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

Abstract

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.

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

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 of Cetto and de la Peña.

The two celebrated touchstones of quantum physics, the harmonic oscillator and the hydrogen problem, 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.

Our studies of the H-problem go back two decades. In earlier work, 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 de la Peña and by Puthoff. The numerics of the hydrogen ground state were performed in 2002 by Cole and Zou 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 and with the inclusion of relativistic corrections. 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 reported that quantum interferences do not show up in the SED version of a double-slit-like quantum model.

Nieuwenhuizen 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 Claverie and Soto.

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. Lengths are expressed in terms of the Bohr radius, times in the Bohr time, speeds in the Bohr speed, energy in the Bohr energy, and angular momentum in terms of the reduced Planck constant. Here the fine structure constant is approximately 1/137, Z is the atomic number, and the electron mass and the speed of light complete the units.

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

r̈ = −r / r³.

The Kepler orbit is solved in the parametric forms given as Equation (2.2) in the source, in which the angle of the orbit with respect to the x-axis, a time-like parameter, the ellipticity, and the quantity κ equal to the square root of one minus the ellipticity squared all appear. The orbit lies on an ellipse (Equation 2.3). Its perihelion lies at the location of the nucleus, and its aphelion on the negative x-axis at twice the ellipticity divided by the squared binding parameter.

Time and a second time are parameterized through the relation between the true time and the time-like parameter, the parameter minus the ellipticity times its sine (Equation 2.4). For circular orbits this scaled time equals the parameter itself; in general the relation exhibits an oscillation on top of it. The angle of the orbit is related to that parameter by the standard tangent half-angle formula of the Kepler problem (Equations 2.5 and 2.6). It exhibits the ongoing revolutions; for circular orbits it equals the parameter, which is the binding parameter cubed times the time. For general ellipticity, the orbit and the time are thus explicit in terms of that parameter.

In Bohr units, the energy is minus one half of the squared binding parameter, and κ equals the binding parameter times the angular momentum in units of the reduced Planck constant. The period is 2π divided by the binding parameter cubed. While the quantum-mechanical ground state corresponds to a binding parameter of one, in SED it takes any value between zero and infinity, that is, ranging from loosely to strongly bound. In the philosophy of SED, the time average of the energy produces the ground state energy of minus one half as the average of the energy over the stationary distribution of energy values. Presuming that it exists, its form has been determined in our earlier work.

Linear perturbations to the Kepler orbit satisfy a second-order equation driven by the curvature matrix W, which equals the identity minus three times the outer product of the unit radius vector with itself, all divided by the cube of the radius (Equation 2.7). In earlier work we presented a set of eigenmodes in the rotating frame; a linear combination of these solutions in the laboratory frame is listed as the six modes of Equation (2.8). The benefit of these modes is that the limits of vanishing ellipticity or vanishing κ can be taken in each of them. The Greens function satisfies the pair of equations (2.9), with dots denoting derivatives to the first time and primes to the second, and with the standard jump conditions (2.10). Following the approach of our earlier paper, we verify that for the second time earlier than the first, the Greens function reads explicitly as a sum over the odd-indexed modes of the antisymmetric combinations divided by the binding parameter cubed, while causality imposes that it vanishes otherwise.

3. Stochastic Electrodynamics

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

r̈ = −r / r³ − βE + D.

The small parameter β is related to the fine structure constant,

β = the square root of two-thirds, times the fine structure constant to the power three halves, times Z, which is approximately Z / 1965,

with charge Z equal to one for hydrogen. The damping has been analyzed in full detail in our earlier work; here the standard approximation in which the damping is the coupling squared times the third time derivative of the position suffices. The stochastic field satisfies

E(t) = −Ȧ(t) = −C̈(t).

It has zero average and correlation functions given in Equation (3.4): the field-field correlator is the real part of six times the identity divided by π times the fourth power of the time difference shifted by minus i times the Compton time; the potential-potential correlator is the real part of minus one divided by π times the square of the same shifted time difference; the mixed correlator has the same form; and the doubly integrated correlator is minus one over π times the real part of the logarithm of a low-frequency cutoff times that shifted time difference. Here the Compton time in Bohr units is the fine structure constant squared times Z squared, and the low frequency cutoff is of order the fine structure constant cubed times the logarithm of its inverse. These correlators are large at coincident times.

The energy radiation is well-understood. Per revolution there is an energy loss equal to minus the coupling squared, times the binding parameter to the fifth, times π, times the quantity three minus κ squared, divided by κ to the fifth (Equation 3.5).

The theme of the present work is the average energy gained from the field. It occurs at a rate given by the coupling squared times the time integral of the correlation of the stochastic field with the Greens function derivative acting on the field at the earlier time (Equation 3.6), where the lower limit is taken to minus infinity. Integrated over a period, it brings the energy gain per period as a double integral over the integrand named I1 (Equations 3.7 and 3.8), which is the field-field correlator times the trace of the Greens function derivative.

This expression has been studied in our previous work. The second-time integral has potentially dangerous behavior where the two times coincide, since there the Greens function derivative is the identity and the field-field correlator is very large. But the shape of that correlator implies that this high frequency effect has a vanishing contribution. Just leaving it out corresponds to a motivated short-time, or high frequency, renormalization. The remaining integrand, the trace of the Greens function derivative minus three, is of fourth order in the time difference (Equation 3.9) and decays rapidly enough to set the Compton time to zero in the correlator 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. Leaving out the subleading damping, it amounts to replacing the field by a renormalized field whose frequency components are the original ones multiplied by the squared resonance frequency divided by the squared frequency (Equation 3.10). At the resonance frequency the two coincide. For nonlinear potentials this demands a generalization. The definition of the Greens function in the hydrogen problem carries the curvature matrix W where the harmonic case carries the squared resonance frequency times the identity. A natural and simple generalization is therefore

Ē(t) = W(t)·C(t),

which indeed reduces to the harmonic case. With the renormalized field inserted in the equation of motion, there will now appear the renormalized integrand I2, equal to the doubly integrated correlator times the trace of W at the first time, the Greens function derivative, and W at the second time (Equation 3.12). We also consider the expressions with one renormalized and one bare field, which result in the integrands I3 and I4 (Equation 3.13).

By partial integration we can generally relate the second-time integral over I1 to one over I4 (Equation 3.14). In the boundary terms, we used the vanishing of the mixed derivative of the Greens function at coincident times and inserted the identity relating its second derivative to the curvature matrix. But when we do the same to relate I3 to I2, we cannot omit the boundary terms at large negative initial time (Equation 3.15), where we took the Compton time to zero on the right-hand side.

The main reason for the complication is that the Greens function, as well as its derivatives, contains an explicit factor of the time difference arising from the secular part of the second eigenmode. With the left-hand side well-behaved as the initial time goes to minus infinity, it follows that the integral on the right hand side must have a divergency of the form of the initial time plus the initial time times the logarithm of its magnitude in this limit. This is confirmed by inspection and implies that the short-time regularization creates a long-time divergency. It is related to the inverse-square-frequency factor in the renormalization 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.

Though the doubly integrated correlator involves a low frequency cutoff, the relation is valid for any such frequency. But even the awkward choice of a cutoff scaling as the inverse initial time 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 the energy vanishes. In our previous works, we showed that this amounts to studying the orbits in the limit where κ vanishes, at fixed angular momentum of order unity. From the radiation formula, one has the energy loss by radiation per orbit

(ΔE)rad ≈ −3π β² / L⁵.

To study this limit, a scaling of the two orbital parameters proportional to κ is introduced, expressing that the main contribution comes from the part of the Kepler orbit near the pericenter. Indeed, in that limit the position and the radius take the simple rational forms of Equation (3.17). Clearly, this part of the orbit is in its zero-binding limit, while the farthest point, lying at approximately twice the inverse squared binding parameter along the x-axis, exhibits a self-ionization in the same limit. For further details of the method we refer to our earlier paper; we reproduce its equations for vanishing κ and multiplied by the period, as the double integral of Equation (3.18).

Continuing along these lines, we find that the third case is equal to this, while the fourth comes out with the second line replaced by the expression of Equation (3.19). Its integral over the second scaled variable is linearly divergent with logarithms, as it is for the second case. This all results in the four outcomes of Equation (3.20): the first and third cases give sixteen times the square root of three, times the coupling squared, divided by five times the sixth power of the angular momentum, while the second and fourth cases are divergent. 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 = 3π (β² / L⁶) (Lc − L), with Lc = 16 / (5π√3) = 0.588057.

Orbits that have achieved a small binding parameter and an angular momentum below the critical value will gain energy on average, which explains the self-ionization observed in all our numerics.

3.3. Other Renormalization Schemes

The renormalization by the curvature matrix involves a numerator with eigenvalues minus two and one (twice). One may wonder whether the “absolute value”, with the eigenvalues plus two and one (twice), fares better. Inspection shows that the divergence does not disappear; if anything, it becomes worse.

A renormalization with a broken power of that absolute value fares better at large times. One may replace the field by minus the square root of the absolute value of the curvature matrix acting on the vector potential, using the explicit square root given in the source, which squares to the absolute value. Like the previous scheme, this approach softens the short time behavior, but it does not ruin the long time regime. This leads to a contribution to the rate of energy gain of the form of Equation (3.22), a boundary term proportional to the logarithm of the Compton time plus a well-behaved logarithmic integral.

Using the Greens function jump conditions at coincident times, the boundary term leads to a rate of energy gain proportional to the logarithm of the Compton time times the time derivative of the trace of the absolute value of the curvature matrix, which equals minus six times the coupling squared times the radial velocity divided by two π times the fourth power of the radius (Equation 3.23). It expresses energy gain — that is, 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 the energy gain per period. In the scaling limit, the two time integrals become integrals over the scaled variables, of which the latter can be performed analytically. Its boundary term at coincident scaled variables vanishes upon integration, while the integral over the boundary term at minus infinity leads to a finite result,

the energy gain per period equals 2.99842 times the first-case result.

Hence it also leads to self-ionization.

The combination of the bare field and the square-root-renormalized field, weighted by a parameter x and its complement, involves from the cross terms a new contribution of the form of Equation (3.25), with a lengthy function vanishing at coincident times. The logarithm of the Compton time again drops out when integrated over a full period. After scaling, the integral over the second variable can be performed analytically; now the primitive at minus infinity is odd in the first variable, while the result comes from the coincident term. This ends up in a quadratic form in x (Equation 3.26): sixteen times the square root of three, times the coupling squared, divided by five times the sixth power of the angular momentum, multiplied by one minus x squared, minus 0.876444 times x times its complement, plus 2.99842 times x squared. Its minimum at x equal to 0.295028 leads to a critical angular momentum of 0.33855, smaller than the 0.58808 found above. For all x, this still leads to self-ionization.

The above “absolute” value looks unnatural, but it was necessary to define a real valued square root of the curvature matrix. The third roots are real however, and are given explicitly in Equation (3.27): the first is the identity minus the quantity two to the one-third plus one times the outer product of the unit radius vector, divided by the radius; the second is the identity plus the quantity two to the two-thirds minus one times that outer product, divided by the squared radius. It is easily verified that the square of the first is the second and its cube is the curvature matrix. They thus permit a renormalization in which the field is replaced by a weighted combination of the first third root acting on one stochastic field and the second third root acting on another (Equation 3.28), for some real valued x, with the two stochastic fields defined as fractional time-integrals of the electric field (Equations 3.29 and 3.30).

In the notation of our earlier work, their correlation functions emerge as the expressions of Equation (3.31), each the real part of an inverse fractional power of the shifted time. The difficulty is again to deal with the singularities as the Compton time goes to zero. To proceed, we perform partial integrations, introducing derivative representations of the two diagonal correlators (Equation 3.32), where we took the Compton time to zero.

In view of the earlier definition of the critical angular momentum we define its four components as double time integrals of the correlation functions times the corresponding traces of fractional powers of the curvature matrix with the Greens function derivative (Equation 3.33). For the first component we perform a partial integration with respect to the second time. Next we write the first time integral as the difference between two integrals starting at the initial time and switch the two integrals. Then we do a partial integration with respect to the first time, switch back and do a final one with respect to the second. This leads to a double integral over a second-derivative term. One boundary term at coincident times is non-trivial, namely the expression of Equation (3.34), proportional to the integral over a period of the radial velocity divided by the cube of the radius. 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 the second diagonal component we perform a partial integration with respect to the second time and evaluate the double integral in the limit of vanishing Compton time. The boundary term at coincident times vanishes identically. With the two off-diagonal traces going as the square of the time difference at coincidence, their integrands are already regular in that limit.

We are interested in these results in the scaling limit of vanishing κ at fixed angular momentum. The resulting integrals are of the type met above. Numerical evaluation yields

L¹¹ = 8.5191, L²² = 2.1944, L¹² = 0.3182, L²¹ = −0.5615.

The combined critical angular momentum corresponding to the mixed renormalization is the quadratic form built from these four numbers (Equation 3.36). It has a minimum at x equal to 0.7886,

Lc(min) = 1.7048,

which sets the boundary for self-ionizing orbits because the quadratic form exceeds this for other values of x.

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 zero can only be achieved for a scheme in which the critical angular momentum 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 of vanishing energy.

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 adds a repulsive term proportional to the inverse square of the radius to the Newton potential. It induces an effective angular momentum equal to the square root of the sum of the squared angular momentum and the squared coefficient, which, if that coefficient is of order six or larger, leads to a stable system without self-ionization. Then the effective angular momentum, 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.

(The reference list and journal end matter are omitted; the complete text, with all displayed equations, is at the source.)