The Lagrangian and Hamiltonian analysis of some relativistic electrodynamics models and their quantization
The work is devoted to the study of the Lagrangian and Hamiltonian properties of some relativistic electrodynamics models and is a continuation of our previous investigations. Based on the vacuum field theory approach, the Lagrangian and Hamiltonian reformulation of some classical electrodynamics...
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| Zitieren: | The Lagrangian and Hamiltonian analysis of some relativistic electrodynamics models and their quantization / N.N. Bogolubov (Jr.), A.K. Prykarpatsky // Condensed Matter Physics. — 2009. — Т. 12, № 4. — С. 603-616. — Бібліогр.: 38 назв. — англ. |
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irk-123456789-1203332017-06-12T03:05:07Z The Lagrangian and Hamiltonian analysis of some relativistic electrodynamics models and their quantization Bogolubov (Jr.), N.N. Prykarpatsky, A.K. The work is devoted to the study of the Lagrangian and Hamiltonian properties of some relativistic electrodynamics models and is a continuation of our previous investigations. Based on the vacuum field theory approach, the Lagrangian and Hamiltonian reformulation of some classical electrodynamics models is devised. The Dirac type quantization procedure, based on the canonical Hamiltonian formulation, is developed. Within the approach proposed in the work a possibility of the combined description both of electrodynamics and gravity is analyzed. Робота присвячена вивченню лагранжевих та гамiльтонових властивостей деяких релятивiстичних моделей електродинамiки i продовжує нашi попереднi дослiдження. Базуючись на теоретикопольовому пiдходi, розроблено Лагранжеве та Гамiльтонове формулювання деяких класичних моделей електродинамiки. Розвинуто процедуру квантування типу Дiрка, що базується на класичному гамiльтоновому формулюваннi. В рамках запропонованого пiдходу аналiзується можливiсть комбiнованого опису електродинамiки i гравiтацiї. 2009 Article The Lagrangian and Hamiltonian analysis of some relativistic electrodynamics models and their quantization / N.N. Bogolubov (Jr.), A.K. Prykarpatsky // Condensed Matter Physics. — 2009. — Т. 12, № 4. — С. 603-616. — Бібліогр.: 38 назв. — англ. 1607-324X PACS: 11.10.Wx, 05.30.-d DOI:10.5488/CMP.12.4.603 http://dspace.nbuv.gov.ua/handle/123456789/120333 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The work is devoted to the study of the Lagrangian and Hamiltonian properties of some relativistic electrodynamics
models and is a continuation of our previous investigations. Based on the vacuum field theory
approach, the Lagrangian and Hamiltonian reformulation of some classical electrodynamics models is devised.
The Dirac type quantization procedure, based on the canonical Hamiltonian formulation, is developed.
Within the approach proposed in the work a possibility of the combined description both of electrodynamics
and gravity is analyzed. |
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Bogolubov (Jr.), N.N. Prykarpatsky, A.K. |
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Bogolubov (Jr.), N.N. Prykarpatsky, A.K. The Lagrangian and Hamiltonian analysis of some relativistic electrodynamics models and their quantization Condensed Matter Physics |
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Bogolubov (Jr.), N.N. Prykarpatsky, A.K. |
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Bogolubov (Jr.), N.N. |
| title |
The Lagrangian and Hamiltonian analysis of some relativistic electrodynamics models and their quantization |
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The Lagrangian and Hamiltonian analysis of some relativistic electrodynamics models and their quantization |
| title_full |
The Lagrangian and Hamiltonian analysis of some relativistic electrodynamics models and their quantization |
| title_fullStr |
The Lagrangian and Hamiltonian analysis of some relativistic electrodynamics models and their quantization |
| title_full_unstemmed |
The Lagrangian and Hamiltonian analysis of some relativistic electrodynamics models and their quantization |
| title_sort |
lagrangian and hamiltonian analysis of some relativistic electrodynamics models and their quantization |
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Інститут фізики конденсованих систем НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/120333 |
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The Lagrangian and Hamiltonian analysis of some relativistic electrodynamics models and their quantization / N.N. Bogolubov (Jr.), A.K. Prykarpatsky // Condensed Matter Physics. — 2009. — Т. 12, № 4. — С. 603-616. — Бібліогр.: 38 назв. — англ. |
| series |
Condensed Matter Physics |
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| first_indexed |
2025-07-08T17:40:42Z |
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2025-07-08T17:40:42Z |
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1837101422306394112 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 4, pp. 603–616
The Lagrangian and Hamiltonian analysis of some
relativistic electrodynamics models and their
quantization
N.N. Bogolubov (Jr.)∗1,2, A.K. Prykarpatsky†3,4
1 The V.A. Steklov Mathematical Institute of RAN, Moscow, Russian Federation
2 The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
3 The AGH University of Science and Technology, Kraków 30-059, Poland
4 The Ivan Franko State Pedagogical University, Drohobych, Lviv region, Ukraine
Received June 19, 2009, in final form July 15, 2009
The work is devoted to the study of the Lagrangian and Hamiltonian properties of some relativistic elec-
trodynamics models and is a continuation of our previous investigations. Based on the vacuum field theory
approach, the Lagrangian and Hamiltonian reformulation of some classical electrodynamics models is de-
vised. The Dirac type quantization procedure, based on the canonical Hamiltonian formulation, is developed.
Within the approach proposed in the work a possibility of the combined description both of electrodynamics
and gravity is analyzed.
Key words: Lagrangian and Hamiltonian formalism, relativistic electrodynamics models, Dirac type
quantization
PACS: 11.10.Wx, 05.30.-d
1. Introduction
The classical relativistic electrodynamics of a freely moving charged point particle in the
Minkovski space-time M4 := E3×R is, as well known, based [5,9,11,31] on the Lagrangian formalism
assigning to it the following Lagrangian function
L := −m0(1 − u2)1/2, (1)
where m0 ∈ R is the so-called particle rest mass and u ∈ E3 is its spatial velocity in the Euclidean
space E3, expressed here and throughout further in the light speed units (that is the light speed c
units). The least action Fermat principle in the form
δS = 0, S := −
∫ t2
t1
m0(1 − u2)1/2dt (2)
for any fixed temporal interval [t1, t2] ⊂ R gives rise to the well known relativistic relationships for
the mass of the particle
m = m0(1 − u2)−1/2, (3)
the momentum of the particle
p := mu = m0u(1 − u2)−1/2 (4)
∗E-mail: nikolai bogolubov@hotmail.com
†E-mail: pryk.anat@ua.fm, prykanat@cybergal.com
c© N.N. Bogolubov (Jr.), A.K. Prykarpatsky 603
N.N. Bogolubov (Jr.), A.K. Prykarpatsky
and the energy of the particle
E0 = m = m0(1 − u2)−1/2. (5)
The origin of Lagrangian (1), owing to the reasonings from [11,31], can be extracted from the
action expression
S := −
t2
∫
t1
m0(1 − u2)1/2dt = −
τ2
∫
τ1
m0dτ, (6)
on the suitable temporal interval [τ1,τ2] ⊂ R, where, by definition,
dτ := dt(1 − u2)1/2 (7)
and τ ∈ R is the so-called proper temporal parameter assigned to a freely moving particle with re-
spect to the “rest” reference system Kr. The action (6) looks a bit controversial from the dynamical
point of view, since it is physically defined with respect to the “rest” reference system Kr, giving
rise to the constant action S = −m0(τ2−τ1), as limits of integrations τ1 < τ2 ∈ R were taken to be
fixed from the very beginning and no other constraints were imposed. Moreover, considering this
particle as charged with a charge q ∈ R and moving in the Minkovski space-time M4 under the
action of an electromagnetic field (ϕ,A) ∈ R × E3, the corresponding classical (relativistic) action
functional is chosen (see [5,9,11,31]) as follows:
S :=
τ2
∫
τ1
[
−m0dτ + q〈A, ṙ〉dτ − qϕ(1 − u2)−1/2dτ
]
, (8)
with respect to the so-called “rest” reference system, parameterized by the Euclidean space-time
variables (r, τ) ∈ E4, where as before, 〈·, ·〉 is the standard scalar product in the related Euclidean
subspace E3 and there is denoted ṙ := dr/dτ in contrast to the definition u := dr/dt. The action
(8) can be rewritten as follows, with respect to the reference system moving with velocity vector
u ∈ E3
S =
t2
∫
t1
Ldt, L := −m0(1 − u2)1/2 + q〈A, u〉 − qϕ, (9)
on the suitable temporal interval [t1, t2] ⊂ R, giving rise to the following [5,9,11,31] dynamical
expressions
P = p+ qA, p = mu, (10)
for the particle momentum and
E0 = [m2
0 + (P − qA)2]1/2 + qϕ (11)
for the particle energy, where, by definition, P ∈ E3 means the common momentum of the particle
and the ambient electromagnetic field at a space-time point (r, t) ∈ M4.
The obtained expression (11) for the particle energy E0 also looks a bit controversial, since the
potential energy qϕ, entering additively, has no impact onto the particle mass m = m0(1−u2)−1/2.
As it was already mentioned [14] by L. Brillouin, the fact that the potential energy has no impact
on the particle mass tells us that “. . . any possibility of the existence of the particle mass related
with an external potential energy, is completely excluded”. This and some other special relativity
theory and electrodynamics problems, as is well known, urged many other prominent physicists of
the past [4,14,19,31,32] and of the present [18,20–24,27–30,33,34,37,38] to make significant efforts
aiming to develop alternative relativity theories based on completely different space-time and
matter structure principles.
There also is another controversial inference from the action expression (9). As one can easily
show [5,9,11,31], the corresponding dynamical equation for the Lorentz force is given as follows:
dp/dt = F := qE + qu×B, (12)
604
The Lagrangian and Hamiltonian analysis of relativistic electrodynamics models
where the operation “×” denotes, as before, the standard vector product and we put, by definition,
E := −∂A/∂t−∇ϕ (13)
for the related electric field and
B := ∇×A (14)
for the related magnetic field, acting on the charged point particle q; the operation “∇” is here, as
before, the standard gradient. The obtained expression (12) means, in particular, that the Lorentz
force F depends linearly on the particle velocity vector u ∈ E3, giving rise to its strong dependence
on the reference system with respect to which the charged particle q moves. Namely, the attempts
to reconcile this and some related controversies [4,14,18,26] forced A. Einstein to devise his special
relativity theory and proceed further to creating his general relativity theory trying to explain
the gravity by means of a geometrization of space-time and matter in the Universe. Here we
must mention that the classical Lagrangian function L in (9) is written by means of the mixed
combinations of terms expressed by means of both the Euclidean “rest” reference system variables
(r, τ) ∈ E
4 and an arbitrarily chosen reference system variables (r, t) ∈ M
4.
These problems were recently analyzed from a completely another “no-geometry” point of view
in [6,7,18], where new dynamical equations were derived, being free of the controversy mentioned
above. Moreover, the devised approach allowed to avoid the introduction of the well known Lorentz
transformations of the space-time reference systems with respect to which the action functional (9)
is invariant. From this point of view there are very interesting reasonings of work [22], in which there
are reanalyzed Galilean invariant Lagrangians, possessing the intrinsic Poincare-Lorentz group
symmetry. Below we will reanalyse the results obtained in [6,7] from the classical Lagrangian and
Hamiltonian formalisms, which will shed a new light on the related physical backgrounds of the
vacuum field theory approach to the common study of electromagnetic and gravitational effects.
2. The vacuum field theory electrodynamics equations: Lagrangian
analysis
2.1. A freely moving point particle – an alternative electrodynamical model
Within the vacuum field theory approach to the common description of the electromagnetism
and the gravity, devised in [6,7], the main vacuum potential field function W̄ : M
4→ R, related to
a charged point particle q, satisfies, in the case of the “rested” external charged point objects, the
following [6] dynamical equation
d
dt
(−W̄u) = −∇W̄ , (15)
where, as above, u := dr/dt is the particle velocity with respect to some reference system.
To analyse the dynamical equation (15) from the Lagrangian point of view we will write the
corresponding action functional as
S := −
t2
∫
t1
W̄dt = −
τ2
∫
τ1
W̄ (1 + ṙ2)1/2 dτ, (16)
expressed with respect to the “rest” reference system Kr. Having fixed proper temporal parameters
τ1 < τ2 ∈ R, from the least action condition δS = 0 one easily finds that
p := ∂L/∂ṙ = −W̄ ṙ(1 + ṙ2)−1/2 = −W̄u, ṗ := dp/dτ = ∂L/∂r = −∇W̄ (1 + ṙ2)1/2, (17)
where, owing to (16), the corresponding Lagrangian function
L := −W̄ (1 + ṙ2)1/2. (18)
605
N.N. Bogolubov (Jr.), A.K. Prykarpatsky
Recalling now the definition of the particle mass
m := −W̄ (19)
and the relationships
dτ = dt(1 − u2)1/2, ṙdτ = udt, (20)
from (17) we easily obtain an exact dynamical equation (15). Moreover, one easily obtains that
the dynamical mass, defined by means of expression (19), is given as
m = m0(1 − u2)−1/2,
coinciding with the result (3) of the preceding section. Thereby, based on the above obtained
results, one can formulate the following proposition.
Proposition 2.1 The alternative freely moving point particle electrodynamical model (15) allows
the least action formulation (16) with respect to the “rest” reference system variables, where the
Lagrangian function is given by expression (18). Its electrodynamics is completely equivalent to
that of a classical relativistic freely moving point particle, described in section 2.
2.2. A moving charged point particle – an alternative electrodynamical model
Proceed now to the case where our charged point particle q moves in the space time with
velocity vector u ∈ E3 and interacts with another external charged point particle, moving with
velocity vector uf ∈ E3 subject to some common reference system K. As was shown in [6,7], the
corresponding dynamical equation on the vacuum potential field function W̄ : M4→ R is given as
d
dt
[
−W̄ (u− uf)
]
= −∇W̄ . (21)
As the external charged particle moves in the space-time, it generates the related magnetic field
B := ∇ × A, whose magnetic vector potential A : M4→ E
3 is defined, owing to the results of
[6,7,18], as
qA := W̄uf . (22)
Since, owing to (17), the particle momentum p = −W̄u, equation (21) can be equivalently rewritten
as
d
dt
(p+ qA) = −∇W̄ . (23)
To represent the dynamical equation (23) within the classical Lagrangian formalism, we start from
the following action functional naturally generalizing functional (16):
S := −
τ2
∫
τ1
W̄ (1 + |ṙ − ξ̇|2)1/2dτ, (24)
where we denoted by ξ̇ = ufdt/dτ, dτ = dt(1 − |u− uf |
2)1/2, which take into account the relative
velocity of our charged point particle q with respect to the reference system K′, moving with
velocity vector uf ∈ E3 subject to the reference system K. In this case, evidently, our charged
point particle q moves with velocity vector u−uf ∈ E3 subject to the reference system K′, and the
external charged particle is, respectively, in rest.
Compute now the least action variational condition δS = 0, taking into account that, owing to
(24), the corresponding Lagrangian function is given as
L := −W̄ (1 + |ṙ − ξ̇|2)1/2. (25)
Thereby, the common particle momentum
P := ∂L/∂ṙ = −W̄ (ṙ − ξ̇)(1 + |ṙ − ξ̇|2)−1/2
= −W̄ ṙ(1 + |ṙ − ξ̇|2)−1/2 + W̄ ξ̇(1 + |ṙ − ξ̇|2)−1/2 = mu+ qA := p+ qA (26)
606
The Lagrangian and Hamiltonian analysis of relativistic electrodynamics models
and the dynamical equation is given as
d
dτ
(p+ qA) = −∇W̄ (1 + |ṙ − ξ̇|2)1/2. (27)
Taking into account that dτ = dt(1 − |u− uf |
2
)1/2 and (1 + |ṙ − ξ̇|2)1/2 = (1 − |u− uf |
2)−1/2, we
finally obtain from (27) exactly the dynamical equation (23). Thus, we can formulate our result as
the next proposition.
Proposition 2.2 The alternative classical relativistic electrodynamical model (21) allows the least
action formulation (24) with respect to the “rest” reference system variables, where the Lagrangian
function is given by expression (25).
2.3. A moving charged point particle – a dual to the classical alternative electrodynamical
model
It is easy to observe that the action functional (24) is written taking into account the classical
Galilean transformations of reference systems. If we now consider the action functional (16) for
a charged point particle, moving with respect the reference system Kr, and take into account its
interaction with an external magnetic field, generated by the vector potential A : M4 → E3, it can
be naturally generalized as
S :=
t2
∫
t1
(
−W̄dt+ q〈A, dr〉
)
=
τ2
∫
τ1
[
−W̄
(
1 + ṙ2
)1/2
+ q〈A, ṙ〉
]
dτ, (28)
where we accepted here that dτ = dt(1 − u2)1/2.
Thus, the corresponding common particle-field momentum looks as follows:
P := ∂L/∂ṙ = −W̄ ṙ(1 + ṙ2)−1/2 + qA = mu+ qA := p+ qA, (29)
satisfying the equation
Ṗ := dP/dτ = ∂L/∂r = −∇W̄ (1 + ṙ2)1/2 + q∇〈A, ṙ〉
= −∇W̄ (1 − u2)−1/2 + q∇〈A, u〉(1 − u2)−1/2, (30)
where
L := −W̄ (1 + ṙ2)1/2 + q〈A, ṙ〉 (31)
is the corresponding Lagrangian function. Taking now into account that dτ = dt(1 − u2)1/2, one
easily finds from (30) that
dP/dt = −∇W̄ + q∇〈A, u〉. (32)
Upon substituting (29) into (32) and making use of the well known [11] identity
∇〈a, b〉 = 〈a,∇〉b+ 〈b,∇〉a+ b× (∇× a) + a× (∇× b), (33)
where a, b ∈ E3 are arbitrary vector functions, we finally obtain the classical expression for the
Lorentz force F, acting on the moving charged point particle q :
dp/dt := F = qE + qu×B, (34)
where, by definition,
E := −∇W̄q−1 − ∂A/∂t (35)
is the corresponding electric field and
B := ∇×A (36)
is the corresponding magnetic field.
We formulate the obtained result as the next proposition.
607
N.N. Bogolubov (Jr.), A.K. Prykarpatsky
Proposition 2.3 The classical relativistic Lorentz force (34) allows the least action formulation
(28) with respect to the “rest” reference system variables, where Lagrangian function is given by
expression (31). Its electrodynamics, described by the Lorentz force (34) is completely equivalent to
the classical relativistic moving point particle electrodynamics, described by means of the Lorentz
force (12) in section 2.
Concerning the previously obtained dynamical equation (27) we can easily observe that it can
be equivalently rewritten as follows:
dp/dt = (−∇W̄ − qdA/dt+ q∇〈A, u〉) − q∇〈A, u〉. (37)
The latter, owing to (32) and (34), finally takes the following Lorentz type force form
dp/dt = qE + qu×B − q∇〈A, u〉, (38)
found earlier in [6,7,18].
Expressions (34) and (38) are equal to each other up to the gradient term Fc := −q∇〈A, u〉,
which allows to reconcile the Lorentz forces acting on a charged moving particle q with respect
to different reference systems. This fact is important for our vacuum field theory approach since
it needs to use no special geometry and makes it possible to analyse both electromagnetic and
gravitational fields simultaneously, based on a new definition of the dynamical mass by means of
expression (19).
3. The vacuum field theory electrodynamics equations: Hamiltonian
analysis
It is well know [1,2,8,9,17] that any Lagrangian theory allows the equivalent canonical Hamil-
tonian representation via the classical Legendrian transformation. As we have already formulated
above our vacuum field theory of a moving charged particle q in the Lagrangian form, we proceed
now to its Hamiltonian analysis making use of the action functionals (16), (25) and (28).
Take, first, the Lagrangian function (18) and the momentum expression (17) for defining the
corresponding Hamiltonian function
H := 〈p, ṙ〉 − L = −〈p, p〉W̄−1(1 − p2/W̄ 2)−1/2 + W̄ (1 − p2/W̄ 2)−1/2
= −p2W̄−1(1 − p2/W̄ 2)−1/2 + W̄ 2W̄−1(1 − p2/W̄ 2)−1/2
= −(W̄ 2 − p2)(W̄ 2 − p2)−1/2 = −(W̄ 2 − p2)1/2. (39)
As a result, we easily obtain [1,2,8,9] that the Hamiltonian function (39) is a conservation law of
the dynamical field equation (15), that is for all τ, t ∈ R
dH/dt = 0 = dH/dτ, (40)
which naturally allows to interpret it as the energy expression. Thus, we can write that the particle
energy
E = (W̄ 2 − p2)1/2. (41)
The suitable Hamiltonian equations, equivalent to the vacuum field equation (15), look as follows:
ṙ := dr/dτ = ∂H/∂p = p(W̄ 2 − p2)−1/2,
ṗ := dp/dτ = −∂H/∂r = W̄∇W̄ (W̄ 2 − p2)−1/2. (42)
Thereby, based on the above obtained results, one can formulate the following proposition.
Proposition 3.1 The alternative freely moving point particle electrodynamical model (15) allows
the canonical Hamiltonian formulation (42) with respect to the “rest” reference system variables,
where the Hamiltonian function is given by expression (39). Its electrodynamics is completely equiv-
alent to the classical relativistic freely moving point particle electrodynamics, described in section 2.
608
The Lagrangian and Hamiltonian analysis of relativistic electrodynamics models
Based now on the Lagrangian expression (25) one can construct, the same way as above,
the Hamiltonian function for the dynamical field equation (23), describing the motion of charged
particle q in external electromagnetic field in the canonical Hamiltonian form:
ṙ := dr/dτ = ∂H/∂P, Ṗ := dP/dτ = −∂H/∂r, (43)
where
H := 〈P, ṙ〉 − L = 〈P, ξ̇ − PW̄−1(1 − P 2/W̄ 2)−1/2〉 + W̄
[
W̄ 2(W̄ 2 − P 2)−1
]1/2
= 〈P, ξ̇〉 + P 2(W̄ 2 − P 2)−1/2 − W̄ 2(W̄ 2 − P 2)−1/2 = −(W̄ 2 − P 2)(W̄ 2 − P 2)−1/2 + 〈P, ξ̇〉
= −(W̄ 2 − P 2)1/2 − q〈A,P 〉(W̄ 2 − P 2)−1/2. (44)
Here we took into account that, owing to definitions (22) and (26),
qA := W̄uf = W̄dξ/dt = W̄
dξ
dτ
·
dτ
dt
= W̄ ξ̇(1 − |u− v|
2
)1/2
= W̄ ξ̇(1 + |ṙ − ξ̇|2)−1/2 = −W̄ ξ̇(W̄ 2 − P 2)1/2W̄−1 = −ξ̇(W̄ 2 − P 2)1/2, (45)
or
ξ̇ = −qA(W̄ 2 − P 2)−1/2, (46)
where A : M4→ R
3 is the related magnetic vector potential, generated by the moving external
charged particle.
Thereby we can state that the Hamiltonian function (44) satisfies the energy conservation
conditions
dH/dt = 0 = dH/dτ, (47)
for all τ, t ∈ R, that is the suitable energy expression
E = (W̄ 2 − P 2)1/2 + q〈A,P 〉(W̄ 2 − P 2)−1/2 (48)
holds. The result (48) essentially differs from that obtained in [11], which makes use of the well
known Einsteinian Lagrangian for a moving charged point particle q in external electromagnetic
field. Thereby, our result can be formulated as follows.
Proposition 3.2 The alternative classical relativistic electrodynamical model (21) allows the Hamil-
tonian formulation (43) with respect to the “rest” reference system variables, where the Hamiltonian
function is given by expression (44).
To make this difference more clear, below we will analyse the Lorentz force (34) from the Hamil-
tonian point of view based on the Lagrangian function (31). Thus, we obtain that the corresponding
Hamiltonian function
H := 〈P, ṙ〉 − L = 〈P, ṙ〉 + W̄ (1 + ṙ2)1/2 − q〈A, ṙ〉 = 〈P − qA, ṙ〉 + W̄ (1 + ṙ2)1/2
= −〈p, p〉W̄−1(1 − p2/W̄ 2)−1/2 + W̄ (1 − p2/W̄ 2)−1/2
= −(W̄ 2 − p2)(W̄ 2 − p2)−1/2 = −(W̄ 2 − p2)1/2. (49)
Since p = P − qA, expression (49) takes the final “no interaction” [11,31,35,36] form
H = −[W̄ 2 − (P − qA)2]1/2, (50)
being conservative with respect to the evolution equations (29) and (30), that is
dH/dt = 0 = dH/dτ (51)
for all τ, t ∈ R. The latter are simultaneously equivalent to the following Hamiltonian system:
ṙ = ∂H/∂P = (P − qA)[W̄ 2 − (P − qA)2]−1/2,
Ṗ = −∂H/∂r = (W̄∇W̄ −∇〈qA, (P − qA)〉)[W̄ 2 − (P − qA)2]−1/2, (52)
609
N.N. Bogolubov (Jr.), A.K. Prykarpatsky
which can be easily checked by direct calculations. Really, the first equation
ṙ = (P − qA)
[
W̄ 2 − (P − qA)2
]−1/2
= p(W̄ 2 − p2)−1/2
= mu(W̄ 2 − p2)−1/2 = −W̄u(W̄ 2 − p2)−1/2 = u(1 − u2)−1/2, (53)
holds, owing to the condition dτ = dt(1 − u2)1/2 and definitions p := mu, m = −W̄ , postulated
from the very beginning. Similarly we obtain that
Ṗ = −∇W̄ (1 − p2/W̄ 2)−1/2 + ∇〈qA, u〉(1 − p2/W̄ 2)−1/2
= −∇W̄ (1 − u2)−1/2 + ∇〈qA, u〉(1 − u2)−1/2, (54)
exactly coinciding with equation (32) subject to the evolution parameter t ∈ R. Our result we now
formulate as the next proposition.
Proposition 3.3 The dual to the classical relativistic electrodynamical model (34) allows the
Hamiltonian formulation (52) with respect to the “rest” reference system variables, where the
Hamiltonian function is given by expression (50).
4. The quantization of electrodynamics models within the vacuum field
theory no-geometry approach
4.1. The problem setting
In our recent works [6,7] there was devised a new regular no-geometry approach to deriving
from the first principles the electrodynamics of a moving charged point particle q in external elec-
tromagnetic field. This approach has, in part, to reconcile the existing mass-energy controversy
[14] within the classical relativistic electrodynamics. Based on the vacuum field theory approach
proposed in [6,7,18] we reanalysed this problem in the above sections both from Lagrangian and
Hamiltonian points of view having derived crucial expressions for the corresponding energy func-
tions and Lorentz type forces, acting on the moving charge point particle q.
Since all our electrodynamics models were represented here in the canonical Hamiltonian form,
they are suitable for applying to them the Dirac type quantization procedure [10,3,13] and for
regular obtaining the related Schrödinger type evolution equations. Namely, to this problem there
is devoted this section.
4.2. Free point particle electrodynamics model and its quantization
The charged point particle electrodynamics models, discussed in detail in sections 2 and 3, were
also considered in [7] from the dynamical point of view, where an attempt was made to apply the
quantization Dirac type procedure to the corresponding conserved energy expressions. Nevertheless,
within the canonical point of view, the true quantization procedure should be based on the suitable
canonical Hamiltonian formulation of the models, which in the case under consideration looks as
(42), (43) and (52).
In particular, consider a free charged point particle electrodynamics model, governed (42) by
the following Hamiltonian equations:
dr/dτ := ∂H/∂p = −p(W̄ 2 − p2)−1/2, dp/dτ := −∂H/∂r = −W̄∇W̄ (W̄ 2 − p2)−1/2, (55)
where we denoted, as before, by W̄ : M4 → R the corresponding vacuum field potential, char-
acterizing the medium field structure, by (r, p) ∈ E
3 × E
3 the standard canonical coordinate-
momentum variables, by τ ∈ R the proper “rest” reference system Kr time parameter, related to
our moving particle, and by H : E3 × E3 → R the Hamiltonian function
H := −(W̄ 2 − p2)1/2, (56)
610
The Lagrangian and Hamiltonian analysis of relativistic electrodynamics models
expressed here and throughout further in the light speed units. The “rest” reference system Kr,
parameterized by variables (r, τ) ∈ E4, is related to any other reference system K subject to which
our charged point particle q moves with velocity vector u ∈ E3, and which is parameterized by
variables (r, t) ∈ M
4, via the following Euclidean infinitesimal relationship:
dt2 = dτ2 + dr2, (57)
which is equivalent to the Minkovskian infinitesimal relationship
dτ2 = dt2 − dr2. (58)
The Hamiltonian function (56) satisfies, evidently, the energy conservation conditions
dH/dt = 0 = dH/dτ (59)
for all t, τ ∈ R. This means that the suitable energy value
E = (W̄ 2 − p2)1/2 (60)
can be treated by means of the Dirac type quantization scheme [10] to obtain, as ~ → 0, (or the
light speed c → ∞) the governing Schrödinger type dynamical equation. To do this, similarly to
[6,7], we need to make canonical operator replacements E → Ê := −~
i
∂
∂τ , p→ p̂ := ~
i
∇, as ~ → 0,
in the following energy determining expression:
E2 := (Êψ, Êψ) = (ψ, Ê2ψ) = (ψ, Ĥ+Ĥψ), (61)
where, by definition, owing to (60),
Ê2 = W̄ 2 − p̂2 = Ĥ+Ĥ (62)
is a suitable operator factorization in the Hilbert space H := L2(R3; C) and ψ ∈ H is the corre-
sponding normalized quantum vector state. Since the following elementary identity
W̄ 2 − p̂2 = W̄ (1 − W̄−1p̂2W̄−1)1/2(1 − W̄−1p̂2W̄−1)1/2W̄ (63)
holds, we can put, by definition, following (62) and (63) that the operator
Ĥ := (1 − W̄−1p̂2W̄−1)1/2W̄ . (64)
Having calculated the operator expression (64) as ~ → 0 up to the operator accuracy O( ~4), we
can easily obtain
Ĥ =
p̂2
2m(u)
+ W̄ := −
~2
2m(u)
∇2 + W̄ , (65)
where we took into account the dynamical mass definition m(u) := −W̄ (in the light speed units).
Thereby, based now on (61) and (65), we obtain, up to the operator accuracy O( ~4), the following
Schrödinger type evolution equation
i~
∂ψ
∂τ
:= Êψ = Ĥψ = −
~2
2m(u)
∇2ψ + W̄ψ (66)
with respect to the “rest” reference system Kr evolution parameter τ ∈ R. Concerning the related
evolution parameter t ∈ R, parameterizing a reference system K, the equation (66) takes the
following form:
i~
∂ψ
∂t
= −
~2m0
2m(u)2
∇2ψ −m0ψ. (67)
Here we took into account that, owing to (60), the classical mass relationship
m(u) = m0(1 − u2)−1/2 (68)
holds, where m0 ∈ R+ is the corresponding rest mass of our point particle q.
The obtained linear Schrödinger equation (67) for the case ~/c → 0 really coincides with that
well-known [5,10,11] from classical quantum mechanics.
611
N.N. Bogolubov (Jr.), A.K. Prykarpatsky
4.3. Classical charged point particle electrodynamics model and its quantization
We start here from the first vacuum field theory reformulation of the classical charged point
particle electrodynamics, considered in section 3 and based on the conserved Hamiltonian func-
tion (50)
H := −
[
W̄ 2 − (P − qA)2
]1/2
, (69)
where q ∈ R is the particle charge and (W̄ , A) ∈ R × E
3 is the corresponding electromagnetic field
potentials and P ∈ E3 is the common particle-field momentum, defined as
P := p+ qA, p := mu, (70)
and satisfying the well known classical Lorentz force equation. Here m := −W̄ is the observable
dynamical mass, related to our charged particle, u ∈ E
3 is its velocity vector with respect to a
chosen reference system K, being all expressed here, as before, in the light speed units.
Since our electrodynamics, based on (69), is canonically Hamiltonian, the Dirac type quantiza-
tion scheme
P → P̂ :=
~
i
∇, E → Ê := −
~
i
∂
∂τ
(71)
should be applied to the suitable energy expression
E :=
[
W̄ 2 − (P − qA)2
]1/2
, (72)
following from the conservation conditions
dH/dt = 0 = dH/dτ, (73)
satisfied for all τ, t ∈ R.
Doing now the same way as above, we can factorize the operator Ê2 as follows:
W̄ 2 − (P̂ − qA)2 = W̄ [1 − W̄−1(P̂ − qA)2W̄−1]1/2[1 − W̄−1(P̂ − qA)2W̄−1]1/2W̄ := Ĥ+Ĥ,
where, by definition, (here as ~/c→ 0, ~c = const)
Ĥ :=
1
2m(u)
(
~
i
∇− qA
)2
+ W̄ (74)
up to operator accuracy O(~4). Thereby, the related Schrödinger type evolution equation in the
Hilbert space H = L2(R3; C) looks like
i~
∂ψ
∂τ
:= Êψ = Ĥψ =
1
2m(u)
(
~
i
∇− qA
)2
ψ + W̄ψ (75)
with respect to the rest reference system Kr evolution parameter τ ∈ R. The corresponding
Schrödinger type evolution equation with respect to the evolution parameter t ∈ R looks, re-
spectively, like
i~
∂ψ
∂t
=
m0
2m(u)2
(
~
i
∇− qA
)2
ψ −m0ψ. (76)
The Schrödinger type evolution equation (75) ( as ~/c → 0, ~c = const) completely coincides
[10,12] with that well known from the classical quantum mechanics.
612
The Lagrangian and Hamiltonian analysis of relativistic electrodynamics models
4.4. Modified charged point particle electrodynamics model and its quantization
Consider now, from the canonical point of view of the true quantization procedure, the electro-
dynamics model, which looks like (27) and whose Hamiltonian function (44) is
H := −(W̄ 2 − P 2)1/2 − q〈A,P 〉(W̄ 2 − P 2)−1/2. (77)
This means that the suitable energy function
E := (W̄ 2 − P 2)1/2 + q〈A,P 〉(W̄ 2 − P 2)−1/2, (78)
where, as before,
P := p+ qA, p := mu, m := −Ẇ , (79)
is a conserved quantity for (27), which we will canonically quantize via the Dirac procedure (71).
To make this, let us consider the quantum condition
E2 := (Êψ, Êψ) = (ψ, Ê2ψ), (ψ, ψ) := 1, (80)
where, by definition, Ê := −~
i
∂
∂t and ψ ∈ H = L2(R3; C) is a suitable normalized quantum state
vector. Making now use of the energy function (78), one can easily obtain that
E2 = W̄ 2 − (P − qA)2 + q2[〈A,A〉 + 〈A,P 〉(W̄ 2 − P 2)−1〈P,A〉], (81)
which upon the canonical Dirac type quantization P → P̂ := ~
i
∇ transforms into the symmetrized
operator expression
Ê2 = W̄ 2 − (P̂ − qA)2 + q2〈A,A〉 + q2〈A, P̂ 〉(W̄ 2 − P̂ 2)−1〈P̂ , A〉. (82)
Having factorized operator (82) in the form Ê2 = Ĥ+Ĥ, we obtain that up to the operator accuracy
O(~4) (as ~/c→ 0, ~c = const)
Ĥ :=
1
2m(u)
(
~
i
∇− qA
)2
−
q2
2m(u)
〈A,A〉 −
q2
2m3(u)
〈
A,
~
i
∇
〉 〈
~
i
∇, A
〉
, (83)
where we put, as before, m(u) = −W̄ in the light speed units. Thus, owing to (80) and (83), the
resulting Schrödinger evolution equation takes the form
i~
∂ψ
∂τ
:= Ĥψ =
1
2m(u)
(
~
i
∇− qA
)2
ψ −
q2
2m(u)
〈A,A〉ψ −
q2
2m3(u)
〈
A,
~
i
∇
〉 〈
~
i
∇, A
〉
ψ (84)
with respect to the “rest” reference system proper evolution parameter τ ∈ R. Similarly one
also obtains the related Schrödinger type evolution equation with respect to the time parameter
t ∈ R which we will not deal here with. The result (84) essentially differs from the corresponding
classical Schrödinger evolution equation (75) which, thereby, forces us to more thoroughly reanalyse
the main physically motivated principles, put into the backgrounds of classical electrodynamical
models, described by the Hamiltonian functions (69) and (77) giving rise to different Lorentz type
force expressions. This analysis we are to do in detail in a next work under preparation.
5. Conclusion
Thereby, we can claim that all the dynamical field equations discussed above are canonical
Hamiltonian systems with respect to the corresponding proper “rest” reference systems, parame-
terized by suitable time parameters τ ∈ R. Owing to the passing to the basic reference system K
with the time parameter t ∈ R the related Hamiltonian structure is naturally lost, giving rise to
a new interpretation of the real particle motion as the one having an absolute sense only with re-
spect to the proper “rest” reference system and being completely relative with respect to all other
613
N.N. Bogolubov (Jr.), A.K. Prykarpatsky
reference systems. Concerning the Hamiltonian expressions (39), (44) and (50) obtained above,
one observes that all of them strongly depend on the vacuum potential field function W̄ : M4→ R,
thereby dissolving the mass problem of the classical energy expression, pointed out earlier [14] by
L. Brillouin. It is necessary to mention here that the procedure subject to the canonical Dirac type
quantization can be applied only to the corresponding dynamical field systems considered with
respect to their proper “rest” reference systems.
Remark 5.1 Some comments can be also made concerning the classical relativity principle. Namely,
we have obtained our results without using any Lorentz transformations of reference systems but
only the natural notion of the “rest” reference system and its suitable parametrization with respect
to any other moving reference systems. It seems reasonable since, in reality, the true state changes
of a moving charged particle q are exactly realized only with respect to its proper “rest” reference
system. Thereby, the only question, still left open here, is that about the physical justification of the
corresponding relationship between time parameters of the moving and “rest” reference systems.
This relationship, being accepted throughout this work, looks like
dτ = dt(1 − u2)1/2, (85)
where u := dr/dt ∈ E3 is the velocity vector with which the “rest” reference system Kr moves with
respect to another arbitrarily chosen reference system K. The expression (85) means, in particular,
that there holds the equality
dt2 − dr2 = dτ2, (86)
which exactly coincides with the classical infinitesimal Lorentz invariant. Its appearance is, evi-
dently, not casual here, since all our dynamical vacuum field equations were successively derived
[6,7] from the governing equations on the vacuum potential field function W : M4→ R in the form
∂2W/∂t2 −∇2W = ρ, ∂W/∂t+ ∇(vW ) = 0, ∂ρ/∂t+ ∇(vρ) = 0, (87)
being a priori Lorentz invariant, where we denoted the charge density by ρ ∈ R and the suitable
local velocity of the vacuum field potential changes by v := dr/dt. Thereby, the dynamical infini-
tesimal Lorentz invariant (86) reflects this intrinsic structure of equations (87). Being rewritten in
the following nonstandard Euclidean form:
dt2 = dτ2 + dr2 (88)
it gives rise to quite another time relationship between reference systems K and Kr :
dt = dτ(1 + ṙ2)1/2, (89)
where, as earlier, we denoted by ṙ := dr/dτ the related particle velocity with respect to the “rest”
reference system. Thus, we observe that all our Lagrangian analysis completed in section 2 is based
on the corresponding functional expressions written in these “Euclidean” space-time coordinates
and with respect to which the least action principle was applied. So, we see that there exist
two alternatives – the first is to apply the least action principle to the corresponding Lagrangian
functions expressed in the Minkovski type space-time variables with respect to an arbitrarily chosen
reference system K, and the second is to apply the least action principle to the corresponding
Lagrangian functions expressed in the space-time Euclidean type variables with respect to the
“rest” reference system Kr.
A slightly amusing but exciting inference following from our analysis in this work is the fact that
all the classical special relativity results related with electrodynamics of charged point particles
can be obtained one-to-one making use of our new definitions of the dynamical particle mass
and the least action principle with respect to the associated Euclidean type space-time variables
parameterizing the “rest” reference system.
One more remark is appropriate here concerning the quantization procedure of the proposed
electrodynamic models. If the dynamical vacuum field equations are expressed in the canonical
614
The Lagrangian and Hamiltonian analysis of relativistic electrodynamics models
Hamiltonian form, there remain only technical problems in order to quantize them and obtain the
corresponding Schrödinger type evolution equations in suitable Hilbert spaces of quantum states.
There exists still another important inference from the approach devised in this work, consisting
in a complete loss of the essence of the well-known Einsteinian equivalence principle [4,5,11,26,31],
which becomes superfluous for our vacuum field theory of electromagnetism and gravity.
Based on the canonical Hamiltonian formalism devised in this work, concerning the alternatively
charged point particle electrodynamic models, we succeeded in treating their Dirac type quanti-
zation. The obtained results were compared with the classical ones, but the physically motivated
choice of a true model is left for the future studies. Another important aspect of the developed vac-
uum field theory no-geometry approach to combining the electrodynamics with the gravity consists
in singling out the decisive role of the related “rest” reference system Kr. Namely, with respect to
the “rest” reference system evolution parameter τ ∈ R, all our electrodynamic models make both
the Lagrangian and Hamiltonian formulations suitable for the canonical quantization. The physical
nature of this phenomenon remains insufficiently understandable so far. There is [11,26,29,31,30]
no physically reasonable explanation of this decisive role of the “rest” reference system, except for
the very interesting reasonings by R. Feynman who argued in [5] that the relativistic expression for
the classical Lorentz force (12) has physical sense only with respect to the “rest” reference system
variables (r, τ) ∈ E4. In the sequel of our work we are to analyse the quantization scheme more in
detail and make a step toward the vacuum quantum field theory of infinite many particle systems.
Acknowledgements
The Authors are cordially thankful to the Abdus Salam International Centre for Theoretical
Physics in Trieste, Italy, for the hospitality during their research 2007–2008 scholarships, and
to the Institute for Condensed Matter Physics for the invitation to take part in the International
Conference “Statistical Physics” held in Lviv, Ukraine, during 23–30 June 2009. A. P. is, especially,
grateful to Profs. P.I. Holod (Kyiv, UKMA), J.M. Stakhira (Lviv, NUL), U. Taneri (Cyprus, EMU),
J. S lawianowski (Warsaw, IPPT), Z. Peradzyński (Warsaw, UW) and M. B laszak (Poznań, UP)
for fruitful discussions, useful comments and remarks. Last but not least thanks go to academician
Prof. A.A. Logunov for his interest to the work, as well as to Mrs. Dilys Grilli (Trieste, Publications
office, ICTP) and Natalia K. Prykarpatska for professional help in preparing the manuscript for
publication.
References
1. Arnold V.I. Mathematical Methods of Classical Mechanics. Springer, New York, 1978.
2. Abraham R., Marsden J. Foundation of Mechanics. The Benjamin/Cummings Publ. Co, Masachusets,
1978.
3. Bogoliubov N.N., Shirkov D.V. Quantum Fields. Nauka, Moscow, 1984.
4. Feynman R.P.M. Lectures on Gravitation. Notes of California Inst. of Technology, 1971.
5. Feynman R., Leighton R., Sands M. The Feynman Lectures on Physics. Electrodynamics, vol. 2.
Addison-Wesley Publ. Co., Massachusetts, 1964.
6. Prykarpatsky A.K., Bogolubov N.N. (Jr.), Taneri U. Theor. Math. Phys., 2009, 160, 1079–1095.
7. Prykarpatsky A.K., Bogolubov N.N. (Jr.), Taneri U. Preprint ICTP, Trieste, IC/2008/051
(http://publications.ictp.it).
8. Prykarpatsky A., Mykytiuk I. Algebraic Integrability of Nonlinear Dynamical Systems Onmanifolds:
Classical and Quantum Aspects. Kluwer Academic Publishers, Netherlands, 1998.
9. Thirring W. Classical Mathematical Physics. Third Edition. Springer, 1992.
10. Dirac P.A.M. The Principles of Quantum Mechanics. Second edition. Clarendon Press, Oxford, 1935.
11. Landau L.D., Lifshitz E.M. Field Theory, vol. 2. Nauka, Moscow, 1973.
12. Landau L.D., Lifshitz E.M. Quantum Mechanics, vol. 6. Nauka, Moscow, 1974.
13. Bogolubov N., Shirkov D. Introduction to the Theory of Quantized Fields. Interscience, New York,
1959.
14. Brillouin L. Relativity reexamined. Academic Press Publ., New York and London, 1970.
15. Faddeev L.D. Russian Physical Surveys, 1982, 136, No. 3, 435–457 (in Russian).
615
N.N. Bogolubov (Jr.), A.K. Prykarpatsky
16. Marsden J., Chorin A. Mathematical Foundations of the Mechanics of Liquid. Springer, New York,
1993.
17. Holod P.I., Klimyk A.I. Mathematical Foundations of Symmetry Theory. Naukova Dumka, Kyiv, 1992
(in Ukrainian).
18. Repchenko O. Field Physics. Galeria Publ., Moscow, 2005.
19. Brans C.H., Dicke R.H. Phys. Rev., 1961, 124, 925.
20. Deser S., Jackiw R. Preprint arXiv:hep-th/9206094, 1992.
21. Dunner G., Jackiw R. Preprint arXiv:hep-th/92004057, 1992.
22. Jackiw R., Polychronakos A.P. Preprint arXiv:hep-th/9809123, 1998.
23. Wilchek F. Ann. Henry Poincare, 2003, 4, 211–228.
24. Bialynicky-Birula I. Phys. Rev., 1967, 155, 1414; Bialynicky-Birula I. Phys. Rev., 1968, 166, 1505.
25. Kleinert H. Path Integrals. Second Edition, p. 685. World Scientific, 1995.
26. Klymyshyn I.A. Relativistic Astronomy. Naukova Dumka, Kyiv, 1980 (in Ukrainian).
27. Logunov A.A., Mestvirishvili M.A. Relativistic Theory of Gravitation. Nauka, Moscow, 1989 (in Rus-
sian).
28. Logunov A.A. The Theory of Gravity. Nauka, Moscow, 2000.
29. Logunov A.A. Relativistic Theory of Gravitation. Nauka, Moscow, 2006 (in Russian).
30. Logunov A.A. Lectures on Relativity Theory and Gravitation. Nauka, Moscow, 1987.
31. Pauli W. Theory of Relativity. Oxford Publ., Oxford, 1958.
32. Weinstock R. Am. J. Phys., 1965, 33, 640–645.
33. Mermin N.D. Am. J. Phys., 1984, 52, 119–124.
34. Mermin N.D. It’s About Time: Understanding Einstein’s Relativity. Princeton University Press, Pri-
nceton, NJ., 2005.
35. Kupershmidt B.A. Diff. Geom. & Appl., 1992, 2, 275–293.
36. Prykarpatsky Y.A., Samoilenko A.M., Prykarpatsky A.K. Opuscula Mathematica, 2005, 25, No. 2,
287–298.
37. Green B. The Fabric of the Cosmos. Vintage Books Inc., New York, 2004.
38. Newman R.P. Comm. Mathem. Physics, 1989, 123, 17–52.
Лагранжiв та Гамiльтонiв аналiз деяких релятивiстичних
моделей електродинамiки та їх квантування
М.М. Боголюбов (мол.)1,2, А.К. Прикарпатський3,4
1 Математичний iнститут iм. В.А. Стєклова РАН, Москва, Росiйська Федерацiя
2 Мiжнародний центр теоретичної фiзики iм. Абдуса Салама, Трiєст, Iталiя
3 Гiрничо-Металургiйна академiя iм. Ст. Сташiца, 30–059 Кракiв, Польща
4 Державний педагогiчний унiверситет iм. I. Франка, Дрогобич, Україна
Отримано 19 червня 2009 р., в остаточному виглядi – 15 липня 2009 р.
Робота присвячена вивченню лагранжевих та гамiльтонових властивостей деяких релятивiстич-
них моделей електродинамiки i продовжує нашi попереднi дослiдження. Базуючись на теоретико-
польовому пiдходi, розроблено Лагранжеве та Гамiльтонове формулювання деяких класичних мо-
делей електродинамiки. Розвинуто процедуру квантування типу Дiрка, що базується на класичному
гамiльтоновому формулюваннi. В рамках запропонованого пiдходу аналiзується можливiсть комбi-
нованого опису електродинамiки i гравiтацiї.
Ключовi слова: Лагранжiв i Гамiльтонiв формалiзм, релятивiстичнi моделi електродинамiки,
квантування типу Дiрака
PACS: 11.10.Wx, 05.30.-d
616
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