Geometry of the Space of Solutions

The representation (3) puts us at an important mathematical juncture: We shall forego the usual picture of viewing this solution as an element of Hilbert space ($C^2\otimes \cal H$ ) with the Klein-Gordon inner product,

 $$\begin{eqnarray}\lefteqn{ {i\over 2}\int^0_{\infty} \psi^{\ast}_{II}{\stackre... ...langle\psi_\omega,\psi_{\omega '}\rangle~ d\omega d\omega'\quad . \end{eqnarray}$$

Instead, we shall adopt a qualitatively new and superior viewpoint. Each Klein-Gordon solution is a spinor field over the Rindler frequency domain. This is based on the vector bundle $C^2\times R$ . Here C² is the complex vector space of two-spinors, which is the fiber over the one-dimensional base manifold $R=~\{ \omega:~-\infty < \omega < \infty \}$ , the real line of Rindler frequencies in the mode integral, Eq.(3).

We know that one can add vectors in the same vector (fiber) space. However, one may not, in general, add vectors belonging to different vector spaces at different $\omega$ 's. The exception is when vectors in different vector spaces are parallel. In that case one may add these vectors. The superposition of modes, Eq.(3), demands that one do precisely that in order to obtain the two respective total amplitudes of Eq.(3). In brief, we are about to show that that the linear superposition principle determines a unique law of parallel transport.

The mode representation of Eq.(3) determines two parallel basis spinor fields over R ,

$$\lbrace \left( \begin{array}{c} 1 \\ 0 \end{array}\right) \... ...right) \phi_\omega :~-\infty < \omega < \infty \rbrace \quad ,$$ 6

one corresponding to ``spin up'' ($\psi$ has zero support in Rindler II ), the other to ``spin down'' ($\psi$ has zero support in Rindler I ). This parallelism is dictated by the superposition principle, Eq.(3): The total amplitude at a point $(\tau,\xi)$ in Rindler I (resp. II ) is obtained by adding all the contributions from Rindler I (resp. II ) only.

One can see from Eq.(6) that, relative to this basis, the fiber metric over each $\omega$ is given by

$$(a^*_\omega b_\omega )\left[ \begin{array}{cc} 1&0 \\ 0&-1 ... ...mega -b_\omega b^*_\omega;~~~-\infty < \omega < \infty \quad .$$ 7

It is evident that the law of parallel transport defined by Eq.(7) is compatible with this (Klein-Gordon induced) fiber metric. This is because the two parallel vector fields, which are represented by

$$\left( \begin{array}{c} 1 \\ 0 \end{array}\right) ~~\hbox{a... ...ray}{c} 0 \\ 1 \end{array}\right); ~-\infty < \omega < \infty$$ 8

relative to the basis (7), are orthonormal with respect to the the fiber metric Eq.(8) at each point $\omega$ of the base space R . It is not difficult to verify that these two spinor fields are (Klein-Gordon) orthonormal in each fiber over R . The spinor field

$$\{ \left( \begin{array}{c} a_\omega \\ b_\omega^* \end{array}\right) : ~-\infty < \omega < \infty \} \quad$$ 9

is a section of the fiber bundle $C^2\times R$ and it represents a linear combination of the two parallel vector fields. It is clear that there is a one-to-one correspondence between $\Gamma(C^2\times R)$ , the $\infty$ -dimensional space of sections of this spinor bundle, and the space of solutions to the Klein-Gordon equation.