Dirac-Kahler Theory and Massless Fields

0.14 MB·English

by  V. A. Pletyukhov

Dirac-K¨ahler Theory and Massless Fields
V. A. Pletyukhov
Brest State University, Brest, Belarus
V. I. Strazhev
Belarusian State University, Minsk, Belarus
Three massless limits of the Dirac-K¨ahler theory are considered. It is shown that the Dirac-
Ka¨hlerequation for massive particles can be represented as aresult of thegauge-invariant mixture
(topological interaction) of the abovemassless fields.
I. INTRODUCTION
In the 1980’s, it was shown in [1] with the use of differential forms apparatus that the the Dirac-Ka¨hler equation
(DKE) is correct in describing Dirac particles (quarks) in the lattice formulation of QCD. After that the DKE has
become to attract attention of many theorists. Note that the name ”Dirac-K¨ahler equation” was introduced in [1]
0
although the vector form of the DKE was discovered by Darwin [2]. His aim was to find an equation of motion of
1
an electronthat wouldbe equivalentto the Dirac equation but without using spinors. Fundamental properties of the
0
2 DKEwereestablishedbyK¨ahler[3]. Lateron,theDKEhasbeenrediscoveredindifferentmathematicalformulations
(see, e.g., ref. [4, pp.38,51] and references therein).
n
One should emphasize that up to now only massive DKE have been studied. The massless limit of the DKE has
a
J not been investigated in detail yet. The present paper is aimed at making up such a deficiency.

4
1
II. THE DIRAC-KA¨HLER EQUATION

]
h
t The DKE is equivalent to the following tensor system
-
p
∂ ψ +mψ =0, (1a)
e µ µ 0
h ∂ ψ˜ +mψ˜ =0, (1b)
µ µ 0
[
∂ ψ +∂ ψ +mψ =0, (1c)
ν [µν] µ 0 µ
1
1
v ε ∂ ψ +∂ ψ˜ +mψ˜ =0, (1d)
3 2 µναβ ν [αβ] µ 0 µ
0 ∂ ψ +∂ ψ +ε ∂ ψ˜ +mψ =0, (1e)
µ ν ν µ µναβ α β [µν]
4 −
2 where ψ˜ = 1 ε ψ is an axial vector, ψ˜ = 1 ε ψ is a pseudoscalar, and ε is the Levi-Civita
. µ 3! µναβ [ναβ] 0 4! µναβ [µναβ] µναβ
1 tensor (ε = i).
1234
0 The set of equ−ations (1) can be brought to the following form
0
1
(Γ ∂ +m) Ψ=0, (2)
µ µ
:
v
i where Ψ is the 16-component wave function
X
r Ψ ΨA :ψ0,ψ˜0,ψµ,ψ˜µ,ψ[µν], (3)
a ≡
consisting of the Dirac-Ka¨hler (DK) field components which form the full set of antisymmetric tensor fields in the
spaceofdimensiond=4. The16 16matricesΓ satisfythe anticommutationrulesanalogousto theseforthe Dirac
µ
×
matrices
Γ Γ +Γ Γ =2 δ . (4)
µ ν ν µ µν
In respect to the theory of relativistic wave equations, the system of the DKE describes a particle with a single
mass m, variable spin 0 or 1, and with double degeneration of states over an additional quantum number (internal
parity). Atthe sametime, the Lagrangianofthe DK fieldis invariantunder atransformationofthe groupofinternal
(dial) symmetry SO (4, 2) [4, pp.28,35]. Group generators have the following form
′ ′ ′ ′ ′ ′
Γ , Γ Γ , Γ Γ , Γ . (5)
µ [µ ν] 5 µ 5
2
′
Here Γ is second set of 16 16 matrices satisfying the Dirac matrix algebra and commuting with Γ . The above
µ × µ
properties of the symmetry are easily checked if one takes into account that in the so-called fermion basis (see, e.g.,
′
the book [4, p.72]) the matrices Γ and Γ can be written as
µ µ
′
Γ =I γ , Γ =γ I , (6)
µ 4 ⊗ µ µ µ ⊗ 4
where γ are the Dirac matrices, I is the unit 4 by 4 matrix.
µ 4
Internal(”dial”)symmetrytransformationsrelatewitheachothertensorsofdifferentranks. Thereby,thetheoretical
groupground is established for associationof a Dirac particle with the DK field. This particle (”geometric fermion”)
in additionto the spin 1 has its inner degreesof freedomwhich areofspace-time origin. For instance,if one turns on
2
electromagnetic interaction in the standard way, i.e. by the replacement ∂ ∂ ieA (A is an electromagnetic
µ µ µ µ
→ −
potential) then the DKE will have solutions equivalent to these for the Dirac equation because the matrices in both
equations have the same algebraic properties.
Sincewearegoingtostudymasslesslimitsofthe DKEitisconvenienttotransformthe system(1)toanotherform
replacing the common mass m by two new mass parameters, namely a parameter m in (1a), (1b) and (1e), and a
1
parameter m in (1c) and (1d). One has the new system
2
∂ ψ +m ψ =0, (7a)
µ µ 1 0
∂ ψ˜ +m ψ˜ =0, (7b)
µ µ 1 0
∂ ψ +∂ ψ +m ψ =0, (7c)
ν [µν] µ 0 2 µ
1
ε ∂ ψ +∂ ψ˜ +m ψ˜ =0, (7d)
2 µναβ ν [αβ] µ 0 2 µ
∂ ψ +∂ ψ +ε ∂ ψ˜ +m ψ =0. (7e)
µ ν ν µ µναβ α β 1 [µν]
−
Matrix form of this system is as follows
(Γ ∂ + m P + m P ) Ψ=0, (8)
µ µ 1 1 2 2
where P and P are the projection operators with the properties
1 2
P2 =P , P2 =P , P +P =1, P P =0,
1 1 2 2 1 2 1 2 (9)
P Γ +Γ P =Γ , P Γ +Γ P =Γ .
1 µ µ 1 µ 2 µ µ 2 µ
A second order equation equivalent to the system (7) is
((cid:3) m m ) ψ =0. (10)
1 2 A
−
This means that the system (7) at m = 0 and m = 0 describes a particle with a single mass m = √m m , i.e. it
1 2 1 2
6 6
does not differ from the usualDKE.But if any ofthe mass parameters(or both simultaneously)is equalto zero then
the system (7) and the equation (8) correspondto the massless case. We proceed now to the study of this important
case.
III. ”ELECTROMAGNETIC” MASSLESS LIMIT
Let us suppose in (7) and (8) that
m =0. (11)
2
If m =0, then without loosing of generality one can put m =1 so that the system (7) is now
1 1
6
∂ ψ +ψ =0, (12a)
µ µ 0
∂ ψ˜ +ψ˜ =0, (12b)
µ µ 0
∂ ψ +∂ ψ =0, (12c)
ν [µν] µ 0
1
ε ∂ ψ +∂ ψ˜ =0, (12d)
2 µναβ ν [αβ] µ 0
∂ ψ +∂ ψ +ε ∂ ψ˜ +ψ =0, (12e)
µ ν ν µ µναβ α β [µν]
−
3
and correspondingly
(Γ ∂ +P ) Ψ=0. (13)
µ µ 1
Note that the transition to the massless limit by means of introduction of projective operators into (2) is a typical
procedure for fields with integer spins. For example, it is carried out when one wants to pass on to massless vector
field (i.e. electromagnetic field) from the massive Duffin-Kemmer equation.
In order to establish the physical meaning of the system (12), we note, first of all, that the field functions Ψ obey
A
the D’Alembert equation
(cid:3)Ψ =0, (14)
A
i.e. thissystemdescribesamasslessfield. Thevectorψ (x)andthepseudovectorψ˜ (x)playroleofpotentialsin(12),
µ µ
and the antisymmetric tensor ψ is an intensity tensor. Physical meaning of the scalar, ψ (x), and pseudoscalar,
[µν] 0
ψ˜ (x), functions will be clarified later on.
0
The system (12) is invariant under gauge transformations of the potentials
δψ (x)=∂ λ(x), δψ˜ (x)=∂ λ˜(x), (15)
µ µ µ µ
where λ(x) and λ˜(x) satisfy the following conditions
(cid:3) λ(x)=0, (cid:3) λ˜(x)=0. (16)
To find independent physicalstates of the field system under consideration,we performthe Fourier transformation
ψ (x) = ψ p eipxd3p+h.c., (17a)
µ Z µ
(cid:0) (cid:1)
ψ˜ (x) = ψ˜ p eipxd3p+h.c., (17b)
µ Z µ
(cid:0) (cid:1)
ψ (x) = ψ p eipxd3p+h.c., (17c)
0 Z 0
(cid:0) (cid:1)
ψ˜ (x) = ψ˜ p eipxd3p+h.c. (17d)
0 Z 0
(cid:0) (cid:1)
Let us expand the amplitudes ψ p and ψ˜ p over the complete basis e(1), e(2), p , n with properties [5]
µ µ µ µ µ µ
(cid:0) (cid:1) (cid:0) (cid:1)
e(i)e(j) =δ , e(i)p =0, e(i)n =0,
µ µ ij µ µ µ µ
p2 =0, n2 = 1. (18)
µ µ −
Note thatthe basis (18) is notorthogonalbecause itcontains anisotropic vectorp . The desireddecompositions can
µ
be written as
2
ψ p = a e(i) + bp + cn ,
µ i µ µ µ
(cid:0) (cid:1) iP=1 (19)
2
ψ˜ p = a˜ e(i) + ˜bp + c˜n .
µ i µ µ µ
(cid:0) (cid:1) iP=1
Now let us take into account that due to (16) the gauge functions λ(x) and λ˜(x) have the form analogous to (17c)
and (17d)
λ (x) = λ p eipxd3p+h.c.,
R (cid:0) (cid:1) (20)
λ˜ (x) = λ˜ p eipxd3p+h.c,
R (cid:0) (cid:1)
where λ p and λ˜ p are arbitrary amplitudes. Inserting decompositions (17a)–(17d) in (15) we obtain gauge
transform(cid:0)a(cid:1)tions for (cid:0)the(cid:1) amplitudes of the potentials
δψ p =iλ p p ,
µ µ
(cid:0) (cid:1) (cid:0) (cid:1) (21)
δψ˜ p =iλ˜ p p ,
µ µ
(cid:0) (cid:1) (cid:0) (cid:1)
4
which mean that the amplitudes ψ (p) and ψ˜ p are defined up to unessential terms iλ p p and iλ˜ p p ,
µ µ µ µ
respectively. Role of such terms in the decompo(cid:0)sit(cid:1)ions (19) play bp and ˜bp . Omitting th(cid:0)em(cid:1) we obtain (cid:0)for(cid:1)the
µ µ
amplitudes ψ (p) and ψ˜ p
µ µ
(cid:0) (cid:1)
2
ψ p = a e(i) + cn ,
µ i µ µ
(cid:0) (cid:1) iP=1 (22)
2
ψ˜ p = a˜ e(i) + c˜n .
µ i µ µ
(cid:0) (cid:1) iP=1
Note that longitudinal oscillations (degrees of freedom) are absent in (22).
Scalar degreesof freedom are eliminated at the second quantizationprocedure when the equations (12a) and(12b)
for quantized field are formulated in the form of conditions imposed on wave functions Ψ in the state space
phys
∂ ψˆ (x)+ψˆ (x) Ψ =0,
µ µ 0 phys
(cid:16) (cid:17)+
(23)
∂ ψˆ˜ (x)+ψˆ˜ (x) Ψ =0,
µ µ 0 phys
(cid:16) (cid:17)+
where the index ”+” means that the corresponding operator contains the positive-frequency part only. Keeping in
mind the relations (17), (18) and (22) we obtain from (23)
ω dˆ cˆ eipxd3p Ψ =0,
phys
(cid:16)R (cid:16) − (cid:17) (cid:17)
(24)
ω dˆ˜ cˆ˜ eipxd3p Ψ =0,
phys
(cid:16) (cid:16) − (cid:17) (cid:17)
R
where
ψ (p) ψ˜ (p)
d= 0 , d˜= 0 (25)
ω ω
play a role of the amplitudes of the scalar fields ψ and ψ˜ . It is follows from (24) that for all p, the function Ψ
0 0 phys
has to satisfy the following conditions
dˆ cˆ Ψ =0, dˆ˜ cˆ˜ Ψ =0, (26)
phys phys
(cid:16) − (cid:17) (cid:16) − (cid:17)
A standardprocedure usedto eliminate longitudinalandscalaroscillationsat quantizationof electromagneticfield
[6,pp.56,68] leads to relations [7]
Ψ , dˆ+dˆ+cˆ+cˆ Ψ =0,
phys phys
(cid:16) (cid:16) (cid:17) (cid:17)
(27)
Ψ , dˆ˜+ dˆ˜+cˆ˜+ cˆ˜ Ψ =0.
phys phys
(cid:16) (cid:16) (cid:17) (cid:17)
Due to (27) the mean values disappear in the part of the energy operator that contains scalar oscillations of both
types. Therefore, the system (12) describes a massless vector field with double degeneration of states. A special
case of such a field is the usual electromagnetic field. As is follows from the previous analysis, there are no physical
states corresponding to the scalar and pseudoscalar functions ψ (x) and ψ˜ (x). These functions serve as gauge fields
0 0
(“ghosts”).
IV. NOTOPH (KALB-RAMOND FIELD)
Let us consider the following case of the system (7):
m =0, m =1. (28)
1 2
5
In this case one has for (7)
∂ ψ =0, (29a)
µ µ
∂ ψ˜ =0, (29b)
µ µ
∂ ψ +∂ ψ +ψ =0, (29c)
ν [µν] µ 0 µ
1
ε ∂ ψ +∂ ψ˜ +ψ˜ =0, (29d)
2 µναβ ν [αβ] µ 0 µ
∂ ψ +∂ ψ +ε ∂ ψ˜ =0. (29e)
µ ν ν µ µναβ α β
−
Here ψ , ψ˜ and ψ serve as potentials and the vectors ψ and ψ˜ serve as intensities. The equations (29c) and
0 0 [µν] µ µ
(29d) are definitions of intensities via potentials and the equations (29a), (29b), and (29e) are equations of motion.
The matrix form of the system (29) is
(Γ ∂ + P ) Ψ=0. (30)
µ µ 2
Using either tensor or matrix formulation of the field system under consideration, one can easily show that all
components of the wave function obey the D’Alembert equation (14), i.e. again we deal with the massless field.
Further discussion becomes to be more convenient if one introduces an auxiliary tensor ψ˜
[µν]
1
ψ˜ = ε ψ . (31)
[µν] 2 µναβ [αβ]
One has for (29d)
∂ ψ˜ + ∂ ψ˜ + ψ˜ =0. (32)
ν [µν] µ 0 µ
This equation will be used below together with (29d).
In order to pass on to the momentum representation, we write down the scalar potentials in form of (17c), (17d)
and tensor potentials in the following form
ψ (x)= ψ p eipxd3p+h.c.,
[µν] [µν]
R (cid:0) (cid:1) (33)
ψ˜ (x)= ψ˜ p eipxd3p+h.c.
[µν] [µν]
R (cid:0) (cid:1)
We expand now the amplitudes ψ p and ψ˜ p over the complete basis (18)
[µν] [µν]
(cid:0) (cid:1) (cid:0) (cid:1)
ψ p = f e(1)e(2) e(1)e(2) +
[µν] (cid:0) (cid:1) (cid:16) µ ν − ν µ (cid:17)
2
+ g e(i)p e(i)p +
X i (cid:16) µ ν − ν µ(cid:17)
i=1
2
+ h e(i)n e(i)n +
X i (cid:16) µ ν − ν µ(cid:17)
i=1
+ e (p n p n ), (34a)
µ ν ν µ
−
ψ˜ p = f˜ e(1)e(2) e(1)e(2) +
[µν] (cid:0) (cid:1) (cid:16) µ ν − ν µ (cid:17)
2
+ g˜ e(i)p e(i)p +
X i (cid:16) µ ν − ν µ(cid:17)
i=1
2
+ h˜ e(i)n e(i)n +
X i (cid:16) µ ν − ν µ(cid:17)
i=1
+ e˜(p n p n ). (34b)
µ ν ν µ
−
Further we take into account that the system (29) is invariant at the gauge transformations
δψ (x) = ∂ λ (x) ∂ λ (x)+
[µν] µ ν ν µ
−
+ ε ∂ λ˜ (x), (35)
µναβ α β
6
where the gauge finctions λ (x) and λ˜ (x) satisfy the conditions
µ µ
(cid:3)λ ∂ ∂ λ =0, (cid:3)λ˜ ∂ ∂ λ˜ =0. (36)
µ µ ν ν µ µ ν ν
− −
Keeping in mind the symmetry between the tensors ψ and ψ˜ one can replace (35) by
[µν] [µν]
δψ (x)=∂ λ (x) ∂ λ (x),
[µν] µ ν ν µ
−
(37)
δψ˜ (x)=∂ λ˜ (x) ∂ λ˜ (x),
[µν] µ ν ν µ
−
where λ (x) andλ˜ (x) still satisfy (36). As inthe caseofthe equation(14), the solutions of(36) aresuperpositions
µ µ
of the plane waves
λ (x)= λ p eipxd3p+h.c., (38)
µ Z µ
(cid:0) (cid:1)
λ˜ (x)= λ˜ p eipxd3p+h.c., (39)
µ Z µ
(cid:0) (cid:1)
with the only difference is that now the amplitudes λ p and λ˜ p being expanded over the basis (18) have the
µ µ
structure (cid:0) (cid:1) (cid:0) (cid:1)
2
λ p = α e(i) + βp ,
µ i µ µ
(cid:0) (cid:1) iP=1 (40)
2
λ˜ p = α˜ e(i) + β˜p ,
µ i µ µ
(cid:0) (cid:1) iP=1
which does not contain terms with n (due to terms ∂ ∂ λ (x) and ∂ ∂ λ˜ (x) in (36)). Inserting (33) and (38)–
µ µ ν ν µ ν ν
(40) in (37) we obtain the following form of the gauge transformations for the potentials ψ and ψ˜
[µν] [µν]
2
δψ p = i α e(i)p e(i)p ,
[µν] i µ ν ν µ
(cid:0) (cid:1) iP=1 (cid:16) − (cid:17) (41)
2
δψ˜ p = i α˜ e(i)p e(i)p .
[µν] i µ ν ν µ
(cid:0) (cid:1) iP=1 (cid:16) − (cid:17)
The expressions (41) show that terms containing g and g˜ in (34a) and (34b) are inessential. Therefore, one can
i i
eliminate them by an appropriate choice of the parameters α and α˜ (α = ig and α˜ = ig˜), i.e. if one puts
i i i i i i
− −
ψ =ψ =ψ˜ =ψ˜ =0. (42)
[23] [31] [23] [31]
Using the definition (31) for the tensor ψ˜ and a relation following from this definition
[µν]
1
ψ = ε ψ˜ , (43)
[αβ] −2 αβµν [µν]
we obtain from (42)
ψ =ψ =ψ˜ =ψ˜ =0. (44)
[14] [24] [14] [24]
As a result, the decompositions (34a) and (34b) take the form
ψ p =f e(1)e(2) e(1)e(2) +e(p n p n ),
[µν] µ ν ν µ µ ν ν µ
(cid:0) (cid:1) (cid:16) − (cid:17) − (45)
ψ˜ p =f˜ e(1)e(2) e(1)e(2) +e˜(p n p n ).
[µν] µ ν ν µ µ ν ν µ
(cid:0) (cid:1) (cid:16) − (cid:17) −
The expressions(45) show that the tensor-potential ψ contains only two independent components either corre-
[µν]
sponding to the state of the massless spin-1 field with the longitudinal polarization. In the literature, such a field is
known as “notoph” [5] or “Kalb-Ramond field” [8]. Because the system (29) contains also the potentials ψ (x) and
0
ψ˜ (x), we conclude that this system (or matrix equation (30) that is equivalent to it) describes the Kalb-Ramond
0
field and the massless scalar field with a doubled set of states degenerated over an additional quantum number.
The massless field systems (12) and (29) like the DKE for a massive particle have an internal symmetry. Making
the use of the matrix form of these systems (13) and (30) and explicit form of the matrices Γ , Γ′, P , and P one
µ µ 1 2
can show that the system symmetry narrows up to the group SO(3,1) of which generators are determined by the
′ ′ ′ ′
matrices Γ Γ and Γ Γ .
[i k] 5 k
7
V. MASSLESS “FERMION” LIMIT
The DKE is an equation describing a free massive Dirac particle with mass m. Therefore, a natural way to pass
on to the massless limit in (7) is to put there m =m =0. We will denote such a transition as “fermion” limit. As
1 2
applied to the system of the tensor equations (7), this transition leads the systems
∂ ψ + ∂ ψ =0, (46a)
ν [µν] µ 0
1
ε ∂ ψ + ∂ ψ˜ =0, (46b)
2 µναβ ν [αβ] µ 0
∂ ψ =0, (46c)
µ µ
∂ ψ˜ =0, (46d)
µ µ
∂ ψ + ∂ ψ + ε ∂ ψ˜ =0. (46e)
µ ν ν µ µναβ α β
−
This system disintegrates over the Lorenz group into the two subsystems (46a),(46b) and (46c)–(46e).
It is evident that each of these subsystems can be written down in a matrix form with field functions U(x) and
V (x) composed from ψ , ψ˜ , ψ , and ψ , ψ˜ , respectively. In addition, if relatively to the Lorenz group, the
0 0 [µν] µ µ
function U(x) is transformed as a direct sum T of presentations for a bivector, a scalar, and a pseudoscalar then an
equationforU(x) is transformedas adirectsumRofpresentationsfor avectorandapseudovector. Forthe function
V(x) and an equationfor it the presentations T and R are interchanged. This means that for eachof the subsystems
(46a),(46b) and (46c)–(46e) taken separately,a massless analogis absent (a field function for a massive equationand
the equation itself are transformed over the same presentation of the Lorenz group). Therefore, only bilinear forms
like U¯(x)β ∂ V (x) and V¯ (x) β ∂ U(x) are Lorenz-invariant. Here β are 8 by 8 matrices of equations for U(x)
µ µ µ µ µ
and V(x). Thus, although the subsystems (46a),(46b) and (46c)–(46e) are independent algebraically the Lagrangian
formulation for them is impossible. The requirements of the Lagrangian formulation of a theory and existence of
its massive analog lead to the need for combined consideration of the subsystems (46a),(46b) and (46c)–(46e). This
circumstance is of crucialimportance because the energy-momentumdensity turns out to be zero for such a massless
system.
Indeed, the Lagrangianfor the system (46) can be written as
1
L = ψ ∂ ψ ψ (∂ ψ ∂ ψ )+
− µ µ 0 − 2 [µν] µ ν − ν µ
1
+ ψ˜ ∂ ψ˜ + ε ψ ∂ ψ˜ . (47)
µ µ 0 2 µναβ [µν] α β
Inserting (47) in the expression for the energy-momentum
∂L ∂Ψ
A
T = δ L, (48)
µν µν
∂ ∂ΨA ∂xν −
(cid:16)∂xµ(cid:17)
using further the obtained formula and equations (46) and keeping in mind that terms like full divergency can be
omitted, one finally has
T =0. (49)
µν
Hence,thefermionmasslesslimitoftheDKEtreatedasequationsdescribingaclassicalbosonfield(aparticlewith
variable spin 0,1) leads to zero energy-momentum density.
VI. GAUGE-INVARIANT MIXING OF MASSLESS FIELDS
In papers [8, 9] a non-Higgs mechanism to generate masses was proposed. The method relies on gauge-invariant
mixing (the topological interaction) an electromagnetic field and a massless vector field with the zero helicity (the
so-called Bˆ Fˆ-theory). The final result of that theory is the Duffin-Kemmer equation for a massive spin-1 particle.
∧
Such an approach is very important for the string theory (see [10–13] and references there in). It is, therefore,
important to generalize this approach for the case of massless systems of the DK type involving the complete set of
antisymmetric tensor fields in the space with dimension d=4.
8
At first, we will proceed from the matrix formulation (13) and (30) of the systems (12) and (29). We replace the
notation Ψ in (13) by Φ, ϕ , ϕ˜ , ϕ , ϕ˜ , ϕ . Now the Lagrangianof the matrix equations (13) and (30) takes the
0 0 µ µ [µν]
form
L = Φ (Γ ∂ + P ) Φ Ψ (Γ ∂ + P ) Ψ , (50)
0 µ µ 1 µ µ 2
− −
where Φ=Φ+Γ Γ′ , Ψ = Ψ+Γ Γ′. We add now to L a term
4 4 4 4 0
L = mΦP Ψ mΨP Φ, (51)
int 2 1
− −
whichdoes notdestroythe gaugeinvarianceofthe Lagrangian(50) relativelyto transformations(15), (16), (35), and
(36). One can obtain then the following matrix equations from the total LagrangianL=L +L
0 int
Γ ∂ Φ+P Φ+mP Ψ = 0, (52)
µ µ 1 2
Γ ∂ Ψ+P Ψ+mP Φ = 0. (53)
µ µ 2 1
Multiplying (52) from the left by the matrix P and using relations (9), we obtain an equation
1
Γ ∂ P Φ+mP Ψ=0. (54)
µ µ 1 2
Analogously, multiplying (53) from the left by the matrix P one has
1
Γ ∂ P Ψ+mP Φ=0. (55)
µ µ 2 1
If one puts together (54) and (55) and introduces a notation
Ψ′ =P Φ+P Ψ= ϕ , ϕ˜ , ψ , ψ˜ , ϕ , (56)
1 2 0 0 µ µ [µν]
(cid:16) (cid:17)
one arrives at the following equation
′
(Γ ∂ +m) Ψ =0, (57)
µ µ
that ut to the notation of the components of the wave function coincides with the DKE (2) and (3).
Therefore,theDKEformassiveparticlescanberepresentedasaresultofagauge-invariantmixtureoftwomassless
systems, namely dial symmetric generalizations of electromagnetic field and the Kalb-Ramond field (notoph). All
knownin the literature constructionsofthe analogousmechanismfortensor fields (anabeliancase)ofdifferentranks
in the d=4 space are particular cases of the considered approach.
Now let us consider the tensor field system (46) together with another system of the same kind
∂ ϕ +∂ ϕ =0,
ν [µν] µ 0
1
ε ∂ ϕ +∂ ϕ˜ =0,
2 µναβ ν [αβ] µ 0
∂ ϕ =0, (58)
µ µ
∂ ϕ˜ =0,
µ µ
∂ ϕ +∂ ϕ +ε ∂ ϕ˜ =0.
µ ν ν µ µναβ α β
−
The Lagrangianof these systems takes the structure
L =L +L , (59)
0 01 02
where LagrangiansL and L have the form (47). The topologicalinteraction of both systems can be described by
01 02
the Lagrangian
L = m aϕ ψ +bϕ˜ ψ˜ +cϕ ψ +
int 0 0 0 0 µ µ
(cid:16)
+ dϕ˜ ψ˜ +eϕ ψ . (60)
µ µ [µν] [µν]
(cid:17)
9
From the total LagrangianL=L +L one can obtain the following equations
0 int
∂ ϕ +amψ =0,
µ µ 0
∂ ϕ˜ bmψ˜ =0,
µ µ 0
−
∂ ϕ +∂ ϕ cmψ =0,
ν [µν] µ 0 µ
−
1
ε ∂ ϕ +∂ ϕ˜ +dmψ˜ =0,
2 µναβ ν [αβ] µ 0 µ
∂ ϕ +∂ ϕ +ε ∂ ϕ˜ +2emψ =0, (61)
µ ν ν µ µναβ α β [µν]
−
∂ ψ +amϕ =0,
ν µ 0
∂ ψ˜ bmϕ˜ =0,
µ µ 0
−
∂ ψ +∂ ψ cmϕ =0,
ν [µν] µ 0 µ
−
1
ε ∂ ψ +∂ ψ˜ +dmϕ˜ =0,
2 µναβ ν [αβ] µ 0 µ
∂ ψ +∂ ψ +ε ∂ ψ˜ +2emϕ =0.
µ ν ν µ µναβ α β [µν]
−
Then we can make transition from the equations to the two systems
∂ Λ +amΛ =0,
µ µ 0
∂ Λ˜ bmΛ˜ =0,
µ µ 0
−
∂ Λ +∂ Λ cmΛ =0, (62)
ν [µν] µ 0 µ
−
1
ε ∂ Λ +∂ Λ˜ +dmΛ˜ =0,
2 µναβ ν [αβ] µ 0 µ
∂ Λ +∂ Λ +ε ∂ Λ˜ +2emΛ =0
µ ν ν µ µναβ α β [µν]
−
and
∂ Ω amΩ =0,
µ µ 0
−
∂ Ω˜ +bmΩ˜ =0,
µ µ 0
∂ Ω +∂ Ω +cmΩ =0, (63)
ν [µν] µ 0 µ
1
ε ∂ Ω +∂ Ω˜ dmΩ˜ =0,
2 µναβ ν [αβ] µ 0− µ
∂ Ω +∂ Ω +ε ∂ Ω˜ 2emΩ =0,
µ ν ν µ µναβ α β [µν]
− −
where
Λ =ϕ +ψ , Λ˜ =ϕ˜ +ψ˜ , Λ =ϕ +ψ ,
0 0 0 0 0 0 µ µ µ
Λ˜ =ϕ˜ +ψ˜ , Λ =ϕ +ψ ,
µ µ µ [µν] [µν] [µν] (64)
Ω =ϕ ψ , Ω˜ =ϕ˜ ψ˜ , Ω =ϕ ψ ,
0 0 0 0 0 0 µ µ µ
− − −
Ω˜ =ϕ˜ ψ˜ , Ω =ϕ ψ .
µ µ µ [µν] [µν] [µν]
− −
In the case of choosing
1
a=d=1, b=c= 1, e= (65)
− 2
we obtain two types of the DKE.
The matrix form of the Lagrangianof the equations (46) and (58) is
L = Φ¯Γ ∂ Φ Ψ¯Γ ∂ Ψ. (66)
0 µ µ µ µ
− −
We add to (66) the term
L = mΦ¯Ψ mΨ¯Φ. (67)
int
− −
10
As a result, we obtain the matrix equations
Γ ∂ Φ+mΨ=0, (68)
µ µ
Γ ∂ Ψ+mΦ=0. (69)
µ µ
Putting together (68) and (69) and introducing the notation
Λ=Φ+Ψ, Ω=Φ Ψ, (70)
−
one arrives at the equations
(Γ ∂ +m) Λ=0, (71)
µ µ
(Γ ∂ m) Ω=0, (72)
µ µ
−
which are the matrix analogs of the tensor systems (62)–(65).
VII. CONCLUSION
WehaveinvestigatedtheremasslesslimitsoftheDKE.Itisshownthatfirstofthelimitsleadstoatwo-potentialfor-
mulationofthe electrodynamicsinthe Feynmangauge. SecondonegivesageneralizeddescriptionoftheOgievetsky-
Polubarinovnotoph. Third limit corresponds to the massless field with zero energy density. The method to generate
masses through the gauge-invariant mixing (the topological interaction) of massless fields (Bˆ Fˆ-theory) is gener-
∧
alized for the case of above massless systems of the DK type. As a result, the DKE of two types for particles with
masses are found. One of these equations after quantization can obey Bose-Einstein statistics and second one can
obey Dirac-Fermi statistics [14, 15]. This means that from the topological interaction of the massless DK fields we
can proceed to the tensor and Dirac fields with masses.
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