Extension of the Sp(2,C) group for description of a three-body system
We propose a new approach to the three-body problem that is based on the extension of the Sp(2,C) group, which is the universal covering group for the Lorentz group, to the Sp(4,C) one. Angular momenta of the particles in the phase space of a system with an inner interaction are obtained. This resul...
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irk-123456789-1070572016-10-13T03:02:12Z Extension of the Sp(2,C) group for description of a three-body system Yaroshenko, A.P. Uvarov, I.V. Section C. Theory of Elementary Particles. Cosmology We propose a new approach to the three-body problem that is based on the extension of the Sp(2,C) group, which is the universal covering group for the Lorentz group, to the Sp(4,C) one. Angular momenta of the particles in the phase space of a system with an inner interaction are obtained. This result can be used to obtain eigenfunctions of angular momenta, and exact quantum mechanical solutions for the system defined by Dirac-like equations, e.g. a system of three zero-spin particles or Regge trajectories of N-baryons. Предлагается новый подход к описанию трёхчастичной системы, основанный на расширении группы Sp(2,C), которая является универсальной накрывающей группы Лоренца, до группы Sp(4,C). В фазовом пространстве системы с внутренним взаимодействием получены угловые моменты частиц. На основе этого результата будут получены собственные функции углового момента, с помощью которых можно найти точное квантово-механическое решение системы, определённой уравнениями типа уравнений Дирака, например, системы трёх бесспиновых частиц, или определить траектории Редже барионов. Пропонується новий підхід до опису тричастинкової системи, що базується на розширенні групи Sp(2,C), яка є універсальною накриваючою групи Лоренця, до групи Sp(4,C). У фазовому просторі системи зі внутрішньою взаємодією отримані кутові моменти частинок. На основі цього результату будуть знайдені власні функції кутового моменту, за допомогою яких можна отримати точний квантово-механічний розв'язок системи, яка визначається рівняннями типа рівнянь Дірака, наприклад, системи трьох безспінових частинок, або визначити траєкторії Редже баріонів. 2012 Article Extension of the Sp(2,C) group for description of a three-body system / A.P. Yaroshenko, I.V. Uvarov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 163-165. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 03.65.Fd, 11.30.Cp, 04.20.Gz http://dspace.nbuv.gov.ua/handle/123456789/107057 en Вопросы атомной науки и техники Dnipropetrovsk National University |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Section C. Theory of Elementary Particles. Cosmology Section C. Theory of Elementary Particles. Cosmology |
| spellingShingle |
Section C. Theory of Elementary Particles. Cosmology Section C. Theory of Elementary Particles. Cosmology Yaroshenko, A.P. Uvarov, I.V. Extension of the Sp(2,C) group for description of a three-body system Вопросы атомной науки и техники |
| description |
We propose a new approach to the three-body problem that is based on the extension of the Sp(2,C) group, which is the universal covering group for the Lorentz group, to the Sp(4,C) one. Angular momenta of the particles in the phase space of a system with an inner interaction are obtained. This result can be used to obtain eigenfunctions of angular momenta, and exact quantum mechanical solutions for the system defined by Dirac-like equations, e.g. a system of three zero-spin particles or Regge trajectories of N-baryons. |
| format |
Article |
| author |
Yaroshenko, A.P. Uvarov, I.V. |
| author_facet |
Yaroshenko, A.P. Uvarov, I.V. |
| author_sort |
Yaroshenko, A.P. |
| title |
Extension of the Sp(2,C) group for description of a three-body system |
| title_short |
Extension of the Sp(2,C) group for description of a three-body system |
| title_full |
Extension of the Sp(2,C) group for description of a three-body system |
| title_fullStr |
Extension of the Sp(2,C) group for description of a three-body system |
| title_full_unstemmed |
Extension of the Sp(2,C) group for description of a three-body system |
| title_sort |
extension of the sp(2,c) group for description of a three-body system |
| publisher |
Dnipropetrovsk National University |
| publishDate |
2012 |
| topic_facet |
Section C. Theory of Elementary Particles. Cosmology |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/107057 |
| citation_txt |
Extension of the Sp(2,C) group for description of a three-body system / A.P. Yaroshenko, I.V. Uvarov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 163-165. — Бібліогр.: 10 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-07T19:26:19Z |
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2025-07-07T19:26:19Z |
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| fulltext |
EXTENSION OF THE Sp(2,C) GROUP FOR DESCRIPTION
OF A THREE-BODY SYSTEM
A.P. Yaroshenko and I.V. Uvarov ∗
Dnipropetrovsk National University, 49050, Dnipropetrovs’k, Ukraine
(Received October 31, 2011)
We propose a new approach to the three-body problem that is based on the extension of the Sp(2, C) group, which
is the universal covering group for the Lorentz group, to the Sp(4, C) one. Angular momenta of the particles in the
phase space of a system with an inner interaction are obtained. This result can be used to obtain eigenfunctions of
angular momenta, and exact quantum mechanical solutions for the system defined by Dirac-like equations, e.g. a
system of three zero-spin particles or Regge trajectories of N-baryons.
PACS: 03.65.Fd, 11.30.Cp, 04.20.Gz
1. INTRODUCTION
Description of a relativistic three-body system is
an important issue in modern nuclear and particle
physics. Statistical approaches or even many-body
technics are not efficient for three-body problems,
and a good treatment of the center of mass is neces-
sary and internal coordinates must be employed. Var-
ious approaches to the solution of this problem exist.
For example, the system can be described by means
of relativistic quantum mechanics (using the Lorentz-
invariant wave equations) [1–3] or quantum field the-
ory (Faddeev equation for three particles) [4, 5]. An
important research direction is connected with exten-
sions of the Minkowski and spinor spaces [6, 7].
Using the method of extension of the Sp(2, C)
group in a minimal manner, we develop the model
that describes the three-body system with an inner
interaction. Expressions for positions, momenta, an-
gular momenta and their eigenfunctions are found.
Furthermore, this new method of spacetime exten-
sion may be used for description of interacting stan-
dart model particles, e.g., N-baryons.
2. SPACETIME EXTENSION
The following is the general overview of the exten-
sion of the two-dimensional spinor space and corre-
sponding groups of symmetry. The proposed method
of extension of two-dimensional spinor space has an
important advantage over the method of the space-
time extension by addition of spatial variables only,
as number of dimensions increases only quadratically,
not exponentially. Therefore, a significantly smaller
number of subsidiary conditions is required.
The Lorentz group SO(1, 3) is covered by the
SL(2, C) ≡ Sp(2, C) group. There is an one-to-
one correspondence between the Sp(2, C) Hermitian
spin-tensors of second rank and the Minkowski four-
vectors that describe the space-time position of a rela-
tivistic particle. The Sp(2, C) group can be extended
to the Sp(4, C) one for the description of few-particle
systems. Similar to the Sp(2, C) group case, there is
a correspondence between Sp(4, C) 4 × 4 Hermitian
spin-tensor and a real 16-vector [8,9]. We will choose
the matrices of the basis as follows:
μM ≡ μ(a,m) = Σa ⊗ σm, (1)
Values of M = 1..16 can be represented through
4 × 4 combinations of two indices (a, m = 0..3).
The Σa and σm matrices can be explicitly expressed
through 2 × 2 unitary matrix I and the Paupli ma-
trices τ1, τ2, τ3:
σ0 = σ̃0 = Σ2 = Σ̃2 = I,
σ1 = σ̃1 = Σ1 = Σ̃1 = τ1,
σ2 = −σ̃2 = Σ0 = −Σ̃0 = τ2,
σ3 = −σ̃3 = Σ3 = Σ̃3 = τ3, (2)
Now we obtain the metrics of 16-dimensional real
space:
gMN =
1
4
Tr(μM μ̃N ) = ĥabhmn, (3)
where hmn = diag (1,−1,−1,−1) is the Minkowski
metrics and ĥab is the Minkowski metrics with the
opposite sign caused by the group extension. The
discussed real space is a sum of four Minkowski met-
rics. The 16-dimensional momentum operator in the
considered space can be represented as four relativis-
tic four-momenta. Since ten of the 16 dimensions
are spacelike, whereas six other are time-like and we
need only one time for a physical interpretation, it is
necessary to exclude five of six time-like components.
∗Corresponding author E-mail address: wvanj@pisem.net
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 163-165.
163
Let us introduce the quadratic function of the mo-
mentum PM = i∂M :
(PP̃ )αβ = Pαα̃P̃ ββ̃ = μM
αα̃PMμNββ̃PN , (4)
where α, ᾱ, β, β̄ = 1..4 are spinoral indices.
The matrix Pαβ is antisymmetric under the in-
terchange of the α and β indices. Let us consider
the antisymmetric part, which has six linearly inde-
pendent matrices in the four-dimensional space. One
of them is proportional to the metric, and, conse-
quently, is invariant under the Sp(4, C). The remain-
ing five matrices create a five-dimensional complex
space. In such way the five-dimensional representa-
tion of the Sp(4, C) group, i.e. the special orthogonal
group SO(5, C), is obtained. These groups are locally
isomorphic.
We introduce linkage to the system by requiring
five of complex quadratic combinations of momenta
to be equal to zero. In this case they are invariant
under the transformations of the Sp(4, C) group. In
the absence of interaction these momenta have the
form of conventional derivatives, commute with each
other and do not create secondary linkage. The sys-
tem of subsidiary conditions, written in the form of
ten real equations, is:
(s, p) = (s, q) = 0, rs0 − r0s = [p,q], (5)
(r, p) = (r, q) = 0, pq0 − p0q = [s, r], (6)
where sm = P3m, pm = P1m, qm = P2m, rm = P0m
are the components of the momentum vector Pa,m,
round brackets denote the scalar products of four-
vectors as (a, b) = a0b0 − ab, and square brackets
denote the vector products of three-vectors.
Equations (6) are a consequence of (5), therefore
we obtained five independent conditions (5), and it
can be seen from their form that rm is the pseudovec-
tor. The 16-momentum has the maximum amount of
independent components when sm is time-like. In
that case pm and qm are space-like, and in the frame
where s = 0, we have p0 = q0 = 0, and p, q are in-
dependent. Thus we can treat this frame as a center-
of-momentum one. Then p and q are thought to be
three-momenta of the relative motion, that have the
corresponding x and y coordinates.
3. ANGULAR MOMENTA
The next step is to include an inner interaction in
the system. We introduce the interaction that leads
to the violation of the Sp(4, C) symmetry, but rel-
ativistic invariance remains. The subsidiary condi-
tions are primary constraints [10], and they can cre-
ate secondary constraints. We demand that there is
no secondary linkage. This strict requirement com-
pletely defines the form of interaction potentials up to
gauge transformations. In the center-of-momentum
frame instead of two sets of inner coordinates and
corresponding momenta we obtain two independent
coordinate spaces Qa, Q′
a and two corresponding mo-
menta spaces were obtained.:
a) P(λ) = p − λy, Q(λ) = q + λx,
b) P(−λ) = p + λy, Q(−λ) = q − λx. (7)
The commutation relations between these vectors
are
[Qa(λ), Pb(λ)] = 2iλδab,
[Q′
a(λ), P ′
b(λ)] = 2iλδab, (8)
where Q′(λ) = P(−λ) and P′(λ) = Q(−λ); δab is the
Kronecker delta (a, b = 1, 2, 3 are Cartesian indices).
The pairs P,Q and P′,Q′ commute with each other.
Using the two proposed sets of coordinates and
momenta, P(λ),Q(λ) and P′(λ),Q′(λ), we can in-
troduce two angular momentum operators:
M = M(λ) =
1
2λ
[Q(λ),P(λ)],
N = M(−λ) =
1
2(−λ)
[Q′(−λ),P′(λ)]. (9)
M and N obey the usual commutation relations
and they are independent.
4. CONCLUSIONS
In the present work, we have investigated the quan-
tum system with nine degrees of freedom using a
method based on the extension of the Sp(2, C) group.
After the introduction of inner interaction in a min-
imal manner, we redefined the coordinates and mo-
menta of the system, that resulted in two indepen-
dent angular momenta, which eigenfunctions will be
found elsewhere. This new result lays the groundwork
for determining the stationary states of a quantum
three-body system with an inner interaction defined
by Dirac-like equations. The theory can be applied,
for example, to analysis of the system of three spinless
particles or to find Regge trajectories of N-baryons.
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