Wikipedia current has a debate on whether the Copernican system was heliocentric. See the Talk pages for Nicolaus Copernicus and Copernican heliocentrism.

This is a diversion from the usual Copernicus argument, which is whether he was Polish or German. He is customarily called Polish, but nation-states were not well defined, and there is an argument that he was more German than Polish.

The main point of confusion is that Copernicus did not really put the Sun at the center of the Earth's orbit. It was displaced by 1/25 to 1/31 of the Earth's orbit radius.

It appears that the center of the Earth's orbit revolved around the Sun, but you could also think of the Sun as revolving around the center of the Earth's orbit.

So is it fair to say that the Sun is at the center of the universe? Maybe if you mean that the Sun is near the center of the planetary orbits. Or that the Sun was at the center of the fixed stars. Or that the word "center" is used loosely to contrast with an Earth-centered system.

In Kepler's system, and Newton's, the Sun is not at the center of any orbit, but at a focus of an ellipse. It was later learned that the Sun orbits around a black hole at the center of the Milky Way galaxy.

I am not sure why anyone attaches such great importance to these issue. Motion is relative, and depends on your frame of reference. The Ptolemy and Copernicus models had essentially the same scientific merit and accuracy. They mainly differed in their choice of a frame of reference, and in their dubious arguments for preferring those frames.

People act as if the Copernicus choice of frame was one of the great intellectual advances of all time.

Suppose ancient map makers put East at the top of the page. Then one day a map maker put North at the top of the page. Would we credit him for being a great intellectual hero? Of course not.

There is an argument that the forces are easier to understand if you choose the frame so that the center of mass is stationary. Okay, but that was not really Copernicus's argument. There were ancient Greeks who thought it made more sense to put the Sun at the center because it was so much larger than the Earth. Yes, they very cleverly figured out that the Sun was much larger. There is a good logic to that also, but it is still just a choice of frame.

## Monday, July 30, 2018

## Friday, July 27, 2018

### How physicists discovered the math of gauge theories

We have seen that if you are looking for theories obeying a locality axiom, and if you adopt a geometrical view, you are inevitably led to metric theories like general relativity, and gauge theories like electromagnetism. Those are the simplest theories obeying the axioms.

Formally, metric theories and gauge theories are very similar. Both satisfy the locality and geometry axioms. Both define the fields as the curvature of a connection on a bundle. The metric theories use the tangent bundle, while the gauge theories use an external group. Both use tensor calculus to codify the symmetries.

The Standard Model of particle physics is a gauge theory, with the group being U(2)xSU(3) instead of U(1), and explains the strong, weak, and electromagnetic interactions.

Pure gauge theories predict massless particles like the photon. The Standard Model also has a scalar Higgs field that breaks some of the symmetries and allows particles to have mass.

The curious thing, to me, is why it took so long to figure out that gauge theories were the key to constructing local field theories.

Newton published his theory of gravity in 1682, but was unhappy about the action-at-a-distance. He would have preferred a local field theory, but he could not figure out how to do it. Maxwell figured it out for electromagnetism in 1865, based on experiments of Faraday and others.

Nobody figured out the symmetries to Maxwell’s equations until Lorentz and Poincare concocted theories to explain the Michelson-Morley experiment,in 1892-1905. Poincare showed in 1905 that a relativistic field theory for gravity resolves the action-at-a-distance paradoxes of Newtonian gravity. After Minkowski stressed the importance of the metric, the geometry, and tensor analysis to special relativity in 1908, Nordstrom, Grossmann, Einstein, and Hilbert figured out how to make gravity a local geometrical theory. Hermann Weyl combined gravity and electromagnetism into what he called a “gauge” theory in 1919.

Poincare turned electromagnetism into a geometric theory in 1905. He put a non-Euclidean geometry on spacetime, with its metric and symmetry group, and used the 4-vector potential to prove the covariance of Maxwell’s equations. But he did not notice that his potential was just the connection on a line bundle.

Electromagnetism was shown to be a renormalizable quantum field theory by Feynman, Schwinger and others in the 1940s. ‘tHooft showed that all gauge theories were renormalizable in 1970. Only after 1970 did physicists decide that gauge theories were the fundamental key to quantum field theory, and the Standard Model was constructed in the 1970s. They picked SU(3) for the strong interaction because particles had already been found that closely matched representations of SU(3).

It seems to me that all of relativity (special and general) and gauge theory (electromagnetism and the Standard Model) could have derived mathematically from general principles, with little or no reference to experiment. It could have happened centuries ago.

Perhaps the mathematical sophistication was not there. Characterizing geometries in terms of symmetry transformations and their invariants was described in Klein's Erlangen Program, 1872. Newton did not use a vector notation. Vector notation did not become popular until about 1890. Tensor analysis was developed after that. A modern understanding of manifolds and fiber bundles was not published until about 1950.

Hermann Weyl was a brilliant mathematician who surely had an intuitive understanding of all these things in about 1920. He could have worked out the details, if he had understood how essential they were to modern physics. Why didn’t he?

Even Einstein, who was supposedly the big advocate of deriving physics from first principles, never seems to have noticed that relativity could be derived from geometry and causality. Geometry probably would not have been one of his principles, as he never really accepted that relativity is a geometrical theory.

I am still trying to figure out who was the first to say, in print, that relativity and electromagnetism are the inevitable consequences of the geometrical and locality axioms. This seems to have been obvious in the 1970s. But who said it first?

Is there even a physics textbook today that explains this argument? You can find some of it mentioned in Wikipedia articles, such as Covariant formulation of classical electromagnetism, Maxwell's equations in curved spacetime - Geometric formulation, and Mathematical descriptions of the electromagnetic field - Classical electrodynamics as the curvature of a line bundle. These articles mention that electromagnetics fields can be formulated as the curvature of a line bundle, and that this is an elegant formulation. But they do not explain how general considerations of geometry and locality lead to the formulation.

David Morrison writes in a recent paper:

String theorists like to believe that useful physical theories can be derived from first principles. Based on the above, I have to say that it is possible, and it could have happened with relativity. But it has never happened in the history of science.

If there were ever an example where a theory might have been developed from first principles, it would be relativity and electromagnetism. But even in that case, it appears that a modern geometric view of the theory was only obtained many decades later.

Formally, metric theories and gauge theories are very similar. Both satisfy the locality and geometry axioms. Both define the fields as the curvature of a connection on a bundle. The metric theories use the tangent bundle, while the gauge theories use an external group. Both use tensor calculus to codify the symmetries.

The Standard Model of particle physics is a gauge theory, with the group being U(2)xSU(3) instead of U(1), and explains the strong, weak, and electromagnetic interactions.

Pure gauge theories predict massless particles like the photon. The Standard Model also has a scalar Higgs field that breaks some of the symmetries and allows particles to have mass.

The curious thing, to me, is why it took so long to figure out that gauge theories were the key to constructing local field theories.

Newton published his theory of gravity in 1682, but was unhappy about the action-at-a-distance. He would have preferred a local field theory, but he could not figure out how to do it. Maxwell figured it out for electromagnetism in 1865, based on experiments of Faraday and others.

Nobody figured out the symmetries to Maxwell’s equations until Lorentz and Poincare concocted theories to explain the Michelson-Morley experiment,in 1892-1905. Poincare showed in 1905 that a relativistic field theory for gravity resolves the action-at-a-distance paradoxes of Newtonian gravity. After Minkowski stressed the importance of the metric, the geometry, and tensor analysis to special relativity in 1908, Nordstrom, Grossmann, Einstein, and Hilbert figured out how to make gravity a local geometrical theory. Hermann Weyl combined gravity and electromagnetism into what he called a “gauge” theory in 1919.

Poincare turned electromagnetism into a geometric theory in 1905. He put a non-Euclidean geometry on spacetime, with its metric and symmetry group, and used the 4-vector potential to prove the covariance of Maxwell’s equations. But he did not notice that his potential was just the connection on a line bundle.

Electromagnetism was shown to be a renormalizable quantum field theory by Feynman, Schwinger and others in the 1940s. ‘tHooft showed that all gauge theories were renormalizable in 1970. Only after 1970 did physicists decide that gauge theories were the fundamental key to quantum field theory, and the Standard Model was constructed in the 1970s. They picked SU(3) for the strong interaction because particles had already been found that closely matched representations of SU(3).

It seems to me that all of relativity (special and general) and gauge theory (electromagnetism and the Standard Model) could have derived mathematically from general principles, with little or no reference to experiment. It could have happened centuries ago.

Perhaps the mathematical sophistication was not there. Characterizing geometries in terms of symmetry transformations and their invariants was described in Klein's Erlangen Program, 1872. Newton did not use a vector notation. Vector notation did not become popular until about 1890. Tensor analysis was developed after that. A modern understanding of manifolds and fiber bundles was not published until about 1950.

Hermann Weyl was a brilliant mathematician who surely had an intuitive understanding of all these things in about 1920. He could have worked out the details, if he had understood how essential they were to modern physics. Why didn’t he?

Even Einstein, who was supposedly the big advocate of deriving physics from first principles, never seems to have noticed that relativity could be derived from geometry and causality. Geometry probably would not have been one of his principles, as he never really accepted that relativity is a geometrical theory.

I am still trying to figure out who was the first to say, in print, that relativity and electromagnetism are the inevitable consequences of the geometrical and locality axioms. This seems to have been obvious in the 1970s. But who said it first?

Is there even a physics textbook today that explains this argument? You can find some of it mentioned in Wikipedia articles, such as Covariant formulation of classical electromagnetism, Maxwell's equations in curved spacetime - Geometric formulation, and Mathematical descriptions of the electromagnetic field - Classical electrodynamics as the curvature of a line bundle. These articles mention that electromagnetics fields can be formulated as the curvature of a line bundle, and that this is an elegant formulation. But they do not explain how general considerations of geometry and locality lead to the formulation.

David Morrison writes in a recent paper:

In the late 1960s and early 1970s, Yang got acquainted with James Simons, ... Simons identified the relevant mathematics as the mathematical theory of connections on fiber bundles ... Simons communicated these newly uncovered connections with physics to Isadore Singer at MIT ... It is likely that similar observations were made independently by others.I attended Singer’s seminars in the 1970s, so I can confirm that it was a big revelation to him that physicists were constructing a standard model based on connections on fiber bundles, a subject where he was a leading authority. He certainly had the belief that mathematicians and physicists did not know they were studying the same thing under different names, and he knew a lot of those mathematicians and physicists.

String theorists like to believe that useful physical theories can be derived from first principles. Based on the above, I have to say that it is possible, and it could have happened with relativity. But it has never happened in the history of science.

If there were ever an example where a theory might have been developed from first principles, it would be relativity and electromagnetism. But even in that case, it appears that a modern geometric view of the theory was only obtained many decades later.

## Wednesday, July 25, 2018

### Clifford suggested matter could curve space

A new paper:

Almost half a century before Einstein expounded his general theory of relativity, the English mathematician William Kingdon Clifford argued that space might not be Euclidean and proposed that matter is nothing but a small distortion in that spatial curvature. He further proposed that matter in motion is not more than the simple variation in space of this distortion. In this work, we conjecture that Clifford went further than his aforementioned proposals, as he tried to show that matter effectively curves space. For this purpose he made an unsuccessful observation on the change of the plane of polarization of the skylight during the solar eclipse of December 22, 1870 in Sicily.I have wondered why some people credit Clifford, and this article spells it out. He was way ahead of everyone with the idea that our physical space might be non-Euclidean, as an explanation for gravity.

## Friday, July 20, 2018

### Vectors, tensors, and spinors on spacetime

Of the symmetries of spacetime, several of them are arguably not so reasonable when considering locality.

The first is time reversal. That takes time t to −t, and preserves the metric and some of the laws of mechanics. It takes the forward light cone to the backward light cone, and vice versa.

But locality is based on causality, and the past causes the future, not the other way around. There is a logical arrow of time from the past to the future. Most scientific phenomena are irreversible.

The second dubious symmetry is parity reversal. That is a spatial reflection that reverses right-handedness and left-handedness.

DNA is a right-handed helix, and there is no known life using left-handed DNA. Certain weak interactions (involved in radioactive decay) have a handedness preference. The Standard Model explains this by saying all neutrinos are massless with a left-handed helicity. That is, they spin left compared to the velocity direction. (Neutrinos are now thought to have mass.)

Charge conjugation is another dubious symmetry. Most of the laws of electricity will be the same if all positive charges are changed to negative, and all negative to positive.

Curiously, if you combine all three of the above symmetries, you get the CPT symmetry, and it is believed to be a true symmetry of nature. Under some very general assumptions, a quantum field theory with a local Lorentz symmetry must also have a CPT symmetry.

The final dubious symmetry is the rotation by 360°. It is a surprising mathematical fact that a rotation thru 720° is homotopic to the identity, but a rotation thru 360° is not. You can see this yourself by twisting your belt.

Thus the safest symmetry group to use for spacetime is not the Lorentz group, but the double cover of the connected component of the Lorentz group. That is, the spatial reflections and time reversals are excluded, and a rotation by x° is distinguished from a rotation by x+360°. This is called the spin group.

Geometric theories on spacetime are formulated in terms of vectors and tensors. Vectors and tensors are covariant, so that they automatically transform under a change of coordinates. Alternatively, you can say that spacetime is a geometric coordinate-free manifold with a Lorentz group symmetry, and vectors and tensors are the well-defined on that manifold.

If we consider spacetime with a spin group symmetry instead of the Lorentz group, then we get a new class or vector-like functions called spinors. Physicists sometimes think of a spinor as a square root of a vector, as you can multiply two spinors and get a vector.

The basic ingredients for a theory satisfying the geometry and locality axioms are thus: a spacetime manifold, a spin group structure on the tangent bundle, a separate fiber bundle, and connections on those bundles. The fields and other physical variables will be spinors and sections.

This is the essence of the Standard Model. Quarks, electrons, and neutrinos are all spinor fields. They have spin 1/2, which means they have 720° rotational symmetry, and not a 360° rotation symmetry. Such particles are also called fermions. The neutrinos are left-handed, meaning they have no spatial reflection symmetry. The Lagrangian defining the theory is covariant under the spin group, and hence well-defined on spacetime.

Under quantum mechanics, all particles with the same quantum numbers are identical, and a permutation of those identical particles is a symmetry of the system. Swapping two identical fermions introduces a factor of −1, just like rotating by 360°. That is what separates identical fermions, and keeps them from occupying the same state. This is called the Pauli exclusion principle, and it is the fundamental reason why fermions can be the building blocks of matter, and form stable objects.

Pauli exclusion for fermions is a mathematical consequence of having a Lorentz invariant quantum field theory.

All particles are either fermions or bosons. The photon is a boson, and has spin 1. A laser can have millions of photons in the same state, and they cannot be used to build a material substance.

The point here is that under general axioms of geometry and locality, one can plausibly deduce that spinor and gauge theories are the obvious candidates for fundamental physical theories. The Standard Model then seems pretty reasonable, and is actually rather simple and elegant compared to the alternatives.

In my counterfactual anti-positivist history of physics, it seems possible that someone could have derived models similar to the Standard Model from purely abstract principles, and some modern mathematics, but with no experiments.

The first is time reversal. That takes time t to −t, and preserves the metric and some of the laws of mechanics. It takes the forward light cone to the backward light cone, and vice versa.

But locality is based on causality, and the past causes the future, not the other way around. There is a logical arrow of time from the past to the future. Most scientific phenomena are irreversible.

The second dubious symmetry is parity reversal. That is a spatial reflection that reverses right-handedness and left-handedness.

DNA is a right-handed helix, and there is no known life using left-handed DNA. Certain weak interactions (involved in radioactive decay) have a handedness preference. The Standard Model explains this by saying all neutrinos are massless with a left-handed helicity. That is, they spin left compared to the velocity direction. (Neutrinos are now thought to have mass.)

Charge conjugation is another dubious symmetry. Most of the laws of electricity will be the same if all positive charges are changed to negative, and all negative to positive.

Curiously, if you combine all three of the above symmetries, you get the CPT symmetry, and it is believed to be a true symmetry of nature. Under some very general assumptions, a quantum field theory with a local Lorentz symmetry must also have a CPT symmetry.

The final dubious symmetry is the rotation by 360°. It is a surprising mathematical fact that a rotation thru 720° is homotopic to the identity, but a rotation thru 360° is not. You can see this yourself by twisting your belt.

Thus the safest symmetry group to use for spacetime is not the Lorentz group, but the double cover of the connected component of the Lorentz group. That is, the spatial reflections and time reversals are excluded, and a rotation by x° is distinguished from a rotation by x+360°. This is called the spin group.

Geometric theories on spacetime are formulated in terms of vectors and tensors. Vectors and tensors are covariant, so that they automatically transform under a change of coordinates. Alternatively, you can say that spacetime is a geometric coordinate-free manifold with a Lorentz group symmetry, and vectors and tensors are the well-defined on that manifold.

If we consider spacetime with a spin group symmetry instead of the Lorentz group, then we get a new class or vector-like functions called spinors. Physicists sometimes think of a spinor as a square root of a vector, as you can multiply two spinors and get a vector.

The basic ingredients for a theory satisfying the geometry and locality axioms are thus: a spacetime manifold, a spin group structure on the tangent bundle, a separate fiber bundle, and connections on those bundles. The fields and other physical variables will be spinors and sections.

This is the essence of the Standard Model. Quarks, electrons, and neutrinos are all spinor fields. They have spin 1/2, which means they have 720° rotational symmetry, and not a 360° rotation symmetry. Such particles are also called fermions. The neutrinos are left-handed, meaning they have no spatial reflection symmetry. The Lagrangian defining the theory is covariant under the spin group, and hence well-defined on spacetime.

Under quantum mechanics, all particles with the same quantum numbers are identical, and a permutation of those identical particles is a symmetry of the system. Swapping two identical fermions introduces a factor of −1, just like rotating by 360°. That is what separates identical fermions, and keeps them from occupying the same state. This is called the Pauli exclusion principle, and it is the fundamental reason why fermions can be the building blocks of matter, and form stable objects.

Pauli exclusion for fermions is a mathematical consequence of having a Lorentz invariant quantum field theory.

All particles are either fermions or bosons. The photon is a boson, and has spin 1. A laser can have millions of photons in the same state, and they cannot be used to build a material substance.

The point here is that under general axioms of geometry and locality, one can plausibly deduce that spinor and gauge theories are the obvious candidates for fundamental physical theories. The Standard Model then seems pretty reasonable, and is actually rather simple and elegant compared to the alternatives.

In my counterfactual anti-positivist history of physics, it seems possible that someone could have derived models similar to the Standard Model from purely abstract principles, and some modern mathematics, but with no experiments.

## Wednesday, July 18, 2018

### Electric charge and local fields

Studying electric charge requires mathematical structures distinct from spacetime. Charge is conserved, but that conservation law is not related to any transformation of space or time. To measure electric fields, we need functions that measure something other than space and time.

The simplest type of such function would be from spacetime to the complex numbers. But such a function has an immediate problem with the locality axiom. A function like that would allow comparing values at spatially separated points, but no such comparison should be possible. It doesn't make sense to relate physics at one spacetime point to anything outside its light cones.

So we need to have a way of doing functions on spacetime, such that function values can only be compared infinitesimally, or along causal curves. The mathematical construction for this is called a line bundle.

For electromagnetism, imagine that every point in spacetime has a phase, a complex number with norm one. The phase does not mean anything by itself, but the change in phase along a causal curve is meaningful, and is the key to describing the propagation of an electromagnetic field.

A bundle connection is, by definition, the mathematical info for relating the phases along a curve. It is usually given in infinitesimal form, so the total phase change along the curve is given by integrating along the curve.

It turns out that a bundle can be curved, just as spacetime can be curved. The curvature is a direct measure of how the phase can change around a closed loop.

You cannot make a flat map of the surface of the Earth, without distorting distances, and you can make a flat map of a curved bundle either. You can choose some function (called a bundle section) that defines a preferred phase at each point. Then any other section is this one multiplied by some complex function on spacetime. Thus any section can be written

For derivatives, we expect a product formula like

The difficulty is in interpreting

The

Maxwell’s equations reduce to interpreting the derivative of the curvature as the charge/current density.

The whole theory of electromagnetism is thus the inevitable consequence of looking for a geometric theory obeying locality.

One can also construct field theories with higher dimensional bundles, replacing the complex numbers

When a field theory is quantized, notorious infinities arise. For field theories based on connections on bundles as above, there is a theory of renormalization to cancel the infinities. It is a generalization of the system for quantum electrodynamics. Other theories are difficult or impossible to renormalize. Physicists settled on gauge theories because they are the only ones to give finite predictions in a quantum theory.

Historical note. This formulation of Maxwell’s equations appears to have been known by Hermann Weyl and other by 1920 or so, but it may not have been explicitly published until 1970 or so.

The simplest type of such function would be from spacetime to the complex numbers. But such a function has an immediate problem with the locality axiom. A function like that would allow comparing values at spatially separated points, but no such comparison should be possible. It doesn't make sense to relate physics at one spacetime point to anything outside its light cones.

So we need to have a way of doing functions on spacetime, such that function values can only be compared infinitesimally, or along causal curves. The mathematical construction for this is called a line bundle.

For electromagnetism, imagine that every point in spacetime has a phase, a complex number with norm one. The phase does not mean anything by itself, but the change in phase along a causal curve is meaningful, and is the key to describing the propagation of an electromagnetic field.

A bundle connection is, by definition, the mathematical info for relating the phases along a curve. It is usually given in infinitesimal form, so the total phase change along the curve is given by integrating along the curve.

It turns out that a bundle can be curved, just as spacetime can be curved. The curvature is a direct measure of how the phase can change around a closed loop.

You cannot make a flat map of the surface of the Earth, without distorting distances, and you can make a flat map of a curved bundle either. You can choose some function (called a bundle section) that defines a preferred phase at each point. Then any other section is this one multiplied by some complex function on spacetime. Thus any section can be written

*fs*, where*f*is an ordinary complex function on spacetime, and*s*is a special section that gives a phase at each point. You can think of the phase as a norm-1 complex value, but that is misleading because the values at one point cannot be directly compared to the values at other points.For derivatives, we expect a product formula like

*d(fs) = f ds + s df*.The difficulty is in interpreting

*ds*, as a derivative requires comparing values to nearby points. A bundle connection is what allows infinitesimal comparisons, so in place of*ds*there is some*ω*for which we have a formula:*d(fs) = f ω + s df*.The

*ω*is what mathematicians call a connection 1-form, and what physicists call a 4-vector potential. It is not necessarily the derivative of a function. The derivative*dω*is what mathematicians call the curvature 2-form, and what physicists call the electromagnetic field. It does not depend on the choice of the section*s*.Maxwell’s equations reduce to interpreting the derivative of the curvature as the charge/current density.

The whole theory of electromagnetism is thus the inevitable consequence of looking for a geometric theory obeying locality.

One can also construct field theories with higher dimensional bundles, replacing the complex numbers

**C**with**C**^{n}for some*n*, and the phase with*U(n)*, the group of*n*x*n*unitary matrices. Other groups are possible also, such as*SU(n)*, the group of such matrices with determinant 1. The 1-form*ω*has values in the Lie algebra, which is the skew symmetric complex matrices, if the group is*U(n)*.When a field theory is quantized, notorious infinities arise. For field theories based on connections on bundles as above, there is a theory of renormalization to cancel the infinities. It is a generalization of the system for quantum electrodynamics. Other theories are difficult or impossible to renormalize. Physicists settled on gauge theories because they are the only ones to give finite predictions in a quantum theory.

Historical note. This formulation of Maxwell’s equations appears to have been known by Hermann Weyl and other by 1920 or so, but it may not have been explicitly published until 1970 or so.

## Friday, July 13, 2018

### Infinitesimal analysis of geometry and locality

The geometric axiom would appear to be in conflict with the locality axiom. The geometries are defined by symmetry transformations that relate points to distant points. Under locality, points only relate to nearby points, not distant points.

The resolution of this conflict is that the geometric operations actually act infinitesmally. The Lorentz transformations do not actually act on spacetime, but on the tangent space at a given point.

Here “infinitesimal” is a mathematical shorthand for the limits used in computing derivatives and tangents. Locality allows a point to be related to an infinitesimally close point in its light cone, and that is really a statement about the derivatives at the point.

In general relativity, matter curves spacetime, and spacetime does not necessarily have rotational or other symmetries. But locally, to first order in infinitesimals, a point looks like the Minkowski space described earlier. That is, the tangent space is linear with the metric

A causal path in a curved spacetime is one that is within the light cones at every point. As the light cones are in the tangent space, this is a statement about the tangents to the curve. In other words, the velocity along the path cannot exceed the speed of light.

There is a mathematical theory for curved spaces. A manifold has a tangent space at each point, and if also has a metric on that, then one can take gradients of functions to get vector fields, find the shortest distance between points, compare vectors along curves, and calculate curvature. All of these things can be defined independently of coordinates on the manifold.

Spacetime (Riemann) curvature decomposes as the Ricci plus Weyl tensors. Technically, the Ricci tensor splits into the trace and trace-free parts, but that subtlety can be ignored for now. The Ricci tensor is a geometric measure of the presence of matter, and is zero in empty space.

There are whole textbooks explaining the mathematics of Riemann curvature, so I am not going to detail it here. It suffices to say that if you want a space that looks locally like Minkowski space, then it is locally described by the metric and Riemann curvature tensor.

The equations of general relativity are that the Ricci tensor is interpreted as mass density. More precisely, it is a tensor involving density, pressure, and stress. In particular, solving the dynamics of the solar system means using the equation Ricci = 0, as the sun and planets can be considered point masses in empty space. Einstein's calculation of the precession of Mercury's orbit was based studying spacetime with Ricci = 0.

Next we look at electric charges.

The resolution of this conflict is that the geometric operations actually act infinitesmally. The Lorentz transformations do not actually act on spacetime, but on the tangent space at a given point.

Here “infinitesimal” is a mathematical shorthand for the limits used in computing derivatives and tangents. Locality allows a point to be related to an infinitesimally close point in its light cone, and that is really a statement about the derivatives at the point.

In general relativity, matter curves spacetime, and spacetime does not necessarily have rotational or other symmetries. But locally, to first order in infinitesimals, a point looks like the Minkowski space described earlier. That is, the tangent space is linear with the metric

*−dx*, in suitable coordinates.^{2}−dy^{2}−dz^{2}+c^{2}dt^{2}A causal path in a curved spacetime is one that is within the light cones at every point. As the light cones are in the tangent space, this is a statement about the tangents to the curve. In other words, the velocity along the path cannot exceed the speed of light.

There is a mathematical theory for curved spaces. A manifold has a tangent space at each point, and if also has a metric on that, then one can take gradients of functions to get vector fields, find the shortest distance between points, compare vectors along curves, and calculate curvature. All of these things can be defined independently of coordinates on the manifold.

Spacetime (Riemann) curvature decomposes as the Ricci plus Weyl tensors. Technically, the Ricci tensor splits into the trace and trace-free parts, but that subtlety can be ignored for now. The Ricci tensor is a geometric measure of the presence of matter, and is zero in empty space.

There are whole textbooks explaining the mathematics of Riemann curvature, so I am not going to detail it here. It suffices to say that if you want a space that looks locally like Minkowski space, then it is locally described by the metric and Riemann curvature tensor.

The equations of general relativity are that the Ricci tensor is interpreted as mass density. More precisely, it is a tensor involving density, pressure, and stress. In particular, solving the dynamics of the solar system means using the equation Ricci = 0, as the sun and planets can be considered point masses in empty space. Einstein's calculation of the precession of Mercury's orbit was based studying spacetime with Ricci = 0.

Next we look at electric charges.

## Wednesday, July 11, 2018

### Bee finds free will in reductionism failure

Sabine Hossenfelder writes on the Limits of Reductionism:

The universe is not really made of particles. The Heisenberg Uncertainty Principles shows that there are no particles. There are no laws that determine trajectories of particles. We have theories for how quantum fields evolve, and even predict bubble chamber tracks like the ones on the side of this blog. But it is just not true that the state tomorrow is encoded in the state today.

Look at radioactive decay. We have theories that might predict half-life or properties of emissions and other such things, but we cannot say when the decay occurs. For all we know, the atom has a mind of its own and decays when it feels like decaying.

The known laws of physics are simply not deterministic in the way that she describes.

She goes on to argue that world might still be indeterministic because of some unknown failure to reductionism.

In a way, chaos theory is such a failure of reductionism, because deterministic laws give rise in indeterministic physics. But she denies that sort of failure.

What she fails to grasp is that quantum mechanics is already compatible with (libertarian) free will.

The advocates of many-worlds (MWI) are nuts, but at least they concede that quantum mechanics does not determine your future (in this world). It makes for a range of future possibilites, and your choices will affect which of those possibilities you will get.

Of course the MWI advocates believe that every time you make a choice, you create an evil twin who gets the world with the opposite choice. That belief has no scientific substance to it. But the first part, saying that quantum mechanics allows free choices, is correct.

The other FQXI contest winners also have dubious physics, and perhaps I will comment on other essays.

Almost forgot to mention I made it 3rd prize in the 2018 FQXi essay contest “What is fundamental?”I don't know how she can get a Physics PhD and write stuff like that.

The new essay continues my thoughts about whether free will is or isn’t compatible with what we know about the laws of nature. For many years I was convinced that the only way to make free will compatible with physics is to adopt a meaningless definition of free will. The current status is that I cannot exclude it’s compatible.

The conflict between physics and free will is that to our best current knowdge everything in the universe is made of a few dozen particles (take or give some more for dark matter) and we know the laws that determine those particles’ behavior. They all work the same way: If you know the state of the universe at one time, you can use the laws to calculate the state of the universe at all other times. This implies that what you do tomorrow is already encoded in the state of the universe today. There is, hence, nothing free about your behavior.

The universe is not really made of particles. The Heisenberg Uncertainty Principles shows that there are no particles. There are no laws that determine trajectories of particles. We have theories for how quantum fields evolve, and even predict bubble chamber tracks like the ones on the side of this blog. But it is just not true that the state tomorrow is encoded in the state today.

Look at radioactive decay. We have theories that might predict half-life or properties of emissions and other such things, but we cannot say when the decay occurs. For all we know, the atom has a mind of its own and decays when it feels like decaying.

The known laws of physics are simply not deterministic in the way that she describes.

She goes on to argue that world might still be indeterministic because of some unknown failure to reductionism.

In a way, chaos theory is such a failure of reductionism, because deterministic laws give rise in indeterministic physics. But she denies that sort of failure.

What she fails to grasp is that quantum mechanics is already compatible with (libertarian) free will.

The advocates of many-worlds (MWI) are nuts, but at least they concede that quantum mechanics does not determine your future (in this world). It makes for a range of future possibilites, and your choices will affect which of those possibilities you will get.

Of course the MWI advocates believe that every time you make a choice, you create an evil twin who gets the world with the opposite choice. That belief has no scientific substance to it. But the first part, saying that quantum mechanics allows free choices, is correct.

The other FQXI contest winners also have dubious physics, and perhaps I will comment on other essays.

## Friday, July 6, 2018

### All physical processes are local

After the geometry axiom, here is my next axiom.

Locality means that physical processes are only affected by nearby processes. It means that there is no action-at-a-distance. It means that your personality is not determined by your astrological sign. It means that the Moon cannot cause the tides unless some force or energy is transmitted from the Moon to the Earth at some finite speed.

Maxwell's electromagnetism theory of 1861 was local because the effects are transmitted by local fields at the speed of light. Newton's gravity was not. The concept of locality is closely related to the concept of causality. Both require a careful understanding of time. Quantum mechanics is local, if properly interpreted.

Time is very different from space. You can go back and forth in any spatial direction, but you can only go forward in time. If you want to go from position

Locality demands that there is some maximum speed for getting from one point to another. Call that speed

Thus a physical event at a particular point and time can only influence events that are within a radius

Since the world is geometrical, there should be some geometry underlying this causal structure. The geometry is not Euclidean. The simplest such geometry to assume a metric where distance squared equals

The symmetries of spacetime with this metric may be found by an analogy to Euclidean space. Translations and reflections preserve the metric, just as with Euclidean space. A spatial rotation is a symmetry:

Imaginary time seems like just a sneaky math trick with no physical consequences, but there are some. If a physical theory has functions that are analytic in spacetime variables, then they can be continued to imaginary time, and sometimes this has consequences for real time.

Thus locality has far-reaching consequences. From the principle, it appears that the world must be something like Minkowski space, with Lorentz transformations.

Next, we will combine our two axioms.

**Locality axiom: All physical processes are local.**Locality means that physical processes are only affected by nearby processes. It means that there is no action-at-a-distance. It means that your personality is not determined by your astrological sign. It means that the Moon cannot cause the tides unless some force or energy is transmitted from the Moon to the Earth at some finite speed.

Maxwell's electromagnetism theory of 1861 was local because the effects are transmitted by local fields at the speed of light. Newton's gravity was not. The concept of locality is closely related to the concept of causality. Both require a careful understanding of time. Quantum mechanics is local, if properly interpreted.

Time is very different from space. You can go back and forth in any spatial direction, but you can only go forward in time. If you want to go from position

*(0,0,0)*to*(1,1,1)*, you can go first in the*x*-direction, and then in the*y*-direction, or go in any order of directions that you please. Symmetries of Euclidean space guarantee these alternatives. But if you want to go from*(0,0,0)*at time*t=0*to*(1,1,1)*at*t=1*, then your options are limited. Locality prohibits you from visiting two spatially distinct points at the same time. Time must always be increasing along your path.Locality demands that there is some maximum speed for getting from one point to another. Call that speed

*c*, as it will turn out to be the (*c*for constant) speed of light. No signal or any other causation can go faster than*c*.Thus a physical event at a particular point and time can only influence events that are within a radius

*ct*at a time*t*later. The origin*(0,0,0)*at*t=0*can only influence future events*(x,y,z,t)*if*x*. This is called the forward light cone. Likewise, the origin can only be influenced by past events in the backward light cone,^{2}+y^{2}+z^{2}≤ c^{2}t^{2}, t > 0*x*. Anything outside those cones is essentially nonexistent to someone at the origin. Such is the causal structure of spacetime.^{2}+y^{2}+z^{2}≤ c^{2}t^{2}, t < 0Since the world is geometrical, there should be some geometry underlying this causal structure. The geometry is not Euclidean. The simplest such geometry to assume a metric where distance squared equals

*−dx*. This will be positive inside the light cones. It does not make sense to relate to something outside the light cones anyway. Four-dimensional space with this non-Euclidean geometry is called Minkowski space.^{2}−dy^{2}−dz^{2}+c^{2}dt^{2}The symmetries of spacetime with this metric may be found by an analogy to Euclidean space. Translations and reflections preserve the metric, just as with Euclidean space. A spatial rotation is a symmetry:

*(x,y,z,t) → (x cos u − y sin u, x sin u + y cost u, z, t).*We can formally find additional symmetries by assuming that time is imaginary, and the metric is a Euclidean one on*(x,y,z,ict)*. Poincare used this trick for an alternate derivation of the Lorentz transformations in 1905. Rotating by an imaginary angle*iu*, and converting the trigonometric functions to hyperbolic ones:This is a Lorentz (boost) transformation with rapidity(x,y,z,ict) → (x, y, z cos iu − ict sin iu, z sin iu + ict cos iu)

= (x, y, z cosh u + ct sinh u, iz sinh u + ict cosh u)

*u*, or with velocity*v = c tanh u*. The more familiar Lorentz factor is*cosh u =*1/√*(1 − v*.^{2}/c^{2})Imaginary time seems like just a sneaky math trick with no physical consequences, but there are some. If a physical theory has functions that are analytic in spacetime variables, then they can be continued to imaginary time, and sometimes this has consequences for real time.

Thus locality has far-reaching consequences. From the principle, it appears that the world must be something like Minkowski space, with Lorentz transformations.

Next, we will combine our two axioms.

## Tuesday, July 3, 2018

### The world is geometrical

In my anti-positivist counterfactual history, here is the first axiom.

The most familiar geometry is Euclidean geometry, where the world is

Newtonian mechanics is a geometrical theory. Space is represented by

It is also a geometrical object because there is a large class of transformations that preserve the metric, and hence also the lines and circles determined by the metric. Those transformations are the rotations, reflections, and translations. For example, the transformation (x,y,z) → (x+5,y,z) preserves distances, and takes lines to lines and circles to circles.

Mathematically, a geometry can be defined in terms of some structure like a metric, or the transformations preserving that structure. This view has been known as the Klein Erlangen program since 1872.

The laws of classical equations can be written in geometrical equations like

Classical mechanics does not specify where you put your coordinate origin (0,0,0), or how the axes are oriented. You can make any choice, and then apply one of the symmetries of Euclidean space. Formulas like

One can think of Euclidean space and

The strongly geometrical character of classical mechanics was confirmed by a theorem of Neother that symmetries are essentially the same as conservation laws. Momentum is conserved because spacetime has a spatial translational symmetry, energy is conserved because of time translation symmetry, and angular momentum is conserved because of rotational symmetry.

Next, we look at geometries that make physical sense.

**Geometry axiom: The world is geometrical.**The most familiar geometry is Euclidean geometry, where the world is

**R**^{3}and distance squared is given by*dx*. There are other geometries.^{2}+ dy^{2}+ dz^{2}Newtonian mechanics is a geometrical theory. Space is represented by

**R**^{3}, with Euclidean metric. That is a geometrical object because of the linear structure and the metric. Lines can be defined as the short distance between points. Plane, circles, triangles, and other geometric objects can be defined.It is also a geometrical object because there is a large class of transformations that preserve the metric, and hence also the lines and circles determined by the metric. Those transformations are the rotations, reflections, and translations. For example, the transformation (x,y,z) → (x+5,y,z) preserves distances, and takes lines to lines and circles to circles.

Mathematically, a geometry can be defined in terms of some structure like a metric, or the transformations preserving that structure. This view has been known as the Klein Erlangen program since 1872.

The laws of classical equations can be written in geometrical equations like

*F=ma*, where*F*is the force vector,*a*is the acceleration vector, and*m*is the mass. All are functions on Euclidean space. What makes*F=ma*geometrical is not just that it is defined on a geometrical space, or that vectors are used. The formula is geometrical because all quantities are covariant under the relevant transformations.Classical mechanics does not specify where you put your coordinate origin (0,0,0), or how the axes are oriented. You can make any choice, and then apply one of the symmetries of Euclidean space. Formulas like

*F=ma*will look the same, and so will physical computations. You can even do a change of coordinates that does not preserve the Euclidean structure, and covariance will automatically dictate the expression of the formula in those new coordinates.One can think of Euclidean space and

*F=ma*abstractly, where the space has no preferred coordinates, and*F=ma*is defined on that abstract space. Saying that*F=ma*is covariant with respect to a change of coordinates is exactly the same as saying*F=ma*is well-defined on a coordinate-free Euclidean space.The strongly geometrical character of classical mechanics was confirmed by a theorem of Neother that symmetries are essentially the same as conservation laws. Momentum is conserved because spacetime has a spatial translational symmetry, energy is conserved because of time translation symmetry, and angular momentum is conserved because of rotational symmetry.

Next, we look at geometries that make physical sense.

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