Algebra of generalized functions (Shirokov)

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An associative algebra 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 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.


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 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 , the model is superrenormalizable, but for potentials contacting , , 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.


We introduce the notation and definitions needed to formulate the basic assumptions. Let be the space of complex-valued functions, including the necessary generalized functions, be the dual space of test functions, and denote a bilinear functional on and . We shall denote elements in by , , ...; elements in by , , ...


Our basic assumptions are as follows.

A. The set is a topological vector space.

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

C. The set is also a topological vector space and forms a dual pair with (see, for example,[8]).

D. Each element in can act on each element in as an operator. This means that there is defined the bilinear product . Such multiplication satisfies the condition of associativity

for all , belonging to and . The relation (1) shows that, in general, the space also contains generalized functions. On products of the type , we assume that they coincide with the ordinary products of functions for nongeneralized functions and . The same we assume with respect to multiplication by any C-number.

E. For all functions in and , 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 and , there is defined an operation of involution that coincides with complex conjugation for ordinary functions and commutes with differentiation.

Algebras containing were studied in [9,10] under different systems of basic assumptions (which did not allow to include their derivatives).

Construction of Algebra [edit]

We begin the construction of the required algebra with the spaces and and the functional . We first introduce the smaller spaces and . Both spaces consist of complex functions of a single variable that are finite, with all their derrivatives, everywhere except at the point , and at have finite limits from the left and the right for the functions themselves and for all derivatives.

Slow rise and fast decay[edit]

At infinity, functions in , together with all their derivatives, can have slow (i.e., not faster than polynomial) growth, while functions in decrease rapidly with all their derivatives (i.e., faster than any polynomial). The expression for a bilinear functional has the standard form


The spaces and 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.

Construct Functionals with compact support[edit]

We now begin construct functlonals over with support at the point . We first introduce the notations

Note that for all real the functionals are defined. This allows to define the functionals and as follows:

Eash of these functionals has supports at .

We complete the space with the generalized functions and . We denote the completed space by . It contains all functions of the form

Here, and are non-negative integers and and are complex numbers. (Here, the superscript not an exponentiation.)

The functional is now defined for all in and for all in . The topology is first introduced separately for each space with fixed and , after which the topology on is obtained through the rigorous inductive limit with respect to and .

Functions with compact support[edit]

Our next task is to complete the space with generalized functions with support at . Note that on functions of the space {A} for all real , there is defined the functional , which corresponds to the generalized function (where for and for .

Regilarizing functionals[edit]

Functions allow to introduce new functionals on functions of the space (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 . We denote these functlonals by and ; we define them as follows:



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,

where is the Kronecker diagonal tensor. Similarly,

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

Thus, the functionals and 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.


Using the functionals and , we can construct a regularizing operator, which we denote by . This operator is applied to the left and maps from to ; define it with

Both summations in fact always terminate, since the number of singular terms in is finite. It follows from (8)-(11) that


so that

Now it is seen, why we call "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 . Namely, on we define the functlonals and by

Accordingly, we complete the space by the functions and . We denote the resulting space by . It contains all functions of the form


where, in analogy with (5), and are non-negative integers, and amd are complex numbers.

Concentrated functionals[edit]

On functions in , we now introduce concentrated functionals and , defining them in analogy to (6) and (7):


All the comments made after the dual formulas (6) and (7) apply to (18) and (19). In analogy with (13),


As the dual to , we have the operator , which acts to the right and maps from to :


As in (12), the summations here are also in fact always finite. In accordance with (20),


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

Generalized bilinear form[edit]

Only now that we have the possibility of defining the definition of the required renormalized bilinear functional of the type , suitable for all functions in the spaces and , respectively:


Assume, the functions and are decomposed into regular and singular parts:


then the definition (22) can be rewritten in the more perspicuous form


We emphasize that the associativity condition(l) is not satisfied for the regularizing operators and . Fortunately, these operators are not included in the set of elements of the renormalized algebra.

Associative multiplication[edit]

This section, constructs a renormalized associative operation of multiplication for functions in . Usually, such operation is believed to not exist, to, we begin with the most important part.

First note, that the operators and form a dual pair; it follows from

for all (all ) such that that (respectively, ). It therefore follows from the associativity condition (1) that all renormalized products of the type are defined.


The operator renormalized products of the type that do not take one out of are defined. We begin with the construction of these last.

We give first two definitions of two new operators and such that for them there exist the products , :

Where are binimial coefficients.

The operators and do not belong to the algebra, but, in contrast to and , they are assotiative; for example, .

Simplification of notations[edit]

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 and will be denoted in the same way as the operation of ordinary multiplication of functions, i.e., , , 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 is defined by


and the renormalized product by

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[edit]

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 . At the same time,


It is therefore convenient to denote

and assume that the functions and behave in accordance with (3) and (4) in functionals of type and in accordance with (9), (10), and (30) in functionals of the type . In such a way, (29) becomes an identity.

The following simplification is suggested by the fact that for any infinitely smooth functions in and in , we have

where is the ordinary delta function. In such a way, there is therefore no possibility of confusion in writing


Finally, it follows from (27), that . In the new notation, the renormalized product has the same form as :

To conclude this section we note that (27) (or, which is now the same thing (33)) implies that

In addition to anti-commuting signum and delta, the square of signum happens to be identically unity,

and this identity holds for all , even at . 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 berformed approximating functions with C-numbers; equaiton (34) explicitly prohinits such approximaiton. Practically, the only the regular part of a function (smooth part) can be aproximated; all singular parts must remain symbolic.

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


In this section, we define differentiation of functions in and . In domains in which a function is differentiable, it is differentiated in the usual manner. The functions , , , 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 , which are defined on functions that are infinitely smooth at the point , so that the operation of differentiation is defined and leads to the result

in such a way there is no problem with differentiation of , following the common rules.

Differentiation of Signum[edit]

It remains to differentiate the function . On smooth functions, Therefore, in accordance with (3),

For this expression, the Integration by parts is possible, because the function and its derivative are smooth at . Notation means non-singular part of the expression in parenthesis, it is regular part of the derrivative of . It is just for , and at the origin has the corresponding finite limits form the left and the right.

In the limit, we find from (38) that

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

Equation (37), (39), and (40) in conjunction with (30) completely define the operation of differentiation for functions in and ~.

Differentiation rules[edit]

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 , we obtain (34) for , 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 . 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[edit]

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


In this way, .

The space can be completed with functions , which, it is true, requires the introduction into the completed space of also the functions

With such elements, we get a new algebra containing functions , , and their products.

Algebra [edit]

It remains to introduce into the algebra the involution , . The involution operation must satisfy the standard requirements


, , ,

Here, 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 and :


The relations (42) justify the notations , introduced in (15) and (16). If functions in are treated as operators of multiplication by functions in , then the involution coincides with Hermitian conjugation. This last is also defined for the operators , and

which justifies the notations , .

It follows from (42) that the space of functions consists of the same functions as . And since , for any pair there is defined the renormalized bilinear form which is obtained by the substitution in (23):

The bilinear form (44) is 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.


If functions in are regarded as operators of multiplication by functions in , the algebra can be extended by completing it with the differentiation operators , which also act on functions in .

Algebraic operations and differentiation[edit]

We have constructed the algebra of all differential operators of finite orders with coefficients in {A}. The typical element of such an algbra can be written as . In the algebra , an involution will be defined by setting

All the operators of are defined on the complete space and do not take one out of this space.

Physical sense[edit]

We obtain a theory of the type of one-dimensional quantummechanics; in this case, se treat as the space of the state vectors and as the algebra of local observables. In such a theory, the potential may show bretty singular behavior:

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 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].


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