A Is there an alternative form of the zero point energy?

[SeM]

Hi, is there an alternative form of the zero point energy for free electrons, where there is no space interval L to be quantized in? The zero point energy for electrons in an atom can be simplified to a variant where Z^2 is present in the nominator, however, these are not free electrons.

Can a ground state energy for a free electron be proposed at all without L?

 

[BvU]

Is taking the limit(s) of the non-alternative form such a bad idea ?

 

[Iliody]

I don't think so... You can try using a different basis for the wavefunctions, but the zero point energy will end being equivalent to the standard formula... You can renormalize it adding a constant that has the same kind of divergence, and it won't mess up anything more on your theory, so... It's a good thing to do.

It's ok my answer?

 

[SeM]

I don't think so... You can try using a different basis for the wavefunctions, but the zero point energy will end being equivalent to the standard formula... You can renormalize it adding a constant that has the same kind of divergence, and it won't mess up anything more on your theory, so... It's a good thing to do.

It's ok my answer?

Hi, renormalizing is a necessity, but the problem is the interval L. For an electron traveling freely, that interval is pretty much infinity? Even worse, if that electron is oscillating in an area which is unknown for its dimensions, then how can one treat L? As a parameter?

 

[SeM]

Is taking the limit(s) of the non-alternative form such a bad idea ?

Not sure what you mean. Do you mean the limits of E_0 as L goes for from 0 to infinity?

 

[Iliody]

...the problem is the interval L. For an electron traveling freely, that interval is pretty much infinity? ... oscillating in an area which is unknown for its dimensions, then how can one treat L?

You regularize first, restricting the base of first-quantized wavefunctions with which you make the base of second-quantized wavefunctions, being them normalized, then you diagonalize your Hamiltonian (plus a renormalizing constant that depends on the regularization), and then you look for the lower eigenval. That eigenval is the zero energy, and (my opinion) you can set it to zero by renormalization (when there are no symmetries in tour theory forbiding it).
Good look!

Pd: I am not sure about how common is that this is opinion is shared in the physics comunity, so it's maybe wrong that I put this answer here.

 

[SeM]

You regularize first, restricting the base of first-quantized wavefunctions with which you make the base of second-quantized wavefunctions, being them normalized, then you diagonalize your Hamiltonian (plus a renormalizing constant that depends on the regularization), and then you look for the lower eigenval. That eigenval is the zero energy, and (my opinion) you can set it to zero by renormalization (when there are no symmetries in tour theory forbiding it).
Good look!

Pd: I am not sure about how common is that this is opinion is shared in the physics comunity, so it's maybe wrong that I put this answer here.

Thanks , I think setting the eigenenergy to zero gives a somewhat unrealistic initial condition. However, I would like to thank you for your comments in any case!

 

[Iliody]

If de don't include gravity in our discussion, it's ok to ser it to any value, if you include gravity, you need to set it to the cosmological constant. I never did perturbative QG calculations (they can be done to one loop if we don't put matter loop corrections in gravity-gravity scattering), so I don't know how many quantities depende on the cosmo-constant.

 

[SeM]

If de don't include gravity in our discussion, it's ok to ser it to any value, if you include gravity, you need to set it to the cosmological constant. I never did perturbative QG calculations (they can be done to one loop if we don't put matter loop corrections in gravity-gravity scattering), so I don't know how many quantities depende on the cosmo-constant.

This is a particle in a Quantum Hall. cosmological constants- would they affect it? Which cosmological constant do you mean?

 

[Iliody]

This is a particle in a Quantum Hall.

Sorry, I thinked that you were talking about other kind of system. I don't remember anything about Quantum Hall... And I remembered now that my affirmations were limited to QFT correlation functions without boundaries (infinite volume limit). There are observables like presure that depend on the zero point energy in a nontrivial way (even in that case, the thing that I said about zero energy renormalization-regularization).

 

[SeM]

Sorry, I thinked that you were talking about other kind of system. I don't remember anything about Quantum Hall... And I remembered now that my affirmations were limited to QFT correlation functions without boundaries (infinite volume limit). There are observables like presure that depend on the zero point energy in a nontrivial way (even in that case, the thing that I said about zero energy renormalization-regularization).

Hi, QFT may have indeed some important common themes with the Quantum Hall. However, is there a "starting" point, or some initial condition one uses in QFT for wavefunction study?

Cheers

 

[Iliody]

Is there a "starting" point, or some initial condition one uses in QFT for wavefunction study?

Mmm... It's usually taken a vacuum state at the infinite past & at the infinite future, and you change that state inserting operators add particle exitations (first-quantized non-interacting wavefunctions) of the different fields. That has a lot of troubles, in some sense.

 

[SeM]

Mmm... It's usually taken a vacuum state at the infinite past & at the infinite future, and you change that state inserting operators add particle exitations (first-quantized non-interacting wavefunctions) of the different fields. That has a lot of troubles, in some sense.

how does this vacuum state look like in terms of function/value/expression?

 

[Iliody]

how does this vacuum state look like in terms of function/value/expression?

It's the state annihilated by all field anihillation operators, that are Fourier modes of the field operators.

 

[SeM]

Thanks! How do the annihilation operators prevent an infinitesimal value of the averages? Do they converge to some minimal value?