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What class of theories/physical systems own finite/infinite complex eigenvalues? I do know that e.g., quasinormal modes of BH do have complex eigenvalues, but are they finite or infinite in number? What about Conformal Field Theories with complex eigenvalues? What are the general spectral properties of CFT? By the other hand, what kind of non-hermitian operators could provide spectra with infinite complex eigenvalues in pairs ( complex-conjugated)? What about the problems in the "new"/"recent" quantum graph theory?Is there some "infinite" graph with complex eigenvalues? Remark: I am not an expert on spectral graph theory but I do know a little bit about its methods.

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There are a lot of questions in one post there. I think it would benefit from reorganizing into a smaller number of questions. If the gist is a more general "how are complex eigenvalues used in physics" it may get more response if phrased that way. –  twistor59 Apr 14 '13 at 8:19
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Just doing it!Thank you for the suggestion. –  riemannium Apr 14 '13 at 21:20

1 Answer 1

Complex-valued eigenvalues can be used to introduce the concept of electromagnetic mass and charge.

Below I will provide an example of the Lorentz invariant model that uses complex-valued eigenvalues as a key component, but nevertheless allows for well defined momentum density with real valued mass density square. The “complexity” of the eigenvalues in this model is used for defining the complex-valued charge density that can further be split into electric and magnetic charge densities.

Let’s start with the free Dirac equation in spinorial form:

$\begin{array}{ cc}{{\partial{}}^{\mu{}\dot{\nu{}}} \ {\eta{}}_{\dot{\nu{}}}\ =-\ im{\xi{}}^{\mu{}}}\\ \\{{\partial{}}_{\mu{}\dot{\nu{}}} \ {\xi{}}^{\mu{}}=-\ im{\eta{}}_{\dot{\nu{}}}}\end{array} \ \ \ \ \ \ \ \ \ \ (i)$

where

$ \left({\partial{}}^{\mu{}\dot{\nu{}}}\right)=\ \left[\begin{array}{ cc} {\partial{}}_0+{\partial{}}_3 & {\partial{}}_1-i{\partial{}}_2 \\ {\partial{}}_1+{i\partial{}}_2 & {\partial{}}_0-{\partial{}}_3 \end{array}\right]=\ {\partial{}}_0 +{\partial{}}_1{\sigma{}}_1 + {\partial{}}_2{\sigma{}}_2+{\partial{}}_3{\sigma{}}_3 $

$ \left({\partial{}}_{\mu{}\dot{\nu{}}}\right)=\ \left[\begin{array}{ cc} {\partial{}}_0-{\partial{}}_3 & -{\partial{}}_1-i{\partial{}}_2 \\ -{\partial{}}_1+{i\partial{}}_2 & {\partial{}}_0+{\partial{}}_3 \end{array}\right]=\ {\partial{}}_0 - {\partial{}}_1{\sigma{}}_1^T - {\partial{}}_2{\sigma{}}_2^T - {\partial{}}_3{\sigma{}}_3^T $

Now let’s consider modification of the free Dirac equation $(i)$ by replacing constant “mass terms” with the variable electromagnetic field spinors:

$ \begin{array}{cc}{{\partial{}}^{\mu{}\dot{\nu{}}}{\eta{}}_{\dot{\nu{}}}=\ +f_{\nu{}}^{\mu{}}{\xi{}}^{\nu{}}} \\ \\ {{\partial{}}_{\mu{}\dot{\nu{}}}{\xi{}}^{\mu{}}=\ -{\dot{f}}_{\dot{\nu{}}}^{\dot{\mu{}}}{\eta{}}_{\dot{\mu{}}}} \end{array} \ \ \ \ \ \ \ \ \ \ (ii) $

Here second-rank spinors of electromagnetic field $ f_{\nu{}}^{\mu{}}$ are defined as follows:

$ f_{\nu{}}^{\mu{}}=\ \left[\begin{array}{ cc} f_1^1 & f_2^1 \\ f_1^2 & f_2^2 \end{array}\right]=\ \left[\begin{array}{ cc} F^3 & F^1-iF^2 \\ F^1+iF^2 & -F^3 \end{array}\right]=F^k{\sigma{}}_k,\ \ k=1,2,3 \ $

where $F^k=E^k-iB^k $

and complex-conjugated spinor $\dot{f}^{\dot \mu}_{\dot \nu}$ is defined as

$\dot{f}^{\dot \mu}_{\dot \nu}=\bar{ f_{\nu}^{\mu}}$

Equation $(ii)$ will replicate the form of the free Dirac equation $(i)$ if we require that spinors $\xi$ and $\eta$ are eigenvectors of the electromagnetic field spinors $ { f_{\nu}^{\mu}}$ and $\dot{f}^{\dot \mu}_{\dot \nu}$:

$ \begin{array}{ ccc} f_{\nu{}}^{\mu{}}{\xi{}}^{\nu{}}=\ \lambda{}\ {\xi{}}^{\mu{}} \\ \\ {\dot{f}}_{\dot{\nu{}}}^{\dot{\mu{}}}{\eta{}}_{\dot{\mu{}}}=\bar{\lambda{}}\ {\eta{}}_{\dot{\nu{}}} \end{array} \ \ \ \ \ \ \ \ \ \ \ \ (iii) $

Indeed, by applying $(iii)$ in $(ii)$ we obtain

$ \begin{array}{cc}{\partial{}}^{\mu{}\dot{\nu{}}}{\eta{}}_{\dot{\nu{}}}=+\ \lambda{}\ {\xi{}}^{\mu{}}\\ \\{\partial{}}_{\mu{}\dot{\nu{}}}{\xi{}}^{\mu{}}=\ -\bar{\lambda{}}\ {\eta{}}_{\dot{\nu{}}} \end{array} \ \ \ \ \ \ \ \ \ \ \ \ (iv)$

Eigenvalues $\lambda$ and $\bar\lambda$ in $(iii)$ and $(iv)$ are the well known invariants of the electromagnetic field:

$ \begin{array}{ ccc} {\lambda{}}_{\pm{}}=\ \pm{}\sqrt{E^2-B^2-2i\bf{EB}} \\ \\ {\overline{\lambda{}}}_{\pm{}}=\ \pm{}\sqrt{E^2-B^2+2\bf{EB}} \end{array} $

The momentum density $P_\mu$ can now be constructed by usual way as a sum of two isotropic 4-vectors $p_\mu$ and $\hat p^\mu$:

$P_{\mu{}}=\ p_{\mu{}}+\ g_{\mu\nu}\hat{p}^\nu$

where

$\begin{array}{cc} p_{\mu{}}=\ \frac{1}{2}\ \left({\xi{}}^+{\sigma{}}_{\mu{}}\xi{}\right) \\ \\ {\hat{p}}^{\mu{}}=\ \frac{1}{2}\left({\dot{\eta{}}}^+{\acute{\sigma{}}}^{\mu{}}\dot{\eta{}}\right)\end{array}$

In spite of the complex values of the “mass densities” $\lambda$ and $\bar\lambda$, the momentum densities of the matter fields $P_\mu$ are:

  • real valued
  • satisfy the continuity equation $ {\partial{}}_{\mu{}}P^{\mu{}}=0$ as a consequence of equation $(ii)$

  • “mass density square” equals to $P^{\mu{}}P_{\mu{}}=4{\left\vert{}\lambda{}\right\vert{}}^2$ and hence is always real and positively defined

  • World vector $P_\mu$ is an eigenvector of the stress-energy tensor of the electromagnetic field

It is worth noting that in this model the momentum density vector $P_\mu$ is always time-like, and its time-like component $P_0$ is always positive, hence no solutions with negative energies are allowed.

So we have shown that using complex eigenvalues $\lambda$ and $\bar\lambda$ the mass density can be expressed via electromagnetic field strengths $\bf{E}$ and $\bf{B}$.

Now we can develop the concept of the charge density that can also be expressed via $\bf{E}$ and $\bf{B}$.

We can do it by requiring that matter field equations $(ii)$ are equivalent (or reduced) to Maxwell equations. Physically that means that particle’s own electromagnetic field evolution is dynamically balanced with evolution of its source – particle’s spinorial field, so that the total field configuration remains stable in time.

As shown here, in the case of transverse plane waves the Maxwell and matter field equations become equivalent if the following relationship between the mass and charge densities is satisfied:

$ J^{\mu{}}=\bar{\lambda{}}\ P^{\mu{}} $

From this we conclude that, in the case of the transverse plane waves, electromagnetic field invariant $\bar\lambda$ plays the role of the charge density (while $\lambda$ plays the same role for anti-particles). Generally $\bar\lambda$ is complex valued, hence allowing for both non-zero electric and magnetic charge densities.

It is also demonstrated that in the case of the transverse plane waves the Lorentz force self-action on the matter field is zero when $E=B$, i.e. when real parts of the squared electromagnetic fields invariants ${\lambda{}}^2$ and ${\bar{\lambda{}}}^2$ are zero.

Particularly, this applies to electromagnetic waves in "vacuum" ($\bf{E}\perp{}\bf{B}$, $E=B$), where we have $\lambda{}=\bar{\lambda{}}=0$, and matter field equations coincide with Maxwell equations for "source-free" electromagnetic plane waves. In this case the momentum density $P^{\mu{}}$ of the matter field is non-zero, while the charge density $J^{\mu{}}$ is zero. In this sense the photons are not actually "source-free" electromagnetic waves.

For more details read the main article, where this model is fully described and applied to 3 types of stable elementary particles:

  • photons
  • charged fermions, and
  • neutrinos
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