Friday, May 29, 2026

Solving the Hubble tension: slow down the expansion of the universe during the first 500 million years

Supernovae of type Ia, and Cepheids suggest that the universe is right now expanding at a rate 73 km/s / Mpc, and, furthermore, the expansion rate is accelerating (dark energy).

But the mapping the cosmic microwave background, and combining it to a ΛCDM model says that the expansion rate is only 67 km/s / Mpc.

This called the Hubble tension. In our previous blog post we suggested a simple solution to the Hubble tension: slow down the expansion rate of the universe by a factor about 1/2X, when the universe is 150 million ... 500 million years old.

We do not know why the expansion of the universe started to speed up some 5 billion years ago. We cannot take for granted that the universe has obeyed the ΛCDM formula during the previous 8.7 billion years, either.

Let us investigate how people derive the figure 67 km/s / Mpc.


What we know of the cosmic microwave background


Literature seems to be certain that the "last scattering" temperature of the CMB was 3,000 K, at a precision +-0.5%. Since the temperature today is 2.73 K, we know that the redshift since the last scattering is

       z  =  3,000 K / 2.73 K

           = 1,100.


The spatial size of the sound horizon in comoving coordinates



In the link there is a textbook by Dr. Hannu Kurki-Suonio, in which the spatial size of the sound horizon is calculated:






where cs(t) is the sound speed (about 60% of c), and a(t) is the scale factor. The integral formula assumes a standard ΛCDM model. The speed of sound before the recombination (last scattering, or decoupling) is:








where ωb is the baryon density and ωγ is the radiation density. We know their approximate values by observing the current universe, and from knowing that the current temperature of the CMB is 2.725 K.

The expansion speed of the universe before recombination follows the formula:






There Ωr is the share of the radiation density of the critical density and Ωm is the share of matter of the critical density. Since |a| is very small, we can ignore the a² and a⁴ terms.

The value of the integral is:













The sound horizon size r* depends on the density of baryons and matter in the current universe. Its dependence on the Hubble constant H₀ in the current universe is not significant: if we calculate time backwards, the final collapse speed of the universe does not depend much on the collapse speed right now.

Apparently, the baryon density and the matter density in the current universe are known quite well. The radiation density we know very well, too. The redshift at the decoupling, zdec, is known well. Thus, if we assume the ΛCDM, we know the sound horizon size r* quite precisely.


The Hubble tension


Our best knowledge of the expansion speed of the current universe come from type Ia supernovae and Cepheids. The data from supernovae and Cepheids agrees about the value 73 km/s / Mpc.

If we assume the ΛCDM model, and calculate backward from the current universe, the cosmic microwave backround rises to the temperature 3,000 K at the redshift 1,100, when the scale factor a of the universe is 1/1,100th of its current value.

The last scattering / recombination / decoupling / end of baryon drag almost certainly happens at that temperature.

The Hubble H₀ constant now has the value 73 km/s / Mpc. The matter density of the current universe is Ωm = 0.31 times the critical density. We can proceed to calculate backward ΛCDM, and can determine the distance D(1,100) in comoving coordinates to the last scattering. That is, a photon scattered then moves the distance D(1,100) in comoving coordinates, before hitting our eyes.

   
                            /|
                         /   |
                      /      |
                   /         |
                / θ*       | r*
          o  -------------    last scattering
          |    D(1,100)                       
         /\                       comoving coordinates


An observer on Earth should see the sound horizon length in the cosmic microwave background to subtend an angle

       θ*  =  r* / D(1,100).

Recall that r* is given in comoving coordinates, too.

We can analyze the CMB very precisely from the Planck data, and obtain a value for θ* which is significantly smaller than the value which we calculated above. This is the Hubble tension. "Features" in the CMB look smaller in the sky than they should be.

The Hubble tension. The tension is between the assumption:

- we have measured H₀ correctly in the current universe, as well as other relevant parameters, and ΛCDM is correct,

and the fact:

- features in the CMB look smaller in the sky than what we calculated.


It is an oversimplification to say that the tension is just in the value of H₀.

A simple way to remove the tension is to assume that the universe expanded slower between the last scattering and the current time than assumed by ΛCDM. If the universe expands slower, then a photon sent from the last scattering has more time to travel in comoving coordinates, and the distance D(1,100) is longer. Consequently, the calculated angle θ* is smaller. We can make the angle so small that it agrees with the angle which we see in the sky.

Dark energy already showed us that the universe may expand surprisingly fast. It is not a stretch to claim that it can also expand surprisingly slowly.


The Hubble tension really is a tension in the angle θ* calculated from the current universe versus what we see in the CMB in the sky?


Let us measure the mass density Ωm in the current universe, as well as the value of the Hubble constant H₀ from type Ia supernovae.

Assuming ΛCDM, we can then calculate how the uneven distribution of the current galaxy clusters should show up in the CMB.

Let us check if galaxy surveys, combined with H₀ = 73 km/s / Mpc, really conflict with the CMB we see in the sky.


Zhiwei Yang et al. (2025) derive H₀ from DESI, H0LiCOW, and Pantheon supernovae. DESI has measured the redshifts of millions of galaxies and built a 3D model of the current universe (actually, it is the universe in the past 10 billion years, since the redshifts z are up to 2 in the survey).


H0LiCOW measures gravitational lensing of quasars and derives a value for H₀ from them.


Pantheon+ contains the light curves of 1,550 type Ia supernovae. They can be used to determine the distances to galaxies, as well as Ωm.

Zhiwei Yang et al. calculated H₀ = 73 km/s / Mpc and r* = 138 Mpc. The size of the sound horizon r* is given in the current universe.

The sound horizon size derived from the first 380,000 years of the universe is 147 Mpc. Why the difference?

Let us check what values for r* have other authors derived from DESI and other galaxy surveys.


The Sloan Digital Sky Survey team got a value r* = 150 Mpc. The value differs a lot from the Yang et al. value.


E. A. Zaborowski et al. (2025) derive the value H₀ = 70 km/s / Mpc from DESI.


Tonghua Liu et al. (2024) list values obtained for r* in various studies: 143 Mpc, 137 Mpc, 144 Mpc. Their own result is 140 Mpc.

There seems to be too much variation in the estimates of the sound horizon size r* in the current universe. We cannot draw any conclusions. The Hubble constant H₀ derived in these calculations varies a lot, too. It is not clear if a Hubble tension exists in the calculated values.

Thus, currently, the tension is between the Hubble value 73 km/s / Mpc derived from Cepheids and supernovae versus the Hubble value calculated from the first 380,000 years of the universe plus a ΛCDM model.


What would the sound horizon angular size θ* be in ΛCMD if we tune the universe expansion rate?


In our blog we have brought forward an idea that the universe on the average follows a FLRW model, but there are "bounces" which make it to expand slower or faster. The current epoch of dark energy is a bounce to a faster expansion.

Let us check if there would be a Hubble tension in a CDM model where Λ would be zero "on the average", when the universe expands.

How much has dark energy recently accelerated the expansion relative to CDM?

We can model the expansion with newtonian gravity.


If we assume that H₀ = 67.8 km/s / Mpc, then the critical density is now

       ρc  =  8.4 * 10⁻²⁷ kg/m³.

The ratio of matter and dark matter to the critical density is believed to be

       Ωm  =  0.31.

Note that the estimate 0.31 depends on the cosmological model that we use. It is based on how much we believe that the gravity of the mass has slowed down the expansion of the universe since the Big Bang.

Let us assume that the expansion of the universe has slightly accelerated in the past 5 billion years. We calculate the amount of energy involved.

Let us have a spherical volume of the current universe. Let the radius be R.

If the mass density of the universe would be 1/0.31 = 3.2-fold, then newtonian gravity would asymptotically stop the expansion. Let us have a test mass m on the surface of the sphere R. The negative potential energy of m is linearly proportional to the mass M of R, or the density of R.


                     ×                   • ---> v
                               R        m


We conclude that the negative potential energy of m is -0.31 times its kinetic energy E = 1/2 m v².

Suppose that the velocity v of m has stayed the same for the past 9 billion years. Then dark energy has done ~ 0.31 * E of work on m, against the gravity potential. Quite a lot of work!

Let us use the "epoch" scale from the previous blog post. The scale factor is a ~ 1 / 2ⁿ during epoch n.

The time lengths of various epochs:

   t₀ : 9 billion years
   t₁ : 3 billion years
   t₂ : 1 billion years
   t₃ : 350 million years
   t₄ : 100 million years
   ...
   t₁₀: 140,000 years.

The contribution to the "angular distance" D of each epoch – this measures how much a photon propagates, in comoving spatial coordinates, during each epoch:

  D₀ : 1
  D₁ : 0.7
  D₂ : 0.5
  D₃ : 0.35
  D₄ : 0.25
  ...
  D₁₀: 0.03.

The sum D = 3.3.

Let us calculate the kinetic energy of our test mass m during each epoch. Recall that the current gravity potential of m is -0.31 * E, where E is the current kinetic energy of m. When a < 1, the gravity potential energy has added to the kinetic energy:

  0 : E
  1 : 1.31 E
  2 : 2 E
  3 : 3 E
  4 : 6 E
  5 : 11 E
  ...
  10 : 300 E.

If we want to make the angular distance D 8% larger, so that the Hubble tension would be removed, we have to reduce the kinetic energy of m a lot during epochs 2, 3, and 4. The expansion rate should only be 75%. That is, the kinetic energy must be reduced by E during epoch 2, and by 3 E during epoch 4.

Let us think. What does this mean? Initially, during epoch 10, the test mass m has a lot of kinetic energy. As m climbs up from the gravity potential well, it loses kinetic energy. The energy goes to the energy of the gravity field. In earlier blog posts we have suggested that retardation of gravity stores extra energy into the gravity field. Could that extra energy be so large that it could be 3 E during epoch 4?

Note that we have a fine-tuning problem here. How did nature "know" that the kinetic energy of m during epoch 10 must be set almost exactly to the negative gravity potential -300 E? As if nature would be fine-tuning the process, so that it is almost exactly a collapse of a cloud run backwards.


Entropy of the universe as a fine-tuning problem


The universe had a low entropy in the Big Bang. A low entropy is unlikely for a random system. Here we have another fine-tuning problem.

The entropy of the universe grows after the Big Bang, as the universe expands. Suppose that the universe follows an FLRW model, and starts to contract again. Suppose that energy is conserved. Then the macroscopic state of the universe in a Big Crunch should be very similar to the corresponding state at the Big Bang. The state is very hot, and particles have random motion. That is, the entropy has to start decreasing at some point before the Big Crunch. This would break the rule that entropy has to increase. What is going on?

Let us imagine an explosion in almost-newtonian mechanics and newtonian gravity. Let the explosion start from a very small volume and be almost uniform. The entropy of the system starts growing. In particular, photons will carry away a large amount of entropy. Let us assume that gravity will pull back most of the mass which exploded. Gravity will compress it back into a small volume. The end result might be a crystal which has a very low entropy. All entropy created after the explosion is carried away by photons.

Photons cannot escape from an FLRW model because there is nothing outside the model. Thus, an FLRW model leads us to a paradox: entropy should magically start decreasing at some point.

Let us assume that black holes do not exist in the FLRW model. It is not clear how they would behave in a Big Crunch.

Entropy bans a Big Crunch? Maybe a Big Crunch is not possible at all? Since FLRW models do allow a Big Crunch, we here have an argument against the correctness of FLRW models: they break the laws of entropy.


Suppose that we have an isolated physical system whose volume first increases and then decreases. A big volume contains more "degrees of freedom". Entropy can then increase. But if some strange law of physics (like an FLRW model) makes the volume to decrease, then there will be fewer degrees of freedom, and entropy may be forced to decrease.

How do we avoid the paradox of decreasing entropy? Either the system is:

1. not isolated, or

2. the system will expand forever.


Bounce-back (cyclic) models of the universe require that entropy has to decrease at some point. If one believes that entropy cannot decrease, then bounce-back models are ruled out.



***  WORK IN PROGRESS  ***
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Wednesday, May 13, 2026

Dark energy: a balloon model

In December 2025 we had problems finding a reasonable mathematical model for retarded gravitation in an asymptotic Minkowski space. Let us try it again, this time for a spherical balloon model. The spacetime is then topologically different from a plane.














Picture publicdomainpictures.net



Can the equation of spacetime be local for a balloon model?


Let us think of the Schwarzschild solution. The solution extends through the entire asymptotically Minkowski space. The central mass M affects locations very far away.

For a spherical rubber balloon, the equation governing its stretching is local, if we assume that the pressure inside is uniform.

Friedmann equations in general relativity look local, but that is only because we assume a uniform mass distribution. Any retardation effects are masked.


A rubber string and a weight


                 rubber string
           ●----------------------------●
      point                            point


Let us draw points uniformly around a sphere, and connect them with tense rubber strings which run on the surface of the sphere. The rubber strings simulate the gravity between the mass points. If the balloon is static, or inflates or deflates at a constant rate, this model looks like an ordinary balloon: the tension in the strings is uniform along the string.


A rubber string and a weight which the string slows down


Let us consider the simplest possible model of a tense rubber string plus weight:
          

                                              force
                                         F  <------ 
         ==| ~~~~~~~~~~~~~~ ● ---> v
    wall         rubber string            M weight


We are interested in what kind of longitudinal waves form in the string when the weight moves to the right while the tension in the string pulls the weight to the left.

Free waves in the string are sine waves. Since the speed of the waves is finite, there is "retardation".

As the weight slows down, it creates new waves. The weight perturbs the string. Calculating the precise form of the wave probably requires a computer. But we are only interested in very crude estimates.


Do binary pulsars prove that there are no longitudinal gravitational waves? Are longitudinal waves confined between masses?


Binary pulsars have confirmed that transverse waves and linearized Einstein field equations explain the energy loss of a binary pulsar up to a precision of 0.1%.

In an earlier blog post we suggested that longitudinal waves must be "absorbed into" matter quickly, because they cannot propagate in empty space. This hypothesis would explain why binary pulsars match the transverse wave model so well.

Another hypothesis: longitudinal wave effects can only exist in the field between two masses M₁ and M₂ and can never escape to empty space.


Longitudinal electromagnetic waves exist in plasma, but they do not exist in empty space. Could this be analogous to our model of retarded gravity? Langmuir waves exist in a plasma which is, on the average, neutral. Since gravity charges always are positive and gravity always pulls, Langmuir waves cannot exist in a gravitating system.


A model with solid rubber inside


Let us have two masses M₁ and M₂ on the surface of the balloon. Visualizing a rubber string between the masses, and running on the surface of the balloon, is somewhat hard. Let us imagine that the rubber string runs inside the balloon.

The simplest case is when M₁ and M₂ are on the opposite sides of the balloon. A rubber string runs through the center of the balloon and connects them.

The balloon expands. The rubber string is not immediately aware of the expansion slowing down. This will cause oscillation in the expansion rate.










Photo publicdomainpictures.net

Let us have a solid rubber ball. The rubber will resist the expansion in most cases. But if there is a shock wave moving from the center toward the surface, the expansion of the rubber ball may actually accelerate at a certain time. "Dark energy" in this case is the energy of the shock wave.

This model predicts that the expansion rate of the universe will oscillate, while it, on the average, is quite similar to an FLRW model.

But we should find a mathematical formula, so that we could compare it to the observed dark energy.


The strength of dark energy seems to be weakening right now




The DESI project measures the 3-dimensional mass density variations of the universe right now, and compares it to the baryon acoustic oscillations after the Big Bang. The results of DESI, when combined to other observations, suggest that dark energy is weakening right now. Wikipedia says that dark energy density right now might be 10% less than 4.5 billion years ago.

This is consistent with our own rubber ball model: on the average, the expansion should be like an FLRW model, but there should be "bounces" from the retardation of gravity.


In the link we have the definition for the CPL parametrization of the dark energy equation of state, w(a), where a is the universe scale factor. The current value of a is set to 1.


Abdul-Karim et al. (2025) write that the energy density of dark energy may have increased in the past. This is called a "phantom crossing". In our rubber ball model, a phantom crossing is expected: at some points the bouncing slows down the expansion, at other points speeds up the expansion.


The Hubble tension


Let us measure the baryon sound horizon size in the cosmic microwave background, and match it to the measured density fluctuations in the current universe. Assuming, for example, the standard ΛCDM model, we end up with a Hubble "constant" current value of

       67 km/s / Mpc.

But a direct measurement, based on Cepheids and type Ia supernovae yields a value

       73 km/s / Mpc.

The values differ too much to be just a statistical fluctuation.


Marco Raveri (2023) tries to use dark energy to make these values compatible. Let us check what he suggests. His conclusion is that dark energy modifications at "late" times, 0.01 < z < 3 cannot solve the Hubble tension.


Early dark energy (EDE)




Xuejian Shen et al. (2024) write about an early dark energy (EDE) model, where an unknown field boosts the expansion of the universe just before the recombination (last scattering) which happens when the universe is 380,000 years old in a standard ΛCDM model.

Shen et al. say that this model can solve the Hubble tension. Is it so?

Let us start from the observed uneven matter distribution in the recent universe, and match it to the the uneven matter distribution in the cosmic microwave background (CMB). The baryon acoustic oscillations (BAO) make bumps into the uneven distribution. Let us match the bumps in recent matter and in the CMB.

Let us calculate time backward from the present time, and model matter distribution at earlier times. We use a standard ΛCMB model. When the universe has shrunk to one 1,100th of its size, the CMB has a temperature of 3,000 K, and the last scattering occurs.

The Hubble tension is that the BAO bumps in this calculation have a larger angular scale than in the measured and mapped CMB in the sky. That is, the "features" in the actual CMB map are smaller than our calculation backward in time predicts.

The early dark energy (EDE) hypothesis talks about what happened before the last scattering. The hypothesis does not help in any way in solving the Hubble tension. We have to check the literature. Can the EDE hypothesis be so much wrong?


Karsten Jedamzik et al. (2020) write that EDE cannot resolve the Hubble tension.





 
Above, θ* is the angular size of the sound horizon, that is, the angular size of "features" in the CMB. We use coordinates comoving with matter. Then r* is the spatial size of the sound horizon (in those coordinates), and D(z*) is the spatial distance to the last scattering.

Jedamzik et al. say that only the quantity

       Ωm h²

affects the spatial distance to the last scattering, in a flat ΛCDM model. There, Ωm is the matter fraction of the critical density, and h is the value of the Hubble constant now.

The authors state that the Dark Energy Survey and the Kilo-Degree Survey have set strict limits on Ωm h², so that any attempt to solve the Hubble tension through manipulating Ωm h² will fail.

Let us check what they say about manipulating r*. Suppose that we calculate the development of a ΛCDM model backward in time. The matter density variations of galaxy clusters eventually turn into variations of the CMB as the temperature rises.


Solving the Hubble tension with a very slow expansion during the first billion years of the universe: also the James Webb paradox of too mature galaxies is solved?


The James Webb telescope has observed galaxies which in a standard ΛCDM model are only 500 million years old, but look much more mature, as if they were 1.5 billion years old.

Let us assume that the expansion during the first billion years of the universe was much slower than the ΛCDM says. We add an "extra" 1 billion years to the age of the universe, to the early phases of the universe. Adding that extra means that the universe expands much slower than in the ΛCDM model, during the first 1 billion years or so.

Then galaxies have much more time to mature. This would solve the James Webb paradox.

During that slow expansion during the first 1 billion years, photons from the last scattering had time to move a "longer" path than in the ΛCDM model. Let us explain what "longer" here means. If we model the expanding universe with coordinates comoving with matter, then the ruler of the spatial metric (say, a ruler 1 meter long) shrinks as time progresses. The "length" of the path of the photons is measured in these comoving coordinates.

Since the photons moved a longer path before coming to our eyes, the features in CMB will have smaller angular diameters in our eyes. This resolves the Hubble tension.


A quantitative calculation of making the universe to expand slower during the first 1 billion years


Let us make a quantitative calculation. We assume a matter-dominated universe with a zero pressure. Then we can use newtonian mechanics to calculate the development of the universe backwards from the present time.


              "feature" size
               <----->
              ●        ●        ●       ●

              ●        ●        ●       ●
                             o ~~~~~  last scattering
                             |     D
                            /\ 
                      observer       c ~ 1 / a


We set a = 1 as the current scale factor of the universe. In comoving coordinates, the speed of light

       c  ~  1 / a.

Let us look at eleven "epochs" where

       a  ≈  1 / 2ⁿ,

0 ≤ n ≤ 10.

The epoch 10 corresponds to the time of the last scattering, that is z ≈ 1,100.

In ΛCDM, the kinetic energy of a particle m in static (not co-moving coordinates) is very roughly

       1/2 m v²  ~  1 / a  =  2ⁿ.

That is, v ~ sqrt(2ⁿ).

This implies that the time the model spends in an epoch n is

       tₙ  ~  (1 / 2ⁿ)  /  sqrt(2ⁿ)

              = 1 / 2^(1.5 n)

              = 1 / 2.8ⁿ

              = 0.35ⁿ.

The sum of tₙ is t ~ 1 / (1 - 0.35) = 1.5.

The time lengths of various epochs:

   t₀ : 9 billion years
   t₁ : 3 billion years
   t₂ : 1 billion years
   t₃ : 350 million years
   t₄ : 100 million years
   ...
   t₁₀: 140,000 years.

The speed of light c ~ 1 / a = 2ⁿ. Light in that time propagates a distance in comoving coordinates:

       Dₙ  ~  2ⁿ  *  1 / 2^(1.5 n)

              = 1 / sqrt(2ⁿ)

              = 1 / 1.4ⁿ

              = 0.7ⁿ.

We are interested in the distance

               10
       D  =  ∑ Dₙ
              n = 0

(in comoving coordinates), which light travels from the last scattering, into the eye of the observer. If the contribution of epoch D₀ is 1, then the sum of the series is

       D  =  1 / (1 - 0.7) = 3.3.

The contributions of various epochs:

  D₀ : 1
  D₁ : 0.7
  D₂ : 0.5
  D₃ : 0.35
  D₄ : 0.25
  ...
  D₁₀: 0.03.

In particular, the contribution of epoch D₁₀ is only 0.03. We see that we have to modify the physics drastically during epoch 10, if we want to increase D by 8%. This shows that solving the Hubble tension by tampering only the first 400,000 years of the universe is an implausible solution.

The first one billion years of the universe corresponds to epochs 2, ..., 10. The sum of D₂, ..., D₁₀ is 1.6.

We see that if we make the universe to expand 16% slower during epochs 2, ..., 10, we can increase D by 8%. Then the angular size of "features" will be properly smaller to match how Planck and other probes saw the CMB.

But now we realize that we only have to add 160 million years to the early history of the universe. That would not explain James Webb observations.

If we double the time length of epoch 3 to 700 million years, we increase D by 11%, and add 350 million years to the early history of the universe. Could this be enough to solve the James Webb paradox?

Dark energy means that the universe has been expanding surprisingly fast during the past 7 billion years. Maybe the expansion rate fluctuates, and was surprisingly slow during the first 500 million years of the universe?


Conclusions


We still do not have a mathematical model which we could use to calculate the effects of retarded gravity. Intuitively, retardation should cause the universe to expand surprisingly slowly at certain times, and surprisingly fast at other times. That is, like a solid ball whose elastic interior can contract and expand.

We will write a new blog post about the Hubble tension. Literature strangely ignores solutions where the expansion of the universe is slowed down during the first 350 million years. Literature seems only to talk about modifying physics at z < 1 (the last 7 billion years), or at z ~ 3,000 (the first 100,000 years).

Why does literature ignore the most obvious solution? Slowing down the first 350 million years could solve also the James Webb paradox of "too" mature galaxies at z ~ 14.
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Thursday, March 26, 2026

de Broglie and Bohm: photon creation from a radio transmitter – the problem of new particles

Let us consider a simple semiclassical process which creates photons. Let us have two charged particles (charge Q) whose mass M is macroscopic. We let the particles pass close to each other, so that they send a radio wave with many photons.


                                                                    screen
                                                                          |
                                                                          |
                                                                          |
                              )   )   )   )   )  radio wave
           ● ---->              
          M, Q                 <---- ●
                                         M, Q


The radio wave will hit a screen far away, and we can determine the positions of individual photons.

We can calculate the shape of the radio wave classically, but we cannot calculate where the individual photons will hit the screen.

The process creates new particles. Is there a way to extend the de Broglie-Bohm model to this case?

Could this work: the photons "pre-exist" in the electric fields of the charges Q, and are "freed" in the scattering process? Then all the particles could have a hidden variable already at the outset of the experiment, and would sail in a deterministic way to the screen – just as in the double-slit experiment.

The outgoing photons have an uncountable number of possible momenta p. Even if we would have pre-existing photons in the field of Q, how would these obtain the required p?


Time asymmetry in the process


The initial configuration of two charges Q flying at each other is quite different from the final configuration where we have the wave function for the two charges Q plus many alternative configurations of photons. The entropy of the system grows, and there are more degrees of freedom in the final configuration.

What if there would be no growth of entropy? Then the initial and the final configurations should look similar. Maybe the number of the degrees of freedom does not grow any more?

What about inverting the direction of time? Then the degrees of freedom will shrink as time progresses.


Absorption of a photon by a hydrogen atom


An excited hydrogen atom can decay and send an individual photon. The atom can absorb the required momentum from the reaction as it launches the photon.

What about the reverse process? A wave function of the photon contains many cases in which the photon does not hit the atom at all. The photon passes the atom.

Can we somehow add more "worlds" in such a way that the the graph of possible worlds would look more time-symmetric?

In the case of a hydrogen atom absorbing a single photon, the degrees of freedom only shrink in the case of the absorption. In other cases, there is no change. A "wave function collapse" only happens in one branch of many-worlds.


Ignore photons and only consider (charged) particles which existed at the outset?


The de Broglie-Bohm approach might work if we only had undestructible particles. Let us have N such particles. Let us have a foliation of spacetime such that spacelike folios define "simultaneousness". Then we could replace the N particles with one particle moving in a space with 3 N spatial dimensions.

But we have a problem already with electrons: pair production and pair annihilation. There are no undestructible particles.


Give up determinism?


Albert Einstein said that God does not play dice. That is, the "next", measured, state of a physical system should follow deterministically from the "previous" state.

Since simultaneousness is not well defined in relativity, we maybe should say this: what happens at some point in time and space is determined by the light cone which precedes that point.

Why not require determinism also in the "initial" states of the particles? Why can the initial states be random? Does God play dice, after all?

Why is the time dimension different from spatial dimensions?


Defining a "branch" in many-worlds


Let us have a human living in spacetime. Most of the massive elementary particles in his brain survive for a long time. They are not annihilated. However, quite a few particles are replaced with other similar particles over his lifetime. How could we define the "branch" in many-worlds, in which he is living?

The brain consists mainly of electrons and of nuclei. These particles are, in most cases, conserved over 100 years.

Then we can apply the de Broglie-Bohm model to these particles, and obtain well defined "branches" of their configurations.

We do not try to define a branch in a more general sense: we do not define, e.g., which photons on Mars belong to his branch.

What if two photons collide and form an electron and a positron, and the electron subsequently is used as a new building block in his brain?

Could we simply accept a model in which the branch of this brain chooses "by random" the initial state of the electron, as it is integrated into the brain?


A new problem: which paths of separate particles are in the "same" branch of many-worlds?


Suppose that two electrons are created in pair productions, and these two electrons are integrated to the brain of the observer.

We can use the de Broglie-Bohm model to describe the paths of the two electrons. But what paths are in the "same" branch of many-worlds?

This might be the solution: we simply choose one de Broglie-Bohm path for each electron. Then the combination of those paths is taken to be in the "same" branch as the brain of the observer.

When a new particle is included into the brain, the brain performs a "quantum jump" to include a certain path of that particle.


Do we need precisely defined branches at all?


Definition of a branch does not change anything in the wave function. In the de Broglie-Bohm model, the path of a particle does not affect the wave function in any way.

However, picking a certain path may define the "real world".

Let us have an observer. Maybe he lives in the "real world", and the copies of him in other branches are just "empty" wave functions?

This does not sound natural. Why should a conscious observer always exist in the "real world" and not in the "empty" branches?

In the de Broglie-Bohm model, the empty branches do physically exist, too. They are just different branches of the wave function.

Maybe it is necessary to drop the concept of the "real world" and treat all the branches equal. This means that Schrödinger's cat is both dead and alive at the same time.


Do we have a reasonable definition of a "branch" now? A robot


Instead of an observer we may consider a computer or a robot which possesses a camera and other sensors. It also has arms for manipulating external objects.

Above we suggested that we can use the de Broglie-Bohm model to define a "branch" of many-worlds, such that an instance of the robot lives in that branch.

The robot will encounter various random (or pseudorandom) quantum phenomena, and adjust the contents of its memory accordingly. Its camera may register random photons hitting its CCD sensor. 

The robot is an almost classical machine. It probably has relatively exactly definable states, and it in most cases functions deterministically.

Is the de Broglie-Bohm model an awkward way to define a branch in this case? Why not use a traditional Copenhagen model where the next state of the machine depends probabilistically on the previous state?

Assigning hidden variables is awkward?


The observing subject


Suppose that we have a reasonable way to define a branch for the robot. The computer in the robot, in that branch, will have a similar world-view as a human observer. It will remember its history. In almost every branch, the robot notices that physical processes in its history behaved according to the probabilistic rules of quantum mechanics.

According to Rene Descartes, cogito ergo sum. That is, I as an observing subject do exist.

My subject might be attached to one branch of a robot, my physical body, or it might "jump" from branch to branch. I would not notice anything if my subject jumps around the branches of many worlds.

It sounds strange if I alternate between different branches. I very much feel that I am in just one branch. This paradox is similar to another paradox about time. In what sense is now different from moments in the past or in the future?


Conclusions


It would be nice if we could define a single "branch" in the many-worlds interpretation. Then we could imagine that a human observer lives in that branch.

The de Broglie-Bohm model gives us a way to define a branch for a robot. Simply plot the paths of the nuclei and the electrons in the mechanism of the robot.

However, that model is not very natural. Why should the branch exactly follow the strange paths of de Broglie-Bohm model?

Maybe a more natural way to define a branch is more fuzzy: it is the collection of those states of the robot which are "similar". For example, we may require that the bits in the memory of the robot have identical values in the collection of the states. This would be a "fuzzy branch".

We still have the question if an "observing subject" is somehow attached to a certain branch. This is a metaphysical problem which is hard to clarify and solve. What makes a certain branch "real" for the subject? How would the subject know if it jumps from branch to branch?

A similar metaphysical problem is in what sense is the world "now" more real than the past or the future. We have no answer.
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Thursday, March 19, 2026

de Broglie-Bohm model and the many-worlds interpretation of quantum mechanics

In the Copenhagen interpretation, the wave function collapses. That destroys information and spoils the time symmetry of the laws of physics.

The many-worlds interpretation would be better. If my "subject", that is, I as an observer, is located in a certain "branch" of possible world histories, then no collapse is required. It just happens that I as a subject am in a certain branch. Then all the other laws of physics can be time-symmetric. If we further assume that the path of the subject through the tree is determined by a time-symmetric method, then we have a model which is completely time-symmetric, and does not destroy information. This would be desirable.


A freely readable .pdf file:


The de Broglie-Bohm model might provide us with a deterministic path for the observing subject, if we can extend it to cover the creation of new particles. The classic Bohm model is limited in that that it only covers a fixed number of particles, and uses the non-relativistic Schrödinger equation.


What constitutes a branch in the many-worlds interpretation?




The definition of a branch is a question which comes up regularly on the Physics Stack Exchange.

Suppose that a single photon hits a charge-coupled device (CCD), and starts a cascade which registers the location which the photon hit. A quantum mechanical object photon "decoheres" into a macroscopic object or into a macroscopic process.

Many people seem to believe that this explains what a branch means in many-worlds. It is a local process of decoherence. However, this definition of a branch is not exact, nor mathematical. If the photon happens to hit the CCD at a slightly (1 micrometer) different location, is that another branch? Do branches actually form a continuum?

The de Broglie-Bohm model might give us a mathematical definition of a branch: it simply is the time development for fixed initial hidden variable values.


We might define a branch with certain initial hidden variable values the "true state of the material world". Albert Einstein wrote that "God does not play dice". The branch would be deterministic, and Einstein would be happy. Does this approach lead to any contradictions?


de Broglie-Bohm as a flow of probability?




        |      |                        |
        |      |                                )        )
        |      |                        |
        |      |                                )        )
        |      |                        |

           --->
      incoming             double   diffracted wave
         wave                    slit       and interference


Let us think about the double-slit experiment. The incoming wave can be understood as a probability distribution, where the probability density is the squarw of the wave function value.

We may imagine a flow of probability, a flux, which takes a "probability parcel" forward from the double slit. That would give us a deterministic "path" for an incoming particle? As is well known, a de Broglie-Bohm particle will move along a strange, curvy path.

In a Feynman path integral, also fully unrealistic paths of a particle are calculated and added. If a de Broglie-Bohm path can be designated as the "true world", what about an unrealistic Feynman path?


Werner Heisenberg writes in his 1958 book Physics and philosophy that the de Broglie-Bohm model has problems handling stationary states. The flow of probability is zero in a typical stationary state, say, for a particle in a box. That suggests that a particle would remain in a fixed place.

A particle in a box, intuitively, has two flows of probability, to opposite directions. The sum of these flows is zero.

    
             wave packet
     <--    ~~~~~~~~~~
                       ~~~~~~~~~~    -->
                       wave packet


Let us have a very long wave packet which describes a single particle. We use a beam splitter to divide the packet into two wave packets and let the packets collide so that they travel to opposite directions.

The flow of probability in the overlapping part of the two packets is zero. But the natural many-worlds interpretation for the process is that a single particle is moving either to the left or to the right in the diagram? 

No. In the many-worlds interpretation, "branching" means a process where the waves for various alternatives will not interfere much. The branching has not happened yet when the wave packets overlap.


Do other branches "exist"?


If we designate one de Broglie-Bohm branch as the "true world", the other branches still exist, in the sense that they interfere with the true world, and affect the probabilities how the true world develops in time. This is quite different from, say, newtonian mechanics.

Certain people have criticized the fact that all branches exist, also those that are not the "true world". But we cannot help it: if we prune some branches, we lose information, and the model no longer corresponds to quantum mechanics.


Generalizing to relativistic wave functions and creation and destruction of particles: quantum field theory


The de Broglie-Bohm model is based on the Schrödinger equation, which is nonrelativistic. A generalization to relativistic particle physics requires that we develop a de Broglie-Bohm model for quantum field theory.

Quantum field theory, in its present form, contains dubious methods like regularization and renormalization. It is unlikely that we can develop a satisfactory deterministic model for quantum field theory unless we can remove regularization and renormalization. In fact, a de Broglie-Bohm model might help us to solve the problems.


Hrvoje Nikolić (2009) describes a model where particles can be created and destroyed.














Nikolić assumes that a particle can have a finite lifetime. In the diagram, x⁰ is the time coordinate. "Virtual particles" pop up and disappear in a very short time. They are denoted with dots.

For the model to be deterministic, the position of those dots, (x⁰, x¹) should depend on the initial conditions (data) at some earlier time. Does it?

We can, of course, formally define that the locations of the dots are decided in the initial data, but that looks ugly.


Flow of probability when new particles are created


                        ~~~~~~~~~~~  real photon
                      /
       e- ----------------------------------
                           |
                           | virtual photon
       e- ----------------------------------


Let us consider a collision of two electrons. Photons and pairs can be created. We can draw Feynman diagrams and calculate the probability of each outcome. Above is a diagram for bremsstrahlung.

In the de Broglie-Bohm model of the double-slit experiment, there is a beautiful way to define the "path" of the particle, along the flow of probability in the Schrödinger equation. The path is meandering, but it is defined in a beautiful way.

But for a Feynman diagram, there is no such beautiful way. The real photon which appears above "breaks" the system into more degrees of freedom. If we have a probability flow in 6 spatial dimensions, there is no obvious way to make it to branch into a flow in 9 dimensions. What would decide the path for an arbitrary initial configuration of two electrons? How does the path suddenly branch to more dimensions?

In our blog we have several times written about the problem of "breaking" into more degrees of freedom. This seems to be associated with regularization and renormalization problems for Feynman diagrams.

Let us assign an arbitrary (irrational) real number r for the initial two electrons. We could design an ad hoc way to reproduce the correct Feynman probability distribution from the infinite number of decimals in r, and obtain a deterministic model. But that would be a very ugly solution.


Schrödinger's cat


The de Broglie-Bohm model for the Schrödinger equation solves a paradox associated with Schrödinger's cat, for the treatment with the Schrödinger equation.

The cat always is either dead or alive, in the "true world" determined by the hidden variables. Opening the box does not change anything. The cat does not suddenly "collapse" into either state.

Let us assume that we find the cat alive.

What remains of the paradox is the "empty" branch in which the cat is dead. The branch does exist in the de Broglie-Bohm model, though that branch is not "occupied" by the values of the hidden variables.

Schrödinger's cat involves a radioactive decay process which decides if the poison is released inside the box. Such a decay must be treated in quantum field theory, since new particles are created. And the de Broglie-Bohm model has no beautiful extension there. Thus, the paradox remains.


Conclusions


The de Broglie-Bohm model is a beautiful way to introduce determinism into the Schrödinger equation. We can define in a beautiful way what is a branch in the many-worlds interpretation. We avoid the "measurement problem" and many paradoxes in quantum mechanics.

But extending the de Broglie-Bohm model into quantum field theory depends on if we can find a beautiful model quantum field theory. Presumably, divergences, regularization, and renormalization should be removed. The problem of breaking into more degrees freedom may be a mathematical problem. Can we solve it?
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