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.
Wednesday, July 25, 2018
Clifford suggested matter could curve space
A new paper:
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 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 Cn for some n, and the phase with U(n), the group of nxn 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.
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 Cn for some n, and the phase with U(n), the group of nxn 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 −dx2−dy2−dz2+c2dt2, in suitable coordinates.
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 −dx2−dy2−dz2+c2dt2, in suitable coordinates.
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 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 x2+y2+z2 ≤ c2t2, t > 0. This is called the forward light cone. Likewise, the origin can only be influenced by past events in the backward light cone, x2+y2+z2 ≤ c2t2, t < 0. Anything outside those cones is essentially nonexistent to someone at the origin. Such is the causal structure of spacetime.
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 −dx2−dy2−dz2+c2dt2. 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.
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:
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 x2+y2+z2 ≤ c2t2, t > 0. This is called the forward light cone. Likewise, the origin can only be influenced by past events in the backward light cone, x2+y2+z2 ≤ c2t2, t < 0. Anything outside those cones is essentially nonexistent to someone at the origin. Such is the causal structure of spacetime.
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 −dx2−dy2−dz2+c2dt2. 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.
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:
(x,y,z,ict) → (x, y, z cos iu − ict sin iu, z sin iu + ict cos iu)This is a Lorentz (boost) transformation with rapidity u, or with velocity v = c tanh u. The more familiar Lorentz factor is cosh u = 1/√(1 − v2/c2).
= (x, y, z cosh u + ct sinh u, iz sinh u + ict cosh u)
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.
Geometry axiom: The world is geometrical.
The most familiar geometry is Euclidean geometry, where the world is R3 and distance squared is given by dx2 + dy2 + dz2. There are other geometries.
Newtonian mechanics is a geometrical theory. Space is represented by R3, 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.
Geometry axiom: The world is geometrical.
The most familiar geometry is Euclidean geometry, where the world is R3 and distance squared is given by dx2 + dy2 + dz2. There are other geometries.
Newtonian mechanics is a geometrical theory. Space is represented by R3, 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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