9 We now consider the potential energy between the static source at x = 0 and another static source located at | x | = r , also at θ = 0. Identifying α = q $^{2}$/ (8 π $^{2}$L ), we reproduce Yndur´ ain’s result (apart from an overall sign): where j ( k ) is the Fourier representation of the source. Following Yndur´ ain [10], we consider a point source of strength q at rest with respect to physical time and in the second time dimension. The source is given by is solved by where ξ $^{µ}$( s ) is the position of the point source in 5-space- time, parameterized by the proper length s . If the source is fixed at the spatial origin and at θ = 0, we have ξ $^{ν}$( s ) = ( s, 0 , 0 , 0 , 0), and the Fourier transform of the source distribution is simply Inserting this result into the expression (A.4) for the po- tential and using the propagator (A.2) yields This shows that a compactified second time dimension introduces an undesirable imaginary component into the scalar potential generated by a static source. Because the same is true for the Coulomb potential generated by a static charge, the energy levels of, e. g., the hydrogen atom will be complex, implying a violation of the con- servation of probability. In other words, a compactified second time dimension leads to violations of unitarity in quantum mechanics, excluding extended space-time models of this kind. We note that Dvali et al. [11] con- sidered a slightly different model, where all fields except gravity are constrained to a specific point in the θ -time. We now switch to spherical coordinates and integrate out the angular parts, obtaining for the integral over k : Now consider the alternate case where we assume that the momentum (energy) component χ conjugate to θ is thermalized with temperature T$_{5}$ . The “thermal” aver- age of the physical-time Feynman propagator is obtained by replacing the integral over χ with a summation over the Matsubara frequencies χ$_{n}$ = 2 πinT$_{5}$ . The evaluation of the contour integral then yields the result The integral over k is easily evaluated by closing the integration contour in the positive complex half-plane, i. e. only the poles with Im k > 0 contribute. In order to perform the integral, we need to specify the values of the conjugate momentum χ , which contribute. The case considered by Yndur´ ain [10], which we examine first, is that the second time coordinate θ is compactified on a circle S 1 with radius L , i. e. that θ is cyclical with period 2 πL . The single-valuedness of the plane wave solution in θ then implies the discrete frequencies χ$_{n}$ = n/L , where n ∈ Z . For real χ$_{n}$ the contour integral (A.8) becomes Returning to the expression (A.7) for the field, we obtain: which is purely real. This shows that all corrections to the potential from the thermal dynamics in θ fall off ex- ponentially with the distance from the source. [2] O. Klein, Z. Phys. 37 , 895 (1926) [Surveys High Energ. [1] T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1921 , 966 (1921). φ ( x ) = q 8 π $^{2}$| x | L ∑ n ∈ Z e i ( | n | | x |− nθ ) /L (A.10) = q 8 π $^{2}$| x | L [ 1 + 2 ∞ ∑ n =1 e in | x | $^{/L}$cos nθ L ] . 2 π 2 | x | e i | χ$_{n}$ | | x | = 2 π 2 | x | e i | n | | x | /L . (A.9) φ ( x ) = q 8 π $^{2}$| x | T$_{5}$ ∑ n ∈ Z e ( nθ −| n | | x | ) T$_{5}$ (A.12) = q T$_{5}$ 8 π $^{2}$| x | [ 1 + 2 ∞ ∑ n =1 e − n | x | T$_{5}$ cosh nθT$_{5}$$^{]}$ = q T$_{5}$ 16 π $^{2}$| x | [ coth | x | − θ 2 T$_{5}$ + coth | x | + θ 2 T$_{5}$$^{]}$ , ∫ d $^{3}$k e i k · x k 2 − χ 2 − iϵ = 2 π i | x | ∫ ∞ −∞ dk k e ik | x | k 2 − χ 2 − iϵ . (A.8) φ ( x ) = − q (2 π ) 4 ∫ d $^{3}$k dχ e i ( k · x − χθ ) χ 2 − k 2 + iϵ (A.7) j ( k ) = 2 π q δ ( ω ) . (A.6) j ( x ) = q ∫ ∞ −∞ ds δ ( ξ $^{µ}$( s ) − x $^{µ}$) , (A.5) φ ( x ) = i (2 π ) 5 ∫ d $^{5}$k e − $^{ikx}$D ( k ) j ( k ) , (A.4) V ( r ) = − q φ ( x ) = − α r [ 1 + 2 ∞ ∑ n =1 e inr/L ] = − iα r tan r 2 L . (A.11)