- #1
knobelc
- 14
- 0
Why are my formulas not displayed correctly?
I'm not sure how to help you here, and nobody else seems to really do so either. First I noted thatknobelc said:So let me repete my question: How can I show in case of the Klein-Gordon field, that both, real part and imaginary part of [itex]\Phi(\vec{k},t)[/itex] are independent Gaussian distributed? I don't expect it to be very difficult, but I don't see yet the formal calculation.
refers to a real KG (neutral), and this is also the case in your notes. Complex KG (charged) would have hadknobelc said:[tex]\pi(x) = \frac{\partial \mathcal{L}}{\partial \dot{\Phi}} = \dot{\Phi}.[/tex]
(or whatever you use to note hermitian conjugate, but the operator will not be equal to [itex]\dot{\Phi}[/itex] unless you have a real field)[tex]\pi(x) = \frac{\partial \mathcal{L}}{\partial \dot{\Phi}} = \dot{\Phi^{\dagger}}[/tex]
There is no "[itex]\vec{k}[/itex] in the ground state", is there ? Your HO-like construction creates plane waves with [itex]\vec{k}[/itex] on the vacuum [itex]\left|0\right\rangle[/itex].knobelc said:From what I have written here, how can I prove that for a given [itex]\vec{k}[/itex] in the ground state [itex]\left|0\right\rangle[/itex] both the real and imaginary part of [itex]\Phi(\vec{k},t) = \int \Phi(\vec{x},t) e^{-i\vec{k}\vec{x}} dx^3[/itex] are independent Gaussian distributed with zero mean, i.e. by means of repeted measurements I would find the values of the real and imaginary part to be independent Gaussian random variables?
The vacuum state of the Klein-Gordon field is the lowest energy state of the quantum field, where there are no particles present and no external forces acting on the field. It is considered the "ground state" of the system.
The vacuum state of the Klein-Gordon field is a quantum state, while the classical vacuum is a state of empty space with no quantum fluctuations. In the quantum vacuum state, there are still fluctuations and virtual particles present due to the uncertainty principle.
The vacuum state is crucial in quantum field theory as it serves as the starting point for calculating the behavior of particles and their interactions in the field. It also plays a role in the concept of spontaneous symmetry breaking and the Higgs mechanism.
The vacuum state of the Klein-Gordon field can be affected by the presence of matter through interactions between particles and the field. This can lead to changes in the energy and properties of the vacuum state.
No, the vacuum state of the Klein-Gordon field cannot be directly observed or measured. However, its effects can be observed through various physical phenomena and experimental results in quantum field theory.