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Original Research ARTICLE

Front. Phys., 23 October 2020 |

Stochastic Electrodynamics: Renormalized Noise in the Hydrogen Ground-State Problem

  • Institute for Theoretical Physics, University of Amsterdam, Amsterdam, Netherlands

The hydrogen ground-state problem is a touchstone for the theory of Stochastic Electrodynamics. Recently, we have shown numerically and theoretically that the H-atom self-ionizes after a characteristic time. In another approach, we reconsidered the harmonic oscillator and renormalized the stochastic force in order to suppress high-frequency tails so that all frequency integrals are dominated by the physical resonances. In the present work, we consider the regularization of the noise in the hydrogen ground-state problem. Several renormalization schemes are considered. Some are well-behaved, whereas in others the high frequency renormalization induces pathologies at low frequencies. In no situation did we find a way to escape from the previously signaled self-ionization.

1. Introduction

Stochastic electrodynamics (SED) is a classical theory that aims to explain quantum phenomena. Particles move in classical orbits. The basic assumption is the existence of a physical stochastic electromagnetic force that fills the universe and acts as an environment on charged particles and causes their quantum behavior at a statistical level. There is much literature on this field, and it can be summarized in the excellent books [1, 2].

The two celebrated touchstones of quantum physics, the harmonic oscillator [36] and the hydrogen problem [79], have received much attention within SED. The harmonic oscillator leads to a reasonable agreement, though not all details coincide. While the outcomes of various frequency integrals were routinely taken from their resonances, we have recently introduced a renormalization of the stochastic force such that high-frequency pathologies do not occur [10].

Our studies of the H-problem go back two decades. In [11], we showed how a classical phase space distribution can produce the shape of the quantum ground state, even Dirac's square-root shape, including relativistic corrections.

Stability of circular orbits was demonstrated by [1214]. The numerics of the hydrogen ground state were performed in 2002 by [15] with a modestly optimistic outlook. With the aim to reconsider the problem, new simulations were performed in our group in 2016. Various schemes for treating the stochastic force numerically were formulated analytically. Liska employed video cards and a modern computer code, speeding up the simulations significantly. They were carried out for the non-relativistic problem [16] and with the inclusion of relativistic corrections [17]. Many CPU hours were spent to achieve long run times and to incorporate many frequency modes. Ongoing findings of self-ionization led to simulation of a variety of formulations of the problem. The bottom line was that there was always self-ionization, suggesting that SED is not a basis for quantum mechanics.

On another track, Huang and Batelaan [18] reported that quantum interferences do not show up in the SED version of a double-slit-like quantum model.

Nieuwenhuizen [19] showed analytically for the H atom that there is a trend for self-ionization when the energy of the elliptic orbit is close to zero and the dimensionless angular momentum lies below a critical value of order unity, thus supporting the numerics and the non-recurrence of orbits found by [8].

The question of whether a proper definition of SED can describe the hydrogen atom is of fundamental interest. It is the purpose of the present work to reinspect stability in the hydrogen ground-state problem, inspired by our recent renormalization of the stochastic force for the harmonic oscillator. In section 2, we recall some properties of elliptic orbits in the Kepler problem. In section 3, we consider energy absorption from the stochastic field for various renormalizations of the force. We close with a discussion.

2. Kepler Orbits

We consider an electron bound to a nucleus with charge Ze and employ the notation of our recent work [19]. Lengths are expressed in terms of the Bohr radius ℏ/αZmec, times in the Bohr time /α2Z2mec2, speeds in the Bohr speed of αZc, energy in the Bohr energy α2Z2mec2, and angular momentum in terms of ℏ. Here, ℏ is the reduced Planck constant, α ≈ 1/137 the fine structure constant, Z the atomic number, me the electron mass, and c the speed of light.

We start recalling the essential details of the dynamics. In Bohr units the classical Newton equation reads

r¨=-rr3.    (2.1)

The Kepler orbit is solved in the parametric forms

r=1-ε cos ak2(cos ϕ, sin ϕ,0)   =(ca-ε,κ sa,0)k2.    (2.2)

Here, ϕ is the angle of the orbit with respect to the x-axis, a is a time-like parameter, ε is the ellipticity, and κ=1-ε2. Furthermore, ca is a shorthand for cos a and sa for sin a. The orbit lies on the ellipse

(k2x+ε)2+k4κ2y2=1    (2.3)

Its perihelion lies at r = 0, the location of the nucleus, and the aphelion at (−2ε/k2, 0, 0).

Time t and a second time s are parameterized as

t=τak3,     τa=a-ε sin a,s=τbk3,     τb=b-ε sin b.    (2.4)

For circular orbits (ε = 0), τa = a is a scaled time. In general, (2.4) exhibits an oscillation on top of this.

The angle ϕ is related to the variable a as

cϕ=ca-ε1-εca,     sϕ=κ sa1-εca,    (2.5)

and reads explicitly

ϕ=2 arctan(1+ε1-εtana2).    (2.6)

It exhibits the ongoing revolutions; for circular orbits (ε = 0) it equals ϕ = a = k3t. For general ε, the orbit and t are thus explicit in terms of a.

In Bohr units, the energy is E=-12k2 and κ = kL with L being the angular momentum in units of ℏ. The period reads P = 2π/k3. While the QM ground state corresponds to k = 1, in SED, k takes any value between 0 and ∞, that is, ranging from loosely to strongly bound, respectively. In the philosophy of SED, the time average of E produces the ground state energy E0=-12 as the average of E over the stationary distribution of E-values. Presuming that it exists, its form has been determined in [11].

Linear perturbations h to the Kepler orbit satisfy

h¨(t)=-W(t)·h(t),  W=1-3r^r^r3.    (2.7)

In [10], we presented a set of eigenmodes in the rotating frame. A linear combination of these solutions reads, in the laboratory frame,

h(1)(t)=1ρa(-sa,κca,0),h(2)(t)=2(ε-ca,-κ sa,0)+3τah(1)(t),h(3)(t)=12ρa(-κ s2a,3-4εca+c2a),h(4)(t)=κ2ρa(3-2εca-c2a,2εsa-s2aκ,0),h(5)(t)=(0,0,sa),h(6)(t)=(0,0,ca-ε).    (2.8)

The benefit of these modes is that the limits ε → 0 or κ → 0 to be taken in each of them.

The Greens function satisfies

G¨(t,s)+W(t)·G(s,t)=1 δ(t-s),G(t,s)+G(t,s)·W(s)=1 δ(t-s),    (2.9)

where dots denote derivatives to t and primes to s. Generally, it holds that

G.(t,t-)=-G(t,t-)=1,  G.(t,t-)=0.    (2.10)

Following the approach of [19], we verify that for s < t, the Greens function reads explicitly


while causality imposes G = 0 for st.

3. Stochastic Electrodynamics

In SED the Kepler orbit is perturbed by the stochastic electric field E and the damping D,

r¨=-rr3-βE+D,         (3.1)

The small parameter β is related to the fine structure constant

β=23α3/2ZZ1965.    (3.2)

with charge Z = 1 for hydrogen. The damping D(t) has been analyzed in full detail in [10]; Here, the standard approximation D=β2r suffices. The stochastic field satisfies

E(t)=-A.(t)=-C¨(t).    (3.3)

It has zero average and correlation functions

CEE(t-s)=E(t)E(s)=R6×1π(t-s-iτc)4,CAA(t-s)=A(t)A(s)=R-1π(t-s-iτc)2,CCE(t-s)=E(t)C(s)=R-1π(t-s-iτc)2,CCC(t-s)=C(t)C(s)                              =-1πR log ωc(t-s-iτc),    (3.4)

where τc=α2Z2 is the Compton time ℏ/mc in Bohr units and ωc~α3 log 1/α is a low frequency cutoff. These correlators are large at s = t.

The energy radiation is well-understood. Per revolution there is an energy loss

(ΔE)rad=-β2k5π3-κ2κ5.    (3.5)

The theme of the present work is the average energy gained from the field. It occurs at the rate

E.field=β2s0tdsE(t)·G.(t,s)·E(s)    (3.6)

where we must take s0 → −∞. Integrated over a period P = 2π/k3, it brings

ΔEfield=β2-P/2P/2dts0tds I1(t,s)    (3.7)


I1(t,s)=CEE(t-s) tr G.(t,s),    (3.8)

This expression has been studied in our previous work. The s-integral has potentially dangerous behavior at s = t where G.=1 and CEE(0) is very large. But the shape (3.4) of CEE implies that this high frequency effect has a vanishing contribution. Just leaving it out corresponds to a motivated short-time (ts) or high frequency renormalization. The remaining integrand

g.(t,s)=tr G.(t,s)-3=O[(t-s)4],    (3.9)

decays rapidly enough to set τc → 0 in CEE so that the integral is well behaved in this limit.

3.1. Short-Time Regularization

In our recent study of the harmonic oscillator we introduced a high-frequency regularization of the ultraviolet contributions [10]. Leaving out the subleading damping D, it amounts to replace EE¯, where the frequency components are related as

E¯ωω02ω2Eω=ω02Cω,    (3.10)

with the equality from E(t)=-C¨(t). At the resonance frequency ω = ω0, the E¯ω and Eω coincide. For nonlinear potentials this demands a generalization. The definition of G is G¨+W·G=1 δ(t-s) in the hydrogen problem, while W(t)1ω02 in the harmonic case. A natural and simple generalization is therefore

E¯(t)=W(t)·C(t).    (3.11)

Indeed, this reduces to (3.10) for the harmonic case. With E¯ instead of E inserted in (3.1), there will now appear in (3.7) the renormalized integrand

I2=E¯(t)·G.(t,s)·E¯(s)     =CCC(t-s) tr W(t)·G.(t,s)·W(s).    (3.12)

We also consider the expressions with one E and one E¯, which result in

I3=CCE(t-s) tr G.(t,s)·W(s),I4=CCE(t-s) tr W(t)·G.(t,s).    (3.13)

By partial integration we can generally relate the s-integral over I1 to one over I4.

-tds G.(t,s)R6π(t-s+iτc)4=-tds G.(t,s)W(s)R-1π(t-s+iτc)2.    (3.14)

In the boundary terms, we used G.(t,t)=0 and inserted G.(t,s)=-G.(t,s)·W(s). But when we do the same to relate I3 to I2, we cannot omit the boundary terms at large negative s0,

s0tds R-G.(t,s)π(t-s+iτc)2=G.(t,s0)π(t-s0)+G.(t,s0)log ω1(t-s0)π-s0tds G.(t,s)·W(s)log ω1(t-s)π,    (3.15)

where we took τc → 0 in the right-hand side. The main reason for the complication is that G(t, s), as well as its derivatives, contain an explicit factor ts arising from the secular part 3k3t h(1)(t) of the h(2)(t) mode, see (2.8). With the left-hand side of (3.15) well-behaved for s0 → −∞, it follows that the integral in the right hand side must have an s0 + s0 log |s0| divergency in this limit. This is confirmed by inspection and implies that the short-time regularization (2.8) creates a long-time divergency. It is related to the 1/ω2 factor in (3.10) and already led for the harmonic oscillator to a divergency; this was, however, subdominant. For the hydrogen problem it is more cumbersome and leads to an ill-defined leading order integral over I2. Similar computational methods of integral calculation have been used in other settings (see e.g., [20]).

Though CCC in Equation (3.4) involves a cutoff ωc, Equation (3.15) is valid for any ω1. But even the awkward choice ω1 ~ −1/s0 would not eliminate the boundary terms that regularize the integral.

3.2. Nearing the Self-Ionization

The important question of whether the H ground state is stable in SED is analyzed for orbits in the limit where E=-12k2 vanishes. In our previous works, we showed that this amounts to studying the orbits in the limit where κ = kL vanishes, at fixed L, in an order unity. From (3.5), one has the energy loss by radiation per orbit

(ΔE)rad-3πβ2L5.    (3.16)

To study this limit, the scaling a → κu, b → κv for κ → 0 is introduced, expressing that the main contribution, described by u and v of the order unity, comes from the part of the Kepler orbit near the pericenter at u = 0. Indeed, it holds that

r=L22(1-u2,2u,0), r=L22(1+u2).    (3.17)

Clearly, this part of the orbit is in its k → 0 limit, while the farthest point, lying at ((1 + ε)/k2, 0, 0) ≈ (2/k2, 0, 0), exhibits a self-ionization for k → 0. For further details of the method we refer to [19]. We reproduce its equations (2.24)–(2.26) for κ → 0 and multiplied by P,

ΔEfield(1)=1445πβ2L6-du-udv×2725+3u2+4(2+u2)uv+(u2-1)v2(1+u2)2(3+u2+uv+v2)4.    (3.18)

Continuing along these lines, we find that ΔEfield(3) is equal to this, while ΔEfield(4) comes out with the second line replaced by

15+20u2+3u4+4(5+8u2+u4)uv(1+u2)5(3+u2+uv+v2)2+5+u2+8u4+2(5+u2)uv+(u2-1)v2(1+u2)5(3+u2+uv+v2)2v2.    (3.19)

Its v-integral is linearly divergent with logarithms, as it is for ΔEfield(2). This all results in

ΔEfield(1)=1635β2L6ΔEfield(2)=divergentΔEfield(3)=1635β2L6ΔEfield(4)=divergent    (3.20)

The equality of the first and third case yields some justification for the renormalization method we investigated.

In case 1 and 3, the average total energy change per orbit thus comes out as

ΔE=3πβ2L6(Lc-L),  Lc=165π3=0.588057.    (3.21)

Orbits that have achieved a small k and L < Lc will gain energy on average, which explains the self-ionization observed in all our numerics.

3.3. Other Renormalization Schemes

The renormalization EĒ=W(t)·C(t) involves W=(1-3r^r^)/r3, of which the numerator has eigenvalues −2 and 1 (twice). One may wonder whether the “absolute value” |W|(1+r^r^)/r3, with the eigenvalues +2 and 1 (twice), fares better. Inspection shows that the divergence does not disappear; if anything, it becomes worse.

A renormalization with a broken power of |W| fares better at large times. One may replace E=-A. by E¯(t)=-|W|(t)·A(t) with the expression |W|=(1+(2-1)r^r^)/r3/2 squaring to |W|. Like (3.11), this approach softens the short time behavior, but it does not ruin the long time regime. This leads to a contribution to 〈E.field of the form

    -tds R-f(t,s)(t-s+iτc)2=f(t,t)|log τc|+-tds f(t,s) log(t-s)+O(τc).    (3.22)

Using G.=1 and G.=0 at s = t, the boundary term leads to

δE.field|log τc|=β22πddttr |W|=-6β22πr.r4.    (3.23)

It expresses energy gain (i.e., the electron becomes less bound, on the average) on the approach to the pericenter, and loss (becoming more bound) on departure. This cutoff dependence is unexpected. Nevertheless, when integrated over a full period, the effect averages out.

Next, we calculate, in analogy with (3.7), the energy gain per period. In the scaling limit, the t, s integrals become u, v integrals, of which the latter can be performed analytically. Its v = u boundary term vanishes upon u-integration, while the integral over the v = −∞ boundary term leads to a finite result,

ΔEfield=2.99842ΔEfield(1).    (3.24)

Hence it also leads to self-ionization.

The combination E¯=(1-x)E-x|W|·A involves from the AE and EA cross terms, a new contribution of the form

    -tds R-2f(t,s)(t-s+iτc)3=f(t,t) log τc+-tds f(t,s) log (t-s)+O(τc).    (3.25)

with a lengthy f having f (t, t) = 0. The log τc again drops out when integrated over a full period. After scaling, the v-integral can be performed analytically; now, the primitive for v → −∞ is odd in u, while the result comes from the v = u term. This ends up in

ΔEfield(x)=1635β2L6×[(1-x)2-0.876444(1-x)x+2.99842x2].    (3.26)

Its minimum at x = 0.295028 leads to Lcmin=0.33855, smaller than Lc = 0.58808 from (3.21). For all x, this still leads to self-ionization.

The above “absolute” value |W| looks unnatural, but it was necessary to define a real valued version of W. The third roots are real however:

W1/3=(1-(21/3+1)r^r^)/r,W2/3=(1+(22/3-1)r^r^)/r2,    (3.27)

It is easily verified that (W1/3)2 = W2/3 and (W1/3)3 = W. They thus permit the renormalization

EĒ=(1-x)W1/3·B1+xW2/3·B2,    (3.28)

for some real valued x, with the stochastic fields

B1=t-2/3E,      B2=t-4/3E,    (3.29)

defined by having eiωt frequency components

(B1)ω=Eω(-iω)2/3,     (B2)ω=Eω(-iω)4/3.    (3.30)

In the notation of [10], their correlation functions 〈Bi(t)Bj(s)〉 = 1Bij(ts) emerge as

B11(t)=-dω2π|ω|3|ω|4/3e-iωt-|ω|τc              =1πΓ8/3R1(it+τc)8/3,B22(t)=1πΓ4/3R1(it+τc)4/3, B12(t)=B21(-t)=1πRe-πi/3(it+τc)2.    (3.31)

The difficulty is again to deal with the singularities in the limit τc → 0. To proceed, we perform partial integrations. We introduce B11(3) and B22(1) to get

B11=B11(3),     B11(3)(t)=-2720πΓ8/3t1/3,B22=B.22(1),     B22(1)(t)=32πΓ4/3t-1/3,B12(t)=B21(t)=-12πt2,    (3.32)

where we took τc → 0. In view of (3.21) we define

Lcij=L63π-P/2P/2dts0tds Bij(t-s)Γij(t,s),Γij(t,s)=tr Wi/3(t)·G.(t,s)·Wj/3(s),    (3.33)

for i, j = 1, 2. For Lc11 we perform a partial integration w.r.t. s. Next we write the t-integral as the difference between two integrals starting at s0 and switch the t and s integrals. Then we do a partial integration w.r.t. t, switch back and do a final one w.r.t. s. This leads to a t, s integral over -Γ.11B11(3). One boundary term at t = s is non-trivial, namely

δLc11=3Γ8/3L610π2τc2/3-P/2P/2dsΓ11(s,s)           =-3(1-2-1/3)Γ8/3L65π2τc2/3-P/2P/2ds r(s)r(s)3.    (3.34)

This vanishes again since it involves a total derivative integrated over a full period. But the integrand itself is moderately large, so that, as before, the average rate of energy exchange with the field results in gain on approach to the pericenter and loss on departure. While weakened by the prefactor and canceling over a period, this cutoff dependence is unexpected.

For Lc22 we perform a partial integration w.r.t. s and evaluate the double integral in the limit τc → 0. The boundary term at s = t vanishes identically. With Γ12 ~ Γ21 ~ (ts)2 for st, the Lc12 and Lc21 integrands are already regular for τc → 0.

We are interested in these results in the scaling limit κ = kL → 0 at fixed L. The resulting integrals are of the type (3.18). Numerical evaluation yields

Lc11=8.5191,     Lc22=2.1944,Lc12=0.3182,     Lc21=-0.5615.    (3.35)

The combined Lc corresponding to (3.28) reads

Lc11(1-x)2+(Lc12+Lc21)x(1-x)+Lc22x2.    (3.36)

It has a minimum at x = 0.7886,

Lcmin=1.7048,    (3.37)

which sets the boundary for self-ionizing orbits because (3.36) exceeds this for other x-values.

4. Discussion

Previous studies, both analytical and numerical, have pointed out that the hydrogen problem in Stochastic Electrodynamics leads to a self-ionization of the electron. The present work investigates whether “easy fixes” of the stochastic force may improve the situation. We consider a short time or high frequency renormalization of the stochastic force that we recently proposed for the harmonic oscillator problem and generalized it for the hydrogen ground-state problem. To achieve this, we consider several options, of which some do, and some do not, lead to a well-defined approach. We find that the renormalization does not help to stabilize the situation, and that its impact on long time behavior actually makes the situation worse.

Next, we study various further renormalization schemes which lead to well behaved dynamics, but neither heal the self-ionization problem. Our approach generally puts forward that stability of orbits with energy near E = 0 can only be achieved for a scheme in which the parameter Lc in (3.21) vanishes. On physical grounds one expects that it can be proven that this quantity is positive. However, we are not aware of such a proof, not even in the scaling limit E → 0.

In our view, the problem does not lie in the Kepler orbits but in the close enough approach to the nucleus where a relatively high amount of energy is absorbed from the stochastic force. Indeed, Kepler orbits can be stable in SED. Nieuwenhuizen [19] adds an L02/2r2 potential to the −1/r Newton potential. It induces an effective angular momentum Leff=(L2+L02)1/2, which, if L0 ≳ 6 is large enough, leads to a stable system without self-ionization. Then Leff, and with it the distance between the pericenter and the nucleus, is large enough to prevent orbits that keep on gaining energy on the average.

In the absence of such an extra potential, we confirm previous findings that the hydrogen self-ionizes in Stochastic Electrodynamics. When the orbit has nearly zero energy and the angular momentum lies below some critical value, then, on the average, more energy gets absorbed from the field than is radiated away, making the orbit more and more delocalized so that ultimately self-ionization occurs. To circumvent this, a fundamental reformulation of Stochastic Electrodynamics seems to be necessary.

Author Contributions

The author confirms being the sole contributor of this work and has approved it for publication.

Conflict of Interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


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Keywords: stochastic electrodynamics, hydrogen problem, hydrogen ground state, self-ionization, renormalization

Citation: Nieuwenhuizen TM (2020) Stochastic Electrodynamics: Renormalized Noise in the Hydrogen Ground-State Problem. Front. Phys. 8:335. doi: 10.3389/fphy.2020.00335

Received: 09 March 2020; Accepted: 20 July 2020;
Published: 23 October 2020.

Edited by:

Ana Maria Cetto, Universidad Nacional Autónoma de México, Mexico

Reviewed by:

Luis De La Peña, National Autonomous University of Mexico, Mexico
Yilun Shang, Northumbria University, United Kingdom
Daniel C. Cole, Boston University, United States

Copyright © 2020 Nieuwenhuizen. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Theo M. Nieuwenhuizen,