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’ (3x2 u2 ’ x1 v 2 ’ x3 v 1 + x4 v 3 ) 2
‚u

’ (3x2 u3 + x1 v 3 ’ x3 v 4 ’ x4 v 2 ) 3
‚u

’ (3x2 u4 ’ x1 v 4 ’ x3 v 3 + x4 v 1 ) 4
‚u

’ (3x2 v 1 ’ x1 u1 + x3 u2 + x4 u4 ) 1
‚v

’ (3x2 v 2 + x1 u2 + x3 u1 ’ x4 u3 ) 2
‚v

’ (3x2 v 3 ’ x1 u3 + x3 u4 + x4 u2 ) 3
‚v

’ (3x2 v 4 + x1 u4 + x3 u3 ’ x4 u1 ) 4 ,
‚v
‚ ‚ ‚ ‚
’ (x2 + x2 ’ x2 ’ x2 )
X14 = 2x1 x3 + 2x2 x3 + 2x3 x4
1 2 3 4
‚x1 ‚x2 ‚x3 ‚x4

’ (3x3 u1 + x2 v 2 + x1 u2 ’ x4 u3 ) 1
‚u

’ (3x3 u2 + x2 v 1 ’ x1 u1 + x4 u4 ) 2
‚u

’ (3x3 u3 + x2 v 4 + x1 u4 ’ x4 u1 ) 3
‚u

’ (3x3 u4 + x2 v 3 ’ x1 u3 + x4 u2 ) 4
‚u

’ (3x3 v 1 ’ x2 u2 + x1 v 2 ’ x4 v 3 ) 1
‚v

’ (3x3 v 2 ’ x2 u1 ’ x1 v 1 + x4 v 4 ) 2
‚v

’ (3x3 v 3 ’ x2 u4 + x1 v 4 ’ x4 v 1 ) 3
‚v

’ (3x3 v 4 ’ x2 u3 ’ x1 v 3 + x4 v 2 ) 4 ,
‚v
‚ ‚ ‚ ‚
+ (x2 + x2 + x2 + x2 )
X15 = 2x1 x4 + 2x2 x4 + 2x3 x4 1 2 3 4
‚x1 ‚x2 ‚x3 ‚x4

’ (3x4 u1 ’ x2 v 4 ’ x3 u3 ’ x1 u4 ) 1
‚u

’ (3x4 u2 + x2 v 3 + x3 u4 ’ x1 u3 ) 2
‚u

’ (3x4 u3 ’ x2 v 2 ’ x3 u1 ’ x1 u2 ) 3
‚u
42 1. CLASSICAL SYMMETRIES


’ (3x4 u4 + x2 v 1 + x3 u2 ’ x1 u1 )
‚u4

’ (3x4 v 1 + x2 u4 ’ x3 v 3 ’ x1 v 4 ) 1
‚v

’ (3x4 v 2 ’ x2 u3 + x3 v 4 ’ x1 v 3 ) 2
‚v

’ (3x4 v 3 + x2 u2 ’ x3 v 1 ’ x1 v 2 ) 3
‚v

’ (3x4 v 4 ’ x2 u1 + x3 v 2 ’ x1 v 1 ) 4 ,
‚v
‚ ‚ ‚ ‚ ‚
X16 = u1 1 + u2 2 + u3 3 + u4 4 + v 1 1
‚u ‚u ‚u ‚u ‚v
‚ ‚ ‚
+ v2 2 + v3 3 + v4 4 ,
‚v ‚v ‚v
‚ ‚ ‚ ‚ ‚
X17 = u2 1 ’ u1 2 ’ u4 3 + u3 4 ’ v 2 1
‚u ‚u ‚u ‚u ‚v
‚ ‚ ‚
+ v1 2 + v4 3 ’ v3 4 ,
‚v ‚v ‚v
‚ ‚ ‚ ‚ ‚
X18 = u3 1 + u4 2 + u1 3 + u2 4 + v 3 1
‚u ‚u ‚u ‚u ‚v
‚ ‚ ‚
+ v4 2 + v1 3 + v2 4 ,
‚v ‚v ‚v
‚ ‚ ‚ ‚ ‚
X19 = u4 1 ’ u3 2 ’ u2 3 + u1 4 ’ v 4 1
‚u ‚u ‚u ‚u ‚v
‚ ‚ ‚
+ v3 2 + v2 3 ’ v1 4 ,
‚v ‚v ‚v
‚ ‚ ‚ ‚ ‚
X20 = v 1 1 + v 2 2 + v 3 3 + v 4 4 ’ u1 1
‚u ‚u ‚u ‚u ‚v
‚ ‚ ‚
’ u2 2 ’ u3 3 ’ u4 4 ,
‚v ‚v ‚v
‚ ‚ ‚ ‚ ‚
X21 = v 2 1 ’ v 1 2 ’ v 4 3 + v 3 4 + u2 1
‚u ‚u ‚u ‚u ‚v
‚ ‚ ‚
’ u1 2 ’ u4 3 + u3 4 ,
‚v ‚v ‚v
‚ ‚ ‚ ‚ ‚
X22 = v 3 1 + v 4 2 + v 1 3 + v 2 4 ’ u3 1
‚u ‚u ‚u ‚u ‚v
‚ ‚ ‚
’ u4 2 ’ u1 3 ’ u2 4 ,
‚v ‚v ‚v
‚ ‚ ‚ ‚ ‚
X23 = v 4 1 ’ v 3 2 ’ v 2 3 + v 1 4 + u4 1
‚u ‚u ‚u ‚u ‚v
‚ ‚ ‚
’ u3 2 ’ u2 3 + u1 4 . (1.92)
‚v ‚v ‚v
The result is in full agreement with that of Ibragimov [5].
6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 43

5.2. Case 2: = 0, »’1 = 0. The complete Lie algebra of symmetries
for the Dirac equations with nonvanishing rest mass is spanned by four-
teen generators, including ten in¬nitesimal generators of the Poincar´ group
e
X1 , . . . , X10 and the generators X19 , X20 , X23 , X16 . There is also a contin-
1 4
uous part generated by the functions F u , . . . , F v dependent on x1 , . . . , x4 ,
which satisfy Dirac equations (1.83) with nonvanishing rest mass.

5.3. Case 3: = 0, »’1 = 0. The complete Lie algebra in this situation
is spanned by fourteen generators. These generators are X1 , . . . , X10 , X19 ,
X20 , X23 , and X11 ’ X16 /2.

5.4. Case 4: = 0, »’1 = 0. The complete Lie algebra of symmetries
for the nonlinear Dirac equations with nonvanishing rest mass is spanned
by thirteen generators. The generators in this case are the ten in¬nitesimal
generators of the Poincar´ group, X1 , . . . , X10 , and X19 , X20 , X23 . This
e
result generalizes the result by Steeb [94] where X20 was found as additional
symmetry to the generators of the Poincar´ group.
e

6. Symmetries of the self-dual SU (2) Yang“Mills equations
We study here classical symmetries of the self-dual SU (2) Yang“Mills
equations. Two cases are considered: the general one and of the so-called
static gauge ¬elds. In the ¬rst case we obtain two instanton solutions (the
Belavin“Polyakov“Schwartz“Tyupkin [6] and ™t Hooft instantons [84]) as
invariant solutions for a special choice of symmetry subalgebras. In a similar
way, for the second case we derive a monopole solution [83].
We start with a concise description of the SU (2)-gauge theory referring
the reader to the survey paper by M. K. Prasad [83] for a more extensive
exposition.

6.1. Self-dual SU (2) Yang“Mills equations. Let M be a 4-dimen-
sional Euclidean space with the coordinates x1 , . . . , x4 . Due to nondegen-
erate metric in M , we make no distinction between contravariant and co-
variant indices, xµ = xµ . The basic object in the gauge theory is the Yang“
Mills gauge potential. The gauge potential is a set of ¬elds Aa ∈ C ∞ (M ),
µ
a = 1, . . . , 3, µ = 1, . . . , 4. It is convenient to introduce a matrix-valued
vector ¬eld Aµ (x), by setting
σa
a
Aa , a
Aµ = gT T= , a = 1, . . . , 3, µ = 1, . . . , 4, (1.93)
µ
2i
where σ a are the Pauli matrices
0 ’i
01 10
σ1 = σ2 = σ3 =
, , , (1.94)
0 ’1
10 i0
g being a constant, called the gauge coupling constant. Throughout this
section we shall use the Einstein summation convention when an index oc-
curs twice. From the matrix gauge potential Aµ dxµ one constructs the
44 1. CLASSICAL SYMMETRIES

matrix-valued ¬eld strength Fµν (x) by
‚ ‚
Aν ’
Fµν = Aµ + [Aµ , Aν ], µ, ν = 1, . . . , 4, (1.95)
‚xµ ‚xν
where [Aµ , Aν ] = Aµ Aν ’ Aν Aµ . If one de¬nes the covariant derivative

Dµ = + Aµ , (1.96)
‚xµ
then (1.95) is rewritten as
Fµν = [Dµ , Dν ]. (1.97)
In explicit component form, one has
Fµν = gT a Fµν ,
a
(1.98)
where
‚a ‚a
a bc
Aν ’
Fµν = A +g abc Aµ Aν (1.99)
‚xν µ
‚xµ
and
±
+1 if abc is an even permutation of (1,2,3),

= ’1 if abc is an odd permutation of (1,2,3), (1.100)
abc


0 otherwise.
We shall use the expression static gauge ¬eld to refer to gauge potentials
that are independent of x4 (x4 to be considered as time), i.e.,

Aµ (x) = 0, µ = 1, . . . , 4. (1.101)
‚x4
For gauge potentials that depend on all four coordinates x1 , . . . , x4 , the
action functional is de¬ned by
1
Fµν Fµν d4 x,
a a
S= (1.102)
4
the integral taken over R4 , while for static gauge ¬elds we de¬ne the energy
functional by
1
Fµν Fµν d3 x,
a a
E= (1.103)
4
whereas in (1.103) the integral is taken over R3 .
The extremals of the action S (or of the energy E for static gauge ¬elds)
are found by standard calculus of variations techniques leading to the Euler“
Lagrange equations

Fµν + [Aµ , Fµν ] ≡ [Dµ , Fµν ] = 0, (1.104)
‚xµ
or in components
‚a
Fµν + g abc Ab Fµν = 0.
c
(1.105)
µ
‚xµ
6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 45

Equations (1.105) is a system of second order nonlinear partial dif-
ferential equations for the twelve unknown functions Aa , a = 1, . . . , 3,
µ
µ = 1, . . . , 4, that seems hard to solve.
Then one introduces the dual gauge ¬eld strength — Fµν as
1

Fµν = µν»ρ F»ρ , (1.106)
2
where µν»ρ is the completely antisymmetric tensor on M de¬ned by
±
+1 if µν»ρ is an even permutation of (1,2,3,4),

µν»ρ = (1.107)
’1 if µν»ρ is an odd permutation of (1,2,3,4),


0 otherwise.
Since the ¬elds Dµ (1.96) satisfy the Jacobi identity
[D» , [Dµ , Dν ]] + [Dµ , [Dν , D» ]] + [Dν , [D» , Dµ ]] = 0, (1.108)
multiplication of (1.108) by and summation result in
µν»ρ
[Dµ , — Fµν ] = 0. (1.109)
If we compare (1.104) with (1.109), we see that any gauge ¬eld which is
self-dual , i.e., for which

Fµν = Fµν , (1.110)
automatically satis¬es (1.101). Consequently, the only equations to solve are
(1.110) with — Fµν given by (1.106). This is a system of ¬rst order nonlinear
partial di¬erential equations.
Instanton solutions for general Yang“Mills equations and monopole so-
lutions for static gauge ¬elds satisfy (1.110) under the condition that S
(1.102) or E (1.103) are ¬nite.
Written in components, (1.110) takes the form
F13 = ’F24 ,
F12 = F34 , F14 = F23 . (1.111)
So in components, the self-dual Yang“Mills equations are described as a
system of nine nonlinear partial di¬erential equations,
’A1 + A1 ’ A1 + A1 ’ g(A2 A3 ’ A2 A3 + A2 A3 ’ A2 A3 ) = 0,
4,1 3,2 2,3 1,4 14 23 32 41
’A2 + A2 ’ A2 + A2 + g(A1 A3 ’ A1 A3 + A1 A3 ’ A1 A3 ) = 0,
4,1 3,2 2,3 1,4 14 23 32 41
’A3 + A3 ’ A3 + A3 ’ g(A1 A2 ’ A1 A2 + A1 A2 ’ A1 A2 ) = 0,
4,1 3,2 2,3 1,4 14 23 33 41
A1 + A1 ’ A1 ’ A1 + g(A2 A3 + A2 A3 ’ A2 A3 ’ A2 A3 ) = 0,
3,1 4,2 1,3 2,4 13 24 31 42
A2 + A2 ’ A2 ’ A2 ’ g(A1 A3 ’ A1 A3 ’ A1 A3 ’ A1 A3 ) = 0,
3,1 4,2 1,3 2,4 13 24 31 42
A3 + A3 ’ A3 ’ A3 + g(A1 A2 + A1 A2 ’ A1 A2 ’ A1 A2 ) = 0,
3,1 4,2 1,3 2,4 13 24 31 42
A1 ’ A1 ’ A1 + A1 + g(A2 A3 ’ A2 A3 ’ A2 A3 + A2 A3 ) = 0,
2,1 1,2 4,3 3,4 12 21 34 43
A2 ’ A2 ’ A2 + A2 ’ g(A1 A3 ’ A1 A3 ’ A1 A3 + A1 A3 ) = 0,
2,1 1,2 4,3 3,4 12 21 34 43
A3 ’ A3 ’ A3 + A3 + g(A1 A2 ’ A1 A2 ’ A1 A2 + A1 A2 ) = 0,
2,1 1,2 4,3 3,4 12 21 34 43
(1.112)
46 1. CLASSICAL SYMMETRIES

whereas in (1.112)
‚a
Aa = A, a = 1, . . . , 3, µ, ν = 1, . . . , 4. (1.113)
µ,ν
‚xν µ
Thus, we obtain a system E ‚ J 1 (π) for π : R12 — R4 ’ R4 .
6.2. Classical symmetries of self-dual Yang“Mills equations. In
order to construct the Lie algebra of classical symmetries of (1.112), we start
at a vector ¬eld V given by
‚ ‚ 1‚ 3‚
V = V x1 + · · · + V x4 + V A1 + · · · + V A4 . (1.114)
‚A1 ‚A3
‚x1 ‚x4 1 4
The condition for V to be a symmetry of equations (1.112) now leads to an
overdetermined system of partial di¬erential equations for the components
1 3
V x1 , . . . , V x4 , V A1 , . . . , V A4 , which are functions dependent of the variables
x1 , . . . , x4 , A 1 , . . . , A 3 .
1 4
The general solution of this overdetermined system of partial di¬erential
equations constitutes a Lie algebra of symmetries, generated by the vector
¬elds
1‚ 1‚ 1‚ 1‚
1
V1f = fx1 + f x2 + f x3 + f x4
‚A1 ‚A1 ‚A1 ‚A1
1 2 3 4
‚ ‚ ‚ ‚
+ f 1 gA3 2 + f 1 gA3 2 + f 1 gA3 2 + f 1 gA3 2
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
’ f 1 gA2 3 ’ f 1 gA2 3 ’ f 1 gA2 3 ’ f 1 gA2 3 ,
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
2
V2f = ’f 2 gA2 1 ’ f 2 gA2 1 ’ f 2 gA2 1 ’ f 2 gA2 1
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
+ f 2 gA1 2 + f 2 gA1 2 + f 2 gA1 2 + f 2 gA1 2
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
2‚ 2‚ 2‚ 2‚
’ f x1 ’ f x2 ’ f x3 ’ f x4 ,
‚A3 ‚A3 ‚A3 ‚A3
1 2 3 4
‚ ‚ ‚ ‚
3
V3f = f 3 gA3 1 + f 3 gA3 1 + f 3 gA3 1 + f 3 gA3 1
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
3‚ 3‚ 3‚ 3‚
’ f x1 ’ f x2 ’ f x3 ’ f x4
‚A2 ‚A2 ‚A2 ‚A2
1 2 3 4
‚ ‚ ‚ ‚
’ f 3 gA1 3 ’ f 3 gA1 3 ’ f 3 gA1 3 ’ f 3 gA1 3 ,
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
V4 = , V5 = , V6 = , V7 = ,
‚x1 ‚x2 ‚x3 ‚x4
‚ ‚ ‚ ‚
+ A1 1 ’ A1 1
’ x1
V8 = x 2 2 1
‚x1 ‚x2 ‚A1 ‚A2
‚ ‚ ‚ ‚
+ A2 2 ’ A2 2 + A3 3 ’ A3 3 ,
2 1 2 1
‚A1 ‚A2 ‚A1 ‚A2
6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 47

‚ ‚ ‚ ‚
’ A1 1 + A1 1
V9 = ’x3 + x1 3 1
‚x1 ‚x3 ‚A1 ‚A3
‚ ‚ ‚ ‚
’ A2 2 + A2 2 ’ A3 3 + A3 3 ,
3 1 3 1
‚A1 ‚A3 ‚A1 ‚A3
‚ ‚ ‚ ‚
’ A1 1 + A1 1
V10 = ’x4 + x1 4 1
‚x1 ‚x4 ‚A1 ‚A4
‚ ‚ ‚ ‚
’ A2 2 + A2 2 ’ A3 3 + A3 3 ,
4 1 4 1
‚A1 ‚A4 ‚A1 ‚A4
‚ ‚ ‚ ‚
’ A1 1 + A1 1
V11 = ’x3 + x2 3 2
‚x2 ‚x3 ‚A2 ‚A3

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