# In Klein-Gordon, why should infinite downwards photon cascades be possible?

Here is a simple point about the standard interpretation of the Klein-Gordon equation that for the life of me I've never been able to understand:

Why would the existence of true negative energy states result in an infinite downward cascade of photon emissions?

I believe it was Dirac who originated this concept, but no matter how I turn it around in my head, I don't understand the logic of it.

A suite of true negative energy particle solutions to Klein-Gordon (not antimatter, which experimentally shows positive energy) would by exact symmetry behave identically to their positive energy equivalents. Thus they would radiate negative energy photons until they reach their own ground states.

So, if viewed from that true negative energy perspective, a downward cascade of positive energy photons would look like an ordinary electron raising its energy indefinitely by emitting negative energy photons. But we don't see that from within out positive particle perspective, so why should we expect the downward version to exist, either?

Is there some critical conceptual and/or mathematical rationale that I'm missing for why infinitely-downward photon cascades should exist?

I guess by implication that I may also be asking this: Why is Klein-Gordon not simply interpreted as having four solutions, rather than just two? They would include the known positive energy "pro" and "anti" matter solutions that share positive energy photons, but also a set of true negative energy versions of matter, antimatter, and photons. Surely this idea has at least been explored and then dismissed for some specific reason? I've looked some, but have not had any luck finding references.

To balance out the total energy of the universe, most current theories seem to assign the negative energy to space itself, again for reasons I truly do not understand. Such an approach leads unavoidably to complex balancing acts of trying to get the very differently structured positive and negative energies always to match in magnitude. Wouldn't it be both easier and simpler to have a true, exactly symmetric negative-energy universe scurrying away from us in the $t^-$ direction, with space simply as the zero-energy balancing point between the two?

Again, there must be some specific analysis that led to the current reliance on the more complicated "no true negative energy particles allowed" approach to balancing the energy of the universe (multiverse really), but I've had no luck uncovering it. What am I missing? Who made the decision to go to negative-energy-space balancing, and why?

2013-02-26 - Some relevant references

To my surprise, Paul Dirac's 1933 Nobel Lecture does a clear and remarkably succinct job of explaining his logic not just about negative energy states, but how he wound up at antimatter by contemplating them. At one point I immediately thought of tachyons. After checking a bit, sure enough, those ideas later became at least an aspect of tachyon theory. Odd how that feature of his negative energy sea is almost never mentioned.

As you can likely guess from the tachyon connection, Dirac interpreted the kinetics of negative energy states in an intriguing way that I was not expecting at all:

If one looks at Eq. (1), $[\frac{W^2}{c^2} - p_r^2 - m^2c^2]\psi = 0$, one sees that it allows the kinetic energy $W$ to be either a positive quantity greater than $mc^2$ or a negative quantity less than $-mc^2$. This result is preserved when one passes over to the quantum equation (2), $[\frac{W}{c} - {\alpha_r}p_r - {\alpha_0}mc]\psi = 0$, or (3) {$\alpha_\mu=I$; $\alpha_{\mu}\alpha_{\nu} + \alpha_{\nu}\alpha_{\mu} = 0$; $\alpha_{\mu}\neq\alpha_{\nu}$; $\mu,\nu = 0, 1, 2, 3$}. These quantum equations are such that, when interpreted according to the general scheme of quantum dynamics, they allow as the possible results of a measurement of $W$ either something greater than $mc^2$ or something less than $-mc^2$.

Now in practice the kinetic energy of a particle is always positive. We thus see that our equations allow of two kinds of motion for an electron, only one of which corresponds to what we are familiar with. The other corresponds to electrons with a very peculiar motion such that the faster they move, the less energy they have, and one must put energy into them to bring them to rest.

(I must remark that it is delightful to observe the little mutations in notation, with $W$ in place of $E$ and $\alpha$ in place of $\gamma$. Einstein's famous $E=mc^2$ is another example; it looked quite different in the original paper.)

Seeing the original version also reminds me of the remarkable number of choices that Dirac had to make to get to his matrices. The reasons for some of his choices were obvious, but for others they were not. In any case, his choice of an inverted-velocities interpretation of the kinematics provides some opportunity for understanding his original logic more meaningfully.

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Negative energy means that there is no ground state for these particles, in the sense that they would eventually stop cascading at some energy, say $-E_0<0$. In actuality we generally only care that the spectrum be bounded below, since we can add a constant to all energies to set that bound equal to zero anyway. So a negative energy state is more or less defined as a state that is unstable against this cascading process towards lower and lower energy.
Αρβανιτάκης Αλέξανδρος, thanks again for addressing this; I see you are well versed in topics such as Banach spaces. I am indeed proposing (without having intended to) a full symmetry that cannot be expressed with a single number line. I do not know a good terminology for that; sorry. Yet the idea remains very simple conceptually: Each side of zero perceives the other side as negative, and the sum of two particles is not energy, but a null state. It would be akin to a virtual electron/positron pair. The two universes moving apart along $t+$ and $t-$ would quite literally form a virtual pair. –  Terry Bollinger Feb 24 '13 at 2:37