content_image/402180.jpg content_image/402179.jpg content_image/402178.jpg content_image/402177.jpg content_image/402176.jpg content_image/402174.jpg content_image/402173.jpg content_image/402181.jpg content_image/402175.jpg
is solved by 

<img src='content_image/402180.jpg'>

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 

<img src='content_image/402179.jpg'>

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 

<img src='content_image/402178.jpg'>

Inserting this result into the expression (A.4) for the po- tential and using the propagator (A.2) yields 

<img src='content_image/402177.jpg'>

We now switch to spherical coordinates and integrate out the angular parts, obtaining for the integral over k : 

<img src='content_image/402176.jpg'>

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 

<img src='content_image/402174.jpg'>

Returning to the expression (A.7) for the field, we obtain: 

<img src='content_image/402173.jpg'>

[1] T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1921 , 966 (1921). 

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): 

<img src='content_image/402181.jpg'>

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. 

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 

<img src='content_image/402175.jpg'>

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. 

