SimpleTensor_example.nb

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In this chapter we will examine applications of SimpleTensor on the example \
of relativistic electromagnetism.

We start by defining a Minkowski space with mostly negative metric:\
\>", "ExampleText",
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We define a Levi-Civita tensor and a time direction vector. This can also be \
done using the full function name:\
\>", "ExampleText",
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Individual components of a tensor can be extracted by contracting it with a \
corresponding unit vector. For example, by contracting EM potential with a \
time direction vector we get the corresponding zeroth component - the scalar \
potential:\
\>", "ExampleText",
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Using the partial derivative tensor operator we can calculate the \
electromagnetic tensor and assign it to a tensor:\
\>", "ExampleText",
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The antisymmetry of EM tensor can be shown either elementwise or by summing \
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\>", "ExampleText",
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From the electromagnetic tensor we can extract the vectors of the electric \
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Let us now derive the field part of Maxwell equations. First we define the \
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The field part of the homogeneous Maxwell equations then reads (for \
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The above calculation can also be done using the exterior calculus. Let us \
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The field part of the inhomogeneous Maxwell equations (for potentials) can be \
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We also define the time direction vector in the moving space:\
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Now we can easily derive the electric and magnetic fields in the moving frame:\
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When the space is no longer needed the following command effectively deletes \
it and all definitions connected to it:\
\>", "ExampleText",
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Spinors\
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In this chapter we will examine applications of SimpleTensor to Dirac and \
Weyl particles.

We start again by defining a Minkowski space with mostly negative metric, \
Levi-Civita tensor and time-direction 4-vector:\
\>", "ExampleText",
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Let us also introduce a convenient conjugate-transpose and adjoint operations:\
\>", "ExampleText",
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Next we define a 4-vector of gamma (Dirac) matrices, the \
\[OpenCurlyDoubleQuote]fifth\[CloseCurlyDoubleQuote] gamma matrix and a set \
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\>", "ExampleText",
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It is easy to check, for example, that the trace of Gamma matrices product is \
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\>", "ExampleText",
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Next let us find a plane-wave solution to the Dirac equation for a particle \
with mass m and momentum p. First we define the 4-momentum and write the \
matrix part of the Dirac equation. Note that when adding a list-valued tensor \
to an explicit Mathematica list one should use the TensorPlus (\[CirclePlus]) \
operator. Otherwise Mathematica automatically threads the tensor inside the \
explicit list before any computation can take place yielding a list of lists \
in the result.\
\>", "ExampleText",
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Let us now find the dispersion. A non-trivial solution exists only when the \
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This is a doubly-degenerate dispersion for Dirac particles and \
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Let us check that this spinor has the right current (the factor 2 is because \
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As a more advanced result, the Berry connection of the bi-spinor is zero \
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\>", "ExampleText",
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Cell["\<\


Let us now quickly calculate the Berry potential for a Weyl particle, as it \
will be needed in the next chapter. First we define a 4-vector of Pauli \
matrices and write the equation for a right-handed Weyl spinor:\
\>", "ExampleText",
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Now we find the right-handed Weyl spinor and normalize it to 1:\
\>", "ExampleText",
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Finally we calculate the Berry potential and check that it indeed gives the \
monopole in momentum space at the coordinate origin:\
\>", "ExampleText",
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Let us now check that the Dirac spinor transforms correctly under Lorentz \
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We can also check the transformation properties of the Dirac matrices \
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Lastly we clear the Minkowski space to not interfere with the next chapter:\
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Chiral Kinetic Theory\
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In this chapter we will examine applications of SimpleTensor to elementary \
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This chapter is based mostly on the excellent papers by Stephanov, Yin, and \
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Note, that because of the partially deliberate lack of optimization in favor \
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In its simplest form chiral particles are living in the Euclidean phase space \
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Let us also introduce a uniform electric and magnetic fields both pointing in \
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The dispersion of a relativistic chiral particle gets modified in the \
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Now we have all ingredients to define the Lagrangian 1-form of a chiral \
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\>", "ExampleText",
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Let us prove that this Lagrangian is indeed Lorentz invariant, at least up to \
the second order in \[HBar]. Let us for simplicity consider a boost in the \
z-direction. In this case the electric and magnetic fields do not change and \
therefore the Lagrangian is invariant with respect to the transformation if \
its Lie derivative with the vector field that generates this transformation \
is zero.  The vector field of the Lorentz transformation is then:\
\>", "ExampleText",
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The Lagrangian 1-form (without the dt term) works as a soldering form in our \
phase space and glues the particle velocity to its momentum. The exterior \
derivative of a soldering form gives the symplectic form of the corresponding \
phase space. However, our Lagrangian has an extra time coordinate and a term \
connected to it. The exterior derivative dL gives what is called a \
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system evolution and the quotient space with respect to this vector gives the \
phase-space. The projection of dL to this phase space is the symplectic form. \
Let us therefore calculate the exterior derivative of L.\
\>", "ExampleText",
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ExteriorDerivative with empty brackets is just a shortcut to take the \
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This system of equations indeed has the unique solution for the system \
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\>", "ExampleText",
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Due to complexity of the evolution it is hard to actually find the canonical \
coordinates of the phase space and project dL onto them. It is still easy, \
however, to find the phase space volume, as it can be found from the highest \
non-trivial power of the symplectic form:\
\>", "ExampleText",
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Cell["\<\
Let us finally check that the Liouville theorem in this theory still holds \
and the volume form stays invariant during the evolution. In other words, we \
show that its Lie derivative in the direction of evolution is zero:\
\>", "ExampleText",
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Lastly, after all the calculations we tidy up after ourselves:\
\>", "ExampleText",
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