The Conformal Universe II

8/13/2019 The Conformal Universe II

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arXiv:1201.3343v3

[hep-th]21Dec

2013

THE CONFORMAL UNIVERSE II:

Conformal Symmetry, its Spontaneous Breakdown

and Higgs Fields in Conformally Flat Spacetime

(revised, amended and improved version)

Renato Nobili

E-mail: [email protected]

Padova, 21 December 2013

Abstract

This is the second of three papers on Conformal General Relativity (CGR). The con-

formal group is here introduced as the invariance group of the partial ordering of causal

events in nD spacetime. Its general structure and field representations are described

with particular emphasis on the remarkable properties of orthochronous inversions.

Discrete symmetries and field representations in flat 4D spacetime are described in

details. The spontaneous breakdown of conformal symmetry is then discussed and

the roles played in this regard by the dilation field and one or more physical scalar

fields of zero mass are evidenced. Hyperbolic and conformalhyperbolic coordinates

are introduced in 4D spacetime for the purpose of providing different but equivalent

mathematical representations of CGR. These are respectively the Riemann and the

Cartan representation, the first of which is manifestly conformal invariant and the

second is not, but in compensation preserves the formal structure of standard General

Relativity (GR) in which all local quantities are multiplied by suitable scale factors.

The equivalence of these representations in regard to scale relativity is also discussed.

In the end, the action integrals, as well as the motion equations and total energy

momentum tensors, of a Higgs field interacting with the dilation field are described in

both the above mentioned representations, in view of the detailed study of Higgsfield

dynamics carried out in the third paper.

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R.Nobili, The Conformal Universe II 2

Contents

1 The conformal group and its representations 3

1.1 Conformal group as the invariance group of causality . . . . . . . . . . . . . 3

1.2 Remarkable geometric properties of orthochronous inversions . . . . . . . . 6

1.3 Conformal transformations of local fields . . . . . . . . . . . . . . . . . . . . 8

1.4 Discrete symmetries of the conformal group in 4D . . . . . . . . . . . . . . 9

1.5 Conformal transformations of tensors in 4D . . . . . . . . . . . . . . . . . . 11

1.6 Orthochronous inversions of vierbeins, currents and gauge fields . . . . . . . 12

1.7 Actionintegral invariance under orthochronous inversionI0 . . . . . . . . . 14

2 Spontaneous breakdown of conformal symmetry 15

2.1 Possible spontaneous breakdowns of global conformal symmetry . . . . . . . 16

2.2 The NG bosons of spontaneously broken conformal symmetry . . . . . . . . 18

3 Relevant coordinate systems in CGR 19

3.1 Hyperbolic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Conformal hyperbolic coordinates . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Two equivalent representations of CGR action integrals 26

4.1 Riemann picture in hyperbolic coordinates . . . . . . . . . . . . . . . . . . . 29

4.2 Cartan picture in conformalhyperbolic coordinates . . . . . . . . . . . . . . 30

5 The Higgs field in CGR 31

5.1 The Higgs field in the Riemann picture . . . . . . . . . . . . . . . . . . . . . 32

5.2 The Higgs field in the Cartan picture . . . . . . . . . . . . . . . . . . . . . . 34

References 38


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1 The conformal group and its representations

As outlined in 2 of Part I [1], both General Relativity (GR) and Conformal GeneralRelativity (CGR) describe the universe as grounded on a differentiable manifold covered

by a continuous set of local tangent spaces, all of which support isomorphic representations

of a finite continuous group, called the fundamental group. The fundamental group of GR

is the Poincare group and that of CGR is its conformal extension.

In the next three subsections, the structure and the most important representations of

the conformal group in nD spacetime are described in details. In the subsequent subsec-

tions, the conformal group in 4D spacetime will be only considered.

As hereafter shown, the importance of the finite group of conformal transformations

lies in the fact that it is the widest invariance group of the partial ordering of causal

events in flat spacetime. We may then infer that the infinite group of local conformal

diffeomorphisms is the widest invariance group of partial ordering of causal events in a

differentiable, generally curved, spacetime manifold.

1.1 Conformal group as the invariance group of causality

Let us define an nD Minkowski spacetimeMn as the Cartesian product of an (n-1)dimensional Euclidean spaceRn1 by a real time axis T. The former is intended torepresent the set of possible inertial observers at rest inRn1 equipped with perfectlysynchronized clocks, and the latter is intended to represent common time x0 Tmarked bythe clocks. All observers are allowed to communicate with one another by signals of limited

speed, the upper limit of which is conventionally assumed not to exceed 1 (the speed of light

inRn1). Let us indicate by x1, x2,...,xn1 the coordinates ofRn1. Hence, regardlessof any metrical considerations, we can write

Mn =

Rn1

T, which means that

Mn is

equivalent to the ndimensional affine space. Clearly, pointsx ={x0, x1, x2,...,xn1} Mn also represent the set of all possible pointlike events, observable in Rn1 at timex0,partially ordered by the relation x y defined as follows:

x can influencey if and only ify0 x0

(y1 x1)2 + + (yn1 xn1)2.In contrast to causality in Newtonian spacetime, this partial ordering equipsRn1 with anatural topology, the basis of which may be formed by the open sets of points y Rn1,


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wherex2 =xx, x = x,= diag{1, 1, . . . , 1}, which we call the orthochronous

inversion with respect to event x = 0 Mn [9]. The equalities (I0)2 = 1, I0()I0 =() and I0S()I0 = S() are manifest. By translation, we obtain the orthochronousinversion with respect to any point a Mn, which acts on x as follows:

Ia: x x

a(x a)2.

Clearly, the causal ordering is also preserved by the transformations E(b) =I0T(b)I0,

which manifestly form an Abelian group and act on x as follows

E(b) : x

x bx2

1 2bx + b2

x2

, (4)

where bx stands for bx. These are commonly known as special conformal transforma-

tions, but we will call them elations, since this is the name coined for them by Cartan in

1922[10].

In conclusion, the topologically connected component of the complete invariance group

of causal ordering in the nD Minkowski spacetime is the ndimensional conformal group

C(1, n1) formed of transformations (1)(4), comprehensively depending onn(n+3)/2+ 1real parameters.

From an invariantive point of view, C(1, n 1) can be regarded as the more generalcontinuous group generated by infinitesimal transformations of the form x x +u(x),where is an infinitesimal parameter and u(x) are arbitrary real functions, satisfying the

equation

ds2 gdxdx (x) gdxdx ,where (x) is an arbitrary real function [11].

Indicating byP,M,D and K the generators ofT(a), (), S() andE(b), respec-

tively, we can easily determine their actions on x

Px = i , Mx =i(x x) ,

Dx = ix , Kx =i(x2 2xx) ,

where is the Kronecker delta. Indicating by the partial derivative with respect to

x, their actions on arbitrary differentiable functions f ofx are

Pf(x) = if(x) ; Mf(x) =i(x x)f(x) ;D f(x) = ixf(x) ; Kf(x) =i(x2 2xx) f(x) ;


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the Lie algebra of which satisfies the following commutation relations

[P, P] = [K, K] = 0 ; [P, K] = 2i(gD+ M) ; (5)

[D, P] =iP; [D, K] = iK; [D, M] = 0 ; (6)[M, P] =i(gP gP) ; [M, K] =i(gK gK) ; (7)[M, M] =i(gM + gM gM g M) , (8)

which form the prototype of the abstract Lie algebra ofC(1, n 1) [12][13].For the sake of clarity and completeness, we add to Eqs. (5)(8) the discrete transforms

I0PI0 = K; I0KI0 = P; I0DI0= D ; I0MI0= M. (9)

Note that I0 and P alone suffice to generate C(1, n 1). In fact, using Eqs. (5)(7)and the first of Eqs. (9), we can define all other group generators as follows:

K = I0PI0 , D= i

8g[K, P] , M=

i

2[K, P] gD .

This tells many things about the partial ordering of causal events inMn. Indeed, wemay think ofI0as representing the operator that performs the partial ordering of causally

related events as seen by a pointlike agent located at x= 0, which receives signals from

its own past and sends signals to its own future, ofT(a) as the operator that shifts the

agent fromx = 0 to x = a in Mnand ofIaas a continuous set of involutions which imparta symmetricspace structure to the lattice of causal event.

Lastly note that, provided that n is even, we can include parity transformation P :

{x0, x} {x0, x} as a second discrete element of the conformal group. Timereversalmust be instead excluded, since it does not preserve the causal ordering of events. This

makes a difference between GR and CGR. Indeed, time reversal, so familiar to GR, is

replaced by orthochronous inversionI0conventionally centered at an arbitrary pointx= 0.

1.2 Remarkable geometric properties of orthochronous inversions

Orthochronous inversion Ia, where a is any point ofMn, has the following properties:1) It leaves invariant the double cone centered at a, swapping the interiors of the

future and past cones so as to preserve the time arrow and the collineation of all points

on straight lines through a, as shown in Fig. 1.


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a T

x

Iax

rIar

S

-1/

H+()

H-(-1/)

Figure 1: The orthochronous inversion cen-

tered at a point a of the Minkowski space-

time interchanges the events lying in the in-

terior of the double cone stemming from a

in such a way that spacetime regions of the

pastcone close to a are mapped onto re-

gions of the futurecone far from a (and vice

versa). T= time axis; S= unit (n 1)D

sphere centered at a and orthogonal to T.

2) It partitions the events of the past and future cones stemming from ainto a twofold

family of (n 1)-D hyperboloids parameterized by kinematic time

=

(x0 a0)2 + + (xn1 an1)2 .

3) It maps future region H+a(), extending from conevertex a to the (n 1)D hyper-boloid at , into region Ha(1/) of the past cone defined by the (n 1)D hyperboloidatand the hyperboloid at = 1/, and vice versa (Fig.1, gray areas). Similarly, itmaps the settheoretic complements ofH+a(), H

a (1/) onto each other.

4) It performs polar inversion of points r internal to the spacelike unit (n1)Dsphere S centered at a and orthogonal to time axis T into points r =Iar external to S,

and vice versa.

5) Functions which are invariant underIadepend upon kinematic timeonly. Thus, if

they vanish near the vertex of the past cone, they also vanish at the infinite kinematic time

of the future cone, and vice versa. This property has an important implication in that,

ifI0 is invariant under the spontaneous breakdown of conformal symmetry, in an infinite

time the universe reaches the same physical conditions in which it existed just a moment

before the symmetry breaking event. In other words, the actionintegral invariance under

I0 is compatible with assuming that the time course of the universe may be described as

a transition from the state of an unstable initial vacuum to that of a stable final vacuum

with same physical properties.


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1.3 Conformal transformations of local fields

When a differential operator g is applied to a differentiable function f ofx, the function

changes asgf(x) =f(gx), which may be interpreted as the form taken by fin the reference

frame of coordinates x = gx. When a second differential operator g acts on f(gx), we

obtain gf(gx) = f(gg x), i.e., we have ggf(x) = f(gg x), showing that g and g act on

the reference frame in reverse order.

The action ofg on a local quantum field (x) of dimension, or weight, w, bearing a

spin subscript, has the general form g(x) =F(g1, x)(gx), whereF(g1, x) is a

matrix obeying the composition law

F(g12 , x) F(g11 , g2x) = F(g12 g11 , x) .

These equations are consistent with coordinate transformations, since the product of two

transformations g1, g2 yields

g2g1(x) =F[(g1g2)1, x](g1g2x) ,

with g2, g1 always appearing in reverse order on the righthand member.

According to these rules, the action ofC(1, n 1) generators on an irreducible unitaryrepresentation (x) of the Poincare group, describing a field of dimension w and spin

subscript, may be summarized as follows

[P, ] = i ; (10)[K, ] =i[x

2 2x(x w)]+ ix(); (11)[D, ] = i(x w) ; (12)[M, ] =i(x x) i() ; (13)

where are the spin matrices, i.e., the generators of Lorenz rotations on the spin space.

The corresponding set of finite conformal transformations are

T(a) : (x) (x + a) ; (14)

E(b) : (x) E(b, x) x bx2

1 2bx+ b2x2

; (15)

S() : (x) ew (ex) ; (16)() : (x) L()[() x] ; (17)


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where E(b, x), L() are suitable matrices which perform the conformal transformationsof spin components, respectively for elations and Lorentz rotations.

As regards the orthochronous inversion, we generally have

I0: (x) I0(x) (x/x2) , (18)

where matrixI0(x) obeys the equation

I0(x)I0(x/x2) = 1 . (19)

For consistency with (14), (15) and Eqs. E(b) =I0(x) T(b) I0(x), we also have

E (b, x) =I0(x)I0(x b) . (20)

For the needs of a Langrangian theory, the adjoint representation of must also be

defined. It is indicated by = B, whereB is a suitable matrix, or complex number,chosen so as to satisfy equation = which impliesBB = 1 and the selfadjointnesscondition of the Hamiltonian. Therefore, under the action of a group element g, the adjoint

representation (x) is subject to the transformation

g: (

x)

(gx

)

F(g

1

, x),

where F(g1, x) = BF(g1, x)B.

1.4 Discrete symmetries of the conformal group in 4D

In this and the next two subsections we focus on the transformation properties of a spinor

field (x) since those of all other fields can be inferred by reducing direct products of

spinor field representations.

As is wellknown in standard field theory, the algebra of spinor representation contains

the discrete group formed by the parity operator P, the charge conjugation C and the

time reversal T. The latter commutes with P, and C, and the elicity projectors P,

P++ P = 1, which are defined by equations R =P+ and L = P, where R and L

stand respectively for the righthanded and the lefthanded spinor elicities. However, as

already pointed out at the end of 1.1,passing from the Poincare to the conformal group,we must exclude T, which violates causal ordering, and transfer the role of this operator

to I0.


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Let us normalize Dirac matrices in such a way that 0 = (0) = (0)T and 2 =

(2) = (2)T, where superscript T indicates matrix transposition, and 5 = i0123.

Then, the equalities L = 12(1 +

5) and R = 12(1 5) hold. As is wellknown in

basic Quantum Mechanics, in this representation P and Cact on L,R as follows:

P :R,L(x) PL,R(P x) ; C: R,L(x) CL,R(x) ;CP :R,L(x) CPR,L(P x) ; (21)

whereP x P{x0, x} = {x0, x},P=0,C= i20,CP=i2 andR,Lare the complex

conjugates ofR,L. We can easily verify the equations

PC = CP, C1 = C = C , C5 = 5C , P5 = 5P,

the last two of which show that bothP andC interchange L with R (whilstPC leavesthem unchanged).

Note that, despite their anticommutativity,P andC act commutatively on fermionbilinears, of which all spinor observables are made of.

The general form ofI0(x) introduced in Eq.(18) is determined by requiring that I0commutes withP,C, P, as time reversal Tdoes, as the heuristic principle of preservation

of formal laws suggests, and that it satisfies Eq.(19). In summary,

0I0(P x) =I0(x) 0 ; 20I0(x) =I0(x) 20 ;5I0(x) =I0(x) 5 ; I0(x)I0(x/x2) = 1 .

Of course, we also require that I0 is not equivalent to 1 and C P. We can easily verify that

all these conditions lead to the general formula

I0(x) = (x2) /x|x| 20 = (x2)1/220/x , (22)

where:|x| =

x2 =

/x2, is a real number and /x= x, with

= diag(1, 1, 1, 1) .

The sign being arbitrary, we take it equal to -1 for the sake of convenience. Note that /x

is a pseudoscalar, sinceP/xP1 =00(P x) = /x.


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To find , we impose the condition that the conformal invariant freefield Feynman

propagator

0|T{(x1), (x2)}|0= i

22/x1 /x2

[(x1 x2)2 + i]2 ,

where T{ } indicates time ordering, be invariant under I0, i.e.,

0|T{I0(x1)I10 , I0(x2)I10 }|0= 0|T{(x1), (x2)}|0 .

Using Eq.(18) we find the adjoint transformation

(x) =(x)0 t0

(x2)1/2(

x/x2) /x020 = (x2)1/2(

x/x2) /x20,

where equalities 20 =02 and 0/x0 = /x were used. Thus, the invariant conditiontakes the form

(x21x22)

1/2 02 /x1(x

21/x2 /x1x22)/x220

[(x1 x2)2 + i]2 = (x21x

22)

+3/2 /x1 /x2[(x1 x2)2 + i]2,

which yields =3/2, i.e., just the dimension of hus, Eq.(22) can be factorized asfollows

I0(x) = x2

3/2

i /x|x

|

i 20

= (x2)DS0(x)C = (x2)DCS0(x) , (23)

whereD is the dilation generator,C the charge conjugation matrix andS0(x) = i /x/|x|xis a spin matrix satisfying the equation S0(x) S0(x) = S0(x) S0(x) = 1.

Note thatS0(x) acts as a selfadjoint reflection operator which transforms the spinorfield to its mirror image with respect to the 3D space orthogonal to timelike 4vector x

at x = 0. Indeed, we have

S0(x) /y S(x) =/y 2(xy)x2

/x , S0(x) S0(x) = 2x/x

x2 . (24)

1.5 Conformal transformations of tensors in 4D

Since /x/y= xy ixy, where = i2 [, ], we obtain from Eqs. (20) and (23)

E (b, x) =(1 + bx ixb)

(1 + 2bx+ b2x2)2 . (25)

Noting that (xb)2 =x2b2 (bx)2, we define

= xb

x2b2 (bx)2 ,


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so2 = 1. We can then pose

1 + bx ixb=A

cos

2 i sin

2

= Aei/2 ,

with

A=

1 + 2bx+ b2x2 and tan

2=

x2b2 (bx)2

1 + 2bx . (26)

In conclusion, we can write

E(b, x) = (1 + 2bx + b2x2)3/2ei/2 ,

showing that E(b, x) is the product of a local dilation and Lorentz rotation ei/2 actingon the spinor space, both of which depend on x.

Since the transformation properties of all possible tensors can be derived by reducing

direct products of spinorfield representations, we can infer the general form of inversion

and elations spinmatrices for fields (x) of any spin inM4 as

I0(x) = (x2)D S0(x) C , E(b, x) = (1 + 2bx + b2x2)D R0(b, x) , (27)

whereDis the dilation generator,

S0(x) is the spinreflection matrix for (x),

Cthe charge

conjugation matrix andR0(b, x) =ei, with defined as in Eq.(26) and

= xb

b2x2 (xb)2,

as the spinrotation matrix for (x) at x = 0.

As an application of the results so far achieved, let us study the transformation prop-

erties of fermionic bilinear forms and corresponding boson fields under the action of the

conformal group.

1.6 Orthochronous inversions of vierbeins, currents and gauge fields

In the last two subsections, the behavior of the fields under orthochronous inversion was

studied in the particular case of flat or conformally flat spacetime (cf. 4.1 of Part I),basing on the transformation properties of spinor fields. As soon as we try to transfer the

very same concepts to Dirac Lagrangian densities and equations, we encounter immediately

the problem of that Diracs matrices , as dealt with so far, have no objective meaning


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and must be replaced by expressions of the form (x) = ea(x) a, where ea(x) is the

vierbeintensor and a a standard representation of Diracs matrices in 4D [14].

Since ea(x) is related to the metric tensor by the equation ea(x) ea(x) = g(x), it

has dimension1 and therefore is transformed by I0 as follows

ea(x) I0 1

x2

ea(x/x2)

xxx2

ea(x/x2)

.

consistently with Eq.(24).

Let a be us a spinor field of dimension 3/2 satisfying canonical anticommutationrelations, where a is some family superscript, and consider the hermitian bilinear forms

J(a,b)(x) =1

2ea(x) [

a(x), a b(x)] , J5(a,b)(x) =1

2ea(x)[

a(x), a 5b(x)],

Ja,b(x) =1

2[a(x), b(x)] , J5a,b(x) =

1

2[a(x), 5b(x)] . (28)

These may be respectively envisaged as the currents and axialvector currents of some

local algebra, and scalar and pseudoscalar densities of a; all of which being expected

to be coupled with appropriate boson fields in some Langrangian density. We leave as an

exercise for the reader to prove thatI0 act on them as follows:

J(a,b) (x) I0 1

(x2)4

J(a,b)(I0x) x

xx2

J(a,b)(I0x)

; (29)

J5(a,b) (x) I0 1

(x2)4

J5(a,b)(I0x) x

xx2

J5(a,b)(I0x)

; (30)

J(a,b)(x) I0 1

(x2)3J(a,b)(I0x) ; J

5(a,b)(x) I0 1

(x2)3J5(a,b)(I0x) . (31)

where I0x= x/x2.As explained in the beginning of 6 Part I, covariant vector fields A(x) and covari-

ant axialvector fields A

5

(x) have dimension 0, while scalar fields (x) and pseudoscalarfields (x) have dimension1. Therefore, for consistency with field equations, they aretransformed by I0 as follows

A(x) I0

A(I0x) xx

x2 A(I0x)

; (32)

A5(x) I0

A5(I0x)

xx

x2 A5(I0x)

; (33)

(x) I0 1

x2(I0x) ; (x)

I0 1x2

(I0x) . (34)


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Combining Eqs. (29)(31) with Eqs. (32)(34) and using the replacements

g(x) I0 (x2)2g(I0x) ,

g(x) I0 (x2)4

g(I0x) ,

we obtain the transformation laws

g(x) J(a,b) (x) A(x) I0

g(I0x) J(a,b) (I0x) A(I0x);

g(x) J5(a,b) (x) A5(x) I0

g(I0x) J5(a,b) (I0x) A(5)(I0x) ;g(x) J(a,b)(x) (x) I0

g(I0x) J(a,b)(I0x) (I0x) ;

g(x) J5(a,b)(x) (x)

I0

g(I0x) J5(a,b)(I

0x) (I

0x) ;

showing that all Lagrangian densities of interest are transformed by I0 as

g(x) L(x) I0

g(I0x) L(I0x) . (35)

1.7 Actionintegral invariance under orthochronous inversion I0

On the 3D Riemann manifold, the action ofI0in the past and future cones H+0 H+0(+)

andH0 H0 (0) satisfies the selfmirroring linear properties described in 1.2,providedthat the geometry is conformally flat. If the geometry is appreciably distorted by the

gravitational field, selfmirroring properties are still found provided that the points of

cones are parameterized by polar geodesic coordinates{, x}, as described in 4.3 of PartI, with > 0 for H+0 and


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2 Spontaneous breakdown of conformal symmetry

Let us assume that the action integral of a system and the fundamental state, i.e., vacuum

state|, are invariant under a finite or infinite group G of continuous transformations.Then, the spontaneous breakdown of the symmetry associated with this invariance is char-

acterized by four main facts: 1) the loss of the groupinvariance of|; 2) the residualinvariance of| under a subgroup S G, called the stability subgroup; 3) the creationof one or more fields called the NambuGoldstone(NG) fieldsof nonzero VEV and gap-

less energy spectrum (one field per each groupgenerator that does not preserve|); 4)

the contraction of the settheoretic complement ofS G into an invariant Abelian sub-group of| transformations, called the contraction subgroup, which produces amplitudetranslations of NGfields and parametric rearrangements of group representations[15]. If

|is invariant under spacetime translations, the energy spectrum bears zeromass poles,implying that the NGfields bear massless quanta, here called the NGbosons.

If G is a finite group of continuous transformations, the symmetry is global and the

symmetry breaking is equivalent to a global phase transition. If the system is not invariant

under spacetime translations the energy spectrum, gapless though, is free of zeromass

poles, implying that NGbosons do not exist.

If G is an infinite group of local continuous transformations, as is indeed the case

with local gauge transformations and conformal diffeomorphisms, the stability group too

is infinite and local, and, because of locality, NGbosons appear. However, these can be

sequestered by zeromass vector bosons to yield massive vector bosons [16] [17] [18], or

condense at possible topological singularities ofS (pointlike singularities, vortical lines,

boundary asymmetries etc.) to produce extended objects. In the latter case, NGfield

VEVs appear to depend upon manifold coordinates.

In Part I, the creation of the universe was imagined as a process generated by the

spontaneous breakdown of a symmetry occurred at a point x = 0 of an empty spacetime,

which primed a nucleating event followed by a more or less complicated evolution in the

interior of the future cone stemming from 0. It is now clear that the symmetry which we

are concerned with is expected to reflect the invariance of the total action integral with

respect to the group of conformal diffeomorphisms. Since this is an infinite group of local

continuous transformations, we also expect that: 1) the stability subgroup Sbe infinite,


16/39


local and possibly provided with a topological singularity including the vertex of future

cone; 2) NGboson condensation takes place at this singularity, with the formation of an

object extended in spacetime. This implies that NGboson VEVs depend on hyperbolic

spacetime coordinates.

The search for spontaneous breakdown with these characteristics is considerably sim-

plified if we focus on the possible stability subgroups of the fundamental group C(1, 3)

of conformal Cartan connections rather than of the full group of diffeomorphisms. This

simplification is legitimated by the following considerations: since we presume that, before

the nucleating event, the manifold was conformally flat or of constant curvature (cf.

7.1

of Part I), we can assume that the evolution of the system immediately after this event

was isotropic and homogeneous. This means that the metric remained conformally flat or

of constant curvature during a certain kinematictime interval, until gravitational forces

began to enter into play, favoring the aggregation of matter. Since during this initial

period the symmetry is virtually global, we may limit ourselves to searching for subgroups

ofC(1, 3) endowed with topological singularities.

The mechanism of the spontaneous breaking of the global conformal invariance was

investigated by Fubini in 1976. We report here his main results.

2.1 Possible spontaneous breakdowns of global conformal symmetry

It is known that the 15parameter Lie algebra of the conformal group in 4D spacetime

C(1, 3) is isomorphic with that of the hyperbolicrotation group O(2, 4) on the 6D linear

space{x0, x1, x2, x3, x4, x5}of metric (x0)2 + (x5)2 (x1)2 (x2)2 (x3)2 (x4)2. Assum-ing that the dilation invariance of the vacuum is broken spontaneously, the sole possible

candidates for a stability subgroup turn out to be:

- SP: the Poincare group, i.e, the 10parameter Lie algebra generated by M and

P. With this choice, possible NGbosons VEVs are invariant under translations

and are therefore constant.

- O(1, 4): the de Sitter groupgenerated by the 10parameter Lie algebra which leaves

invariant the fundamental form (x0)2 (x1)2 (x2)2 (x3)2 (x4)2 of de Sitter


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spacetime [19]. Its generators are M and

L =1

2

P K

,

which anticommute with orthochronous inversion I0 and satisfy the commutation

relations

[L, L] = i M; [M, L] =i(gL gL) .

Any NGboson VEV (x) associated with this stability subgroup must satisfy the

equation

L(x) = 0 , M(x) i(x x) (x) = 0 ,

the second of which implies that (x) depends on x2 2 only.

- O(2, 3): theantideSitter groupgenerated by the 10parameter Lie algebra which

leaves invariant the fundamental form (x0)2 + (x4)2 (x1)2 (x2)2 (x3)2 of theantideSitter spacetime. Its generators are M and

R= 1

2P+ K ,

which commute with I0 and satisfies the commutation relations

[R, R] =i M; [M, R] =i(gR gR) .

Any NGboson VEV +(x) associated with this stability group must satisfy the

equations

R+(x) = 0 , M+(x) i(x x) +(x) = 0 ,

the second of which implies that +(x) too depends on x2 =2 only.

Comparing the results obtained for the de Sitter and antide Sitter groups, we note that

L and D are the generators of the settheoretic complement ofO(3, 2) and that R and

D are the generators of the settheoretic complement ofO(1, 4). Now, from equations

[R, D] =i L , [L, D] =i R ,


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we derive

[R, D] () =i L () = 0 , [L, D] +() =i R +() = 0 ,

showing that the settheoretic complements of O(3, 2) and O(1, 4) act respectively on

VEVs +() and () as invariant Abelian subgroups of transformations, as is indeed

expected. Since the explicit expressions for R and L acting on a scalar field (x) of

dimension1 are respectively

R(x) i1 x2

2 + x(x

+ 1)

(x) ,

L(x) i

1 + x22

x(x+ 1)

(x) ,

as established by Eq.(11) of 1.3, we can easily verify that the above equations for NGboson VEVs are equivalent to

(x) =() , 2() c2 3() = 0 ,

where

2f() f() = 2+

3

f() ,

the general solutions of which are, respectively,

+() = (0)

1 + 2, () =

(0)

1 2, where (0) =

8

c.

However, these expressions are not uniquely determined, since by an arbitrary change of

scale /0, 0> 0, they become

+() = (0)

1 + ( /0)2, () =

(0)

1 ( /0)2 , where (0) = 1

0

8

c. (37)

2.2 The NG bosons of spontaneously broken conformal symmetry

The energy spectra of functions (37) described at the end of the previous subsection

are manifestly gapless and free of zeromass poles. It is thus evident that, on a suitable

kinematic-time scale,+() and() are the classic Lorentzrotationinvariant solutions

of the motion equations coming from action integrals

A=

1

2()()

c24

4

d4x


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(with positive potential energy), so that they can be interpreted respectively as a zero

mass physical scalar field and a zeromass ghost scalar field, which are precisely the

ingredients of the Higgs field action integral discussed in 6 of Part I. Since the VEV of aclassic field coincides with the field itself, we can think of() as the VEV|(x)|of a possible zeromass scalar ghost field (x), and of+() as the VEV|(x)| of apossible zeromass physical scalar field(x). It is therefore also natural to expect that the

spontaneous breakdown of the full group of conformal diffeomorphisms is characterized by

the generation of NGfields of this kind. This means that we must be prepared to introduce

into the conformalinvariant Lagrangian density on the Riemann or Cartan manifold one

or more such scalar fields and interpret the classic fields as extended macroscopic objects

formed of zeromass NGbosons condensed into stability subgroup singularities.

However, we cannot ignore the fact that the conformal invariance of the action integrals

may be illusory, because of possible ultraviolet cutoff parameters brought into play by

renormalization procedures. Since we are working in the semiclassical approximation we

can ignore this problem, being confident that, in its quantum mechanical version, it may

be solved by the conformal-invariant renormalization procedures suggested by Englert et

al. (1976)[20].

Comparing these results with those discussed in Part I, we immediately realize that, for

small , in the conformal flatness approximation and after a suitable change of scale, the

NGfield(), associated with the deSitter stability subgroup O(4, 1), may be regarded

as the seed of the ghost scalar field (x) introduced in the geometricLagrangian density

described in Part I. This is consistent with the view of Gursey, according to which the

cosmic background has the properties of a de Sitter spacetime (cf. 8.1 of Part I).Similarly, the NGfields of type +(x), associated with antideSitter stability subgroup

O(2, 3), may be regarded as the seeds of a possible multiplet of zeromass scalar fields,with nonzero VEVs, belonging to the matterLagrangian density.

3 Relevant coordinate systems in CGR

If the gravitational field is negligible, or can be represented as a linear perturbation of

the metric tensorg, and the spacetime is conformally flat (see 8.1 of Part I), the mostconvenient choice of coordinates is suggested by the symmetry properties of orthochronous


20/39


inversion I0. On the Riemann manifold, this leads naturally to hyperbolic coordinates. On

the Cartan manifold, the scale factor of the fundamental tensor must be also considered,

and we are naturally led to conformal hyperbolic coordinates.

3.1 Hyperbolic coordinates

After the spontaneous breaking of conformal symmetry, the partial ordering of causal

events is more conveniently parameterized by adimensional contravariant coordinates x,

reflecting the actionintegral invariance under orthochronous inversion I0, that is, hyper-

bolic coordinates centered at x = 0. These are: the kinematic time , the hyperbolic

angle and the Euler angles , , all of which are related to the Lorentzian coordinates

x= {x0, x1, x2, x3} by the equations

x0 = cosh ; x1 = sinh sin cos ;

x2 = sinh sin sin ; x3 = sinh cos .

The reader can verify that the hyperboliccoordinate expression of the squared line

element ds2 = (dx0)2 (dx1)2 (dx2)2 (dx3)2 is

ds2 =d2 2d2 + sinh )2d2 + sinh sin 2d2 ,showing that the hyperbolic coordinates too are orthogonal because the covariant metric

tensor is diagonal

g(x) = diag

1, 2, (sinh )2, (sinh sin )2

as well as its contravariant companion

g(x) = diag

1, 12

, 1(sinh )2

, 1(sinh sin )2

.

Thence the volume element is

g(x) dx4 =3(sinh )2 sin d d d d .

Defining x= {,,}, we can write x =x(, x) and the 4velocity along the directionofx as u x =x/. So we have uu = uu = (u0)2 |u|2 = 1.


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1

0 sinhr

3

1 ( ) ( )dV x dV x

1 ( )dV x

Figure 2: The future cone H+ of de Sit-

ter spacetime in hyperbolic coordinates. The

position of a point on a hyperboloid is deter-

mined by the kinematic time and the hy-

perbolic and Euler angles x= {,,}. Thespacetime volume element atx = {,,,}can be written as d4x = dV(x) d =

3dV1(x) d, wheredV1(x) is the volume el-

ement on the unit hyperboloid.

Fig.2 illustrates the future cone H+ together with the profiles of the unit hyperboloid

and of the hyperboloid at kinematic time . The 3D volumeelements of these hyperboloids

are respectively

dV1(x) (sinh )2 sin ddd ; dV(x) 3dV1(x) . (38)

Therefore, we can write g(x) d4x= 3dV1(x) d

dV(x) d.

The squared gradient of a scalar function f(, x) is

g(f)(f) ()2 12

|1f|2 , (39)

where

|1f|2 = (f)2 + (f)2

(sinh )2+

(f)2

(sinh sin )2. (40)

The covariant derivatives in hyperbolic coordinates act on a scalar function f(x) as

follows

Df =f; DDf=Df=f f , (41)where the Christoffel symbols constructed out from g. From the definition

=

12g

(g + g g), we obtain the following nonzero Christoffel symbols

011= ; 022 = sinh

2; 033= (sinh sin )2; 110=

220=

330=

1

; 332 = cot ;

221= 331= coth ;

233= sin cos; 122= sinh cosh; 133 = sinh cosh sin2 ;

[cf. Eqs.(A-2) in the Appendix of Part I].


22/39


The hyperbolic BeltramidAlembert operator acts on a scalar field f(x) as

D2f 1g

gg f

= 2f+

3

f 1

21 f , where (42)

1 f 1(sinh )2

(sinh )2 f

+

1

sin (sin f) +

1

(sin )22f

(43)

is the 3D Laplacian operator on the unit hyperboloid.

This operator has a complete set of orthonormalized eigenfunctions k(x) labeled by

the hyperbolic momentum eigenvalues

k=

{k,l ,m

}, where 0

k


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Let us define d = e(,x)d as the proper time element of the Cartan manifold and

dx= {d , dx} as the differentials of a new coordinate system x= { ,x}, calledconformalhyperbolic coordinates. Then, the square of the new line element takes the form

ds2 =d2 e2(x)2(x)d2 + sinh )2d2 + sinh sin 2d2 ,where (x) =[x(x)]. Here,x(x) {(x),x} are the hyperbolic coordinatesxas functionsof x. Accordingly, scalar functions of xwill be denoted by f(x), g(x) etc.

The conformalhyperbolic 4D volumeelement of the Cartan manifold and 3D volume

element of the spacelike surface at proper time are respectively represented by

d4x= dV(x) d =e4(x)3(x) dV1(x) d; dV(x) =e

3(x)3(x) dV1(x) . (44)

The latter can be thought of as the small region of the distorted hyperboloid H+(),

belonging to the Cartan manifold of the inflated de Sitter spacetime, which corresponds to

the volume elementdV(x) of the hyperboloidH+() described in the previous subsection.

Covariant conformalhyperbolic derivatives act on a scalar function f(x) as follows

Df= f e(x)f(x) ; DDf= Df= ff , (45)

where = +

+

g are the conformal Christoffel symbols con-

structed out from g=e2(x)g(x) [cf. Eq.(A-10) in the Appendix of Part I].

By replacing kinematic time with proper time in Eqs.(39) and (42), we find that

the other conformalhyperbolic differential operators act on f as follows

g

f

f

=

f2 1

e22(x)|1f|2 ,

D2f 1g

g gf

= 2f e

2

2(x)1f+ 3 ln

e22(x)

(f) =

2f e2

2(x)1f+ 3

1()

+

f 3 e2

2(x)

1 e22(x)1f . (46)

Here, = g are the contravariant components of gradient operator, the dot in

between the round brackets on the right stands for scalar product and () is the kinematic

time of the Riemann manifold that corresponds to proper time on the Cartan manifold.

The quantity

ln

e(x)(x)

=(x)

(x) + (x) = H0( ,x) ,


24/39


may be identified as the Hubble constant during the inflationary epoch at proper time

and hyperbolicangle x and measured in propertime units. Note that we cannot define

H0 as the expansion rate of the universe radius at proper time because the spacelike

surfaces of synchronized comoving observers are not spheres but hyperboloids.

Since we presume that, in the postinflationary era, (x) is virtually zero and the

gravitational forces surpass the inflationary repulsion, we expect that in the course of

time H0(x) joins smoothly to the Hubble constant HRW(x) of the FriedmanRobertson

Walker cosmology[22].

If the fundamental tensor is conformally flat or homogeneous, which means that the

Ricci curvature R of the Riemann manifold is zero or constant and that the universe is

homogeneous and isotropic, then , fand kinematic time depend only on proper time

and the conformalhyperbolic BeltramidAlembert operator simplifies to

D2f =2f+ 3

1()

+

f (47)

2

2L

2

1

( )

2 1 e d

1

2

1

( )

2 2( )

1 1

( )

( )

L e

L e

1L

Figure 3: Proper time and hyperbolic con-

formal coordinates on a conformally flat Car-

tan manifold. Passing from proper time 1to

proper time 2 the length Lof a comoving

radial interval, as well as the length unit, in-

creases by a factor of(2)e(2)/(1)e

(1).

In this case, as illustrated in Fig.3, passing from proper time 1 to proper time 2, the

ratio between the lengths L1, L2 of two radial intervals, comoving with the 3Dspaces

of synchronized observers, isL2

L1=

(2) e(2)

(1) e(1).

The measurement units of all physical quantities are instead preserved. Note that,

in hyperbolic coordinates, the lengths would instead increase by the ratio of 2/1 and

the measurement unit of a physical quantity of dimension n would increase by a factor of


25/39


en[(2)(1)]. As shown in Fig.3, all world lines of the inflated de Sitter spacetime are the

geodesics stemming from the originating event. All comoving reference frames, which on

the Riemann manifold are synchronized and at rest in a 3D space labeled by kinematic

time, are also synchronized and at rest in a 3D space of the Cartan manifold labeled by

proper time .

If the fundamental tensor is not conformally flat, the axial symmetry of the de Sitter

spacetime, in both the Riemann and Cartan representations, is no longer valid because

the world lines of heavy bodies also depend on the local details of gravitational field,

and ultimately on the time course of mass distribution. Therefore, neither the hyperbolic

coordinates nor the conformalhyperbolic coordinates can exactly represent the geometry

of the manifolds independently of the dynamical history of the system.

Since in this case, depends on both and x and is a function of both and x. In

fact, the equation holds

(, x) =

0

e( ,x)d .

2

1

,

2 1

xe d

2

1

x

x

. ( , ),x x

Figure 4: Proper time and hyperbolic confor-

mal coordinates on curved Cartan manifold. Theworld line of a particle comoving with the expand-

ing universe is a geodesics (x). This is uniquely

determined by hyperbolic and Euler angles x

measured at the origin of the future cone (cf. 8.3of Part I). Proper time , which is in general a

function of hyperbolic coordinates{, x}, labelsthe 3D spaces of synchronized comoving observers

on the Cartan manifold. Here, 1and 2stand re-

spectively for (1,x) and (2,x).

In Fig.4, the main features of the inhomogeneous de Sitter spacetime described by

conformal hyperbolic coordinates are summarily sketched. As the gravitational forces

take the upper hand on the repulsive forces generated by rapid scalefactor variations, the

3D spaces labeled by proper time become more and more irregular. Correspondingly,

the geodesics become more and more meandering and may even cross at certain points.


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4 Two equivalent representations of CGR action integrals

In the following, the representations of a physical system on the Riemann and Cartan

manifolds are respectively called the Riemann pictureand the Cartan picture.

The Riemann picture suits the need to describe the universe from the viewpoints

of synchronized comoving observers living in the postinflation era and provided with

kinematictime clocks. Looking back to the past, these observers interpret the events

occurred during the inflation epoch as subject to the action of the dilation field. In par-

ticular, all dimensional quantities, both geometric and physical, are imagined to undergo

considerable changes of scale. Possible nonuniformities of this process are explained as co-

ordinate distortions caused by the gravitational field. If this is negligible, or of sufficiently

small amplitude, the best description is obtained in terms of hyperbolic coordinates. Oth-

erwise, the metric tensor should be modified by the inclusion of appropriate terms aij(x),

as described in 7.3 of Part I.In contrast, the Cartan manifold picture suits the need to describe the universe as might

have be seen from within its expanding structure by hypothetical comoving and coeval

observers, at rest in the inertial reference frames of expanding 3D portions of spacetime,

equipped with synchronized propertime clocks and coexpanding measurement devices.

Such ideal observers would see the measurement units of all physical quantities to be

preserved. If the gravitational field has a sufficiently small amplitude, the best description

is obtained in terms of conformalhyperbolic coordinates. Otherwise, the fundamental

tensor should be modified by the inclusion of appropriate terms aij(x) as specified above.

These two descriptions, both of which are characteristic of CGR, fully express the

concept of relativity in regard to scale change. They are perfectly equivalent, since we can

move from one to the other by replacing all local quantities grounded, for instance, on the

Riemann manifold with their tilde counterparts grounded on the Cartan manifold, or vice

versa, and converge to each other in the postinflation era.

Let us be more precise about this point. As explained in 2.2 of Part I, the fundamentaltensor of the Cartan manifold is related to the metric tensor of the Riemann manifold by

equation g(x) = e2(x)g(x), where x are Riemann manifold coordinates. However,

in passing from the Riemann to the Cartan picture, it is more appropriate to replace

g(x) with g(x) g[x(x)], wherex(x) are the hyperbolic coordinates as functions of


27/39


conformalhyperbolic coordinates x, as described in 3.2. Conversely, in passing from theCartan to the Riemann picture, it is more appropriate to replace g(x) withe2(x)g(x)

by putting x= x(x). Here is the list of main replacements

x x(x); ; g(x) g(x); g(x) g(x); R(x) R(x);g(x)d4x

g(x)d4x; n(x)n(x) =en(x)n[x(x)]; (x)0; (48)

where is the partial derivative with respect to x, R(x) is the Ricci scalar and R(x)

its Cartan counterpart, n(x) is a field of dimension n, n(x) its tilde counterpart, and

(x) =[x(x)].

The replacement of R(x) with R(x) deserves a comment as it is a little tricky. The

point is that in the Riemann picture these Ricci scalars, which have dimension 2, are notrelated to each other by an equation of the form n(x) = e

n(x)n(x), as those relating

the sides of all other replacements listed in (48), namely by equation R(x) =e2(x)R(x).

They are instead related by the Eq.(A-16) described in the Appendix of Part I, namely,

R(x) =e2(x)R(x) 6 e(x)D2e(x) , (49)

where D2 is the BeltramidAlembert operator in hyperbolic coordinates. This depends

on the fact that R(x) is constructed out from the conformal Christoffel symbols

(x) = 1

2g(x)

g(x) + g(x) g(x)

=

(x) + e

(x) + e(x) ge(x) ,

where (x) = 12g

(x)

g(x) + g(x) g(x)

are the Christoffel symbols of

the Riemannianmanifold metric tensor [cf. Eqs.(A-1) and (A-10) in Appendix of Part I].

Note that, ifR vanishes, we are left with R(x) =6 e(x)D2e(x), which in generaldoes not vanish. One should bear this in mind to avoid the unpleasant pitfall coming from

the fact that, if the Riemann manifold is conformally flat then R(x) disappears from the

action integral. So, in passing from the Riemann to the Cartan picture, one should never

forget to preserve the term proportional to R(x). It is therefore advisable to never get rid

of R(x) in action integrals grounded on a conformally flat manifold and possibly put it

equal to zero only when spacetime integrations are performed.

Now let us consider the relationship between the Riemann and the Cartan picture

with regard to conformal invariance. Let A0 be a conformal-invariant GR actionintegral


28/39


29/39


preted as the effect of the spontaneous breakdown of local conformal symmetry consequent

to the fact that in the Riemann picture the vacuum expectation value of(x) is not zero.

Because, as explained in 1.7, this spontaneous breakdown confines the geometry tothe interior a de Sitter spacetime, we can restate the actionintegral equivalence in the

form

A=

H+

g(x) L(x)(x) d4x A=

H+

g(x)L0(x) d4x ,

where H+ and H+ stand for the interiors of the future cones respectively stemming from

x= 0 in the Riemann picture and from x= 0 in the Cartan picture.

Since, for the reasons discussed in 7.4 of Part I, we presume that matter distributionremains homogeneous and isotropic during the inflation epoch, we infer that in these

circumstances the two manifolds remain conformally flat or of constant curvature.

In which case, we have d = e()d. Consequently, the spacetimevolume elements

of the Riemann and Cartan manifold are respectively

g(, x) d4x= dV(x)d 3 dV1(x) d;g( ,x) d4x= dV(x) d e3(x)3() dV1(x) d;

where dV1(x) = (sinh )2 sin ddd, as described by Eqs.(38) and (44) of 3.2.

4.1 Riemann picture in hyperbolic coordinates

On account of the conclusions reached in the previous section, the general form of a

complete action integral of matter and geometry on the Riemann manifold spanned by

hyperbolic coordinates is

A= +

0

dV

g L(g,R,, ) dV(x)

with the following identifications:

- g = +h, with the hyperbolic metric tensor and h the gravitational

field, is the metric tensor of the Riemann manifold subjected to the unimodularity

conditiong= 1 (cf. 2.2 of Part I); ifhis negligible we put =g;

- R, the Ricci scalar tensor; if the gravitational field is zero or small, R too is;


30/39


- d, the kinematic time differential;

- dV(x) = 3dV1(x), the volume element at kinematic time,dV1(x) being the volume

element of the unit hyperboloid;

- (x) = 0e(x), the dilation field, where 0 =

6 MrP, MrP being the reduced

Planck mass;

- , matter fields of various types and dimensions.

This representation is particularly useful from a mathematical standpoint as, in the

absence of the cosmologicalconstant term descending from the unimodularity condition

discussed in 2.5 of Part I, the conformal invariance of the action is manifest.It is less useful for physical interpretations as both the geometric and the physical

quantities are subjected to local changes of scale. As described in 6 of Part I, is theghost scalar field of positive potential energy that, interacting with a zeromass physical

scalar field, mediate a positive energy transfer from geometry to matter while bounding

of the energy of the system from below (what saves Smatrix unitarity[23]).

4.2 Cartan picture in conformalhyperbolic coordinates

The general form of the approximate action integral on the Cartan manifold spanned by

conformal hyperbolic coordinates x= { ,x} has the general form

A=

+0

d

V

L(g,, R, 0, ) dV(x)

with the following identifications:

- g(x) = e2(x)g[x(x)], the fundamental tensor of the Cartan manifold, with g

andg as in the previous section, and h replaced byh ;

- dV(x) = e3(x)3()dV1(x), as described by Eq.44of 3.2;

- d=e(x)dis the propertime differential discussed in 3.2;

- R, the conformal Ricci scalar; if the fundamental tensor is conformally flat or ap-

proximately conformally flat, R does not vanish for the reasons explained in 4.


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- 0

6MrP, the conformalsymmetry breaking parameter;

- , matter fields of various types and dimensions.

This representation is particularly useful from a physical standpoint, as the action

integral has the formal properties of the corresponding action integral ofGR. In this case

the conformal symmetry appears manifestly broken by the dimensional constant 0, which

gives mass to all fields and introduces the correct gravitational coupling constant.

5 The Higgs field in CGR

In the next subsections we introduce in detail the Lagrangian formalism which is necessary

to describe the mechanism of matter generation in CGR. Namely, the interaction of the

scalar ghost field with a physical massless scalar field, which thereby becomes a Higgs field

of dynamically varying mass capable of mediating a huge energy transfer from geometry

to matter, as explained in6 of Part I. The motion equations, the EM tensors, along withtheir respective traces, and their separation into energydensity and pressure components,

are derived from the action integral in both the Riemann and the Cartan picture.

Unfortunately, in carrying out this task we encounter a difficulty which jeopardizes

the simplicity of the subject. In 2.4 of Part I, we stated the conformal invariance ofthe CGR actionintegral in the Riemann picture. But, in the subsequent 2.5, we con-tradicted this statement by showing that, if the conformal symmetry is spontaneously

broken, the conformal invariance is violated by a cosmological constant c imposed by

the unimodularity condition. Astronomical data confirm this theoretical prediction by

showing thatc is equivalent to a vacuum energydensity vacof about 1047 GeV4. This

means that the Ricci scalar R(x) is not zero on average but is affected by an additional

term R= c 1.685 1084 GeV2, where = 6/20 1.685 1037 GeV2 is thegravitational coupling constant. Because of its smallness, R has negligible effects on the

inflation process, but in the long run it becomes important on cosmological scales.

An annoying consequence of this state of affairs is that during the inflationary epoch

the de Sitter spacetime cannot be regarded as exactly flat. To avoid this, in the next

subsections we assume instead the conformal flatness and postpone dealing with the

cosmologicalconstant problem only in Part III.


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5.1 The Higgs field in the Riemann picture

In the Riemann picture, the interaction of a massless physical scalar field (x) with the

dilation field (x) is described by the action integral A= AM + AG, where

AM =

+0

d

V1

3

2

g

2

2

2H

2

2

20

2+

R

62

dV1; (50)

AG = +0

d

V1

3

2

g

+

R

62

dV1 , (51)

represent respectively the matter and the geometry action integrals. Here H = 126.5

GeV is the mass of the Higgs boson in the postinflation era (cf. 6 of Part I).At variance with the notation stated in 3.4 of Part I, the interaction term is

absorbed into AM. Here, g(f)(f) (f)2 2|1f|2, is the (adimensional)selfinteraction constant of and 20 = 6 M

2rP, where MrP is the reduced Planck mass,

henceH/0 21.11018. The expression ofAM is dictated by the condition that in theCartan picture the selfinteraction potential of (x) = e(x)(x) has the form

2

2H/22

, typical of Higgs Lagrangian densities. Recall that the presence of terms R2/6

andR2/6 ensures the conformal invariance of A provided that R itself is conformalcovariant ( 3.3, 3.4 of Part I), which means that a possible cosmological constant isexcluded from R.

The motion equations for and are respectively

D2 +

2 2H

2

2

20

R

6 = 0 , D2+

2H220

2

2H

2

2

20

R

6 = 0 , (52)

and the corresponding EM tensors of matter and geometry, obtained by variation ofAM

and AG with respect to g(x), are respectively

M =

g

2

(

2

2

2H

2

2

20

2+

16

gD2 DD2 + 2

6

Rg2 R

, (53)

G =

+g

2

16

gD

2 DD

2 2

6

Rg

2 R

. (54)

According to 2.5 of Part I, the gravitational equation of the Higgs field interacting withthe dilation field should be

(x) = M(x) +

G(x) =c g(x) , (55)


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but here we assume c= 0 for the reasons explained in the end of the previous section.

Contracting the indices of tensors (53), (54) and Eq.(55) withc= 0, and using motion

Eqs.(52) together with Eq.(55), we find the EMtensor traces

M =2H

2

2

20

2H2

2

20 2

=

2H2

e42H

2 e22

; G = M . (56)

If we regard the Higgs field as a fluid of energy density H, pressurepHand 4velocity

u = dx/ds = (x)(d/ds), where ds/d =

1 |u|2, we can put M =

H +

pH

uu gpH. Contracting the indices of this tensor with uu in the coordinates ofsynchronous comoving reference frames (where

|u|= 0), we obtain uuM

= H and,

contracting with g, the equation pH=

H M

/3. Hence, using Eq.(53), we obtain

H = 1

2

2 12

|1|2

+

4

2

2H

2

2

20

2+

1

2

1

132

1

2 +

2

6

R R

2

,

pH = 1

6

2 12

|1|2

+

12

2

2H

2

2

20

2 +

3 2H2

220

+

1

61

1

321

2 +2

18RR

2 ,where, R uuRand the formulas

uf=f , g

f

(f

= (2 1

2|1|2 ,

D2 uuDD

f=

3

f 1

21f ,

introduced in Eqs. (40) and (41) of 3.1were used.As explained in 7.3, 7.4 and 7.5 of Part I, the current uniformity and homogeneity

of matter on the large scale follows from the fact that, during the acute stage of inflation

and until the beginning of the photonproduction epoch the enormous viscosity and

repulsivity exerted by the dilation field in expansion forced the Higgs bosons created by the

energy transfer from geometry to matter to be at rest in the reference frame of synchronous

comoving observers. In these primeval conditions, the gas was cold and electrically neutral.

After this period we presumably have the following scenario. Towards the end of the

inflationary epoch, as fermions and antifermions are produced by Higgs boson decay and

fermionantifermion annihilations gave rise to an intense production of photons, the gas


34/39


became a very hot plasma. Then, because of its enormous temperature and pressure, its

uniformity and homogeneity distribution persisted for a certain period and terminated

only when the gravitational attraction took the upper hand on plasma pressure.

Finally, in the postinflation era, as (x) converged to 0 and CGR to GR, in conse-

quence of the spontaneous symmetrybreakdowns promoted by the corollary NG bosons

associated with the Higgs field (see 6 of Part I), of all the decay products descendingfrom the primeval Higgs bosons, only the more stable particles and particle aggregations

of the Standard Model survived.

5.2 The Higgs field in the Cartan picture

On the Cartan manifold, the action integral A= AM+AG of a scalar field interacting with

the dilation field in the future cone of the inflated de Sitter spacetime can be obtained

from that on the Riemann manifold, A = AM + AG described in the previous subsection

by performing the following replacements

A A; V V; dV(x) d dV(x)d; g(x) g(x) ;

; (x)

0; (x)

(x); R(x)

R(x) , (57)

where xare conformalhyperbolic coordinates (cf. 3.2) andare partial derivatives withrespect to x. In regard to the last replacement, it is important to notice that, providing

that the cosmological constant c coming from the unimodularity condition be zero, R is

related to R by equation

R(x) =e2(x)R(x) 6 e(x)D2e(x) = 20

2(x)R(x) 6

(x)D2(x) , (58)

where D2 is the BeltramidAlembert operator in hyperbolic coordinates described by

Eq.(A-8) in the Appendix of Part I.

Substitutions (57) yield the action integrals of matter and geometry in the future cone

AM =

+0

d

V

1

2

g

2

2

2H

2

2+

R

62

dV(x) ;

AG = 20

12

+0

d

V

R dV(x) ,

with d and dV(x) as defined in 4.2.


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The partition ofAinto AM and AG is performed in such a way that the potential energy

of the Higgs field is zero for = H/2. Note that the manifest conformal invarianceofA has disappeared and an explicit breakdown of conformal symmetry instead appears.

For the sake of clarity, let us expand the kinetic term of the matter Lagrangian density

as described by the first of Eqs. (46) in 3.2

g()() = ()2 1

2()e2|1|2 .

The motion equation for is thus

D2 +

2 2H

2

R

6= 0 , (59)

where, on account of second of Eqs.(46) in 3.2,we have

D2= 2 + 3

ln

() e

3

1

() e1)

3() e3 1

2() e2. (60)

Correspondingly, the EM tensor decomposes into

M = 1

2

g2

g

+ g

4

2

2H

2

2+

16

gD

2 DD

2 + 26

R1

2gR

; (61)

G = 206

R1

2gR

. (62)

Hence, always assuming c= 0, we obtain the CartanEinstein gravitational equation

R12

gR= M, where

1

=

206

=M2rP. (63)

Then, by contraction with g and using motion equation (59), we obtain from Eqs. (61),

(62) and the gravitational equation in the Cartan picture

M

+

G

= 0. The traces ofM and

G are

M = 2H

2

2H2

2

; G = 1

R= M ; R= 3

2H

20

2

2H

2

. (64)

Inserting the last of these into Eq.(59), we obtain the motion equation

D2 +

2H

2 20

2

2H

2

= 0 , (65)


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which differs from Eq.(59) by the simple replacement of selfcoupling constant with

= 2H/220. Since H is about 126.5 GeV and is expected to be in the orderof magnitude of 101, while 0 6 1018GeV, we see that the selfcoupling constantcorrection coming from R is absolutely negligible. Note that the asymptotic value (x)

is still H/

2. However, if we add to M a contribution coming from other matter

fields, Eq.(65) becomes

D2 +

2 2H

2

+

20= 0 , (66)

clearly exhibiting a dependence of the asymptotic value of on

.Noting that the minimum of the potential energy density of (x) occurs at =

H/

2, it is more suitable to define the true Higgsfield as (x) = (x)H/

2,

which in the linear approximation satisfies the motion equation

D2+ 2H H

2 20 .

This shows that (x) behaves as a potential source of Higgs bosons.

The expressions for energy density and pressure in the comoving synchronous coordi-

nates of the Cartan picture are obtained as in the previous section, which yields

H = uuM=

1

2

2 12(x)

|1|2

+

4

2

2H

2

2+

1

2

1(x)

+

13 2(x)

1

2 +

2

6

uuR R

2

;

pH = H M

3 =

1

2

2 12(x)

|1|2

+

12

2

2H

2

2 +

3 2H2

+

1

2

1(x)

+

13 2(x)

1

2 +

2

18

uuR R

2

;

where , u = dx/d = e(x)u is the velocity fourvector in comoving syn-chronous coordinates, (x) is the value of kinematic time as a function of conformal

hyperbolic coordinates x and Eq.(47)

Action integrals AM, AG simplify considerably if we assume that the geometry is con-

formally flat and does not depend on spacelike coordinates x. We have already seen

that on the Riemann manifold this implies R = 0. Correspondingly, on the Cartan mani-

fold, the dilation field is a constant while and R depend on only. In this case, Atakes


37/39


the form

A= AM + AG =

+0

d

V

1

2

2 2

2

2H

2

2+

R

6

2 20

dV(x) .

Since depends only on , D2 takes the form stated by Eq.(47) of 3.2 and thereforeEq.(60) simplifies to

D2= 2 + 3 1

()+

.

Consequently, using the last of Eqs. (64) and putting () = s1() s(), motion

equation (59) simplifies to

2() + 3

1()

+

()+

2H220

2() 2H

2

() = 0 , (67)

which, because of the negligibility of2H/220 compared to , can be safely replaced by

2() + 3 1

()+

() +

2()

2H

2

() = 0 . (68)

Here, the equation for the dilation field is only apparently disappeared. Actually, it is

replaced by the relation ofR, as given by the last of Eqs.(64), with scale factor s() e(),where () =[()] is given by Eq.(58) with R(x) = 0, i.e.,

R() = 6s()

D2s() = 6s()

D2s() ,

Thus, by using the last of Eqs.(64), we obtain the following motion equation for the scale

factor in conformal hyperbolic coordinates

D2s() =2s() + 3 1

()+

s()

s

s() =

2H220

2H2

2()

s() . (69)

Since the boundary conditions for (x) assumed in 8.1 of Part I require that () =() converges to 1 for

or, which is the same, for

, we see from Eq.(69) that

to this limit () e()() converges to () =() =H/2, which is consistentwith Eqs. (67) and (68). Furthermore, towards the end of the inflationary epoch, all

the equations that govern the dynamics of the matter fields grounded on the Cartan

manifold converge to the EinsteinHilbert equations of matterfield dynamics grounded

on the Riemann manifold, in which the conformal symmetry appears explicitly broken. In

other terms, in the post inflation era all physical properties depending on the underlying

conformal symmetry of CGR dissolve and all matter fields follow the laws of standard GR.


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References

[1] Nobili, R.: The Conformal Universe I: Physical and Mathematical Basis of Conformal

General relativity.arXiv:1201.2314v2 [hep-th](2013).

[2] Zeeman, E.C.: Causality implies the Lorentz Group.J. Math. Phys.5, 490493 (1964).

[3] Alexandrov, A.D. and Ovchinnikova, V.V.: Notes on the foundations of the theory

of relativity. Vestnik Leningrad Univ. 11, 94110 (1953).

[4] Alexandrov, A.D.: A contribution to chronogeometry.Can. J. of Math.19, 11191128

(1967).

[5] Borchers, H.J. and Hegerfeldt, G.C.: The structure of spacetime transformations.

Commun. Math. Phys. 28, 259266 (1972).

[6] Gobel, R.: Zeeman Topologies on Space-Times of General Relativity Theory. Comm.

Math. Phys. 46, 289307 (1976).

[7] Hawking, S.W., King, A.R. and McCarthy, P.J.: A new topology for curved space

time which incorporates the causal, differential, and conformal structures. J. Math.

Phys. 17, 174181 (1976).

[8] Popovici, I. and Constantin Radulsecu, D.: Characterization of conformality in a

Minkowski space. Ann. Inst. Henri Poincare. 25, 131148 (1981).

[9] Nobili, R.: Conformal covariant Wightman functions.Il Nuovo Cimento.13, 129143

(1973).

[10] Cartan, E.: Sur les espaces conformes generalises et lUniverse optique. Comptes

Rendus de lAcademie des Sciences. 174, 734736 (1922).

[11] Haag, R.: Local Quantum Physics: Fields, Particles, Algebras. pp 1317, Springer

Verlag, Berlin (1992).

[12] Mack, G. and Salam, A.: FiniteComponent Field Representations of the Conformal

Group. Annals of Physics. 53, 174202 (1969).
http://lanl.arxiv.org/abs/1201.2314http://lanl.arxiv.org/abs/1201.2314


39/39


[13] Fubini, S.: A New Approach to Conformal Invariant Field Theories. Il Nuovo Ci-

mento. 34A, 521554 (1976).

[14] Weinberg, S.: Gravitation and Cosmology. pp. 365373. John Wiley & Sons, NY

(1972)

[15] Umezawa, H., Matsumoto, H. and Tachiki, M.: Thermo Field Dymamics and Con-

densed States. NorthHolland Pub. Co., Amsterdam (1982)

[16] Englert, F. and Brout, R.: Broken Symmetry and the Mass of Gauge Vector Mesons.

Phys. Rev. Lett. 13, 321323 (1964).

[17] Higgs P.W.: Broken Symmetries and the Masses of Gauge Bosons. Phys. Rev. Lett.

13, 508509 (1964).

[18] Kibble, T. W. B.: Symmetry Breaking in NonAbelian Gauge Theories. Phys. Rev.

135, 15541561 (1967).

[19] Moschella, U.: The de Sitter and antide Sitter Seightseeing Tour. Seminaire

Poincare. 1, 112 (2005).

[20] Englert, F., Truffin, C. and Gastmans, R.: Conformal invariance in quantum gravity.

Nuclear Physics B. 117, 407432 (1976).

[21] Grib, A.A., Mamayev, S.G. and Mostepanenko, V.M.: Particle Creation from Vacuum

in Homogeneous Isotropic Models of the Universe. Gen. Rel. and Grav. 7, 535547

(1976).

[22] Gorbunov, D.S. and Valery, A.R.: Introduction to the Theory of the Early Universe:

Cosmological Perturbations and Inflationary Theory. World Scientific (2011).

[23] Ilhan, I.B, and Kovner, A.: Some Comments on Ghosts and Unitarity: The Pais-

Uhlenbeck Oscillator Revisited.arXiv:1301.4879v1 [hep-th] (2013).

[24] Eisenhart, L.P.: Riemannian Geometry. p. 84, Princeton University Press (1949).
http://lanl.arxiv.org/abs/1301.4879http://lanl.arxiv.org/abs/1301.4879

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The Conformal Universe II