Problems of virtual pairs in empty space
Postulating virtual pairs in "empty" space leads to many problems:
1. Lorentz invariance: if space is filled with virtual pair matter, what is our speed relative to it?
2. What is the density of virtual pairs per cubic meter? If it is infinite, then tunneling should happen at an infinite rate and vacuum polarization should render all charges effectively zero.
3. What is the energy density of the virtual pair matter? If it is large, then space should immediately collapse into a black hole.
4. If a virtual pair can absorb a virtual photon like in the simple Feynman diagram, physics becomes "non-causal" in the sense that a potentially infinite number of virtual pairs can take part in the process.
5. Virtual pair loops cause divergences in Feynman integrals.
Our blog aims to show that processes of real particles create their own environment - there is no virtual pair matter floating around in empty space. What may look like virtual pairs popping out from empty space is really phenomena which are produced by the real particles that entered the physical experiment.
Energy flow between particles and the electromagnetic field
e- ~~~~~~~ e+
In a Feynman diagram, no arrow is drawn into a line which describes the interaction of two particles via a virtual photon. That makes sense because the interaction is mutual, it equally happens to both directions.
An exception to this is when system A emits a real photon which later interacts with system B. In that case we could specify the direction of the interaction.
When an electron approaches a positron, the pair draws energy from the electric field. The pair acquires kinetic energy from the field. We cannot uniquely say how the extra kinetic energy is split between the particles because the numerical value depends on our choice of the inertial coordinate system. It is the pair which acquires kinetic energy relative to each other and not an individual particle. Similarly, it is the pair whose electric field gives up that energy.
Since we are dealing with a pair, maybe we should work in a 6-dimensional space plus the time coordinate? Then it is a single particle moving in a potential which has an infinitely deep well at the diagonal x = x', y= y', z = z'.
How does energy flow create real pairs?
Real pair production in quantum field theory has a classical analogue in an electric breakdown of a medium under an intense electric field. In a dynamic system, an intensifying electric field may produce an electric breakdown.
When an electron and a positron approach each other they receive kinetic energy from the electric field. That means that the electric field grows weaker in most of the space. Looking from far away, the charges of the electron and the positron cancel out each other and the electric field becomes almost zero.
But the electric field does grow in magnitude in the zone between the electron and the positron. At the middle point it doubles relative to just having one of the particles present. Nature has an option to reduce the growing electric field by creating a pair which cancels out some of the electric field. The electric field behaves then nonlinearly because the resulting field is not the sum of the two original component fields.
When the electron and the positron recede fron each other, the extra kinetic energy that they drew from the electric field returns back to the electric field. The electric field grows stronger almost everywhere except in the zone in between the electron and the positron.
Suppose that we have a growing electric field:
-------------> E electric field
Pair production reduces the electric field:
-------------> E
e- <---- e+ field of the pair
We assume that the electron and the positron are born close to each other. They need to tunnel far enough of each other so that the energy reduction of the electric field offsets the energy of the pair. The energy of the original electric field E flows into the energy of the pair. The creation of the pair opens a new degree of freedom into the system and energy flows to that new degree of freedom.
It is the spatial energy flow which is important in breaking in a new degree of freedom. It is not the static electric field.
If we have a plane wave, then conservation of momentum prevents the conversion of electromagnetic energy to real pairs. There is no energy flow in a plane wave. There is a uniform energy density throughout the whole space.
How can we quantify the magnitude of the "dynamics" of a system? If we are working in the center of mass coordinates of a colliding electron and a positron, then we can divide space into cells and determine the electromagnetic field energy in a cell as a function of time.
Another way is to do a Fourier decomposition of the field against time. What kind of a decomposition does a "dynamic" field have? What is the decomposition of a static field? Since local flow of energy is important in pair production, a global decomposition into Fourier modes may not help us in any way?
If we have several electrons flying at random directions far away from each other, there is energy flow in the center of mass coordinates, but no pair production. What is the difference from a collision? Why the energy flow in a collision produces pairs?
In the far away case, we can ignore interactions. For each flying electron, there is an inertial coordinate system where its field is static close to the electron. And a static field does not produce pairs.
In the collision case, there is no inertial coordinate system where the field is static. There is flow of energy regardless of the inertial coordinate system.
When an electron and a positron approach each other they receive kinetic energy from the electric field. That means that the electric field grows weaker in most of the space. Looking from far away, the charges of the electron and the positron cancel out each other and the electric field becomes almost zero.
But the electric field does grow in magnitude in the zone between the electron and the positron. At the middle point it doubles relative to just having one of the particles present. Nature has an option to reduce the growing electric field by creating a pair which cancels out some of the electric field. The electric field behaves then nonlinearly because the resulting field is not the sum of the two original component fields.
When the electron and the positron recede fron each other, the extra kinetic energy that they drew from the electric field returns back to the electric field. The electric field grows stronger almost everywhere except in the zone in between the electron and the positron.
Suppose that we have a growing electric field:
-------------> E electric field
Pair production reduces the electric field:
-------------> E
e- <---- e+ field of the pair
We assume that the electron and the positron are born close to each other. They need to tunnel far enough of each other so that the energy reduction of the electric field offsets the energy of the pair. The energy of the original electric field E flows into the energy of the pair. The creation of the pair opens a new degree of freedom into the system and energy flows to that new degree of freedom.
It is the spatial energy flow which is important in breaking in a new degree of freedom. It is not the static electric field.
If we have a plane wave, then conservation of momentum prevents the conversion of electromagnetic energy to real pairs. There is no energy flow in a plane wave. There is a uniform energy density throughout the whole space.
Measuring the magnitude of energy flow
How can we quantify the magnitude of the "dynamics" of a system? If we are working in the center of mass coordinates of a colliding electron and a positron, then we can divide space into cells and determine the electromagnetic field energy in a cell as a function of time.
Another way is to do a Fourier decomposition of the field against time. What kind of a decomposition does a "dynamic" field have? What is the decomposition of a static field? Since local flow of energy is important in pair production, a global decomposition into Fourier modes may not help us in any way?
If we have several electrons flying at random directions far away from each other, there is energy flow in the center of mass coordinates, but no pair production. What is the difference from a collision? Why the energy flow in a collision produces pairs?
In the far away case, we can ignore interactions. For each flying electron, there is an inertial coordinate system where its field is static close to the electron. And a static field does not produce pairs.
In the collision case, there is no inertial coordinate system where the field is static. There is flow of energy regardless of the inertial coordinate system.
Self-energy as energy flow
If an electron is in an accelerating motion, then there is no inertial coordinate system where its electric field is static.
Conjecture 1. The self-energy Feynman diagram, where an electron seems to send a virtual photon to itself, describes the energy flow in the electric field of an accelerating electron.
We will address later the divergence of the self-energy Feynman integral. The self-energy loop can circulate an arbitrary amount of momentum p, which causes a logarithmic divergence in the integral value.
A freely flying electron does not have the self-energy phenomenon, because it is static in an inertial coordinate system. People sometimes draw an electron flying inside a cloud of self-energy photons. In our opinion, that is wrong.
---------------------> E electric field
Quantizing the energy flow process
We now have some classical intuition about the energy flow in a collision. A great conundrum is how on earth can Feynman simplify the complex process into the flight of a few virtual particles, and get accurate numerical results!
Quantum mechanics does exhibit similar simplicity in the hydrogen atom. When a hydrogen atom becomes excited, a very complex orbital change can be summarized as absorption of one photon.
The simplest Feynman diagram that describes an electron flyby of a nucleus, contains just a virtual photon line which transfers momentum p to the electron but no energy.
But the process does involve flow of energy. The kinetic energy of the electron draws on the energy of the field. Also, there is flow of energy into the zone between the electron and the nucleus. When the electron recedes, the extra kinetic energy flows back to the field.
A potentially forming real electron-positron pair tries to take its toll on the increasing energy in various places in the field.
The energy flow in the flyby happens slower than the speed of light. If we conjecture that the flow is transferred by quasi-particles, those particles have a non-zero rest mass. This is the reason why the conversion to a real pair (which has a non-zero rest mass) can happen, while it cannot happen when energy is carried by light-speed photons.
We have found a new interpretation for a virtual photon: it is a quasi-photon with a non-zero rest mass. Because of the rest mass, it can move at a slow speed and carry more momentum than a photon.
Real pair production can be interpreted as a tunneling process where an energy flow into a zone of space tunnels into the creation of a real pair. The energy must arrive to the zone at a speed less than light because otherwise we cannot conserve momentum in pair creation.
A virtual pair is a failed creation of a real pair.
What is the energy quantum of a quasi-photon? To create a real pair, it should be larger than 1.022 MeV.
The simplest Feynman diagram that describes an electron flyby of a nucleus, contains just a virtual photon line which transfers momentum p to the electron but no energy.
But the process does involve flow of energy. The kinetic energy of the electron draws on the energy of the field. Also, there is flow of energy into the zone between the electron and the nucleus. When the electron recedes, the extra kinetic energy flows back to the field.
A potentially forming real electron-positron pair tries to take its toll on the increasing energy in various places in the field.
The energy flow in the flyby happens slower than the speed of light. If we conjecture that the flow is transferred by quasi-particles, those particles have a non-zero rest mass. This is the reason why the conversion to a real pair (which has a non-zero rest mass) can happen, while it cannot happen when energy is carried by light-speed photons.
We have found a new interpretation for a virtual photon: it is a quasi-photon with a non-zero rest mass. Because of the rest mass, it can move at a slow speed and carry more momentum than a photon.
Real pair production can be interpreted as a tunneling process where an energy flow into a zone of space tunnels into the creation of a real pair. The energy must arrive to the zone at a speed less than light because otherwise we cannot conserve momentum in pair creation.
A virtual pair is a failed creation of a real pair.
What is the energy quantum of a quasi-photon? To create a real pair, it should be larger than 1.022 MeV.
Classical limit
Our analysis has finally brought us some understanding of the classical limit of pair production.
The virtual photon in the simplest Feynman diagram signifies a complex energy flow between the kinetic energy of the particles and the electric field, as well as energy flow within the field.
Those zones where the electric field grows will tend to produce pairs. Classically, we may imagine that the increase of the electric field somehow destabilizes a local electric dipole and may cause it to break apart, releasing a free electron and a positron. A static electric field, on the other hand, cannot break the dipole. As if an electron would be in a hole. A static electric field cannot pull it out, but an increasing electric field can.
---------------------> E electric field
______ ______
|_e-_|
^
| disturbance
A growing electric field is a disturbance which can pop the electron out of the hole. The hole is actually the hole of the Dirac hole theory. It is immune to a static electric field. It does not polarize under the static electric field, in contrast to a more familiar electron "hole", the hydrogen atom.
If electron holes have an infinite density per cubic meter, then any small polarization of a hole under an electric field would cancel out all electric fields altogether.
Pair production reduces the increase of an electric field. But why it cannot reduce the field which was there before the increase started, the static electric field?
We have been claiming in this blog that a static electric field cannot produce pairs. Suppose that we are wrong.
Let us calculate an example.
The electric field strength of an electron reaches the Schwinger limit at a radius of 3 * 10^-14 m. The classical electron radius is 3 * 10^-15 m and the Compton wavelength is 2 * 10^-12 m.
Suppose that we have an electron at a distance 1 meter from a positron. How close should we move the electron to the positron to recover its rest mass of 511 keV in mechanical energy?
The potential of the positron-electron system is
k e^2 / r.
The above should be equal to 511,000 e. We get
r = 9 * 10^9 * 1.6 * 10^-19 / 511,000 m
= 3 * 10^-15 m.
It is the classical electron radius.
The zitterbewegung of an electron has an amplitude equal to the Compton wavelength 2 * 10^-12 m.
If there exists tunneling phenomena in the electron field, it should happen at the scale of the classical radius 3 * 10^-15 m. Could tunneling at such short distances explain the light-speed zitterbewegung at a distance scale 700 times larger?
Suppose that a virtual pair is created close to an electron. The virtual positron comes within less than 3 * 10^-15 m of the electron and annihilates. Then the virtual electron becomes real. The location of a 511 keV electron cannot be determined with a better accuracy than the Compton wavelength. The electron appears to have jumped a distance of approximately one Compton wavelength.
That could be the explanation of the zitterbewegung. But zitterbewegung is not present in Dirac solutions which contain purely positive frequency waves. How would we explain the absence of zitterbewegung in that case?
In this blog we have been claiming that a static electric field cannot produce pairs. But if we have an electron present, then it is a system of an electron plus its static electric field. Maybe we could allow pair production in such a case?
Can we avoid non-causality if we allow pair production close to a free electron?
If electron holes have an infinite density per cubic meter, then any small polarization of a hole under an electric field would cancel out all electric fields altogether.
Zitterbewegung through tunneling in the field of the electron?
Pair production reduces the increase of an electric field. But why it cannot reduce the field which was there before the increase started, the static electric field?
We have been claiming in this blog that a static electric field cannot produce pairs. Suppose that we are wrong.
Let us calculate an example.
The electric field strength of an electron reaches the Schwinger limit at a radius of 3 * 10^-14 m. The classical electron radius is 3 * 10^-15 m and the Compton wavelength is 2 * 10^-12 m.
Suppose that we have an electron at a distance 1 meter from a positron. How close should we move the electron to the positron to recover its rest mass of 511 keV in mechanical energy?
The potential of the positron-electron system is
k e^2 / r.
The above should be equal to 511,000 e. We get
r = 9 * 10^9 * 1.6 * 10^-19 / 511,000 m
= 3 * 10^-15 m.
It is the classical electron radius.
The zitterbewegung of an electron has an amplitude equal to the Compton wavelength 2 * 10^-12 m.
If there exists tunneling phenomena in the electron field, it should happen at the scale of the classical radius 3 * 10^-15 m. Could tunneling at such short distances explain the light-speed zitterbewegung at a distance scale 700 times larger?
Suppose that a virtual pair is created close to an electron. The virtual positron comes within less than 3 * 10^-15 m of the electron and annihilates. Then the virtual electron becomes real. The location of a 511 keV electron cannot be determined with a better accuracy than the Compton wavelength. The electron appears to have jumped a distance of approximately one Compton wavelength.
That could be the explanation of the zitterbewegung. But zitterbewegung is not present in Dirac solutions which contain purely positive frequency waves. How would we explain the absence of zitterbewegung in that case?
In this blog we have been claiming that a static electric field cannot produce pairs. But if we have an electron present, then it is a system of an electron plus its static electric field. Maybe we could allow pair production in such a case?
Can we avoid non-causality if we allow pair production close to a free electron?
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