I Finding ##\partial^\mu\phi## for a squeezed state in QFT

Sciencemaster
Messages
129
Reaction score
20
TL;DR Summary
I'm trying to apply an operator to a massless and minimally coupled squeezed state, I'm having trouble calculating ##\partial^\mu\phi## but due to a sum over k and the ladder operators.
I'm trying to apply an operator to a massless and minimally coupled squeezed state. I have defined my state as $$\phi=\sum_k\left(a_kf_k+a^\dagger_kf^*_k\right)$$, where the ak operators are ladder operators and fk is the mode function $$f_k=\frac{1}{\sqrt{2L^3\omega}}e^{ik_\mu x^\mu}$$ (assuming periodic boundary condition in a three-dimensional box of side L where k is the wave number).
However, I'm having trouble calculating ##\partial^\mu\phi## due to the sum over k and the ladder operators. I would very much appreciate it if someone could help me through the math of this step!
 
Physics news on Phys.org
Well, ##\partial_\mu## is a linear operator, so you can apply it term by term in the sum. As for the ladder operators, if they're independent of ##x## you can just treat them like constants during the partial differentiation.

Btw, you'll need to use a different dummy summation index in the exponent so as not to conflict with the free index ##\mu## on ##\partial_\mu##. E.g., change ##k_\mu x^\mu## to ##k_\alpha x^\alpha##.
 
  • Like
Likes topsquark and vanhees71
One should also note that this doesn't describe a state but a field operator in terms of free-field energy eigenmodes or a neutral scalar field. The ##\hat{a}_k## are annihilation and ##\hat{a}_k^{\dagger}## in Fock space.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
Back
Top