Symmetry Breaking in the Static Coordinate System of De Sitter

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2 ysics Lab oratory en, Connecticut 06520-8120 and Alan Cho dos v 1 ysics, Sloane Ph , New Ha as Kaiser y ale.edu ale.edu ersit Andr ysics.y of de Sitter Spacetime ysics.y View metadata,citationandsimilarpapersatcore.ac.uk ale Univ ter for Theoretical Ph Y alph2.ph 〉 Cen PostScript processed by the SLAC/DESY Libraries on 29 Jan 1995. HEP-TH-9501130 ho dos@y c [email protected] 2 1 Symmetry Breaking in the Static Co ordinate System provided byCERNDocumentServer brought toyouby CORE

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

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)

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

x = (1 r = ) sinh(t= )

x = r sin # sin '

x = r sin # cos '

(2:5)

x = r cos #

2 2 1=2

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:

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:

1 1 @ L () m

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

@# @# @'

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

1 r =

cosh (~r= )

r~ :[0;1) r:[0; )

we can bring the action to the following form:

1 @ L () m 2 1 1

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

@ @ @ L () m

0= tanh(r= ~ ) tanh(r= ~ ) + (2:13)

2 2

@t @r~ @r~

cosh (~r= ) cosh (~r= )

Forr= ~ 1 the action b ecomes

1 @ 1

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

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 4 2 2 4 2 4 2 2 4 i!

2 2

(1 r = ) =

3 2

2 (l + )(i! )

r r

l 3 i! 1 9 l 3 i! 1 9 3 r

2 2 2 2

F + + + m ; + + m ; l + ; Y (#; ')

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:

r sin(#)

^ ^

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:

(r;#;') (2:19) (r;#;')= d!

!lm !lm

l;m

then express the action:

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

i! t ! !

2 2

( ;t)= e e (2:21)

!lm

from which immediately follows:

(2:22) < >=

!lm

3.The correlation functions

Using this, and the eld expansion, we can easily express the equal-time 2-p oint function:

0 0 0 0 0 0

^ ^

<(r ;# ;' );(r;#;') >= (r ;# ;' ) (r;#;') (3:1)

!lm

l;m

If we set r = 0 , the l 6= 0 mo des can b e neglected, since they are zero at the origin. We don't have

to care ab out # or ' either, b ecause of the spherical symmetry of the l = 0 mo des.

^ ^

<(0) ;(r)>= (0) (r )=

! 00

q q

3 3 i! 1 9 i! 1 9

2 2 2 2

+ m m

4 2 2 4 4 2 2 4

p p

(3:2) =

)(i! ) 2 4 (

q q

3 3 i! 1 9 i! 1 9

2 2 2 2

+ + m + m

i! 4 2 2 4 4 2 2 4

2 2

p p

(1 r = )

2 4 ( )(i! )

r r

3 i! 1 3 i! 1 3 9 9

2 2

2 2 2 2

F + + m ; + m ; ; r =

4 2 2 4 4 2 2 4 2

Examine the r =0 case rst. The hyp ergeometric function gives 1 at zero argument, and since

,wehave an expression with a p owerlike ultraviolet divergence. In the massless (iy )(iy )=

ysinh (y)

case we also have an infrared divergence, due to the square ro ot cancelling the 3=4 in the argumentofthe

gamma function.

Now examine r ! ( r = is the event horizon, as discussed in section 2). Use the following

transformation of the hyp ergeometric function [10]:

F (a; b; c ; z)=

(c)(c a b)

F(a; b; a+bc+1 ; 1z)+ (3:3)

(c a)(c b)

(c)(a + b c)

cab

+(1z) F(c a; cb; cab+1 ; 1 z)

(a)(b)

which gives:

<(0) ;(r)>=

q q

(

i! 1 3 i! 1 3 9 9

2 2 2 2

+ + m + m

4 2 2 4 4 2 2 4

d! i!

2 2

= (1 r = )

16!

( )(i! )

r r

3 i! 1 9 3 i! 1 9

2 2

2 2 2 2

F + + m ; + m ; i! +1 ; 1r = + (3:4)

4 2 2 4 4 2 2 4 5

q q

3 i! 1 9 i! 1 9 3

2 2 2 2

+ m m

4 2 2 4 4 2 2 4

2 2

+ (1 r = )

( )(i! )

)

r r

i! 1 9 3 i! 1 9 3

2 2

2 2 2 2

F + + m ; + m ; i! +1 ; 1r =

4 2 2 4 4 2 2 4

As r ! , the arguments of the hyp ergeometric functions approach zero, therefore their values

approach1,sowe set them to 1 for the purp oses of this discusssion. In the next section we will need to

know the b ehaviour of some of these expressions near the r = limit, having the freedom to cho ose the

radial co ordinate arbitrarily close to . In the following analysis we will havetomake some approximations

to b e able to investigate certain prop erties of this expression. These approximations will cause some nite

error in the result, but it will not b e crucial for two reasons. The rst reason is that we will study certain

divergences in the obtained expressions, for which the nite error is irrelevant. The second reason is that

the error can b e made arbitrarily small bycho osing the radial co ordinate suciently close to , when we

evaluate the expressions.

2 2

i! !

2 2 i ln (1r = )

2 2

On the other hand, the (1 r = ) term can b e expressed as e , which results

in an in nitely fast oscillation of this term with resp ect to ! ,as r! .

Fo cus on the m ! 0 case, taking the m ! 0 limit b efore the r ! limit, b ecause this is the

2 2

3 m i!

case that we will need in the next section. As m ! 0, we can substitute for

2 6 2

q q

2 2

m 3 i! 1 9 i! 3 i! 1 9

2 2 2 2

+ m , and m . As we take for

4 2 2 4 6 2 4 2 2 4

the m ! 0 limit, the error resulting from this substitution will also go to zero.

After these substitutions wehave:

(

2 2 2 2

3 m i! m i!

+ +

i! d!

2 2

2 6 2 6 2

(1 r = ) +

16!

( )(i! )

)

2 2 2 2

3 m i! m i!

2 2

2 6 2 6 2

+ (1 r = ) (3:5)

( )(i! )

Now break up the ! integral into three parts:

0 | m = 3

2 2 2

m = 3 | 2=( ln(1 r = )) (3:6)

2 2

2=( ln(1 r = )) | 1

Taking the m ! 0 limit faster than the r ! limit assures this order and the m =3

2 2

2=( ln(1 r = ) condition.

2 2

. As the Consider the rst part. The ab ove condition allows us to substitute 1 for (1 r = )

2 2

m ! 0 limit is taken faster than the r ! limit, ! ln(1 r = ) go es to zero, justifying the ab ove

approximation. Also, all the errors that the following approximations will cause go to zero with m ! 0.

2 2 2 2

3 3 m i! ! m

Substitute ( ) for ( ) , since m ! 0 , and in this interval, and the

2 2 6 2 2 6

gamma function do es not have a p ole at 3=2. Then substitute 1=z for (z ) , wherever jz j 1 , which

gives us

m =3

d! i! i!

+ (3:7)

2 2 2 2

m i! m i!

16!

6 2 6 2

which shows that this rst of the three parts go es to a ( nite) constantasm!0.

2 2

Next consider the third interval of integration, ! : 2=( ln(1 r = )) | 1 . Here we can

2 2 2 2

3 m i! 3 i! m i! i!

replace ( ) with ( ) , and ( ) with ( ) in the

2 6 2 2 2 6 2 2

1 1

1 i! i!

2 2

m ! 0 limit. Then use (i! )=(2) 2 ( )( )

2 2 2 6

This waywe get

(

i! i! 3

d! i!

2 2

2 2 2

+ (1 r = )

1 1

3 i! 1 i!

16!

2 2

( )(2) 2 +

2 2 2 2

2 2

2=( ln (1r = ))

)

3 i! i!

2 2

2 2 2

+ (1 r = ) (3:8)

1 1

i! 3 i! 1

2 2

)(2) ( 2

2 2 2 2

then use (z +1)= z(z ) , cancel the appropriate terms, and rearrange:

( )

p p

2 2 2 2

2 2

d! (1 i! ) (1 + i! ) 1 r = 1 r =

+ (3:9)

3 3

16! 4 4

( ) ( )

2 2

2=( ln(1r = ))

break it into twointegrals:

( )

p p

2 2 2 2

2 2

1 r = 1 r = d!

+ +

3 3

16! 4 4

( ) ( )

2 2

2=( ln (1r = ))

( )

p p

2 2 2 2

2 2

1 r = 1 r = i i d!

+ + (3:10)

3 3

16 4 4

( ) ( )

2 2

2=( ln (1r = ))

where the rst integral is nite b ecause of the following. As a consequence of the oscillation, the integral

will consist of contributions with p erio dically alternating sign, and decreasing magnitude. Therefore the rst

contribution gives an upp er b ound to the integral. We get an upp er b ound to this rst contribution, if we

2 2

1r =

)] ), with the supremum of the function multiply the length of the interval (half p erio d, 2=[ ( ln

in the interval, which gives

3 3

2 2

=( ) ( )

2 ln(1 r = )

2 2

= (3:11)

2 2

1r = 2 2

16 ln(1 r = ) ln(4)

2 2

( ln )

ln (1r = )

which is nite ( 0

The second integral is the integral of a p erio dic oscillating function, whichwe regulate with an e

typ e ultraviolet cuto , then take ! 0 limit, obtaining

) (

4 4 ! !

i! i ln( +i! i ln ( ) )

2 2 2 2

2 2

1r = 1r =

ie = ie

16 ( )

2 2

2=( ln (1r = ))

4 !

i ln( ) i!

2 2

1r =

= e +

16 ( ) i ln( i )

2 2

2 1r = 2

4 !

+i! i ln ( )

2 2

1r =

= (3:12) + e

i ) + i ln(

2 2

2 2 1r = 2

2=( ln (1r = ))

2 2

ln(1r = )ln (4)

2 2 2 2

ln(1r = ) ln(1r = )

= p e +

3 4

16 ( ) ln( ) i

2 2

2 1r = 2 7

2 2

ln(1r = )ln (4)

i +

2 2 2 2

ln(1r = ) ln(1r = )

+ e

ln( ) + i

2 2

1r = 2

which is, once again, nite, and in the ! 0 limit yields

2 2

ln(1 r = ) ln(4) 1

cos (3:13)

3 4

2 2

ln(1 r = )

4 ( )ln( )

2 2

2 1r=

from whichwe can see that this interval also gives a nite contribution to the total integral.

2 2 2

Now consider the second interval of integration, ! : m =3 | 2=( ln(1 r = )) . Here

2 2

we can neglect the oscillation of the (1 r = ) term without changing the magnitude of the

result, b ecause the upp er limit of integration was determined such that the phase change of this term

3 3 1 i!

2 2

always stays less than . Also we can substitute ( ) for ( m ), b ecause

2 2 6 2

2 2 2 2 2

m =3 2=( ln(1 r = )) , and from r follows 2= ln(1 r = ) 1, so we

2 2

m !

have b oth 1 and 1. From this also follows that we can use the 1=z substitution for

6 2

2 2

m i!

(z ) for the ( ) and (i! ) terms.

6 2

As we see, we nally arrive at the same expression as in the rst part, but here wehave di erent limits

for the integration.

2 2

2=( ln (1r = ))

i! i! d!

2 2 2 2

2 m i! m i!

16!

6 2 6 2

m =3

2 2

2=( ln(1r = ))

d! (! )

2 2

m !

2 2

16!

( ) +( )

6 2

m =3

2 2

2=( ln(1r = ))

= (3:14)

2 2

! m

2 2

) +( ) (

6 2

m =3

2 2

2=( ln (1r = ))

2 2

1 ! m

= ln +

8 2 6

! =m =3

2 2

ln(1r = ) 6

= ln

2 2 2

whichisadivergent expression in the m ! 0 limit. We will need this expression in the next section.

4.Symmetry breaking

Now apply these results to the problem of symmetry breaking. Consider a mo del with complex scalar

eld and a symmetry-breaking p otential. We will use the approximation of a xed radial comp onent

of the complex scalar eld at the classical minimum, jj = , which leaves us with a massless minimally

coupled scalar eld as the angular comp onent. We give a summary of this treatment. For a fuller discussion

see [1].

In a broken-symmetry vacuum state the < (0) (r ) > correlation do es not go to zero as we let

r approach the event horizon, so a vanishing correlation is evidence for symmetry restoration. Express

< (0) (r ) > with the angular eld:

(r) (0)

i i

0 0

e > (4:1) < (0) (r ) >=

0 8

Divide the angular eld op erator into creation and annihilation parts, and commute the former to the

left, the latter to the right, to annihilate the vacuum states at the resp ective ends. Then the remaining

commutators give us:

(0) (0) (0) (r)

> < > <

2 2

0 0

(4:2) < (0) (r ) >= e

whichwe will have to renormalize due to the ultraviolet divergence in the second term (the two terms

of the exp onent are also infrared divergent separately, but these divergences cancel each other, so we can

take nite mass, calculate the exp onent then let the mass go to zero). We can renormalize by dividing the

whole expression by < (0) (r ) > , where wekeep r at any xed value while letting r go to the event

0 0

horizon.

(0) (r )

(0) (r)

< > < >

< (0) (r ) >

2 2

0 0

= e (4:3)

< (0) (r ) >

As wehave seen previously, for r ! , <(0) (r ) > is given by an infrared divergentinte-

gral (it diverges as we let the mass that regulates the infrared b ehaviour go to zero), which divergence

2 2

also has a ln( ln(1 r = )) dep endence. From this follows, that as r ! , the exp onent,

(0) (r ) (0) (r )

2 2

< > < > will diverge as ln( ln(1 r = )) in other words it will go to 1

2 2

0 0

. (It mightbeofinterest to note, that in terms of the conformal radiusr ~ this expression for the exp onent

takes the form (in the r ! limit): ln(2r= ~ ) , a single logarithmic b ehaviour instead of a double.)

<(0) (r )>

correlation function go es to zero, as r approaches the event horizon. This means, that the

<(0) (r )>

This leads us to the conclusion that the O (2) symmetry of our mo del is not broken in the vacuum state of

the static system, in other words the vacuum state of a geo desic observer.

5.Discussion

In Minkowski spacetime there is not muchambiguity ab out the concept of symmetry breaking. The rst

step, the choice of the co ordinate system for the eld quantization is natural, b ecause the usual rectangular

system has all p ossible attractive features: all co ordinate lines are geo desics and translational invariance

is manifest. The only symmetries which need more complex co ordinate-transformations are the rotational

symmetries and Lorentz b o osts. However when we quantize the eld, the vacuum state turns out to b e also

invariant under these symmetries. It is invarianteven under the Lorentz b o osts, although in the quantization

pro cess the equal-time hyp ersurfaces play a crucial role at the prescription of the commutation relations, and

these hyp ersurfaces are certainly not invariant under Lorentz-b o osts. Also, the n-particle states are mapp ed

into n-particle states, only the particle 4-momenta transform corresp ondingly under the symmetries. Thus

all inertial observers have the same vacuum, and corresp onding multi-particle states, which can b e used for

studying symmetry breaking, and p ossible restoration pro cesses of the symmetric vacuum state, for example

in the case of nite volume.

As wemove to a non- at spacetime, these natural choices disapp ear. Wehave studied eld theory

in a de Sitter background, which has constant spacetime curvature, and a 10-element symmetry (like the

Minkowski symmetry group, whichisacontraction of the de Sitter group). But there is no co ordinatization

of the manifold that would consist of geo desics and would have translational invariance in all of its co ordinate

parameters. Furthermore there is no unique vacuum for all geo desic observers. Wehave to make our choices

for equal-time surfaces, which will then a ect the quantization through the commutator-prescriptions, that

will lead to the corresp onding vacuum states and Fock-spaces (for discussions of this asp ect see for example

[5] and [7]), and nally determine the existence or absence of symmetry-restoration pro cesses (which are due

to certain infrared divergences, therefore crucially dep end on the prop erties of the Fock-space).

Since it is not only a p ossibility to arrive to di erent answers in di erent co ordinate systems, but indeed

the answer was found to b e di erent using di erent systems previously [1,4], we think that it is useful to study

the problem through a system that is selected to corresp ond to the measurements of a geo desic observer,

leading to a Fock-space where the vacuum state is the one in which the observer do es not detect particles,

etc. (see also [8] and [9]). The co ordinate system that satis es these requrements is the \static" system

of the observer, in which the spatial sections are geo desic hyp ersurfaces orthogonal to the world line of the

observer, with the spatial co ordinates translated along the world line of the observer. 9

As discussed in [1], the metho d wehave employed searches for sp ontaneous symmetry breakdown by

examining < (0) (r ) > as r gets large. If this quantity tends to a p ositive constant, it would imply

symmetry breaking. On the other hand if it tends to zero (which is what wehave found), than that is

consistent with the restoration of symmetry. It is still conceivable, however, that symmetry breaking do es

take place in this co ordinate system, but manifests itself in a more subtle form. Studying the structure of the

Fock-space might give more insightinto this p ossibility,aswell as into the physical reasons and the meaning

of the di erent results that were obtained in refs. [1] and [4] on the sub ject of symmetry restoration studied

in di erent co ordinate systems of the same manifold. Weintend to study the problem from this direction in

a later pap er.

Acknowledgements

We are grateful to Hisao Suzuki and Atsushi Higuchi for helpful discussions.

Research supp orted in part by DOE grant no. DE-AC02-ERO3075.

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[5] N. D. Birrell and P.C.W.Davies, Quantum elds in curvedspace,

Cambridge monographs on mathematical physics, 1985.

[6] A. Higuchi, Class. Quantum Grav. 4 , 721 (1987);

Haru-Tada Sato and Hisao Suzuki, OU-HET 202, LMU TPW 94-13,

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