<< . .

. 22
( : 58)



. . >>

6.2, and the results of Subsection 7.2, leading to the direct sum of two Lie
algebras, each of which having a similar structure to the Lie algebra for
the massive Thirring model. So we are forced to search for nonlocal higher
symmetries, including the nonlocal variables (3.87) associated to the vector
¬elds V1 , V2 in (3.85).
Surprisingly, carrying through the huge computations, the nonlocal vari-
ables dropped out automatically from intermediate results, ¬nally leading
to local (x, t)-dependent higher symmetries. So, for simplicity we shall dis-
cuss the search for creating and annihilating symmetries, assuming from the
beginning that they are local.
The formulation of creating and annihilating symmetries will follow from
the Lie brackets of these symmetries with Yi± , meaning going up or down in
the hierarchy. The symmetries Y0+ , Y0’ are of degree 0, Y1+ , Y’1 , Y1’ , Y’1 ’
+

are of degree 2, while the symmetries Y2+ , Y’2 , Y2’ , Y’2 are of degree 4, see

+

(3.105).
We now search for an (x, t)-dependent higher symmetry of second order,
linear with respect to x, t, and of degree 2, i.e., for a vector ¬eld V of the
form
V = xH1 + tH2 + C — , (3.107)
where H1 , H2 are higher symmetries of degree four and, due to the fact that
m1 , m2 are of degree two, H1 , H2 are assumed to be linear with respect
to Y0+ , Y0’ , . . . , Y2+ , Y’2 , Y2’ , Y’2 , while V in (3.107) has to satisfy the

+

symmetry condition. From these conditions we obtained the following result.
The symmetry condition is satis¬ed under the special assumption for V ,
(3.107), leading to the following four higher symmetries:
1 1
X11 = x ’Y’2 + m2 Y0+ + t Y’2 + m2 Y0+ + C11 ,
+ +
41 41
1 1
X12 = x Y2+ ’ m2 Y0+ + t Y2+ + m2 Y0+ + C12 ,
1
41
4
1 1
X13 = x ’Y’2 + m2 Y0’ + t Y’2 + m2 Y0’ + C13 ,
’ ’
42 42
1 1
X14 = x Y’2 ’ m2 Y0’ + t Y’2 + m2 Y0’ + C14 .
’ ’
(3.108)
2
42
4
where in (3.108) the functions C11 , . . . , C14 are given by the following ex-
pressions
1 ‚
C11 = 2v1x + m1 u2 + »u1 (R3 + R4 )
2 ‚u1
136 3. NONLOCAL THEORY

1 ‚
’ 2u1x + m1 v2 + »v1 (R3 + R4 )
+ ,
2 ‚v1
1 ‚
’ 2v2x + m1 u1 ’ »u2 (R3 + R4 )
C12 =
2 ‚u2
1 ‚
2u2x + m1 v1 ’ »v2 (R3 + R4 )
+ ,
2 ‚v2
1 ‚
2v3x + m2 u4 ’ »u3 (R1 + R2 )
C13 =
2 ‚u3
1 ‚
’ 2u3x + m2 v4 ’ »v3 (R1 + R2 )
+ ,
2 ‚v3
1 ‚
’ 2v4x + m2 u3 + »u4 (R1 + R2 )
C14 =
2 ‚u4
1 ‚
+ 2u4x + m2 v3 + »v4 (R1 + R2 ) . (3.109)
2 ‚v4
From (3.108) and (3.109) we de¬ne
’ ’
+ +
Z’1 = X11 , Z1 = X12 , Z’1 = X13 , Z1 = X14 . (3.110)

Computation of the commutators of Z’1 , Z1 , Z’1 , Z1 and Yi± , where
’ ’
+ +

i = ’2, . . . , 2, leads to the following result:
1
[Z’1 , Y2+ ] = ’ m2 Y1+ ,
+
[Z1 , Y2+ ] = Y3+ ,
+
21
1
[Z’1 , Y1+ ] = m2 Y0+ ,
+
[Z1 , Y1+ ] = Y2+ ,
+
41
[Z’1 , Y0+ ] = 0,
+
[Z1 , Y0+ ] = 0,
+

1
+ + +
[Z1 , Y’1 ] = ’ m2 Y0+ , nonumber
+ +
[Z’1 , Y’1 ] = ’Y’2 , (3.111)
41
1
+ + + + + +
[Z1 , Y’2 ] = m2 Y’1 , nonumber
[Z’1 , Y’2 ] = Y’3 , (3.112)
1
2
1
[Z’1 , Y2’ ] = ’ m2 Y1’ ,

[Z1 , Y2’ ] = Y3’ ,

22
1
[Z’1 , Y1’ ] = m2 Y0’ ,

[Z1 , Y1’ ] = Y2’ ,

42
[Z’1 , Y0’ ] = 0,

[Z1 , Y0’ ] = 0,


1
’ ’ ’
[Z1 , Y’1 ] = ’ m2 Y0’ , nonumber
’ ’
[Z’1 , Y’1 ] = ’Y’2 , (3.113)
42
1
’ ’ ’ ’ ’ ’
[Z1 , Y’2 ] = m2 Y’1 ,
[Z’1 , Y’2 ] = Y’3 , (3.114)
2
2
while
1 1
’ ’ ’
+ + +
[Z’1 , Z1 ] = ’ m2 Z0 , [Z’1 , Z1 ] = ’ m2 Z0 . (3.115)
1 2
2 2
7. SYMMETRIES OF THE FEDERBUSH MODEL 137

All other commutators are zero. The vector ¬eld Y3+ is given by

m1
Y3+,u1 = ’ 4v2xx ’ 4»R34 u2x + 2m1 u1x ’ 4»u2 (R34 )(1)
4
+ m2 v2 ’ »m1 R34 v1 + »2 R34 v2 ,
2
1
m1
Y3+,v1 = + 4u2xx ’ 4»R34 v2x + 2m1 v1x ’ 4»v2 (R34 )(1)
4
’ m2 u2 + »m1 R34 u1 ’ »2 R34 u2 ,
2
1
1
Y3+,u2 = ’ 8u2xxx ’ 4m1 v1xx + 12»R34 v2xx + 8»v2 (R34 )(2)
4
+ 24»v2x (R34 )(1) + 8»v2 (R34 )(1,1) + u2x (4m2 + 6»2 R34 )
2
1
’ 4»m1 R34 u1x + 12»2 u2 R34 (R34 )(1) ’ 4»m1 u1 (R34 )(1)
+ m3 v1 ’ 2»m2 R34 v2 + »2 m1 R34 v1 ’ »3 R34 v2 ,
2 3
1 1
1
Y3+,v2 = ’ 8v2xxx + 4m1 u1xx ’ 12»R34 u2xx + 8»u2 (R34 )(2)
4
’ 24»u2x (R34 )(1) ’ 8»u2 (R34 )(1,1) + v2x (4m2 + 6»2 R34 )
2
1
’ 4»m1 R34 v1x + 12»2 v2 R34 (R34 )(1) ’ 4»m1 v1 (R34 )(1)
’ m3 u1 + 2»m2 R34 u2 ’ »2 m1 R34 u1 + »3 R34 u2 ,
2 3
1 1

» »
Y3+,u3 = Y3+,v3 = ’ u3 L,
v3 L,
4 4
» »
Y3+,u4 Y3+,v4 = ’ u4 L,
= v4 L, (3.116)
4 4

where in (3.116)

R34 = R3 + R4 ,
(R34 )(1) = u3 u3x + v3 v3x + u4 u4x + v4 v4x ,
(R34 )(2) = u3 u3xx + v3 v3xx + u4 u4xx + v4 v4xx ,
(R34 )(1,1) = u2 + v3x + u2 + v4x ,
2 2
3x 4x
L = 8(u2 u2xx + v2 v2xx ) ’ 4(u2 + v2x ) + 12»R34 (u2x v2 ’ v2x u2 )
2
2x
+ 4m1 (u1 v2x ’ v1 u2x + u2 v1x ’ v2 u1x ) ’ m2 (2R2 + R1 )
1
+ 4m1 »R34 (u1 u2 + v1 v2 ) ’ 3»2 R2 R34 .
2



The results for the vector ¬elds Y’3 , Y3’ , Y’3 are similar to (3.116) and

+

are not given here, but are obtained from discrete symmetries σ and „ , to
be described in the next section.
’ ’
+ +
From the above it is clear now, why the vector ¬elds Z’1 , Z1 , Z’1 , Z1
are called creating and annihilating operators.
138 3. NONLOCAL THEORY

We thus have four in¬nite hierarchies of symmetries of the Federbush

+ ’
+
model, i.e., Y’n , Yn , Y’n , Yn , n ∈ N. A formal proof of the in¬niteness of
the hierarchies is given in Subsection 7.5.3.
7.4. Discrete symmetries. In deriving the speci¬c results for the
symmetry structure of the Federbush model, we realised that there are dis-
crete transformations which transform the Federbush model into itself and
by consequence transform symmetries into symmetries. Existence of these
disrete symmetries allow us to restrict to just one part of the Lie algebra of
symmetries, the discrete symmetries generating the remaining parts. These
discrete symmetries σ, „ are given by
σ : u1 ” u3 , v1 ” v3 , u2 ” u4 , v2 ” v4 , m1 ” m2 , » ” ’», t ” t;
„ : u1 ” u2 , v1 ” v2 , u3 ” u4 , v3 ” v4 , » ” ’», x ” ’x, t ” t. (3.117)
The transformations satisfy the following rules:
σ 2 = id,
„ 2 = id,
σ —¦ „ = „ —¦ σ.
Physically, the transformation σ denotes the exchange of two particles.
The action of the discrete smmetries on the Lie algebra of symmetries
is as follows:
σ(Yi+ ) = Yi’ ,
„ (Yi+ ) = Y’i ,
+

„ (Yi’ ) = Y’i ,



where i = 0, 1, 2,

+
σ(Z1 ) = Z1 ,
+ +
„ (Z1 ) = Z’1 ,
’ ’
„ (Z1 ) = Z’1 , (3.118)
while Y’3 , Y3’ , Y’3 , arising in the previous section, are de¬ned by

+


Y3’ = σ(Y3+ ), ’
Y’3 = „ (Y3+ ),
+
Y’3 = „ σ(Y3+ ). (3.119)
7.5. Towards in¬nite number of hierarchies of symmetries. In
this subsection, we demonstrate the existence of an in¬nite number of
hiearchies of higher symmetries of the Federbush model. We shall do this
by the construction of two (x, t)-dependent symmetries of degree 0 which
are polynomial with respect to x, t and of degree 2. This will be done in
Subsection 7.5.1.
Then, after writing the Federbush model as a Hamiltonian system, we
show that all higher symmetries obtained thusfar are Hamitonian vector
¬elds; this will be done in Subsection 7.5.2. Finally in Subsection 7.5.3
7. SYMMETRIES OF THE FEDERBUSH MODEL 139

we give a proof of a lemma from which the existence of in¬nite number of
hierarchies of Hamiltonians becomes evident, and from this we then obtain
the obvious result for the symmetry structure of the Federbush model.
7.5.1. Construction of Y + (2, 0) and Y + (2, 0). First, we start from the
presentation of these vector ¬elds, which is assumed to be of the following
structure
Y + (2, 0) = x2 (±1 Y2+ + ±2 m1 Y1+ + ±3 m2 Y0+ + ±4 m1 Y’1 + ±5 Y’2 )
+ +
1
+ 2xt(β1 Y2+ + β2 m1 Y1+ + β3 m2 Y0+ + β4 m1 Y’1 + β5 Y’2 )
+ +
1
+ t2 (γ1 Y2+ + γ2 m1 Y1+ + γ3 m2 Y0+ + γ4 m1 Y’1 + γ5 Y’2 )
+ +
1
+ + +
+ xC1 + tC2 + C0 , (3.120)

In (3.120), the ¬elds Yi+ , i = ’2, . . . , 2, are given in previous sections, ±1 ,
+ + +
βi , γi , i = 1, . . . , 5, are constant, while C1 , C2 , C0 , which are of degree 2,
2 and 1 respectively, have to be determined.
From the symmetry condition (2.29) on p. 72 we obtained the following
result: There does exist a symmetry of presentation (3.120), which is given
by
1
Y + (2, 0) = x2 (Y2+ ’ m2 Y0+ + Y’2 ) + 2xt(Y2+ ’ Y’2 )
+ +
1
2
1
+ t2 (Y2+ + m2 Y0+ + Y’2 ) + xC1 + tC2 ,
+ + +
(3.121)
1
2
whereas in (3.120) and (3.121),
‚ ‚
+
C1 = (’2v1x ’ m1 u2 ’ »R34 u1 ) + (2u1x ’ m1 v2 ’ »R34 v1 )
‚u1 ‚v1
‚ ‚
+ (’2v2x + m1 u1 ’ »R34 u2 ) + (2u2x + m1 v1 ’ »R34 v2 ) ,
‚u2 ‚v2
‚ ‚
+
C2 = (2v1x + m1 u2 + »R34 u1 ) + (’2u1x + m1 v2 + »R34 v1 )
‚u1 ‚v1
‚ ‚
+ (’2v2x + m1 u1 ’ »R34 u2 ) + (2u2x + m1 v1 ’ »R34 v2 ) ,
‚u2 ‚v2
+
C0 = 0. (3.122)
In a similar way, motivated by the structure of the Lie algebra obtained
thusfar, we get another higher symmetry of a similar structure, i.e.,
1
Y ’ (2, 0) = x2 (Y2’ ’ m2 Y0’ + Y’2 ) + 2xt(Y2’ ’ Y’2 )
’ ’
2
2
1
+ t2 (Y2’ ’ m2 Y0’ + Y’2 ) + xC1 + tC2 ,
’ ’ ’
(3.123)
2
2
whereas in (3.123),
‚ ‚

C1 = (’2v3x ’ m2 u4 + »R12 u3 ) + (2u3x ’ m2 v4 + »R12 v3 )
‚u3 ‚v3
140 3. NONLOCAL THEORY

‚ ‚
+ (’2v4x + m2 u3 + »R12 u4 ) + (2u4x + m2 v3 + »R12 v4 ) ,
‚u4 ‚v4
‚ ‚

C2 = (2v3x + m2 u4 ’ »R12 u3 ) + (’2u3x + m2 v4 ’ »R12 v3 )
‚u3 ‚v3
‚ ‚
+ (’2v4x + m2 u3 + »R12 u4 ) + (2u4x + m2 v3 + »R12 v4 ) ,
‚u4 ‚v4

C0 = 0. (3.124)

To give an idea of the action of the vector ¬elds Y + (2, 0), Y ’ (2, 0), we
compute their Lie brackets with the vector ¬elds Y1+ , Y0+ , Y’1 , Y1’ , Y0’ ,
+

Y’1 , yielding the following results

[Y ’ (2, 0), Y1’ ] = 2Z1 ,

[Y + (2, 0), Y1+ ] = 2Z1 ,
+

[Y ’ (2, 0), Y0’ ] = 0,
[Y + (2, 0), Y0+ ] = 0,
’ ’
[Y ’ (2, 0), Y’1 ] = 2Z’1 ,
+ +
[Y + (2, 0), Y’1 ] = 2Z’1 ,
[Y + (2, 0), Yi’ ] = 0, [Y ’ (2, 0), Yi+ ] = 0, (3.125)

where i = ’1, 0, 1. These results suggest to set

Y ± (1, i) = Zi± , Y ± (0, i) = Yi± , i ∈ Z. (3.126)

The complete Lie algebra structure is obtained in Subection 7.5.3.
7.5.2. Hamiltonian structures. We shall now discuss Hamiltonians (or
conserved functionals) for the Federbush model described by (3.84),

u1,t + u1,x ’ m1 v2 = »(u2 + v4 )v1 ,

<< . .

. 22
( : 58)



. . >>