How Is Vacuum Energy Calculated for a Scalar Field?

[tunafish]

Hi guys!
I searchead a lot for this but i couldn't find in anywhere: what calculation is made to esimate the vacuum energy of a scalar field??

I red that it starts considering the Hamiltonian in the form
$$\hat H=\frac{1}{2}\int d^{n-1}k[\hat n_k+\frac{1}{2}\delta^{n-1}(0)]\omega$$
and then letting it act on a ground state, such that $\hat n_k |0\rangle=0|0\rangle$
and so
$$\hat H|0\rangle=\frac{1}{4}\int d^{n-1}k\delta^{n-1}(0)\omega=\frac{1}{4}\sum_k\omega$$
and putting a $\omega_{max}$ should get the result.

Now i as 2 things:
1) what precisely are the $k,\omega$ termu in the integration? I mean, they doesen't seem to be component of the four vector $k^\mu=(\omega,\vec{k})$!

2) explicitly, how does the result is obtained?


thanks!

 

[eljose79]

Hello there!

The calculation for estimating the vacuum energy of a scalar field involves using the Hamiltonian operator and the ground state of the system. The Hamiltonian operator is used to calculate the energy of the system and the ground state is the lowest energy state of the system. The process is as follows:

1. Start with the Hamiltonian operator, which is given by:
\hat H=\frac{1}{2}\int d^{n-1}k[\hat n_k+\frac{1}{2}\delta^{n-1}(0)]\omega

2. Apply the operator to the ground state, which is denoted by |0\rangle. This will give us the energy of the ground state, which is the vacuum energy.

3. To calculate the energy, we use the commutation relations for the creation and annihilation operators, which are given by: [\hat a_k,\hat a^\dagger_k]=\delta^{n-1}(0). This gives us the result:

\hat H|0\rangle=\frac{1}{4}\int d^{n-1}k\delta^{n-1}(0)\omega=\frac{1}{4}\sum_k\omega

4. Finally, we can put a \omega_{max} to get the final result for the vacuum energy.

To answer your questions:

1. The k and \omega terms in the integration represent the momentum and energy of the scalar field, respectively. They are not components of the four vector k^\mu=(\omega,\vec{k}) because we are only considering the spatial components in this calculation.

2. The result is obtained by applying the Hamiltonian operator to the ground state and using the commutation relations for the creation and annihilation operators. This gives us the expression for the vacuum energy, which can be further simplified by putting a \omega_{max}.

I hope this helps clarify the calculation for estimating the vacuum energy of a scalar field. Let me know if you have any other questions!