Much of modern physics rests on the assumption that electromagnetism has a gauge symmetry, and that the electromagnetic field is a gauge field.
https://en.wikipedia.org/wiki/Introduction_to_gauge_theory
The very first gauge symmetry introduced in Wikipedia is the option to set the zero electric potential at any value we like.
Let us then consider the electromagnetic field with charges, say, electrons and positrons.
As long as the inertial mass of an electron does not change substantially with the potential, we can indeed fix V = 0 at any voltage we like.
However, this is not true if the potential, relative to far-away space, is significant compared to the rest mass 511 keV of the electron.
We can say that the electric potential of far-away space is a preferred electric potential. All potentials were not created equal.
The Klein-Gordon equation is derived from the energy-momentum relation
E^2 = p^2 + m^2,
(where we have set c = 1).
The relation holds in vacuum. But it does not hold if the particle has significant potential energy.
If we in a potential write the relation as
(E - V)^2 = p^2 + m^2,
that relation holds approximately if the potential |V| is small. It does not hold in a relativistic setting.
The "minimal coupling", which is used to introduce an electromagnetic vector potential A to the Klein-Gordon equation or the Dirac equation, assumes that |V| is small.
The error in the minimal coupling explains the Klein paradox of the Dirac equation.
https://meta-phys-thoughts.blogspot.com/2018/10/we-solved-klein-paradox.html
The minimal coupling is derived in various sources from an assumed gauge symmetry of electromagnetism. The symmetry does not hold in relativistic settings. That explains why the minimal coupling does not work reasonably with large potentials.
We have shown that relativistic electromagnetism with charges is not a gauge theory, at least in the sense which is traditionally assumed. This means that much of the standard model is based on a shaky assumption.
If the magnetic field is strong, is the minimal coupling formula wrong, in the same fashion as for the electric potential? No. The total energy of a charge is not affected by a strong magnetic field.
Saturday, August 24, 2019
Tuesday, August 20, 2019
The Navier-Stokes millennium problem and the existence of solutions for the Einstein equations
In this blog we have brought up the question whether the Einstein field equations have a solution for any realistic physical system.
The Einstein equations are nonlinear. Therefore we cannot build complex solutions by summing simple solutions.
The Einstein equations are very strict: they imply results like the Birkhoff theorem, which dictates energy conservation for all matter fields.
https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
The Navier-Stokes equation is nonlinear, too.
https://arxiv.org/abs/1402.0290
Terence Tao suspects in his 2015 paper that at least some smooth initial problems "blow up", that is, develop singularities.
The problem in proving the existence of solutions for Navier-Stokes is in turbulence. We currently lack tools to understand turbulent behavior mathematically. Turbulence might create a singularity very quickly for almost all initial conditions.
The (assumed) generation of singularities is another common feature of Navier-Stokes and Einstein.
We do not know if solutions of the Einstein equations might contain turbulent behavior. The problem in finding solutions for Einstein seems to lie in the strictness of the equations. Solving the equations requires that "ends meet" when we assemble a puzzle on the FLRW model finite space. How do we know there exists any configuration of pieces such that the ends meet?
Terence Tao remarks that turbulence may be analogous to problems of complexity theory, for instance, to the P = NP millennium problem. We believe that P = NP will never be solved, because the collection of all polynomial-time algorithms is too complicated a system to be tamed and analyzed.
The Einstein equations are nonlinear. Therefore we cannot build complex solutions by summing simple solutions.
The Einstein equations are very strict: they imply results like the Birkhoff theorem, which dictates energy conservation for all matter fields.
https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
The Navier-Stokes equation is nonlinear, too.
https://arxiv.org/abs/1402.0290
Terence Tao suspects in his 2015 paper that at least some smooth initial problems "blow up", that is, develop singularities.
The problem in proving the existence of solutions for Navier-Stokes is in turbulence. We currently lack tools to understand turbulent behavior mathematically. Turbulence might create a singularity very quickly for almost all initial conditions.
The (assumed) generation of singularities is another common feature of Navier-Stokes and Einstein.
We do not know if solutions of the Einstein equations might contain turbulent behavior. The problem in finding solutions for Einstein seems to lie in the strictness of the equations. Solving the equations requires that "ends meet" when we assemble a puzzle on the FLRW model finite space. How do we know there exists any configuration of pieces such that the ends meet?
Terence Tao remarks that turbulence may be analogous to problems of complexity theory, for instance, to the P = NP millennium problem. We believe that P = NP will never be solved, because the collection of all polynomial-time algorithms is too complicated a system to be tamed and analyzed.
Wednesday, August 14, 2019
How was the fractal-like anisotropy of the cosmic microwave background born?
http://www.astro.ucla.edu/~wright/CMB-DT.html
https://www.physicsoverflow.org/19466/why-is-the-cmb-nearly-scale-invariant
The differences in temperature are in the range 25 - 70 microkelvins if we measure a feature size between 0.2 degrees and 20 degrees in the sky. The overall temperature is 3 K.
The features are roughly "scale-invariant" in the sense that a magnified map shows similar-looking anisotropies to an unmagnified map.
The inflation hypothesis explains the scale invariance by "random quantum fluctuations" which were inflated to a cosmic scale by the rapid expansion of the universe. There is a huge amount of energy in the anisotropies, even though the temperature difference is just around 1 / 100,000.
The energy had to come from somewhere. In the inflation hypothesis, the expansion of the universe produces energy from nothing to a scalar inflaton field. In our previous blog post we remarked that the non-conservation of energy may contradict basic principles of quantum mechanics. New energy quanta would pop up from nothing.
In classical physics, turbulence of, say hot smoke which rises from a smoke stack, creates a fractal-like structure. But it is hard to see how an expanding universe would have turbulence. In the case of the smoke, the turbulence is a result of the hot smoke colliding with cool, static air.
In a Big Bang model without inflation, there are roughly 6,000 points in the night sky which were not in a causal contact at the time when the microwave backround was born. Such points are roughly at a distance 3 degrees from each other.
How can the patterns of the anisotropies be coordinated over 20 degrees of the sky if there was no causal contact between different parts of that area? The inflation hypothesis solves this by asssuming a very fast expansion phase which caused a pattern to expand faster than light.
Could there be a mechanism which produces large patterns without a causal contact?
In a big bounce model, the causal contact would have occurred in the contraction phase.
Is there any other way to explain the large patterns? Maybe the original signal is at a constant temperature, but the space between the signal and us contains a fractal-like phenomenon which can change the apparent temperature. But how do we explain galaxy formation in that case? The fractal-like phenomenon should cause matter to collapse into galaxy clusters.
Yet another explanation is that, for an unknown reason, the matter in the Big Bang was created into a fractal-like structure. Since the Big Bang is beyond our understanding, there is no reason why the original matter content should have been smooth. If there was some process which created the original Big Bang universe, and that process was not constrained by the light speed in our universe, then the fractal-like structure could be a result of that process.
https://www.physicsoverflow.org/19466/why-is-the-cmb-nearly-scale-invariant
The differences in temperature are in the range 25 - 70 microkelvins if we measure a feature size between 0.2 degrees and 20 degrees in the sky. The overall temperature is 3 K.
The features are roughly "scale-invariant" in the sense that a magnified map shows similar-looking anisotropies to an unmagnified map.
The inflation hypothesis and the patterns of the microwave background
The inflation hypothesis explains the scale invariance by "random quantum fluctuations" which were inflated to a cosmic scale by the rapid expansion of the universe. There is a huge amount of energy in the anisotropies, even though the temperature difference is just around 1 / 100,000.
The energy had to come from somewhere. In the inflation hypothesis, the expansion of the universe produces energy from nothing to a scalar inflaton field. In our previous blog post we remarked that the non-conservation of energy may contradict basic principles of quantum mechanics. New energy quanta would pop up from nothing.
In classical physics, turbulence of, say hot smoke which rises from a smoke stack, creates a fractal-like structure. But it is hard to see how an expanding universe would have turbulence. In the case of the smoke, the turbulence is a result of the hot smoke colliding with cool, static air.
In a Big Bang model without inflation, there are roughly 6,000 points in the night sky which were not in a causal contact at the time when the microwave backround was born. Such points are roughly at a distance 3 degrees from each other.
How can the patterns of the anisotropies be coordinated over 20 degrees of the sky if there was no causal contact between different parts of that area? The inflation hypothesis solves this by asssuming a very fast expansion phase which caused a pattern to expand faster than light.
Could there be a mechanism which produces large patterns without a causal contact?
In a big bounce model, the causal contact would have occurred in the contraction phase.
Is there any other way to explain the large patterns? Maybe the original signal is at a constant temperature, but the space between the signal and us contains a fractal-like phenomenon which can change the apparent temperature. But how do we explain galaxy formation in that case? The fractal-like phenomenon should cause matter to collapse into galaxy clusters.
Yet another explanation is that, for an unknown reason, the matter in the Big Bang was created into a fractal-like structure. Since the Big Bang is beyond our understanding, there is no reason why the original matter content should have been smooth. If there was some process which created the original Big Bang universe, and that process was not constrained by the light speed in our universe, then the fractal-like structure could be a result of that process.
Why there are no "domain walls" or other defects in the visible universe?
It is assumed that the symmetry breaking of various fields, for example, the Higgs field and the electroweak field caused the universe to form "crystals" where a certain field has a constant value. The value in the neighbor crystal may be different.
We have not found domain walls in the visible universe. It looks like the whole visible universe is inside a single crystal.
Here we again have the problem how the whole visible universe is inside a single crystal if different parts have not been in a causal contact since the Big Bang.
The inflation hypothesis solves the problem by assuming that every crystal expanded immensely during the inflation.
Big bounce models might assume that the large crystal formed in the contraction phase.
We might also imagine that the original creation process of the Big Bang universe already had the symmetries broken, and the corresponding fields have a single value throughout the universe.
We might also imagine that the original creation process of the Big Bang universe already had the symmetries broken, and the corresponding fields have a single value throughout the universe.
The quantum measurement problem and "quantum fluctuations" in the inflaton field
In standard quantum mechanics, if we try to measure the energy content of some object, we will get varying values because of the uncertainty principle. Some people call these varying values "quantum fluctuations".
In a measurement, the measuring apparatus interacts with the object. If we claim that the variations of temperature in the sky are "quantum fluctuations", what is the measuring apparatus in that case? Is the scalar field itself and the metric a "measuring apparatus" which measures the energy content of the scalar field?
An analogous process is crystallization of a cooled liquid. If the temperature happens to drop within a small group of molecules, they will form a small initial crystal, a seed, which will grow larger. The measuring apparatus of the temperature is the liquid itself. A small random variation of the temperature downward will start a cascading process where at the end, it is the large crystal which interacted with the original temperature variation. The crystal is the measuring apparatus.
But is the inflaton field really analogous to crystal formation? A crystal is formed from a finite number of atoms. What are the "atoms" in the case of the inflaton field?
Monday, August 12, 2019
How does a scalar field behave in an expanding spatial metric?
The inflation hypothesis assumes the existence of a scalar field whose scalar value throughout the space "rolls" to a lower energy value.
The field may initially have, say, the value 0 in all space, and it somehow moves slowly to a lower energy value of, say, 1.
What does all this mean? We do have some experience of scalar fields in classical systems, but such fields are not the lowest level primitive fields. Such scalar fields are a high-level abstraction in a system consisting of elementary particles.
For example, if we have a drum skin, its vertical displacement is a scalar value in the 2D space of the skin. The lowest level description of the skin consists of atoms. The vertical displacement is a high-level concept.
If the 2D space expands, then the vertical displacement is affected in various ways.
Why would the inflaton field in an expanding space keep its scalar value, so that the energy of the field increases as the space stretches? Such a field certainly is "exotic matter" if its energy grows as the space grows.
The energy of an individual photon decreases in an expanding universe relative to "static" matter. The decrease can be attributed to the redshift that the static matter sees relative to the matter from which the photon originated.
The number of stable elementary particles, like the proton or the electron, probably does not grow in the expansion.
The energy content of the electron field preserves the energy in the rest mass of the electrons, but loses the kinetic energy of the electrons, because an electron keeps moving to a location where its velocity is less relative to static matter.
We face the dilemma why a scalar field would behave differently. Why would its energy grow in the expansion, if the energy of the electron field does not grow?
If we have an electric or a magnetic field which differs from zero, there is a quantum associated with it: a photon, or a quantum of some other field, like the electron field. The electromagnetic field does not have a life of its own independent of these quanta.
If we have a scalar field, why would it have a non-zero value which is not associated with a quantum of some field?
We need to study the logic behind the Higgs field. The Higgs boson is an excitation of the field. The Higgs field is assumed to be non-zero throughout the space. The non-zero value of the field is not born out from some quanta.
The vacuum is often defined as the lowest energy state of fields. There are no quanta present then.
We might then claim that quanta must be present if the fields are not in the lowest energy state. If the inflaton field initially is at a metastable higher energy state, then there must exist quanta which are responsible for that. If space expands, the number or energy of those quanta does not grow. Rather, the quanta are diluted in a larger volume of space. This would imply that the inflation mechanism cannot work. The value of the inflaton field will approach the minimum energy value as the quanta are diluted.
The reasoning above forbids exotic matter whose energy would grow as space expands. It also predicts a new particle, the charge particle of a scalar field. The charge particle would be responsible when the field has a value which is not the minimum energy value. Can the charge particle decay to other particles, so that the charge is lost? If that is the case, then the scalar charge is not conserved.
The field may initially have, say, the value 0 in all space, and it somehow moves slowly to a lower energy value of, say, 1.
What does all this mean? We do have some experience of scalar fields in classical systems, but such fields are not the lowest level primitive fields. Such scalar fields are a high-level abstraction in a system consisting of elementary particles.
For example, if we have a drum skin, its vertical displacement is a scalar value in the 2D space of the skin. The lowest level description of the skin consists of atoms. The vertical displacement is a high-level concept.
If the 2D space expands, then the vertical displacement is affected in various ways.
Does the energy of a primitive field grow as the space expands?
Why would the inflaton field in an expanding space keep its scalar value, so that the energy of the field increases as the space stretches? Such a field certainly is "exotic matter" if its energy grows as the space grows.
The energy of an individual photon decreases in an expanding universe relative to "static" matter. The decrease can be attributed to the redshift that the static matter sees relative to the matter from which the photon originated.
The number of stable elementary particles, like the proton or the electron, probably does not grow in the expansion.
The energy content of the electron field preserves the energy in the rest mass of the electrons, but loses the kinetic energy of the electrons, because an electron keeps moving to a location where its velocity is less relative to static matter.
We face the dilemma why a scalar field would behave differently. Why would its energy grow in the expansion, if the energy of the electron field does not grow?
Is a non-zero value of a field always associated with some quantum?
If we have an electric or a magnetic field which differs from zero, there is a quantum associated with it: a photon, or a quantum of some other field, like the electron field. The electromagnetic field does not have a life of its own independent of these quanta.
If we have a scalar field, why would it have a non-zero value which is not associated with a quantum of some field?
We need to study the logic behind the Higgs field. The Higgs boson is an excitation of the field. The Higgs field is assumed to be non-zero throughout the space. The non-zero value of the field is not born out from some quanta.
The vacuum is often defined as the lowest energy state of fields. There are no quanta present then.
We might then claim that quanta must be present if the fields are not in the lowest energy state. If the inflaton field initially is at a metastable higher energy state, then there must exist quanta which are responsible for that. If space expands, the number or energy of those quanta does not grow. Rather, the quanta are diluted in a larger volume of space. This would imply that the inflation mechanism cannot work. The value of the inflaton field will approach the minimum energy value as the quanta are diluted.
A charge particle of a scalar field?
The reasoning above forbids exotic matter whose energy would grow as space expands. It also predicts a new particle, the charge particle of a scalar field. The charge particle would be responsible when the field has a value which is not the minimum energy value. Can the charge particle decay to other particles, so that the charge is lost? If that is the case, then the scalar charge is not conserved.
Saturday, August 10, 2019
Problems with the cosmological inflation hypothesis
In the past week we have been studying the FLRW model and the inflation hypothesis.
https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric
The FLRW model is an exact solution of the Einstein field equations. It requires that the mass-energy density ϱ and the pressure p are uniform throughout the space.
In our previous post we stressed that it is not known if the Einstein equations have a solution for a realistic, nonuniform mass-energy density. That is, we do not know if a modified FLRW model can describe a realistic universe.
However, if we ignore the Einstein equations, we may simply assume that
1. the uniform, symmetric FLRW metric is a practical approximate model for our universe, and
2. we may combine the newtonian gravity to that model,
and we get a hybrid FLRW & Newton model.
Cosmologists use such hybrid models to simulate the evolution of the universe.
We do not know if any solution of the Einstein equations is close to the hybrid model. Thus, it may be that the hybrid model has nothing to do with general relativity.
Simulations with hybrid models have been moderately successful in explaining galaxy formation. Lots of open problems remain, though.
https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric
The FLRW model is an exact solution of the Einstein field equations. It requires that the mass-energy density ϱ and the pressure p are uniform throughout the space.
In our previous post we stressed that it is not known if the Einstein equations have a solution for a realistic, nonuniform mass-energy density. That is, we do not know if a modified FLRW model can describe a realistic universe.
A hybrid FLRW & Newton model might not be a general relativistic model at all
However, if we ignore the Einstein equations, we may simply assume that
1. the uniform, symmetric FLRW metric is a practical approximate model for our universe, and
2. we may combine the newtonian gravity to that model,
and we get a hybrid FLRW & Newton model.
Cosmologists use such hybrid models to simulate the evolution of the universe.
We do not know if any solution of the Einstein equations is close to the hybrid model. Thus, it may be that the hybrid model has nothing to do with general relativity.
Simulations with hybrid models have been moderately successful in explaining galaxy formation. Lots of open problems remain, though.
Dark energy
Observations suggest that the expansion of the universe has accelerated in the past 5 billion years or so. The standard ΛCDM cosmological model explains this with a positive cosmological constant Λ.
The FLRW model includes a cosmological constant, the same which appears in the Einstein equations. The model is an exact solution of the equations then.
The need to set the cosmological constant non-zero can be seen as a weakness of the ΛCDM model: the more adjustable parameters you have, the less predictive is the theory.
The inflation hypothesis
The inflation hypothesis claims that there was a scalar field in a higher energy state in the first moments of the universe. The field caused an exponentially fast expansion.
The field decayed or "rolled" to a lower energy state after a short period of time.
The energy density of the field was equal to the negative pressure that the field caused.
The FLRW model allows the energy density as well as the pressure to vary during the evolution of the universe. However, they have to be uniform throughout the space.
Let us list some of the problems involved with the inflaton field:
1. The inflaton field is "exotic matter". The negative pressure causes negative gravity which exceeds the positive gravity of the energy density. It breaks the strong energy condition:
http://cds.cern.ch/record/424782/files/0001099.pdf
(Visser and Barcelo 1999).
(Visser and Barcelo 1999).
2. Suppose that we in a Minkowski space have matter which breaks the strong energy condition. If such matter allows light to propagate faster than in the surrounding Minkowski space, then we face the paradoxes of closed timelike curves. Visser and Barcelo write about this problem in connection with wormholes, but the problem of closed timelike curves is possible in an asymptotic Minkowski space, too.
3. Why would the pressure be uniform throughout space when the inflaton field decays to a lower energy state? If the pressure is not uniform, does there exist a solution of the Einstein field equations?
4. What is the status of the "quantum fluctuations" which the fast expansion of the universe supposedly magnified to astronomical dimensions? Such quantum fluctuations are not a concept of standard quantum mechanics.
5. Can an expanding universe magnify quantum phenomena? We would need a theory of quantum gravity to answer that question. Why would a growing spatial metric cause the scalar field to expand? If we have an electric field of an electron, the electric field probably does not expand with the spatial metric. Why would a scalar field be different from an electric field?
6. The only known scalar field in the standard model is the Higgs field. We do not have any empirical observation of other scalar fields. Why would there exist a scalar field which can roll down to a lower energy state?
https://www.scientificamerican.com/article/cosmic-inflation-theory-faces-challenges/
Paul Steinhardt explained in a 2017 Scientific American article some other problems of the inflation hypothesis. It looks like the inflaton field has to be fine-tuned to make it consistent with observations. A prime motivation for the inflation hypothesis was to do away with the fine tuning required to get a flat spatial metric in the current observable universe. If the fine tuning problem comes back in the fine tuning of the inflation process, what is the benefit?
The flatness is at least partially explained by the anthropic principle: if the universe would collapse in a short time, or expand so fast that stars cannot form, then there would not exist intelligent observers in the universe.
4. What is the status of the "quantum fluctuations" which the fast expansion of the universe supposedly magnified to astronomical dimensions? Such quantum fluctuations are not a concept of standard quantum mechanics.
5. Can an expanding universe magnify quantum phenomena? We would need a theory of quantum gravity to answer that question. Why would a growing spatial metric cause the scalar field to expand? If we have an electric field of an electron, the electric field probably does not expand with the spatial metric. Why would a scalar field be different from an electric field?
6. The only known scalar field in the standard model is the Higgs field. We do not have any empirical observation of other scalar fields. Why would there exist a scalar field which can roll down to a lower energy state?
https://www.scientificamerican.com/article/cosmic-inflation-theory-faces-challenges/
Paul Steinhardt explained in a 2017 Scientific American article some other problems of the inflation hypothesis. It looks like the inflaton field has to be fine-tuned to make it consistent with observations. A prime motivation for the inflation hypothesis was to do away with the fine tuning required to get a flat spatial metric in the current observable universe. If the fine tuning problem comes back in the fine tuning of the inflation process, what is the benefit?
The flatness is at least partially explained by the anthropic principle: if the universe would collapse in a short time, or expand so fast that stars cannot form, then there would not exist intelligent observers in the universe.
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