{(m-n)+}(x)  \right) {C_m}^n \langle \eta_{\rm s}^{(m)} (x),\psi(x) \rangle \\ \\ \end{align} $$ Where $$ {C_m}^n=\frac{m!}{n!(m-n)!}$$ are binimial coefficients.

The operators $$g(x)$$ and $$g^+(x)$$ do not belong to the algebra, but, in contrast to $$P_{\rm f} (x)$$ and $$ p_{\rm f}^{+}(x)~$$, they are assotiative; for example, $$ \Big( A(x)g(x)\Big)\psi_{\rm f}(x)=A(z)\Big(g(x)\psi_{\rm f} (x)\Big)$$.

Simplification of notations
Formulas (25) and (26) already appear too long. In order to continue, some compact notations are necessary. The associativity discussed in the previous section allows such simplification, and we introduce a notation convention. The renormallzed multiplication operations $$\{ {\rm A}\} \otimes \{\Psi\} \rightarrow \{\Psi\} $$ and $$\{ {\rm A}\} \otimes \{ {\rm A}\} \rightarrow \{ {\rm A}\}$$ will be denoted in the same way as the operation of ordinary multiplication of functions, i.e., $$~A(x)\psi(x)~$$, $$~A(x)B(x)~$$, etc. This convention cannot lead to confusion provided the assumptions B and D of Sec.1 are satisfied. It is immediately verified that B and D will be satisfied if the renormallzed product $$~A(x)\psi(x)~$$ is defined by

$$(27) A(x)\psi(x)= \Big( A(x) p_{\rm f}(x)   \Big) \Big( p_{\rm f}(x) \psi(x) \Big) + A(x) g(x) \Big( P_{\rm f}(x) \psi(x) \Big) +\Big( A(x) p_{\rm f}^+(x)\Big) g^+(x)\psi(x) $$,

and the renormalized product $$~A(x)B(x)~$$ by $$ (28) \big(A(x)B(x)\Big) \psi(x) = A(x) \big( B(x)\psi(x) \big) $$

There is great seduction to suggest even more "naive" simplification. One can show that in the framework of these assumptions, the definition above is unique, apart from the possibility of replacing both minus signs on the right-hand side of (25) by plus signs. However, we shall show below

that such a replacement is not compatible with the operation of differentiation. In such a way, there are not so many different ways to construct the algebra of generalized functions.

More simplification of notations; summary on the algebra
We now make small simplifications of the notation. For this, we note that the product (27) (in contrast to the bilinear form (23)) is also defined for functions that do not decrease at infinity and, in particular, for $$\psi(x) = 1$$. At the same time, $$(29) A(x)\cdot 1 =A_{\rm f}(x) + \sum_{m=0}^M \delta_{\rm s}^{(m)+}A_m \sum_{n=0}^N \delta_{\rm a}^{(n)+}A^n $$. It is therefore convenient to denote $$(30) \delta_{\rm s}^{(m)+}(x)=\delta_{\rm s}^{(m)}(x), \delta_{\rm a}^{(m)+}(x)=-\delta_{\rm s}^{(m)}(x) $$

and assume that the functions $$\delta_{\rm s}(x)$$ and $$\delta_{\rm a}(x)$$ behave in accordance with (3) and (4) in functionals of type $$\langle \delta \psi \rangle $$ and in accordance with (9), (10), and (30) in functionals of the type $$\langle A,\psi\rangle $$.

In such a way, (29) becomes an identity.

The following simplification is suggested by the fact that for any infinitely smooth functions $$A_{\rm sm}(x)$$ in $$\{ {\rm A}\}$$ and $$\psi_{\rm sm}(x)$$ in $$\{ \Psi \}$$, we have $$  \langle A_{\rm sm}(x),\delta_{\rm s}^{(n)}(x) \rangle =\langle A_{\rm sm}(x),\delta^{(n)}(x) \rangle ~,~ \langle A(x),\delta_{\rm s}^{(n)}(x) \rangle =\langle A_{\rm sm}(x),\delta^{(n)}(x) \rangle$$ where $$\delta(x)$$ is the ordinary delta function. In such a way, there is therefore no possibility of confusion in writing $$(31)~ \delta_{\rm s}^{(n)}(x)=\delta^{(n)}(x) $$. Finally, it follows from (27), that $$(32)~ \delta_{\rm a}^{(n)}(x)=\delta^{(n)}(x) \varepsilon(x)

$$. In the new notation, the renormalized product $$A(x) B(x)$$ has the same form as $$A(x)\psi(x)$$: $$(33)~ A(x)B(x)~=~ \Big( A(x) p_{\rm f}(x) \Big) \Big(P_{\rm f}(x) B(x) \Big) ~+~

A(x)g(x) \Big(P_{\rm f}(x) B(x)\Big) ~+~ \Big(A(x)p_{\rm f}(x) \Big) g^{+}(x)B(x) . $$

To conclude this section we note that (27) (or, which is now the same thing (33)) implies that $$(34) ~ \varepsilon(x) \delta^{(m)}(x) + \delta^{(m)}(x) \varepsilon(x) = 0 $$

$$(35) ~ \delta^{(m)}(x) \delta^{(n)}(x) = 0 $$

$$(36) \delta^{(m)}(x) A_{\rm sm}(x) = \sum_{n=0}^{m} ~{C_m}^n ~(-1)^n~ \delta^{(m-n)}(x)~A_{\rm sm}^{(n)}(0) . $$ In addition to anti-commuting signum and delta, the square of signum happens to be identically unity, $$ \varepsilon(x)\varepsilon(x)=1 $$ and this identity holds for all $$x$$, even at $$c\!=\!0$$. However, such exotic properties do not violate any of properties declared in the Introduction.

The relations (34)-(36) completely characterize the renormalized operation of multiplication in the algebra * of generalized function. In particular, they show that no implementation of the algebra can be performed approximating functions with C-numbers; equation (34) explicitly prohibits such approximation. Practically, the only the regular part of a function (smooth part) can be approximated; all singular parts must remain symbolic.

It remains to introduce differentiation and involution in this algebra. This will be subject of the next section.

Differentiation
In this section, we define differentiation of functions in $$\{ {\rm  A} \}$$ and $$ \{ {\rm \Psi} \}$$. In domains in which a function is differentiable, it is differentiated in the usual manner. The functions $$\delta_{\rm s}^{(n)} $$, $$\delta_{\rm a}^{(n)} $$, $$\delta_{\rm s}^{(n)+} $$, $$\delta_{\rm a}^{(n)+} $$ are differentiated in accordance with (3), (4), (15), and (16) before the passages to the limit indicated on their roght-hand sides. In this case, we must differentiate the functions $$\delta^{(n)}(x-y)$$, which are defined on functions that are infinitely smooth at the point $$~x=y~$$, so that the operation of differentiation is defined and leads to the result

$$(37) \frac{{\rm d}x}\delta_{\rm s}^{(n)}(x) = \delta_{\rm s}^{(n+1)}(x) ~,~ \frac{{\rm d}x}\delta_{\rm a}^{(n)}(x) = \delta_{\rm a}^{(n+1)}(x); $$ in such a way there is no problem with differentiation of $$\delta(x)$$, following the common rules.

Differentiation of Signum
It remains to differentiate the function $$~\varepsilon(x)~$$. On smooth functions, $$ \frac{{\rm d}x}\varepsilon(x) = 2\delta(x).$$ Therefore, in accordance with (3), $$(38)~ \left\langle \delta_{\rm s}(x), \psi_{\rm f}(x) \right\rangle ~=~\frac{1}{2} \left\langle \frac{1}{2}\frac{{\rm d} \varepsilon(x\!-\!0)}{{\rm d}x}+\frac{1}{2}\frac{{\rm d} \varepsilon(x\!+\!0)}{{\rm d}x} \delta_{\rm s}(x), \psi_{\rm f}(x) \right\rangle ~=~\frac{-1}{2} \left\langle~ \varepsilon(x\!-\!0) + \varepsilon(x\!+\!0) , \left(\frac{{\rm d} \psi_{\rm f}(x)}{{\rm d} x}\right)_{\!\rm f} +\delta(x) \psi_{\rm a}(x) \right\rangle $$ For this expression, the Integration by parts is possible, because the function $$\varepsilon(x\! -\! 0) + \varepsilon(x\! +\! 0)$$ and its derivative are smooth at $$~x\!=\!0~$$. Notation $$\left(\frac{{\rm d} \psi_{\rm f}(x)}{{\rm d} x}\right)_{\!\rm f} $$ means non-singular part of the expression in parenthesis, it is regular part of the derivative of $$~\psi~$$. It is just $$\psi_{\rm f}'(x) $$ for $$ x\!\ne\! 0$$,

and at the origin has the corresponding finite limits form the left and the right.

In the limit, we find from (38) that $$(39)~ \delta_{\rm s}(x)~=~\frac{1}{2}~{\rm d}\varepsilon(x)/{\rm d}x $$ which confirms the validity of the simplifying notation (31) (and corresponds to the common sense of differentiation of the signum function).

From (39), we readily obtain $$(40) \frac{{\rm d}A_{\rm f}(x)} {{\rm d}x}= \left( \frac{{\rm d}A_{\rm f}(x)}{{\rm d}x} \right)_{\!\!\rm f} +\delta(x) A_{\rm a}(x),~ \frac{{\rm d}\psi_{\rm f}(x)} {{\rm d}x}= \left( \frac{{\rm d}\psi_{\rm f}(x)}{{\rm d}x} \right)_{\!\!\rm f} +\delta(x) \psi_{\rm a}(x),~ $$

Equation (37), (39), and (40) in conjunction with (30) completely define the operation of differentiation for functions in $$~\{ {\rm A}\}~$$ and $$~\{ {\rm \Psi}\}~$$~.

Differentiation rules
By direct verification we can show that the renormalized operations of multiplication (27) and (33) satisfy the rule for differentiating a product. In particular, differentiating the equation $$ \varepsilon(x)\varepsilon(x) = 1 $$, we obtain (34) for $$ m\! =\! 1~$$, and it is this that prohibits the replacement of the minus sighs by plus signs on the right-hand side of (25). We have introduced differentiation on the algebra $$~\mathcal{A}~ $$. We emphasize that the rule for differentiating a product can be applied only to renormalized multiplication. Perhaps, it is the only possible way to differenticate elements of the algebra of generalized functions. Any attempts to differentiate the right-hand sides of (27) and (33) as we differentiate a products or regular funcitons may lead to expressions, for which we have no definitions.

The regularization of the differentiation rules and anti-commuting signum and delta is the price we pay to get the associative algebra.

Calculation of derivatives
To conculude this section, we note that, using the definitions (6) and (18), we can calculate the derivatives of the function

$$(41) {\rm d}\eta^{(0)+}(x)/{\rm d} x= -{\rm d}\eta^{(0)}(x)/{\rm d} x= -2\varepsilon(x) \delta(x) ~$$.

In this way, $$ \eta_{\rm s}^{(n)+}(x) = - \eta_{\rm s}^{(n)}(x) ~$$.

The space $$\{\Psi \}$$ can be completed with functions $$\eta_{\rm s}(x)$$, which, it is true, requires the introduction into the completed space of also the functions

$$ \varepsilon(x)\eta_{\rm s}^{(n)}(x), \delta^{(n)}(x)\eta_{\rm s}^{(m)}(x), \varepsilon(x)\delta^{(n)}\eta_{\rm s}^{(n)}(x), $$ With such elements, we get a new algebra containing functions $$~A_{\rm f}(x)~$$, $$~ \delta^{m}(x)~$$, $$~ \eta^{(n)}(x)~$$ and their products.

Algebra $$\mathcal{A}$$
It remains to introduce into the algebra $$\mathcal{A}$$ the involution $$A(x)\rightarrow A^+(x)$$, $$\psi(x)\rightarrow \psi^(x)$$. The involution operation must satisfy the standard requirements

$$~(A^+)^+=A~$$, $$~(\psi^+)^+=\psi~$$, $$~(\lambda A)^+=\lambda^*A^+~$$, $$~(\lambda \psi)^+=\lambda^*\psi~$$,

Here, $$~\lambda~$$ is a complex number and the asterisk denotes the complex conjugate. In conjunction with the basic assumption F of Sec.1, the listed requirements uniquely define the involution for all functions in $$~\{{\rm A}\}~$$ and $$~\{{\rm \Psi}\}~$$: $$(42)~ A_{\rm f}^+(x)=A_{\rm f}^*(x), \psi_{\rm f}^+(x)=\psi_{\rm f}^*(x), \Big(\delta^{(n)}(x)\Big)^+=\delta^{(n)}(x), \Big(\varepsilon(x)\delta^{(n)}(x)\big)=\delta^{(n)}(x) \varepsilon(x)= -\varepsilon(x)\delta^{(n)}(x)~ $$.

The relations (42) justify the notations $$\delta_{\rm s}^{(n)+}(x)$$, $$\delta_{\rm a}^{(n)+}(x)$$ introduced in (15) and (16). If functions in $$~\{{\rm A}\}~$$ are treated as operators of multiplication by functions in $$~\{{\rm \Psi}\}~$$, then the involution coincides with Hermitian conjugation. This last is also defined for the operators $$~p_{\rm f}^+(x)~$$ $$~P_{\rm f}~(x)$$ $$~g_{\rm f}(x)~$$ $$~g_{\rm f}^+(x)~$$, and $$(43)~ \big( P_{\rm f}(x)\Big)^+=p_{\rm f}^+(x), \big( g_{\rm f}(x)\Big)^+=g_{\rm f}^+(x) $$ which justifies the notations $$~p_{\rm f}^+(x)~$$, $$~p_{\rm f}^+(x)~$$.

It follows from (42) that the space $$~\{{\rm \Psi}^+\}~$$ of functions $$~\psi^+(x)~$$ consists of the same functions as $$~\{{\rm \Psi}\}~$$. And since $$~\{{\rm \Psi}\}~\subset ~\{{\rm A}\}~$$, for any pair $$~\psi(x),\chi(x) \in \{{\rm \Psi}\}~$$ there is defined the renormalized bilinear form $$~\langle \psi(x),\chi(x)\rangle~$$ which is obtained by the substitution $$~A(x)\rightarrow\chi(x)$$ in (23):

$$(44)~ \left\langle \chi^+(x),\psi(x) \right\rangle = \left\langle \chi^+(x)p_{\rm f}^+(x),\psi(x) \right\rangle + \left\langle \chi^+(x),P_{\rm f}\psi(x) \right\rangle - \left\langle \chi^+(x)p_{\rm f}^+,P_{\rm f}\psi(x) \right\rangle . $$

The bilinear form (44) is real:

$$(45)~ \langle           \psi^{+}(x),\psi(x)        \rangle   ={\rm Real}. $$

Interesting that this form is not positive definite. The absence of a positive definite renormallzed bilinear form seems to ba a general property of all algebras of generalized functions with differentiation and Involution (of [2]). The construction of the algebra of generalized functions with differentiation and Involution is completed.

Conclusions
If functions in $$\{ {\rm A }\}$$ are regarded as operators of multiplication by functions in $$~\{ {\rm \Psi }\}$$, the algebra $$~\mathcal{A}$$ can be extended by completing it with the differentiation operators $$~\left( {\rm d}/{\rm d}x\right)^n$$, which also act on functions in $$\{ {\rm \Psi }\}$$.

Algebraic operations and differentiation
We have constructed the algebra $$\{ \mathfrak{A}\}$$ of all differential operators of finite orders with coefficients in {A}. The typical element of such an algbra can be written as $$A(x)+B(x){\rm d}/{\rm d}x+C(x)({\rm d}/{\rm d}x)^2+...+G(x)({\rm d}/{\rm d}x)^n $$. In the algebra $$~\{ \mathfrak{A}\}~$$, an involution will be defined by setting

$$(46) ({\rm d}/{\rm d}x)^+ = - ({\rm d}/{\rm d}x) .$$

All the operators of $$~\{ \mathfrak{A}\}~$$ are defined on the complete space $$~\{ \Psi\}~$$ and do not take one out of this space.

Physical sense
We obtain a theory of the type of one-dimensional quantummechanics; in this case, se treat $$~\{ \Psi \}~$$ as the space of the state vectors and $$~\{ \mathfrak{A}\}~$$ as the algebra of local observables. In such a theory,

the potential may show bretty singular behavior: $$(47) V(x)=V_{\rm f}(x)+ \sum_{m=0}^M\delta^{(m)}(x)V_m \sum_{n=0}^N\delta^{(n)}(x)\varepsilon(x)V^n $$

Such a theory does not have a direct physical meaning because the metric is indefinite. The inescapability of an indefinite metric in a quantum theory with strongly singular potentials was pointed out by Berezin [7].

However, in the case of the algebra $$~\{ \mathfrak{A}\}~$$ it is possible to go over to smaller algebras on smaller spaces in each of which the metric is positive and all the observables contain singular terms of the types defined. This makes it possible to obtain physically meaningful quantummodels with strongly singular con- centrated potentials. The description of such models requires a separate paper [12].