Characterisation and representation of non-dissipative electromagnetic medium with a double light cone

M.F. Dahl

Journal of Mathematical Physics, Vol. 54, 2013, 011501 (23 pages).

Journal page
preprint,
full text pdf. By permission. Copyright 2013 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.

The paper relies on some computations that require computer algebra. Below are Mathematica toolbooks for these computations.

Notebooks for paper

Theorem 3.1: Coordinate transformations for twisted normal forms. [notebook, pdf]
Theorem 3.5: Coordinate transformations for permutating constants in Metaclass I. [notebook, pdf]
Theorem 3.5: Explicitly verify that converse direction. [notebook, pdf]
Theorem 3.6 (i): Non-birefringence implies A=0 or rho=0: [notebook, pdf]
Theorem 3.6 (ii): Symbolic expressions for det kappa and det H, proof that g and H can not be proportional: [notebook, pdf]
Theorem 4.4: Hodge + Axion is not decomposable for Lorentz signature [notebook, pdf]
Theorem 4.4: Hodge + Axion is not decomposable for (--++) signature [notebook, pdf]
Section 4.3: Compute epsilon matrix: [notebook, pdf]
Discussion at end of Section 4.3 regarding factorisability for the Fresnel polynomial for algebraically decomposable medium tensors with gamma != 0 and beta²-alpha gamma=0:

5-parameter family with at least one linear factor: [notebook, pdf]
Example with exactly one linear factor: [notebook, pdf]
3-parameter family with exactly two linear factors: [notebook, pdf]
1-parameter family with four linear factors: [notebook, pdf]
notebook for finding algebraically decomposable medium tensors with beta²-alpha gamma=0: [notebook, pdf]

Discussion at end of Section 4.3 regarding factorisability for the Fresnel polynomial for algebraically decomposable medium tensors when gamma != 0, beta²-alpha gamma!=0 and the nonlinear equation for D has no solutions:

beta²-alpha gamma<0 and Fresnel polynomial is product of quadratic forms of signatures (++--): [notebook, pdf]
4-parameter family for which beta²-alpha gamma>0 and Fresnel polynomial is product two linear factors and a quadratic form: [notebook, pdf]
1-parameter family for which beta²-alpha gamma<0 and Fresnel polynomial is product two quadratic forms: [notebook, pdf]
Notebook to find tensors for which the equation for D is not solvable. [notebook, pdf]

Theorem 5.1: Proof of (i) => (ii):

Metaclass I, Case alpha1 = alpha2: [notebook, pdf]
Metaclass I, Case alpha1 != alpha2: [notebook, pdf]
Metaclass II: [notebook, pdf]
Metaclass IV, Case alpha1 = alpha3: [notebook, pdf]
Metaclass IV, Case alpha1 != alpha3: [notebook, pdf]

Theorem 5.1: Proof of (ii) => (i):

Metaclass I representation of Hodge+Axion medium: [notebook, pdf]
Metaclass I: [notebook, pdf]
Metaclass II: [notebook, pdf]
Metaclass III: [notebook, pdf]
Metaclass IV: [notebook, pdf]
Metaclass V: [notebook, pdf]
Metaclass VI: [notebook, pdf]
Metaclass VII: [notebook, pdf]

Theorem 5.1: Proof of (i) => (iii):

Metaclass I: [notebook, pdf]
Metaclass II: [notebook, pdf]
Metaclass IV: [notebook, pdf]

Not all medium can be represented by a bivector A that is simple: [notebook, pdf]
Script helper.m: [script]

Additional notebooks

These are supplementary notebooks.

For Lorentz metrics g and h, N(g) subset N(h) implies that g and h are confromally related: [notebook, pdf]
Doublecheck identities before Lemma 2.3: [notebook, pdf]
Doublecheck the factorisation of the Fresnel polynomials in I. Lindell, L. Bergamin, A. Favaro, Decomposable Medium Conditions in Four-Dimensional Representation, IEEE Transactions on Antennas and propagation, Vol. 60, No. 1, 2012 [preprint, print]:

QDCM medium: [notebook, pdf]
PDCM medium: [notebook, pdf]
SDCM medium: [notebook, pdf]

kappaLib: A library for manipulation of electromagnetic medium

To run the above notebooks, you will need kappaLib version 1.2, which is a collection of Mathematica routines for manipulating electromagnetic medium tensors.