NeroKid said:I have been reading Franz Gross book, he define the time translation operator as U0 UI, with U0 stands for free particle time translation operator , so under this definition the field O transform as O = UIO0 UI-1 and under the coulomb gauge the the free field V =0, so V in present of interaction should equals 0, but in the books he present it as coulomb self energy so I have been wondering whether we can do the same for other components of field
Im pretty sure that I was using Coulomb gauge not Weyl Gauge, what I meant was if other ##A_\mu## you have the transform in the interaction is UI-1 A##\mu## UI(the field in between is free field) but not for ##A_0## in the presence of photon and electron the author suddenly add the coulomb self energy which, in my understanding of EM field, should be $$A_0 = \int d^3 x \frac{\bar{\psi} \gamma^0 \psi}{r}$$ and the self Halmitonian is $$ H_s = \int \int d^3 x_1 d^3 x_2 \frac{[ \bar{\psi} \gamma^0 \psi]_1 [ \bar{\psi} \gamma^0 \psi]_2}{|r_1 - r_2 |}$$ I don't understand why , I thought the field suppose to transform from free field like UI -1 O0 UI how can a 0 field transfom in this way to a non zero field they are both supposed to describe the EM field and in turn the field quanta and I don't want to make the interaction vanish, what I want to ask is that whether we can generalize both ##\psi## and ##\vec{A}## to the global operator for both electron and photonfzero said:I am not certain, but I think you are confusing two different uses for the letter ##V##. In one case, ##V## is the temporal component of the gauge field ##A_0= V##. There is a partial gauge fixing in which one takes ##A_0=0##, but this is called the temporal or Weyl gauge, and is different from the Coulomb gauge.
The other use of ##V## is as the interaction potential in a general Hamiltonian, ##H = H_0 + V##. Then ##H_0## appears in ##U_0##, while ##V## appears in ##U_I##, if I have your definitions correct. In QED, ##V## is not the same as ##A_0##, but involves
$$ V \sim e \int d^4 x A_\mu \bar{\psi} \gamma^\mu \psi . $$
There is no gauge in which we can make every component of ##A_\mu## vanish, so it is impossible for the interaction potential to vanish by a gauge choice.
If I am misunderstanding your question, please carefully elaborate.
A quantized non free EM (electromagnetic) field is a theoretical concept in physics that describes the behavior of electromagnetic radiation at a very small scale. It is based on the idea that electromagnetic fields can only exist in discrete packets, or "quanta," rather than being continuous.
A classical EM field, as described by classical physics, is continuous and can have any energy level. In contrast, a quantized non free EM field only exists in discrete energy levels, and the energy of each level is determined by the frequency of the electromagnetic radiation.
The concept of a quantized non free EM field is important in understanding the behavior of electromagnetic radiation at the quantum level. It helps to explain phenomena such as the photoelectric effect and the emission and absorption of light by atoms.
No, a quantized non free EM field cannot be directly observed in experiments. However, its effects can be observed through experiments such as the double-slit experiment and the measurement of the energy levels of atoms.
The quantized non free EM field is a key concept in the field of quantum mechanics, which describes the behavior of particles at the atomic and subatomic level. It is a fundamental aspect of the quantum theory of light and plays a crucial role in understanding the behavior of matter and energy at the quantum level.