A Explicit construction of Galilean-invariant space

Adgorn
Messages
133
Reaction score
19
I'm trying to explicitly find a projective unitary scalar representation of the Galilean group. I'll denote a generic element of the group by ##(a, {\bf b},R, {\bf v})##, corresponding respectively to time translation, space translation, rotation and boosts. In a representation with central charge ##M##, the commutation relations of the boost and space translation genrations are:

$$\tag{1} [K_i,P_j]=-iM\delta_{ij}.$$

The candidate I'm considering is the most obvious one. Take the space of square integrable functions on ##\mathbb R^3## (momentum space) and give it the action:

$$\tag {2}(a, {\bf b},R, {\bf v})\phi({\bf p})=e^{i[-a(\frac {p^2} {2M}+E_0)+{\bf b}\cdot{\bf p}]}\phi(R{\bf p}+M{\bf v}).$$

Another candidate which I saw in a paper was the inverse transformation:

$$\tag {3}(a, {\bf b},R, {\bf v})\phi({\bf p})=e^{i(aE-{\bf b}\cdot{\bf p})}\phi(R^{-1}({\bf p}-M{\bf v})).$$

At any rate, I'm having a hard time reconciling any option for the phase with ##(1)##. The Baker-Campbell-Hausdorff formula along with ##(1)## implies that for a composition of a translation and a boost:

$$e^{-i{\bf v} \cdot {\bf K}}e^{-i{\bf b} \cdot {\bf P}}=e^{iM\frac {{\bf v}\cdot {\bf b}} 2}e^{-i({\bf v} \cdot {\bf K}+{\bf b} \cdot {\bf P})}.$$

But if I actually compose these operations in either ##(2)##or ##(3)##, I don't get the ##\frac 1 2## factor. So are these really projective representations? If not, what choice of phase factor is appropriate?

The papers I'm refering to are:

1.https://doi.org/10.1007/BF01646020

2.https://doi.org/10.1007/BF01645427
 
Last edited:
Physics news on Phys.org
Adgorn said:
[...]

The papers I'm refering to are:

1.https://doi.org/10.1007/BF01646020

2.https://doi.org/10.1007/BF01645427

These are classical papers. I believe CMP was freely available on the NumDam server.

1. Lévy-Leblond, J.-M. (1967). Nonrelativistic particles and wave equations. Communications in Mathematical Physics, 6(4), 286–311. doi:10.1007/bf01646020

2. Lévy-Leblond, J.-M. (1967). Galilean quantum field theories and a ghostless Lee model. Communications in Mathematical Physics, 4(3), 157–176. doi:10.1007/bf01645427
 
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