ñòð. 17 |

Theorem 2, and we ?nd:

Theorem 3. (i) System (1), (2) is invariant under the algebra Pµ , Jµ? .

(ii) System (1), (2) is invariant under Pµ , Jµ? , D with D = x? ?? ? k R?R , k = 0

2

if and only if

Rµ Rµ ?µ ?µ ?µ Rµ

F = Rk G , , ,

R2+k Rk R1+k

where G is an arbitrary continuously di?erentiable function.

(iii) System (1), (2) is invariant under Pµ , Jµ? , D0 with D0 = x? ?? if and only

if

?µ ?µ ?µ Rµ

F = Rµ Rµ G R, , ,

Rµ Rµ Rµ Rµ

where G is an arbitrary continuously di?erentiable function.

(iv) System (1), (2) is invariant under Pµ , Jµ? , D, Kµ with D = x? ?? ? R?R

and Kµ = 2xµ D ? x2 ?µ if and only

?µ ?µ

F = R2 G ,

R2

where G is an arbitrary continuously di?erentiable function of one variable.

Implicit and parabolic ansatzes: some new ansatzes for old equations 75

The last case contains, as expected, case (i) of Theorem 2 when we choose G(?) =

? ? ?R2 . Each of the resulting equations in the above result is invariant under the

operator M and so one can use the ansatz de?ned by M to reduce the equation but

we do not always obtain a nice Schr?dinger equation. If we ask now for invariance

o

under the operator L = R?R (it is the operator L of case (v), Theorem 2, expressed

in the amplitude-phase form), then we obtain some other types of restrictions:

Theorem 4. (i) System (1), (2) is invariant under Pµ , Jµ? , L if and only if

Rµ Rµ Rµ ?µ

F =G , ?µ ? µ , .

R2 R

(ii) System (1), (2) is invariant under Pµ , Jµ? , D0 , L with D0 = x? ?? if and only

if

R2 ?µ ?µ R?µ Rµ

Rµ Rµ

F= G , .

R2 Rµ Rµ Rµ Rµ

(iii) System (1), (2) is invariant under Pµ , Jµ? , Kµ , L , where Kµ = 2xµ x? ?? ?

x2 ?µ ? 2xµ R?R , if and only if

F = ??µ ?µ ,

where ? is a constant.

The last case (iii) gives us the wave equation

jµ jµ

2? = (? ? 1) ?,

|?|4

1? ?

where jµ = 2i ??µ ? ??µ , which is the current of the wave-function ?. For ? = 1,

we recover the free complex wave equation. This equation, being invariant under

both M and N , can be reduced by the ansatzes they give rise to. In fact, with the

ansatz (obtained with L)

? = e( x)/2

?(?x, ?x, ?x)

with , ? isotropic 4-vectors with ? = 1, and ?, ? two space-like orthogonal 4-vectors,

the above equation reduces to the equation

j·j

? ? (? ? 1)

?? = ?,

|?|4

= ? 2 /?y1 + ? 2 /?y2 with y1 = ?x, y2 = ?x, and we have

2 2

where ? = ?x and

1? ?

[??? ? ???].

j=

2i

These results show what nonlinearities are possible when we require the invariance

under subalgebras of the conformal algebra in the given representation. The above

equations are all related to the Schr?dinger or heat equation. There are good reasons

o

for looking at conformally invariant equations, not least physically. As mathematical

reasons, we would like to give the following examples. First, note that the equation

2p,q ? = 0, (3)

76 P. Basarab-Horwath, W.I. Fushchych

where

2p,q = g AB ?A ?B , A, B = 1, . . . , p, p + 1, . . . , p + q

with g AB = diag(1, . . . , 1, ?1, . . . , ?1), is invariant under the algebra generated by

p q

the operators

JAB = xA ?B ? xB ?A , KA = 2xA xB ?B ? x2 ?A ? 2xA ??? + ??? ,

?A ,

M = i ??? ? ??? ,

D = xB ?B , L = ??? + ??? ,

L1 = i ??? ? ??? , L2 = ??? + ??? ,

namely the generalized conformal algebra AC(p, q) ? M, L, L1 , L2 which contains

the algebra ASO(p, q). Here, ? denotes the direct sum. Using the ansatz which the

operator M gives us, we can reduce equation (3) to the equation

i?? ? = 2p?1,q?1 ?. (4)

This equation (4) is known in the literature: it was proposed by Feynman [7] in

Minkowski space in the form

i?? ? = (?µ ? Aµ )(? µ ? Aµ )?.

It was also proposed by Aghassi, Roman and Santilli [8] who studied the representation

theory behind the equation. Fushchych and Seheda [9] studied its symmetry properties

in the Minkowski space. The solutions of equation (4) give solutions of (3) [14]. We

have that equation (4) has a symmetry algebra generated by the following operators

JAB , GA = ? ?A ? xA M,

T = ?? , PA = ? A ,

p+q?2 i ? ?

D = 2? ?? + xA ?A ? L, M = (??? ? ??? ), L = (??? + ??? ),

2 2

? (p + q ? 2)

x2

S = ? 2 ? ? + ? xA ? A ? M ? L

2 2

and this algebra has the structure [ASL(2, R) ? AO(p ? 1, q ? 1)] L, M, PA , GA ,

where denotes the semidirect sum of algebras. This algebra contains the subalgebra

AO(p ? 1, q ? 1) T, M, PA , GA with

[JAB , JCD ] = gBC JAD ? gAC JBD + gAD JBC ? gBD JAC ,

[PA , PB ] = 0, [GA , GB ] = 0, [PA , GB ] = ?gAB M,

[PA , JBC ] = gAB PC ? gAC PB , [GA , JBC ] = gAB GC ? gAC GB ,

[PA , D] = PA , [GA ; D] = GA , [JAB , D] = 0, [PA , T ] = 0, [GA , T ] = 0,

[JAB , T ] = 0, [M, T ] = [M, PA ] = [M, GA ] = [M, JAB ] = 0,

It is possible to show that the algebra with these commutation relations is contained

in AO(p, q): de?ne the basis by

1

(P1 ? Pq ), M = P1 + Pq ,

T= GA = J1A + JqA ,

2

(A, B = 2, . . . , q ? 1),

JAB

Implicit and parabolic ansatzes: some new ansatzes for old equations 77

and one obtains the above commutation relations. We see now that the algebra

AO(2, 4) (the conformal algebra AC(1, 3)) contains the algebra AO(1, 3) M, PA , GA

which contains the Poincar? algebra AP (1, 3) = AO(1, 3) Pµ as well as the Galilei

e

algebra AG(1, 3) = AO(3) M, Pa , Ga (µ runs from 0 to 3 and a from 1 to 3). This

is re?ected in the possibility of reducing

22,4 ? = 0

to

i?? ? = 21,3 ?

which in turn can be reduced to

21,3 ? = 0.

4 Two nonlinear equations

In this ?nal section, I shall mention two equations in nonlinear quantum mechanics

which are related to each other by our ansatz. They are

|?|2? ? ?2|?| = ??|?|? (5)

and

|u|

(6)

iut + u= u.

|u|

We can obtain equation (6) from equation (5) with the ansatz

? = ei(?? ?( x)/2)

u(?, ?x, ?x),

where ? = ?x = ?µ xµ and , ?, ?, ? are constant 4-vectors with ?2 = 2 = 0,

? 2 = ? 2 = ?1, ?? = ?? = ? = ? = 0, ? = 1.

Equation (5), with ? = m2 c2 / 2 was proposed by Vigier and Gu?ret [11] and by

e

Guerra and Pusterla [12] as an equation for de Broglie’s double solution. Equation (6)

was considered as a wave equation for a classical particle by Schiller [10] (see also [13]).

For equation (5), we have the following result:

Theorem 5 (Basarab-Horwath, Fushchych, Roman [3, 4]). Equation (5) with

? > 0 has the maximal point-symmetry algebra AC(1, n+1)?Q generated by operators

(1)

Pµ , Jµ? , Pn+1 , Jµn+1 , D(1) , Kµ , Kn+1 , Q,

(1)

where

? ?

= i(u?u ? u? ?u? ),

, Jµ? = xµ P? ? x? Pµ , Pn+1 =

Pµ = µ n+1

?x ?x

n

Jµn+1 = xµ Pn+1 ? xn+1 Pµ , D(1) = xµ Pµ + xn+1 Pn+1 ? (??? + ?? ??? ),

2

Kµ = 2xµ D ? (xµ x + xn+1 x

(1) (1) µ n+1

)Pµ ,

Q = ??? + ?? ??? ,

(1)

Kn+1 = 2xn+1 D(1) ? (xµ xµ + xn+1 xn+1 )Pn+1 ,

78 P. Basarab-Horwath, W.I. Fushchych

where the additional variable xn+1 is de?ned as

??

i

= v ln

= ?xn+1

n+1

x , ? > 0.

?

2?

For ? < 0 the maximal symmetry algebra of (9) is AC(2, n) ? Q generated by the

same operators above, but with the additional variable

??

i

xn+1 = xn+1 = v ln , ? < 0.

2 ?? ?

In this result, we obtain new nonlinear representations of the conformal algebras

AC(1, n + 1) and AC(2, n). It is easily shown (after some calculation) that equation

(5) is the only equation of the form

2u = F (?, ?? , ??, ??? , ?|?|?|?|, 2|?|)?

invariant under the conformal algebra in the representation given in Theorem 5. This

raises the question whether there are equations of the same form conformally invariant

in the standard representation

?

Jµ? = xµ P? ? x? Pµ ,

Pµ = ,

?xµ

n?1

(??? + ?? ??? ),

D = xµ Pµ ? Kµ = 2xµ D ? x2 Pµ .

2

There are such equations [3] and [4], for instance:

2? = |?|4/(n?1) F |?|(3+n)/(1?n) 2|?| ?, n = 1,

2|u|

2u = 2|u|F , |u| u, n = 1,

(?|u|)2

2|?| (2|?|)n

42? = +? ?, n arbitrary,

|?| |?|n+4

2|?|

2? = (1 + ?) ?,

|?|

2|?| ?

2? = 1+ ?,

|?| |?|4

2|?| ?

2? = 1+ ?.

|?| 1 + ?|?|4

Again we see how the representation dictates the equation.

We now turn to equation (6). It is more convenient to represent it in the amplitude-

phase form u = Rei? :

?t + ?? · ?? = 0, (7)

? + 2?? · ?R = 0. (8)

Rt +

Its symmetry properties are given in the following result:

Implicit and parabolic ansatzes: some new ansatzes for old equations 79

Theorem 6 (Basarab-Horwath, Fushchych, Lyudmyla Barannyk [5, 6]). The

maximal point-symmetry algebra of the system of equations (7), (8) is the algebra with

basis vector ?elds

1

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