Is the Dirac free equation actually free

[ftr]

I mean the equation shows the particle could have any momentum, how did that came about. If it is truly free it should have only an energy of mc^2, shouldn't it.

 

[A. Neumaier]

I mean the equation shows the particle could have any momentum, how did that came about. If it is truly free it should have only an energy of mc^2, shouldn't it.

No; this is only the rest energy. You also need to count the (relativistic) kinetic energy! A free nonrelativistic particle can also move with any momentum and then has an energy of $p^2/2m$.

 

[Nugatory]

"Free" means not subject to any forces, so the momentum is not changing. It does not follow that the momentum and hence the kinetic energywill be zero.

 

[ftr]

No; this is only the rest energy. You also need to count the (relativistic) kinetic energy! A free nonrelativistic particle can also move with any momentum and then has an energy of $p^2/2m$.

In both cases the momentum or speed should be against/relative to something else, but that cannot happen for "single" particle, correct or not.

 

[A. Neumaier]

In both cases the momentum or speed should be against/relative to something else, but that cannot happen for "single" particle, correct or not.

It is against a fixed Lorentz frame in space-time. The frame is not a material entity but a way to describe space-time independent of its contents. It is needed even to describe the vacuum state - without it there is no meaning to the standard requirement that the vacuum state should be Lorentz invariant.

 

[ftr]

"Free" means not subject to any forces, so the momentum is not changing. It does not follow that the momentum and hence the kinetic energywill be zero.

At fundamental quantum level particles are subject to the "four" known forces, and these are relentless. Any particle subjected to them they will be under the influence no matter how small, but if they become so small then you are back to only mc^2. It seems that Newton's first law is ambiguous for quantum particles.

 

[A. Neumaier]

At fundamental quantum level particles are subject to the "four" known forces

A single electron alone in the universe will be free. Note that free particles are (like almost everything in physics) an abstraction form the all-too-complex reality. One simplifies and abstracts in order to be able to understand and classify.

 

[ftr]

A single electron alone in the universe will be free. Note that free particles are (like almost everything in physics) an abstraction form the all-too-complex reality. One simplifies and abstracts in order to be able to understand and classify.

Of course I understand that, I have a Master's in EE. But from following up on materials that spawned from recent threads by demystifier and Morgan it seems to me that simplification has led to confusion including the Klein paradox , a no solution relativistic particle in a box(or simple potentials) and ... so on with all of that carried to QFT as discussed in the mentioned threads. It just seems to me the interpretation or the "simplification" of original Dirac equation is the culprit.

 

[ftr]

To add: it seems to me to have that momentum realistically another identical particle has to exist and in that case the interaction might change the form of the dispersion relation. But I guess that will put us in QFT renormalization ..

 

[Nugatory]

Any particle subjected to them they will be under the influence no matter how small, but if they become so small then you are back to only mc^2.

That is simply not true, because both the momentum $p$ and the kinetic energy $p^2/2m$ are frame-dependent - you can make them take on any value you please simply by choosing coordinates that make them come out the way you want no matter what forces are present or not.

It seems that Newton's first law is ambiguous for quantum particles.

It would be better to say that it doesn't apply - it's a classical description that assumes that both position and momentum can be known simultaneously.

 

[Demystifier]

I mean the equation shows the particle could have any momentum, how did that came about. If it is truly free it should have only an energy of mc^2, shouldn't it.

What does the first Newton law say?

 

[ftr]

momentum ppp and the kinetic energy p2/2mp2/2mp^2/2m are frame-dependent

That was my concern in the OP. That is why I said under realistic condition of interacting with another particle/s can give more realistic results.

 

[ftr]

What does the first Newton law say?

In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

In QM settings it is usually particles interacting via potential or some scattering that are interesting and revealing/useful. Add to that particle trajectory issues and the UP.

 

[ftr]

That was my concern in the OP. That is why I said under realistic condition of interacting with another particle/s can give more realistic results.

Actually my concerns have been taken care of long time ago. although I knew about the equations but never thought about them in that context. Here is a reference as a clarification.

Bethe-Salpeter Equation -- The Origins

https://arxiv.org/ftp/arxiv/papers/0811/0811.1050.pdf

https://en.wikipedia.org/wiki/Bethe–Salpeter_equation