1 A general method for obtaining degenerate solutions to the Dirac and Weyl equation s and a discussion on the

2025-04-28 0 0 542.97KB 21 页 10玖币
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1
A general method for obtaining degenerate solutions to the
Dirac and Weyl equations and a discussion on the
experimental detection of degenerate states
Georgios N. Tsigaridas1,*, Aristides I. Kechriniotis2, Christos A. Tsonos2 and
Konstantinos K. Delibasis3
1Department of Physics, School of Applied Mathematical and Physical Sciences,
National Technical University of Athens, GR-15780 Zografou Athens, Greece
2Department of Physics, University of Thessaly, GR-35100 Lamia, Greece
3Department of Computer Science and Biomedical Informatics, University of Thessaly,
GR-35131 Lamia, Greece
*Corresponding Author. E-mail: gtsig@mail.ntua.gr
Abstract
In this work we describe a general method for obtaining degenerate solutions to the
Dirac equation, corresponding to an infinite number of electromagnetic 4-potentials
and fields, which are explicitly calculated. In more detail, using four arbitrary real
functions, one can automatically construct a spinor which is solution to the Dirac
equation for an infinite number of electromagnetic 4-potentials, defined by those
functions. An interesting characteristic of these solutions is that, in the case of Dirac
particles with non-zero mass, the degenerate spinors should be localized, both in
space and time. Our method is also extended to the cases of massless Dirac and Weyl
particles, where the localization of the spinors is no longer required. Finally, we
propose two experimental methods for detecting the presence of degenerate states.
Keywords: Dirac particles; Weyl particles; Degenerate solutions; Electromagnetic 4-
potentials; Electromagnetic fields
1. Introduction
We consider the Dirac equation in the form
0i a m



 +  − =
(1)
where
,
0,1,2,3
=
, are the standard Dirac matrices,
m
is the mass of the particle
and
a qU

=
where
q
is the electric charge of the particle and
U
is the
electromagnetic 4-potential. It should also be noted that Eq. (1) is written in natural
units, where the speed of light in vacuum
and the reduced Planck constant are
both set equal to one.
In a recent article [1] we have shown that all solutions to the Dirac equation satisfying
the conditions
† 0
 =
and
20
T
 
, where
0 1 2 3
i
 
=+
, are degenerate,
2
corresponding to an infinite number of electromagnetic 4-potentials which are
explicitly calculated through Theorem 5.4. We have also shown that all solutions to
the Weyl equations are degenerate. In this case, the corresponding electromagnetic
4-potentials are calculated through Theorem 3.1. In [2-5] we have extended these
results providing several classes of degenerate solutions to the Dirac and Weyl
equations for massive [2, 5] and massless [3, 4] particles, and describing their physical
properties and potential applications. Furthermore, in [4] we discuss some very
interesting properties of Weyl particles, mainly regarding their localization.
In this work, we provide a general method for obtaining degenerate solutions to the
Dirac equation for real 4-potentials, which are explicitly calculated. The method is
described in detail in section 2, and in section 3 is extended to massless Dirac and Weyl
particles. In section 4 we discuss two experimental techniques for detecting the
presence of degenerate states and the transition between these states and the non-
degenerate ones. Our conclusions are presented in section 5. We have also added
two appendices for providing the necessary mathematical background.
2. Formulation of the method and description of the degenerate spinors and
the corresponding electromagnetic 4-potentials in the case of massive Dirac
particles
It is easy to verify that any spinor of the form
cos cos
1 sin 1 sin
cos cos
1 sin 1 sin
TR




−
 
 
−+
 
 = +
 
 
− − −
 
(2)
where
,TR
are arbitrary complex functions of the spatial coordinates and time and
2, nn

 +
is an arbitrary real constant is degenerate.
Substituting the spinor given by Eq. (2) into the Dirac equation, we obtain the
following system of equations
( ) ( )
1 3 0 1 3 0
cos sin cos sinR i a a a R
  
+ + = + +
(3)
( ) ( )
1 3 0 1 3 0
cos sin cos sinT i a a a T
  
+ + = + +
(4)
( ) ( ) ( )
2 3 0 2 3 0
cos sin cos sin 1 sini R a ia ia R im T
  
− − − = − − + −
(5)
( ) ( ) ( )
2 3 0 2 3 0
cos sin cos sin 1 sini T a ia ia T im R
  
− + + = + + + +
(6)
Defining the matrix
3
cos 0 0 0
0 cos cos 0
sin 1 1 0
1 sin sin 1
ii




−−

=
−

−

(7)
and setting
11
22
33
00
T
D
D
D
D
 
 
 
=
 
 
 
(8)
where
T
is the transpose of
, the system of equations (3)-(6) can be written as
11
D R A R=
(9)
11
DT AT=
(10)
( )
22 1 sinD R A R im T
= + −
(11)
( )
33 1 sinD T A T im R
= + +
(12)
where
( )
1 1 3 0
cos sinA i a a a

= + +
(13)
2 2 3 0
cos sinA a ia ia

= − −
(14)
3 2 3 0
cos sinA a ia ia

= + +
(15)
As shown in Appendix A, using the following transformation of the coordinates
0 1 2 3
, , ,x x x x
11
22
33
00
xs
xs
xs
xs
 
 
 
=
 
 
 
(16)
the linear differential operators
, 1,2,3
i
Di=
can be written as
i
si
= 
.
Consequently, the system of equations (9)-(12) takes the form
11
R A R=
(17)
4
11
T AT=
(18)
( )
22 1 sinR A R im T
 = + −
(19)
( )
33 1 sinT AT im R
 = + +
(20)
where
1 2 3
, , , ,A A A R T
are the functions
1 2 3
, , , ,A A A R T
expressed in the coordinates
0 1 2 3
, , ,s s s s
.
Multiplying equations (17), (18) with
( )
11
exp Ads−
we obtain that
( )
( )
1 1 1
exp 0R Ads − =
(21)
( )
( )
1 1 1
exp 0T Ads − =
(22)
Consequently, the functions
,RT
can be written as
( )
11
exp R
R Ads g=
(23)
( )
11
exp T
T Ads g=
(24)
where
,
RT
gg
are arbitrary complex functions of the coordinates
0 2 3
,,s s s
. Substituting
equations (23), (24) into (19), (20) and supposing that
1
A
depends only on
01
,ss
we
obtain the following system of equations for the functions
,
RT
gg
:
( )
( )
22 1 sin
RT
A g im g
 − = −
(25)
( )
( )
33 1 sin
TR
A g im g
 − = +
(26)
Multiplying Eq. (25) by
( )
1 sinim
+
and Eq. (26) by
( )
1 sinim
−
, yields that
( )
( )
( )
22
22 1 sin cos
RT
A im g m g

− + = −
(27)
( )
( )
( )
22
33 1 sin cos
TR
A im g m g

− − = −
(28)
which, according to equations (25), (26), can be written as
( )( )
22
2 2 3 3 cos
TT
A A g m g
− − = −
(29)
5
( )( )
22
3 3 2 2 cos
RR
A A g m g
− − = −
(30)
Multiplying equations (29), (30) by
( )
2 2 3 3
exp A ds A ds−−

and assuming that
32 0A=
and
23 0A=
, the above system of equations takes the following form:
( )
( )
( )
22
2 3 2 2 3 3 2 2 3 3
exp cos exp
TT
A ds A ds g m A ds A ds g
− − = − − −
   
(31)
( )
( )
( )
22
2 3 2 2 3 3 2 2 3 3
exp cos exp
RR
A ds A ds g m A ds A ds g
− − = − − −
   
(32)
Consequently, the functions
R
g
,
T
g
can be written as
( )
2 2 3 3
exp
RR
g A ds A ds W=+

(33)
( )
2 2 3 3
exp
TT
g A ds A ds W=+

(34)
where
( )
0 2 3
,,
R
W s s s
,
( )
0 2 3
,,
T
W s s s
are solutions to the differential equation
22
23 cos W m W
= −
(35)
Here, we have also assumed that
12 0A=
and
13 0A=
, because the functions
,
RT
gg
depend only on
0 2 3
,,s s s
.
Thus, assuming that
21 0A=
,
31 0A=
,
12 0A=
,
32 0A=
,
13 0A=
,
23 0A=
(36)
the functions
,RT
can be written as
( )
1 1 2 2 3 3
exp R
R Ads A ds A ds W= + +
 
(37)
( )
1 1 2 2 3 3
exp T
T Ads A ds A ds W= + +
 
(38)
Finally, substituting the above expressions into Eq. (20), yields that the functions
,
TR
WW
should be related through the following formula:
( )
31 sin
TR
W im W
 = +
(39)
摘要：

1AgeneralmethodforobtainingdegeneratesolutionstotheDiracandWeylequationsandadiscussionontheexperimentaldetectionofdegeneratestatesGeorgiosN.Tsigaridas1,*,AristidesI.Kechriniotis2,ChristosA.Tsonos2andKonstantinosK.Delibasis31DepartmentofPhysics,SchoolofAppliedMathematicalandPhysicalSciences,NationalT...

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