- #1
LAHLH
- 409
- 1
Hi,
So in a general curved spacetime we have no preferred choice of modes and the Bogolubov transformations allow us to convert between the fields expanded in the various complete sets of modes.
If we have one set of modes [tex]f_{i} [/tex] and another [tex] g_i [/tex] both normalized like normalized as [tex] (f_i,f_j)=\delta_{ij} [/tex] and [tex] (f^{*}_i,f^{*}_j)=-\delta_{ij} [/tex], then we can tranform between the modes as:
[tex] f_i=\sum_j \left(\alpha^{*}_{ji}g_j-\beta_{ji}g^{*}_j\right)[/tex], where the Bogolubov coeffs are given as [tex] \alpha_{ij}=(g_i,f_j) [/tex] and [tex] \beta_{ij}=-(g_i,f^{*}_j) [/tex]
Thus it follows that
[tex] \delta_{ij}=(f_i,f_j)=\left(\sum_m \left(\alpha^{*}_{mi}g_m-\beta_{mi}g^{*}_m\right),\sum_n \left(\alpha^{*}_{nj}g_n-\beta_{nj}g^{*}_n\right) \right) [/tex]
expanding this and using the normalization of the g modes [tex] (g_i,g_j)=\delta_{ij} [/tex] and [tex] (g^{*}_i,g^{*}_j)=-\delta_{ij} [/tex], others zero:
[tex] \delta_{ij}= \sum_m \left( \alpha^{*}_{mi}\alpha^{*}_{mj}-\beta_{mi}\beta_{mj} \right)[/tex]
Giving the normalization of the coefficients. However this answer differs to the one quoted in say Birrell and Davies:
[tex] \delta_{ij}= \sum_m \left( \alpha_{im}\alpha^{*}_{jm}-\beta_{im}\beta^{*}_{jk} \right)[/tex]
Just wondering if anyone can spot how to get this normalization?
thanks alot
So in a general curved spacetime we have no preferred choice of modes and the Bogolubov transformations allow us to convert between the fields expanded in the various complete sets of modes.
If we have one set of modes [tex]f_{i} [/tex] and another [tex] g_i [/tex] both normalized like normalized as [tex] (f_i,f_j)=\delta_{ij} [/tex] and [tex] (f^{*}_i,f^{*}_j)=-\delta_{ij} [/tex], then we can tranform between the modes as:
[tex] f_i=\sum_j \left(\alpha^{*}_{ji}g_j-\beta_{ji}g^{*}_j\right)[/tex], where the Bogolubov coeffs are given as [tex] \alpha_{ij}=(g_i,f_j) [/tex] and [tex] \beta_{ij}=-(g_i,f^{*}_j) [/tex]
Thus it follows that
[tex] \delta_{ij}=(f_i,f_j)=\left(\sum_m \left(\alpha^{*}_{mi}g_m-\beta_{mi}g^{*}_m\right),\sum_n \left(\alpha^{*}_{nj}g_n-\beta_{nj}g^{*}_n\right) \right) [/tex]
expanding this and using the normalization of the g modes [tex] (g_i,g_j)=\delta_{ij} [/tex] and [tex] (g^{*}_i,g^{*}_j)=-\delta_{ij} [/tex], others zero:
[tex] \delta_{ij}= \sum_m \left( \alpha^{*}_{mi}\alpha^{*}_{mj}-\beta_{mi}\beta_{mj} \right)[/tex]
Giving the normalization of the coefficients. However this answer differs to the one quoted in say Birrell and Davies:
[tex] \delta_{ij}= \sum_m \left( \alpha_{im}\alpha^{*}_{jm}-\beta_{im}\beta^{*}_{jk} \right)[/tex]
Just wondering if anyone can spot how to get this normalization?
thanks alot