Sunday, November 12, 2017

The beginning of the end of Trump-ism

Today I am going to skip the physics post and talk instead of a very important event, the elections in Virginia which signals the beginning of the end for Trump-ism. I live in Virginia, I voted in this election and my experience is shared by many other people.

The current outgoing governor of Virginia was a close friend/allied of Hillary Clinton who heavily promoted him over better democrat candidates in order to control the swing state of Virginia for Hillary during last presidential elections. Terry McAuliff was a slimy politician who took money from dishonest corporations but he managed to lay low and stayed outside attention during his term.

At the end of McAuliff's term, his second in command (Ralph Northam) ran for office and from the republican party was opposed by Ed Gillespie. Northam generated just as much enthusiasm as Hillary Clinton (meaning close to absolute zero) but started the race with a sizable lead. Gillespie was closing the gap when he made a huge miscalculation: he became a Trump clone. Gillespie's thinking was to energize the Trump base but it backfired big time on him. 

Originally I planned to stay home and not vote - I did not care for either Northam or Gillespie, but when Gillespie became "Trump 2" I became determined to vote against him. According to local polls, 50% of Northam's voters did so exactly like me and their vote was not pro Northam but against Trump. I think if democrats would have put a monkey on the ballot, the monkey would have still won. 

One day before the vote Gillespie was projected to win, and I voted without enthusiasm. I was very surprised to learn Northam won in a landslide. Moreover republican's large control of the local legislature was wiped off. 

Now is the time to undo Republican's district gerrymandering in Virginia. Gerrymandering is the dirty trick of carving up the districts to ensure your party maintains control even if you lose the majority of the voters. Here is how gerrymandering works:

As I stated above, I did not vote with enthusiasm because I was expected to loose, but now once it became clear what happened, there is renewed energy and drive to kick Trump and his clones out. Next step is to get rid of Nancy Pelosi. She made a deal with Trump and is now protecting Trump from impeachment. Next year the republicans will lose control of the House and the impeachment road will be clear. 

If you agree we should end the Trump nightmare, please sign this petition. This is run by by Tom Stayer

Sunday, October 29, 2017

The electromagnetic field

Continuing from last time, today I will talk about the electromagnetic field as a gauge theory.

1. The gauge group

In this case the gauge group is \(U(1)\) - the phase rotations. This group is commutative. This can be determined if we start from Dirac's equation and we demand that the group leaves the Dirac current of probability density:

\(j^\mu = \Psi^\dagger \gamma^0 \gamma^\mu \Psi\)


2. The covariant derivative giving rise to the gauge group

Here the covariant derivative takes the form:

\(D_\mu = \partial_\mu - i A_\mu\)

To determine the gauge connection \(A_\mu\) we can substitute this expression in Dirac's equation:

\(i\gamma^\mu D_\mu \Psi = m\Psi\)

and require the equation to be invariant under a gauge transformation:

\(\Psi^{'} = e^{i \chi}\Psi\)

which yields:

\(A^{'}_{\mu} = A_\mu + \partial_\mu \chi\)

This shows that:
- the general gauge field for Dirac's equation is an arbitrary vector field \(A_\mu (x)\)
-The part of the gauge field which compensates for an arbitrary gauge transformation of the Dirac field \(\Psi (x)\) is the gradient of on an arbitrary scalar field.

3. The integrability condition

Here we want to extract a physically observable object out of a given vector field \(A_\mu (x)\). From above it follows that there is no external potential if \(A_\mu = \partial_\mu \chi\) and this is the case if and only if:

\(\partial_\mu A_\nu - \partial_\nu A_\mu = 0\)

4. The curvature

The curvature measures the amount of failure for the integrability condition and by definition is the left-hand side of the equation from above:

\(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\)

and this is the electromagnetic field tensor.

5. The algebraic identities

There is only one algebraic identity in this case stemming from the curvature tensor antisymmetry:

\(F_{\mu\nu} +F_{\nu\mu} = 0\)

6. The homogeneous differential equations

If we take the derivative of \(F_{\mu\nu}\) and we do a cyclic sum we obtain:

\(F_{\mu\nu , \lambda} + F_{\lambda\mu , \nu} + F_{\nu\lambda , \mu} = 0\)

which is analogous with the Bianchi identity in general relativity.

This identity can be expressed using the Hodge dual as follows:

\(\partial_\rho {* F}^{\rho\mu} = 0\)

and this is nothing but two of the Maxwell's equations:

\(\nabla \cdot \overrightarrow{B} = 0\)
\(\nabla \times \overrightarrow{E} + \frac{\partial}{\partial t} \overrightarrow{B} = 0\)

7. The inhomogeneous differential equations

If we take the derivative of \(\partial_\beta F^{\alpha\beta}\) we get zero because the F is antisymetric and \(\partial_{\alpha\beta} = \partial_{\beta\alpha}\). and so the vector  \(\beta F^{\alpha\beta}\) is divergenless. We interpret this as a current of a conserved quantity: the source for the electromagnetic field and we write:

\(\partial_\rho F^{\mu\rho} = 4\pi J^\mu\)

where the constant of proportionality comes from recovering Maxwell's theory (recall that last time \(8\pi G\) came from similar arguments.

From this we now get the other two Maxwell's equations:

\(\nabla \cdot \overrightarrow{E} = 4\pi \rho\)
\(\nabla \times \overrightarrow{B} - \frac{\partial}{\partial t} \overrightarrow{E} = 4\pi \overrightarrow{j}\)

Now we can compare general relativity with electromagnetism:

Coordinate transformation - Gauge transformation
Affine connection \(\Gamma^{\alpha}_{\rho\sigma}\) - Gauge connection \(iA_\mu\)
Gravitational potential \(\Gamma^{\alpha}_{\rho\sigma}\) - electromagnetic potential \(A_\mu\)
Curvature tensor \(R^{\alpha}_{\beta\gamma\delta}\) - electromagnetic field \(F_{\mu\nu}\)
No gravitation \(R^{\alpha}_{\beta\gamma\delta} = 0\) - no electromagnetic field \(F_{\mu\nu} = 0\)

Sunday, October 8, 2017

The gravitational field

Today we will start implementing the 7 point roadmap in the case of the gravitational field. Technically gravity does not form a gauge theory but since it was the starting point of Weyl's insight, I will start with this as well and next time I will show how the program works in case of the electromagnetic field.

1. The gauge group

The "gauge group" in this case is the group of general coordinate transformations in a real four-dimensional Riemannian manifold M. Now the argument against Diff M as a gauge group comes from locality. An active diffeomorphism can move a state localized near the observer to one far away which can be different. However, for the sake of argument I will abuse this today and considered Diff M as a "gauge group" because of the deep similarities (which will explore in subsequent posts) between this and proper gauge theories like electromagnetism and Yang-Mills.

2. The covariant derivative giving rise to the gauge group

For a vector field \(f^\alpha\) the covariant derivative is defined as follow:

\(D_\rho f^\alpha = \partial_\rho f^\alpha +{\Gamma}^{\alpha}_{\rho\sigma} f^\alpha\)

where \({\Gamma}^{\alpha}_{\rho\sigma}\) is called an affine connection. If we demand that the metric tensor is a covariant constant under D we can find that the connection is:

\({\Gamma}^{\sigma}_{\mu\nu} = \frac{1}{2}[g_{\rho\mu,\nu} + g_{\rho\nu,\mu} - g_{\mu\nu,\rho}]\)

where \(f_{\rho,\sigma}  = \partial_\sigma f_\rho\) 

3. The integrability condition

We define this condition as the commutativity of the covariant derivative. If we define the notation: \(D_\mu D_\nu f_\sigma = f_{\sigma;\nu\mu}\) we can write this condition as:

\(f_{\rho;\mu\nu} - f_{\rho;\nu\mu} = 0\)

Computing the expression above yields:

\(f_{\rho;\mu\nu} - f_{\rho;\nu\mu} = f_\sigma {R}^{\sigma}_{\rho\mu\nu}\)
\({R}^{\sigma}_{\rho\mu\nu} = {\Gamma}^{\tau}_{\rho\mu}{\Gamma}^{\sigma}_{\tau\nu} - {\Gamma}^{\tau}_{\rho\nu}{\Gamma}^{\sigma}_{\tau\mu} + {\Gamma}^{\sigma}_{\rho\mu,\nu} - {\Gamma}^{\sigma}_{\rho\nu,\mu}\)

4. The curvature

From above the integrability condition is \({R}^{\sigma}_{\rho\mu\nu} = 0\) and R is called the Riemann curvature tensor.

5. The algebraic identities

The algebraic identities come from the symmetry properties of the curvature tensor which reduces the 256 components to only 20 independent ones. I am too tired to type the proof of the reduction to 20, but you can easily find the proof online.

6. The homogeneous differential equations

If we take the derivative of the Riemann tensor we obtain a differential identity known as the Bianchi identity:

\({R}^{\sigma}_{\rho\mu\nu;\tau} + {R}^{\sigma}_{\rho\tau\mu;\nu} + {R}^{\sigma}_{\rho\nu\tau;\mu} = 0\)

7. The inhomogeneous differential equations

This equation is of the form:

geometric concept = physical concept

And in this case we use the stress energy tensor \(T_{\mu\nu}\) and we find a geometric object with the same mathematical properties: symmetric and divergenless build out of curvature tensor. The left-hand side is the Einstein tensor:

\(G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R\)

The constant of proportionality comes from recovering Newton's gravitational equation in the nonrelativistic limit. In the end one obtains Einstein's equation:

\(G_{\mu\nu} = 8\pi G T_{\mu\nu}\)

Next time I will go through the same process for the electromagnetic field and map the similarities between the two cases. Please stay tuned.

Sunday, September 24, 2017

The Math of Gauge Theories

With a bit of a delay I am resuming the posts on gauge theory and today I will talk about the math involved. 

In gauge theory you consider the base space-time as a manifold and you attach at attach point an object or what is called a fiber forming what it is called a fiber bundle. The picture which you should have in mind is that if a rug.

The nature of the fibers is unimportant at the moment, but they should obey at least the properties of a linear space. 

Physically think of the fibers as internal degrees of freedom at each spacetime point, and a physical configuration would correspond to a definite location at one point long the fiber for each fibers. 

The next key concept is that of a gauge group. A gauge group is the group of transformations which do not affect the observables of the theory. 

Mathematically, the gauge symmetry depends on how we relate points between nearby fibers and to make this precise we only need (only) one critical step: define a covariant derivative.

Why do we need this? Because an arbitrary gauge transformation does not change the physics and the usual ordinary derivative sees both infinitesimal changes to the fields, and the infinitesimal changes to an arbitrary gauge transformation. Basically we need to compensate for the derivative of an arbitrary gauge transformation.

If d is the ordinary derivative, let's call D the covariant derivative and their difference (which is a linear operator) is called either a differential connection, a gauge field, or a potential:

A(x) = D - d

D and d act differently: d "sees" the neighbourhood behaviour but ignores the value of the function on which it acts, and D acts on the value but is blind to the neighbourhood behaviour.   

The condition we will impose on D is that is must satisfy the Leibniz identity because it is derivative:

D(fg) = (Df)g+f(Dg)

which in turn demands:

A(fg) = (Af)g+f(Ag)

In general only one part of A may be used to compensate for gauge transformations, and the remaining part represent an external field that may be interpreted as potential. When no external potentials are involved, A usually respects integrability conditions. Those conditions depend on the concrete gauge theory and we will illustrate this in subsequent posts.

When external fields are present, the integrability conditions are not satisfied and this is captured by what is called a curvature. The name comes from general relativity where lack of integrability is precisely the space-time curvature.

The symmetry properties arising out of curvature construction gives rise to algebraic identities.

Next in gauge theories we have the homogeneous and inhomogeneous differential equations. As example of homogeneous differential equations are the Bianchi identities in general relativity and the two homogeneous Maxwell's equations. The inhomogeneous equations are related to the sources of the fields (current in electrodynamics, and stress-energy tensor in general relativity).

So to recap, the steps used to build a gauge theory are:

1. the gauge group
2. the covariant derivative giving rise to the gauge field
3. integrability condition
4. the curvature
5. the algebraic identities
6. the homogeneous equations
7. the inhomogeneous equations

In the following posts I will spell out this outline first for general relativity and then for electromagnetism. Technically general relativity is not a gauge theory because diffeomorphism invariance cannot be understood as a gauge group but the math similarities are striking and there is a deep connection between diffeomorphism invariange and gauge theory which I will spell out in subsequent posts. So for now please accept this sloppiness which will get corrected in due time.

Monday, September 4, 2017

The Bohm-Aharaonov effect

Today we come back to gauge theory and continue on Weyl's ideas. With the advent of quantum mechanics Weyl realized that he could reinterpret his change in scale as a change in the phase of the wavefunction. Suppose we make the following change to the wavefunction:

\(\psi \rightarrow \psi s^{ie\lambda/\hbar}\)

The overall phase does not affect the Born rule and we did not change the physics (here \(\lambda\) does not depend on space and time and it is called a global phase transformation). Let's make this phase change depend on space and time: \(\Lambda = \Lambda (x,t) \) and see where it leads. 

To justify this assume we are studying charged particle motion in an electromagnetic field and suppose that \(\Lambda\) corresponds to a gauge transformation for the electromagnetic field potentials \(A\) and \(\phi\):

\(A\rightarrow A + \nabla \Lambda\)
\(\phi \rightarrow \phi - \partial_t \Lambda\)

This should not change the physics and in particular it should not change Schrodinger's equation. To make Schrodinger's equation invariant under a local \(\Lambda\) change we need to add  \(-eA\) to the momentum quantum operator:

\(-i\hbar \nabla \rightarrow -i\hbar \nabla -eA\)

And the Schrodinger equation of a charged particle in an electromagnetic field reads:

\([\frac{1}{2m}{(-i\hbar\nabla -eA)}^2 + e\phi +V]\psi = -i\hbar\frac{\partial \psi}{\partial t}\)

But why do we have the additional \(eA\) term to begin with? It's origin is in Lorentz force. If \(B = \nabla \times A\) and \(E = -\nabla \phi - \dot{A}\), the Lagrangian takes the form:

\(L = \frac{1}{2} mv^2 - e\phi + ev\cdot A\)

which yields the canonical momenta to be:

\(p_i = \partial{\dot{x}_i} = mv_i + eA_i\)

and adding \(-eA\) to the momenta in the Hamiltonian yields Lorentz force from Hamlton's equations of motion. 

Coming back to Schrodinger's equation we notice that the electric and magnetic fields E and B do not enter the equation, but instead we have the electromagnetic potentials. Suppose we have a long solenoid which has inside a non zero magnetic field B, and outside zero magnetic field. Outside the solenoid, in classical physics we cannot detect any change if the current flows or not through the wire. However the vector potential is not zero outside the solenoid (\(\nabla\times A = 0\) does not imply \(A=0\)) and the Schrodinger equation solves differently when \(A = 0\) and \(A\ne 0\). 

From this insight Bohm and Aharonov came up with a clever experiment to put this to the test: in a double slit experiment, after the slits they proposed to add a long solenoid. Record the interference pattern with no current flowing through the solenoid and repeat the experiment with the current creating a magnetic field inside the solenoid. Since the electrons do not enter the solenoid, from classical physics we should expect no difference, but in quantum mechanics the vector potential is not zero and the interference pattern shifts. Unsurprisingly the experiment confirms precisely the theoretical computation.

There are several important points to be made. First, there is no classical explanation of the effect: E and B are not fundamental, but \(\phi\) and \(A\) are. It is mind boggling that even today there are physicists who do not accept this and continue to look for effects rooted in E and B. Second, the gauge symmetry is not just a accidental symmetry of Maxwell's equation but a basic physical principle which turns out to govern all fundamental forces in nature. Third, the right framework for gauge theory is geometrical and we will explore this in depth in subsequent posts. Please stay tuned.

Due to travel, the next post is delayed 2 days.

Sunday, August 20, 2017

Impressions from Yellowstone

I was on vacation for a week in Yellowstone and I will put the physics post on hold want to share what I saw. First, the park is simply amazing and I highly recommend to visit if you have the chance. You need at least 3 days as a bare minimum. The main road is like the number 8 and on the west (left) side you get to see lots of fuming hot spots ejecting steam and sulfur.

The colors are due to bacteria and different bacteria live at different temperatures giving the hot spots rings of color.

On the south side you get the geysers and Old Faithful which erupts every 90 minutes.

You need to be there approximately 1 hour before the eruption to get a sit on the benches which surround Old Faithful. There are other geysers but you don't know when they erupt.

On the east side at the bottom of the 8 there is Yellowstone lake which gives rise to Yellowstone river and the Yellowstone canyon. Not much to do at the lake, the water is very cold.  The river forms two large waterfalls and you can visit them on both sides.

Coming north on the east side, you encounter more waterfalls and a bit of bisons. If you are lucky you get to see in the distance bears usually eating a dead moose. By the way, there is a big business ripoff in terms of bear sprays. You can buy one for $50, but you should rent one for $10/day when you hike in the forest. Even better just buy a $1 bell to wear to let the wildlife you are there (bears avoid people if they can hear them coming).

You can hike Mt. Washburn (4 hour round trip hike) to get a panoramic view of the park 50 miles in any direction.

There is nothing to see in the east-west part of the road at the middle of the 8, and on the the north of the east road there is another road leading east in Lamar's valley. Here is where you see a ton of wildlife: bisons, moose, wolves. Literally there are thousands of bisons in big herds which often cross the road.

Driving in the park is slow (25 mph) due to many attractions on the side and the traffic jams caused by animals. You need one day for north part, one day for the south loop, and one day for Lamar valley.

Yellowstone is at the spot of a supervolcano which erupted 7 times in the past: when it erupts it covers half of US with volcanic ash. There is a stationary hot spot of magma and because the tectonic plate moves different eruptions occur in different places. The past eruption locations trace a clear path on the map.

Yellowstone park is located in the caldera (the volcano crater) of the last eruption.

Sunday, August 6, 2017

The origins of gauge theory

After a bit of absence I am back resuming my usual blog activity. However I am extremely busy and I will create new posts every two weeks from now on. I am starting now a series explaining gauge theory and today I will start at the beginning with Hermann Weyl's proposal.

In 1918 Hermann Weyl attempted to unify gravity with electromagnetism (the only two forces known at the time) and in the process he introduce the idea of gauge theory. He espouse his ideas in his book "Space Time Matter" and this is a book which I personally find hard to read. Usually the leading physics people have crystal clear original papers: von Neumann, Born, Schrodinger, but Weyl's book combines mathematical musings with metaphysical ideas in an unclear direction. The impression I got was of a mathematical, physical and philosophical random walk testing in all possible ways and directions and see where he could make progress. He got lucky and his lack of cohesion saved the day because he could not spot simple counter arguments against his proposal which could have stopped him cold in his tracks. But what was his motivation and what was his approach?

Weyl like the local character of general relativity and proposed (from pure philosophical reasons) the idea that all physical measurements are relative. I particular, the norm of a vector should not be thought as an absolute value, but as a value that can change at various point of spacetime. To compare at different points, you need a "gauge", like a device used in train tracks to make sure the train tracks remained at a fixed distance from each other. Another word he used was "calibration", but the name "gauge" stuck.

So now suppose we have a norm \(N(x)\) of a vector and we do a shift to \(x + dx\). Then:

\(N(x+dx) = N(x) + \partial_{\mu}N dx^{\mu}\)

Also suppose that there is a scaling factor \(S(x)\):

\(S(x+dx) = S(x) + \partial_{\mu}S dx^{\mu}\)

and so to first order we get that N changes by:

\(( \partial_{\mu} + \partial_{\mu} S) N dx^{\mu} \)
Since for a second gauge \(\Lambda\), \(S\) transforms like:

\(\partial_{\mu} S \rightarrow \partial_{\mu} S  +\partial_{\mu} \Lambda \)

and since in electromagnetism the potential changes like:

\(A_{\mu}  \rightarrow A_{\mu} S  +\partial_{\mu} \Lambda \)

Weyl conjectured that \(\partial_{\mu} S = A_{\mu}\).

However this is disastrous because (as pointed by Einstein to Weyl on a postcard) it implies that the clocks would change their frequencies based on the paths they travel (and since you can make atomic clocks it implies that the atomic spectra is not stable).

Later on with the advent of quantum mechanics Weyl changed his idea of scale change into that of a phase change for the wavefunction and the original objections became mute. Still more needed to be done for gauge theory to become useful.

Next time I will talk about Bohm-Aharonov and the importance of potentials in physics as a segway into the proper math for gauge theory. 

Please stay tuned.