- #1
jfy4
- 649
- 3
Hi,
I was wondering what the expression would be for the energy operator in general relativistic quantum mechanics. So I attempted it for strictly diagonal metrics. Here it is:
Assume a stationary observer. Then the expression for a particle is
[tex]E=-g_{\alpha\beta}p^\alpha u_{obs}^{\beta}.[/tex]
However, the observer is stationary, and must obey the time-like condition of 4-velocity, hence
[tex]g_{\alpha\beta}u_{obs}^{\alpha}u_{obs}^{\beta}=g_{tt}\left(u_{obs}^{t}\right)^2=-1.[/tex]
Then
[tex]u_{obs}=\sqrt{-g^{tt}}[/tex]
which implies
[tex]E=-g_{\alpha\beta}p^\alpha u_{obs}^{\beta}\rightarrow \sqrt{-g_{tt}}p^t.[/tex]
Then substituting the expression for the energy operator in Minkowski space-time
[tex]\boxed{\hat{E}=i\hbar \sqrt{-g_{tt}}\frac{\partial}{\partial t}.}[/tex]
This is the final form for the energy operator in an arbitrary diagonal metric. Would anyone have an idea of how to construct the momentum operator for a general metric (it can be diagonal too, anything will make me happy). Corrections and comments entirely welcomed!
I was wondering what the expression would be for the energy operator in general relativistic quantum mechanics. So I attempted it for strictly diagonal metrics. Here it is:
Assume a stationary observer. Then the expression for a particle is
[tex]E=-g_{\alpha\beta}p^\alpha u_{obs}^{\beta}.[/tex]
However, the observer is stationary, and must obey the time-like condition of 4-velocity, hence
[tex]g_{\alpha\beta}u_{obs}^{\alpha}u_{obs}^{\beta}=g_{tt}\left(u_{obs}^{t}\right)^2=-1.[/tex]
Then
[tex]u_{obs}=\sqrt{-g^{tt}}[/tex]
which implies
[tex]E=-g_{\alpha\beta}p^\alpha u_{obs}^{\beta}\rightarrow \sqrt{-g_{tt}}p^t.[/tex]
Then substituting the expression for the energy operator in Minkowski space-time
[tex]\boxed{\hat{E}=i\hbar \sqrt{-g_{tt}}\frac{\partial}{\partial t}.}[/tex]
This is the final form for the energy operator in an arbitrary diagonal metric. Would anyone have an idea of how to construct the momentum operator for a general metric (it can be diagonal too, anything will make me happy). Corrections and comments entirely welcomed!