10.2. Temperature of horizons
There exists a natural definition of QFT in the original (D + 1)-dimensional space; in particular, we can define a vacuum state for the quantum field on the Z^{0} = 0 surface, which coincides with the t = 0 surface. By restricting the field modes (or the field configurations in the Schrodinger picture) to depend only on the coordinates in , we will obtain a quantum field theory in in the sense that these modes will satisfy the relevant field equation defined in . In general, this is a complicated problem and it is not easy to have a choice of modes in which will lead to a natural set of modes in . We can, however, take advantage of the arguments given in the last section - that all the interesting physics arises from the (Z^{0}, Z^{1}) plane and the other transverse dimensions are irrelevant near the horizon. In particular, solutions to the wave equation in which depends only on the coordinates Z^{0} and Z^{1} will satisfy the wave equation in and will depend only on (t, r). Such modes will define a natural s-wave QFT in . The positive frequency modes of the above kind (varying as exp(- i Z^{0}) with > 0.) will be a specific superposition of negative (varying as e^{i t}) and positive (varying as e^{-i t}) frequency modes in leading to a temperature T = (g / 2) in the 4-dimensional subspace on one side of the horizon. There are several ways of proving this result, all of which depend essentially on the property that under the transformation t t ± (i / g) the two coordinates Z^{0} and Z^{1} reverses sign.
Consider a positive frequency mode of the form F_{}(Z^{0}, Z^{1}) exp[- i Z^{0} + iPZ^{1}] with > 0. These set of modes can be used to expand the quantum field thereby defining the creation and annihilation operators A_{}, A_{}^{†}:
(166) |
The vacuum state defined by A_{}| vac > = 0 corresponds to a globally time symmetric state which will be interpreted as a no particle state by observers using Z^{0} as the time coordinate. Let us now consider the same mode which can be described in terms of the (t, r) coordinates. Being a scalar, this mode can be expressed in the 4-dimensional sector in the form F_{}(t, r) = F_{} [Z^{0}(t, r), Z^{1}(t, r)]. The Fourier transform of F_{}(t, r) with respect to t will be:
(167) |
Thus a positive frequency mode in the higher dimension can only be expressed as an integral over with ranging over both positive and negative values. However, using the fact that t t - (i / g) leads to Z^{0} - Z^{1}, it is easy to show that
(168) |
This allows us to write the inverse relation to (167) as
(169) |
The term with K_{}^{*} represents the contribution of negative frequency modes in the the 4-D spacetime to the pure positive frequency mode in the embedding spacetime. A field mode of the embedding spacetime containing creation and annihilation operators (A_{}, A_{}^{†}) can now be represented in terms of the creation and annihilation operators (a_{}, a_{}^{†}) appropriate to the (t, r) coordinates as
(170) |
where N_{} is a normalization constant. Identifying a_{} = N_{}(A_{} + e^{- / g} A_{}^{†}) and using the conditions [a_{}, a_{}^{†}] = 1, [A_{}, A_{}^{†}] = 1 etc., we get N_{} = [1 - exp(- 2 / g)]^{-1/2}. It follows that the number of a -particles in the vacuum defined by A_{}| vac > = 0 is given by
(171) |
This is a Planckian spectrum with temperature T = g / 2. The key role in the derivation is played by equation (168) which, in turn, arises from the analytic properties of the spacetime under Euclidean continuation.