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non-Lie symmetry) which cannot be reduced to the algebra AP (1, 3) (5.10) [13, 29].
I shall say here briefly about one of such symmetries.
Theorem 7. (Fushchych 1971, 1974 [30, 31]. The Dirac equation (5.9) is invariant
with respect to the following representation of the Poincar? algebra
e
?
(2)
Pa = ?i
(2)
(5.14)
P0 = H = ?0 ?a pa + ?0 m, , a = 1, 2, 3,
?xa
i
(2)
Jab = xa pb ? xb pa + Sab , (?a ?b ? ?b ?a ), (5.15)
Sab =
4
1
(2)
J0a = x0 pa ? (xa H + Hxa ). (5.16)
2
Thus we have two different representations of the Poincar? algebra AP (1, 3) (5.10)
e
and (5.14)–(5.16).
The representation (5.15) and (5.16) generates nonlocal transformations
(2) (2)
xa > xa = exp{iJab vb }xa exp{iJ0c vc } = Lorentz transform, (5.17)
(2) (2)
t > t = exp{iJ0b vb }t exp{?iJ0c vc } = t. (5.18)

Thus, time does not change in relativistic physics. Time is absolute in relativistic
physics.
352 W.I. Fushchych

There are two nonequivalent possibilities (duality) for transformations of coordi-
nates and time: Lorentz transformation (5.11), (5.12) and non-Lorentz transformation
(5.17), (5.18).
The Maxwell and Klein–Gordon–Fock equations are also invariant under nonlocal
transformations (5.17), (5.18) when time does not change. However energy and
momentum are transformed by the Lorentz law [31,32]. We have new relativity prin-
ciple (5.17), (5.18).
What is the reason of such a paradoxical statement? The reason is that the
(2)
operators J0a are non-Lie symmetry operators and the standard relation (S. Lie’s
theorems) between Lie groups and Lie algebras is broken.
So, physics is not equivalent to geometry and geometry is not physics. Physics is
Nature. Theoretical Physics is only a Model of Nature!


6 On some new motion equations
Some new motion equations are adduced in this section. These equations are generali-
zations of known classical equations. Symmetry of these equations has not been
investigated.


6.1 High order parabolic equation in quantum mechanics
The Schr?dinger equation (5.1) is not the only equation compatible with the Galilei
o
relativity principle. A more general equation was suggested in [1, 2]

(?1 S + ?2 S 2 + · · · + ?n S n )u = ?u,
(6.1)
p2
S ? p0 ? a , S 2 = S · S, S n = S n?1 S,
2m
?, ?1 , ?2 , . . . , ?n are arbitrary parameters. Equation (6.1) as well as the classical
equation (5.1) is invariant with respect to the Galilei transformations but it is not
invariant with respect to scale and projective transformations.
A new equation for two particles (waves):
12
p0 u 1 = p u1 + V1 t, x1 , x2 , . . . , x6 , u1 , u2 ,
2m1 a
12
p0 u 2 = p u2 + V2 t, x1 , x2 , . . . , x6 , u1 , u2 ,
2m2 a+3
u1 = u1 (t, x1 , x2 , x3 ), u2 = u2 (t, x4 , x5 , x6 ), V1 and V2 are potentials.

6.2 Nonlinear generalization of Maxwell equations
If we assume that the light velocity is not constant [34], we can suggest some
generalizations of the Maxwell equations

?H
= rot {c(E 2 , H 2 , E H)H}, = ?rot {c(E 2 , H 2 , E H)E},
?E
(6.2)
?t ?t
div {a(E , H , E H)E} = 0, div {b(E 2 , H 2 , E H)H} = 0,
2 2
Ansatz ’95 353

where a, b and c are some functions of electromagnetic field;

?E ?D
= rot {c(B 2 , D2 , B D)B} + j, = rot {c(E 2 , H 2 , E H)E} + j,
?t ?t
or (6.3)
?H ?B
= ?rot {c(B 2 , D2 , B D)D}, = ?rot {c(E 2 , H 2 , E H)E},
?t ?t

?1 D + ?2 2D = F1 (E 2 , H 2 , E H)E + F2 (E 2 , H 2 , E H)H,
(6.4)
?3 B + ?4 2B = R1 (E 2 , H 2 , E H)E + R2 (E 2 , H 2 , E H)H,

(6.5)
div D = ?, div B = 0,

where F1 , F2 , R1 , R2 are functions of fields E and H, c in equations (6.2), (6.3) can
be a function of (t, x), c = c(t, x), or depend on the gravity potential V , c = C(V ).
Nonlinear wave equations for E and H have form

?2E ?2H
? c2 ?E = 0, ? c2 ?H = 0, (6.6)
2 2
?t ?t
or
?2E ?2H
? ?(c2 E) = 0, ? ?(c2 H) = 0; (6.7)
2 2
?t ?t
or
?2 ?2
1 1
? ?E = 0, ? ?H = 0, (6.8)
E H
?t2 c2 ?t2 c2
with one of the conditions
?E
+ ?H
1 ?t ?t
2
(6.9)
c=
2 (rot H)2 + (rot E)2
or
?c2 ?c2
(6.10)
= 0.
?xµ ?xµ
or
?c2
= ?(E 2 H 2 , E H)F?? c? , (6.11)

?xµ

c? is the four-velocity of the light (electromagnetic field), c2 = c? c? .
Equations of hydrodynamical type for electromagnetic field have form

?E
= a1 ? ? (c ? H) + a2 ? ? (c ? E) ,
?t
?H (6.12)
= b1 ? ? (c ? E) + b2 ? ? (c ? H) ,
?t
?c
+ (c?)c = R1 E + R2 H,
?t
354 W.I. Fushchych

c is the three-velocity of the light, where a1 , a2 , b1 , b2 , R1 , R2 are functions of E 2 ,
H 2 , E H.
Maxwell’s equations in a moving frame with the velocity can be generalized in
such forms
?E ?E ?H ?H
+ ? 1 vk + ?2 rot H = 0, + ? 3 vk + ?4 rot E = 0,
?t ?xk ?t ?xk
or
?E ?H ?H ?E
+ ? 1 vk + ?2 rot H = 0, + ? 3 vk + ?4 rot E = 0,
?t ?xk ?t ?xk
?vk
+ vl ?vk = 0.
with the conditions ?t ?xl


6.3 Equations for fields with the spin 1/2
Fields with the spin 1/2 are described, as a rule, by first-order equations, by the Dirac
equation. However, such fields can be also described by second-order equations. Some
of such equations are adduced below:
? ? ?
pµ pµ ? = F1 (??)?, ??µ pµ ? = F2 (??); (6.13)
? ? ?
pµ pµ ? = R1 (??)?, (??µ ?)pµ ? = F2 (??)?; (6.14)
? ? ? ?
pµ pµ ? = F1 (??)?, (??µ ?)(?pµ ?) = F3 (??); (6.15)
?
pµ pµ ? + ??µ pµ ? = F (??)?; (6.16)
? ? ? ?
p0 ? = {(??0 ?)(??k ?)pk + m??0 ?}?.
pµ pµ ? = F1 (??)?,

6.4 How to extend symmetry of an equation
with arbitrary coefficients?
Let us consider the a second-order equation
?2u ?u
(6.17)
aµ? (x) µ ? + bµ (x) µ + F (u) = 0.
?x ?x ?x
Equation (6.17) with arbitrary fixed coefficients has only a trivial symmetry (x >
x = x, u > u = u). However, if we do not fix coefficient functions aµ? (x), bµ (x),
such an equation can have wide symmetry. E.g., if aµ? , bµ satisfy the equations
?u ?u
2aµ? = (6.18)
F1 (u)
?xµ ?x?
or
?2u
?u
2bµ = F2 (u) 2aµ? (6.19)
, = F3 (u),
?xµ ?xµ ?x?
then the nonlinear system (6.17), (6.18), (6.19) is invariant with respect to the Poin-
car? group P(1,3). Let us emphasize that here even if we put F1 = 0, F2 = 0,
e
Ansatz ’95 355

equations (6.17), (6.18), (6.19) are a nonlinear system of equations. With some parti-
cular functions F1 and F2 , it is possible to construct ansatzes which reduce system
(6.17), (6.18), (6.19) to the system of ordinary differential equations.
So, considering (6.17) as a nonlinear equation with additional conditions for aµ? ,
b? , we can construct the exact solution for eqation (6.17). The adduced idea about
extension of the symmetry of (6.17) can be used for construction of exact solutions
for motion equations in gravity theory.
The second example of equations which have wide symmetry is

? 2 F??
(6.20)
vµ v? µ ? = 0,
?x ?x
?v?
(6.21)
vµ = 0.
?xµ
If in (6.20) vµ are fixed functions the equation, as a rule, has trivial symmetry.

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