# PROBLEM CONCERNING THE FUNDAMENTAL EQUATIONS OF THE RELATIVISTIC QUANTUM THEORY OF THE FIELD

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Collection:

Document Number (FOIA) /ESDN (CREST):

CIA-RDP82-00039R000100230012-3

Release Decision:

RIPPUB

Original Classification:

R

Document Page Count:

4

Document Creation Date:

December 22, 2016

Document Release Date:

May 10, 2012

Sequence Number:

12

Case Number:

Publication Date:

March 6, 1952

Content Type:

REPORT

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CIA-RDP82-00039R000100230012-3.pdf | 1.23 MB |

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Declassified in Part - Sanitized Copy Approved for Release 2012/05/10: CIA-RDP82-00039R000100230012-3
STAT
Problem Concerning the Fundamental Equations
of the Relativistic Quantum Theory of the Field
N. N. Bogolyubov (Corresponding Member of Academy
of Sciences USSR);, Doklady Akademii Nauk SSSR,
Volume LIKKI, No 5, pages 757-760.
Moscow/Leningrad: 11 December 1951.
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Declassified in Part - Sanitized Copy Approved for Release 2012/05/10 : CIA-RDP82-00039R000100230012-3
tos
"Problem Concerning the Ftindamental Equations
of the Relativistic Quantum Theory of the Field"
N. N. Bogolyubov
/gote: The following report appeared in the regular Mathematical Physics
*section of the thrice-monthly journal Doklady Akademii Nauk SSSR, Volume
81, No 5, 11 December 1951, pages 757-760,7
In present works on the quantum theory, fields proceed from the
representation for the interaction in which the wave vector M is considered
assumes the form:
as a functional of the spatially similar surface a- andSchr8dinger's equation
ina
do- 1
x I
It is very clear, however, from considerations of covaancy that the introduction
of arbitrary spatially similar surfaces is in general superfluous and that it is
limited completely sufficiently by the class of spatially similar hyperplanes.
Such hyperplanes can be given by the equation
xt
L.
= x00
with unit temporally similar vectorrtl by characterizing them by the scalar T
and by the three spatial components (IF- In this way the wave vector becomes a
function of the four variables indicate .
If we desire to preserve the most characteristic features of the present
theory -- the strict determinization of the evolution of the wave vector and the
preservation of its norm during transition from one hyperplane to another -- then
we must write the fundamental wave equations, for example, in the following form:
. .
aM(T
=
am(TA)
ih = H (r,O(TA) oc=
af C4 C)(
1)
in which the operators H, H must be Hermitian. Here we can consider, as is
usual, that H, H, are exprehions depending upon the operators of generation and
annihilation of rree particles. The conditions governing the compatibility of the
system of equations (1) will be:
?
it15H HH =
qco(
(2)
a or r3 HigHot
In order to formulate the requirement of the relativistic covariancy of
these equations, we introduce the unitary operator UT with whose aid we transform
the operators of the free particles during transformgtion of spaces by the Lorentz
transformation L = ;obit, (Ltr x = x+a). This condition then can be written down
in the following form:
*6.0
U H(T tft oN tA ? 1.$ 0 4
r'
+
UTH(
??.0,
LsLk
H (t )(gL
Tit 0(4,3 ) r r ot,
.4.4.V41 '41
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(3)
1'4
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This remark, however, is vitiated.. ,by the fact that in all physically
interesting cases such an operator asIS(T,t) does not exist.
More accurately, if such an operator does exist, then all energy levels
of the considered system will with the presence of interaction be the same as during
the complete absence of interaction.
We can attempt to bypass the indicated dif4culj,I by noting that we do not
need at all that there should exist an operator,...itselfr?Vywhich reduces the wave
vector of free nor1,74Kterp:0,q&particles to11011(T;E); it-ii-only necessary that the
symbolic product 18(TA.)S(r,Oshould possess meaning, which repFesnts the operator
of the transformatro-fi"the-wave vector with the hyperplanel(Tlf, ,)?to the hyper-
plane t&,t).
Let us condider the formal expansion:
1 1 n
Tti-S(T,t) 460 Sn(r,) ? ? (7)
such that
and
Then the expressions following satisfy formally our conditions
as 1 as as .+
1 H = ?I + .?(-2_ ?18 ) +
at if. 'at at a.
i
! H
= ..........? 2(2. -----?'i)k .4.s
i) N
, 0 aft (X in
? ? ?
? ? 4
(. )11 2:
(Iler.n)5r n-k
1,n as
L
(1/4Kn)at. n-k
)n7-1(
? 0 0
The expressions for H, H are thus obtained in the form of series,,
even ordinary equations, however, the quantum theory of fields contain
expansions of a number of quantities, for example field mass and charge,
which are employed for renormalization.
Now arises the problem of selectingS ()such that the series (10)
should possess meaning and the following seri6sn shOuld converge
s(-1.7,,)s(T,t) = 1 ?
? ?
)31 li(-1)n-ksk( T1 31 An_k(T4)
1
(Mtn)
(a)
if only ferit',t1 sufficiently close tok,t.' This latter condition would ensure
1
thepossibilityoftheintegration of the amental equations (1).
Mathematical Institute
imeni V. A. Steklov,
Academy of Sciences USSR.
-E N D-
-3.
Submitted 15 October 1951.
Declassified in Part - Sanitized Copy Approved for Release 2012/05/10: CIA-RDP82-00039R000100230012-3