In our previous blog post we asked what is the relationship of the wave and the particle interpretation of quantum mechanics.
proton ●
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e-
2 MeV kinetic energy
A proton is so heavy that we may treat it as a classical particle. If an electron passes the proton at a distance, say, 10^-15 m, it can emit a bremsstrahlung photon whose energy is 1 MeV. The wavelength of such a photon is 10^-12 m.
We face the "length scale" dilemma: how can a sharp turn whose length scale is 10^-15 m produce a wave whose length scale is 1,000 times larger?
It looks like that an electromagnetic wave can behave like a "particle": all the energy in the wave is concentrated into a small volume. In our example, all the energy in the wave originates from a small area around the nucleus. In a converse process, all the energy in a wave is absorbed inside a small volume.
Classically, the Fourier decomposition of the electromagnetic wave which is emitted from a sharp turn within 10^-15 m, should contain waves whose length is ~ 2 π * 10^-15 m. If the electron turns 180 degrees, it is like half a cycle of a dipole antenna where the charge moves at almost the speed of light for a distance π * 10^-15 m.
If the charge would keep oscillating at that speed, then it would really send a wave whose wavelength is 2 π * 10^-15 m. Somehow, the incomplete cycle allows the antenna to send a wave whose wavelength is much larger.
The Fourier decomposition of a sharp "pulse" like a Dirac delta contains all frequencies. The Fourier decomposition of a nicely oscillating dipole wave contains just one frequency. This might be the reason why an abrupt turn can produce a long wave?
In an earlier blog post we introduced the "reversed time" model which claims that a nice smooth 10^-12 m sine wave can concentrate its energy into a 10^-15 m area and induce a reaction, which if time were reversed, could produce such a 10^-12 m wave.
A large momentum virtual pair corresponds to a very sharp turn or acceleration?
We have been asking if classical virtual pairs which are created very close to each other might be responsible for high momentum virtual electrons. They are under very large acceleration, and according to the nonrelativistic Larmor formula lose all their energy very quickly. Does this happen in the relativistic case, too?
Suppose that the electron and the positron are moving at opposite directions at almost the light speed. What is the radiated power?
Let us move to a comoving inertial frame for the electron. In that frame, length (and the potential of the positron) is contracted in one direction by the factor
γ = sqrt(1 - v^2 / c^2).
In the comoving frame, the electron is nonrelativistic.
Time t is slowed down by the factor γ in the comoving frame relative to a static frame. Energy E in the comoving frame has to be multiplied by 1 / γ to get the energy in the static frame. Thus, the power P which an observer measures in the comoving frame is equal to the power which a static observer measures.
If the electron is deep within the potential well of the positron, the comoving observer will measure enormous acceleration and radiated power from the electron. A static observer will see the same enormous power dissipation. We conclude that such electrons radiate away their energy very quickly, in classical physics. The outgoing electromagnetic wave is extremely sharp because the slowdown happens under a very short distance.
In quantum mechanics, the sharp wave mysteriously loses its sharp edges and appears as a quite smooth 10^-13 m wave, according to the energy 1 MeV it is carrying.
The analysis suggests: a collision of photons can create a virtual pair where the particles are very close to each other and have very large opposite momenta.
This, in turn, suggests that large momentum virtual pairs do contribute to photon-photon scattering.
A classical wave analogue of virtual particles
Recall the two string & rubber membrane model of two interacting fields:
1. a massless Klein-Gordon field (the electromagnetic field);
2. a massive Klein-Gordon field (the Dirac field).
Is there any classical mechanism through which nice smooth sine waves in the massless field could somehow create very sharp localized waves in the massive field?
How does an external observer detect these sharp waves? He measures some scattering which he attributes to hypothetical particles.
Maybe there is some classical mechanism which can be interpreted as resulting from high-momentum virtual particles?
Generally, any scattering in the classical setting is a result of nonlinearity of the system. Maybe some of this nonlinearity can be explained with high-momentum virtual particles?
Suppose that we have a standing wave in the massless Klein-Gordon field, and its frequency is too low to create a particle in the massive field.
Let us assume that the standing wave disturbs a massive Klein-Gordon field. It is a source in the massive Klein-Gordon equation.
We may try to determine what kind of transient waves the source generates in the massive field. To accomplish that, let us use the Green's function for a Dirac delta source spike, and sum over all spacetime points.
There is a problem though: the Green's function determines the behavior of the free Klein-Gordon equation. Our massive Klein-Gordon field is coupled, and probably transfers all energy quickly back to the massless field.
How to decompose the movement of the massive Klein-Gordon field?
What is the steady state of the two fields? The massless field probably keeps doing the standing wave.
The massive field receives a source which oscillates like the standing wave. How does the massive field return the energy back to the massless field? How quickly?
We may try to decompose the movement of the massive field into plane wave solutions of the massive field. Those solutions correspond to real particles of the massive field.
But since no real particles can be created, we have to devise a way to return the energy in those particles quickly back to the massless field.
Another way to decompose is to allow "virtual particles", that is, plane waves
exp(-i (E t - p x)),
where
E^2 != p^2 + m^2.
The Green's function solution to the problem uses this kind of a decomposition. The components are not solutions of the massive Klein-Gordon equation, but their sum is a solution.
Hypotheses. 1. The particle interpretation of the particles created in a collision of quanta of the massless field is really some kind of a decomposition of the disturbance of the massive field into plane waves.
2. If no real particles in the massive field are created, then none of those plane waves satisfies the energy-momentum relation.
The hypotheses might explain why we think that virtual particles are created in a collision of quanta: the intensity of a virtual particle wave where the momentum |p| > C may be something like ~ 1 / C^2, which we may interpret that quanta of the massless field passed each other at a very short distance.
If the hypothesis is true, we can treat particle collisions purely in a classical wave model without a need to resort to a particle model.
We would still need the particle model in measurements: we observe quanta, not classical waves. Also, a state transition of a hydrogen atom is hard to explain without quanta.
But in collisions, the particles that we think are colliding are just a way to explain the nonlinear behavior of the interacting fields.
Can this interpretation work in the context of charged particles like electrons and positrons? We had hard time trying to figure out a wave interpretation for the Coulomb force. The force is easy to define for particles, but hard for waves.
Anyway, the wave model may be a way to prove that no divergence happens in a vacuum polarization loop.
Scattering of photons from a block of glass
Let us aim a coherent laser beam of 500 nm photons to a glass block. Roughly 10% of the light will reflect, that is, scatter from the surfaces of the block.
Glass is amorphous material. Each atom or molecule acts as a polarizable dipole which scatters some of the coherent wave. But why is there no apparent scattering inside the glass?
The reason is destructive interference. Let us analyze the light which is scattered 90 degrees to the right.
__________
glass
------- ----------> scattered wave
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------- crests of light waves
__________
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| laser beam
There is total destructive interference for scattered light, except for a narrow, less than wavelength thick, vertical layer in the diagram. We can imagine that scattering only happens at a thin layer at the surface of the glass block.
If we double the dimensions of the glass block, the scattered flux grows by a factor 4, while the volume of the block grows by a factor 8.
Compare this to a pure particle model of light: if each atom would scatter a small number of particles, then making the volume of the block 8-fold would make the scattered flux 8-fold.
We can reduce reflection by making the refractive index go gradually to 1 at the surface of the glass. In principle, we can cut scattering in the above experiment to zero.
Why there is no destructive interference in a particle accelerator or atmosphere?
Why there is no significant destructive interference in a collision of two particles in a particle accelerator?
Because, in principle, we can observe both particles after the collision, and distinguish between different positions of the collision. For example, if we let a nucleus collide with a photon, then we can afterwards measure the position of the nucleus very precisely.
What is the difference if we let a photon collide with a block of amorphous glass versus letting it collide with atoms and molecules in the atmosphere?
Atoms and molecules in the atmosphere move in a random way. There is no such extensive destructive interference that is present when the photon collides with static atoms in glass.
Is there a classical analogue of the quantum interference of a system of two particles?
Let us describe two colliding particles with wave packets. We mentioned in the previous section that we can measure the position of both particles afterwards, and the interference happens between pairs of the measured final positions of the particles.
In classical physics, interference happens for a wave independently of other waves.
-------------------------------- plane
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| coupling from large
| amplitude waves
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____/\_________/\______ string
---> <---
wave X wave Y
Let us imagine the classical system in the diagram. We have two waves X and Y colliding in a 1-dimensional string. A coupling to a plane moves energy from a large amplitude wave to the plane.
The generated wave in the plane can be understood as some kind of scattering of the waves in the string. Interference in the plane happens for a 2D point (x, y). It does not happen for x or y individually.
We may say that particles X and Y got "entangled" in the plane and one cannot treat them as independent particles.
If we measure that the particle (or two 1-dimensional particles) in the plane is located at a position (x, y), we may return the two scattered particles back to the 1-dimensional world of the string.
Conclusions
This blog posting contained analysis of the wave-particle duality from various points of view. Let us next study if a classical wave model casts light on photon-photon scattering. We have the simple model of couples massive and massless Klein-Gordon fields. We need to study it carefully.
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