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

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1. Dec 9, 2017

### 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?

2. Dec 9, 2017

### BvU

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

3. Dec 10, 2017

### 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 wont mess up anything more on your theory, so... It's a good thing to do.

4. Dec 11, 2017

### SeM

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?

5. Dec 11, 2017

### SeM

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

6. Dec 11, 2017

### Iliody

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.

7. Dec 11, 2017

### SeM

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!

8. Dec 11, 2017

### 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.

9. Dec 11, 2017

### SeM

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

10. Dec 11, 2017

### Iliody

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).

11. Dec 12, 2017

### SeM

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

12. Dec 12, 2017

### Iliody

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.

13. Dec 12, 2017

### SeM

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

14. Dec 12, 2017

### Iliody

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

15. Dec 13, 2017

### SeM

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