Schrödinger equation and a moving screen: it is Galilean covariant

In our previous blog post we asked if the Schrödinger equation is Galilean covariant. That is, if the equation has a solution in a frame, can we – in a beautiful way – transform the solution and obtain the solution in a moving frame?

Newtonian mechanics is Galilean covariant.  If we have a history of mechanical system, we can – in a simple way – transform the solution to a frame moving at a constant velocity v. The transformed solution satisfies newtonian mechanics in the moving frame.

Huygens's principle

Let us have a wave. Using Huygens's principle, we can easily construct an approximation for the diffraction pattern created by a pinhole in a screen.

  

                                     |

          |     |     |     |          )      )      )

                                     | pinhole

                                     |

         wave --> v    screen     diffracted wave

      ^  y

      |

       ------> x   moving frame

       --> v      

https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle

Let the velocity of the wave be v. Let us switch to a moving frame which comoves with the incoming wave.

In the comoving frame, the incoming wave is static. Can we use Huygens's principle for a static wave?

The screen is moving against a static wave. How can the screen create the diffraction pattern?

If we are looking at water waves, then the frame where water does not flow horizontally, can be defined as a preferred frame.

Does it make sense to demand that Huygens's principle should work in any other frame than the preferred frame?

For water waves, a disturbance at a location r will spread to every direction at some fixed speed v relative to the water. It makes little sense to use any other frame than the frame in which water is static.

In the case of the Schrödinger equation, there is no self-evident preferred frame. In principle, we should be able to use any inertial frame.

Richard Feynman derived the Schrödinger equation from a path integral, i.e., Huygens's principle

https://www.physics.utah.edu/~starykh/phys7640/Lectures/FeynmansDerivation.pdf

David Derbes (1996) describes how Richard Feynman used a path integral approach to derive the Schrödinger equation. The path integral has much the same idea as in Huygens's principle. If we know the wave function

       ψ(t, r)

at a time t₀, we can construct the wave at a later time t₁ by summing the contributions of ψ(t₀, r) for each point r in space.

Huygens said that each point r acts as a new "source" of a new mini-wave. At a later time, the wave's crests are where there is a constructive interference of the mini-waves.

Huygens's principle works right for light

                                         |

      |     |     |  ---> c              )      )      )

                                         | pinhole

 

         ^

         |

          ------->   ---> v

         moving frame

Let us assume that the incoming wave is electromagnetic. Let us switch to a moving frame. We assume that v << c, so that we can ignore time dilation.

In the moving frame, there is a redshift of the incoming wave. Its frequency is lower. But the redshift is canceled by the blueshift for the screen and the pinhole moving to the left.

The pinhole "sees" the frequency of the wave identical in the laboratory frame and the moving frame.

Let us then switch back to the laboratory frame. The pinhole works as a source of waves (a transmitter).

The Merzbacher textbook contains a proof of Galilean covariance for Schrödinger

The textbook by Eugen Merzbacher, Quantum mechanics (1961, printed in 1998) contains a proof of Galilean invariance for the Schrödinger equation, on pages 75 - 78.

The key observation is that one can transform a plane wave

      ψ(t, r)  =  exp(i (p • r  -  E t) / ħ)

to a frame moving at a velocity v by the formula

       ψv(t, r)  =  exp(i (m v • r  -  m v² t / 2))

                         * ψ(t, r  -  v t).

That is one can use the simple transformation ψ(t, r - v t) if one multiplies it with a factor which does not depend on p.

An arbitrary sum Ψ of plane waves can be transformed by multiplying all of them by the same factor. Relations like Ψ = Φ are preserved in the transformation. Everything behaves very well.

https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation

If ψ satisfies the Schrödinger equation

in the laboratory frame, then Merzbacher shows that ψv satisfies the corresponding equation in a moving frame whose velocity is v. The potential V is mapped in the trivial way to the moving frame.

The Green function or path integral approach

Let us have a plane wave where the momentum p is very precisely 0.

         particle

              •                     v <--- | mirror 

           p = 0

  

The wave function ψ of the particle is essentially a constant complex number,

      ψ  =  C.

What happens when a steep potential wall (mirror) plows into the constant wave function?

We had problems figuring out how Huygens's principle behaves in this case. The wave function is constant. There is no propagating wave. How could we apply Huygens's principle if there is no wave.

Green's functions give us a clue. Let us have a point r. We imagine a sharp impulse "hitting" the Schrödinger equation at the point r. The impulse response (Green's function) contains waves of all wavelengths.

Previously, we have used the allegory that a "sharp hammer" hits a table.

The hammers hits simultaneously all points to the left of the mirror. Short wavelegths are canceled by destructive interference. Very long wavelengths remain.

How is a very long wavelength reflected from the mirror?

Newtonian physics says that the reflected particle should have a velocity 2 v to the left. This gives us the momentum p of the wave.

The phase velocity of the reflected wave is v to the left. The wave has a constant value on the surface of the mirror. This constant value must match C. This determines the phase of the reflected wave.

We were able to determine the momentum and the phase of the reflected wave.

Even though the constant value C of ψ does not have a "phase", it does determine the phase of the reflected wave. This resolves the conceptual problem we had with Huygens's principle.

Diffraction and Huygens's principle

  

                                     |

          |     |     |     |       )     )     )

                                     | pinhole

                                     |

We had problems understanding how Huygens's principle can handle diffraction if a screen with a pinhole plows into a wave function ψ where p = 0.

In the comoving frame of the screen, the pinhole creates half-spherical waves propagating from the pinhole.

But in the laboratory frame, p = 0, and the screen plows into a constant wave function. How can it then create waves moving into many directions? The "input" to the pinhole is constant and does not contain any oscillation.

In the case of the mirror, the momentum of the reflected particle and the continuity of the wave function determined the wavelength and the phase of the generated wave.

The solution to the mystery probably is the same for the pinhole: the momentum of the diffracted particle gives p and continuity gives the phase.

Conclusions

The Schrödinger equation is Galilean covariant.

We can use Green's functions (path integral) to build solutions for the equation.

Deriving the Schrödinger equation from zitterbewegung? Galilean covariance

People say that the Schrödinger equation cannot be derived. It has to be guessed. Let us try to derive it from the following hypothesis:

- the electron is a light-speed particle or a wave, similar to the photon, bouncing inside a small "box".

In a wave packet, which is built from solutions to the Dirac equation, the expected position of the electron makes a small circle at the speed of light. The movement is called the zitterbewegung. The role of zitterbewegung, if any, is not currently understood. Anyway, in a sense, the electron is a light-speed particle in a box.

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

The double-slit experiment

           ________________________  screen

                        |

                        | \

                        |α \  angle

                        |     \

                        |       \

                        |         \

                        |           \

         ----------   |    -----   \   -----------   two slits

                        |             |

                        □             □     

                        ^             ^

                        |             | v

  

                         electron

The boxes □ represent two alternative "paths" of an electron moving vertically at the speed v. The box contains a particle bouncing back and forth at the speed of light.

We should determine the relative phase of the rightmost box when it meets the leftmost box on the screen.

The phase difference determines the interference pattern on the screen. Are we able to reproduce the pattern predicted by the Schrödinger equation?

According to de Broglie, the wavelength of the electron is

       λ  =  h / p,

where p is the momentum:

       p  = me v.

The crucial thing is what is the length of the path of the light-speed particle in the box.

         -----------------

        |                  |

        |                  |  v

         -----------------

        |                  |

        |                  |  v

         -----------------

                  c

Let us assume that, before meeting the double-slit, the light-speed particle moves along the diagonals of a rectangle. It zigzags upward in the diagram above.

We assume that

       v  <<  c.

The sides of the rectangle have relative lengths c and v.

             

           |\ 

           |α\

What happens if we tilt the vertical lines in the diagram by some small angle α to the left?

No, we are not able to explain the de Broglie wavelength in this way.

The Schrödinger equation is Galilean covariant?

        

                                                            |

           particle  •  --> p            

                                                            |

      

                                                            |

                                              

                                                  double slit

           ^ y

           |

            ------> x 

            ---> v  moving frame

The wavelength of the particle is

      λ  =  h / p.

Now switch to a moving frame, in which the momentum of the particle is smaller. Its wavelength then can be much larger. Can it produce the same interference pattern as in the static frame?

Let us then replace the particle with a laser beam. In the moving frame, the Doppler effect makes the wavelength of the beam longer. Can the interference pattern remain the same? A moving double slit does produce an interference pattern which is different from a static double slit?

Special relativity probably makes the laser interference patterns to match in different moving frames. Electromagnetism is Lorentz covariant.

https://physics.stackexchange.com/questions/56024/galilean-covariance-of-the-schrodinger-equation

The Schrödinger equation is Galilean covariant, says the Physics Stack Exchange post. But is it?

Various people on the Internet claim that it is not Galilean covariant. 

A particle flux reflected by a potential wall and Galilean covariance

                            v

              |            <--- p flux

              |

              |           -p ---> reflected flux

                                   -v

       potential

           wall

               ^ y

               |

                ------> x

               <--- v moving frame

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

The free particle solution for the Schrödinger equation is

where p is the momentum of the particle, r is the spatial coordinate of the wave, the energy

       E  =  p² / (2 m),

m is the mass of the particle, and t is the time coordinate of the wave.

The velocity of the particle is

       v  =  p / m.

The phase velocity of the wave ψ:

       p Δr      =  E Δt

=>

       Δr / Δt  =  E / p

                    =  p² / (2 m)  *  1 / p

                    =  1/2 v.

Let us switch to a frame where the momentum p of the flux arriving from the right is almost zero. Then the energy E is almost zero.

In the moving frame, the wave function of the incoming flux is almost constant with respect to the new time and spatial coordinates. The wave function at the wall is almost constant.

The wave function of the outgoing flux is approximately

       ψ(r, t)  ~  exp(i (2 p • r  -  4 E t)).

The phase velocity of the outgoing wave is v, since the velocity of the particle is 2 v. The wall moves to the right at a speed v. Thus, the wave function at the wall is almost constant.

We can find a solution where the sum of the incoming and outgoing wave functions is constant at the wall. Galilean covariance is satisfied.

But is there a problem here? When we transform a wave function to a moving frame, we must transform also p and E. It does not suffice to transform r and t. What determines the phase of the transformed wave function? In the wall example, we were able to find a solution. The solution requires that the wave functions of the incoming flux and the outgoing flux have matching phases. Can we always find such matching phases? Probably yes. We have to check the proofs in the literature.

Does a gravity field or an accelerating potential break the Schrödinger equation?

On February 12, 2025 we were able to show that Maxwell's equations do not have a solution for an accelerating system. This is because linear equations cannot capture the accelerating process and conserve energy.

Can we do the same thing with the Schrödinger equation? The equation is linear and very simple. Can it handle accelerating systems?

Conclusions

We were not able to find a derivation of the Schrödinger equation from zitterbewegung.

Instead, our attention turned to Galilean covariance of the Schrödinger equation. Is it Galilean covariant?

In the next blog post we will investigate this, and also check if an accelerating system can have solutions for the Schrödinger equation.

Is it problematic if the Schrödinger equation is not Galilean covariant? Not really. Since the equation is not relativistic, we know that the equation is only approximate. We know that the equation matches extremely well various empirical tests. It works in practice.

Suppose that the Schrödinger equation does not have solutions for accelerating systems. That is not problematic, either. The equation is approximate, and it is enough to have approximate solutions.

There are problems in adding a potential V to the relativistic Dirac equation. The Klein paradox produces nonsensical results. We can say that we do not know an equation which would accurately describe quantum mechanics.

Retardation and dark energy

Let us have a collapsing spherical shell of dust. In this blog we have been claiming that a clock at the center cannot know the gravity potential right "now" in the coordinate time, and will tick faster than predicted by general relativity. We claim that the rate of a clock is an "effect" of the gravity field, and the effect cannot propagate faster than light.

We have been struggling to calculate how this will affect the collapse of a uniform dust ball.

In FLRW models, the expansion of the (dust-filled) universe will slow down as if gravity would be newtonian: we can study a small spatially spherical volume of the universe, and we obtain the deceleration from newtonian gravity.

We believe that the dust-filled universe in an FLRW model behaves like an expanding dust ball in an asymptotically Minkowski space.

We have been struggling to incorporate the retardation effect to this newtonian model of expansion (or collapse) of a dust ball.

Let us, once again, try to figure out what happens. We will study a collapse of a uniform dust ball in an asymptotically Minkowski space.

Retardation makes the gravity potential well shallower: a rubber sheet model

We believe that a clock inside a collapsing dust ball is not aware of the acceleration of individual dust particles far away. Therefore, the potential, as defined by the clock rate, is higher than one would calculate if one would assume that the clock knows the configuration "right now" in the coordinate time.

                          bulge

      ___                                        ___  rubber sheet

            \•______----------______•/

   weight  --->                  <--- weight

      

In a rubber sheet model of gravity, a contracting ring of weights will have the rubber sheet bulged upward at the center, because the rubber at the center of the ring does not yet know that the weights in the ring have been accelerated as they fall lower.

As the weights slide lower, potential energy is released. Most of the potential energy will go to the kinetic energy of the weights. Some will go to a "longitudinal" wave in the rubber sheet, that is, to the stretching of the bulge, and the kinetic energy of the rubber sheet.

In this model, the "gravity" simulated by the rubber sheet is somewhat weaker because not all released potential energy goes to the kinetic energy.

Presumably, the weakening effect grows stronger when the ring contracts. But could it entirely cancel the acceleration of the simulated "gravity" at some point? (Dark energy seems to be accelerating the expansion of the universe.)

Then all the potential energy released would at some point go to the deformation of the rubber sheet and the kinetic energy of the rubber sheet.

      

                   • -----______----- •

                              pit

Could it be that the bulge has time to flip into a "pit" at some point? Then the simulated "gravity" would appear stronger.

Modeling the retarded gravity field with a rubber sheet: a singularity appears

Let us have a mass M which is initially static. Then we start to accelerate it at a constant acceleration a.

                                                 observers

               ● ---> a      R              ×     r     ×

              M                                2            1

             ----->

      

Let observer 1 be such that he is not yet aware of the acceleration of M. 

Let us use the standard retardation rule: observer 2 sees the location of M as if M would have moved at the constant speed v, where v is the last observation that observer 2 made.

The gravity potential V(x) seen by observer 2 is continuous in x, but the derivative dV / dx in not continuous. The derivative should be continuous in a rubber sheet model?

                   ____

                  /               --> v

           -----

        rubber sheet

If we have a sharp turn moving in the rubber sheet, then the acceleration of the sheet is infinite at the turn. It is a singularity – nonsensical.

An electric charge which is suddenly accelerated

https://physics.weber.edu/schroeder/mrr/mrrtalk.html

How does the Edward M. Purcell calculation handle the analogous case for the electromagnetic field?

The electric field lines above are continuous, but make sharp turns. Can we produce those turns with a time-varying magnetic field which does not contain a singularity?

https://en.wikipedia.org/wiki/Maxwell%27s_equations

We can determine the curl of B from the time-varying field E. Is there a guarantee that B will not contain sources, and that the curl of E satisfies the upper equation?

If we would use the standard retardation formula for the electric potential V, then at the rightmost point of the circle above,

       dE / dt

would be infinite. But the requirement that the electric field lines are continuous means that we cannot apply the retardation rule to the potential V at that point.

What about the sharp turns of the electric field lines at other places? There, dE / dt is infinite and the curl of B is infinite. That seems nonsensical. Suppose then that we make the turns smooth.

When the charge q moves to the right, the magnetic field B takes the well known for of field lines circling the trajectory of q.

Let us accelerate q to the right. The second formula above is, at least approximately, satisfied. The same holds for the first formula.

In electromagnetism, the electric potential V does not have any direct effect on anything. It does not make clocks to tick slower. Therefore, the rubber sheet model does not describe electromagnetism. The potential V can change infinitely fast inside a contracting or expanding shell of charges.

We conclude that the electromagnetic analogy is not useful for us.

A new take: a tense rubber sheet

Let us try to prove that the bulge can turn into a pit. 

                          bulge

      ___                                         ___ rubber sheet

             \•______----------______•/

   weight --->                  <--- weight

Let the rubber sheet be very tense. We can assume that the weights move much slower than the waves in the rubber sheet.

Let us assume that the weights start moving. A bulge forms. Could it turn into a pit as the rubber sheet tries to straighten itself up at the center area?

No, it is not possible.

The clock on the bulge runs fast: the observer at the bulge sees the contraction of the dust shell to slow down mysteriously?

This could finally explain why the collapse of a dust shell appears to slow down. If we reverse time, then the expansion would speed up mysteriously, just like in dark energy.

But in the expansion model, if the expansion has accelerated recently, then far away galaxies should have the redshift surprisingly small relative to the brightness of the galaxy.

Thus, this will not work in a collapse of a dust ball.

However, in an expansion of a dust ball, this fits the scheme: the "bulge" in this case is downward. Clocks run slower in the bulge.

Retardation in a balloon model of the universe

Let us inflate a spherical balloon by blowing air into it. There are no retardation effects because the force keeping the rubber sheet tense is between adjacent molecules. It is a very short-range force.

But in the universe, the gravity force is predominantly a very long-range force. There might exist retardation effects which are due to long distances between galaxies.

An "expanding" rubber sheet model

          --------             --------    rubber sheet

                     \         /

            v <--   • --- •   --> v

                     ring of 

                     weights

Let us assume that the initial state is an expanding ring of weights sliding on the rubber sheet.

                     bulge

                      ____

         ------ • --       -- • ------   rubber sheet

        v <---                 ---> v

The weights slow down as they climb higher on the sheet. The circular region between them bulges upward. Eventually, the bulge may grow higher that the plane of the rubber sheet. The bulge pushes the weights outward: the expansion of the universe is accelerated.

Did we finally find an explanation for dark energy?

Conclusions

This blog post was left incomplete before the summer holiday of the author of the blog. Let us close the post now.

We finally came up with an idea about what could accelerate the expansion of the universe. From where does the energy come to the weights when they start to slide faster in the diagram above?

It comes from the kinetic energy of the rubber sheet on its way upward.

What is the interpretation of this in retardation terms?

The rubber sheet is a way to implement retardation in such a way that energy is conserved.

Problems in the anthropic principle

People appeal to the anthropic principle to explain the following observation:

- the laws of nature seem to be fine-tuned to allow biological life.

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

Anthropic reasoning goes this way: if the laws would not be fine-tuned, then there would exist no observer who would be wondering the fine-tuning.

Let us analyze what exactly is involved in the reasoning. What are the assumptions?

If there can only exist one universe, then the anthropic principle does not explain anything

Suppose that laws of nature dictate that exactly one universe must exist. Not zero or 2. 

Why are the laws of nature fine-tuned to allow biological life and sentient observers in that one universe?

The anthropic principle in this case does not explain anything. It could well be that the only universe is not suitable for life.

The existence of observers in our universe is an a posteriori observed fact. There is nothing a priori which requires observers.

Religious assumptions: the one universe must be fine-tuned for life

Let us assume that laws of nature dictate that exactly one universe must exist.

There might be a law of nature which requires that observers must exist in that one universe, and that I must be born as a creature which as an adult will become an observer. This would explain my a posteriori observation.

These assumptions resemble a religion: the laws of nature must be fine-tuned for life. It is like the creation myth in the Bible.

John D. Barrow and Frank J. Tipler formulated a hypothetical law of nature: the universe must be constructed in such a way that intelligent observers will arise.

A multiverse and the anthropic principle

Let us then assume that there exists a vast number of universes, with different laws of nature. Some models in "string theory" have this assumption: the multiverse.

Then there might exist a very large number of universes suitable for life. Many universes will contain observers.

Let us assume the following:

1.   My "soul" is predestined to be born inside a creature which will be sentient and become an observer as an adult.

2.   There are many universes with such creatures.

Then it is not surprising at all that I was born into a universe which contains observers.

Note that we need assumption 1, too, in addition to 2. If I would be predestined to be born either as a human, a rock, or an electron, then 2 would not explain why I happen to be an observer. Why am I not an electron?

Why do I find myself being one of the first intelligent observers on Earth?

There have been a huge number of vertebrates on Earth in the past 530 million years. Only about 10 billion humans have, so far, been aware of other galaxies besides the Milky Way.

Can an anthropic principle explain why I am among the first such observers on Earth?

If Earth could not harbor life, there would exist no such observers. This is the standard anthropic argument.

But there is no reason why I should be among the first. Why not the 10²⁰'th such observer?

We come back to the doomsday argument.

Observers could exist in the universe, but I could be a fish?

The a posteriori observation which I have made is that intelligent observers exist in a universe which contains billions of galaxies.

If I were a fish living in the year 2025, I would not have made such observation, even though it is true currently.

What do anthropic principles say about this?

They say that life must exist in our universe. But do they say anything about why I am not a fish?

An analogous question: suppose that on some exoplanet there exists an observer who is far superior to humans. A man is like a fish to this superobserver. Why was I born as a human (= fish), and not as the superobserver?

Weak anthropic principles basically say that for an observer of a type A to exist, the universe must be such that A can exist. That is close to a tautology. Obviously, it cannot answer to more complex questions, like "why was I not born as a fish?"

A human fetus may not be much more intelligent than an adult fish. If my "soul" would choose its birthplace by random from creatures which have some rudimentary intelligence, it is extremely unlikely that I would be born as a human fetus, which as an adult will be one of the first 10 billion humans to know about about galaxies.

We need stronger principles than the weak anthropic principles, to explain why "I" am among the first intelligent observers on Earth

Let us look at so-called strong anthropic principles.

Barrow and Tipler (1986) proposed at least the following variants:

1.   laws of nature require that there must exists exactly one universe, and that universe must contain observers;

2.   laws of nature require that any existing universe must contain observers (e.g., in quantum mechanics, an observer is required to make the wave function to collapse);

3.   laws of nature require that there must exist many universes (and some must contain observers).

All these imply that there exists at least one universe with observers. But they do not explain why I was not born as a fish. They do not explain why I was able to observe that this universe contains observers. Furthermore, they do not explain why I am among the ~ 10 billion first intelligent observers on Earth.

The Copernican principle

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

The Copernican principle can be stated like this: the physical location of the Solar system is "typical" in the universe. It is not the center of any important cosmological structure. The principle is strictly in contradiction with the Ptolemaic model of the Solar system where Earth is the center of everything.

Empirical observations strongly support the Copernican principle. 

What about the Copernican principle in the time dimension?

The doomsday argument is a Copernican principle with respect to time.

Do we exist at a "typical" time in the history of universe? No. The universe is expected to be very much suitable for life for ar least 1,000 billion years. We are living in a "young" universe.

The Copernican principle does not seem to hold on the surface of Earth. I am not a fish.

Conclusions

"Weak" anthropic principles are almost tautologies.

"Strong" anthropic principles contain a very brave hypothetical law of nature: a universe must necessarily produce "observers" at some point of time.

The Copernican principle is true for the spatial location of the Solar system, but it is not true with respect to the time dimension. Furthermore, on Earth the spatial Copernican principle does not hold at all: I was born a human and not a fish, even though there are more instances of a fish than a human.

Can we conclude that there must exist a mysterious law of nature "outside our universe", which places us to the current epoch and into conscious observers called humans?

Yes, that is the natural conclusion. Note that even if we would be placed at a random epoch, also that would a constitute a law of nature: you can expect to exist at a random epoch.

Ordinary laws of physics do not say anything about where and when we can expect to exist as an observer. Laws of quantum physics do talk about a collapse of a wave function caused by an "observer". Usually, people assume that the "observer" can be any large object which causes the wave function to "decohere". That does not require that I am the observer.

We can talk about natural laws of the placement of the subject. The placement seems to be non-random.

Suppose that you buy a new computer game and choose the character you are going to play in the game. The game is the "universe" and the character is the "observer". Obviously, the character will probably not be a random character in the game.

The hypothesis that our universe is a computer game, explains why we find ourselves living during a very special epoch as very special observers. The hypothesis implies strong anthropic principles: a computer game always contains "observers" if it has players.

What other hypothesis could explain our special position?

The doomsday argument is refuted because we are not born as random humans. The big oversight in the doomsday argument is that it assumes that we somehow know the prior probabilities of how we would be placed as observers. We do not know.

Suppose then that we would find ourselves as random observers. Why would we be random? What hypothesis could explain that?

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

In Plato's allegory of the cave, people have spent their entire life chained to the wall of a cave. They see shadows projected on the opposite wall. They do not see the real world, only shadows. However, through philosophy, one can learn about the real world.

Plato's allegory is somewhat similar to the computer game hypothesis. We are living inside a computer game. But through mathematics and logic we can learn something of the real world which exists outside the game.

When I started this blog in 2013, I, unfortunately, named it "metaphysical thoughts". The blog had been about physics, not metaphysics. The current blog post and the previous one can honestly be called "metaphysics". We finally have some content which fits the name of this blog!

The doomsday argument

The doomsday argument was popularized in the 1980s by the astrophysicist Brandon Carter, a colleague of Stephen Hawking.

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

The argument is so simple that it has been invented several times in the course of history.

People estimate that ~ 100 billion individuals of homo sapiens have lived on Earth in the past 250,000 years. Suppose that I was "chosen" by some mechanism to be a random individual among all the homo sapiens who will ever live.

Then it is likely that the total number of homo sapiens who will ever live will be something like ~ 200 billion. If we assume that the population of Earth will stay at 10 billion, then homo sapiens is expected to go extinct in ~ 700 years.

How could the number future homo sapiens could be so low, only 10¹¹? If humans will colonize exoplanets, there might be 10¹⁷ humans living in the next 10,000 years.

The doomsday argument is an example of anthropic reasoning.

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

Could it be that humans will be replaced by artificial intelligence machines?

What if artificial intelligence machines will replace humans in the future? Communication between exoplanets is slow. Each colonized exoplanet must have at least one instance of AI, autonomous from other instances. If the AI sends space probes, each probe must have an autonomous AI instance.

If humans are replaced by AI machines, we expect a very large number of autonomous, individual AI machines to exist in the future.

Why was I born as an instance of homo sapiens and not an instance of an AI machine? Or is it so that only ~ 100 billion AI machines will exist in the future?

Why was I born as homo sapiens, and not a random vertebrate?

Vertebrates appeared on Earth 530 million years ago. There has been a huge number of individuals in the past. The 100 billion instances of homo sapiens is an extremely tiny fraction of all instances of vertebrates which have lived.

Why am I not a random instance of a vertebrate, living some time in the past 530 million years?

Maybe I was destined to be born as a "self-conscious" being? But I was not born that way: a fetus or a newborn child is not self-conscious in the way that an adult is.

The self-indication assumption objection refutes the doomsday argument

The self-indication assumption is one way to refute the doomsday argument. Suppose that I am one of an almost infinite number N of "souls" who may be born as homo sapiens, or not be born at all. If I then find myself born as homo sapiens, I cannot deduce anything about the number of homo sapiens who will ever live.

Let us have two possible worlds:

- A: 200 billion homo sapiens will live;

- B: 10¹⁰⁰ homo sapiens will live.

Let, a priori, the probability of A and B be both 0.5.

Let me find myself living as the 100 billionth instance of homo sapiens in the world. Can I now deduce anything more about the probabilities of A and B?

No. In both cases, A and B, the probability of me being born as the 100 billionth instance is the same.

Self-indication assumption and vertebrates

The self-indication assumption does not answer the question: why was I born as an intelligent vertebrate (homo sapiens), and not as an average vertebrate (a small fish)?

There may exist a huge number of homo sapiens in the future, but that does not explain why I was born as homo sapiens at a time when only a tiny fraction of all instances of vertebrates have been homo sapiens.

I was not born as a random instance of a vertebrate?

This is the natural hypothesis: I was somehow predestined to be born as a being which becomes self-aware as an adult. There was no randomness in this.

Then it is also natural to assume that I was not born as a random instance of homo sapiens.

The doomsday argument is refuted because there is no randomness.

If there is no randomness, then there is some law of nature outside our universe. That law of nature brought me here.

The doomsday argument itself assumes that there is a law of nature outside our universe: that a "random process" causes me to be born as an instance of homo sapiens.

Criticism of the self-indication assumption

https://en.wikipedia.org/wiki/Self-indication_assumption_doomsday_argument_rebuttal

Nick Bostrom (2002) argued that the self-indication assumption leads to absurd consequences. Bostrom's argument is called the "presumptuous philosopher".

Suppose that we have two possible cosmologies:

- A: there are only 10¹¹ conscious observers in the universe;

- B: there are 10¹⁰⁰ conscious observers in the universe.

A priori, A and B have a probability 0.5. I am a conscious observer. Can a I deduce if I am in A or B?

Let us use the self-indication assumption. Let us assume that "souls" are assigned at random from the very large pool of N souls to the universe. Let us assume that I am a random soul in the pool. I realize that I was born. This implies that I am almost certainly in the universe B, not in A. Let us call the reasoning in this paragraph P.

Nick Bostrom is quoted in Wikipedia saying that it is "absurd" if one can draw the conclusion above in the reasoning P. He believes that the absurdity discredits the self-indication assumption.

We in our blog do not think that the reasoning P is absurd – rather, P is the natural conclusion if the randomization model and the pool of N souls is correct.

The weakness in P is that we do not know if the assumptions are correct. It certainly does not look like that my soul was assigned randomly into the universe.

Conclusions

The doomsday argument probably does not hold. It is easily refuted by the self-indication assumption. It is also refuted if there is no randomness. We cannot claim that only ~ 200 billion humans will ever live.

There seems to be a law of nature which is outside our universe. That law of nature decides in what role I am born.

In quantum mechanics, the collapse of the wave function, or in the many-worlds interpretation, how we choose the branch in which we live, is an unsolved mystery. The mystery looks like the question "why I was born as this instance of homo sapiens?"

We have to think about this: what kind of a law of nature outside our universe could possible solve the mystery in quantum mechanics?

John Bell's inequality: did John von Neumann derive it in 1932? No

John von Neumann's 1932 book about the foundations of quantum mechanics contains a proof, which, according to von Neumann, shows that quantum mechanics cannot be simulated with a "hidden variable" theory.

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

Grete Hermann in 1935 claimed that the proof contains a flaw, and John Bell in 1966 came up with a similar criticism.

Who is right?

Quantum mechanics is about waves: can we model quantum mechanics with "non-wavelike" phenomena?

A wave packet which models the position ans the momentum of a particle, is a typical quantum mechanical object.

Can we somehow build a wave packet from "non-wavelike" phenomena?

Intuitively, it is not easy. We can build a wave packet as a sum of two wave packets, but how could we build a wave packet from, say, ten real numbers?

Suppose that we have a sample of 1,000 classical particles, each of which has a definite position and momentum. This sample can approximately model a wave packet, but not exactly.

We have not yet checked the details of the proof by von Neumann. We believe that von Neumann aimed to show that, under certain assumptions, one cannot build a wave from a (finite) number of non-wavelike subsystems.

Criticism by Hermann and Bell: a digital computer

Grete Hermann and John Bell criticized the proof by von Neumann. They said that it might still be possible to simulate wavelike behavior with hidden variables.

Suppose that we have a digital computer which models and calculates a wave packet. Certainly, we can, at least approximately, simulate the wave packet in the computer. A digital computer is not a "wavelike" object. We proved that one can, approximately, simulate a wave with non-wavelike objects.

Could it be that a computer could simulate a wave packet exactly?

The random number generator is a problem. A pure digital computer must contain a deterministic pseudo-random number generator. This implies that the behavior of the computer program does differ from an idealized quantum system, and that we, in principle, can observe that the output of the program is not like that of a true quantum system. We conclude that a digital computer cannot simulate a quantum system.

Maybe this is what von Neumann proved?

The de Broglie-Bohm model contains a "pilot wave"

The model of Louis de Broglie and David Bohm (1952) contains a global "pilot wave". The particle is like a small boat which sails on the pilot wave.

We know that the model can reproduce the standard quantum mechanical results, like the interference pattern in a double-slit experiment.

The model does contain a wave. It is not surprising that the model can describe wave phenomena.

But the model cannot simulate the "true" random behavior of quantum mechanics? If we initialize the particles to certain definite values, then the system certainly will not behave in a truly random way. It is just like in the case of a digital computer.

Bell's inequality (1964)

https://en.wikipedia.org/wiki/Bell%27s_theorem

The famous inequality of John Bell (1964) shows that one cannot model the results of the Einstein-Podolsky-Rosen experiment with a "local hidden variable" theory. The state of the two particles at distant locations cannot be simulated with a model where each particle possesses a determined unique state.

The state of the two particles is "entangled". It cannot be split into two determined parts.

The result by Bell refers to locality, while the result of von Neumann does not. It is now clear that von Neumann did not prove Bell's theorem in 1932.

Did von Neumann prove a triviality?

How possibly could a short argument about expectation values rule out hidden variable models?

It has to be magic – or a triviality.

Suppose that we have a true random generator which outputs either -1 or 1 at random. Then the expectation value for its first output is 0.

But if it is a pseudo-random generator, which uses a complicated mathematical formula to generate a sequence of pseudo-random numbers, then the expectation value of the first output is -1 or 1 – not 0.

Did John von Neumann prove this triviality?

https://youtu.be/vRA62cPdRtY

Tom Harper (2023) has posted a video and a paper where he analyzes arguments by Louis Caruana (1995):

https://philarchive.org/rec/CARJVN

and Mermin and Schack (2018):

https://arxiv.org/abs/1805.10311

According to Caruana, von Neumann proves the following trivial fact:

- If quantum systems A and B are truly indistinguishable, then there cannot exist any hidden variables whose state could differ between A and B.

The criticism by John Bell (1966)

https://csiflabs.cs.ucdavis.edu/~gusfield/Bell-Von-Neuman.pdf

"Dispersionless", informally, means that a system is a classical particle, or some other classical, not quantum, object. It cannot be a quantum object in a superposition state.

Suppose that we have a quantum system which consists of dispersionless subcomponents. In a sense, the quantum system then is determined by hidden variables. An example: a tense plastic string behaves much like a wavelike object (quantum), but at a low level we can decompose it into atoms, which might be non-wavelike.

In section III of the paper, John Bell writes that one should not require the additivity of expectation values of different measurements for dispersionless subcomponents. Bell's statement is suspicious. The parameters of a classical particle can be measured arbitrarily accurately, without disturbing the system. That is, all measurements "commute", in the terminology of quantum mechanics. Then the expectation value of 

       measured position / meter

       + measured momentum / kg m/s

is the sum of the expectation values of each summand!

If we have an arbitrary hidden variable model which outputs values for measurements, then the question what is the expectation value of

       position + momentum

is not well defined. Let the hidden variable model be a computer program which outputs a number when we type either "position" or "momentum". What does the "expectation value" of position + momentum mean? We cannot type that string as the input to the computer. Should we first type "position" and then "momentum", and take the sum?

Conclusions

John von Neumann proved a triviality: something like this:

- You cannot simulate a genuine random number generator with a pseudo-random number generator, because for a pseudo-random generator you can, with a mathematical formula, predict the output of the generator!

Von Neumann used his operator algebra for the proof. The algebra is somewhat complicated, and people have had hard time evaluating the impact and relevance of the proof.

Hidden variable models are not trivial. A trivial result will not illuminate the problem much.

Our own contribution in this blog is noting that since quantum mechanics is about waves, then simplifying a system into a few real numbers is not going to work, in most cases.

This implies the famous result of John Bell: if you assume that the "wave" describing particles A and B has collapsed (i.e., the wave has been reduced into a few real numbers), then you will not be able to produce the phenomena of quantum mechanics. Letting a wave collapse often destroys information – but that information is needed to reproduce quantum phenomena.

To produce the interference pattern in the double-slit experiment, you need a wave. A classical particle which is characterized by a few real numbers (its velocity components), will not do.

Since the de Broglie-Bohm model includes the pilot wave, it can produce the interference pattern with hidden variables, deterministically.

We conclude that the critics of the von Neumann proof were right: the proof is not relevant for hidden variable models.

Chyba, Hand, Hossenfelder and generating electricity from a stationary magnetic field? It does not work

Sabine Hossenfelder (2025) is advertising a strange invention, almost a perpetuum mobile, which is supposed to stand static relative to the surface of Earth and produce an electric current from:

1.   the dipole magnetic field B of Earth, and

2.   the rotation of Earth as the device stands static relative the surface of Earth.

The claim that such a simple device can extract energy is very suspicious.

https://backreaction.blogspot.com/2025/04/this-new-device-creates-electricity.html

The original paper of Chyba and Hand (2016) is here:

https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.6.014017

J. Jeener (2018) refutes the result of Chyba and Hand:

https://arxiv.org/abs/1712.04283

From where would the energy to the electric current come? The dipole magnetic field of Earth does not change anything

Let us assume that Earth is a rigid charged sphere whose magnetic field is produced by charges which are static relative to Earth.

If the device of Chyba and Hand would produce in a circuit loop L an electric current I which can do work, it should be able to tap the rotation energy of Earth. The device must be able to send some of the angular momentum J of Earth to space, through electromagnetic waves.

Specifically, if the current I in the circuit loop does work on a resistor Ω, then the radiation of angular momentum to space should be so much larger that some energy can be extracted locally in the loop.

Let us first assume that Earth is uncharged, rotating, and there is a current I in the loop L.

Let the amount of of angular momentum radiated by the system be

       dJ / dt.

Does it change if we add a static dipole magnetic field B₀ to Earth?

Far away from the system, the electric field E of the radiated electromagnetic wave oscillates or rotates. The Poynting vector

      S  =  1 / μ₀  *  E  ×  B₀

is, on the average, zero. Adding the static dipole field B₀ does not change anything, in terms of the radiation.

The energy extraction of the device does not depend on the magnitude of B₀ in any way. We cannot say that the magnetic field B₀ of Earth helps the device in any way.

The device can extract energy from the rotation of Earth if it contains an electric charge or a permanent magnet. Then the device radiates the angular momentum of Earth slowly away in electromagnetic waves. But this not the mechanism which Chyba and Hand allege to exist.

We conclude that the claim of Chyba and Hand is erroneous.

Electromotive force in a circuit loop

https://phys.libretexts.org/Bookshelves/University_Physics/Physics_(Boundless)/22%3A_Induction_AC_Circuits_and_Electrical_Technologies/22.1%3A_Magnetic_Flux_Induction_and_Faradays_Law

Faraday's law states that the electromotive force (the path integral of the voltage) around a circuit loop is

       dΦ / dt,

where Φ is the magnetic flux through the loop.

The device of Chyba and Hand stands static on the surface of Earth. As time progresses, there, obviously, is no change in Φ through the loop. There is no electromotive force and there is no current.

Empirical measurements by Chyba, Hand, and Chyba (2025)

https://arxiv.org/abs/2503.15790

The measurements supposedly show power generation which almost fits inside the error brackets of the measurement. This is typical for claims about perpetuum mobiles. The device never produces 1 kW or 1 MW of surplus energy. That would be easy to detect.

It is always something which is very difficult to measure – because it does not really generate any power.

The same was true for "cold fusion" experiments.

Conclusions

It is extremely unlikely that the device of Chyba and Hand could produce electric power. Sabine Hossenfelder should correct her video blog to include the refutation by J. Jeener (2018).