View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Symmetry Breaking in the Static Co ordinate System of de Sitter Spacetime 1 2 Andras Kaiser and Alan Cho dos Center for Theoretical Physics, Sloane Physics Lab oratory Yale University, New Haven, Connecticut 06520-8120 processed by the SLAC/DESY Libraries on 29 Jan 1995. 〉 PostScript HEP-TH-9501130 1 [email protected] 2 cho [email protected] Abstract: We study symmetry breaking in the static co ordinate-system of de Sitter space. This is done with the help of the functional-Schrodinger approach used in previous calculations by Ratra [1]. We consider a massless, minimally coupled scalar eld as the parameter of a continuous symmetry (the angular comp onent of an O(2) symmetry). Then we study the correlation function of the massless scalar eld, to derive the correlation function of the original eld, which nally shows the restoration of the continuous symmetry. 1.Intro duction Studying quantum eld theory in de Sitter space has a long history [2], and the interest in the in ationary mo del of the universe brought some attention to the study of symmetry breaking in this spacetime [3]. As is known from previous results [1,4], the question of whether an internal symmetry is sp ontaneously broken in a eld theory quantized on a de Sitter background may dep end on the co ordinate system that is chosen. In this pap er, wecho ose to do the computation for a minimally coupled scalar eld in the so-called static co ordinate system, which is distinguished by the fact that the interpretation of the results is particularly transparent in this case. The co ordinate-system we wish to use has the sp ecial prop erty that its spatial sections are geo desic hyp ersurfaces with identical metric, and the time co ordinate for each slice is given by the prop er time of the observer whose spatial co ordinate is the origin. In this sense this system is the one that describ es the spacetime as a geo desic observer p erceives it. It inherently excludes anything b eyond the event horizon of the observer, and manifests time-translational invariance, b ecause time-translation is generated by an element of the de Sitter group (which is not the case for the systems used previously), namely the one that translates the observer along its world-line ( , rotation in the 0-4 plane according to the co ordinate de nitions of 04 i! t section 2). Only if the eld is quantized in this system, will the eld mo des have e time-dep endence, with the t co ordinate time b eing the prop er time of the observer at the origin. The imp ortant consequence of this prop erty is that this quantization gives us the Fock-space as p erceived by the observer (i.e. the vacuum state of this quantization will b e the state in which the observer do esn't detect particles, etc.). So if wewantto de ne symmetry breaking according to the eld measurements of our geo desic observer in its vacuum state, then wemust calculate the exp ectation value of the eld on the vacuum state of this quantization. 2.The static system De Sitter spacetime can b e emb edded in a 5D Minkowski-space (we use signature < +; ; ; ; >), where it is a 4D hyp ersurface determined by 2 2 2 2 2 2 0 1 2 3 4 x x x x x = (2:1) 2 where is the de Sitter radius, which also determines the spacetime curvature, R =12 [5]. For the static system we will use a time and p olar spatial co ordinates, t; r;#;'. Cho osing the worldline of our geo desic observer to b e x = sinh(t= ) ; 0 ; 0 ; 0 ; cosh (t= ) (2:2) in terms of the co ordinates of the emb edding space determines the geo desic 3-surfaces orthogonal to 0 4 the worldline at every t, which are given by the x = tanh(t= ) x condition within the de Sitter manifold. These spatial hyp ersurfaces meet on the 2-surface 2 2 2 2 0 1 2 3 4 x =0; x +x +x = ; x =0 (2:3) This 2-surface thus forms a co ordinate singularity in the static system, whichmust b e exluded from the range of the co ordinates. The physical reason for the existence of this singularity is that null-geo desics passing through it form the event horizon of our observer. We will use 2 2 2 2 1 2 3 r = x + x + x (2:4) radial co ordinate on the geo desic 3-surface of a time-instant, even though this is not the prop er distance from the origin on the spatial sections. The prop er distance from the origin on these hyp ersurfaces is given 2 by s = arcsin(r= ). We also intro duce p olar co ordinates on the r = const: t = const: 2-spheres in the usual way. The emb edding relations of the static co ordinate system are given by: 2 2 1=2 0 x = (1 r = ) sinh(t= ) 1 x = r sin # sin ' 2 x = r sin # cos ' (2:5) 3 x = r cos # 2 2 1=2 4 x = (1 r = ) cosh (t= ) where the co ordinate ranges are: t :(1; 1) ; r :[0; ) ; # :[0;] ; ' :[0;2] (2:6) As we can see from these relations, the static system covers only a strip on the de Sitter manifold, for 2 2 2 2 1 2 3 which x + x + x < . This is the set of p oints on the manifold, which are in causal relation with the observer in b oth directions. The metric in this co ordinate system is given by: 1 2 2 2 2 2 2 2 2 2 ds =(1r = )dt dr r fd# + sin #d' g (2:7) 2 2 1 r = from whichwe can see that the observer at the origin p erceives an e ect of time dilatation, i.e. time seems to pass slower as one lo oks further away, an e ect due to the spacetime curvature (similarly to decreasing radial co ordinate near a black hole). We write the action for a minimally coupled scalar eld , with mass m: 2 Z 2 1 2 _ jj 1 1 @ L () m #' 2 2 2 2 2 + jj S = dt dr d# d' r sin # (1 r = ) (2:8) 2 2 2 1 r = 2 @r 2 r 2 where L is the angular part of the Laplacian at unit radius: #' 2 2 1 @ @ @ + L = + cot # (2:9) #' 2 2 2 @# @# @' sin # This action leads to the wave equation: 2 2 r @ @ @ 2 2 2 2 2 0= r (1 r = ) L ()+ r m (2:10) #' 2 2 2 1r = @t @r @r Notice that by rede ning of the radial co ordinate + r 1 ln r = tanh(r= ~ ) r~ = 2 r dr dr~ dr~ = dr = (2:11) 2 2 2 1 r = cosh (~r= ) r~ :[0;1) r:[0; ) we can bring the action to the following form: Z 2 2 1 @ L () m 2 1 1 2 #' 2 _ S = dt drd#d'~ tanh(r= ~ ) sin # jj jj + 2 2 2 2 2 @r~ 2 cosh (~r= ) 2 cosh (~r= ) tanh(r= ~ ) (2:12) and the wave equation: 3 2 2 2 2 @ @ @ L () m #' 0= tanh(r= ~ ) tanh(r= ~ ) + (2:13) 2 2 2 @t @r~ @r~ cosh (~r= ) cosh (~r= ) Forr= ~ 1 the action b ecomes Z 2 2 1 @ 1 2 _ S = dt drd#d' ~ sin # jj (2:14) 2 2 @r~ and the wave equation 2 2 2 @ @ @ @ @ 2 2 2 0= = (2:15) 2 2 2 @t @r~ @r~ @t @r~ which yields plane waves with resp ect to ther ~ conformal radial co ordinate in this limit. Lo oking at the relation b etween r andr ~ again, we can check that the conformal co ordinate of a p ointisgiven by the co ordinate time that is needed for a lightraytotravel b etween the origin and the p oint in question. From this we can also see that the plane wave gets in nitely compressed (dr~ =dr !1) as it approaches the event horizon, which is due to the time dilatation discussed previously, i.e. it takes in nite co ordinate time for a lightray to reach the event horizon. Now return to the original (non-conformal) radial parameter r , and lo ok for solutions in the form: i! t i! t m ^ = e (r;#;')= e f (r) Y (#; ') (2:16) !lm !l l The solutions are [6]: ^ (r;#;')= !lm q q 3 i! 1 9 l 3 i! 1 9 l 2 2 2 2 l=2 + + + m + + m 2 2 4 2 2 4 2 4 2 2 4 i! r 2 2 2 p (1 r = ) = 3 2 2 (l + )(i! ) 2 r r 2 l 3 i! 1 9 l 3 i! 1 9 3 r m 2 2 2 2 F + + + m ; + + m ; l + ; Y (#; ') l 2 2 4 2 2 4 2 4 2 2 4 2 (2:17) where F is the hyp ergeometric function. The solutions are orthonormalized with the following measure: Z 2 r sin(#) 0 ^ ^ 0 0 (r;#;') (r;#;')= (! ;!) (2:18) dr d# d' 0 0 0 !lm l ;l m ;m ! l m 2 2 1 r = We can expand the eld in terms of these mo des: Z X ^ (r;#;') (2:19) (r;#;')= d! !lm !lm l;m then express the action: Z 2 X 1 ! 2 2 _ S = dt d! j j j j (2:20) !lm !lm 2 2 l;m Wehave the action of a simple harmonic oscillator for each mo de, which has the ground-state wave- function: 4 1 4 2 i! t ! ! 2 2 ( ;t)= e e (2:21) !lm from which immediately follows: 1 2 (2:22) < >= !lm 2! 3.The correlation functions Using this, and the eld expansion, we can easily express the equal-time 2-p oint function: Z X d! 0 0 0 0 0 0 ^ ^ <(r ;# ;' );(r;#;') >= (r ;# ;' ) (r;#;') (3:1) !lm !lm 2! l;m 0 If we set r = 0 , the l 6= 0 mo des can b e neglected, since they are zero at the origin.