Remarks on Weyl geometry and quantum mechanics

0.3 MB·

by  Robert Carroll

REMARKS ON WEYL GEOMETRY AND QUANTUM
MECHANICS
ROBERTCARROLL
UNIVERSITYOF ILLINOIS,URBANA,IL 61801
Abstract. Ashortsurveyofsomematerialrelatedtoconformalgen-
eralrelativity(CGR),integrableWeylgeometry,andDirac-Weyl(DW)
8
theoryisgivenwhichsuggeststhatCGRisessentiallyequivalenttoDW
0
withquantummasscorrespondingtoconformalmass;furthermorevar-
0
ious actions can be reformulated in terms of the quantumpotential.
2

n
a
J
3 Contents

] 1. INTRODUCTION 1
c
2. THE DIRAC WEYL APPROACH 5
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- 3. CONNECTIONS TO ELECTROMAGNETISM 8
r
g 4. DIVERSE DEVELOPMENTS IN WEYL GEOMETRY 10
[ 5. BOHMIAN QUANTUM GRAVITY 12

6. DARK ENERGY AND DARK MATTER 18
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v 7. SOME SUMMARY REMARKS 22
1 References 24
2
9
3
.
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1. INTRODUCTION
0
7
This is primarily a survey and review article, the main point of which
0
is to make precise an observation of Bonal, Quiros, and Cardenas in [13]
:
v
relating conformal mass to quantum mass. We acknowledge with thanks
i
X the valuable comments of a reviewer, some of which are partially included
r here, and some typos have been corrected. A revised and shortened version
a
is in preparation.
ThuswewillgathertogethersomeformulasinvolvingWeylgeometryand
Weyl-Dirac theoryinvariouswaysandshowconnectionstotheSchro¨dinger
equation (SE)and Klein-Gordon equation (KG). Many aspects of this have
been sketched in [19, 20,21,23](based inparton [6,7,8,24,25, 26, 43,44,
Date: January,2008.
email: [email protected].
1
2 ROBERTCARROLLUNIVERSITYOFILLINOIS,URBANA,IL61801
65, 68, 71]) and related it to Ricci flow in [22] and we now want to deepen
our understanding and rephrase some material with some expansion. The
presentations in [29, 63, 64] are very extensive so we will begin with Is-
raelit [43, 44] to get started (cf. also [52] for semi-Riemannian geometry).
We recall first the standard transformations of contravariant and covariant
vectors via x x¯ in the form
→
∂x¯ν ∂xσ
(1.1) Tν T¯ν = Tσ ; T T¯ = T
→ ∂xσ µ → µ σ∂x¯µ
For parallel transport of a vector along a curve from xν xν +dxν one
→
sets
1
(1.2) dTµ = TσΓµ dxν; Γλ = gλσ[∂ g +∂ g ∂ g ]
− σν µν 2 ν µσ µ νσ − σ µν
where Γλ is the standard Christoffel symbol. Some calculation then shows
µν
that dT = 0 under parallel transport and one obtains also (1A) dT =
µ
T Γσ dxν. In Riemannian geometry one defines also covariant derivatives
σ µν
via
(1.3) Tµ = ∂ Tµ+TσΓµ ; T = ∂ T T Γσ
∇ν ν σν ∇ν µ ν µ− σ µν
From the above one determines easily that (1B) g = 0 = gµν and
λ µν λ
∇ ∇
if the vector Tµ is parallel transported around an infinitesimal closed par-
allelogram it follows that ( ) ∆Tλ = TσRλ dxµdxν where the Riemann-
• σµν
Christoffel tensor is
(1.4) Rλ = ∂ Γλ +∂ Γλ Γα Γλ +Γα Γλ
σµν − ν σµ µ σν − σµ αν σν αµ
On the other hand from dT = 0 the length of the vector ∆T = 0 and one
finds
(1.5) T T = T Rσ
∇ν∇µ λ−∇µ∇ν λ σ λµν
Contracting Rσ gives R = Rλ and the curvature scalar R = gµνR .
µνλ µν µνλ µν
For Weyl geometry one has a metric tensor g = g and a length
µν νµ
connection vector w while both the direction and the length of a vector
µ
changeunderparalleltransport. Thusifavectorisdisplacedbydxν onehas
(1C) dTµ = TσΓˆµ dxν and its length is changed via (1D) dT =Tw dxν.
σν ν
−
Onecan alsowrite(1E) d(T2)= 2T2w dxν andforthistoagree with(1C)
ν
and T2 = g TµTν for arbitrary Tµ and dxν requires
µν
(1.6) g Γˆσ +g Γˆσ = ∂ g 2g w
µσ νλ νσ µλ λ µν − µν λ
Following Weyl one assumes that the connection is symmetric (i.e. Γˆλ =
µν
Γˆλ ) leading to
νµ
(1.7) Γˆλ = Γλ +g wλ δλw δλw
µν µν µν − ν µ− µ ν
REMARKS ON WEYL GEOMETRY AND QUANTUM MECHANICS 3
Using this Weyl connection Γˆλ one forms a covariant Weyl derivative via
µν
(1.8) ˆ Tµ = ∂ Tµ+TσΓˆµ ; ˆ T = ∂ T T Γˆσ
∇ν ν σν ∇ν µ ν µ− σ µν
One can also write then
(1.9) ˆ Tµ = Tµ+Tσ[g wµ δµw δµw ]
∇ν ∇ν σν − ν σ − σ ν
This all generalizes to tensors in an obvious manner and in particular
(1.10) ˆ g = g g Γˆσ g Γˆσ = 2g w
∇λ µν ∇λ µν − σν µλ− µσ νλ µν λ
Similarly one has (1F) Wgµν = 2gµνw .
∇λ − λ
In particular consider a spacetime with a symmetric metric tensor g
µν
and an asymmetric connection Γ˜λ defining a parallel displacement (cf.
µν
(1C)) (1G) dTµ = TσΓ˜µ dxν. The connection can be split into three
σν
−
parts, the Christoffel symbol, the contorsion tensor, and the nonmetricity,
namely
1
(1.11) Γ˜λ = Γλ +Cλ + gλσ[Q +Q Q ]
µν µν µν 2 νµσ µνσ − λµν
The contorsion tensor Cλ may be expressed in terms of the torsion tensor
µν
Γ˜λ = (1/2)[Γ˜λ Γ˜λ ] via (1H) Cλ = Γ˜λ +gλσg Γ˜κ +gλσg Γ˜κ .
[µν] µν − νµ µν [µν] µκ [νσ] νκ [µσ]
The nonmetricity can in turn be represented as minus the covariant deriv-
ative of the metric tensor with respect to Γ˜λ , namely
µν
(1.12) Q = ∂ g +g Γ˜σ +g Γ˜σ
λµν − λ µν σν µλ σµ νλ
ComparingwiththeWeylconnection(1.7)onecansaythattheWeylgeom-
etry is torsionless with nonmetricity (1I) Qˆ = 2g w (in Riemannian
λµν µν λ
−
geometry both torsion and nommetricit vanish). Further in the Weyl ge-
ometry for parallel transport of a vector Tµ around an infinitesimal closed
parallelogram one has (1J) ∆Tλ = TσKλ dxµδxν and using (1D) there
σµν
results for the total length change (1K) ∆T = TW dxµδxν. In (1J) Kλ
µν σµν
is a curvature tensor similar to the Riemannian one but with Γλ replaced
µν
by the Weyl connection Γˆλ . In (1K) one has the Weyl length curvature
µν
tensor (1M) W = w w = ∂ w ∂ w and one sees that in
µν ν µ µ ν ν µ µ ν
∇ −∇ −
Weyl geometry ∆T = 0 unless W = 0.
µν
6
Transporting a vector of length T around a closed loop leads to T =
new
T + TW dSµν where S is the area confined by the loop and dSµν a
S µν
suitable area element. If one assumes that under Weyl gauge transforma-
R
tions (WGT) the components Tµ remain unchanged the length changes via
T T˜ = exp(λ)T where λ(xν) is a differentiable function of the coordi-
→
nates. One has then
(1.13) g g˜ = e2λg ; gµν g˜µν = e−2λgµν
µν µν µν
→ →
4 ROBERTCARROLLUNIVERSITYOFILLINOIS,URBANA,IL61801
In order to have dT = Tw dxν for transformed quantities one has to take
ν
a WGT law (1N) w w˜ = w + ∂ λ. Further a quantity is said to
ν ν ν ν
→
be gauge covariant if under a WGT it is transformed via (1O) ψ ψ˜ =
→
exp(nλ)ψ and n is called the Weyl power of ψ; one writes Π(ψ) = n. Thus
Π(g ) = 2, Π(gµν) = 2, Π(T) = 1, Π(√ g) = 4, etc. If Π is zero one
µν
− −
hasagaugeinvariantquantity andmakinguseof (1.13)and(1N)itfollows
that (1P) Π(Γˆλ ) = 0; quantities are called in-invariant if they are both
µν
coordinate and gauge invariant.
REMARK 1.1. The Dirac modification of Weyl theory involves a
scalar functionβ(xν)satisfying(1Q) β β˜= exp( λ)β (i.e. Π(β) = 1).
→ − −
Since this provides a 1-1 relation between λ and β one calls β the Dirac
gauge function (also Dirac field). Given that λ is intrinsic to Weyl space-
time one can say that β is also a part of the geometric structure (cf. also
[28, 45, 61]). (cid:4)
Nowsomecurvaturetensorsaredevelopedstartingwith(1.4)-(1.5). Con-
tracting Rˆλ one obtains the Ricci tensor
σµν
(1.14) Rˆ = Rˆλ = ∂ Γλ +∂ Γλ Γα Γλ +Γα Γλ
µν µνλ − λ µν ν µλ− µν αλ µλ αν
This leads to the Ricci curvature R = gµνR and the Einstein tensor
µν
(1R) Gν = Rν (1/2)δνR which satisfies the contracted Bianchi iden-
µ µ − µ
tity (1S) Gν = 0. For the Weyl counterpart we have following (1.3)
∇ν µ
(1T) ˆ ˆ T ˆ ˆ T = T Kσ with Weylian curvature tensor Kσ
∇ν∇µ λ −∇µ∇ν λ σ λµν λµν
given by
(1.15) Kλ = ∂ Γˆλ +∂ Γˆλ Γˆα Γˆλ +Γˆα Γˆλ
σµν − ν σµ µ σν − σµ αν σν αµ
Contracting this and using (1.7) yields
(1.16) K = Kλ =
µν µνλ
R g wλ+ w 3 w +2g wλw 2w w
µν µν λ µ ν ν µ µν λ µ ν
− ∇ ∇ − ∇ −
Consequently
(1.17) K K = 4( w w ) = 4W
µν νµ µ ν ν µ µν
− ∇ −∇ −
Contracting in (1.15) one obtains now (1U) Kλ = 4W and from
λµν − µν
(1.16) there is a scalar (1V) K = gµνKλ = R 6 wλ +6wλw . One
µνλ − ∇λ λ
forms also (1W) W2 = W Wλσ and considers a Weyl space with torsion.
λσ
In this case the connection can be written in the form
(1.18) Γ¯λ = Γλ +g wλ δλw δλw +Cλ
µν µν µν − ν µ− µ ν µν
and one defines a covariant derivative (1X) ¯ T = ∂ T T Γ¯σ with
∇ν λ ν λ− σ λν
(1.19) ¯ ¯ T ¯ ¯ T = T K¯σ 2¯ T Γ¯σ
∇ν∇µ λ−∇µ∇ν λ σ λµν − ∇σ λ [µν]
REMARKS ON WEYL GEOMETRY AND QUANTUM MECHANICS 5
InsummaryonenotesthatinRiemanniangeometryunderparalleltrans-
portthedirectionofavectorischangedbutnotitslength. HoweverinWeyl
geometry both the length and direction are changed. In theories based on
Weyl geometry the equations and laws have to be covariant with respect
to both coordinate transformations (CT) and Weyl gauge transformations
(WGT). Both Weyl connections Γˆ and Γ¯ are gauge invariant so the corre-
spondingcontractions and curvatures are covariant with respect to CT and
WGT.
2. THE DIRAC WEYL APPROACH
First, following Israelit [43], we add a few more remarks about Weyl
geometry. Look first at the Weyl geometry without torsion involving Γˆλ
µν
with nonmetricity Q as in (1.12). There are two other possible choices
λµν
of connection of interest here.
(1) For 2Γˆλ = Γλ +g wλ+δλw δλw with (1C) (cf. (1.7)) there
µν µν µν µ ν − ν µ
results d T = Tw dxν and 2Q = 2g w (cf. (1I)) leading to
2 ν λµν µν λ
−
a torsion tensor 2Γˆλ = δλw δλw
[µν] µ ν − ν µ
(2) Anotherconnectionispossiblewiththeform3Γˆλ = Γλ +(1/2)2Cλ
µν µν µν
where2Cλ = 2g wλ 2δλw isthecotorsiontensorfor2Γˆλ . Here
µν µν − ν µ µν
one has d T = 0 = ∆ T and vanishing nonmetricity, but there is
3 3
torsion.
Now the Weyl-Dirac theory was developed in various ways following [28,
43, 44, 61, 65] (cf. also [19, 23] for a survey) and we follow first [43] here.
Thus one can deal with an action integral (cf. [43] for details)
(2.1) I = [WλµW β2R+σβ2wλw +(σ+6)∂ β∂λβ+
λµ λ λ
−
Z
+2σβ∂ βwλ+2Λβ4+L ]√ gd4x
λ M
−
(Λ is a cosmological constant). Note here (L = L )
G geom
(⋆) β2K = β2R+6β2 wλ 6β2wλw
λ λ
− − ∇ −
(⋆⋆) L = WλσW β2R+6β2 wλ 6β2wλw +k(∂ β+βw )2+2Λβ4
G λσ λ λ λ λ
− ∇ −
However β2 wλ = (β2wλ) 2β∂ βwλ and the first term integrates
λ λ λ
∇ ∇ −
out, leading to (2.1). Then to avoid Proca terms one sets σ = k 6 = 0
−
leading to
(2.2) I = [WλµW β2R+6∂ β∂λβ+2Λβ4 +L ]√ gd4x
λµ λ M
− −
Z
(cf. [43]for details). Here β is an additional dynamicalvariable (field)with
Π(β) = 1 and one is interpreting w as an EM field potential with W
µ λµ
−
6 ROBERTCARROLLUNIVERSITYOFILLINOIS,URBANA,IL61801
as the EM field strength. The equations of the theory follow from δI = 0
and first varying g gives
µν
1 8π
(2.3) Gµν = Rµν gµνR = (Mµν +Tµν)+
− 2 −β2
2 1
+ (gµν α β ν µβ)+ (4∂µβ∂νβ gµν∂ β∂αβ) gµνΛβ2
β ∇ ∇α −∇ ∇ β2 − α −
Onehasenergy-momentumtensors(2B) Mµν = (1/4π)[(1/4)gµνWλσW
λσ
−
WµλWν] for the EM field and (2C) Tµν = [(1/8π√ g)[δ(√ gL )/δg ]
λ − − M µν
for the matter. Varying w gives the Maxwell equation (2D) Wµν =
µ ν
∇
4πJµ where Jµ = (1/16π)(δL /δw ) and varying in β one obtains
M µ
1 δL
(2.4) βR+6 λ β 4Λβ3 = 8πB; B = M
λ
∇ ∇ − 16π δβ
(B is a ”charge” conjugate to β). Taking the divergence of (2D) gives the
conservation law for electric charge (2E) Jλ = 0 and taking the diver-
λ
∇
gence of (2.3) one gets (using the Bianchi identity) the equations of motion
for the matter (2F) Tµν T(∂µβ/β) = WµνJ .
ν ν
∇ − −
Consider now the matter part of the in-invariant action (2.2), namely
(2G) I = L (√ gd4x which depends on g , w , β, and perhaps ad-
M M µν µ
−
ditional dynamical variables ψ . Since the latter are not present in L one
µ M
R
has (2H) δ[√ gL ]/δψ = 0 so
M µ
−
(2.5) δI = 8π [Tµνδg +2Jµδw +2Bδβ]√ gd4x
M µν µ
−
Z
Now carry out a CT with an arbitrary infinitesimal vector ηµ, i.e. xµ
→
x¯µ = xµ+ηµ; this yields
(2.6) δg = g ηλ+g ηλ; δw = w ηλ+ w ηλ; δβ = ∂ βηλ
µν λν µ µλ ν µ λ µ λ µ λ
∇ ∇ ∇ ∇
Putting this in (2.5) and integrating by parts gives
(2.7) δI = 16π [ Tλ w Jλ JλW +B∂ β]ηµ√ gd4x
M −∇λ µ − µ∇λ − µλ µ −
Z
HoweversinceI isanin-invariantitsvariationwithrespecttoCTvanishes
M
so from (2.8) one obtains the conservation law (2J) Tλ + w Jλ +
∇λ µ µ∇λ
JλW B∂ β = 0. Next consider a WGT with λ(xµ) infinitesimal. Using
µλ µ
−
(1.13), (1N), and (1Q) one has
(2.8) δg =2g λ; δβ = βλ; δw = ∂ λ
µν µν µ µ
−
Putting these in (2.5) and integrating by parts gives then
(2.9) δI = 16π [T Jλ Bβ]λ√ gd4x
M λ
−∇ − −
Z
REMARKS ON WEYL GEOMETRY AND QUANTUM MECHANICS 7
Given the invariance of (2.5) under WGT one has from (2.9) (2K) T =
Jλ+βB and in view of (2E) this leads to (2L) T =βB. Further using
λ
∇
(2E) and (2L) in (2J) yields (2M) Tλ T(∂ β/β) = JλW which is
∇λ µ − µ − µλ
identical with the equations of motion for the matter (2F). Going back to
the field equations (2.3) one contracts to get
8π 6
(2.10) Gσ = T + λ β 4Λβ2
σ −β2 β∇ ∇λ −
Putting (2L) into (2.10) and using Gσ = Rσ = R one obtains again
σ − σ
the equation for the β field (2.4). Consequently (2.4) is rather a corollary
rather than an independent field equation and one is free to choose the
gauge function - this freedom indicates the gauge covariant nature of the
Weyl Dirac theory.
We gonowtothethirdpaperinIsraelit[44]andlookat integrableWeyl-
Dirac (Int-W-D) theory where w = ∂ w and W = 0. Thus the action
µ µ µν
in (2.1) has no WλµW term and one can introduce an ad hoc gradient
λµ
(2N) b = ∂ β/β with W = w +b and W = w +b (W will be gauge
µ µ µ µ µ
invariant). Replace now k 6= 16πσˆ and varying the action in (2.1) with
−
respect to w yields (2O) 2 (σˆβ2Wν) = S where S is called a Weyl scalar
ν
∇
charge (2P) 16πS = δL /δw. Note here that (2.3) can be rewritten as
M
µ
T 1
(2.11) Gν = 8π ν +16πσˆ WνW δνWσW +
µ − β2 ν − 2 µ σ
(cid:20) (cid:21)
+2(δν bσ bν)+2bνb +δνbσb δνβ2Λ
µ∇σ −∇µ µ µ σ − µ
and (2.4) as
(2.12) Rσ +k( bσ +bσb ) = 16πσˆ(wσw wσ)+4β2Λ+8πβ−1B
σ ∇σ σ σ −∇σ
B is defined as before and looking at CT and WGT transformations as
above leads to
(2.13) Tλ Sw βBb = 0; S+T βB = 0 Tλ Tb = SW
∇λ µ − µ− µ − ⇒ ∇λ µ − µ µ
Going back to (2.11) one can introduce the energy momentum density ten-
sor of the W field
µ
1
(2.14) 8πΘµν = 16πσˆβ2 gµνWλW WµWν
λ
2 −
(cid:20) (cid:21)
Makinguseof(2O)oneobtains(2Q) Θλ Θb = SW andfrom(2.13)
∇λ µ− µ − µ
and (2Q) there results (2R) (Tλ +Θλ) (T +Θ)b = 0. This gives a
∇λ µ µ − µ
mini-survey of the Int-W-D framework. A fascinating W-D cosmology was
developed in [43, 44] butwe omit this here (cf. also [24, 26, 29, 30, 63, 64]).
There are many general theories of relativity involving spin, torsion, non-
metricity, etc. and we refer to [12, 40, 57] for information (with apologies
for omissions).
8 ROBERTCARROLLUNIVERSITYOFILLINOIS,URBANA,IL61801
3. CONNECTIONS TO ELECTROMAGNETISM
We go first to [43, 44, 46] and recall the Maxwell equations in the form
(3.1) E =4πρ; H = 4πj+∂ E; E = ∂ H; H = 0
t t
∇· ∇× ∇× − ∇·
Theasymmetryregardingelectricandmagneticcurrentsisseenmoreclearly
in writing
ǫµναβ
(3.2) Fµν = 4πJµ; F˜µν = 0; F˜µν = F
ν ν αβ
∇ ∇ −2√ g
−
where ǫµναβ is the completely antisymmetric Levi-Civita symbol. From
this one can write (3A) F = A A which in current free regions
µν ν µ µ ν
can be written in Lorentz gaug∇e as (−3B∇) νAµ +AνRµ = 0. In order
ν ν
∇ ∇
to build up a theory admitting intrinsic magnetic and electric currents and
massive photons one asumes σ = k 6 = 0 which for k < 6 leads to a
− 6
Proca field that can be interpreted as an ensemble of massive bosons with
spin one (i.e. massive photons). Thus take a symmetric metric g = g ,
µν νµ
a Weyl vector w , a Dirac gauge function β, and a torsion tensor Γ¯λ as
µ [µν]
in (1.18) with contorsion tensor
(3.3) Cλ = Γ¯λ gλβg Γ¯σ gλβg Γ¯σ
µν [µν]− σµ [βν]− σν [βµ]
and under WGT one has
(3.4) g g˜ = e2λg ; w w˜ = w +∂ λ; β β˜= e−λβ
µν µν µν µ µ µ µ
→ → →
and one assumes (3C) Γ¯λ Γ˜λ = Γ¯λ with a curvature formula as in
[µν] → [µν] µν
(1.15), namely
(3.5) Kλ = ∂ Γ¯λ +∂ Γ¯λ Γ¯α Γ¯λ +Γ¯α Γ¯λ
µνσ − σ µν ν µσ − µν ασ µσ αν
Further there is a formula (1.19) describing geometrical properties invoked
viatorsionwhere(via ¯ g = 2g w )onecanexpressthetorsionalcur-
λ µν µν λ
vature term as (3D) ∇2¯ T Γ¯−α a¯ W Γ¯α where W is the Weyl
− ∇α µ [νσ] ∼ ∇α µλ [νσ] µν
lengthcurvaturetensor. Theactionintegral(2.1)isthenreplacedbyathree
line expression (cf. [43], p. 63) with dynamical variables Γ¯λ , w , g ,
[µν] µ µν
and β (a dependence of L on additional matter field variables is not ex-
M
cluded).
Now varying w in this action gives
µ
β2
(3.6) Wµν 2 Γ¯α[µν] = (k 6)Wµ +2β2Γ¯α[µ+4πJµ
∇ν − ∇α 2 − α]
h i
where (3E) W = w + ∂ log(β) and under WGT (3F) W˜ = W with
µ µ µ µ µ
16πJµ = δL /δw (W is called the gauge invariant Weyl connection
M µ µ
vector and Jµ = 0 if L does not depend on w ). Variation of the action
M µ
REMARKS ON WEYL GEOMETRY AND QUANTUM MECHANICS 9
with respect to Γ¯λ gives another field equation (cf. Israelit [43], p. 64)
[µν]
involving 16πΩ[µν] = δL /δΓ¯λ whose contraction yields
λ M [µν]
(3.7) Wµν = 3β2Wµ+2β2Γ¯ν[µ 4πΩ[µν]
∇ν − ν] − ν
One regards Jµ as the electric current vector and the magnetic current
[µν]
densityvector willbeexpressedintermsofΩ . Onehastwoconservation
λ
laws following from (3.6) and (3.7), namely
(3.8) (k 6) (β2Wµ)+8π Jµ+4 (β2Γ¯ν[µ) = 0
− ∇µ ∇µ ∇µ ν]
(3.9) 3 (β2Wµ)+4π Ω[µν] 2 (β2Γ¯ν[µ)= 0
∇µ ∇µ ν − ∇µ ν]
Further calculation leads to the introduction of a strength tensor for the
EM fields
(3.10) Φ =W 2 Γ¯α W W 2 Γ¯α
µν µν − ∇α [µν] ≡ ∇ν µ−∇µ ν − ∇α [µν]
allowing one to write (3.6) in the form
1
(3.11) Φµν = β2(k 6)Wµ +2β2Γ¯α[µ+4πJµ
∇ν 2 − α]
(note Φ = 0 if W = 0 and Γ¯α = 0). The dual field tensor is now
µν µν [µν]
(3G) Φ˜µν = (1/2√ g)ǫµναβΦ and one can write
αβ
− −
(3.12) Φ + Φ + Φ = 4π(Ω +Ω +Ω ) 4πΘ
λ µν ν λµ µ νλ µ[νλ] λ[µν] ν[λµ] µνλ
∇ ∇ ∇ ≡
with Ω g g Ω[αβ]. This all leads then to (3H) Φ˜µν = 4πLµ
λ[µν] ≡ µα νβ λ ∇ν
where Lµ = (1/6√ g)ǫµλνσΘ with Lµ = 0. Thus one obtains two
λνσ µ
− − ∇
equations (3.11) and (3H) for Φ and Φ˜ . The fields are created by
µν µν
intrinsic magnetic and electric currents andfor k = 6 a Procaterm appears
6
in the Φ equation.
µν
This is developed to considerable extent in Israelit [43, 44, 46] and the
torsionplaysacrucialroleinthatonecangenerateadualfieldtensorhaving
anonvanishingdivergence(makingpossiblemagneticcurrents). Indeedone
can define (3I) Γ¯λ[µν] = (1/√ g)ǫλµνσV leading to
σ
− −
k 6
(3.13) Φµν = νWµ µWν = − β2Wµ+4πJµ;
ν ν ν
∇ ∇ ∇ −∇ ∇ 2
Φ˜µν = νVµ µVν = 2πLµ
ν ν ν
∇ ∇ ∇ −∇ ∇ −
WorkingintheEinsteingaugeβ = 1onehasaProcaequation(3J) νwµ+
ν
∇ ∇
κ2wµ = 0 (κ2 = (1/2)(6 k)) and in the absence of magnetic fields Vµ a
−
massless photon is implied via κ = 0 (cf. [5, 40, 41, 42] for other points of
view).
10 ROBERTCARROLLUNIVERSITYOFILLINOIS,URBANA,IL61801
4. DIVERSE DEVELOPMENTS IN WEYL GEOMETRY
Wegatherinthissection somematerialfromtheliteratureinanattempt
to clarify and unify various approaches (cf. [6, 7, 8, 12, 26, 29, 30, 34, 35,
40, 47, 49, 56, 58, 61, 62, 63, 64, 65, 68, 71]). We coordinate notation by
noting that the notation for the Weyl connection differs in many of these
references but they are all equivalent modulo a factor of 1 or 1/2 in the
± ±
definition of w . Thus e.g. [43] [65] [47], [64] [71] -[43], [29] [35]
µ
≡ ≡ ≡ ≡ ≡
-[6] -[12] (1/2)[43]. There are various approaches to Weyl geometry
≡ ≡ ≡
and we only try to build on the framework already developed in Section
1-2. Thus go to [47] which has a compatible point of view and start with
transformations (4A) g exp(2λ)g with length changes (4B) ℓ
µν µν
→ →
exp(λ)ℓ; then the relation (1.10) holds under Weyl transformations w
µ
→
w +∂ λ. If∂ w ∂ w = 0thereisaWeyltransformationreducingw to
µ µ µ ν ν µ µ
−
zero and the space is Riemannian (or semi-Riemannian). Here Γˆλ is given
µν
by (1.7) and on has curvature K from (1.17) with K as in (1I). There
νµ
is a nice discussion in [47] of attempts to develop EM in the Weyl theory
and its difficulties. In particular one problem is that the Weyl vector does
not seem to couple to spinors (see the last chapter in [47]). The Einstein-
Schro¨dinger approach to EM is also discussed along with Brans-Dicke and
Jordan theories. In terms of conformal geometry one considers conformally
equivalence via g˜ = exp(2λ)g (or what is the same d˜s = exp(λ)ds)
µν µν
with corresponding connection Γˆλ as in (1.7) with w ∂ λ (integrable
µν µ → − µ
Weyl geometry). One gets then (in 4-D)
(4.1) R˜ = R +g 2λ+2( λ+g ∂ λ∂σλ ∂ λ∂ λ)
νρ νρ νρ ρ ν νρ σ ν ρ
∇ ∇ −
leading to
(4.2) R˜ = e−2λ[R+62λ+6∂ λ∂σλ]
σ
where (4C) 2λ = (1/√ g)∂ (√ ggµν∂ λ) as usual. Equation (4.2) has
µ ν
− −
an alternate formsincein4-D thetransformation law for acurvaturescalar
is (4D) R˜ = exp( 2λ)[R+2(2λ+∂ λ∂µλ)] leading to
µ
−
(4.3) R˜ = e−2λ R+6e−λ2eλ
h i
in place of (4.2). One shows also that for φ˜= exp( λ)φ there results
−
(4.4) 2˜ φ˜+ 1R˜φ˜= e−3λ 2φ+ 1Rφ
6 6
(cid:20) (cid:21)
so 2φ+(1/6)Rφ = 0 is conformally invariant.
There is also information on scalar-tensor theories in [47] (cf. also [34]
for a string oriented approach). We recall first (1.10) ˆg = 2g w or in
µν µν λ
∇
the notation of [47] (4E) ˆ g = 2g w . This is invariant under WGT
ρ µν µν ρ
∇