**Maximal-acceleration corrections to the Lamb shift**

**of one-electron atoms (*)**

G. LAMBIASE(1)(**), G. PAPINI(2)(***) and G. SCARPETTA(1)(3)

(1*) Dipartimento di Scienze Fisiche “E. R. Caianiello”, Università di Salerno*
*84081 Baronissi (SA), Italy*

*INFN, Sezione di Napoli - Napoly, Italy*

(2*) Department of Physics, University of Regina - Regina, Sask. S4S 0A2, Canada*
(3*) International Institute for Advanced Scientific Studies - Vietri sul Mare (SA), Italy*
(ricevuto il 5 Settembre 1996; approvato il 10 Dicembre 1996)

**Summary. — The maximal-acceleration corrections to the Lamb shift of **

one-electron atoms are calculated starting from the Dirac equation and splitting the
*spinor into large and small components. The results depend on Z*8_{and a cut-off L.}

*Sizeable values are obtained even at Z 41 for LAa*0*/2, where a*0is the Bohr radius.

These values are compatible with theoretical and experimental results. PACS 04.90 – Other topics in general relativity and gravitation. PACS 12.20.Ds – Specific calculations.

**1. – Introduction**

The notion of a limiting value to the proper acceleration of a particle, advanced on classical and quantum grounds by several authors [1-3], has different aspects and formulations. The importance of its existence, if proven, cannot be underestimated. It would by itself rid quantum field theory of unpleasent divergencies, and make several important theoretical procedures finite.

The idea finds a particularly interesting formulation in the original attempts by Caianiello [1] to incorporate quantum position-momentum uncertainty relations into the geometry of space-time, and to interpret geometrically quantization as a conse-quence of curvature in phase space.

*In this model a particle of mass m accelerating along its world line behaves*

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) E-mail: lambiaseHvaxsa.csied.unisa.it

(***) E-mail: papiniHcas.uregina.ca

dynamically as if subject to an effective gravitational field
*g*
A_{mn}_{4 g}_{mn}

### g

_{1 2}

*Nx*O N2

*A*2

### h

, (1.1)*where Nx*ON2

*4 Ngmnx*O

_{m}*x*O

*n*

N is the square length of the relativistic acceleration
*four-vector, A 4mc*3*/ˇ is the maximal proper acceleration of the particle and gmn* represents

a background gravitational field.

Equation (1.1), the arrival point of an embedding procedure of an eight-dimensional
space-time tangent bundle TM in four-dimensional space-time [4], has several
important implications for relativistic kinematics [5], the energy spectrum of a
uniformly accelerated particle [6], the periodic structure as a function of the
*momentum p of the neutrino oscillations [6], the Schwarzschild horizon [7], the*
expansion of the very early universe [8] and on the value of Higgs-boson mass [9]. It
would make the metric observer dependent, as conjectured by Gibbons and
Hawking [10], and lead in a natural way to hadron confinement [11].

The difficulties of a direct test of eq. (1.1) obviously reside in the extreme large
*value that the normalizing constant A takes for all known particles (A C0.45Q10*30* _{m}*
m s22

_{MeV}21

_{). Nevertheless, a realistic test that makes use of photons in cavities has}been recently suggested [12]. Hopefully, attempts in this direction will lead to some concrete results. In the meantime, indirect evidence may be gathered from a variety of different means.

The purpose of this work is to calculate the corrections to the Lamb shift of hydrogen or hydrogen-like atoms due to the maximal acceleration. Because of their relatively simple configuration and of the consequent precision with which they can be described and observed, these atoms lend themselves admirably to the precise test of physical laws. It is therefore natural to think in this context of the contribution of maximal acceleration, embodied by eq. (1.1).

**The plan of the paper is as follows. Section 2 contains the derivation of the Dirac**
Hamiltonian using the tetrad formalism and the metric (1.1) in a flat background. In
**sect. 3 the Schrödinger equation is obtained from the Dirac equation by splitting the**
*spinor c into large and small components, which is frequently used in calculations of*
**the Lamb shift. In sect. 4 the actual corrections to the levels 2 s**_{1 O2}*, 2 p*_{1 O2}*and 2 p*_{3 O2}are
**calculated. Section 5 contains the Lamb shift, shift, the of the states 2 p**

### (

*d E( 2 p*3 O2) 2

*d E( 2 p*1 O2)

### )

*, 1 s and the Lamb shift corrections in which the maximal acceleration is a*

**universal constant. Section 6 contains summary and discussion.**

**2. – The Dirac Hamiltonian**

If the background metric is flat, eq. (1.1) becomes simply conformally flat:

*g*
A_{mn}_{4 h}_{mn}

### g

_{1 2}

*Nx*O N2

*A*2

### h

*4 s*2

_{(x) h}*mn*, (2.1)

*where Nx*O

*(s) N*2

*4 Nx*O

*mx*O

_{m}*N is the acceleration field of the particle and s (x) is the*
*conformal factor. The dependence of s on the variable x implies that the effective*
geometry defined by (2.1) is no longer flat. This has important consequences. In fact
one must now take into account effects due to the curvature of space-time and its
coupling to the particle itself.

The electron in the hydrogen atom for the Lamb shift problem is described by the Dirac equation in flat space-time. With the introduction of the metric tensor (2.1), the Dirac equation must be generalized to curved space-time. This generalization is not trivial, but can be accomplished by means of vierbeins that connect a generic non-inertial frame to a locally Minkowski frame [13]. Equation (2.1) above provides immediately the vierbeins

*ema(x) 4s(x) dma*,

(2.2)

where Latin indices refer to the local inertial frame and the Greek indices to the generic non-inertial frame. The covariant Dirac equation is written in the form

### (

*iˇgm*

*(x) Dm2 m*

### )

*c(x) 40 ,*

(2.3)

*where the matrices gm*

*(x) satisfy the anticommutation relations ]gm _{(x), g}n*

*(x)( 4*
*2 gAmn*

*(x). The covariant derivative Dm*f ¯*m1 vm* *contains the total connection vm* given

by
*vm*4
1
2 *s*
*ab _{v}*

*mab*, (2.4) where

*sab*4 1 4

*[g*

*a*

_{, g}b_{] ,}(2.5)

*vmab4 (Gmnl*

*ela*2 ¯

*mena) enb*, (2.6) and

*Glmn*4 1 2

*g*

*la*

*(gam , n1 gan , m2 gmn , a*) (2.7)

are the usual Christoffel symbols. The usual flat space-time Dirac matrices are
*represented by ga _{. For conformally flat metrics v}*

*m*takes the form

*vm*4
1
*ss*
*ab _{h}*

*ams, b*. (2.8)

By using the transformations

*gm _{(x) 4e}maga*,

(2.9)

*vm4 emava*,

(2.10)

*and the property em*

*aemb4 dba*, eq. (2.3) can be written in the form

### k

*iˇga*¯

*a1 i*3 ˇ

*2 sg*

*a*

_{(¯}

*aln s) 2m*

### l

*c(x) 40 .*(2.11)

Since in the problem at hand the electron also interacts with the electromagnetic field

*Aa*f*(A0***, A), eq. (2.11) must be rewritten as**

### k

*iˇga*

### g

¯*a1 i*

*e*ˇ

_{c}*Aa*

### h

*1 i*3 ˇ

*2 sg*

*a*

_{( ln s)}*, a2 m*

### l

*c(x) 40 .*(2.12)

By writing eq. (2.12) in the form of a Schrödinger equation one arrives at the Hamiltonian

* H f ca Q p 1eA*0

*2*

**1 ea Q A 1 mc***b 2i*

*3 ˇc*

2 *g*

0_{g}a_{( ln s), a}_{,}

(2.13)

where the last term contains all maximal-acceleration effects and can be written as

*H 8f2 i3 ˇc*
2 *g*
0_{g}a*(2s*21_{)}
*, a4 2 i3 ˇc*
2

### u

*e*0

**s Q e**

**s Q e**

*e*0

### v

, (2.14)*where e0*f

*21*

_{(2s}_{), 0}

*21*

_{and e f ˘(2s}_{).}

The Hamiltonian that can be derived from (2.3) is in general Hermitian neither with respect to the conserved scalar product of the spinors in curved space-time, nor with respect to the flat scalar product [14]. Hermiticity can however be recovered for stationary states of the atom when the gravitational field changes slowly with time in a locally inertial rest frame of the atom.

*H given by (2.13) is also not Hermitian with respect to the flat scalar product. When*

one splits the Dirac spinor into large and small components, the only non-Hermitian
*term is, however, the one proportional to e*0*. If s varies slowly in time, or is time*
independent, this term may be neglected.

**3. – The maximal-acceleration corrections**

The Hamiltonian (2.14) will now be treated perturbatively . The common textbook
*procedure [15] consists in splitting c(x) in large and small components indicated by W*
*and x, respectively. One obtains for the perturbation due to H 8:*

(3.1) * _{d E4 anljmNH 8 Nnljmb 42 i}3 ˇc*
2

_{d}3

*†*

**r c***nljm*

### u

*e*0

**s Q e**

**s Q e**

*e*0

### v

*cnljm*4

*4 2 i3 ˇc*2

_{d}3

*0*

_{r [e}*(W*†

*W 1x*†

*x) 1W*†

*†*

**(s Q e) x 1x***On introducing the large and small components, related to each other by*

**(s Q e) W] .***x 42i*ˇ
*c*

**s Q ˘**

*2 m* *W ,*
(3.2)

and integrating by parts, eq. (3.1) becomes
*d E4d E*02
3 ˇ2
*4 m*

_{d}3

*†*

_{r W}*(3.3) where*

**[ (s Q e)(s Q ˘) 1 (sQ˘)(sQe) ] W ,***d E*0

*4 2 i*

*3 ˇc*2

_{d}3

*†*

**r W***e*0

### y

1 2 ˇ2*4 m*2

*c*2

**(s Q ˘)**2

### z

*W .*(3.4)

Rearranging the term in square bracket in (3.3) one obtains

*d E4d E*02
3 ˇ2
*4 m*

_{d}3

*†*

**r W***(3.5)*

_{[˜ Q e 12eQ˜] W ,}* where for e f ˜(2s*21

_{) one must now substitute}

**e 4** 1
*s*2˜

### o

1 2*Nx*ON2

*A*2 4 2 1

*2 s*3

*2*

_{A}*˜Nx*O N2. (3.6)

*It is now convenient to examine the term Nx*ON2*in detail. For particles of charge q that*
move in electromagnetic fields one has

*Nx*ON24

### g

*gq*

*m*

### h

2**[2(EQb)**2

**1 NEN**2

*2*

**1 2 E Q (b 3 B) 1 NbN****NBN**2

**2 (b Q B)**2] , (3.7)

*where g 41O*

### k

**1 2NbN**2

**and b 4v/c.**In the case of non-relativistic electrons in an electrostatic field, which is of interest here, eq. (3.7) reduces to

*Nx*ON24

### g

*e*

*m*

### h

2

**NE(r) N**2,
(3.8)

while eq. (3.6) becomes

**e 42** 1
*2 s*3

### g

*e*

*mA*

### h

2**˜NE(r) N**2. (3.9)

*On neglecting terms of order A*24 * _{in the expansion s A12Nx}*O
N2

*/2 A*2

1 R , and
* restricting E(r) to central electric fields E(r) 4Ze/r*2, eq. (3.9) becomes

**e 42**

### g

*Ze*2

*mA*

### h

2

_{r}*r*6 . (3.10)

*Moreover, d E*0*4 0 because for electrostatic fields e*04 0. Substituting the result
(3.10) in (3.5) one obtains the maximal-acceleration corrections for an electron in an
electrostatic field

*d E4d E*1*1 d E*2,
(3.11)

where
*d E*14 6 K

*† 1*

_{r W}*r*6

*W ,*(3.12)

*d E*24 2 4 Kd3

*† 1*

_{r W}*r*5 ¯ ¯

*rW ,*(3.13) and

*K f*3 ˇ 2

*4 m*

### g

*Ze*2

*mA*

### h

2 . (3.14)**4. – The correction to the energy levels 2 s and 2 p**

Before calculating the average values of eqs. (3.12)-(3.13), a few preliminary
*considerations are in order. By virtue of (3.8), the conformal factor s (x) may be written*
as
*s (x) 4*

### o

1 2*Nx*O N2

*A*2 4

### o

1 2### g

*r*0

*r*

### h

4 , (4.1) where*r*04

### o

*Ze*2

*mA*Ak

*Z 3.3 Q 10*214

*(4.2)*

_{m .}*Equation (4.1) is real for r Dr0*. However, it was assumed in the expansion of the square
*root (4.1) leading to eq. (3.10) that Nx*ON2*/A*2

b1. This requires that in the following only
*those values of r be chosen that are above a cut-off L, such that for r DLDr0* the
validity of the expansion is preserved. The actual value of L will be selected later.

*In order to calculate the corrections to the energy levels 2 s and 2 p, the explicit*
expression of the corresponding wave functions are useful. These are:

*W*100*(r , u , f) 4R*10(r) Y00*(u , f) 4*
2
*a*03 O2
*e2rOa*0* _{Y}*
00,
(4.3)

*W*200

*(r , u , f) 4R*20(r) Y00

*(u , f) 4*2

*( 2 a0*)3 O2

### g

1 2*r*

*2 a*0

### h

*e2rO2 a*0

*00, (4.4)*

_{Y}### u

*W*211

*W*210

*W*

_{21 21}

### v

*4 R*21

*(r)*

## u

*Y*1 1

*(u , f)*

*Y*10

*(u , f)*

*Y*121

*(u , f)*

## v

4 1 k*3( 2 a*0)3 O2

*r*

*a*0

*e2rO2 a*0

### u

*Y*11

*Y*1 0

*Y*121

### v

, (4.5)and down spin wave functions

*x*_{1 O24}

### u

10

### v

and*x*21 O24

### u

0 1### v

*appear in the average values only in the combinations x*†1 /2*x*1 /2*4 1 and x*21 /2
†

*x*21 /24 1
and have been omitted from (4.3)-(4.5).

*From (4.4) and (3.12), (3.13) one finds for n 42, l40*

*d E*12 , 04
*K*
*4 a*06

### mk

4### g

*a*0 L

### h

3 2 8### g

*a*0 L

### h

2 1 11### g

*a*0 L

### h

### l

*e*

*2L/a*0

*1 11 Ei*

### g

2L*a*0

### h

### n

, (4.6)*d E*22 , 04

*K*

*4 a*06

### mk

4### g

*a*0 L

### h

2 2 10### g

*a*0 L

### h

### l

*e*

*2L/a*0

*2 11 Ei*

### g

2L*a*0

### h

### n

, (4.7)*where K/a0*6

*A Z*81.03 kHz and

*Ei(2m) 42*1 Q

*dxe*

*2mx*

*x*,

*m D0 .*(4.8)

*The correction to the level 2 s is obtained by summing (4.6) and (4.7),*

*d E*2 , 0
4 *K*
*a*06

### k

### g

*a*0 L

### h

3 2### g

*a*0 L

### h

2 1 1 4### g

*a*0 L

### h

### l

*e*

*2L/a*0

_{.}(4.9)

*Likewise, the explicit expression for the correction to 2 p level is:*

(4.10) *d E*2 , 1*4 d E*12 , 1*1 d E*22 , 14
*4 6 K*

*dr*1

*r*4

*[R21(r) ]*2

*2 4 K*L Q

*dr R21(r)*1

*r*3 ¯ ¯

*rR*21

*(r) 4*

*K*

*a*06 1 12

### g

*a*0 L

### h

*e*

*2L/a*0

_{.}

**5. – Lamb shift, shift of the states 2 p and 1 s**

*It is now possible to calculate the contribution to the Lamb shift d E*L*4 d E*2 , 02

*d E*2 , 1_{. The correction is given in our case by}

*d E*L*4 d E*2 , 0*2 d E*2 , 14
*K*
*a*06

### k

### g

*a*0 L

### h

3 2### g

*a*0 L

### h

2 1 1 6### g

*a*0 L

### h

### l

*e*

*2L/a*0

_{.}(5.1)

The cut-off L must now be chosen in a way that is compatible with the value normally
adapted in QED. This is a characteristic length of the system, the Bohr radius for the
hydrogen atom, that cures the divergences introduced by the radiative corrections.
*One may tentatively choose L Aa0*in the present calculation. Then eq. (5.1) yields

*d E*L*A 1Z*80.0631 kHz .
(5.2)

For the sake of completeness we also give the maximal-acceleration corrections to the
*1 s ground state Lamb shift d E*1 , 0

are
*d E*11 , 04
*8 K*
*a*06

### mk

### g

*a*0 L

### h

3 2### g

*a*0 L

### h

2 1 2### g

*a*0 L

### h

### l

*e*

*22 L/a*0

*2 4 E1*

### g

2 L*a*0

### h

### n

, (5.3)*d E*21 , 04

*8 K*

*a*06

### mk

### g

*a*0 L

### h

2 2 2### g

*a*0 L

### h

### l

*e*

*22 L/a*0

_{1 4 E1}### g

2 L*a*0

### h

### n

. (5.4)*By subtracting from d E*1 , 0 *with L Aa*0*the value d E*2 , 0 given by (4.9), one obtains the
desired shift

*d E*1 , 0*2 d E*2 , 0C 0.98 *K*

*a*06

*4 Z*81.01 kHz .
(5.5)

It has been assumed so far that the maximal acceleration is mass dependent. It is also
*possible to take the view [3] that the maximal acceleration be a universal constant A*P4

*m*P*c*3*/ˇ, where m*P4kˇ*c/G is Planck’s mass. In this case r*04

### k

*Ze*2

*/mA*PAk

*Z 2.13 Q*10225

_{m, and AP}A 5.5 Q 1051m/s2*and K/a0*6* _{A Z}*81.28 Q 10245

_{kHz.}

Obviously, the Bohr radius can no longer be chosen as cut-off in this case because the magnitudes of the lengths of physical interest are comparable with Planck’s length

*l*P4

### k

ˇ*G/c*3A 10235m. A better cut-off may be derived from the requirement that in

*the expansion of s terms (r0/r)*4

be at least A1022_{. This leads to L A100}1 /4_{r}

0A 6.76 Q
10225_{m. The corresponding correction follows from (5.1):}

*d E*L*A Z*80.61 Q 1023kHz .
(5.6)

**6. – Summary and discussion**

The present calculation is based on the Dirac Hamiltonian (2.13) with the
maximal-acceleration corrections confined to the term (2.14). This has been treated as a
*perturbation. Legitimate fears about the hermiticity of H have proved unwarranted*
*because of the static nature of the problem which requires e*0*4 (2s*21), 04 0.

The standard non-relativistic procedure to split the Dirac wave function into large and small components has also been followed. This is usually justifiable because the Lamb shift is a non-relativistic effect at low frequencies. The results are represented by (5.2), (5.5) and (5.6). The latter result pertains to the interpretation, expoused by some authors [3], that the maximal acceleration is a universal constant.

*A common feature of all results is the dependence on the eighth power of Z. This*
would be experimentally significant if the present uncertainty of about 4 kHz [16] in
*the measurement of d E*L*for Z 41 could be extrapolated to atoms of moderately high Z,*
*or ionized atoms. Reported theoretical uncertainties from uncalculated terms for Z 41*
*amount to 11.0 kHz, which is well above (5.2) and could drastically increase for high Z.*
The result is nevertheless interesting. Its closeness to (5.6) also indicates that a
Lamb-shift measurement would hardly be the way to distinguish between the
maximal-acceleration models of refs. [1] and [3]. The value (5.2) also is smaller than the
ground-state phase shift, as expected on intuitive grounds. However, the reported [17]
*measurement precision of the 1s Lamb shift in hydrogen is higher than the 2s 2 2p Lamb*
shift and at 0.6 kHz is only a factor 10 higher than (5.2). While, on the other hand, the

TABLE*I. – All values reported are in kHz.*
L *a*0 *a*0/2 *a*0/2.5 Theoretical
value
Experimental
value
*d E*L 0.0631 2.81 6.90 1 057 849(11) 1 057 851(4)
*d E*1 , 0
*2 d E*2 , 0 1.01 20.72 50.95 8 172 802(40) 8 172 874(60)

corresponding theoretical error is 0.4 kHz, agreement between the experimental value
8172.874(60) MHz and the theoretical one 8172.802(40) MHz still leaves much to be
desired. It is a small consolation that the correction (5.2) is in the right direction. This also
applies to the Lamb shift for the He1_{ion [18]. In this case (5.2) predicts a positive shift of}
*15.9 kHz vs. the present QED deficit of A1 MHz. Agreement between theory and*
*experiment at higher values of Z is much worse and so are the corresponding*
uncertainties. The small contribution (5.3) to the fine structure should not be surprising,
given the non-relativistic nature of our approximations.

*The values (5.2), (5.5) and (5.6) refer to LAa0* and are rather conservative. While
Lamb shifts and fine-structure effects are cut-off independent, the values of the
corresponding maximal-acceleration corrections increase when L decreases. This can be
understood intuitively because the electron finds itself in regions of higher electric field at
*smaller values of r.*

*A lower cut-off, still compatible with the critical value r*0given by (4.2), is represented
by the Compton wavelength of the electron. This cut-off, however, yields corrections
which are decidedly too high. Both Lamb shifts become very significant already at values
*of LAa*0*/2. In fact for Z41 one finds d E*L*C2.70 kHz at L4a*0*/2 and d E*LC6.98 kHz
*at L4a0*/2.5. These corrections, added to the theoretical value 1 057 849 kHz, yield,
respectively, 1 057 851.7 kHz and 1 057 855.9 kHz, in very good agreement with the
*experimental value 1 057 851 kHz. Similary for Z41 the values of d E*1, 0*2d E*2, 0become
*20.72 kHz for L4a0/2 and 50.95 kHz for L4a0*/2.5. When added to the theoretical value
8 172 802 kHz, they give respectively 8 172 822.72 kHz and 8 172 852.95 kHz, also in good
agreement with the experimental value 8 172 874(60) kHz. The situation is summarized in
table I.

*It is quite remarkable that both d E*L *and d E*1, 0*2d E*2, 0 have the appropriate signs
and increase with decreasing values of L as intuitively expected. In view of the results
*presented and the approximations used, the cut-off LAa*0/2 is a better choice.

* * *

GP gladly acknowledges the constant research support of Dr. K. DANFORD, Dean of Science, University of Regina. GL wishes to thank the D. DANFORDfor the kind hospitality during a stay at the University of Regina, where part of this work has been done. Research supported by MURST fund 40% and 60%, DPR 382/80.

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