The "lifetime" of a vacuum polarization pair is very short? Then the Feynman integral converges

https://physics.stackexchange.com/questions/53802/what-is-delta-t-in-the-time-energy-uncertainty-principle

The Heisenberg uncertainty relation for energy and time is

       ΔE Δt >= h / (4π),

where h is the Planck constant.

Let us calculate how many wavelengths does a virtual photon with a borrowed energy E = hf live:

       h f Δt = h / (4π)

<=>

       Δt = 1 / (4π f)

It lives for a length

       L = c / f * 1 / (4π)

The wavelength λ = c / f. We see that a virtual photon which has borrowed all its energy lives for a meager 0.1 wavelengths.

In vacuum polarization, the pair borrows all its energy and momentum from the vacuum. The lifetime of the pair is extremely short, and we may speculate that its spatial extension is equally short, typically just 0.1 wavelengths.

The graphical appearance of the Feynman vacuum polarization diagram looks suspicious:

                 virtual photon

        Z+ ~~~~~~~O~~~~~~~~ e-

                        loop of

                        virtual

                        e- and e+

How can a virtual pair which resides midway between a nucleus and an electron affect the Coulomb force felt by the electron?

The weakening of the Coulomb force is supposed to happen because the loop "reflects" the virtual photon and causes a half a wavelength phase shift to the photon. The phase shift does destructive interference to the probability amplitude calculated without the loop:

               virtual photon

       Z+ ~~~~~~~~~~~~~~~ e-

In an earlier blog post we wrote about the the phase shift which a vacuum polarization loop causes to a real photon flying in empty space. We argued that the phase shift would break conservation of the speed of the center of mass.

Additionally, destructive interference in a free real photon would break conservation of energy.

The concept of a photon bouncing from an "object with zero energy and zero momentum" does not sound right. In classical physics, that kind of a process would break conservation laws.

We suggest the following principle:

A photon flying "freely" cannot be affected by vacuum polarization loop pairs.

However, if the vacuum polarization loop is very close to the electron e-, then it can affect the behavior of the system.

                                             ______

                                           /            \

       Z+                             e-     e-     e+

                                           \_______/

                                            virtual

                                              pair

In the diagram, we have an example of classical polarization. The nucleus Z+ attracts the virtual positron e+ and repels the virtual electron e-. The effect is that the electron in the middle feels less Coulomb force.

We noted in an earlier blog post that an interacting classical particle can break the energy-momentum relation and be "virtual" in that special sense. The loop in the diagram has strong interactions present. It would be no suprise that the vacuum polarization effect is felt by the electron in the middle, even though it is not felt when the loop is farther away.

The divergence of the Feynman vacuum polarization integral can be seen as a result from the claim that virtual pairs anywhere in space would contribute to weakening of the Coulomb interaction. Our new rule makes this more sensible: only the pairs which classically could weaken the force are counted.

e- ------------>  3 * 10^-15 m  <------------ e-

                 

Suppose that we have two relativistic electrons which collide and exchange about 500 keV in momentum. Classically, they will come to within 3 * 10^-15 meters of each other (= the classical radius).

The Compton wavelength of 500 keV is 2 * 10^-12 m.

A virtual pair which has borrowed 500 keV will "live" for 0.1 wavelengths, or 2 * 10^-13 m.

We see that we cannot exclude virtual pairs whose 4-momentum is < 33 MeV. They can take part in vacuum polarization.

For larger 4-momenta (E, q), we can put an attenuating factor

       C = 33 MeV / (|E| + |q|)

into the Feynman integral for the loop. Without the attenuating factor, the integral would diverge logarithmically, when we integrate spherical shells E^2 + q^2 = r^2 over r.

The attenuating factor makes the integral to converge.

We have argued that one should implement a smooth cutoff at 67 * the exchanged momentum of the electron collision. We need to calculate what kind of running does that hypothesis cause in the coupling constant.

Is there destructive interference in the vacuum polarization loop in a Feynman diagram?

Suppose that a virtual photon with a spatial momentum p approaches our electron. Suppose that at each point in its flight, it can produce a virtual electron whose 4-momentum is k, and the initial phase of the virtual electron is determined by the photon.

The phase of the virtual electron might at the creation be equal to the phase of the photon.

If the virtual electron has a very big |k| and it could live for many wavelengths, there would be significant destructive interference. But if the lifetime of the virtual electron is just 0.1 wavelenghts, then we can ignore destructive interference. This solves the question we have been pondering for a long time about destructive interference.

A model of real pair production from two photons: a model how a photon acts as a source of the Dirac equation

We can introduce a new model which explains how real pairs are produced.

A photon, virtual or not, always and everywhere disturbs the Dirac field and acts as a source for a new Dirac wave.

The electromagnetic field as a source for the Dirac field. A non-zero electromagnetic field constantly at every point in spacetime produces new virtual electrons using the Green's function for the Dirac equation. The electrons inherit their phase from the photon of the electromagnetic field.

The Green's function is the "impulse" response of the Dirac field: we kind of hit the Dirac field with a sharp hammer and look how it vibrates.

However, since the system the photon & the virtual electron and the virtual positron is off-shell, the "vibration" of the Dirac field dies off very quickly, after about 0.1 wavelenghts.

On-shell electrons can be seen as "resonant vibrations" of the Dirac field. They live forever. Off-shell vibrations die off very quickly.

Feynman diagrams apparently treat photons as sources of the Dirac field, just as we described above.

Real pair production can happen when the photon is colliding to another photon and is very close to it. Then there are strong interactions and a part of an off-shell electron wave can escape as an on-shell wave. The model qualitatively explains the cross-section of pair production from two photons.

How exactly do the on-shell Dirac waves escape from the two colliding photons? We have no model of that. It is a strongly interacting system of the the electromagnetic and Dirac fields. It would be hard to calculate the exact behavior. It is like an oar making waves in water: close to the oar, water and the oar are interacting strongly. It is hard to calculate how water moves.

The converse process - annihilation of an electron and a positron - is equally hard to describe exactly. But we can use conservation of energy and momentum to determine which end results are possible. Dirac in 1930 was able to calculate the cross section using a semiclassical model.

The electromagnetic field as a source for the Dirac field

The lagrangian of QED describes how a classical electromagnetic field A and a non-zero Dirac field interact. Klein and Nishina in 1928 were able to calculate the behavior of an electron field under the classical electromagnetic field of a photon, and derived the formula for Thomson scattering.

But how does field A interact with a zero Dirac field? In pair production, A has to conjure up vibrations into the Dirac field, so that the real pair is produced. How does A do that?

Our hypothesis above says that A is constantly acting as a source for the Dirac field. Can we derive this from the lagrangian? We have to check if anyone has succeeded in that.

Production of a virtual pair can be viewed as "tunneling" of electromagnetic energy or momentum into the Dirac field. The lifetime of the pair is short. This brings up another question: in Schrödinger's equation, a tunneling particle has negative kinetic energy and an imaginary momentum. What effect does an imaginary momentum have on things?

The Dirac field as a source for the electromagnetic field

A classical electron has the Coulomb field around it. The Coulomb field can be modeled as "constant hammering" of the electric potential with a sharp hammer. That is, we can build the 1 / r potential by hitting the electromagnetic field with a Green's function at infinitesimal time intervals at the electron position.

     t

      ^      hammer hits at short intervals

      |            :           

      |            :           #         sharp hammer

      |            :           #=========

      |            :           #

      |            :           v

      |            :

      |            :

       ---------------------------------------> x

                   e-

The time-independent components of the Green's function have constructive interference.

Time-dependent components would have complete destructive interference if their waves would reach to infinity. But the model we introduced above says that the waves only reach out some 0.1 wavelengths.

The time-independent components form the Fourier decomposition of the 1 / r potential.

Do the time-dependent components form the "dressed electron"? Is there any way to see the "dress"?

Anyway, here we have a model how a point particle can act as a source for the electromagnetic field A.

But how can the Dirac field act as a source?

If we have an electron with zero momentum somewhere in a large spatial volume, how do we create the Coulomb potential?

Maybe we should do just as we did in the previous section.

The Dirac field as a source for the electromagnetic field. A non-zero Dirac field constantly at every point in spacetime produces new virtual photons using the Green's function for the electromagnetic wave equation. The photons inherit their phase from the electron.

Feynman diagrams contain the above rule for the production of photons.

Created virtual pair as a "bound state"

We can view a virtual pair as a positronium atom which we have momentarily raised up from the zero energy, zero mometum state. Then it is a "bound state", and Feynman diagrams are known not to work for bound states.

If a virtual pair is created far away from a real electron, then in our model, it cannot pull or push on the real electron.

Why is that? Should we treat the electric field of the pair as the sum of the electron and the positron, and not separarely each field?

A hydrogen atom in the lowest energy state appears as electrically neutral to the outside world. At least in that case, we have to sum the electric field of the proton and the electron.

Or could it be that the short lifetime of the virtual pair prevents it from communicating with the electron far away?

Experimental measurements of the QED coupling constant at different energy scales

A quick Internet search reveals that the QED coupling constant α has been measured at the energy scale 0.6 GeV to 180 GeV. At 180 GeV, the coupling constant is some 7% stronger.

However, effects due to hadrons seem to dominate the change in α in the range from 0.6 GeV up. We cannot test our new hypothesis against that data.

We would need to know the constant at the 1 MeV scale, so that heavier particles do not affect vacuum polarization.

There is a claim that vacuum polarization in QED affects the Lamb shift by some 27 MHz. That contribution would come from low energies. We have to check that.

We can extend our new model to QCD and check if it affects vacuum polarization produced by hadrons.

Some back-of-the-envelope calculations suggest that α in our new model could differ by up to 1% from the value predicted by renormalization procedures, in the energy scale 10 MeV to 100 MeV. This is because in our model the cutoff depends on the energy scale of the collision experiment. In renormalization, the cutoff is often set at a constant large value Λ which does not depend on collision energy.

We need to check what our new model predicts about α at low energies, say, 1 eV to 1 keV. If we lower the cutoff to 30 eV, then the bare charge of the electron would become visible because the effect of vacuum polarization is very small?

The coupling constant α is experimentally a constant at hydrogen atom energies in the 10 eV scale. In our new model, should we put the cutoff at 30 eV or at 30 MeV in those phenomena?

What is the right way to calculate vacuum polarization effects?

We argued above that a Feynman diagram is an incorrect way to calculate the effect of vacuum polarization. Feynman diagrams do not work for bound states.

We were able to get the Feynman vacuum polarization integral to converge by removing those loops which cannot affect the scattering electron.

However, is there any reason why the corrected formula would model vacuum polarization right? The Feynman diagram might be a totally wrong model. Making the integral to converge would not make the formula any more correct.

We need to think about this. In an earlier blog post we introduced the principle that a photon field always acts as a source for the Dirac field. The coupling constant determines how efficient a source the photon field is.

If the photon field or the Coulomb field close to the electron produce virtual pairs, how do we calculate their effect on the electron?

The Feynman way of calculating the effect does look reasonable in the momentum space description of the process.

Honoring conservation of the speed of the center of mass removes divergence in vacuum polarization?

In our previous blog post we argued that the energy E of the the positron and the electron in a vacuum polarization loop has to be greater or equal to zero. This is to conserve the speed of the center of mass.

e-  --------------------------

                   |  photon, no energy

                   | 

                   O vacuum polarization

                   |  loop

                   |

Z+ --------------------------

In the simplest Coulomb scattering diagram where the exchanged momentum has E = 0, both particles in the loop must then have E = 0.

This effectively removes one dimension, the time dimension, from the integral for the loop.

Recall that there is no time dimension in the calculation of the diagram without the loop: the "virtual photon" which pushes the colliding particles is a wave which undulates in spatial dimensions but not in time. The wave is a Fourier component of the static Coulomb potential which is time-independent.

Classically, Coulomb scattering does involve exchange of energy: when the electron approaches the nucleus, potential energy is temporarily converted to kinetic energy of the electron. The simple Feynman diagram ignores all this detail and just calculates the end result: spatial momentum was exchanged.

If there exists hypothetical vacuum polarization, it might be that its effect has to be calculated in the same fashion: just look at the end result, where no energy was exchanged. We may drop out the time dimension from the calculation, just as we did for the simplest diagram.

       + /\/\/\/\/\ -

          spring

Classically, an electric dipole in polarization may be modeled with a harmonic spring device. When a charge approaches, some energy is stored into the spring. When the charge recedes, the spring returns the energy back. The spring does not retain any energy.

Also, maybe we should drop one spatial dimension, too? Classically, the electron and the nucleus move in a plane. Why should we need to calculate in 3 spatial dimensions?

Additionally, the momentum exchange lives in a plane. Why would a single virtual pair whose momentum is not in that plane take part in the process?

We have argued that it makes sense to drop a dimension or two in this case from the corresponding Feynman integrals. Does the vacuum polarization loop diverge in that case? Probably not. Removing one dimension is like doing dimensional regularization. If there are less than 4 dimensions, then the integral converges.

https://canvas.harvard.edu/files/936398/download%3Fdownload_frd%3D1%26verifier%3DPAHa18X0trEAYjwcsmQXrhusjECF3ct6JNxDQ41E

The note by Matthew Schwartz (2012) in the link contains calculations about dimensional regularization of the vacuum polarization loop. We need to find a more detailed calculation and check how it is modified if time is dropped.

https://homepages.dias.ie/ydri/qft_chap_five.pdf

Badis Ydri (2011) has written a more detailed calculation.

If we drop the time dimension, the integral Π^μν_2 in formula (31) in Ydri's paper seems to diverge worse than with the normal 4-dimensional space. That is because in the square of the 4-momentum,

        k^2 = -E^2 + q^2,

the negative term -E^2 cancels positive contributions of spatial momentum q^2. If we drop time, we lose -E^2.

The divergence of the integral in Ydri's formula (31) is only logarithmic, because of cancellation effects, if we integrate one thin spherical shell at |k| = r at a time. The divergence would be worse if we would integrate in some other order. Absolute convergence is what we would like to have.

We need to analyze if we can drop also one spatial dimension. Also, we need to check destructive interference again.

What kind of operations in a Feynman diagram honor conservation of the speed of the center of mass?

NOTE December 14, 2020: We can allow negative masses in classical physics, and still honor conservation laws. A negative mass just moves in the opposite direction to the momentum vector p. The problem in the Feynman vacuum polarization formula seems to be that it allows the antiparticle "magically" make the particle to disappear, wherever the particle is.

An example of "negative mass" in classical physics is a balloon filled with air, submerged into a water pool. If we move the balloon to some direction, the system center of mass moves to the opposite direction.

-------

In an earlier post this week we remarked that if an electron in a vacuum polarization loop has positive energy and it carries some momentum q from a particle A to particle B at some distance, then conservation of center of mass is broken.

What is the root cause of the problem?

If the loop electron has zero energy, then it relays the momentum immediately forward, in zero time. Conservation is ok then.

The problem might be that we allow a negative energy particle, the loop positron. (We do not refer here to the formally negative E in the Dirac positron wave, but negative mass-energy as observed by an outsider.)

In classical mechanics we do not have negative mass particles. Also, a zero mass particle cannot absorb momentum for a time > 0. These recipes ensure that the speed of the center of mass is conserved.

What about particles which do not obey the energy-momentum relation

       E^2 = p^2 + m^2

(where c = 1)?

Can they break conservation laws in classical physics? A week ago we blogged about the analogy of a virtual particle in classical physics. If the particle is interacting, it may disobey the energy-momentum relation. Conservation laws are honored, though. Thus, at least some virtual particles do behave well.

Coulomb scattering

Let us consider the simplest interesting case: Coulomb scattering of an electron e- from a nucleus Z+.

No energy is exchanged. The electron receives a virtual photon from the nucleus. The photon carries pure momentum q.

            Z+ ●   

                            ^

                            |

                            |

                            |

                           e-

If a vacuum polarization electron has E > 0, then its wave travels "forward in time", while the positron wave travels "backward in time" since E < 0. How can the pair then annihilate each other? Time travel maybe?

Let us only allow pairs where both particles have E = 0. Then their waves only undulate in the spatial directions, not to the time direction. These particles are "frozen in time" like the virtual photon that only transfers momentum, no energy.

The virtual photon is a Fourier component of the static Coulomb potential. What kind of a time-independent curve do the frozen electrons represent?

The divergent vacuum polarization integral becomes very much different if we only allow E = 0. We need to calculate what it is then.

The logic of Pauli-Villars renormalization and Kenneth Wilson's hypothesis

https://www.mpi-hd.mpg.de/personalhomes/harman/HCI-WS2013-2014/Lecture-11.pdf 

(Zoltan Harman, 2014)

The mathematical limit of the vacuum polarization loop integral is not well-defined if we integrate over all momenta p. The contributions of various values of p may cancel each other out: the limit may depend on the order of integration.

Or the limit may be infinite for all integration orders.

By the integration order we mean the order in which the contributions of various p are summed to the integral.

The integral can be seen as an infinite series where the terms have various complex values. Some values cancel each other out. The limit of the series may depend on the ordering of the terms. The limit may be infinite for some or all orderings.

The basic idea of Pauli-Villars:

1. We guess that unknown laws of nature suppress the contribution of high |p|. Nature imposes a smooth cutoff for high |p|.

2. We guess that the smooth cutoff is like the one of Pauli-Villars, for some large Λ. Or if the integral converges for arbitrarily large Λ, then we guess that the "right" integration order is the order given by Pauli-Villars.

Kenneth Wilson's scale hypothesis

According to a hypothesis of Kenneth Wilson, microscopic behavior of Nature imposes a cutoff on the ill-defined integral.

There exists some correct microscopic theory T of nature. Vacuum polarization masks the low-level theory T. In a laboratory we measure the sum of the "true" electric potential of an electron (in T) and the masking effect caused by vacuum polarization.

What we measure is the familiar Coulomb potential. One can then speculate what the "true" electric potential of an electron might be in T, behind the mask of vacuum polarization.

One hypothesis is that the coupling constant is stronger for high momenta in the vacuum polarization loop. This is called a running coupling constant. With the running, we can get the vacuum polarization loop integral to stay constant for different large cutoffs Λ' and Λ.

Assume that the unmasked electric potential is much stronger than the measured potential. There is an unknown large Λ, up to which we have to integrate the vacuum loop so that we get the measured Coulomb potential.

If we just integrate the loop over Λ' < |p| < Λ, does the limited integral carry some meaning?

Could the limited integral give us some intermediate low-level theory for an "energy scale" Λ'?

If we assume the running coupling constant hypothesis, then the integrals from 0 up to Λ' and Λ have the same value. The limited integral from Λ' to Λ is zero.

To get concreteness to all this, we should be able to observe scattering so that only virtual pairs whose momentum |p| is in a certain range, need to be summed over the low-level theory T.

But in a Feynman diagram, we need to sum over all |p|.

Suppose that in a collider with very high energies, scattering fails to conform to the simplest Feynman diagram. There might be more scattering than predicted. Then one may speculate that the coupling constant is larger for large energies.

Let us find out what experimental evidence there is about a running coupling constant in QED.

Classically, a surprisingly large cross section for Coulomb scattering with high energies implies that the electric potential is steeper than 1 / r.

https://cds.cern.ch/record/836614/files/arXiv:hep-ex_0505072.pdf

The OPAL Collaboration (2006) measured the coupling constant at 1.81 GeV^2 and 6.07 GeV^2.

In literature, from the LEP there is robust evidence that the cross section really is larger than expected if we would assume that the coupling constant stays constant for large energies.

https://www.nikhef.nl/~h24/qcdcourse/section-6.pdf

In the link, (Michiel Botje, 2013), the running is explained by dividing the diverging integral into one over 1 / z and a converging part which depends on the momentum q transferred in Bhabha scattering. The converging part does change when |q| is raised. The integral over 1 / z is made finite by imposing a cutoff M for large momenta.

We need to think more about this. What kind of vacuum polarization processes can conserve the speed of the center of mass? Can those processes explain the running of the coupling constant in QED?

Feynman vacuum polarization pair breaks conservation of the center of mass in the general case?

A few days ago we remarked that if a virtual pair absorbs a real photon and delays its progress (for half a wavelength), then conservation of the speed of the center of mass is broken.

The same problem seems to exist also with a virtual photon which carries just momentum in Coulomb scattering.

  e-  ----------------------------------------

                        |

                        O   vacuum polarization

                        |    p virtual photon

  Z+  ----------------------------------------

In the diagram, the nucleus Z+ pushes a positron e+ in the vacuum polarization loop and gives it the momentum p. The positron continues its travel, pulls on the electron, and gives the momentum p to the electron.

After that, the positron annihilates with its pair, a virtual electron, and nothing is left of the pair: no energy, no charge, no momentum.

While the momentum p was piggybacking the positron, it moved the center of mass through the movement of the mass of the positron. But when the positron annihilates, the movement of the center of mass is lost.

Feynman's momentum space view enforces conservation of energy and momentum, but it does not enforce conservation of the center of mass.

In a Feynman diagram, the contribution of the virtual loop has a half a wave phase shift relative to the simplest diagram where the virtual photon carries p directly from Z+ to e- and there is no loop. The vacuum polarization contribution causes destructive interference and cancels part of the probability amplitude of the simplest diagram.

Can the phase shift save the center of mass speed?

The contribution of the simplest diagram might be seen as a wave packet where the packet is spread over one meter, or 3 nanoseconds for a relativistic speed. The uncertainty in the momentum of the input electron is 10^-33 kg m/s, which for the electron is one millimeter per second.

If we are monitoring one cubic meter, the flight of the scattered positron may take 3 nanoseconds to give the momentum p to our real electron.

The delay of 3 nanoseconds is significant: the destructive interference only cancels the later part of the electron wave packet. The end result is that in the sum, the electron wave packet has progressed too fast, compared to the simplest diagram, gor which we assume that the speed of the center of mass is conserved.

Obviously, if we have one history where center of mass is ok, and subtract another where it is not ok, then the center of mass is not ok in the sum.

We need to study the Feynman diagram machinery in more detail. Momentum space, which is used in calculations, ignores the location of particles and can lead to errors which break conservation of the speed of the center of mass.

The notion of "zero energy virtual pairs" which pop out of nothing, and affect what happens in the real world, and disappear again, is generally suspect. They easily break conservation laws of newtonian mechanics.

https://journals.aps.org/pr/abstract/10.1103/PhysRev.76.769

In section 7 of the 1949 Space-time paper, Feynman writes that closed loops must be allowed, so that a produced (almost real) pair can be annihilated again.

A resolution of the Abraham-Minkowski controversy about the momentum of a photon in a medium: Abraham is right

https://en.wikipedia.org/wiki/Abraham–Minkowski_controversy

The controversy touches our analysis of the photon phase shift in the real photon vacuum polarization diagram. We blogged last week and claimed that a photon which goes through a glass pane pushes that pane a little bit forward. This is because the speed of the center of mass must stay constant.

Let the refractive index be n = 2. Abraham claims that the photon inside the glass has lost half of its momentum to the glass pane. Minkowski, strangely, claims that the momentum of the photon has doubled!

https://physics.stackexchange.com/questions/3189/is-the-abraham-minkowski-controversy-resolved

The strange claim of Minkowski derives from the photon wave function which inside the glass looks roughly like this:

       exp(-i (E t - 2 p x))

(we have set c = 1 in vacuum, h = 1). The energy E is the same as when the photon was flying in vacuum. The coefficient 2 is there because the speed of light is just half in the glass (n = 2). Thus the wavelength is just a half, and the phase of the wave function must rotate faster when we move along the x axis. If the wave would exist in empty space, then 2 p would really be the momentum.

What would an experiment say? We believe in the conservation of the speed of the center of mass. Then it has to be that the light pulse has lost half of its momentum to the glass pane when the light pulse is inside. The light pulse could be a short laser flash.

Suppose that we have something inside the glass pane which absorbs the light pulse. That something will absorb the rest of the momentum in the light pulse.

Does the same hold for an individual photon? We may imagine that the photon has a very large energy. Then it should be no problem to measure separately the momentum which it loses to the glass pane when it enters, and the momentum which it loses in absorption.

Thus, Abraham is right. This also shows that the energy-momentum relation for an interacting photon is not the same as for a free photon. For a free photon, |p| = E, but inside the glass, |p| = E / n.

Vacuum energy is zero, not infinite?

https://en.wikipedia.org/wiki/Vacuum_energy

Each point in space could act as an oscillator for an electromagnetic wave. In quantum mechanics, the lowest energy state of a harmonic oscillator has kinetic energy 1/2 hf, where f is the resonance frequency.

Does this imply that each point in space contains substantial energy, and consequently, energy density in space is infinite?

Another way to look at the problem is to imagine a cubic vessel whose walls are perfect mirrors. Photons can form standing waves there at an infinite number of frequencies f. If each frequency has some minimum energy, then the energy content is infinite.

A crystal of atomic matter can contain sound waves. Each atom can be treated like a little harmonic oscillator. The minimum kinetic energy of these atoms is significant. If space is analogous to the crystal, then the energy of the "atoms" in space is infinite.

There might be an error in these arguments. A quantum mechanical oscillator contains a massive particle, often an electron. Even if the oscillator would have zero kinetic energy, it would have the mass-energy of the particle.

In quantum mechanics, we must describe the particle as a wave. A wave packet which is confined in a small space must contain high frequencies. These frequencies mean a high momentum and a lot of kinetic energy.

An electron can only have p = 0 if its wave function is spread over an infinite volume of space.

If we take the electron out of the oscillator, then the oscillator can have zero kinetic energy.

What about photons and space as an oscillator? If there is no photon, then the oscillator is empty, and its total energy can be zero. Thus, the energy content of empty space is zero.

The electromagnetic wave function of empty space is everywhere zero. This is a valid solution of the classical electromagnetic wave equation.

On the other hand, if we have one electron confined into a small space, then the classical Dirac wave function must contain p != 0, that is, it must have kinetic energy.

Working with classical fields, we see that a typical harmonic oscillator really must have p != 0. But a field is allowed to be identically zero if it does not contain mass or energy.

What about a harmonic oscillator frame which only gains mass-energy when it is excited? The lowest energy state of such an oscillator is zero. Empty space can be seen as an oscillator frame which only oscillates if mass-energy is put into it.

In this blog we have suggested the following thought experiment: we have a tense string of zero mass. When we feed a wave into it, the string gains mass-energy. The gained mass-energy is the thing which the string is oscillating when the string moves.

Could this be analogous to empty space and electromagnetic wave energy which is fed into it? No. The mass-energy of the string is larger when we grow the amplitude of waves. Then those waves would move slower. The same is not true for electromagnetic waves.

Dark energy in cosmology might be true vacuum energy. A Higgs type field which is non-zero everywhere might contain energy.