Description

Non-Lie and discrete symmetries of the Dirac equation

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Related Documents

Share

Transcript

Nonlinear Mathematical Physics 1997, V.4, N 3–4, 436–444.
Non-Lie and Discrete Symmetriesof the Dirac Equation
Jiri NIEDERLE
∗
and
Anatolii NIKITIN
⋆
* Institute of Physics of the Academy of Sciences of the Czech Republik,Na Slovance 2, Prague 8, Czech Republic
⋆
Institute of Mathematics of the National Academy of Sciences of Ukraina,3 Tereshchenkivs’ka Street, Kyiv-4, Ukraina
Abstract
New algebras of symmetries of the Dirac equation are presented, which are formed bylinear and antilinear ﬁrst–order diﬀerential operators. These symmetries are appliedto decouple the Dirac equation for a charged particle interacting with an external ﬁeld.
I. Introduction
Symmetries of diﬀerential equations have important applications in construction of conser-vation laws [1], separation of variables [2], reduction of nonlinear problems to more simpleones [3], etc. All that causes the continuous interest of physicists and mathematicians inthe classical group-theoretical approach [4] and its modern generalizations.Early in the seventies, W.I. Fushchych proposed the fruitful concept of non-Lie symme-tries. It happens that even such well-studied subjects as the Maxwell and Dirac equationsadmit extended symmetry algebras which cannot be found using the classical Lie approach[5-7]. The distinguishing feature of these algebras is that they have usual Lie structuresin spite of the fact that their basis elements are not Lie derivatives and belong to classesof higher-order diﬀerential operators or even integro-diﬀerential operators.In recent paper [8] a new invariance algebra
D
of the Dirac equation was found. Beingthe algebra of the higher dimension than other known ﬁnite symmetry algebras of thisequation, the algebra
D
is formed by discrete symmetries like parity or charge conjugation.This algebra has useful applications in searching for hidden supersymmetries and reductionof the Dirac equation for a particle interacting with various external ﬁelds [8].In the present paper we continue the analysis of algebraic structures of discrete sym-metries and study their connections with non-Lie symmetries of the Dirac equation. Weﬁnd a ﬁnite dimensional symmetry algebra of the Dirac equation, which unites both thenon-Lie [6, 7] and involutive discrete [8] symmetries. We also apply discrete symmetriesto decouple the Dirac equation for a particle interacting with an external ﬁeld.
Copyright
c
1997 by Mathematical Ukraina Publisher.All rights of reproduction in form reserved.
NON-LIE AND DISCRETE SYMMETRIES 437
II. Lie and non-Lie symmetries of the Dirac equation
Let us start with the free Dirac equation
Lψ
= 0
, L
=
γ
µ
p
µ
−
m.
(2.1)Here
p
µ
=
i ∂ ∂x
µ
,
µ
= 0
,
1
,
2
,
3,
γ
µ
are the Dirac matrices which we choose in the form
γ
0
=
0
I I
0
, γ
a
=
0
−
σ
a
σ
a
0
, γ
4
=
iγ
0
γ
1
γ
2
γ
3
=
I
2
00
−
I
2
,
(2.2)
σ
a
(
a
= 1
,
2
,
3) are the Pauli matrices,
I
2
is the 2
×
2 unit matrix.We say a linear operator
Q
is a
symmetry
of equation (2.1) if there exists such anoperator
α
Q
that[
Q,L
] =
α
Q
L.
(2.3)In the classical Lie approach [4] symmetry operators are searched for in the form
Q
=
a
µ
p
µ
+
b
(2.4)where
a
µ
are functions of
x
= (
x
0
,x
1
,...
)
,b
is a matrix dependent on
x
. The maximalinvariance algebra of equation (2.1) in the class of operators (2.4) is the Poincar´e algebrawhose basis elements are
P
µ
=
p
µ
, J
µν
=
x
µ
p
ν
−
x
ν
p
µ
+
i
4 [
γ
µ
,γ
ν
]
.
(2.5)In other words, any symmetry of the Dirac equation, which has form (2.4), is a linearcombination of generators (2.5) (refer, e.g., to [9]). The related
α
Q
in (2.3) are equal tozero.Supposing that coeﬃcients
a
µ
in (2.4) are matrices, we ﬁnd the simplest non-Lie sym-metry algebra for equation (2.1) which is generated by the following operators [7, 9]Σ
µν
= 12 [
γ
µ
,γ
ν
] + 1
m
(1
−
iγ
4
)(
γ
µ
p
ν
−
γ
ν
p
µ
)
,
Σ
1
=
γ
4
−
im
(1
−
iγ
4
)
γ
µ
p
µ
.
(2.6)Operators (2.6) satisfy relations (2.3) for
α
Σ
µν
= 1
m
(
γ
µ
p
ν
−
γ
ν
p
µ
) and
α
Σ
1
=
−
1
mγ
4
γ
µ
p
µ
.Moreover, operators Σ
µν
commute with Σ
1
and form the Lie algebra isomorphic to so(1,3).We notice that Lie symmetries (2.5) and non-Lie symmetries (2.6) can be united inframes of the 17-dimensional Lie algebra which includes (2.5) and (2.6) as subalgebras [9].
III. Algebras of discrete symmetries of the Dirac equation
It is well known that the Dirac equation is invariant w.r.t. speciﬁc discrete transformationslike parity or charge conjugation. Let us analyze algebraic structures generated by thesesymmetries.
438 J. NIEDERLE, A. NIKITINConsider reﬂections of independent variables
x
= (
x
0
,x
1
,x
2
,x
3
):
θ
0
x
= (
−
x
0
,x
1
,x
2
,x
3
)
, θ
1
x
= (
x
0
,
−
x
1
,x
2
,x
3
)
, θ
2
x
= (
x
0
,x
1
,
−
x
2
,x
3
)
,θ
3
x
= (
x
0
,x
1
,x
2
,
−
x
3
)
, θx
= (
−
x
0
,
−
x
1
,
−
x
2
,
−
x
3
)
.
(3.1)The corresponding symmetry operators for equation (2.1) have the formΓ
0
=
γ
4
γ
0
ˆ
θ
0
,
Γ
1
=
γ
4
γ
1
ˆ
θ
1
,
Γ
2
=
γ
4
γ
2
ˆ
θ
2
,
Γ
3
=
γ
4
γ
3
ˆ
θ
3
,
Γ
4
=
iγ
4
ˆ
θ
(3.2)where ˆ
θ
µ
and ˆ
θ
are operators deﬁned as follows:ˆ
θ
µ
ψ
(
x
) =
ψ
(
θ
µ
x
)
,
ˆ
θψ
(
x
) =
ψ
(
−
x
)
.
(3.3)Let us add the list of symmetries (3.2) by the following
antilinear
operatorΓ
5
=
C
=
iγ
2
c
(3.4)where
c
is the complex conjugation,
cψ
(
x
) =
ψ
∗
(
x
).Operators (3.2), (3.4) generate very interesting algebraic structures. First, they satisfythe Cliﬀord algebraΓ
k
Γ
l
+ Γ
l
Γ
k
= 2
g
kl
(3.5)where
g
00
=
−
g
11
=
−
g
22
=
−
g
33
=
g
44
=
g
55
= 1;
g
kl
= 0,
k
=
l
. Secondly, this Cliﬀordalgebra can be extended by adding the seventh basis elementΓ
6
= Γ
0
Γ
1
Γ
2
Γ
3
Γ
4
Γ
5
=
iC.
(3.6)Finally, the enveloping algebra of this seven-dimensional Cliﬀord algebra is isomorphicto the algebra
gl
(8
,R
). In other words, there are 64 linearly independent products of theoperators Γ
f
(
f
= 0
,
1
,...
6):
Γ
m
,
Γ
m
Γ
n
,
Γ
k
Γ
m
Γ
n
,
ˆ
I
, k,l,m,
= 0
,
1
,...
6 (3.7)(ˆ
I
is the unit operator) which form a basis of the Lie algebra isomorphic to
gl
(8
,R
). Thisisomorphism will be constructed explicitly in Section V.Thus the discrete symmetries of the Dirac equation generate a very extended Lie alge-bra. Restricting ourselves to linear symmetries we come to the 16-dimensional Lie algebraincluding the identity operator ˆ
I
and the following 15 operators
{
Γ
a
,
Γ
b
Γ
c
}
, a,b,c
= 0
,
1
,...
4 (3.8)with Γ
a
deﬁned in (3.2). Operators (3.8) form a basis of the algebra
so
(2
,
4).We notice that the Dirac equation for a charged particle interacting with an externalﬁeld(
γ
µ
π
µ
−
m
)
ψ
= 0
, π
µ
=
p
µ
−
eA
µ
(
x
) (3.9)still admits some of symmetries (3.7) provided functions
A
µ
(
x
) have deﬁnite parities w.r.t.the related reﬂections (3.1) or their combinations. Moreover, if the corresponding symme-try (3.7) is diagonalizable, then equation (3.9) can be reduced to two uncoupled subsystems[8]. We will demonstrate in Section VI that for some classes of vector-potentials
A
µ
theDirac equation can be reduced to
eight
uncoupled equations.
NON-LIE AND DISCRETE SYMMETRIES 439
IV. The maximal present symmetry algebra for the Diracequation
Thus there exist two symmetry algebras for the Dirac equation which are deﬁned byrelations (2.6), (3.7) and which are of diﬀerent srcin. Symmetries (2.6) are of the form of diﬀerential operators whereas (3.7) are functional operators of discrete transformations.Nevertheless, it is possible to ﬁnd an algebraic structure which unify both of them.First let us note that it is impossible to include all symmetries (2.6) and (3.8) into aﬁnite-dimensional Lie algebra. Indeed, commutators of operators (2.6) and (3.2) gener-ate second-order diﬀerential operators whose commutators give fourth-order diﬀerentialoperators, and so on. However the subset of symmetries (3.7) which commute with
γ
4
,i.e.,Γ
k
Γ
l
,
Γ
4
,
Γ
4
Γ
k
Γ
l
, k,l
= 4 (4.1)can be united with operators (2.6) in framework of a 120-dimensional (!) Lie algebra withthe following basis
Q
4
µ,
4
µ
=
C
Γ
µ
, Q
5
µ,
5
µ
=
C
Γ
4
Γ
µ
, Q
5
ν,
5
µ
=
g
µµ
C
Γ
4
Γ
µ
Σ
µν
,Q
4
ν,
4
µ
=
g
µµ
C
Γ
µ
Σ
µν
, Q
4
ν,
5
µ
=
−
γ
4
Q
5
µ,
5
ν
, Q
µν,µν
=
γ
µν
ˆ
θ
µ
ˆ
θ
ν
,Q
5
ν,
4
µ
=
−
γ
4
Q
4
µ,
4
ν
, Q
54
,
54
=
i
Γ
4
, Q
54
,µν
=
ε
µνλσ
γ
µ
γ
ν
ˆ
θ
µ
ˆ
θ
ν
Σ
λσ
,Q
µλ,µσ
=
g
µµ
Σ
µλ
γ
µ
γ
ν
ˆ
θ
µ
ˆ
θ
ν
, Q
σλ,µν
=
ε
µσνλ
g
µµ
g
νν
Σ
1
γ
µ
γ
ν
ˆ
θ
µ
ˆ
θ
ν
,Q
µν,mn
= Σ
µν
, Q
54
,mm
=
i
Γ
4
.
(4.2)Here
Q
kl,mn
are tensors which are antisymmetric w.r.t. permutations of the ﬁrst pairof indices and symmetric w.r.t. permutations of the second pair of indices and whosediagonal elements are equal, i.e.,
Q
kn,ll
=
Q
kn,mm
for any
l
and
m
. The Greek indicesruns over the values 0
,
1
,
2
,
3 and no summation over repeated indices is assumed.Let us notice that algebra (4.2) can be extended by the following 136 elements
Q
4
µ,λσ
=
C
Γ
µ
ˆ
θ
µλσ
, Q
4
µ,
45
=
C
Γ
µ
ˆ
θ
νλσ
, ν,λ,σ
=
µ,Q
5
µ,λσ
=
C
Γ
4
Γ
µ
ˆ
θ
µλσ
, Q
5
µ,
45
=
C
Γ
4
Γ
µ
ˆ
θ
νλσ
, ν,λ,σ
=
µ,Q
µν,
4
λ
=
γ
µ
γ
ν
ˆ
θ
µ
ˆ
θ
ν
ˆ
θ
µνλ
, Q
µν,
5
λ
=
γ
µ
γ
ν
ˆ
θ
µ
ˆ
θ
ν
ˆ
θ
µνλ
Γ
4
Σ
1
,Q
µν,
45
=
γ
µ
γ
ν
ˆ
θ
λσ
, Q
54
,µν
=
i
Γ
4
ˆ
θ
µν
(4.3)whereˆ
θ
µν
= 1 + 1
m
(1
−
iγ
4
)(
γ
µ
p
µ
+
γ
ν
p
ν
) ˆ
θ
µ
ˆ
θ
ν
, µ
=
ν,
ˆ
θ
µνλ
= 1 + 1
m
(1
−
iγ
4
)
γ
µ
p
µ
+
γ
ν
p
ν
+
γ
λ
p
λ
ˆ
θ
µ
ˆ
θ
ν
ˆ
θ
λ
, µ
=
ν,µ
=
λ,ν
=
λ
(no sum over repeated indices).These elements are new symmetries of equation (2.1), which cannot be expressed viacommutators of symmetries (2.6), (4.1).
440 J. NIEDERLE, A. NIKITINOperators (4.2), (4.3) form a basis of the 256-dimensional real invariance algebra of theDirac equation, deﬁned over the ﬁeld of real numbers. This algebra is characterized bythe following commutation relations
Q
kl,mn
,Q
k
′
l
′
,m
′
n
′
=
−
2[
δ
mm
′
g
kk
′
Q
ll
′
,nn
′
+
g
ll
′
Q
kk
′
,nn
′
−
g
kl
′
Q
lk
′
,nn
′
−
g
lk
′
Q
kl
′
,nn
′
+
δ
nn
′
g
kk
′
Q
ll
′
,mm
′
+
g
ll
′
Q
kk
′
,mm
′
−
g
kl
′
Q
lk
′
,mm
′
−
g
lk
′
Q
kl
′
,mm
′
+
δ
mn
′
g
kk
′
Q
ll
′
,nm
′
+
g
ll
′
Q
kk
′
,nm
′
−
g
kl
′
Q
lk
′
,nm
′
−
g
lk
′
Q
kl
′
,nm
′
+
δ
nm
′
g
kk
′
Q
ll
′
,mn
′
+
g
ll
′
Q
kk
′
,mn
′
−
g
kl
′
Q
lk
′
,mn
′
−
g
lk
′
Q
kl
′
,mn
′
]+
g
mnm
′
n
′
gf
g
kk
′
Q
ll
′
,gf
+
g
ll
′
Q
kk
′
,gf
−
g
kl
′
Q
lk
′
,gf
−
g
lk
′
Q
kl
′
,gf
−
12
(
δ
mm
′
δ
nn
′
+
δ
mn
′
δ
nm
′
)
g
kk
′
Q
ll
′
,ss
+
g
ll
′
Q
kk
′
,ss
−
g
kl
′
Q
lk
′
,ss
−
g
lk
′
Q
kl
′
,ss
(4.4)where
m
=
m
′
,
n
=
n
′
,
g
mnm
′
n
′
gf
is the totally symmetric unit tensor whose nonzerocomponents correspond to noncoinciding values of all indices, and no sum over
s,g,f
.For
m
=
n
ve have
Q
kl,nn
,Q
k
′
l
′
,m
′
n
′
=
−
2
g
kk
′
Q
ll
′
,m
′
n
′
+
g
ll
′
Q
kk
′
,m
′
n
′
−
g
kl
′
Q
lk
′
,m
′
n
′
−
g
lk
′
Q
kl
′
,m
′
n
′
.
(4.5)Taking into account both the invariance algebra of the two-component Klein-Gordonequation and equivalence of this equation to the Dirac one, it is possible to shaw thatrelations (4.2), (4.4) present the maximally extended ﬁnite symmetry algebra for equation(2.1).Thus we found the most extended symmetry algebra for the Dirac equation, which in-cludes non-Lie symmetries (2.6), discrete symmetries (4.1) and their combinations. Sym-metries (4.2) form its 120-dimensional subalgebra and also satisfy relations (4.4), (4.5) forthe restricted set of values of indices deﬁned in (4.3). Another its interesting subalgebrais 56-dimensional one formed by the linear (i.e., without complex conjugation) operatorswhich are presented as follows:
{
Q
µν,λσ
, Q
µν,
54
, Q
54
,µν
, Q
54
,
54
}
, µ,ν,λ,σ
= 0
,
1
,
2
,
3
.
(4.6)
V. Commutative discrete symmetries
In order to describe all possible reductions of the Dirac for a charged particle interactingwith an external ﬁeld we shall prove that operators (3.7). It is easy to see that thefour-component complex wave function of (3.9)Ψ =
column
(Ψ
1
,
Ψ
2
,
Ψ
3
,
Ψ
4
)
,
Ψ
k
= Ψ
(1)
k
+
i
Ψ
(2)
k
, k
= 1
,
2
,
3
,
4 (5.1)is equivalent to the eight–component real function˜Ψ =
column
Ψ
(1)1
,
Ψ
(2)1
,
Ψ
(1)2
,
Ψ
(2)2
,
Ψ
(1)3
,
Ψ
(2)3
,
Ψ
(1)4
,
Ψ
(2)4
.
(5.2)In representation (5.2) the Dirac matrices (2.2) are extended to the 8
×
8 real matrices
γ
0
→
˜
γ
0
=
0
I
4
I
4
0
, γ
a
→
˜
γ
a
=
0
−
˜
σ
a
˜
σ
a
0
, γ
4
→
˜
γ
4
=
I
4
00
I
4
,

Search

Similar documents

Tags

Related Search

Economic And Monetary Union Of The European Usocial and cultural history of the Second WorIntellctual and social history of the Mamluk Economic and Social History of the Ottoman EmCultural and Spatial Representations of the UTeachers’ and Students’ Perceptions of the IdFunctional and Evolutionary Morphology of theCultural and Political Activism of the 1960s Ethnohistory and Mesoamerican Cultures of theHealth and safety culture of the Nigerian co

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks

SAVE OUR EARTH

We need your sign to support Project to invent "SMART AND CONTROLLABLE REFLECTIVE BALLOONS" to cover the Sun and Save Our Earth.

More details...Sign Now!

We are very appreciated for your Prompt Action!

x