Friday 5 June 2020

Variance of an interacting quantum field in its vacuum state


A non-interacting quantum field $\hat{\phi}(x)$ can be decomposed into $a_{\textbf{k}}$ and $a_{\textbf{k}}^\dagger$. This enables us to calculate the variance of a free field. For example, the variance of the free real scalar field, in the vacuum $|0\rangle$ of the theory, is computed to be (without a momentum cut-off) $${\rm Var}(\phi)_0=\langle0|\phi^2|0\rangle-\big(\langle0|\phi|0\rangle\big)^2\\=\int\frac{d^3k}{(2\pi)^3}\frac{1}{\sqrt{\textbf{k}^2+m^2}}\rightarrow \infty.$$ Now, consider an interacting quantum field theory described by the Hamiltonian $H$ and $|\Omega\rangle$ is the vacuum state of the interaction theory i.e. $$H|\Omega\rangle=0|\Omega\rangle\hspace{0.5cm}\big(\text{also,}~P^\mu|\Omega\rangle=0|\Omega\rangle\big).$$ Decomposing the field into creation and annihilation operators is no longer possible. So how does one compute the variance $${\rm Var}(\phi)_\Omega=\langle\Omega|\phi^2|\Omega\rangle-\big(\langle\Omega|\phi|\Omega\rangle\big)^2$$ in $\lambda\phi^4$ theory?




No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...