The idea of the positron's existence?

[Lamarr]

Dirac's conception of the Positron?

I have read that Dirac predicted the existence of the positron when trying to combine QM with Relativity.

This doesn't make sense to me. How is the positron related to all this?

 

[Vorde]

The easy way to explain this is that Dirac was figuring out his 'Dirac Equation', which in a really hand-wavy way can be called a relativistic version of the Schrödinger equation which preceded it.

Dirac noticed that his equation (governing electrons) not only allows for the normal electron solution but also a second 'negative' (note: not negative energy) solution at anyone point. Rather than dismiss this negative solution as a problem in his equation, he predicted the existence of what would be known as the positron.

I do not know whether this negative solution was not predicted by the Schrödinger equation or just no one had noticed it, but I suspect the former.

 

[bcrowell]

I do not know whether this negative solution was not predicted by the Schrödinger equation or just no one had noticed it, but I suspect the former.

Right, it wasn't predicted by the Schrödinger equation. When you add a constant to the potential in the S eqn, all it does is shift the spectrum of energies of the solutions up by the same constant. This is sort of what you'd expect from classical Newtonian physics, where potential energies are always arbitrary up to a constant. Because of this, there can never be any special zero of energy picked out by the S eqn, so there's no way it could predict special properties for negative-energy states.

 

[Lamarr]

So how does adding Relativity to the mix lead to negative energies?

 

[Vorde]

I think the best answer is to say the positron is just a fact of the equations that wasn't apparent in the non-relativistic approximation but appears in the more accurate Dirac Equation.

Someone could post here correcting me and saying that the inclusion of relativity is more fundamental to the existence of the positron, but I doubt that will happen. More likely it was just that in the inclusion of relatively led to a more accurate equation which told us about anti-matter.

 

[Lamarr]

So relativity doesn't really have anything to do with it then?

 

[PeterDonis]

In his 1986 Dirac Memorial Lecture at Cambridge, Feynman gave his take on it, which was basically this: first, assume that all energies are positive. Another way to put this is that there is some state of minimum energy, and we can call this state the "vacuum", and any non-vacuum state will have a greater energy than this. (Sometimes you see this expressed as "the Hamiltonian is bounded below".)

If you assume that all energies are positive, then every particle has a nonzero amplitude to move faster than light; or, put another way, the amplitude for a particle to go from one event to another is nonzero if the events are spacelike separated. (This follows from a theorem in Fourier analysis.) But if the events are spacelike separated, then there is some inertial frame in which the second event is earlier than the first--or, put another way, there is some inertial frame in which the particle appears to travel backwards in time. But this "backwards in time" version of the particle *is* the antiparticle.

I don't know how generally accepted this view is; I have seen comments on both sides. But it does at least give a link between relativity and antiparticles, since it is relativity that tells us that "faster than light", in some frames, will appear as "backwards in time".

 

[Lamarr]

In his 1986 Dirac Memorial Lecture at Cambridge, Feynman gave his take on it, which was basically this: first, assume that all energies are positive. Another way to put this is that there is some state of minimum energy, and we can call this state the "vacuum", and any non-vacuum state will have a greater energy than this. (Sometimes you see this expressed as "the Hamiltonian is bounded below".)

If you assume that all energies are positive, then every particle has a nonzero amplitude to move faster than light; or, put another way, the amplitude for a particle to go from one event to another is nonzero if the events are spacelike separated. (This follows from a theorem in Fourier analysis.) But if the events are spacelike separated, then there is some inertial frame in which the second event is earlier than the first--or, put another way, there is some inertial frame in which the particle appears to travel backwards in time. But this "backwards in time" version of the particle *is* the antiparticle.

I don't know how generally accepted this view is; I have seen comments on both sides. But it does at least give a link between relativity and antiparticles, since it is relativity that tells us that "faster than light", in some frames, will appear as "backwards in time".

Are you quoting from that old Superman film where traveling faster than light makes time go backwards?

I don't think that's covered under Lorentz transformations.

 

[PeterDonis]

Are you quoting from that old Superman film where traveling faster than light makes time go backwards?

No, of course not.

I don't think that's covered under Lorentz transformations.

Sure it is (the "backwards in time" thing, not the Superman thing). A Lorentz transformation can change the time ordering of a pair of spacelike separated events. That's the key point.

 

[bhobba]

Its got to do with the the equation for energy allowing positive and negative solutions in relativity. Normally you ignore the negative solutions as unphysical but in QM you can't do that because electrons by emitting photons will jump to a lower energy state - in this case negative energy states. Obviously something is wrong. To get around it Dirac assumed all the negative energy states were filled and that holes in all these filled states were a new kind of particle called a positron.

However further development of the theory into QFT (Quantum Field Theory) showed the sea was unnecessary - they automatically occurred in that formalism without further ado by means of so called creation and annihilation operators that showed the theory was completely symmetrical between the two types of particles - it was not possible to have one without the other.

Thanks
Bill

 

[Lamarr]

But how does Relativity produce negative energies?

 

[bhobba]

But how does Relativity produce negative energies?

In relativity the formula for energy is E^2 = P^2*C^2 + M^2*C^4 which has both positive and negative solutions.

If you want to know why that formula is true you will need to consult a book on relativity.

At rock bottom that formula for energy is the cause of the issues of combining relativity and QM. It means the resulting equation is a square of the derivative in time (E = part deriv wave function wrt to time) so is not linear, but QM demands it to be linear in first derivative of time - if not you end up with absurdities like negative probabilities. Dirac came up with a smart and sneaky way of it obeying the equation and the equation still being linear. However it still implies negative energies.

Thanks
Bill