Clothed particle representation in quantum field theory: boost generators of the Poincaré group
The method of the unitary clothing transformations is used to construct the generators of the Poincaré group in the instant form of relativistic dynamics in the second and third orders in the coupling constant. It is shown that the respective algebra of generators is fulfilled up to the sixth order....
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2007 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Clothed particle representation in quantum field theory: boost generators of the Poincaré group / V.Yu. Korda and P.A. Frolov // Вопросы атомной науки и техники. — 2007. — № 3. — С. 33-37. — Бібліогр.: 6 назв. — рос. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859653589264236544 |
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| author | Korda, V.Yu. Frolov, P.A. |
| author_facet | Korda, V.Yu. Frolov, P.A. |
| citation_txt | Clothed particle representation in quantum field theory: boost generators of the Poincaré group / V.Yu. Korda and P.A. Frolov // Вопросы атомной науки и техники. — 2007. — № 3. — С. 33-37. — Бібліогр.: 6 назв. — рос. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The method of the unitary clothing transformations is used to construct the generators of the Poincaré group in the instant form of relativistic dynamics in the second and third orders in the coupling constant. It is shown that the respective algebra of generators is fulfilled up to the sixth order. The relativistic invariance of the mass and vertex corrections derived is proven.
За допомогою методу унітарних одягаючих перетворень побудовані генератори групи Пуанкаре в миттєвій формі релятивістської динаміки в другому і третьому порядках за константою взаємодії. Показано, що алгебра генераторів групи задовольняється до шостого порядку. Обґрунтовано релятивістську інваріантність розрахованих поправок до мас частинок і константі взаємодії.
С помощью метода унитарных одевающих преобразований построены генераторы группы Пуанкаре в мгновенной форме релятивистской динамики во втором и третьем порядках по константе взаимодействия. Показано, что алгебра генераторов группы удовлетворяется до шестого порядка. Обоснована релятивистская инвариантность рассчитанных поправок массам частиц и константе взаимодействия.
|
| first_indexed | 2025-12-07T13:37:00Z |
| format | Article |
| fulltext |
CLOTHED PARTICLE REPRESENTATION
IN QUANTUM FIELD THEORY:
BOOST GENERATORS OF THE POINCARÉ GROUP
V.Yu. Korda and P.A. Frolov
Institute of Electrophysics and Radiation Technologies NAS of Ukraine,
Kharkov, Ukraine;
e-mail: kvyu@kipt.kharkov.ua
The method of the unitary clothing transformations is used to construct the generators of the Poincaré group in
the instant form of relativistic dynamics in the second and third orders in the coupling constant. It is shown that the
respective algebra of generators is fulfilled up to the sixth order. The relativistic invariance of the mass and vertex
corrections derived is proven.
PACS: 21.45.+v; 24.10. Jv; 11.80.-m
1. INTRODUCTION. GENERATORS
OF THE POINCARÉ GROUP
Poincaré invariance requires that there exists a uni-
tary representation of the Poincaré group defined in a
Hilbert space. Corresponding ten generators fulfill the
set of commutation relations:
[ , ]i k iklN N i Jε= − l l, , , [ , ]i k iklJ N i Nε= [ , ]i k ikl lJ J i Jε=
[ , ]l lH N iP= − , [ , [ , , ]i l ilP N i Hδ= − , ]i k ikl lJ P i Jε=
[ , ] 0iP H = , [ , , ] 0iJ H =
[ , ] 0i kP P = , ( . (1) , , 1, 2,3)i k l =
Here is the Hamiltonian operator, and
are three components of the momentum, angular mo-
mentum and boost operators, respectively. In the instant
form of relativistic dynamics (after Dirac), four opera-
tors and N carry interaction.
H
H
,iP iJ iN
One-particle eigenstates of and differ from
states of their free parts:
H N
†( ) ( )FH Hα α α αΩ ≠ Ω† , (2)
†( ) ( )Fα α α αΩ ≠ ΩN N † . (3)
Here denotes the whole set of creation and destruc-
tion operators of the bare particles with bare masses
which interact by means of the bare coupling.
α
Therefore, one may ask a question whether there ex-
ists such a set of creation and destruction operators α
for which the total Hamiltonian , the boost op-
erators and their free counterparts and
would fulfill the conditions
c
( )cK α
( )cαB
)
( )F cK α
( cαBF
†( ) ( )c c F c cK Kα α α αΩ = Ω† , (4)
†( ) ( )c c F c cα α α αΩ = ΩB B †
R
cα
I cα
α
α
),p
ferm
, (5)
keeping the algebra?
Greenberg and Schweber [1] assumed that such set
of operators α existed and was connected with the
initial set of operators α via the unitary transformation
which kept S-operator intact
c
R R
c e eα α−= , α α , . (6) R R
ce e−= †R = −
Generator of the unitary transformation, called “cloth-
ing”, was chosen in a way the Hamiltonian operator
R
( ) ( ) ( ) ( )( ) R R
c c F c IH e H e K K Kα α α α−= ≡ = + , (7)
satisfied the condition (4).
However, herewith it was not obvious that simulta-
neously the boost operator could be presented in a simi-
lar form
( ) ( ) ( ) ( )( ) R R
c c F ce eα α α α−= ≡ = +N N B B B , (8)
so that the condition (5) would be automatically fulfilled.
2. CLOTHED PARTICLES AND BOOST
GENERATORS
The problem will be analyzed using the quantum
field model in which a spinor fermion (nucleon) field
interacts with a neutral meson (pion) field by means of
the Yukawa-type threelinear pseudoscalar (PS) cou-
pling. Within the model,
( ) ( ) ( )F IH H H Hα α= = + , (9)
( ) ( ) ( )F Iα α= = +N N N N . (10)
The free parts of and have the form H N
† ( ) ( )FH d a aω= ∫ kk k k ( ) (†
r
d E b r b r+ ,∑∫ pp p
( ) ( )†d r d r + , , p p , (11)
( )F mesα = +N N N , (12)
2
2mes
i d d
ω ω µ
ω ω
+ +
= ∫ k k
k k
k k
N k k '
'
''
†( ) ( ) ( )a aδ∂ −
×
∂
k k k k
k
' ' , (13)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (1), p.33-37. 33
'
( ' )'
2ferm
i m md d E
E E
δ∂ −
=
∂∫ p
p p
p pN p p
p
† †( , ) ( , ) ( , ) ( , )
r r
u r u r b r b r
′,
′ ′ ′ ′× ∑ p p p p
† †( , ) ( , ) ( , ) ( , )v r v r d r d r′ ′ ′ ′ − p p p p . (14) M m
Here 2 2E m= +p p , 2 2ω = +k k
p
µ m, and are
nucleon and meson physical masses; and are nu-
cleon and meson momenta; r is the spin projection
index. The creation (destruction) operators of mesons
and the same for fermions
and antifermions
satisfy the commutation relations
µ
, ,dp p
k
( )
( ) ( )(†a ak k
( ) ((† ,b r bp
)
)
p
)p
n mes α
α
n mes α
α
c
),
), rp ( )( )d r r†
( ) ( ) ( )†,a a δ′ ′ = − k k k k ;
( ) ( ){ } ( )†
,, , , r rb r b r δ δ′′ ′ ′= −p p p ;
( ) ( ){ } (†
,, , , r rd r d r δ δ′′ ′ ′= −p p p . (15)
The interaction parts of and have the form H N
( ) ( ) ( )I reH V Mα α ,= +
( ) ( )ren ferm renM Vα,+ + ; (16)
( ) ( ) ( )I reα α ,= +N W N
( ) ( )ren ferm renα,+ +N W . (17) g
The interaction operators V and are written as W
ˆ( ) ( )V d a HVα = + . .,∫ kk k (18) R d
( ) ( ) (†
,
ˆ i i j j
r r i j
d d F r V r r F rV ,
′,
′ ′ ′ ′ ′= , , ; ,∑∑∫k kp p p p p p ,(19)
( )
( )
( )3 2 22
i j
ig mV r r
E E
δ
ωπ
, /
′
′ ′ ′, ; , = + −k
k p p
p p p k p
( ) (5i jr U rU γ′ ′× , ,p )p
),
, (20)
ˆ( ) ( )d a H cα = + . .,∫ kk kW W (21) K H
( ) ( ) (†
,
ˆ ,i i j j
r r i j
d d F r r r F r,
′,
′ ′ ′ ′ ′= , , ; ,∑∑∫k kp p p p p pW W (22)
( )
( )
( )
3 2 22
i j
g mr r
E E
δ
ωπ
, /
′
′∂ + −−′ ′, ; , =
∂
k
k p p
p p
p p
k
k
W
( ) (5i jr U rU γ′ ′× , ,p )p
,
,
) )r
. (23) 3
Here – physical coupling constant and the notations
are as follows
g
( ) ( )
( )
( )
( )
1
2
U r u r
U r
U r v r
, ,
, = = , − ,
p p
p
p p
( ) ( )
( )
( )
( )
1
†
2
F r b r
F r
F r d r
, ,
, = = , −
p p
p
p p
, (24)
where and v are spinors satisfying usual
Dirac equations with physical masses.
(u r,p (− ,p
The mass counterterms are presented as
(
2 2
†0( ) ( ) ( )
4ren mes
dM a
µ µ
α
ω,
−
= ∫
k
k k ka
.
)( ) ( )a a H c+ − + .k k , (25)
( ) ( )†
0
, ,
( )ren ferm i
r r i j
dm m F
E
α,
′
′= − ,∫ ∑∑
p
p p r
)r(, ( ; )i j jM r r F′× , , ,p p p , (26)
( )2 2
0
1( )
4ren mes
i d dα µ µ
ω ω
,
′
′= − ∫
k k
N k k ;
( ) ( )† ( ) ( ) ( ) ( ) ,a a a a H c δ
∂′ ′× + − + . .
∂
k k k k k k
k
′− (27)
( ) ( )†
0
, ,
1( ) ', 'ren ferm i
r r i j
im m m d d F r
E E
α,
′
′= − ∑∑∫
p p
N p p
( )
p
× ( ), ( ; )i j jM r r F r δ
∂′ ′ ′, , , −
∂
p p p p p
p
, (28)
where ( ) (, ( ; )i j i jM r r U r U′ ′ ′ ′, , = , ,p p p p ) 0mr
c−
), ,j
, and
– nucleon and meson bare masses.
0µ
The vertex counterterms V и are determined
by the same formulae (18)-(23) as for V and but
with substituted by δ ≡ where is the
bare coupling constant.
ren
0g g
renW
g−
W
0g g
Total Hamiltonian (9) and boost (10) consist of the
- order operators (18)-(23), called bad, which prevent
satisfaction of the conditions (4)-(5). Their removal by
the first clothing transformation [2] with the generator
1
ˆ ( ) .c cR a H= ∫ kk k . , (29)
( ) ( ) (†
, , ,
, ,
ˆ , , ; ,c c i i j c
r r i j
R d d F r R r r F r
′
′ ′ ′ ′ ′= ∑∑∫k kp p p p p p (30)
( ) ( )
( ) ( )
,
, 1 1
, ; ,
, ; ,
1 1
i j
i j i j
V r r
R r r
E E ω− −
′
′ ′
′ ′ =
− −− −
k
k
p p
p p
p p
k
,
( , 1,2)i j = , (31)
leads to
( ) ( ) ( ) ( ) [ ]1
2c F c ren c ren cM V Rα α α α= + + + V,
[ ] [ ]1
3renR M R R V + , + , , + .. . , (32)
( ) ( ) ( ) ( ) [ ]1
2c F c ren c ren c Rα α α α= + + + ,B N N W W
[ ] [ ]1
renR R R + , + , , + .. N W . (33) .
Thus, and appear free from bad terms up
to second order in .
( )cK α ( )cαB
g
3. MASS AND VERTEX
RENORMALIZATION IN BOOST
GENERATORS
After first clothing, (32) and (33)
contain bad terms of the and higher orders. In
, the operators [ and
( )cK α
2g
( )cαB
]( )cK α R V,
34
ren ren mesM M ,=
R,W
ren fermM ,+
] ren
g
( )
have parts of the -order, as
well as [ and N N in .
2g
ren mes ren ferm, ,= +N
[ ]R V,
ren mesM ,
2
( )cαB
( )
4
4
erm
2 2
0δµ µ≡ −
(p E= p
2 2
2
µ
π
=
),p k
]R V,
µ−
,ren f
2
4 (2
gm
m π
= 2 ,m mδ ≡ −
1( ) dI p ∫ k
k
pk
ω µ
,
2 ( )I p
E
∫ q
q
( )q E= ,q q
R,W N
2
,q
−
2g
d m
m pq
=
ren mes,
p
qµ µ
+
]V
]
renV R+ ,
3g
g
[ ]R R V , ,
renV
3 2
g d
ωπ µ′
∫
q
q
(2 )
gδ =
2
p
µ
′+ −p p q
pq
′ ′
2
d m
+ ∫
q
q p
E m pq µ + −
p k−
kq
′
2
d m
+ ∫
q
q p
m p q′ + −
p k −
q
′
=q q 2ω µ′ +2q
3g
=
( , )p E ′′ ′= p p
[ ]R R , , W
( ,q ω ′′ ′= q q
renW
After normal ordering, involves parts bilinear
in the meson operators that can be cancelled by the re-
spective counterparts from . Then we have
meson mass shift of the -order [2]
2
3 21
4
g d
E pk
µ +
∫ p
p
, (34)
)where and . At the same time,
another part of [ partly cancel with in the
same order, giving the fermion mass shift [3]
(ω= ,k k
M
[ ]0 13 ( ) ( )
)
I p I p+
2 2
1 1
2 2pk pkµ
= −
− +
2 2
2 2 22 2
pq m
m p
− +
− − +
(35)
with .
Having fixed the mass corrections in the -order,
we immediately verify that similar terms in the opera-
tors [ ] , and cancel. ren ferm,N
Operator [ ] [1
3renM R R+ , , contains
terms of the -order which replicate the operator
structure of the interaction operator V and, thus, start
the program of vertex renormalization .
Using normal ordering of fermionic operators, several
part of the commutator is cancelled with
some part of the commutator [ , providing the
meson and nucleon wave function renormalization in the
lowest order in . At the same time, another part of the
commutator is cancelled with the part of
, determining the charge shift in the -order [4,5]:
[ ]R R V , ,
R M,
ren
g3
3 2
2 22
p q m
m p q
′ ′
′− −
2 2
22 2 2
q m p q
µ
′ ′− +
−
2 2
22 2 2
q m p q
E kµµ
′ ′+ −
+
. (36)
Here 2 2E m+ ; ′q ;
; . )
Having fixed the vertex correction in the -order,
we immediately verify that similar terms in the opera-
tors , and [ cancel. ]
)g
)
renR,N
4. RELATIVISTIC INTERACTIONS
AND BOOST GENERATORS
Selecting bad terms of the -order in , we
further remove them using the second clothing trans-
formation. After that, applying this transformation to
, we see that the latter does not contain bad terms
of the -order too. Proceeding in such a way, we re-
move bad terms from and up to third
order in . The remaining bad terms of higher orders
must be removed via successive clothing unitary trans-
formation.
2g ( )cK α
( )cαB
2g
g
( )cK α ( )cαB
Thus, up to -order, we have 4g
( ) ( ) ( ) ((2) (3) 4
I c c cK K K Oα α α= + + ; (37)
( ) ( ) ( ) ((2) (3) 4
I c c c O gα α α= + +B B B . (38)
Here the operators of physical interactions between
physical particles in the second and third orders are as
follows
( ) ( ) ( )(2)
cK K NN NN K NN NNα = → + →
( ) ( )K NN NN K N Nπ π+ → + →
( ) (K N N K NNπ π ππ+ → + ↔ ) , (39)
( ) ( ) ( )(3)
cK K NN NN K NN NNα π= ↔ + ↔π
( ) ( )K NN NN K NNπ πππ+ ↔ + ↔
( ) ( ...K N N K N Nπ ππ π ππ+ ↔ + ↔ )+
)
. (40)
Similarly, the respective operators in have the
form
(I cαB
( ) ( ) ( )(2)
c NN NN NN NNα = → + →B B B
( ) ( )NN NN N Nπ π+ → + →B B
( ) (N N NNπ π ππ+ → + ↔B B ) ; (41)
( ) ( ) ( )(3)
c NN NN NN NNα π= ↔ + ↔B B B π
( ) ( )NN NN NNπ π+ ↔ + ↔B B ππ
( ) ( ...N N N Nπ ππ π ππ+ ↔ + ↔ +B B ) . (42)
In Eqs. (39) – (42), the transparent notations , N N
and are used to distinguish clothed (physical) nu-
cleon, antinucleon and pion, respectively.
π
For brevity, in what follows we omit the spin indi-
ces.
The nucleon-nucleon interaction operator is equal to
( ) 1 2 1 2
1
2
K NN NN d d d d d′ ′→ = ∫ p p p p k
( ) (11 1 1 11 2 2; ;R V′ ′× ⋅k -kp p p p )
c+
( ) ( ) ( ) ( )† †
1 1 1 2 1 1 1 2 . .F F F F H′ ′× p p p p (43)
The antinucleon-antinucleon interaction operator has
the form
35
( ) 1 2 1 2
1
2
K NN NN d d d d d′ ′→ = ∫ p p p p k
( ) (22 1 1 22 2 2; ;R V′ ′× ⋅k -kp p p p )
c+
( ) ( ) ( ) ( )† †
2 1 2 2 2 1 2 2 .F F F F H′ ′× p p p p . (44)
The nucleon-antinucleon interaction operator can be
presented as
( ) 1 2 1 2
1
2
K NN NN d d d d d′ ′→ = ∫ p p p p k
( ) ( ) ( ) ({ 11 1 1 22 2 2 12 1 2 21 2 1; ; ;R V R V′ ′ ′ ′× ⋅ − ⋅k -k k -kp p p p p p p p
( ) ( ) ( ) ( }22 2 2 11 1 1 21 2 1 12 1 2; ; ;R V R V′ ′ ′+ ⋅ − ⋅k -k k -kp p p p p p p p
);
)′
c+
;
( ) ( ) ( ) ( )† †
1 1 2 2 1 1 2 2 . .F F F F H′ ′× p p p p (45)
The pion-nucleon interaction operator can be given as
( )
( )1 2 1 2 2 111
1 , ;
2
K N N
d d d d R V
π π→
= ∫ 1 2k -kp p k k p p
( ) ( ) ( ) ( )† †
1 2 2 1 1 1 .F a F a H c× p k p k . , (46) +
);
where
( ) ( ) (, ; ;im mjij
R V d R V′ ′ = ∫1 2 1 2k k k kp p q p q q p
( ) ( ); ;im mjV R′ −
2 1k kp q q p . (47)
The pion-antinucleon interaction operator has the
form
( ) (1 2 1 2 2 122
1 ,
2
K N N d d d d R Vπ π → = ∫ 1 2k -kp p k k p p
( ) ( ) ( ) ( )† †
2 2 2 2 1 1 . .F a F a H c× +p k p k
);
(48)
The interaction operator for nucleon-antinucleon
pair production (annihilation) by pair of pions is equal
to
( ) (1 2 1 2 2 112
1 ,
2
K NN d d d d R Vππ ↔ = ∫ 1 2k kp p k k p p );
c+
( ) ( ) ( ) ( )†
1 2 2 1 1 2 . .F F a a H× p p k k . (49)
The interaction operators B are determined
by the same formulae (43)-(49) as for but
with V substituted by .
( )(2)
cα
( )cα
( )(2)
cK α
( )cα W
The operators of the third order in have similar
forms but more difficult structure. For example, the
operator of the pion production on the pair of nucleons
has the form [6]
g
( ) 1 2 1 2 1 2
1
3
K NN NN d d d d d dπ ′ ′→ = ∫ p p p p k k
( ) ( ){ 1 1 2
†
11 1 1 2 211
; , ;R R V′ ′ × ⋅
k k kp p p p
( ) (1 2 1
†
11 1 1 2 211
2 ; , ;R R V′ ′ + ⋅
k k kp p p p )
);
)
)
)
)
], 0= 0
l
( ) (†
1 2 1
1 1 11 2 211
, ;R V R− ′ ′ + ⋅
k k kp p p p
( ) (†
2 1 1
1 1 11 2 211
2 , ; ;R V R− ′ ′ + ⋅
k k kp p p p
( ) (1 2 1
†
11 1 1 2 211
; , ;V R R′ ′ + ⋅
k k kp p p p
( ) ( }†
2 1 1
1 1 11 2 2
11
, ; ;R R V − ′ ′+ ⋅
k k kp p p p
( ) ( ) ( ) ( ) († † †
1 1 1 2 1 1 1 2 2F F F F a′ ′× p p p p k . (50)
5. WHETHER THE POINCARÉ ALGEBRA IS
FULFILLED?
In order to answer this question it is necessary to
evaluate explicitly the set of all commutation relations
(1) in the respective orders. It is important to emphasize
that the corresponding verification is not difficult
though quite tedious. Thus, we just give only some nec-
essary explanations. The commutation relations which
do not contain and are automatically fulfilled.
The rotational and translational invariance of the inter-
action operator V leads to the fact that the
commutation relations [ , and
are also satisfied. The relation
is proven by partial integration and
the way of verifying [ ] and
up to the sixth order is similar [5].
K
l
H
ikl lJ
B
iP H [ ],iJ H =
, lH N iP= −
[ ],i k iklJ N i Jε=
[ ],i l ilP N iδ= −
[ ],i kN N iε= −
6. CONCLUSION
Using the method of the unitary clothing transforma-
tion, we have constructed the generators of the Poincaré
group in the instant form of relativistic dynamics in the
second and third orders in the coupling constant. In the
model of the three-linear Yukawa type pseudoscalar
interaction between nucleon and meson fields we have
shown how the generators of the Poincaré group acquire
one and the same sparse structure in the Fock space of
particle states.
The explicit form for the boost generators enables us
to check directly the properties of the bare and clothed
operators and states under the Lorentz transformations.
Contrary to the bare vacuum and bare one-particle
states, the clothed vacuum and the clothed one-particle
states appear invariant under these transformations. The
new states being expressed in terms of the bare ones
have a very difficult structure which witnesses that the
clouds of virtual particles are included in the clothed
operators and states.
It is important to emphasize that the algebra of gen-
erators of the Poincaré group is fulfilled up to the six
order in the coupling constant. As a byproduct of our
clothing procedure, the mass and vertex renormalization
in the second and third orders in the coupling constant
respectively is performed in a relativistic manner.
REFERENCES
1. O. Greenberg and S. Schweber. Clothed particle
operators in simple models of quantum field theory
//Nuovo Cim. 1958, v. 8, p. 378-406.
2. A.V. Shebeko and M.I. Shirokov. Unitary transfor-
mations in quantum field theory and bound states
//Phys. Part. Nuclei. 2001, v. 32, p. 15-79.
3. V.Yu. Korda and A.V. Shebeko. The clothed parti-
cle representation in quantum field theory: mass re-
normalization //Phys. Rev. 2004, v. D70, 085011.
4. V.Yu. Korda and P.A. Frolov. Meson-two-nucleon
clothed generators of the Poincaré group derived
36
from quantum field theory: mass and vertex renor-
malization //Proc of the Int. conf. NPAE-2006. Kyiv.
2006, p. 105.
5. V.Yu. Korda and P.A. Frolov. Clothing particles in
meson-nucleon system: mass and charge renormali-
zation //Proc of the 56th Int. conf.”Nucleus-2006”,
Sarov. 2006, p. 247.
6. V.Yu. Korda, L. Canton, A.V. Shebeko. Relativistic
interactions for the meson-two-nucleon system in
the clothed-particle unitary representation, nucl-
th/0603025.
ПРЕДСТАВЛЕНИЕ ОДЕТЫХ ЧАСТИЦ В КВАНТОВОЙ ТЕОРИИ ПОЛЯ:
ГЕНЕРАТОРЫ БУСТОВ ГРУППЫ ПУАНКАРЕ
В.Ю. Корда, П.А. Фролов
С помощью метода унитарных одевающих преобразований построены генераторы группы Пуанкаре в
мгновенной форме релятивистской динамики во втором и третьем порядках по константе взаимодействия.
Показано, что алгебра генераторов группы удовлетворяется до шестого порядка. Обоснована релятивист-
ская инвариантность рассчитанных поправок массам частиц и константе взаимодействия.
ЗОБРАЖЕННЯ ОДЯГНЕНИХ ЧАСТИНОК В КВАНТОВІЙ ТЕОРІЇ ПОЛЯ:
ГЕНЕРАТОРИ БУСТІВ ГРУПИ ПУАНКАРЕ
В.Ю. Корда, П.О. Фролов
За допомогою методу унітарних одягаючих перетворень побудовані генератори групи Пуанкаре в миттє-
вій формі релятивістської динаміки в другому і третьому порядках за константою взаємодії. Показано, що
алгебра генераторів групи задовольняється до шостого порядку. Обґрунтовано релятивістську інваріант-
ність розрахованих поправок до мас частинок і константі взаємодії.
37
|
| id | nasplib_isofts_kiev_ua-123456789-110910 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T13:37:00Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Korda, V.Yu. Frolov, P.A. 2017-01-06T18:39:16Z 2017-01-06T18:39:16Z 2007 Clothed particle representation in quantum field theory: boost generators of the Poincaré group / V.Yu. Korda and P.A. Frolov // Вопросы атомной науки и техники. — 2007. — № 3. — С. 33-37. — Бібліогр.: 6 назв. — рос. 1562-6016 PACS: 21.45.+v; 24.10. Jv; 11.80.-m https://nasplib.isofts.kiev.ua/handle/123456789/110910 The method of the unitary clothing transformations is used to construct the generators of the Poincaré group in the instant form of relativistic dynamics in the second and third orders in the coupling constant. It is shown that the respective algebra of generators is fulfilled up to the sixth order. The relativistic invariance of the mass and vertex corrections derived is proven. За допомогою методу унітарних одягаючих перетворень побудовані генератори групи Пуанкаре в миттєвій формі релятивістської динаміки в другому і третьому порядках за константою взаємодії. Показано, що алгебра генераторів групи задовольняється до шостого порядку. Обґрунтовано релятивістську інваріантність розрахованих поправок до мас частинок і константі взаємодії. С помощью метода унитарных одевающих преобразований построены генераторы группы Пуанкаре в мгновенной форме релятивистской динамики во втором и третьем порядках по константе взаимодействия. Показано, что алгебра генераторов группы удовлетворяется до шестого порядка. Обоснована релятивистская инвариантность рассчитанных поправок массам частиц и константе взаимодействия. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Quantum field theory Clothed particle representation in quantum field theory: boost generators of the Poincaré group Зображення одягнених частинок в квантовій теорії поля: генератори бустів групи пуанкаре Представление одетых частиц в квантовой теории поля: генераторы бустов группы пуанкаре Article published earlier |
| spellingShingle | Clothed particle representation in quantum field theory: boost generators of the Poincaré group Korda, V.Yu. Frolov, P.A. Quantum field theory |
| title | Clothed particle representation in quantum field theory: boost generators of the Poincaré group |
| title_alt | Зображення одягнених частинок в квантовій теорії поля: генератори бустів групи пуанкаре Представление одетых частиц в квантовой теории поля: генераторы бустов группы пуанкаре |
| title_full | Clothed particle representation in quantum field theory: boost generators of the Poincaré group |
| title_fullStr | Clothed particle representation in quantum field theory: boost generators of the Poincaré group |
| title_full_unstemmed | Clothed particle representation in quantum field theory: boost generators of the Poincaré group |
| title_short | Clothed particle representation in quantum field theory: boost generators of the Poincaré group |
| title_sort | clothed particle representation in quantum field theory: boost generators of the poincaré group |
| topic | Quantum field theory |
| topic_facet | Quantum field theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110910 |
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