proton ● ~~~~~~~~ photon 1 MeV
^
|
|
e- kinetic energy 2 MeV
A prime example of the length scale problem is bremsstrahlung when a 2 MeV electron passes a proton and emits a 1 MeV photon.
The wrinkle in the electric field of the electron is created in a very short distance, ~ 10^-15 m, and time ~ 10^-23 s.
The wavelength of a 1 MeV photon is 10^-12 m and the cycle time 3 * 10^-21 s.
Since a particle can be born in a very small volume in a very short time, it cannot be a wave?
Let us think about an arbitrary wave equation. One can create a pure long wave in a very small volume if the source lasts for a long time. By pure we mean that there are not too many other frequencies created. An example is a finger which presses rhythmically a drum skin.
One can create a pure long wave in a short time if one is allowed to use a source which has a wide spatial extent.
But it is impossible to create a pure long wave in a short time in a small volume. An example is hitting a drum skin with a sharp hammer. The impulse response, or Green's function, contains all frequencies.
This suggests that the 1 MeV photon really is a particle, not a wave. We can create a particle in a short time in a small volume.
This sounds like a Feynman diagram. There particles emit and absorb other particles.
Do particles interact through other particles or through forces?
Should we explain the Coulomb force between an electron and a proton with particles? Not necessarily. A force is a very useful concept when we talk about particles.
We had problems defining the Coulomb force between wave packets when we tried to model charged particles as wave packets. Waves and a force do not match. Particles and forces match very well.
When we tried to explain the bending of a beam of electrons under a static electric field, we had problems explaining it with a Feynman diagram.
Wave phenomena arise from path integrals?
Richard Feynman suggested that in an experiment we should calculate "all possible" paths which give the same measurement result. The probability amplitude of the measurement is calculated by summing
exp(i S / h-bar),
where S is the action calculated along the path. The integration measure (= weight) for a path has to be decided in some "natural" way. There is no formal definition what mathematical measure we should use.
| | |
| | ) ) |
| | | |
| | ) ) | screen
| | |
plane wave diffracted waves
double slit
The double slit experiment is a prime example of the success of this method. A plane wave describes a photon whose precise location we do not know.
The phase of the plane wave tells the "current phase" of the photon if it is located at a spacetime point (t, x).
There are two paths which contribute to a photon which is measured at a specific cell of the image sensor at the screen.
The action S grows by 2 π per the wavelength λ of the photon along the photon path.
The calculation gives the familiar interference pattern of the double slit experiment.
The classical limit of bremsstrahlung
Let us imagine that the proton and the electron are extremely heavy particles and their charges are scaled accordingly up.
Then we can use the classical limit and calculate the electromagnetic wave which the electron outputs with the Edward M. Purcell method, or whatever classical rule.
The existence of a classical limit suggests that, in some sense, the experiment does output a wave with an extremely sharp wrinkle in the electric field.
If we scale down the masses and the charges of the proton and the electron, at what point we lose the wave nature of the wrinkle in the electric field, and should talk about particles like photons?
Maybe we should treat the proton and the electron as classical particles, even though the electron weighs only 10^-30 kg.
Then the "path" includes the particles as well as the electromagnetic field produced by those particles. The path is a process of classical physics.
The output of the experiment is an electron and the electromagnetic field with all its wrinkles.
How to cut off detail which is too sharp for the energy scale?
Suppose that we have a detector which can get excited by a 3 MeV photon. The very sharp wrinkle in the electromagnetic field does contain a Fourier component which could excite the detector. But the energy of the incoming electron was just 2 MeV. The detector cannot get excited.
How do we prevent the detector from getting excited?
We have to cut off the frequencies which would break energy conservation.
If the classical electron at the end of the experiment has 1 MeV of kinetic energy left, then we must assign to a single photon all the energy which was lost to electromagnetic radiation.
In the classical description, the 1 MeV photon is just one frequency of a continuous band of frequencies if we Fourier transform the output wave. We probably should assign the energy of too high frequencies to the 1 MeV photon. What about too low frequencies?
The cutoff procedure makes a "low-resolution" photograph of the classical process which contains very sharp features. The resolution is determined by the energy 2 MeV of the input electron.
In the low-resolution photograph, energy which is present in high frequencies is assigned to a frequency such that energy is conserved.
A path integral is the right way to draw the low-resolution photograph
A path integral may be a method to produce the low-resolution photograph. Sharp detail is removed by destructive interference.
Maybe constructive interference highlights the path where energy is conserved? That is, the energy which is lost in destructive interference of high frequencies heaps up at "the right" frequencies? Then the energy in "too low" frequencies would heap up at the right frequencies, too? This is probably the right solution.
If we describe the incoming electron as a wave packet, then it does contain frequencies which have a much higher energy than 2 MeV. In that case, the output will contain some high frequencies, too.
With a palette of different frequencies we can draw a detailed picture of the physical process. If the input is pure plane waves, then a "momentum space" description of the process might be the right one.
A new type of resonance: the energy output in all frequencies heaps up at the "resonant" frequency
The Larmor formula describes a complex process of classical physics. A classical electron passes a classical proton and sends a very sharp and complex electromagnetic wave whose Fourier decomposition contains a large spectrum of frequencies.
Why does the corresponding Feynman diagram claim that all that energy in the electromagnetic wave is concentrated to a single frequency? And why does the Feynman diagram calculate the energy right?
A single path in the path integral does contain that complex process of classical physics, and does emit a complex electromagnetic wave with many different frequencies.
But destructive interference in the path integral destroys all the fine detail and removes all but one frequency in the output electromagnetic wave. That one frequency gets massive constructive interference and all the energy heaps up at that one frequency.
We may imagine that since the energy cannot escape through the frequencies that are destroyed by interference, it will escape by the one allowed route. That is what happens in the anti-reflective coating of lenses. There is destructive interference for reflection. Energy must take the allowed route: transmission.
The caleidoscope model of interference and conservation of energy
The path integral describes an infinite number of possible sequences of events. In the bremsstrahlung example, the electron may have its closest encounter with the proton at any spacetime point. It is like a caleidoscope showing the same event in a myriad locations.
Destructive interference removes all small detail, and in general, any detail which is not repeated at just the right intervals in the caleidoscope. Only "resonant" waves are preserved.
The input plane waves determine the preferred intervals in the caleidoscope. Fine detail repeats at too short intervals and is removed. Long waves repeat at too long intervals and are removed.
The caleidoscope effect apparently forces conservation of energy. If some event would not conserve energy, that is, it would create a wave with too short a wavelength, then destructive interference would remove it. The effect also enforces conservation of momentum.
This seems to be related to the Noether theorem which states that symmetry in time translations enforces conservation of energy and symmetry in spatial translations enforces conservation of momentum.
Conclusions
Electrons and protons "really" are classical particles which obey classical mechanics and Coulomb's force law.
But the "photograph" of the experiment process must be drawn with the palette of available frequencies. The frequencies are input in the wave descriptions of the incoming particles.
A path integral is the way to produce the "photograph".
A path integral of the process causes destructive interference to all the detail which cannot be described with the palette. The energy in the destroyed detail heaps up at the frequencies which are shown in the final "photograph". Energy is conserved.
We claim that deep down particles do obey classical physics. Wave phenomena is imposed by the "photograph method", that is, path integrals.
Our claim has sweeping consequences for loops in Feynman diagrams. Classical particles have strict rules which they must obey. One cannot just add arbitrary 4-momentum to circulate in a loop.
In an earlier blog posting we noticed that off-shell electrons can actually be quite normal classical electrons which are interacting with other particles and fields. Maybe that is how things really are.
No comments:
Post a Comment