| q momentum
v gained by e-
v ≈ c
e- • --->
R = minimum distance
/ | \
| | | E strong field,
\ | / dense energy
● proton+
When e- passes close to the proton, the total energy of the electric field E drops, but the field between the two particles becomes stronger, its energy grows, and this growing energy creates a repulsion between the particles. The repulsion, of course, cannot cancel the overall attraction, but the repulsion is significant.
Vacuum polarization can reduce the energy of the field between the particles, and make the attraction stronger.
Vacuum polarization adds a new degree of freedom to the system. The field between the particles can use this freedom to reduce the total energy of the system. It is a good guess that if the energy W can escape to the Dirac field, then the total energy of the system electric field & Dirac field is reduced by W.
When the electron passes the proton at a distance R, the spatial size of the extra strong field E close to the proton is ~ R, and the time is ~ R / c. It is a smooth disturbance pulse ("bump") of those dimensions. The wavelength of the momentum exchange q is ~ 861 R.
Semiclassical analysis, with point particles e- and proton+
Let us first analyze this semiclassically in the zone of the size R. We Fourier decompose the bump. The most important components have the wavelength ~R.
The bump is the sum of the field of the electron and the proton. When they are close, the energy density of that part of the electric field E is double the sum of the individual fields.
mildly relativistic
e- • --------------------------------------
|
| q virtual photon
|
/ \ k + q virtual electron-
\ / positron pair e- e+
|
| q virtual photon
|
● ---------------------------------------
proton+
static
The transient electric field E of the bump disturbs the Dirac field, through the coupling. We can construct the disturbance with Green's functions of the Dirac field.
We can imagine that q above designates the bump in E. The bump q hits the Dirac field through the coupling. The hit will create Fourier components k according to the propagator of the Dirac field. Because the bump is smooth, destructive interference will mostly cancel out |k| > |q|.
If q creates a virtual pair, then q will be absorbed to the pair, and the energy of the field E will weaken accordingly.
If the pair is long-lived, then the virtual pair will survive for the whole transit of the electron past the proton. The pair will reduce the energy of the field E for a long time. The pair will not escape as a real pair, because the electron e- is only mildly relativistic and cannot donate 1.022 MeV. The pair eventually must annihilate and give q back to E.
A good guess is that a long-lived virtual pair increases the attraction of the electron and the proton as if an extra momentum exchange q would happen between the electron and the proton.
A long-lived pair is called "almost-bremsstrahlung".
For q = 0, no almost-bremsstrahlung will happen. In the standard vacuum polarization calculation, this is represented by the integral
Π₂(0).
If q ≠ 0, then there will be almost-bremsstrahlung:
Π₂(0) - Π₂(q²).
In the vertex correction we saw that bremsstrahlung appeared as a missing part of the electric form factor F₁(q²) integral. The integral F₁(0) describes the process when the electric field of the electron can keep up with the movement of the electron. Bremsstrahlung is the field which "broke free".
Another way to view q ≈ 0: then R is large, and the transit of the electron past the proton lasts for a long time. Any virtual pair created by the bump will have plenty of time to annihilate, and they will not be able to reduce the average energy of E much.
The role of the coupling constants in the pair loop. We can understand why there is the (small) coupling constant e² / (4 π) in the first vertex which creates the virtual pair from q. But, since the pair necessarily has to annihilate as the electron leaves the proton, why is there the second coupling constant e² / (4 π) at the second vertex? Why is the coupling constant not 1?
One of the reasons might be time symmetry. The process must look the same if we reverse time.
Above we assumed natural units. In them, the coupling constant is e² / (4 π) ≈ 1/137.
The bump in E is at least crudely modeled by a single hit with a blunt hammer. The bump of the increasing E hits the Dirac field, and this hit, at least crudely, is like a hit with a blunt hammer. This partially explains the curious feature of Feynman diagrams that they seem to model complex processes with a single hammer strike. Why not 100 strikes?
Alternatively, the smooth orbit of the electron e- past the proton is a "blunt" disturbance, which can be modeled with a single hit of a blunt hammer.
Quantum magnification hypothesis
Quantum magnification hypothesis. In QED, the Feynman diagram of waves "imitates" the classical process with a
861 X = 2 π / α
"magnification".
Above we analyzed the classical electric field E between the point particles e- and proton+. If the scattering angle of the electron is large, then
R ~ re = 2.8 * 10⁻¹⁵ m.
That is 1/861 of the electron Compton wavelength λe = 2.4 * 10⁻¹² m.
The Fourier decomposition of the bump E has wavelengths which are ~ 1/861 of the electron Compton wavelength. Quantum phenomena only can have a resolution of ~ the Compton wavelength. Therefore, the wave representation of the process in a Feynman diagram must have waves ~ 861 times longer than the classical description. Is this a problem? Does a calculation with a 861 X magnification produce the same results as the classical calculation?
We know that the simplest Feynman diagram calculates elastic scattering probabilities which agree with classical formulae with point charges. At least in that basic case, the magnification works.
Bremsstrahlung. Let us have a mildly relativistic electron. Let us calculate from the Larmor formula the total energy W it loses if it passes at a distance R from a proton.
There,
a = 1 / (4 π ε₀) * e² / R² * 1 / me.
The time that the power P radiates is
t = 2 R / c.
The total radiated energy is
W = 4/3 * 1 / (4 π ε₀)³ * e⁶ / c⁴ * 1 / R³
* 1 / me²
≈ 2 * 10⁻⁵⁷ * 1 / R³
joules. The mass-energy of an electron is 511 keV ≈ 10⁻¹³ J.
If R = re = 2.8 * 10⁻¹⁵ m, then
W ≈ 10⁻¹³ J.
We conclude that the cross section of a mildly relativistic electron losing most of its kinetic energy in bremsstrahlung, classically, is very crudely π re², or
0.3 * 10⁻²⁸ m² = 0.3 barn.
The presumably correct empirical cross section is found from a Wikipedia article. It is something like
0.0008 barn.
We see that quantum mechanics efficiently prevents the electron from radiating when R << λe.
Quantum mechanics suppresses the production of high-energy real photons, but not virtual photons? The simple elastic scattering of an electron from a protons obeys the classical cross sections. But bremsstrahlung is severely suppressed when the minimum distance (impact factor) R is much less than the electron Compton wavelength λe.
Quantum mechanics allowed the momentum exchange q to be classical. The photon in q is virtual, it is pure spatial momentum.
Classical limit of the bump E in vacuum polarization
In the electron-proton scattering, let us make the charges N-fold and the masses N²-fold, where N is a large number. The new particle is a "heavy electron". We do not change the ordinary electron in any way. The heavy electron is a new particle, like a muon.
The heavy electron will still track the same classical path. This time, the bulge in the electric field E is classical. We can analyze the process in a localized spatial volume. We do not need to resort to the "momentum space" of Feynman diagrams.
In this semiclassical treatment, obviously, high 4-momenta |k| are strongly suppressed by destructive interference. The time-dependent electric field E disturbs the Dirac field. We can construct the impact response with Green's functions of the Dirac field. Destructive interference cancels high 4-momenta. It is like hitting a tense rubber membrane with a blunt hammer.
Feynman diagrams do not prohibit macroscopic masses or charges. In this classical limit, we know that destructive interference removes ultraviolet divergences.
The classical limit also demonstrates why Feynman diagrams can do with a single hit or the "sharp hammer", or a single application of a Green's function. When the heavy electron comes close to the proton, the change in the field E can be crudely approximated with a single hit of a blunt hammer. Generally, processes which last a limited, short time, often can be modeled with a single hammer hit.
Moving from the classical limit back to the quantum realm
| q momentum
v gained by e-
v ≈ c
e- • --->
R = minimum distance
/ | \
| | | E strong field,
\ | / the "bump"
● proton+
The crucial question is what happens if we switch from the heavy electron back to the normal electron. Does the system still behave in a classical way?
When the mildly relativistic electron passes the proton, the process lasts a time
t = R / c.
Let us calculate from the energy-time uncertainty relation how much energy W the system is allowed to "borrow" for such a short time.
W t = h / (4 π)
=>
W = 1 / (4 π) * h c / R.
The energy required to create a photon of wavelength R is
W' = h f = h c / R.
We see that we can "almost" borrow enough energy to describe a bump of a size R with a real photon, for the duration of the transit. This suggests that vacuum polarization, which is due to virtual pairs, could be replicated in the classical form, even though we have far too little energy to create real photons which would build the bump. We can borrow enough energy?
Peskin and Schroeder (1995) formula (7.91) has the vacuum polarization correction in QED. If we scale me and q by a factor N², the integral on the right does not change. But scaling the charge of the electron by N adds a factor N² to α = e² / (2 π) (in natural units).
In bremsstrahlung, the Planck constant h has a dramatic effect for the electron, because the electron cannot emit a wave such that h f is larger the kinetic energy of the electron. Apparently, no such dramatic effect is present in vacuum polarization.
Superlinear vacuum polarization as "almost-bremsstrahlung"
We can imagine that a pointlike electron constantly hits a tense rubber sheet with a sharp hammer. The formed pit is the electric field of the electron. Since the electric field interacts with the Dirac field, these hits indirectly also hit the Dirac field, forming some kind of (static) vacuum polarization.
If the electron is free and not under an acceleration, the system is static. The electron is surrounded by an electric field and, presumably, a nonzero Dirac field.
v ≈ c
e- • ---->
| q momentum
v change
R
●
proton+
When the electron flies past a proton, the hammer hits no longer form a static configuration. The electron gains a momentum q.
A new hammer hit fails to "absorb" everything from the previous hit. The failed part escapes as electromagnetic bremsstrahlung.
In the case of the Dirac field, the failed part is "almost-bremsstrahlung" of a virtual pair. This is the "superlinear polarization" we have been talking about. If the virtual pair does not possess 1.022 MeV, it cannot escape as a real pair – but it can linger for the duration of the electron transit.
The momentum gain q reflects the maximum loss W of the energy of the combined electric field E as the the electron flies by. These are roughly linear if we keep R constant:
q ~ W.
e- • --->
| q
|
O virtual pair
|
| q
●
proton+
A formed virtual pair means that some electric field energy escaped to a new "degree of freedom". How much energy? Since the field energy fell by W as the electron came close to the proton, we can guess that the virtual pair drains the same energy W.
At the beginning if this blog post we noted that if an energy W can escape to a new degree of freedom, then the total energy of a system often falls by the same amount W. The falling total energy means that the attraction between the electron and the proton is stronger. The magnitude of the added attractive force comes from the relation q ~ W.
Almost-bremsstrahlung hypothesis. The increased attraction between the electron and the proton comes from the electric field energy escaping, for a while, to a virtual pair. This hypothesis explains why the correction makes the attraction stronger between charges of a different sign, and the correction is:
Π₂(0) - Π₂(q²).
If the charges are of the same sign, the escaping energy makes the repulsion weaker.
Does the almost-bremsstrahlung interpretation solve all ultraviolet divergences in any vacuum polarization?
If in some quantum field theory,
Π₂(0) - Π₂(q²)
ultraviolet diverges, then we have a problem. Classically, it would mean that an infinite amount of energy escapes, for a while, into a new degree of freedom, if the particle absorbs a momentum q. The energy in the almost-bremsstrahlung is infinite and the attractive force becomes infinite.
The same problem could appear as infinite bremsstrahlung if we just have a force field surrounding a particle. No vacuum polarization is required to get an ultraviolet divergence if we ignore the fact that the energy must come from somewhere.
These problems would arise from the fact that the energy of the force field of a point particle is, formally, infinite. The acceleration caused by q can then, potentially, extract an infinite energy.
An obvious solution to the problem is to claim that the force field of the particle cannot contain more mass-energy than the mass of the particle. Does that work?
Another possible solution is to appeal to destructive interference. But in the classic model, the field close to a point particle is a problem. Its Fourier decomposition contains very short wavelengths.
The true "gravity charge" is always positive in a Feynman diagram?
Classically, any disturbance in a field always carries a positive energy. Feynman diagrams allow arbitrary negative energies in a virtual particle, but is this "negative energy" just a formal tool, and does not represent genuine negative energy which would have a negative gravity charge?
A sharp hammer hit (Green's function) always puts positive energy into the disturbed field, even if a Fourier component would formally carry a negative energy.
A static field contains energy, even if the Fourier decomposition is time-independent, and formally has no energy.
In general relativity, people often require that the energy density in space is everywere ≥ zero. Otherwise, we could arrive at time paradoxes. This is called an energy condition.
In a Feynman diagram, the 4-momentum really tells us about a net transfer of 4-momentum during the entire process. The "gravity charge" of a line does not need to be the formal amount of energy specified in the line.
In QED, in the vacuum polarization loop, the electron may carry a positive energy W, and the positron a negative energy -W. But classically, polarization of a material always requires positive energy.
If the gravity charge is always positive, is vacuum polarization possible in gravity at all?
In QED, the electric field of colliding charges pumps energy into the Dirac field, creating electric polarization. What would be an analogous process in gravity? We would need energy to create a positive energy particle and pull it away from a negative energy particle? Opposite gravity charges, presumably, repulse each other. Why would we need energy to pull them apart?
Note that a collision of electric charges can produce a real electron and a real positron. Vacuum polarization can be seen as an "almost-production" of a pair. But we do not believe that the corresponding process is possible in gravity. If a particle has a negative energy and a negative gravity charge, the repulsion from masses could lift its energy close to zero, but not to a positive territory. And we not believe that a real particle can have a negative energy.
Hypothesis: no vacuum polarization in gravity. All particles, virtual or real, have a positive "true" energy and a positive gravity charge. There is no vacuum polarization in gravity.
If the hypothesis is true, it solves all problems with vacuum polarization in gravity. It also removes the awkward possibility that we could use quantum mechanics to create a zone of a negative gravity charge.
The hypothesis bans the existence of any virtual particles in empty space. In this blog, we have repeatedly claimed that there cannot exist virtual particles in empty space. Such particles would break unitarity of quantum mechanics, because they would introduce an unknown causal factor into quantum mechanical processes.
Always positive gravity charge and arbitrary loops in Feynman diagrams of gravity
Above we argued that the gravity charge always has to be positive for a virtual particle, even for a particle which formally has a negative energy in a Feynman diagram.
~~~~~~~~~~~~~~~~~~~ graviton
| |
| | virtual gravitons
| |
~~~~~~~~~~~~~~~~~~~ graviton
--> t
Loops with gravitons are known to ultraviolet diverge in quantum gravity. Let us analyze what our hypothesis would imply for them.
Is the diagram above sensible at all? If a graviton scatters from another graviton (denoted by the first vertical virtual graviton), how could it scatter a second time?
Conclusions
We found an intuitive model for QED vacuum polarization, and an explanation why the renormalized Feynman integral may calculate it correctly.
Vacuum polarization effects have been measured experimentally in high-energy electron-positron scattering experiments, like the LEP of CERN. The precision of the experiments probably is not very good. They do agree with QED predictions.
In low-energy experiments, vacuum polarization probably has a very small effect. We do not know if such experiments have tested vacuum polarization.
In nature, ultraviolet divergences are a concern only in connection with gravity. Our hypothesis of no vacuum polarization in gravity may solve the problems. But let us do more research. What is bremsstrahlung and the vertex correction like in gravity?
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