Vladimirov & Wightman Generalized Functions in Rn

On the Generalization of Vladimirov and Wightman
Generalized Functions
By Susana Elena Trione
In this article we generalize to Rn the Vladimirov generalized functions
and the Wightman-type generalized functions. Our results are the formulae
(I,1;1), (I,1;3), (I,1;4), (I,1;5), (I,2;1), (II,1;2), (II,1;3), (II,1;4), and (II,1;6).
Here, we also introduce an n-dimensional multiplicative products of the
Wightman-type generalized functions (cf. formula (II,2;4)).
Notations: Let x1 ; x2 ; : : : ; xp ; xp+1 ; : : : ; xn ‘ be a point of the ndimensional Euclidean space Rn ; we define
ŽxŽ2 = x21 + · · · + x2p ;
ŽX 2 Ž = x21 + · · · + x2p − x2p+1 − · · · − x2p+q :

1
sgn x0 = 0

−1
1
2X 2 ‘ =
0
if x0 > 0;
if x0 = 0;
if x0 < 0:
if X 2 > 0;
if X 2 < 0:
Here p is the number of positive squares of the form X 2 , q is the number of
negative squares of the form X 2 , and p + q = n.
Address for correspondence: Instituto Argentino de Matemática, Viamonte 1636, 1er. cuerpo, 1er. piso,
1055 Buenos Aires, Argentina. Fax: 54-1-372-5976; E-mail: trione%[email protected].
STUDIES IN APPLIED MATHEMATICS 98:207–211
207
© 1997 by the Massachusetts Institute of Technology
Published by Blackwell Publishers, 350 Main Street, Malden, MA 02148, USA, and 108 Cowley Road,
Oxford, OX4 1JF, UK.
208
Susana Elena Trione
I.1. We define the following generalized function:
lim ŽxŽ2 − xp+1 + iε‘2 − xp+2 + iε‘2 − · · · − xn + iε‘2
ε→+0
−s/2
= ŽxŽ2 − xp+1 + i0‘2 − xp+2 + i0‘2 − · · · − xn + i0‘2
−s/2
;
s = 0; ±1; ±2; : : : : (I,1;1)
If s ≤ 1, this generalized function is a locally integrable function of power
increase at infinite and it may be evaluated without difficulty.
It is clear that
−s/2
lim ŽxŽ2 − xp+1 + iε‘2 − · · · − xn + iε‘2
ε→0
= lim ŽxŽ2 − ’xp+1 + iε‘2 + · · · + xn + iε‘2 “
ε→0
= lim ŽxŽ2 − x2p+1 + 2xp+1 εi + iε‘2 + · · ·
ε→0
+ xn + 2xn εi + iε‘2
−s/2
−s/2
= lim ŽxŽ2 − x2p+1 − · · · − x2n + qiε‘2
ε→0
+ 2iεxp+1 + · · · + xn ‘
−s/2
:
(I,1;2)
If s = 1, we have
lim ŽxŽ2 − xp+1 + iε‘2 − · · · − xn + iε‘2
ε→0
−1/2
= ŽxŽ2 − ’xp+1 + i0‘2 + · · · + xn + i0‘2 “
−1/2
= ’2X 2 ‘X 2 “−1/2 + i sgn xp+1 + · · · + xn ‘
· ’2−X 2 ‘ −X 2 ‘ “−1/2 :
(I,1;3)
If s = 2k + 1, k = 1; 2; : : : ; taking into account (I,1;2), we obtain
2
−k−1/2
ŽxŽ − xp+1 + i0‘2 − · · · − xn + i0‘2
= ’2X 2 ‘X 2 “−k−1/2 + i−1‘k sgn xp+1 + · · · + xn ‘
· ’2−X 2 ‘−X 2 ‘“−k−1/2 :
If s = 2k, k = 0; 1; : : : ; from (I,1;2) we get
2
ŽxŽ − xp+1 + i0‘2 − · · · − xn + i0‘2
=
(I,1;4)
−k
iπ
sgn xp+1 + · · · + xn ‘ δk−1‘ X 2 ‘
k − 1‘ !
+ −1‘k Pf
1
y
X 2 ‘k
where Pf indicates, as usual, the finite part.
(I,1;5)
Vladimirov and Wightman Generalized Functions
209
The formula (I,1;5) can be obtained in the same manner as we proved the
formula (I,13;21), [1, p. 59].
I.2. We define the following generalized function:
lim ln ŽxŽ2 − xp+1 + iε‘2 − · · · − xn + iε‘2
ε→+0
= ln ŽxŽ2 − xp+1 + i0‘2 − · · · − xn + i0‘2
(I,2;1)
= ln ŽXŽ2 + iπ sgn xp+1 + · · · + xn ‘2X 2 ‘ :
Remark: We note that if X 2 = x21 + x22 + · · · + x2n−1 − x20 , the generalized
functions (I,1;4), (I,1;5), and (I,2;1) are the Vladimirov generalized functions
(cf. [2, pp. 298, 299, formulae (133), (136), and (138)]).
Notations: Let x = xν ‘ = x0 ; X‘, X = x1 ; x2 ; : : : ; xn−1 ‘ be a point of
the n-dimensional Minkowski space of real numbers, with
x2 = x2ν = x20 − X 2 ;
X2 =
n−1
X
x2i :
i=1
II.1. We define the Wightman-type generalized functions x2 ∓ ix0 0‘λ for
Re λ > 0 by the following formulas
def
x2 − ix0 0‘λ = lim e−iπλ X 2 − x0 − iε‘2 ‘λ
ε→0
(II,1;1)
= e−iπλ ’eiπλ 2x0 ‘x2 ‘ + e−iπλ 2−x0 ‘x2 ‘“
+ ’2−x20 ‘X 2 ‘λ “;
(II,1;2)
and
def
x2 + ix0 0‘λ = x2 − ix0 0‘λ :
(II,1;3)
We observe that in the formula (II,1;2) the poles cancel and therefore
x2 ∓ ix0 0‘λ are analytic in all λ-complex plane.
For λ = −n, we obtain
x2 ∓ ix0 0‘−n = x2 ‘−n ±
+
iπ−1‘n−1
εx0 ‘ δn−1‘ x2 ‘
n − 1‘ !
π3
2n−3 n
n−2
δx‘
;
− 1‘ ! n − 2‘ !
where we have written, by definition,
εx0 ‘ δ x2 ‘ = 2x0 ‘ δ x2 ‘ − 2−x0 ‘ δ x2 ‘;
(II,1;4)
210
Susana Elena Trione
and
=
p
X
x2i −
i=1
n
X
x2i :
(II,1;5)
i=p+1
We also have
x2 − ix0 0‘−n − x2 + ix0 0‘−n =
2πi−1‘n−1
εx0 ‘ δn−1‘ x2 ‘:
n − 1‘ !
(II,1;6)
The present formulae are defined for the four-dimensional real Minkowski
space in [3, Sect. 6.2].
II.2. In [4, p. 398, formula (3.5)], Güttinger and Pfaffelhuber say textually:
“Products of generalized functions x2 − ix0 ‘λ are defined by
x2 − ix0 0‘λ · x2 − ix0 0‘µ = x2 − ix0 0‘λ+µ :”
(II,2;1)
In this paragraph we generalize the four-dimensional multiplicative product
(II,2;1) and also we justify that product by means of the method used in [1,
Sect. I.3].
We begin by proving the formulae (I,2;1) in the bidimensional space; it is
x1 − ix0 0‘λ · x1 − ix0 0‘µ :
(II,2;2)
Taking into account the definitory formula (II,1;1) we approximate both
terms of (II,2;2), respectively, by the smooth functions, of identical structure,
X 2 − x0 − iε‘2 ‘λ and X 2 − x0 − iε‘2 ‘µ and define (first for Re λ > 0, Re
µ > 0, and then, by analytical continuation, for every λ, µ ∈ C)
x21 − ix0 0‘λ · x21 − ix0 0‘µ
def
= lim ’e−iπλ x21 − x0 − iε‘2 ‘λ · e−i𵠐x1 − x0 − iε‘2 ‘µ “
ε→0
= lim e−iπλ+µ‘ x21 − x0 − iε‘2 ‘λ+µ
ε→0
def
= x21 − x0 − i0‘2 ‘λ+µ :
(II,2;3)
We remark that it can be shown that our definition (II,2;3) is intimately
related with the method used by Antosik et al. [5] to define the multiplicative
product of two generalized functions.
The following proposition is a multidimensional analogue of (II,2;3).
Theorem. The following formula is true for λ, µ ∈ C:
x2 ∓ ix0 0‘λ · x2 ∓ ix0 0‘µ = x2 ∓ ix0 0‘λ+µ ;
(II,2;4)
where x ∈ Rn and x2 ∓ ix0 0‘r are defined by (II,1;2) and (II,1;3), respectively.
Proof : It is identical “mutatis mutandis” with our proof of (II,2;3) and
therefore we omit it.
Vladimirov and Wightman Generalized Functions
211
References
1. S. E. Trione, Distributional Products, Cursos de Matemática, No. 3, IAM, CONICET,
Buenos Aires, 1980.
2. V. S. Vladimirov, Methods of the Theory of Functions of Many Complex Variables, MIT
Press, Cambridge, MA, 1966.
3. A. Rieckers and W. Güttinger, Spectral representations of Lorentz invariant distributions and scale transformation, Commun. Math. Phys. 7:190–217(1968).
4. W. Güttinger and E. Pfaffelhuber, Dynamics of Unrenormalizable Interactions in
Minkowski and Euclidean Spaces, Vol. LII A, No. 2, Il Nuovo Cimento, 1967.
5. P. Antosik, J. Mikusiński, and R. Sikorski, Theory of Distributions: The Sequential Approach, Amsterdam/Warszawa, 1973.
Universidad de Buenos Aires, Argentina
(Received December 21, 1995)