This semester, I am taking a course on relativistic quantum mechanics. Currently we are covering the "hole interpretation" of negative energy solutions to the Dirac equation.

I've done this stuff before, as an undergraduate, in private study, and in various grad-level courses. So I'm used to the interpretation being given. But I decided recently that it is perfectly absurd.

Other schemes have failed |

*always*a lower energy state. So then electrons should be able to cascade down to infinity forever emitting

*infinite energy*!

The oil companies will go bankrupt!

To prevent this absurdity, Dirac made the

*very*rational step of proposing an infinite number of negative-energy electrons everywhere at all points in space filling up the infinitely-many negative energy states, preventing electrons from falling in to them. But because all space is filled with these negative-energy electrons, then we just call them "vacuum" and only pay attention to what is on top of them.

This infinite collection of electrons existing everywhere in every possible negative-energy state is called "the Dirac sea".

The first few times I heard of this, I sort of let it go as a kind of mathematical abstraction to help the equations make sense. But then I started to realize that physicists actually believe in the existence of an infinite "sea" of electrons that fills all space with every possible energy, as a real thing that really ontologically exists.

One thing I want to get cleared away, is that this Dirac sea is where we get the interpretation of spontaneous creation of electron-positron pairs in quantum mechanics. I hear a lot of arts-degree-atheists bring up this sort of quantum behavior of the "vacuum" to explain away Christian assertions that "something cannot come from nothing". And it can't. In the Dirac interpretation, there is an infinite number of electrons everywhere already existing; a photon excites one of them from negative to positive energy, so now we can see not only the electron itself, but the lack of negative-energy electron in the "sea" which we call a positron. So things apparently come from "nothing" only because we

*define*"nothing" to be infinitely-many things. Likewise, in the Heisenberg interpretation of field theory, the mathematics forces us to have a "vacuum" not with infinitely-many particles but infinite energy, and we call this infinite energy "vacuum" (or "nothing") and so it appears that "something" (electron-positron pairs) comes from "nothing" (infinite particles and energy everywhere in all space and time).

After that, is the mathematical absurdity of the hole theory. The negative energy states are said to be all "filled"; all infinity of them are. They're so filled that there is no room for observed electrons to collapse to negative energy. That's the idea. But mathematically, infinity and infinity+1 are the "same" thing. For a neat parable on this, read "Hilbert's Hotel". A basic summary of the story, if you have a hotel with infinitely-many rooms, and all of the rooms are filled -- all infinity of them, filled -- and another guest comes along, you can still make room for him by asking everyone to scootch down one room. If

*infinitely-many*guests come along looking for rooms, you can

*still*make room for all of them in your filled hotel by asking everyone to take their room number, double it, and move to that room. If

*, each of which has*

**infinitely-many busses***shows up at your hotel that is already filled with infinitely-many occupants, then you can*

**infinitely-many passengers***make room for the new-arrivals through a complicated procedure of labeling the rooms.*

**still**Applying Hilbert's Hotel isn't entirely one-to-one because the energy spectrum of negative energy solutions is continuously-infinite and not denumerably infinite... which actually only makes the situation

*even worse*, from a mathematical standpoint; accommodating a single discrete electron should be much more possible for the continuous spectrum than the discrete spectrum of Hilbert's Hotel. After all, every positive integer from 0 to infinity can be mapped to the interval between 0 and 1, and with plenty of elbow room to spare.

So the proposal of infinitely-many electrons filling all the negative energy states does not even solve the problem of cascading; it just obscures it.

turtles --> electrons |

So I've started objecting loudly in class about this. Sadly, there is a language barrier between my professor and I and I do not get my answers across very well. He has recommended, if I don't like it, to find my own interpretation. Maybe I will.

But seriously, what the heck?

Schematic Diagram of Dirac Hole Theory |

## 6 comments:

I assume the "turtles all the way down" stuff is a reference to the intro to "A Brief History of Time".

You lost me at " Currently we are covering the "hole interpretation" of negative energy solutions to the Dirac equation."

I am twisting my brain trying to figure out what turtles have to do with electrons and the speed of light, lol!

"Turtles all the way down" is a joking reference to a story reported by Stephen Hawking. He claims he once gave a lecture on astronomy, and at the end, an elderly lady told him his presentation was very nice, but wrong. "Everyone knows the world really sits on the back of a giant turtle," she told him. He slyly asked what the turtle is standing on. She replied, "Don't play that game. It's turtles all the way down!"

The idea is, there's just an infinite stack of turtles, each holding the next up. So the earth doesn't fall down because infinitely many turtles below it hold it up.

In the study of electrons, we get a weird problem, where our equations predict electrons with negative energy. The negative energies will go from 0 to negative infinity. This is a problem, firstly, because we have never seen a single negative-energy particle. Moreover, it is a problem because electrons like to lose energy and go to lower energies whenever they can. If there are negative energies, then electrons will *always* have a lower energy state they can go to, so electrons should *always* be losing energy.

The result would be awesome - free infinite energy forever! - but it isn't true. So why don't electrons constantly fall to the lower energy levels? In fact, why don't they ever even get slightly negative?

Dirac proposed -- Because there's an infinite stack of electrons filling all the negative energy levels! Each keeps the next from falling down! Electrons don't reach the negative energy levels because they're all filled, all infinity of them!

Which is basically the exact same idea as the turtles.

I also stumbled upon this when I first read about Dirac Sea. I have found something on this matter, hope this resolves your problem.

"...there is no Dirac Sea. This was an early idea that turned out to be incorrect. They were not antiparticles, they were supposed to be normal particles in negative energy states, and antiparticles were the vacancies ("holes"). But Quantum Field Theory replaced these negative energy states with positive energy antiparticles." Bill_K's reply to the thread https://www.physicsforums.com/threads/dirac-sea-of-antiparticles.583318/

In my humble opinion the QFT explanation isn't very much better. It does get rid of the Hilbert Hotel problem in the Dirac Sea, but apart from that, it mostly just replaces the infinitely-many-and-everywhere stack of electrons with infinite energy everywhere.

Asking Bill_K's same question, how does that infinite vacuum energy transform under Lorentz transformations? It would have to be invariant, but since we know energy transforms like a two-tensor under Lorentz transforms, then the the vacuum energy would define a privileged rest frame. So there cant be infinite vacuum energy (I think Bill_K actually says this). However, the infinite vacuum energies are endemic to QFT.

'Pure' math doesn't completely avoid this sort of thing.

I was always bothered by the 'fact' that 0!=1. I get why we made that definition by the various analogies to how we *use* factorials, but I was super excited to learn about the gamma function later on in my undergrad time.

But it turns out that's just even worse. I don't like talking about it any more.

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