Author Topic: Dyon Solutions in Non-Temporal Gauge  (Read 3802 times)

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Dyon Solutions in Non-Temporal Gauge
« on: February 18, 2012, 02:35:06 am »
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Author : Vinod Singh and D C Joshi
International Journal of Scientific & Engineering Research Volume 3, Issue 1, January-2012
ISSN 2229-5518
Download Full Paper : PDF

Abstract
Employing the Cabbibo–Ferrari type non- Abelian field tensor we consider the  gauge theory under the non-temporal gauge conditions and show that the obtained solutions are dyonic and have finite energy.

Index Term: Dyon Solutions, Non-Abilian Field Tensor, Gauge Field Theory

1. Introduction
In 1930s Dirac1 advanced the idea that isolated magnetic poles might exist. The idea of magnetic monopoles got a boost in 1970s when ’t Hooft2 and Polyakov3 showed that in gauge field theories in which the symmetry group is spontaneously broken possess classical solutions with the natural interpretation of magnetic monopoles. Soon the Julia and Zee’s4 conjecture was seen as the non-Abelian analogue of Schwinger’s Abelian dyons5. The interest on monopoles and dyons generated by Dirac1, ’t Hooft2, Polyakov3 and Julia and Zee4 has remained undiminished and extensive theoretical and experimental works on the related topics have been undertaken6-21, 30.
Since, the solutions which were interpreted as magnetic monopoles were originally found in SO(3) gauge group and this group being small for unifying electromagnetic  and  weak  interactions, larger  gauge  groups  like SU(3) were explored8-12, 22, 23 . A key factor of such theories is the twin combination of the choice of gauge and choice of gauge field tensor. Theories have in general followed the approach of Julia and Zee4 and employed usual Yang-Mills type field tensor and have used temporal gauge conditions to arrive at monopole solutions and obtained dyon solutions in non-temporal gauge.
 In 1960s, Cabbibo and Ferrari24 developed a two potential field tensor for developing a theory of Abelian dyons and Yang Mills type field tensor continued to be used for dyon solutions in non-Abelian gauge theories. One of the authors (DCJ) has in earlier papers11 developed a Cabbibo-Ferrari24 type field tensor for non-Abelian fields and employed12-13 it on non-Abelian gauge theories with electric and magnetic sources. Using the same field tensor and the Kyriakopoulos22 technique we show in the previous paper that the dyon solutions be obtained in the temporal gauge (31). The Kyriakopoulos (22) technique under the temporal gauge conditions reduced the gauge field equations into the first order differential equations whose solutions depicted a set of dyon solutions. Extending the analysis in the present paper we examine the  gauge under the non- temporal gauge conditions and find that in this case too we obtain the finite energy dyon solutions but unlike the previous case they emerge as the solutions of second order differential equations. The paper has been divided into six sections. Section 2 defines the Lagrangian density, the gauge group of the theory, field equations and matrix notation .The ansatz for obtaining the solutions has been presented in section 3. The solutions have been shown to have finite energy in section 4.the adjoining solutions be obtained in section 5. That the obtained solutions belong to electric and magnetic charges has been shown in section 6 to which then follow the concluding remarks.
2. The Gauge Group and the Lagrangian Density
 In this section we briefly recapitulate the steps from the previous paper(31).
The system whose gauge group is  , is described by the Lagrangian density
    (1)
where(31)
    (2a)
and its dual is
    (2b)
in which gauge fields   and   transform as
          (3a)
and          (3b)
where   is a gauge function
             (4)
with   the real functions of space-time and   representing the group generators of   group obeying
             (5)
The  are the   structure constants with a, b, c running from 1 to 8.  , where  (a = 1, 2, … 8) are eight Gell-Mann matrices25.
The   in the Lagrangian density (1) indicates the products in which the fields have been assumed mutually non-interacting. As a result of this assumption the mutual interaction terms, i.e. the cross-terms, disappear leaving
          (6)
 
 (7)
and   
             (8)
where
          (9)
and                (10)
with
       (11)
             (12)
                (13)
and                (14)
The covariant derivative   which expressed as
             (15)
transform as
             (16)
The potential energy   in the Lagrangian density (1) describe the self interaction of field   and  has the form
          (17)
in which   and   are real constants with . The fields   may denote the Higgs26, 27 triplet fields.
The Euler-Lagrange variations of the Lagrangian density (1) with respect to  ,  ,   and   lead to the field equations
       (18)
       (19)
          (20)
and          (21)
   Introducing the notation
                (22a)
and                (22b)
and also express the Higgs field  as
    (22c)
where  with (a = 1,2,…..,8) the Gell-Mann matrices (25), we may express the field equations (18) to (21) in matrix notation as
       (23)
       (24)
       (25)
       (26)
respectively. It is obvious from the above that11
3. The Ansatz
 In the previous paper31 the gauge field obeyed the temporal gauge conditions and here temporal parts  and  do not vanish we were required to have the ansatz 28
    
 
    (27)
where  , x1, x2 and x3 being the components of distance three-vector. We also introduce the three-vector functions   and   expressed by28
                (28)
                (29)
                (30)
                (31)
                (31)
             (32)
and29
           (33)
          (34)
             (35)
and                (36)
where    are purely   dependent.
The ansatz for the Higgs fields   as before28, 29
             (37)
and             (38)
where the coefficients   and   too are purely r-dependent. We also introduce the vector
4. Finite energy Solutions.
   In earlier paper defined31
    (39)
 (40)
where   have been defined in equations (28) to (30) and
                (41)
          (42)
                (43)
             (44)
and
       (45)
       (46)
where
             (47)
             (48)
          (49)
          (50)
with similar relations with   and  .
As shown in the following subsection, the ansatz (33), (34), (37) and (38) allow us to write the field equations (18)–(21) in terms of field equations without   indices.
We use the same ansatz and notations as used in the earlier paper(30) for temporal gauge. We also employ the ansatz for non temporal gauge(22)
we can express the space-time component of  and   as(28)
 (51)
Where
             (52)
             (53)
             (54)
             (55)
and
       (56)
where
              (57)
             (58)
             (59)
             (60)
Now we look at the field equations 31 (23) to(26) and separate their space and time components. Using eqyuations (51) and (56) the respective space and timecomponents of (23) and (24) can jbe expressed as
 (61)
     (62)
          (63)
and             (64)
where  and  are (39) and (40) for the space and time parts of eqs (25)and (250, we observe their V = 0 and find that, due to the static nature of fields and the ansatz  (25) and (26) vanish leaving the space parts as
          (65)
          (66)
    Now we first look at the set of eqs. (61), (62) and (65) that contain the space parts  of the gauge field  . Using eqs (34) in these equations we can calculate the individual terms as

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