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simplify calculations.

His previous papers were based on the equations

2(E ? V ) ? 2 ?

?? ? (2)

= 0,

E2 ?t2

8? 2

(E ? V )? = 0, (3)

?? + 2

where ? is a real function, E is energy.

When the potential V does not depend on time, Schr?dinger derives from (2), (3)

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the fourth-order wave equation

2

8? 2 16? 2 ? 2 ?

?? (4)

V ?+ 2 = 0,

2 ?t2

where ? is a real function.

Schr?dinger write about the equation (4): “. . . the equation (4) is the unique

o

and general wave equation for the ?eld scalar ? . . . . the wave equation (4) contai-

ns the dispersion law and can serve as a foundation for the theory of conservati-

ve system which I had developed. Its generalization for the case of time-dependent

potential demands some caution . . . an attempt to generalize the equation (4) for

non-conservative systems encounters the di?culty arising because of the term ?V . ?t

Therefore in the following I will go the other way which is simpler from the point of

view of calculations. I consider this way to be the most correct in principle.”

Further Schr?dinger writes down the equation (1) for the complex function ?. Just

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in this place of the paper [1] Schr?dinger makes a step of genius (and non-logical),

o

writing the equation (1) for a complex function.

As to the equation (1) Schr?dinger writes: “There is certainly some di?culty in

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application of complex wave functions. If they are necessary in principle, and not only

as a way to simplify calculations then it means that in principle two functions exist

which only together can give the description of the state of the system . . . The fact that

in the pair of equations (1) we have only a substitute, which is extremal convenient at

least for calculations. The real wave equation most certainly must be a fourth-order

equation. Though I have not succeeded to ?nd such equation for a non-conservative

system ?V = 0 .”

?t

We can make following conclusions from the above:

Conclusion 1. In 1926 Schr?dinger thought that the correct equation in quantum

o

mechanics has to be a fourth-order equation. For the case when the potential does not

depend on time this equation has the form (4).

Symmetry of equations of nonlinear quantum mechanics 125

Conclusion 2. In June, 1926 Schr?dinger considered that the equation (1) which is

o

?rst order in time and second order in space variables for the complex function is

interim (not principal), which is to be used only to simplify calculations.

Conclusion 3. Schr?dinger considered that in the case when the potential V depends

o

on time, the motion equation has to be also of the fourth order for the given function.

He could not derive such equation.

Now we can undoubtedly say that E. Schr?dinger did a mistake in respect of

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importance (fundamental role) of the equations (1), (4). Really, the equation (1) is

a principal equation of the quantum mechanics, and the equation (4) cannot be a

motion equation as it is not compatible with the Galilei relativity principle.

This statement follows from the symmetry analysis of the equations (1) and (4)

[2, 4]:

the equation (1) is invariant with respect to the Galilei group;

the equation (4) is not invariant with respect to the Galilei group.

With respect to the above we shall answer the following questions below:

1. Which linear equations of second, fourth, n-th order are compatible with the

Galilei relativity principle?

2. Does linear equations which are ?rst-order in time variable, fourth order in space

variables and are compatible with the Galilei relativity principle exist?

Theorem 1 [4] [Fushchych, 1987]. The Euclid algebra AE1 (1, 3) is the maximal

invariance algebra of the equation (4) (V = 0).

We have the following corollaries of the adduced theorems.

Corollary 1. The equation (4) is not compatible with the Galilei relativity principle.

This means that (4) cannot be considered as an equation of particle motion in quantum

mechanics.

2 Derivation of the Schr?dinger equation

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and higher order equations

Let us derive Schr?dinger equation out of the requirement of invariance of an equati-

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on with respect to the Galilei transformations and to the group of space and time

translations.

In [6] is proposed the following generalisation (V = 0) of the Schr?dinger equation

o

(1)

?1 S + ?2 S 2 + · · · + ?n S n ? = ??,

2 n

(5)

p2 p2

p0 ? a p0 ? a

2 n

S= ,...,S = ,

2m 2m

where ?, ?1 , ?2 , . . . , ?n are arbitrary parameters.

The equation (5) is compatible with the Galilei relativity principle and is invariant

with respect to the Galilei algebra AG(1, 3), but it is not invariant with respect to

the scale operator D and projective operator ? (?1 = 0, ?2 = 0).

The complete information on the symmetry of the equation (5) is given by the

following theorem.

126 W.I. Fushchych

Theorem 2 [19] [Fushchych and Symenoh, 1997]. There is only one equation

among linear arbitrary order equations which is invariant with respect to the algebra

AG(1, 3), and that is the equation (5). In the case when ? = ?1 = ?2 = · · · = ?n?1 =

0, the equation (5) is invariant with respect to the algebra AG2 (1, 3).

Thus the class of linear Galilei-invariant equations of arbitrary order is rather

narrow and reduced to the equation (5). All other Galilei-invariant equations are

locally invariant to the equation (5).

3 Nonlocal Galilei symmetry of the relativistic

pseudodifferential wave equation

Let us consider a pseudodi?erential equation

E ? (p2 + m2 )1/2 , (6)

p0 u = Eu, u = u(x0 , x).

a

We may consider the equation (6) as a “square root of the wave operator” for a scalar

complex function u.

We can check by direct calculation that the equation (6) is invariant with respect

to the standard representation of the Poincar? algebra and not invariant with respect

e

to the standard representation of the Galilei algebra.

Theorem 3 [8] [Fushchych, 1977]. The equation (6) is invariant with respect to

the 11-dimensional Galilei algebra with the following basis operators:

p2 ? ?

(2) (2)

=? Pa = pa = ? Jab = xa pb ? xb pa ? Jab ,

(2)

P0 = , ,

2m 2m ?xa (7)

m

= tpa ? mxa , pa ? pa , E = (p2 + m2 )1/2 .

G(2)

a a

E

The proof of the theorem is reduced to checking the invariance condition

[p0 ? E, Ql ]u = 0, (8)

where Ql is any operator from the set (7).

The operators (7) satisfy the commutation relations of the Galilei algebra.

(2)

Ga are pseudodi?erential operators which generate, as distinct from the standard

operators Ga , nonlocal transformations.

So the set of solutions of the motion equation (6) for a scalar particle (?eld) with

positive energy has a nonlocal Galilei symmetry, whose Lie algebra is given by the

operators (7).

4 Nonlocal Galilei symmetry of the Dirac equation

It is well-known that the Dirac equation

(9)

p0 ? = (?0 ?a pa + ?0 ?4 m)? = H(p)?

is invariant with respect to the Poincar? algebra with the basis operators (see [2])

e

? ? i

Pk = ?i Jµ? = xµ p? ? x? pµ + Sµ? , [?µ , ?? ]. (10)

P0 = i , , Sµ? =

?x0 ?xk 4

Symmetry of equations of nonlinear quantum mechanics 127

The Dirac equation, as it was established in our papers (see references in [2]) has wide

nonlocal symmetry.

In this paragraph we shall establish nonlocal Galilei symmetry of the Dirac equa-

tion. For this purpose, using the method described in [2], by means of the integral

operator

1 H

W=v E = (p2 + m2 )1/2 , (11)

1 + ?0 , H = ?0 ?a pa + ?0 ?4 m

a

E

2

we transform the system of four connected ?rst-order di?erential equations to the

system of non-connected pseudodi?erential equations

? ?

10 0 0

?0 1 0?

?? 0

= ?0 E?, ?0 = ? ?, (12)

i ?0 0 ?1 0?

?t

0 ?1

00

?0 E = W HW ?1 . (13)

? = W ?,

Having found additional symmetry of the equation (12), we simultaneously establi-

sh symmetry of the Dirac equation (9).

Theorem 4 [8] [Fushchych, 1977]. The equation (12) is invariant with respect to

the 11-dimensional Galilei algebra with the following basis operators:

p2 ?

(3)

Pa = pa = ?

(3)

P0 = , , I,

2m ?xa (14)

m

(3)

= xa pb ? xb pa + Sab , = tpa ? mxa , pa ? ?0 pa .

G(3)

Jab a

E

The operators (14) satisfy the commutation relations of the Galilei algebra

AG(1, 3).

To prove the theorem is necessary to make sure that the invariance condition

[p0 ? ?0 E, Ql ]? = 0 (15)

is satis?ed for any operator Ql from the set (14).

(3)

Ga are integral operators which generate nonlocal transformations, which do not

coincide with the standard Galilei transformations.

Thus the equation (12), and also the Dirac equation (9), has the nonlocal sym-

metry, which is given by the operators (14). The explicit form of the operators (14)

for the equation (9) is calculated by means of the formula

Ql = W ?1 Ql W. (16)

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2. Fushchych W., Shtelen W., Serov N., Symmetry analysis and exact solutions of eEquations of

nonlinear mathematical physics, Dordrecht, Kluwer Academic Publishers, 1993.

3. Fushchych W., Nikitin A., Symmetries of equations of quantum mechanics, New York, Allerton

Press, 1994.

4. Fushchych W., How to extend symmetry of di?erential equations?, in Symmetry and Solutions

of Nonlinear Mathematical Physics, Kyiv, Inst. of Math., 1987.

128 W.I. Fushchych

5. Fushchych W., Cherniha R., The Galilei relativictic principle and nonlinear partial di?erential

equations, J. Phys. A: Math. and Gen., 1987, 18, 3491.

6. Fushchych W., Symmetry in problems of mathematical physics, in Algebraic-Theortic Studies

in Mathematical Physics, Kyiv, Inst. of Math., Kyiv, 1981.

7. Fushchych W., Seheda Yu., On a new invariance algebra for the free Schr?dinger equation,

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Doklady Acad. Sci. USSR, 1977, 232, ¹ 4, 800.

8. Fushchych W., Group properties of equations of quantum mechanics, in Asymptotic problems

in theory of nonlinear oscilations, Kyiv, Naukova Dumka, 1977, 238.

9. Fushchych W., Nikitin A., Group invariance for quasirelativistic equation of motion, Doklady

Acad. Sci. USSR, 1978, 238, ¹ 1, 46.

10. Fushchych W., Serov M., On some exact solutions of the three-dimensional non-linear

Schr?dinger equation, J. Phys. A: Math. and Gen., 1987, 20, L929.

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11. Fushchych W., Boyko V., Continuity equation in nonlinear quantum mechanics and the Galilei

relativity principle, J. Nonlinear Math. Phys., 1997, 4, 124–128.

12. Schiller R., Quasi-classical transformation theory, Phys. Rev., 1962, 125, 1109.

13. Rosen N., American J. Phys., 1965, 32, 597.

14. Guerra F., Pusterla M., Nonlinear Klein–Gordon equation carrying a nondispersive solitonlike

singularity, Lett. Nuovo Cimento, 1982, 35, 256.

15. Gu?ret Ph., Vigier J.-P., Relativistic wave equation with quantum potential nonlinearity, Lett.

e

Nuovo Cimento, 1983, 38, 125.

16. Doebner H.-D., Goldin G.A., Properties of nonlinear Schr?dinger equations associated with

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di?eomorphism group representations, J. Phys. A: Math. Gen., 1994, 27, 1771.

17. Fushchych W., Cherniha R., Chopyk V., On unique symmetry of two nonlinear generalizations

of the Schr?dinger equation, J. Nonlinear Math. Phys., 1996, 3, 296.

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18. Fushchych W., On additional invariance of the Klein–Gordon–Fock equation, Doklady Acad.

Sci. USSR, 1976, 230, ¹ 3, 570.

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lean relativity, J. Phys. A: Math. Gen., 1997, 30, L131.

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