The Radiating Atom 7: Quantum Electro Dynamics Without Infinities?

The interaction between matter in the form of an atom and light as electro-magnetic wave is supposedly described by Quantum Electro Dynamics QED as a generalization of quantum mechanics into the "jewel of physics" according to Feynman as main creator.  However QED was from start loaded with infinities requiring  "renormalization", which made the value of the jewel as a "strange theory" questionable according to Feynman himself:

• But no matter how clever the word, it is what I call a dippy process! Having to resort to such hocus pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self consistent. ... I suspect that renormalization is not mathematically legitimate. 

Let us see what we can say from the experience of the present series of posts on The Radiating Atom leading to the following Schrödinger equation for a radiating Hydrogen atom subject to exterior forcing:

• $\dot\psi + H\phi -\gamma\dddot\phi = f$,       (1)
• $-\dot\phi + H\psi -\gamma\dddot\psi = g$,      (2)

where $\psi = \psi (x,t)$ and $\phi = \phi (x,t)$ are real-valued functions of space-time coordinates $(x,t)$ (as the real and imaginary parts of Schrödinger's complex-valued electronic wave function $\psi +i\phi$), $\dot\psi =\frac{\partial\psi}{\partial t}$,

• $H=-\frac{h^2}{2m}\Delta + V(x)$

is  the Hamiltonian with $\Delta$ the Laplacian with respect to $x$, $V(x)=-\frac{1}{\vert x\vert}$ the kernel potential, $m$ the electron mass and $h$ Planck's constant,  $-\gamma\dddot\phi$ is a Abraham-Lorentz radiation recoil force with corresponding radiation energy $\gamma\ddot\phi^2$ with $\gamma$ a small positive radiation coefficient and $f=f(x,t)$ and $g=g(x,t)$ express exterior forcing. Note that here the electron wave function is coupled to radiation and forcing through a radiative damping modeled by $(-\gamma\dddot\phi ,-\gamma\dddot\psi )$ and the right hand side $(f,g)$, and not through a time-dependent potential connecting an incoming electric field to an electronic dipole moment, which is a common alternative. An advantage of the above more phenomenological model is simpler mathematical analysis since the potential is kept independent of time.

The system (1)-(2) can be viewed as a generalized harmonic oscillator with small radiative damping subject to exterior forcing similar to the system analyzed in Mathematical Physics of Black Body Radiation. The essence of this analysis is a balance of forcing and radiation (cf. PS5 below):

• $R \equiv\int\gamma (\ddot\psi^2 +\ddot\phi^2)dxdt\approx \int (f^2 + g^2)dxdt$,

which can be viewed to express that $output \approx input$.

A radiating atom with wave function $(\psi ,\phi )$ can be viewed to interact with an electromagnetic $(E,B)$ through the charge density

• $\rho (x,t) =\psi^2(x,t) + \phi^2(x,t)$,

according to Maxwell's equations:

• $\dot B + \nabla\times E = 0$, $\nabla\cdot B =0$,
• $-\dot E + \nabla\times B = J$, $\nabla\cdot E =\rho$,

with $J$ a corresponding current. For a superposition of two pure eigen-states with eigenvalues $E_1$ and $E_2$ the charge density varies in time with frequency $\omega =(E_2 -E_1)/h$ and then as an electrical dipole generates outgoing radiation

• $P\sim\omega^4$,   

which is balanced by the radiation damping in Schrödinger's equation

• $R=\int\gamma (\ddot\psi^2 +\ddot\phi^2)dxdt\sim\omega^4$.

The above QED model combining Schrödinger's equation for an atom with Maxwell's equations for an electro-magnetic field, thus explains the physics of 

1. an electron configuration as a superposition of two pure eigen-states of different energies, 
2. which generates a time variable charge/electrical dipole, 
3. which generates an electro-magnetic field, 
4. which generates outgoing radiation,
5. under exterior forcing.

The analysis in Mathematical Physics of Black Body Radiation shows that in this system 

• $P \approx R\approx \int (f^2 + g^2)dxdt$, that is,
• outgoing radiation $\approx$ radiative damping $\approx$ exterior forcing.  

The fact that outgoing radiation $\approx$ exterior forcing makes it possible to reverse the physics (1) from an atom generating outgoing radiation as an electromagnetic field (emission) into (2) a model of the reaction of an atom subject to an incoming electro-magnetic field (absorption). This is the same reversal that can be made to use a loadspeaker as a microphone (or that an antenna reradiates about half what it absorbs allowing Swedish Television agents to detect individual watchers and check if the TV-license has been paid).

Note that the physics of (1) may be easier to explain/understand than (2), since outgoing radiation/emission can be observed, while atomic absorption of incoming electro-magnetic waves is hidden to inspection.  On the other hand if (2) is just the other side of (1), then explaining/understanding (1) may be sufficient.

The analysis thus offers an explanation of self-interaction without a catastrophy of acoustic feedback between loadspeaker and microphone, which may be at the origin of the infinities troubling Feynman's jewel of physics QED with photons being emitted and possibly directly being reabsorbed in a form of catastrophical photonic feedback.

PS1 The radiation damping $-\gamma\dddot\psi$ may alternatively take the form
$\gamma \vert\dot\rho\vert^2\dot\psi$, with again $R\sim \omega^4$ for a superposition of eigen-states, and $R=0$ for a pure eigen-state with $\dot\rho =0$. Compare PS5 below.

PS2 The basic conservation laws built into (1)-(2) with $f=g=0$ are (with PS1)

• $\frac{d}{dt}\int\rho (x,t)dx =0$   (conservation of charge), 
• $\frac{d}{dt}\int (\psi H\psi +\phi H\phi)dx = -\int(\gamma\vert\dot\rho\vert^2(\dot\psi^2+\dot\phi^2)dx$  (radiative damping of energy).

PS3 Feynman states in the above book: 

• It is very important to know that light behaves like particles, especially for those of you who have gone to school, where you were probably told something about light behaving like waves. I am telling you the way does behave - like particles. ...every instrument (photomultiplier) that has been designed to be sensitive enough to detect weak light has always ended up discovering the same thing: light is made of particles.

We read that Feynman concludes that because the output of a light detector/photo-multiplier under decreasingly weak light input, changes from a continuous signal to an intermittent signal to no signal, light must also be intermittent as if composed of a stream of isolated particles.  But this is a weak argument because it draws a general conclusion about the normal nature of light from an extreme situation where blips on a screen or sound clicks are taken as evidence that what causes the blips also must be blip-like, that is must be particles. But to draw conclusions about normality by only observing extremity or non-normality, is to stretch normal scientific methodology beyond reason. In particular, the infinities troubling QED seems to originate from particle self-interaction. With light and atom instead in the form of waves and their interaction consisting of interference of waves, self-interaction does not seem to be an issue.


PS4 The book Atoms and Light Interactions presents what its author by J. D. Dodd refers to as a semi-classical view of the interaction of electromagnetic radiation and atoms, thus as waves and not particles (which is also my view):

• It may well be that the semiclassical view falls down at some stage and is unable to predict correctly certain phenomena; my own view is that it succeeds much more widely than it is given credit for. Even if it is not justified from the point of view of many physicists, i is still useful for another reason. Even if the quantum nature of radiation (QED) is required, the underlying physics needs a firm understanding of its classical basis.  

Yes, it may well by that also atomistic physics is a form of wave mechanics and thus a form of classical continuum physics, as expressed by Zeh:

• There are no quantum jumps and nor are there any particles.

PS5 The analysis of Mathematical Physics of Black Body Radiation is more readily applicable if (1)-(2) is formulated as a second order in time wave equation of the form

• $\ddot\psi +H^2\psi + \gamma\dot\rho^2\dot\psi = F$,

with the following tentative main result as an extension of the analysis from radiative damping $-\gamma\dddot\psi$ to $\gamma\dot\rho^2\dot\psi$ (with $\gamma >0$ constant):

• $\int\gamma\dot\rho^2\dot\psi^2dxdt\approx\int F^2dxdt$.

Here $\gamma$ may have a dependence on $\psi$ to guarantee charge conservation under forcing.