Classical relativistic system of N charges. Hamiltonian description, forms of dynamics, and partition function
The procedure of reducing canonical field degrees of freedom for a system of charged particles plus field in the constrained Hamiltonian formalism is elaborated up to the first order in the coupling constant expansion. The canonical realization of the Poincare algebra in the terms of particle variab...
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| Cite this: | Classical relativistic system of N charges. Hamiltonian description, forms of dynamics, and partition function / A. Duviryak, A. Nazarenko, V. Tretyak // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 5-14. — Бібліогр.: 10 назв. — англ. |
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irk-123456789-1197562017-06-09T03:04:01Z Classical relativistic system of N charges. Hamiltonian description, forms of dynamics, and partition function Duviryak, A. Nazarenko, A. Tretyak, V. The procedure of reducing canonical field degrees of freedom for a system of charged particles plus field in the constrained Hamiltonian formalism is elaborated up to the first order in the coupling constant expansion. The canonical realization of the Poincare algebra in the terms of particle variables is found. The relation between covariant and physical particle variables in the Hamiltonian description is written. The system of particles interacting by means of scalar and vector massive fields is also considered. The first order approximation in c⁻² is examined. An application to calculating the relativistic partition function of an interacting particle system is discussed. Розроблено процедуру редукції канонічних польових ступенів вільности для системи заряджених частинок з електромагнетним полем у гамільтоновому формалізмі з в’язями у першому порядку за константою взаємодії. Знайдено канонічну реалізацію алгебри Пуанкаре у термінах змінних частинок. Записано співвідношення між коваріянтними та фізичними змінними частинок. Також розглянуто систему частинок, які взаємодіють через скалярне та векторне масивні поля. Досліджено перше наближення за c⁻². Обговорюється застосування до обчислення статистичної суми системи взаємодіючих частинок. 2001 Article Classical relativistic system of N charges. Hamiltonian description, forms of dynamics, and partition function / A. Duviryak, A. Nazarenko, V. Tretyak // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 5-14. — Бібліогр.: 10 назв. — англ. DOI:10.5488/CMP.4.1.5 PACS: 03.30.+p, 05.20.-y http://dspace.nbuv.gov.ua/handle/123456789/119756 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The procedure of reducing canonical field degrees of freedom for a system of charged particles plus field in the constrained Hamiltonian formalism is elaborated up to the first order in the coupling constant expansion. The canonical realization of the Poincare algebra in the terms of particle variables is found. The relation between covariant and physical particle variables in the Hamiltonian description is written. The system of particles interacting by means of scalar and vector massive fields is also considered. The first order approximation in c⁻² is examined. An application to calculating the relativistic partition function of an interacting particle system is discussed. |
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Duviryak, A. Nazarenko, A. Tretyak, V. |
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Duviryak, A. Nazarenko, A. Tretyak, V. Classical relativistic system of N charges. Hamiltonian description, forms of dynamics, and partition function Condensed Matter Physics |
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Duviryak, A. Nazarenko, A. Tretyak, V. |
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Classical relativistic system of N charges. Hamiltonian description, forms of dynamics, and partition function |
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Classical relativistic system of N charges. Hamiltonian description, forms of dynamics, and partition function |
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Classical relativistic system of N charges. Hamiltonian description, forms of dynamics, and partition function |
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Classical relativistic system of N charges. Hamiltonian description, forms of dynamics, and partition function |
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Classical relativistic system of N charges. Hamiltonian description, forms of dynamics, and partition function |
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classical relativistic system of n charges. hamiltonian description, forms of dynamics, and partition function |
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Інститут фізики конденсованих систем НАН України |
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2001 |
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Classical relativistic system of N charges. Hamiltonian description, forms of dynamics, and partition function / A. Duviryak, A. Nazarenko, V. Tretyak // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 5-14. — Бібліогр.: 10 назв. — англ. |
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Condensed Matter Physics |
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AT duviryaka classicalrelativisticsystemofnchargeshamiltoniandescriptionformsofdynamicsandpartitionfunction AT nazarenkoa classicalrelativisticsystemofnchargeshamiltoniandescriptionformsofdynamicsandpartitionfunction AT tretyakv classicalrelativisticsystemofnchargeshamiltoniandescriptionformsofdynamicsandpartitionfunction |
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2025-07-08T16:32:40Z |
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Condensed Matter Physics, 2001, Vol. 4, No. 1(25), pp. 5–14
Classical relativistic system of N
charges. Hamiltonian description,
forms of dynamics, and partition
function
A.Duviryak, A.Nazarenko, V.Tretyak
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received October 3, 2000
The procedure of reducing canonical field degrees of freedom for a sys-
tem of charged particles plus field in the constrained Hamiltonian formal-
ism is elaborated up to the first order in the coupling constant expansion.
The canonical realization of the Poincaré algebra in the terms of particle
variables is found. The relation between covariant and physical particle
variables in the Hamiltonian description is written. The system of particles
interacting by means of scalar and vector massive fields is also considered.
The first order approximation in c−2 is examined. An application to calcu-
lating the relativistic partition function of an interacting particle system is
discussed.
Key words: classical relativistic mechanics, forms of relativistic dynamics,
relativistic statistical mechanics, charged particles
PACS: 03.30.+p, 05.20.-y
1. Introduction
During a long period of time, the development of relativistic statistical mechanics
was limited by insufficient understanding of the relativistic classical and quantum
description of the systems with finitely many particles. At present, the classical rel-
ativistic mechanics of N -particle system with the direct interaction is formulated in
a consistent way within various formalisms and approaches. These formulations are
based on the Poincaré-invariance conditions; the relation of the general expressions
for interaction potentials with the concrete field-theoretical model can be performed
either by means of several approximation (e.g., Darwin’s Lagrangian in electrody-
namics), or with the use of the Fokker-type action integrals (see, e.g., [1]).
The different formalisms of the classical relativistic mechanics are more or less
c© A.Duviryak, A.Nazarenko, V.Tretyak 5
A.Duviryak, A.Nazarenko, V.Tretyak
equivalent, although the Lagrangian formalism has some advantages in its concep-
tual simplicity and the direct relations with the field-theoretical models [1,2]. But
statistical description of the interacting particle system is more transparent in the
Hamiltonian formalism. The transition from Lagrangian to Hamiltonian description
in the classical relativistic dynamics is not simple or direct and demands the use
of various approximations. On the other hand, it is natural to try constructing the
Hamiltonian description of an interacting particle system starting from the Hamil-
tonian formalism for the “particle plus field” system and then eliminating the field
degrees of freedom. Such a program was discussed in the illuminating series of papers
by Lusanna with coworkers (see [3]).
Here we present a simpler approach [4,5] which uses the geometrical forms of
dynamics [2] to impose the gauge fixing conditions concerning the chronometrical
invariance of action. In section 2 we consider the constrained Hamiltonian description
of charged particles with electromagnetic fields and the canonical transformation
which isolate nonphysical (gauge) degrees of freedom of the electromagnetic fields.
We also consider the massive scalar and vector interactions and obtain the generators
of time evolution and Lorentz transformations on the physical phase space. The
elimination of the field degrees of freedom is discussed in section 3 within the linear
approximation in the coupling constant. We obtain the canonical generators of the
Poincaré group for the interactions considered. We demonstrate that the first order
approximation in c−2 agrees with the well known results of various approaches. An
application to calculating the relativistic partition function of an interacting particle
system is discussed in section 4.
2. Hamiltonian description of the “field plus particle” syst em
At the beginning we briefly outline the main steps of the Hamiltonian description
of a system of N point particles with electromagnetic interaction [5]. The particles
are described by their world lines in the Minkowski space-time1 γa : τ 7→ xµ
a(τ).
An interaction between charges is mediated by an electromagnetic field Fµν(x) =
∂µAν(x)−∂νAµ(x) with the electromagnetic potential Aµ(x); ∂ν ≡ ∂/∂xν . An action
for the system is
S = −
N
∑
a=1
∫
dτa
{
ma
√
u2
a(τa) + eau
ν
a(τa)Aν [xa(τa)]
}
−
∫
d4xF 2, (2.1)
where ma and ea are the mass and the charge of particle a, respectively, F 2 ≡
(1/4)Fλσ(x)F
λσ(x), and uµ
a(τa) = dxµ
a(τa)/dτa. The action is manifestly invariant
under two types of gauge transformations: reparametrization of the particle world
1The Minkowski space-time is endowed with a metric ‖ηµν‖ = diag(1,−1,−1,−1). The Greek
indices µ, ν, . . . run from 0 to 3; the Roman indices from the middle of alphabet, i, j, k, . . . run from
1 to 3 and both types of indices are subject of the summation convention. The Roman indices from
the beginning of alphabet, a, b, label the particles and run from 1 to N . The sum over such indices
is indicated explicitly.
6
Classical relativistic system of N charges
lines
τa 7→ φ(τa), φ′ > 0, (2.2)
and ordinary gauge transformation of the electromagnetic potential
Aµ 7→ Aµ + ∂µΛ. (2.3)
Moreover, action (2.1) is invariant under (global) transformations of the Poincaré
group; this invariance results in the conservation of the symmetric energy-momen-
tum tensor [6]:
θµν(x) =
N
∑
a=1
∫
ma
uµ
a(τa)u
ν
a(τa)
√
u2
a(τa)
δ4(x− xa(τa))dτa − F µλF ν
λ + ηµνF 2, (2.4)
θµν(x) = θνµ(x), ∂νθ
µν(x) = 0. (2.5)
We fix the freedom in the parametrization of particle world lines by choosing the
form of relativistic dynamics, which is specified by one-parameter family {Σ t | t ∈ R}
of space-like or isotropic hypersurfaces Σt = {x ∈ M4 | σ(x) = t} foliating the
Minkowski space-time (see [2]). Because the hypersurface equation σ(x) = t can be
solved with respect to x0 in the form:
x0 = f(t,x), x = (x1, x2, x3), (2.6)
the functions xi = xi
a(t), i = 1, 2, 3, completely determine the parametric equations
of the particle world lines in a given form of dynamics:
x0 = f(t,xa(t)), xi = xi
a(t). (2.7)
The variable t serves as a common evolution parameter of the system.
Now we can use the definition of the form of dynamics as a gauge fixing condition
and put action (2.1) into a single-time form [5]
S =
∫
dtL (2.8)
with Lagrangian L(t) depending on the functions xa(t), A
µ(t,x) and their first order
derivatives with respect to evolution parameter, ẋa(t) = dxa(t)/dt and Ȧµ(t,x). The
conservation of the energy-momentum tensor (2.4) gives us ten conserved quantities
in a given form of dynamics:
P µ =
∫
Σt
θµνdσν , Mµν =
∫
Σt
(xµθνρ − xνθµρ) dσρ. (2.9)
The single-time Lagrangian L is invariant under gauge transformation (2.3) and
leads to the constrained Hamiltonian description. As it has been discussed in [5],
the structure of the corresponding constraints depend on the form of dynamics. In
the following we confine ourselves to the most common case of the instant form of
7
A.Duviryak, A.Nazarenko, V.Tretyak
dynamics (x0 = t). The Lagrangian function in this form of dynamics is represented
by:
L = −
N
∑
a=1
{
ma
√
1− ẋ2
a + ea
[
A0(t,xa) + ẋi
aAi(t,xa)
]
}
−
1
4
∫
(2eie
i + FijF
ij)d3x,
(2.10)
where Fij = Aj,i −Ai,j and ei = A0,i − Ȧi.
In the Hamiltonian formulation of our system we start with canonical variables
xi
a(t), Aµ(t,x) and conjugated momenta pai(t), E
µ(t,x) which are subject of the
first class constraints [7]
E0 ≈ 0, Γ ≡ ̺− ∂iE
i ≈ 0, (2.11)
where ≈means ”weak equality” in the sense of Dirac and ̺(t,x) =
N
∑
a=1
eaδ
3(x−xa(t))
is a charge density. Then we perform the canonical transformation that dissect the
field phase space into the physical part described by the gauge invariant variables
aα = (δiα − δi3∂α/∂3)Ai, E
α; α = 1, 2, and unphysical part parametrized by the
canonical pairs (Q,Γ) and (A0, E
0).
The time evolution of the physical degrees of freedom is generated by the Hamil-
tonian
H =
N
∑
a=1
√
m2
a + [pa − eaA⊥(xa)]2 −
1
2
∫
(
A⊥
i ∆A⊥
i − Ei
⊥
Ei
⊥
+ ̺∆−1̺
)
d3x, (2.12)
where
Ei
⊥
= (δiα − δi3∂α/∂3)E
α, A⊥
i = (δαi + ∂i∆
−1∂α)aα. (2.13)
Inverse differential operators are defined by the relations
1/∂3δ
3(x) = (1/2)δ(x1)δ(x2)sgn(x3), ∆−1δ3(x) = −1/(4π|x|). (2.14)
Conserved quantities (2.9) being reexpressed in terms of canonical variables deter-
mine the canonical realization of the Poincaré group. On the physical subspace the
generator P 0 coincide with the Hamiltonian (2.12), and the generator of the Lorentz
transformation is given by
Mk0 =
N
∑
a=1
{
xk
a
√
m2
a + [pa − eaA⊥(xa)]
2 − tpka
}
−
1
2
∫
xk̺∆−1̺d3x
+
∫
xk
(
1
4
F⊥
ij F
⊥
ij +
1
2
Ei
⊥
Ei
⊥
+ El
⊥
∂l∆
−1̺
)
d3x− t
∫
El
⊥
∂kA⊥
l d
3x. (2.15)
where F⊥
ij = A⊥
j,i −A⊥
i,j .
It is instructive to consider in a similar manner the Hamiltonian description of
the system of particles with massive vector and scalar interactions. So, in the first
case we have a system described by the action that differs from (2.1) by a massive
8
Classical relativistic system of N charges
term 1
2
µ2AνAν . The instant form of Hamiltonian description of the system is based
on the canonical variables xi
a(t), Aµ(t,x) and pai(t), E
µ(t,x) with a pair of the
second class constraints
E0 ≈ 0, Γ− µ2A0 ≈ 0, (2.16)
which can be excluded by means of the Dirac bracket. The canonical Hamiltonian
is
H =
N
∑
a=1
√
m2
a + [pa − eaA(t,xa)]
2
+
∫
[
1
4
FijFij +
1
2
EiEi −
1
2
µ2AiA
i + A0
(
Γ−
1
2
µ2A0
)]
d3x. (2.17)
After exclusion of the constraints (2.16) one obtains for the boost generator
Mk0 =
N
∑
a=1
{
xk
a
√
m2
a + [pa − eaA(t,xa)]2 − tpka
}
+
∫
xk
[
1
4
FijFij +
1
2
EiEi −
1
2
µ2AiA
i +
1
2µ2
Γ2
]
d3x
− t
∫
[
Ej∂kAj −
1
2
µ2AkΓ
]
d3x. (2.18)
In the case of a system of particles interacting by means of the scalar field ϕ(x) we
find the standard Hamiltonian description without constraints with the Hamiltonian
H =
N
∑
a=1
√
p2
a + [ma − eaϕ(t,xa)]2 +
1
2
∫
[π2 + (∇ϕ)2 + µ2ϕ2]d3x, (2.19)
and the boost generator
Mk0 =
N
∑
a=1
{
xk
a
√
p2
a + [ma − eaϕ(t,xa)]2 − tpka
}
+
1
2
∫
xk[π2 + (∇ϕ)2 + µ2ϕ2]d3x− t
∫
π∂kϕd3x. (2.20)
We will see that after elimination of the field degrees of freedom all the three con-
sidered cases give the canonical generators of a similar structure.
3. Elimination of the field degrees of freedom
Now we are interested in elimination of the physical field degrees of freedom.
As a result, we shall obtain the description of our system in the terms of particle
variables only. Such a reformulation is especially effective when the free radiation is
not essential.
9
A.Duviryak, A.Nazarenko, V.Tretyak
Our procedure of the field reduction has got three steps [8]. First, we must find
a solution of the field equations of motion. In the Hamiltonian mechanics the field
equations are non-linear, so we use an approximation which is based on the coupling
constant expansion. In general, the problem of choosing the Green’s function arises in
this approach. But in the first-order (linear) approximation in the coupling constant
the advanced, retarded, or symmetric solutions coincide. We use here the time-
symmetric Green’s function G(x2) = G(x2
0 − x2). It is well known [1], that the
Green’s function determines the nonrelativistic potential u(r):
u(r) =
∫
dαG(α2 − r2). (3.1)
The general solution of the field equations is a sum of the source free field Arad
s (s is
the number of the physical field components), which satisfies the homogeneous equa-
tion, and the solution of the inhomogeneous equation As in the terms of canonical
particle variables.
Second, we perform a canonical transformation [8]:
As = Arad
s +As, Es = Es
rad + Es, (3.2)
xi
a = qia +
∫
[(
Arad
s +
1
2
As
)
∂Es
∂kai
−
(
Es
rad +
1
2
Es
)
∂As
∂kai
]
d3x, (3.3)
pai = kai −
∫
[(
Arad
s +
1
2
As
)
∂Es
∂qia
−
(
Es
rad +
1
2
Es
)
∂A⊥
k
∂qia
]
d3x, (3.4)
after that the free field terms (Arad
s , Es
rad) become the new canonical variables. It
is assumed in our problem that the field hasn’t got its own degrees of freedom, so
third step consists in elimination of the field variables by means of constraints
Arad
s ≈ 0, Es
rad ≈ 0. (3.5)
We find the Dirac bracket, which coincides with the particle Poisson bracket
{qia, kbj} = δabδ
i
j . The canonical generators of the Poincaré group for the considered
interactions in the linear approximation are
H = c
N
∑
a=1
k0
a +
c
2
N
∑′
a,b=1
eaeb
f(ωab)
k0
a
u(ρab), k0
a =
√
m2
ac
2 + k2
a, (3.6)
P k =
N
∑
a=1
kk
a , M ij =
N
∑
a=1
(qiak
j
a − qjak
i
a), (3.7)
Mk0 =
N
∑
a=1
(
qka
c
k0
a − tkk
a
)
+
1
2c
N
∑′
a,b=1
eaebq
k
b
f(ωab)
k0
a
u(ρab), (3.8)
where the prime over sum denotes that a 6= b (a = b terms is excluded by means
of mass renormalization); ρ2
ab = q2ab + (kaqab/k
0
a)
2, qab = qa − qb, qab = |qab|, ωab =
10
Classical relativistic system of N charges
kµ
akbµ/mambc
2, and f(ω) = 1 for the scalar interaction and f(ω) = ω for the vector
interaction. It can be easily demonstrated, that the expressions (3.6)–(3.8) satisfy
the commutation relations of the Poincaré group in a given approximation with
arbitrary functions u(r) and f(ω).
According to (3.3), the covariant particle positions xi
a are connected with the
canonical variables as
xi
a = qia +
1
2
∫
[
As
∂Es
∂kai
− Es∂As
∂kai
]
d3x. (3.9)
It can be verified directly that in a given approximation the expression (3.9) satisfies
the world line condition
{xi
a,M
k0} = xk
a{x
i
a, H} − tδik. (3.10)
The Poisson brackets between particle positions do not vanish,
{xi
a, x
j
b} =
∫
(
∂As
∂kbj
∂Es
∂kai
−
∂Es
∂kbj
∂As
∂kai
)
d3x, (3.11)
in a full agreement with the famous no-interaction theorem [9].
Let us examine the generators (3.6), (3.8) up to c−2 approximation. Using the
first order expansions
u(ρab) = u(qab) +
(qabka)
2
2qabm2
ac
2
du(qab)
dqab
, f(ωab) = 1 +
f ′(0)
2c2
(
ka
ma
−
kb
mb
)2
(3.12)
and performing the canonical transformation generated by the function
Λ =
1
4c2
N
∑
a<b
eaebu(qab)
[
qab
(
ka
ma
−
kb
mb
)]
, (3.13)
we obtain the expressions
H = H(0) +H(1), (3.14)
Mk0 =
N
∑
a=1
(qkama − tkk
a) +
1
2c2
N
∑′
a,b=1
eaebq
k
bu(qab), (3.15)
where
H(0) =
N
∑
a=1
(
mac
2 +
k2
a
2ma
)
+ U (0), U (0) =
N
∑
a<b
eaebu(qab), (3.16)
H(1) = −
N
∑
a=1
k4
a
8m3
ac
2
−
N
∑
a<b
eaeb
{
1
2c2mamb
[
kakbu(qab) + (kaqab)(kbqab)
du(qab)
qabdqab
]
−
A
2c2
(
ka
ma
−
kb
mb
)2
u(qab)
}
, (3.17)
and A = f ′(0) − 1. Specifically, A = −1 for the scalar and A = 0 for the vector
interactions. The latter in the massless case corresponds to the Darwin’s Lagrangian
for electromagnetic interaction. Expression (3.17) agrees with the post-Newtonian
Hamiltonians obtained within various approaches [1].
11
A.Duviryak, A.Nazarenko, V.Tretyak
4. Statistical description
Having obtained the Hamiltonian description of interacting particle system in the
term of canonical particle variables we can define the relativistic partition function
in the usual way
Z =
1
N !
∫
e−βH
N
∏
a=1
d3kad
3qa
(2π)3
(4.1)
as an integral over the phase space of the particle system. But, according to (3.9),
we need to define correctly the boundary conditions for the canonical coordinates
qia, while the physical variables xi
a vary into the volume V . It needs calculating the
Jacobian J = ∂(qia, k
i
a)/∂(x
i
a, k
i
a) by means of the expression (3.9). Next we can use
the various approximations as in the nonrelativistic case.
Here we present only the result of the post-Newtonian approximation for the
partition function corresponding to the Hamiltonian (3.14), (3.17). In this approx-
imation Jacobian J = 1. Using the general results of paper [10], we obtain in our
case
Z = Z idQ[1 +R/(βmc2)], (4.2)
R =
15
8
N − 3β
A
Q
∂Q
∂β
, Q = V −N
∫
e−βU (0)
∏
a
d3xa, (4.3)
where ma = m and Z id is the partition function of ideal gas.
Another way of calculating the partition function of the system can be based
on the “field plus particle” Hamiltonians discussed in section 2. Paper [4] discuss-
es the use of Gibbs approach, which is based on the Hamiltonian formulation of
dynamics of such a system. Treating the field and particle variables on an equal lev-
el, the Liouville equation for distribution function and the partition function have
been obtained. If the dynamics contains constraints, which have arisen in the sys-
tem with vector-type interaction, we need to correctly reformulate the equations of
statistical mechanics. Paper [4] demonstrates how one writes the Liouville equation
for distribution function and the partition function for the system with constraints.
Taking into account the nonlinear dependence of the instant form Hamiltonians
(2.12), (2.17) and (2.20) on the physical fields, we cannot perform integration over
fields without using the approximation scheme in the classical partition function.
But it is demonstrated in [5] that the use of the front form of dynamics (given by
x0 = t + x3) allows us to exactly exclude the electromagnetic field variables from
the partition function.
References
1. Gaida R. P. Quasirelativistic systems of interacting particles // Sov. J. Part. Nucl.,
1982, vol. 13, p. 179.
2. Gaida R. P., Kluchkovsky Yu. B., Tretyak V. I. Forms of relativistic dynamics in the
classical Lagrangian description of a system of particles // Theor. Math. Phys., 1983,
12
Classical relativistic system of N charges
vol. 55, No. 1, p. 372–384; Gaida R. P., Kluchkovsky Yu. B., Tretyak V. I. Three-
dimensional Lagrangian approach to the classical relativistic dynamics of directly in-
teracting particles. – In: Constraint’s Theory and Relativistic Dynamics. G. Longhi
and L. Lusanna, eds. Singapore, World Scientific Publ., 1987, p. 210–241; Duviryak
A., Shpytko V., Tretyak V. Isotropic forms of dynamics in the relativistic direct in-
teraction theory // Cond. Matter Phys., 1998, vol. 1, No. 3(15), p. 463–512.
3. Crater H., Lusanna L. The rest-frame Darwin potential from the Lienard-Wiechert
solution in the radiation gauge. SISSA e-preprint: hep-th/0001046, 2000; Alba D.,
Lusanna L. // Int. J. Mod. Phys. A, 1998, vol. 13, p. 2791.
4. Duviryak A., Nazarenko A. The Liouville equation for systems with constraints // J.
Phys. Stud., 2000, vol. 3, No. 4, p. 399–408 (in Ukrainian).
5. Nazarenko A., Tretyak V. Classical relativistic systems of charged particles in the
front form of dynamics and the Liouville equation // Cond. Matter Phys., 2000, vol. 3,
No. 1(21), p. 5–22.
6. Rohrlich F. Classical Charged Particles: Foundations of Their Theory. Addison-
Wesley, New York, 1990; Parrott S. Relativistic Electrodynamics and Differential Ge-
ometry. Springer, New York, 1987.
7. Lusanna L. The N - and 1-time classical descriptions of N -body relativistic kinematics
and the electromagnetic interaction // Int. J. Mod. Phys., 1997, vol. 12, No. 4, p. 645–
722.
8. Nazarenko A. Canonical realization of the Poincaré algebra for a relativistic system of
charged particles plus electromagnetic field // Proceedings of Institute of Mathematics
of NAS of Ukraine, 2000, vol. 30, Part 2, p. 343–349.
9. Currie D. G., Jordan J. F., Sudarshan E. C. G. Relativistic invariance and Hamiltonian
theories of interacting particles // Rev. Mod. Phys., 1963, vol. 35, p. 350–375.
10. Tretyak V. On relativistic models in the equilibrium statistical mechanics // Cond.
Matter Phys., 1998, vol. 1, p. 553–568.
13
A.Duviryak, A.Nazarenko, V.Tretyak
Класична релятивістична система N зарядів.
Гамільтонів опис, форми динаміки та статистична
сума
А.Дувіряк, А.Назаренко, В.Третяк
Інститут фізики конденсованих систем НАН Укpаїни,
79011 Львів, вул. Свєнціцького, 1
Отримано 3 жовтня 2000 р.
Розроблено процедуру редукції канонічних польових ступенів віль-
ности для системи заряджених частинок з електромагнетним полем
у гамільтоновому формалізмі з в’язями у першому порядку за кон-
стантою взаємодії. Знайдено канонічну реалізацію алгебри Пуанка-
ре у термінах змінних частинок. Записано співвідношення між кова-
ріянтними та фізичними змінними частинок. Також розглянуто систе-
му частинок, які взаємодіють через скалярне та векторне масивні по-
ля. Досліджено перше наближення за c−2 . Обговорюється застосу-
вання до обчислення статистичної суми системи взаємодіючих час-
тинок.
Ключові слова: класична релятивістична механіка, форми
релятивістичної динаміки, релятивістична статистична механіка,
заряджені частинки
PACS: 03.30.+p, 05.20.-y
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