Hydrogen Atom Spectrum and the Lamb Shift in Noncommutative QED

A. Tureanu; M. Chaichian; M. M. Sheikh-Jabbari

REVIEW

Hydrogen Atom Spectrum and the Lamb Shift in Noncommutative QED

M. Chaichian, M. M. Sheikh-Jabbari, A. Tureanu

High Energy Physics - Theory · Submitted 20 Oct 2000 · 18 references

Pith's verdict: The tree-level jz splitting in Eq. (3.2) is clean and worth citing; the numerical bound at the end of §IV is internally inconsistent by orders of magnitude and should not be quoted until it is redone.

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CitationA. Tureanu, M. Chaichian, M. M. Sheikh-Jabbari. "Hydrogen Atom Spectrum and the Lamb Shift in Noncommutative QED." arXiv:hep-th/0010175 (2000). https://pith.science/paper/O6EWEFJS

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Editor's note

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The thing worth taking from this paper is Eq. (3.2): a tree-level, jz-dependent shift of the 2P_1/2 level that splits the Lamb line into two components proportional to theta. It falls straight out of the Bopp shift applied to the Coulomb potential, and the physics is transparent — the noncommutative electron carries an electric dipole moment (e/2 hbar) theta x p, and hydrogen is simply a place to look for it. The angular momentum algebra is standard and the coefficient checks: for n=2, l=1, jz=1/2 it gives about -5.0e-6 eV times theta/lambda_e^2, which is right. The Stark cancellation in §V is also correct and neatly argued, and the observation that the tree-level shift mimics spin-orbit with theta/lambda_e^2 in place of S/hbar is a good way to frame it. The one-loop ingredients are imported from the published [4], which is fine; the combination is what is new here, not the pieces. On the reader's worry about integrating the theta-expansion down to r=0: it does not land. For l=1 the integrand of <1/r^3> goes as r dr near the origin, so the region r < sqrt(theta) contributes a fraction of order theta/a_0^2, roughly 1e-18 for any theta in the perturbative regime. The paper's one-line dismissal in §III is substantively right and should have been quantified, but it is not a flaw in the result. Same for the one-loop integral in (4.4)-(4.5): dI/dtheta = 1/(2 pi r theta) is stated and it is correct, so that is a sketch, not an assertion. The real problem is the bound at the end of §IV, the paper's only falsifiable number. The two printed forms are not the same bound: theta/lambda_e^2 <~ 1e-7 alpha is about 7e-10, while theta <~ (1e4 GeV)^-2 is about 2.6e-3 in the same units — six orders of magnitude apart, and the second is barely a bound at all since it exceeds lambda_e^2. Neither matches the stated 1e-6 to 1e-7 criterion. My own arithmetic from (3.2) against the 4.37e-6 eV Lamb shift gives Delta E_NC/Delta E_Lamb ~ 1.15 theta/lambda_e^2, so the criterion implies theta/lambda_e^2 <~ 1e-6, i.e. theta <~ (1e6 GeV)^-2. That is a third number. The bound needs redoing before anyone quotes it. Minor: the Zeeman shift (5.5) has no state dependence, so it shifts every level equally and is not a spectroscopic effect; and the (3 - 2/3) in (4.2) is imported without showing the combination. Who should read it: people working on noncommutative phenomenology who want a concrete atomic observable, and anyone who needs a worked example of the Bopp shift in a bound-state problem. The tree-level part is worth citing; the bound is not. Recommendation: send it out, but the §IV bound has to be fixed first.

Referee report

Significance

The central observation is clean and, if correct, useful: space-space noncommutativity induces a tree-level, j_z-dependent, spin-orbit-like perturbation of atomic levels, so that atomic spectroscopy becomes a probe of θ without requiring loop-level precision. The derivation from the Bopp shift is standard and self-contained, the angular-momentum algebra leading to Eq. (3.2) is elementary and checkable, and the prediction is in principle falsifiable: a polarized splitting of the 2P_{1/2} line whose magnitude scales with θ/λ_e². The paper is one of the early attempts to connect NCQED to a precision atomic observable, and that program is worth pursuing. The weakness is that the paper's only quantitative, falsifiable number — the bound on θ at the end of §IV — is not internally consistent as printed, which is what prevents the phenomenological claim from being usable in its present form.

Major objections (3)

[§IV, final paragraph (bound on θ)] The bound is stated in two forms that are not equivalent, and neither follows from the criterion the authors themselves state. The text requires the tree-level contribution of Eq. (3.2) to be '10^-6 − 10^-7 smaller than the usual one loop result', and then quotes 'θ/λ_e² ≲ 10^-7 α' and 'θ ≲ (10^4 GeV)^-2'. Taking Eq. (3.2) with n = 2, l = 1, j_z = 1/2 (f_{2,1} = 1/24, j_z(1 − 1/3) = 1/3) gives ΔE_NC ≈ −(m_e c² α⁴/288)(θ/λ_e²) ≈ −5.1×10^-6 eV × (θ/λ_e²), while the Lamb shift is 4.37×10^-6 eV, so ΔE_NC/ΔE_Lamb ≈ 1.2 (θ/λ_e²). The stated 10^-6–10^-7 criterion therefore implies θ/λ_e² ≲ 10^-6–10^-7, i.e. θ ≲ (1 GeV)^-2, whereas 'θ/λ_e² ≲ 10^-7 α' corresponds to θ/λ_e² ≈ 7×10^-10 and 'θ ≲ (10^4 GeV)^-2' corresponds to θ/λ_e² ≈ 3×10^-15. The two printed forms differ from each other by about five orders of magnitude in θ/λ_e², and both are far weaker than what the stated criterion gives. Since this bound is the paper's only quantitative phenomenological output, the authors should recompute it explicitly from Eq. (3.2), show the numerical comparison with the measured Lamb shift, and quote a single self-consistent number.
[§IV.B, Eqs. (4.5)–(4.6)] The θ-dependent logarithm in Eq. (4.5), of order α ln(θΛ²), is numerically much larger than the (Zα)² θ/λ_e² term that actually enters the bound, and it is set aside on the grounds that it 'can be understood as the normalization of charge at one loop level'. That step is asserted rather than demonstrated: the physical coupling is not defined here through the Thomson limit, and the cutoff Λ that appears inside the logarithm is never assigned a value, so ln(θΛ²) is not even numerically specified. The paper defers the Thomson-limit analysis to Ref. [18], which is listed as 'Work in preparation'. As it stands, the reader cannot tell whether the θ-dependent log contributes to the Lamb shift, nor whether the bound quoted at the end of §IV is dominated by the term the authors discard. The authors should either carry out the renormalization argument in this paper or state explicitly that the bound is conditional on it.
[§IV.B, Eq. (4.4) and the evaluation of I(r,θ)] The θ-dependent integral in Eq. (4.4) is evaluated in one line by asserting dI/dθ = 1/(2πrθ) and hence I(θ,r) = (1/2πr) ln(θΛ²), with the range 1/(Λ√θ) ≲ q ≲ Λ invoked from IR/UV mixing. The differentiation step, the constant of integration, and the claim that the integral is dominated by that momentum window are all left implicit. Because this integral is the sole source of the leading one-loop θ-dependence in Eq. (4.5), the derivation should be shown, at minimum the q-integral performed with the stated window and the cutoff dependence exhibited explicitly.

Minor objections (7)

[§III, paragraph following Eq. (3.2)] The authors correctly note that the θ-expansion fails for r ≲ √θ and dismiss the issue because the l ≠ 0 wave functions vanish at the origin. The dismissal is substantively right, but it would be more convincing if quantified: for l = 1 the integrand of ⟨1/r³⟩ behaves as r dr near the origin, so the region r < √θ contributes a relative amount of order θ/a_0², which is negligible. A one-line estimate of this kind would close the point.
[§V, Zeeman effect, Eqs. (5.4)–(5.5)] The noncommutative correction to the Zeeman energy in Eq. (5.5) is proportional to θ·B and carries no dependence on the atomic state, so it shifts all levels of a given atom equally and does not produce a Zeeman splitting in transition frequencies. The authors should either state that this is a state-independent shift and hence not an observable Zeeman effect, or identify the state-dependent piece that would make it one.
[§IV.A, Eq. (4.2)] The numerical factor (3 − 2/3) in Eq. (4.2) is not derived in this paper. It presumably follows from combining the vertex and propagator contributions of Ref. [4], but the reader is left to reconstruct it; a short indication of its origin would help.
[§III, discussion of the polarized Lamb shift] The statement that noncommutativity 'increases the widths and splits the Lamb shift line' conflates two different effects. A splitting of the 2P_{1/2} sublevels produces two lines of the same intrinsic width; if the experimental resolution does not separate them, the observed line is broadened, but that is a statement about resolution rather than about the theory. The wording should be tightened.
[§IV.B, Eq. (4.4)] The factors of ħ in the argument of the logarithm, ln(q²q̃²/ħ⁴), and the overall normalization after 'reintroducing ħ, c factors' should be checked and stated explicitly, since the dimensional bookkeeping in this equation is not transparent.
[§IV, comparison with experiment] The paper compares its correction to 'the usual one loop result' without quoting the numerical value of the Lamb shift or the experimental uncertainty used. Giving the measured value and its error would make the bound reproducible.
[§II, Hamiltonian] The proton is treated as infinitely heavy and noncommutative effects in the proton are neglected without comment. A sentence on the reduced-mass and proton contributions would clarify the level of approximation.

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The paper

The paper claims

If space coordinates fail to commute, hydrogen's Lamb shift splits into two direction-dependent lines, and the size of the split bounds the noncommutativity scale.

The reading

The paper asks what happens to the hydrogen atom if the coordinates of space themselves do not commute, $[x_i,x_j]=iθ_{ij}$, as suggested by string theory in a background magnetic field. The authors take the ordinary Coulomb Hamiltonian, rewrite it in noncommutative coordinates, and show that to first order in $θ$ the potential picks up a term proportional to $L·θ/r^3$ — formally identical to spin-orbit coupling with the noncommutativity vector playing the role of spin. The consequence they want to establish is that hydrogen gets energy shifts already at tree level, before any loop is drawn, and that these shifts depend on the magnetic quantum number $j_z$, so the familiar $2P_{1/2}→2S_{1/2}$ Lamb transition is no longer a single line but a pair of lines whose separation grows with $θ$. They then add the one-loop noncommutative corrections to the vertex and the photon propagator and compare the total against measured Lamb shift data to extract a bound, $θ/λ_e^2 ≲ 10^{-7}α$, or $θ ≲ (10^4$ GeV$)^{-2}$. The appeal of the argument is that it turns a piece of high-energy speculation into something an atomic physics experiment could in principle see.

What carries the argument

The load-bearing device is the variable redefinition $x_i = x̂_i + θ_{ij}p̂_j/2ħ$, $p_i = p̂_i$, which maps the noncommutative algebra $[x̂_i,x̂_j]=iθ_{ij}$, $[x̂_i,p̂_j]=iħδ_{ij}$ onto the ordinary canonical algebra. Because the transformation is a canonical one, the Hilbert space and the wave functions are unchanged, and the entire effect of noncommutativity is pushed into the argument of the potential. Expanding the Coulomb potential in the shifted coordinates then produces the $L·θ/r^3$ term that carries all the spectrum results, with the same structure as the spin-orbit interaction that Landau and Lifshitz tabulate. The claim that this potential is the correct non-relativistic limit is supported separately by the tree-level noncommutative QED vertex $Γ^μ = e^{i p·θ·p'/2ħ^2}γ^μ$, whose first-order term is an electric dipole moment coupling to the field.

Extended reading notes

Core claim

The central claim is that space-space noncommutativity leaves a tree-level imprint on the hydrogen spectrum, and that this imprint is directional. Shifting to the commutative variables $x_i = x̂_i + θ_{ij}p̂_j/2ħ$ turns the noncommutative Coulomb potential into $-Ze^2/r - Ze^2 L·θ/(4ħ r^3)$ up to $O(θ^2)$, which is the spin-orbit form with $θ$ where the spin would be. Evaluating this in the usual Coulomb wave functions gives the shift of Eq. (3.2), proportional to $(Zα)^4$, to $θ/λ_e^2$, and to $j_z(1 ∓ 1/(2l+1))$, so that the $2P_{1/2}$ level moves by different amounts for $j_z = ±1/2$ while the $2S_{1/2}$ level does not move at all. The authors call the resulting splitting of the Lamb line a polarized Lamb shift, and note that it opens a new transition channel $2P^{-1/2}_{1/2} → 2P^{+1/2}_{1/2}$ that has no commutative counterpart. Adding the one-loop vertex correction, driven by the noncommutative electric dipole moment $⟨P⟩ = -e(θ×p)/4ħ$, and the one-loop photon-propagator correction, which contributes a $θ$-dependent $1/r$ piece and a conventional $δ^3(r)$ piece, yields Eq. (4.6). Comparing the tree-level term with the measured Lamb shift gives the bound quoted above. The paper also works out two external-field cases: the noncommutative contribution to the Stark effect vanishes at tree level because the relevant commutator with $H_0$ has no diagonal matrix element, while the Zeeman effect acquires a term proportional to $θ·B$.

Load-bearing premise

The expansion of the Coulomb potential to first order in $θ$ is integrated all the way down to $r=0$ when computing $⟨1/r^3⟩$, even though the authors state that the expansion fails for $r$ below $√θ$; their one-line justification that $l≠0$ wave functions vanish at the origin does not show that the coefficient of the leading term survives cutting the integral off at $√θ$.

Editorial extensions

If this is right

The $2P_{1/2}→2S_{1/2}$ Lamb transition becomes two lines separated by an amount proportional to $θ$ and to $j_z$, so high-resolution spectroscopy of that line is a direct search for space noncommutativity.
The same mechanism should shift the fine structure of any hydrogenic ion, scaling as $(Zα)^4$, so heavier ions would amplify the effect.
The noncommutative Stark shift vanishes at tree level, so an external electric field is not a useful lever, while an external magnetic field couples directly through $θ·B$ in the Zeeman term of Eq. (5.5).
The bound $θ ≲ (10^4\ \mathrm{GeV})^{-2}$ follows from existing Lamb shift data and can be tightened by more precise measurements or by other processes such as $e^+e^-$ scattering.
Because $θ$ picks a preferred direction in space, the effect is anisotropic: its size depends on the orientation of the atom relative to $θ$, which is itself a signature to look for.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The jz-dependent splitting is a cleaner experimental target than the overall Lamb shift bound, because a line-shape measurement looking for a resolved doublet does not require the enormous precision needed to detect a small uniform shift.
The same canonical shift that generates $L·θ/r^3$ should also modify the Darwin term and the relativistic kinetic corrections at the same order in $θ$, which the paper does not compute; a complete treatment would need those to make the bound quantitative.
If the effect is real, the anisotropy means that an atomic clock or a hydrogen maser would show a sidereal-time modulation as Earth rotates relative to the $θ$ direction, which is the standard signature used in analogous Lorentz-violation searches.
The paper's bound is weak compared with what a dedicated precision measurement could reach; the interesting question is whether the polarized Lamb shift is the most sensitive atomic observable or merely the first one that was easy to compute.

Original abstract

We have calculated the energy levels of the hydrogen atom and as well the Lamb shift within the noncommutative quantum electrodynamics theory. The results show deviations from the usual QED both on the classical and on the quantum levels. On both levels, the deviations depend on the parameter of space/space noncommutativity.

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We have calculated the energy levels of the hydrogen atom and as well the Lamb shift within the noncommutative quantum electrodynamics theory. The results show deviations from the usual QED both on the classical and on the quantum levels. On both levels, the deviations depend on the parameter of space/space noncommutativity.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We have calculated the energy levels of the hydrogen atom and as well the Lamb shift within the noncommutative quantum electrodynamics theory.	contribution 1.00	0.96	0.13	asserted 1.00
s1	The results show deviations from the usual QED both on the classical and on the quantum levels.	result 0.99	0.92	0.67	asserted 1.00
s2	On both levels, the deviations depend on the parameter of space/space noncommutativity.	result 0.94	0.85	0.28	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.83.

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References

Works cited (18)

[4]

I. F. Riad, M. M. Sheikh-Jabbari, Noncommutative QED and Anomalous Dipole Moments , JHEP 0008 (2000) 045

View on Pith 2000
[18]

Work in preparation. 4

View on Pith
[1]

N. Seiberg, E. Witten, String Theory and Noncommu- tative Geometry , JHEP 09 (1999) 032, and references therein

View on Pith 1999
[2]

L. Alvarez-Gaume, S. R. Wadia, Gauge Theory on a Quantum Phase Space , hep-th/0006219

View on Pith Pith review arXiv
[3]

A. Micu, M. M. Sheikh-Jabbari, Noncommutative Φ4 Theory at Two Loops , hep-th/0008057

View on Pith Pith review arXiv
[5]

I. Ya. Aref’eva, D. M. Belov, A. S. Koshelev, Two-Loop Diagrams in Noncommutative φ44 Theory, Phys. Lett. B476 (2000) 431

View on Pith 2000
[6]

M. Hayakawa, Perturbative analysis on infrared and ul- traviolet aspects of noncommutative QED on R4, hep- th/9912167

View on Pith
[7]

C. P. Martin, F. R. Ruiz, Paramagnetic Dominance, the Sign of the Beta Function and UV/IR Mixing in Non- commutative U(1) , hep-th/0007131

View on Pith Pith review arXiv
[8]

J. Gomis, T. Mehen, Space-Time Noncommutative Field Theories And Unitarity , hep-th/0005129. M. Chaichian, A. Demichev, P. Preˇ snajder, A. Tureanu, Space-Time Noncommutativity, Discreteness of Time and Unitarity , hep-th/0007156

View on Pith Pith review arXiv
[9]

F. Ardalan, H. Arfaei, M. M. Sheikh-Jabbari, Non- commutative Geometry From Strings and Branes , JHEP 9902 (1999) 016

View on Pith 1999
[10]

P-M. Ho, S-C. Chu, Noncommutative Open String and D-brane, Nucl. Phys. B550 (1999) 151

View on Pith 1999
[11]

M. M. Sheikh-Jabbari, Discrete Symmetries (C, P, T) in Noncommutative Field Theories , Phys. Rev. Lett. 84 (2000) 5265

View on Pith 2000
[12]

I. Mocioiu, M. Pospelov, R. Roiban, Low energy Limits on the Antisymmetric Tensor Field Background on the Brane and on the Noncommutativity Scale , Phys. Lett. B489 (2000) 390

View on Pith 2000
[13]

N. Chair, M. M. Sheikh-Jabbari, Pair Production by a Constant External Field in Noncommutative QED , hep- th/0009037

View on Pith
[14]

M. M. Sheikh-Jabbari, Open Strings in a B-ﬁeld Back- ground as Electric Dipoles , Phys. Lett. B455 (1999) 129

View on Pith 1999
[15]

L. D. Landau, E. M. Lifshitz , Quantum mechanics: non- relativistic theory, Pergamon Press, 1977

View on Pith 1977
[16]

C. Itzykson, J-B. Zuber, Quantum Field Theory , Mc Graw-Hill, 1980

View on Pith 1980
[17]

H. D. Abarbanel, M. L. Goldberger, Phys. Rev. 165 (1968) 1594

View on Pith 1968

How the literature uses it

32 arXiv papers cite this in 36 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 20 sentences without a confident label

Uses its method or tool 1 paper

To move along, one introduces the following variables [ref] [equation] It can be easily checked that the above variables will satisfy ([ref]) if the unprimed variables satisfy the ordinary Poisson brackets.

A cosmological viewpoint on the correspondence between deformed phase-space and canonical quantization · Phase-space deformation: a procedure for quantization · 0.89

Compares with it 1 paper

This is of the same order as the bounds arising from Lamb shift and bounds on Lorentz violation [ref] .

Quantum Space-Time and Noncommutative Gauge Field Theories · NC SM based on the no-go theorem · 0.98

Cites it as background 14 papers

The most extensively studied noncommutative space-time is described by the algebra $[X_μ, X_ν] = iθ_μν$, where $θ_μν$ a totally anti-symmetric constant tensor [ref] .

Influences of the coordinate dependent noncommutative space on charged and spin currents · Introduction · 0.97
In last couple of years, the implications of noncommutativity, and in particular that of the Hopf algebra structure of symmetries of such space-time are being studied [ref] .

Newton's Equation on the kappa space-time and the Kepler problem · Introduction · 0.96
The quantum mechanical systems defined on noncommutative spaces [ref] have recently attracted a considerable interest [ref] .[Also noncommutative field theories have been intensely studied [ref] , but they are not the subject of our present discussion.] The simplest such system involves only two dynamic variables $x,y$ with a nontrivial commutator xy-comm [x,y] = iθ .

Comments on noncommutative quantum mechanical systems associated with Lie algebras · Introduction · 0.98

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [4]

I. F. Riad, M. M. Sheikh-Jabbari, Noncommutative QED and Anomalous Dipole Moments , JHEP 0008 (2000) 045

View on Pith 2000

[2] [18]

Work in preparation. 4

View on Pith

[3] [1]

N. Seiberg, E. Witten, String Theory and Noncommu- tative Geometry , JHEP 09 (1999) 032, and references therein

View on Pith 1999

[4] [2]

L. Alvarez-Gaume, S. R. Wadia, Gauge Theory on a Quantum Phase Space , hep-th/0006219

View on Pith Pith review arXiv

[5] [3]

A. Micu, M. M. Sheikh-Jabbari, Noncommutative Φ4 Theory at Two Loops , hep-th/0008057

View on Pith Pith review arXiv

[6] [5]

I. Ya. Aref’eva, D. M. Belov, A. S. Koshelev, Two-Loop Diagrams in Noncommutative φ44 Theory, Phys. Lett. B476 (2000) 431

View on Pith 2000

[7] [6]

M. Hayakawa, Perturbative analysis on infrared and ul- traviolet aspects of noncommutative QED on R4, hep- th/9912167

View on Pith

[8] [7]

C. P. Martin, F. R. Ruiz, Paramagnetic Dominance, the Sign of the Beta Function and UV/IR Mixing in Non- commutative U(1) , hep-th/0007131

View on Pith Pith review arXiv

[9] [8]

J. Gomis, T. Mehen, Space-Time Noncommutative Field Theories And Unitarity , hep-th/0005129. M. Chaichian, A. Demichev, P. Preˇ snajder, A. Tureanu, Space-Time Noncommutativity, Discreteness of Time and Unitarity , hep-th/0007156

View on Pith Pith review arXiv

[10] [9]

F. Ardalan, H. Arfaei, M. M. Sheikh-Jabbari, Non- commutative Geometry From Strings and Branes , JHEP 9902 (1999) 016

View on Pith 1999

[11] [10]

P-M. Ho, S-C. Chu, Noncommutative Open String and D-brane, Nucl. Phys. B550 (1999) 151

View on Pith 1999

[12] [11]

M. M. Sheikh-Jabbari, Discrete Symmetries (C, P, T) in Noncommutative Field Theories , Phys. Rev. Lett. 84 (2000) 5265

View on Pith 2000

[13] [12]

I. Mocioiu, M. Pospelov, R. Roiban, Low energy Limits on the Antisymmetric Tensor Field Background on the Brane and on the Noncommutativity Scale , Phys. Lett. B489 (2000) 390

View on Pith 2000

[14] [13]

N. Chair, M. M. Sheikh-Jabbari, Pair Production by a Constant External Field in Noncommutative QED , hep- th/0009037

View on Pith

[15] [14]

M. M. Sheikh-Jabbari, Open Strings in a B-ﬁeld Back- ground as Electric Dipoles , Phys. Lett. B455 (1999) 129

View on Pith 1999

[16] [15]

L. D. Landau, E. M. Lifshitz , Quantum mechanics: non- relativistic theory, Pergamon Press, 1977

View on Pith 1977

[17] [16]

C. Itzykson, J-B. Zuber, Quantum Field Theory , Mc Graw-Hill, 1980

View on Pith 1980

[18] [17]

H. D. Abarbanel, M. L. Goldberger, Phys. Rev. 165 (1968) 1594

View on Pith 1968