05 - Saturday

In the algebraic approach to QFT one can construct Bogoliubov transforms between different expansions of a given field. The Bogoliubov transform in general is a process for diagonalising a Hamiltonian, but it has found significant use in the case of semiclassical quantum gravity calculations, such as Hawking radiation or the Unruh effect.

The traditional explanation is that one can write a field in terms of positive frequency solutions to the underlying wave equation. For example, given a massless Hermitian scalar field on a curved background described by the classical metric \(g^{\alpha\beta}\) the wave equation takes the form:

\[\nabla_{\alpha}\nabla_{\beta}\phi g^{\alpha\beta}=0\]

And we may decompose the field as:

\[\phi=\sum_i f_i\hat{a}_i+f_i^*\hat{a}_i^{\dagger}\]

Where the \(f_i\) are positive frequency solutions to the above wave equation, and the \(\hat{a}_i\) \(\hat{a}_i^{\dagger}\)may then be interpreted as annihilation and creation operators respectively. We generally take the \(f_i\) to be plane wave solutions with their frequencies defined positive with respect to a suitable timelike killing vector, e.g. If \(\partial_t\) is such a killing vector then:

\[\partial_tf=-i\omega f\]

With \(\omega>0\) defining a positive frequency. In a curved spacetime there are essentially unlimited coordinate systems, different observers can setup their own local coordinates which can be wildly different from one another. Thus there are also near unlimited choices of timelike killing vectors, and so different ways to decompose the field. Consider a second decomposition of the field:

\[\phi=\sum_i p_i\hat{b}_i+p_i^*\hat{b}_i^{\dagger}\]

Both decompositions are complete and define the entire field completely, and so we can write the wave solutions of one in terms of the other:

\[f_i=\sum_j\alpha_{ij} p_j+\beta_{ij}p_j^*\]\[p_i=\sum_j\alpha_{ij}^* f_j-\beta_{ij}^*f_j^*\]

These relations define a Bogoliubov transform, and we can write similar relations for the operators:

\[a_i=\sum_j\alpha_{ij} b_j+\beta_{ij}^*b_j^{\dagger}\]\[b_i=\sum_j\alpha_{ij}^* a_j-\beta_{ij}^*a_j^{\dagger}\]

The coefficients \(\alpha_{ij}\) and \(\beta_{ij}\) are called the Bogoliubov coefficients and calculating them is generally the hardest part of performing physics with this formalism.