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arXiv:hep-th/0507247v2 28 Jul 2005

An ”Accidental” Symmetry Operator for the Dirac Equation

in the Coulomb Potential

Tamari T. Khachidze and Anzor A. Khelashvili1

3 Chavchavadze Avenue, Department of Theoretical Physics,

Iv. Javakhishvili Tbilisi State University, Tbilisi 0128, Georgia

Abstract

On the basis of the generalization of the theorem about K-odd operators (K is the

Dirac’s operator), certain linear combination is constructed, which appears to commute

with the Dirac Hamiltonian for Coulomb field. This operator coincides with the Johnson

and Lippmann operator and is intimately connected to the familiar Laplace-Runge-Lenz

vector. Our approach guarantees not only derivation of Johnson-Lippmann operator, but

simultaneously commutativity with the Dirac Hamiltonian follows.

PACS :

Key words: : Dirac’s operator, hidden symmetry, Hydrogen atom, commutativity, super-

charge, Witten superagebra.

1Email: anzorkhelashvili@hotmail.com

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Recently H.Katsura and H.Aoki [1] considered an exact supersymmetry in the relativistic

hydrogen atom in general dimensions. It was established, that supercharges are connected to

the pseudoscalar invariant, which commutes with the Dirac Hamiltonian in Coulomb field. In

usual 3-dimensions this invariant coincides to the Johnson-Lippmann (J.-L.) operator, which

was introduced by these authors in 1950 in a very brief abstract [2]. As to more detailed settle,

by our knowledge, it had not been published neither then nor after. First respond on relativistic

Kepler problem in this point of view appeared in 60-ies [3]. Main attention was paid to the

ways of deriving simpler solutions of the Dirac equation as far as possible.

After 80-ies the interest was grown from the positions of supersymmetric quantum mechan-

ics. Supersymmetry of the Dirac equation in Coulomb problem was demonstrated, but for this

purpose mainly the radial equations was considered for deriving the best separation of variables.

In the above mentioned paper [1] the supercharges in relativistic case is obtained for the

first time. As it turns out, the decisive part plays the J.-L. operator, the generalization of

which in arbitrary dimensions was introduced by these authors and, therefore they named it

as Johnson-Lippman -Katsura-Aoki operator.

In any article [1] or text books (See, [4] , for example), there is underlined only that its

commutativity with Hamiltonian can be proved by longtime and tedious calculations.

Derivation of Witten superalgebra is based on very simple logic: It is known that the so

called Dirac operator, which has the form [4]

K = β

??Σ?l + 1

?

, (1)

commutes with the Dirac Hamiltonian

H = ? α? p + βm −a

r,

a ≡ Ze2= Zα .(2)

(In fact, K commutes with Hamiltonian for arbitrary central potential). In equation (1)?l is

the angular momentum vector and?Σ is the electron spin matrix

?Σ = ρ1? α = γ5? α =

?

? σ

0 ? σ

0

?

.

The Sommerfeld formula for the Hydrogen atom spectrum reads [5]

E

m=

1 +

(Zα)2

k2− (Zα)2?2

j(j + 1) + 1/4 = j + 1/2. It seems, that the energy

?

n − |k| +

?

−1/2

. (3)

Here k is the eigenvalue of K, |k| =

spectrum is degenerate with respect of signs of k, ±sgn(k).

?

An operator, that anticommute with K, is able to interchange these signs, moreover if

this operator would commute with Hamiltonian at the same time, the supersymmetry algebra

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appears. It was shown in [1], that the J.-L. operator has precisely these properties. The J.-L.

operator looks like [2]

?? α? r

r

A = γ5

−

i

maKγ5(H − βm)

?

. (4)

We are inclined to think that in following importance of this operator will enhance in studies of

dynamical symmetries because of its connection to the Laplace-Runge-Lenz (LRL) vector and

”accidental” degeneracy.

Therefore it seems highly desirable , by our opinion, to derive the J.-L. operator from the

first and simplest principles. Below we propose one of our path of construction this operator,

which is rather simple and very transparent.

First of all we need to extend one important theorem known for Pauli electron [6, 7] to the

Dirac equation.

Theorem: Suppose?V is a vector with respect to the orbital angular momentum?l, i.e.

?l ×?V +?V ×?l = 2i?V .

At the same time?V is to be perpendicular to?l, i.e.,?l?V =?V?l = 0.

Then the Dirac’s K operator anticommutes with?J- scalar operator

??Σ?V

?

, i.e.,

?

K,

??Σ?V

For that it is sufficient to consider the product

??

= 0 .(5)

The proof of this theorem is very easy.

??Σ?l + 1

matrices. It follows that

???Σ?V

?

in direct and reversed orders, making use of commutating properties of Dirac

??Σ?l + 1

???Σ?V

?

= −

??Σ?V

? ??Σ?l + 1

?

.

Remembering β?Σ =?Σβ, this result can be carry over the Dirac’s K operator

K

??Σ?V

?

= −

??Σ?V

?

K .

It is evident, that the class of anticommuting with K(or K- odd) operators is not confined

by these operators only - any operator of typeˆO

otherwise arbitrary, also is K-odd.

??Σ?V

?

, whereˆO is commuting with K, but

Let mention, that in the framework of constraints of above theorem, the following very

useful relation takes place

K

??

Σ?V

?

= −β

?

?Σ ,1

2

??V ×?l −?l ×?V

??

. (6)

We see, that the antisymmetrized vector product, familiar for LRL vector, appears on the right-

hand-side of this relation.

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Important special cases, resulting from the above theorem, include?V =ˆ? r (unit radial

vector),?V = ? p (linear momentum) and?V =?A (LRL vector), which has the following form [8]

?A =ˆ? r −

i

2ma

?

? p ×?l −?l × ? p

?

. (7)

According to (6), there appears one relation between these three odd operators

?Σ?A =?Σˆ? r +

i

maβK

??Σ? p

??

?

. (8)

So far as [β , K] = 0, it follows that

permissible K-odd operator.

?

K , βK

??Σ? p= 0 and K

??Σ? p

?

can be used as a

Our purpose is to construct such combination of K-odd operators, which would be commut-

ing with Dirac Hamiltonian. We can solve this task by step by step. As a first trial expression

let consider the following operator

A1= x1

??Σˆ? r

?

+ ix2K

??Σ? p

?

.

Here the coefficients are chosen in such a way, that A1be Hermitian, when x1, x2are arbitrary

real numbers. These numbers must be determined from the requirement of commuting with

H. Let calculate

[A1,H] = x1

???Σˆ? r

?

,H

?

+ ix2

?

K

??Σ? p

?

,H

?

.

Appearing here commutators can be calculated easily. The result is

[A1,H] = x12i

rβKγ5− x2a

r2K

??Σˆ? r

?

.

One can see, that the first term in right-hand side is antidiagonal, while the second term

is diagonal. So this expression never becomes vanishing for ordinary real numbers x1 , x2.

Therefore we must perform the second step: one has to include new odd structure, which

appeared on the right-hand side of above expression. Hence, we are faced to the new trial

operator

A2= x1

+ ix2K

??Σˆ? r

???Σ? p

?

+ ix3Kγ5f(r) .(9)

Here f(r) is an arbitrary scalar function to be determined. Let calculate new commutator. We

have

[A2,H] = x12i

r2K

Grouping diagonal and antidiagonal matrices separately and equating this expression to zero,

we obtain equation

rβKγ5− x2a

??Σˆ? r

?

− x3f′(r)K

??Σˆ? r

?

− ix32mβKγ5f(r) .

K

??Σˆ? r

??a

r2x2+ x3f′(r)

?

+ 2iβKγ5

?1

rx1− mx3f(r)

?

= 0 .

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This equation is to be satisfied, if diagonal and antidiagonal terms become zero separately, i.e.,

a

r2x2= −x3f′(r),

1

rx1= mx3f(r) .

First of all, let us integrate the second equation in the interval (r , ∞). It follows

x3f(r) = −a

rx2.

Accounting this in the first equation, we have

x2= −1

max1.

Then we obtain also

x3f(r) = −1

mrx1.

Therefore finally we have derived the following operator which commutes with Dirac Hamilto-

nian

A2= x1

?

It is K-odd, in accord with above theorem.

If we turn to usual ? α matrices using the relation?Σ = γ5? α and taking into account the

expression (2) of the Dirac Hamiltonian, A2can be reduced to the more familiar form (x1, as

unessential common factor, may be dropped)

???Σˆ? r−

i

maK

??Σ? p

?

+i

mKγ51

r

?

. (10)

A2= γ5

?

? αˆ? r −

i

maKγ5(H − βm)

?

.

This expression is nothing but the Johnson-Lippmann operator, (4).

In order to clear up its physical meaning, remark, that equation (10) may be rewrite in the

form

A2=?Σ

ˆ? r −

2maβ

It is evident, that in nonrelativistic limit, when β → 1 and γ5→ 0, this operator coincides to

the projection of LRL vector on the electron spin direction, A2=?Σ?A or because of?l?A = 0 it

is a projection on the total?J momentum direction.

?

i

?

? p ×?l −?l × ? p

??

+

i

mrKγ5.

After this it is clear that the Witten algebra can be derived by identifying supercharges as

Q1= A ,Q2= iAK

k

.

It follows that

{Q1,Q2} = 0 , and Q2

1= Q2

2= A2.

This last factor may be identified as a Witten Hamiltonian (N=2 supersymmetry).

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As for spectrum, it is easy to show that the following relation holds [1]

A2= 1 +

?K

a

?2?H2

m2− 1

?

.

Because all operators entered here commute with H, we can replace them by their eigenvalues.

Therefore, we obtain energy spectrum pure algebraically. In this respect it is worth-while to

note full analogy with classical mechanics, where closed orbits were derived by calculating the

square of the LRL vector without solving the differential equation of motion, [10]. In conclusion

we are convinced that the degeneracy of spectrum relative to interchange k → −k is associated

to the existence of conserved J.-L. operator which takes its origin from the LRL vector. It is

also remarkable, that the same symmetry is responsible for absence of the Lamb shift in this

problem. Inclusion the Lamb-shift terms into the Dirac Hamiltonian breaks commutativity of

A with H.

In conclusion we should like to express our cincere appreciation to professors Misha Zvi-

adadze, Gela Devidze and Badri Magradze for their critical remarks.

This work was supported in part by the reintegration Grant No. FEL. REG. 980767.

References

[1] H. Katsura and H. Aoki, quant-ph/0410174.

[2] M. Johnson and B. Lippmann, Phys. Rev. vol.78 (1950) 329(A).

[3] L. Biedenharn, Phys. Rev. vol.126 (1962) 845; L. Biedenharn and N. Swamy, Phys. Rev.

B133 (1964) 1353.

[4] V. Berestekski et al., Quantum Elactrodyamics, (Pergamon, 1982).

[5] P. Dirac, Quantum Mechanics, (fourth ed., Oxford, 1958).

[6] D. Atkinson and P. Johnson, Quantum Field Theory, A Self-Contained Course, (Riton

Press, 2002).

[7] L. Biedenharn and L. Louck. In ”Encyclopedia of Mathematics and Its Application”,

(Addison-Wisley Pub.Comp., 1981).

[8] R. Tangerman and J. Tjon, Phys. Rev. A48 (1993) 1089.

[9] W. Pauli, Zeit. f. Fiz., vol.36 (1926) 336.

[10] H. Goldstein et al., Classical Mechanics, (3rd ed., Pearson Education, 2002).

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