onsdag 10 december 2014

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.

21 kommentarer:

  1. Are there any attempts to calculate the hydrogen spectrum? If so, are you then using gamma as a fitting parameter, or how do you else calculate it?

  2. The spectrum as the difference of eigenvalues of the Hamiltonian/h, does not depend on gamma, but the presence of a positive gamma with corresponding radiative damping is what makes it possible to observe the spectrum as outgoing radiation. The precise value of gamma is insignificant.

  3. Regarding your objections against Feynmans argument under PS3.

    Although Feynman uses an example of a photomultiplier, that is not the only argument for the point he is making. And I'm quite positive Feynman knew that. But remember that the book is an adaption of a set of lectures he gave, aimed at the GENERAL PUBLIC (in the late 70s). There is no need to give any deeper arguments than that to give the big picture.

    Starting in the 70s, and continued until today, there is a tremendous amount of different kinds of experiments that really nails the coffin for any kind of semiclassical (that is a theory of only waves) description of light, since it would be astronomically improbable for such a theory to work, regarding the experimental data.

    I urge you to read the published paper

    Observing the quantum behavior of light in an undergraduate laboratory

    for a concrete example.

    This, and similar experiments, is what you need to argue against if you want to refute the photon model.

    Any objections?

  4. This article states that the photoelectric and Compton effect do not give evidence of the existence of photons, and I agree. Their experimental evidence very difficult to understand, and certainly beyond undergraduate level. To say that something is impossible, like saying that a complicated experiment cannot possibly be explained by wave theory, is easy to do but probably impossible to prove. To show that something is possible is possible by just doing it, while showing impossibility is virtually impossible.

  5. I strongly disagree with you.

    The experiment is not at all difficult to understand. The theory behind it is highly accessible after finishing a modern introductory quantum mechanical course. And on the master level this is not hard at all.

    Coincidence techniques and beam splitters are not rocket science, and the element of field theory is pretty straight forward.

    So I need to ask you, difficulties aside (I don't think the universe cares if undergraduates find the theory difficult or not ;-) ), you don't seem to object to the experimental result? Any theory containing classical waves predicts a g^(2)
    of 1 or larger, so the experimental evidence says that a classical theory can't account for the data. If you disagree, why so?

  6. The evidence is negative: It is claimed that a certain phenomenon is not readily explained by wave theory, but how can we be so sure that an explanation is impossible? Photoelectricity used to be evidence of particle nature of light, but the authors of the article do not buy that argument. To claim that something is impossible favors ignorance and inability, and that is dangerous.

  7. It is claimed that a certain phenomenon is not readily explained by wave theory

    Not really. From any theory with a classical EM-field, there is a prediction on the correlation function in question. Since the observed effect is lower then this prediction (by 377 standard deviations!), the classical theory must be seen as falsified. This doesn't mean that a semi-classical description can't be used. The question then is, when is it justified to use a semi-classical description as an approximation? There is no generality in a semi-classical description, this can be justified to a really high precision judging from experimental data.

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

    I would say that you misjudge this title. We do know from quantum mechanics that there are no particles. That doesn't mean that there are waves. It is a false dichotomy. The general quantity in quantum mechanics is the quantum field, that is neither a wave nor a particle.

    I don't know if you read Zeh's paper in detail, either way, you should really re/read the paragraph in detail.

    It thus appears becoming evident that our classical concepts describe mere shadows on
    the wall of Plato's cave in which we are living. Using them for describing reality must
    lead to 'paradoxes'.

  8. Ok, so we agree that there are no particles and then what remains is waves. To say neither particle nor wave but something else without telling what is not constructive, to me at least.

  9. I did write what, an excitation in a quantum field.

    The general quantity in quantum mechanics is the quantum field, that is neither a wave nor a particle.

  10. And what is it then? In physical terms?

  11. That is kind of a silly question.

    Classically speaking, what is an electro-magnetic field? In physical terms?

  12. A distributed function of space and time satisfying Maxwell's wave equation that is a wave, not a particle.

  13. A distributed function [...]

    But that doesn't say what the electromagnetic field really is, physically.

    You say here how to describe the field mathematically. From a classical point of view the classical electromagnetic field is fundamental, so you can't describe it in much more detail than you do here.

    It is the same with a quantum field. If you want the mathematical formalism, see for instance
    An Introduction To Quantum Field Theory

    If you accept the classical method of defining an electromagnetic field (I assert that you do, since you just used it that way), I can't see how you wouldn't accept the exact same method for the quantum field.

    At the same time, the quantum field must be a more fundamental description, for two (at least) reasons.

    First, it contains the classical fields and equations as a subset.

    Second, it can be used to account for more experimental data (as discussed above).

  14. We are talking about waves or particles. I say electromagnetic fields are waves as real-valued functions of space and time, not particles, which satisfy Maxwell's equations and thus can be understood by many. QFT is loaded with infinities and not understood by many, if any.

  15. We are talking about waves or particles.

    Not really, you are talking about waves OR particles. I tried to mention earlier that doing so is a false dichotomy, false dilemma, false duality or what you want to call it. None the less, it is a logical fallacy since you miss the situation where we have something that is more fundamental than our notions of particles and waves. Heck, you even linked to a paper that had that as a main theme (the Zeh paper and reference to a wall in Plato's cave in the conclusions).

    It is a great misfortune that the name wave-particle duality still remains.

    [...] which satisfy Maxwell's equations and thus can be understood by many.

    I would call this a fallacy as well. Humans ability to understand a theory can of course not have any impact on how well that theory describes reality. Maybe I misunderstand you here. If so, what do you really mean?

  16. Yes, a theory which cannot be understood is not useful, like the special and general theories of relativity. I have the impression that QFT falls in the same category, but in this case I may be wrong. Anyway, I am seeking a continuum model for the radiating atom which can make sense and thus be understood.

  17. In what way cannot special and general relativity be understood? The theoretical frameworks are not that complicated, especially in the case of special relativity. And I don't know of any inconsistencies either. So I must admit that I don't understand what you mean here.

  18. I got a bit curios to what you think cannot be understood about the theory of relativity, looked around on this blog and found your other blog, The World As Computation. And more specific the post
    Questioning Relativity 2: Unphysical Lorentz Transformation

    And I do see a little where your confusion about the theory originates.

    You write there

    However, the figure is misleading: The x'-axis defined by t' =0 is not parallel to the x-axis, since it is given by the line t=vx which is tilted with respect to the x-axis.

    And you then conclude that the transformation must be unphysical.

    This looks really strange, and I can not see how this has to do with relativity at all. In relativity, the primed coordinate system is not given by an equation, it is just another inertial system, it is predefined, or given if you so wish.

    The Lorentz transformation then, is just the passive (no physical change) transformation that relates the physical event (x,t) in one coordinate system with another coordinate system that describes the SAME PHYSICAL event as (x',t'). Given the constraint that both systems should agree numerically on a measurement of the light speed.

    To be honest, this is really simple stuff. General relativity gets trickier, but is not impossible.

  19. If you think that special relativity is a physical theory, then you are fooling yourself. Yes, it is as mathematical theory simple/trivial because it just a simple linear transformation, but it has no physical meaning and thus the simplicity you perceive is just an illusion. A meaningless theory cannot be viewed as simple.

  20. but it has no physical meaning

    I must ask you to be more specific in what you mean. What criteria do you use to call a theory physical and with meaning?

  21. I explain this in detail in Many Minds Relativity. Take it or leave it.