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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 Z8_{and a cut-off L.}

Sizeable values are obtained even at Z 41 for LAa0/2, where a0is 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

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dynamically as if subject to an effective gravitational field g A_mn_{4 g}_mn

g

_{1 2} Nx O N2 A2

h

, (1.1) where NxON24 Ngmnx O_m xOn

N is the square length of the relativistic acceleration four-vector, A 4mc3/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 C0.45Q1030_m m s22_MeV21_{). 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 p3 O2) 2

d E( 2 p1 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 A2

h

4 s 2_{(x) h} mn, (2.1) where NxO(s) N2 4 NxOmx 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.

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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 Dmf ¯m1 vm contains the total connection vm given

by vm4 1 2 s ab_v mab, (2.4) where sab 4 1 4[g a_{, g}b_{] ,} (2.5) vmab4 (Gmnl ela2 ¯mena) enb, (2.6) and Glmn4 1 2g 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}

mtakes the form

vm4 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)

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

Aaf(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 1eA01 ea Q A 1 mc2b 2i 3 ˇc

2 g

0_ga_{( 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_ga (2s21₎ , a4 2 i3 ˇc 2

u

e0 s Q e s Q e e0

v

, (2.14) where e0f_(2s21_{), 0}_{and e f ˘(2s}21_).

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 e0. 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

_d3 r c†nljm

u

e0 s Q e s Q e e0

v

cnljm4 4 2 i3 ˇc 2

_d3_{r [e} 0(W†W 1x†x) 1W†(s Q e) x 1x†(s Q e) W] . On introducing the large and small components, related to each other by

x 42iˇ c

s Q ˘

2 m W , (3.2)

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and integrating by parts, eq. (3.1) becomes d E4d E02 3 ˇ2 4 m

_d3_{r W}† [ (s Q e)(s Q ˘) 1 (sQ˘)(sQe) ] W , (3.3) where d E04 2 i 3 ˇc 2

_d3 r W†e0

y

1 2 ˇ2 4 m2 c2(s Q ˘) 2

z

W . (3.4)

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

d E4d E02 3 ˇ2 4 m

_d3 r W†_{[˜ Q e 12eQ˜] W ,} (3.5)

where for e f ˜(2s21_{) one must now substitute}

e 4 1 s2˜

o

1 2 NxON2 A2 4 2 1 2 s3_A2˜Nx O N2. (3.6)

It is now convenient to examine the term NxON2in detail. For particles of charge q that move in electromagnetic fields one has

NxON24

g

gq m

h

2 [2(EQb)2 1 NEN21 2 E Q (b 3 B) 1 NbN2NBN22 (b Q B)2] , (3.7) where g 41O

k

1 2NbN2 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

NxON24

g

e

d3_{r W}† 1 r6W , (3.12) d E24 2 4 K

v

appear in the average values only in the combinations x†1 /2x1 /24 1 and x21 /2 †

x21 /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

a0 L

h

l

e 2L/a0 1 11 Ei

g

2L a0

h

n

, (4.6) d E22 , 04 K 4 a06

mk

4

g

a0 L

h

2 2 10

g

a0 L

h

l

e 2L/a0 2 11 Ei

g

2L a0

h

n

, (4.7) where K/a06 A Z81.03 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 E2 , 0 4 K a06

k

e 2L/a0_. (4.9)

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

(4.10) d E2 , 14 d E12 , 11 d E22 , 14 4 6 K

L Q dr 1 r4[R21(r) ] 2 2 4 K

L Q dr R21(r) 1 r3 ¯ ¯rR21(r) 4 K a06 1 12

g

a0 L

h

e 2L/a0_.

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 EL4 d E2 , 02

d E2 , 1_{. The correction is given in our case by}

d EL4 d E2 , 02 d E2 , 14 K a06

k

e 2L/a0_. (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 Aa0in the present calculation. Then eq. (5.1) yields

d ELA 1Z80.0631 kHz . (5.2)

For the sake of completeness we also give the maximal-acceleration corrections to the 1 s ground state Lamb shift d E1 , 0

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are d E11 , 04 8 K a06

mk

a06

4 Z81.01 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 AP4

mPc3/ˇ, where mP4kˇc/G is Planck’s mass. In this case r04

k

Ze2/mAPAkZ 2.13 Q 10225_{m, and AP}

A 5.5 Q 1051m/s2and K/a06_{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

lP4

k

ˇG/c3A 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 ELA Z80.61 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 e04 (2s21), 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 ELfor 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

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TABLEI. – All values reported are in kHz. L a0 a0/2 a0/2.5 Theoretical value Experimental value d EL 0.0631 2.81 6.90 1 057 849(11) 1 057 851(4) d E1 , 0 2 d E2 , 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 r0given 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 LAa0/2. In fact for Z41 one finds d ELC2.70 kHz at L4a0/2 and d ELC6.98 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 E1, 02d E2, 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 EL and d E1, 02d E2, 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 LAa0/2 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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Maximal-acceleration corrections to the Lamb shift

of one-electron atoms (*)

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