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WEYL GEOMETRY IN LATE 20TH CENTURY

PHYSICS

ERHARD SCHOLZ

ABSTRACT. Weyl’s original scale geometry of 1918 (“purely inﬁn-

itesimal geometry”) was withdrawn from physical theory in the

early 1920s. It had a comeback in the last third of the 20th cen-

tury in diﬀerent contexts: scalar tensor theories of gravity, founda-

tions of physics (gravity, quantum mechanics), elementary particle

physics, and cosmology. Here we survey the last two segments. It

seems that Weyl geometry continues to have an open research po-

tential for the foundations of physics after the turn of the century.

1. INTRODUCTION

Roughly at the time when his famous book Raum · Zeit · Materie

(RZM) went into print, Hermann Weyl generalized Riemannian ge-

ometry by introducing scale freedom of the underlying metric, in or-

der to bring a more basic “purely inﬁnitesimal” point of view to bear

(Weyl 1918c, Weyl 1918a). How Weyl extended his idea of scale gauge

to a uniﬁed theory of the electromagnetic and gravitational ﬁelds, how

this proposal was received among physicists, how it was given up – in

its original form – by the inventor already two years later, and how it

was transformed into the now generally accepted U(1)-gauge theory of

the electromagnetic ﬁeld, has been extensively studied. 1 Many times

Weyl’s original scale gauge geometry was proclaimed dead, physically

misleading or, at least, useless as a physical concept. But it had sur-

prising come-backs in various research programs of physics. It seems

well alive at the turn to the new century.

Weylian geometry was taken up explicitly or half-knowingly in dif-

ferent research ﬁelds of theoretical physics during the second half of

the 20th century (very rough time schedule):

• 1950/60s: Jordan-Brans-Dicke theory

• 1970s: a double retake of Weyl geometry by Dirac and Utiyamah

• 1970/80s: Ehlers-Pirani-Schild and successor studies

• 1980s: geometrization of (de Broglie Bohm) quantum potential

• 1980/90s: scale invariance and the Higgs mechanism

• 1990/2000s: scale covariance in recent cosmology

Date: 23. 09. 2009 .

1 (Vizgin 1994, Straumann 1987, Sigurdsson 1991, Goenner 2004, O’Raifeartaigh

2000, O’Raifeartaigh 1997, Scholz 2001, Scholz 2004, Scholz 2005a).

E. SCHOLZ

All these topics are worth of closer historical studies. Here we concen-

trate on the last two topics. The ﬁrst four have to be left to a more

extensive study.

With the rise of the standard model of elementary particles (SMEP)

during the 1970s a new context for the discussion of fundamental ques-

tions in general relativity formed. 2 That led to an input of new ideas

into gravity. Two subjects played a crucial role for our topic: scale or

conformal invariance of the known interactions of high energy physics

(with exception of gravity) and the intriguing idea of symmetry reduc-

tion imported from solid state physics to the electroweak sector of the

standard model. The latter is usually understood as symmetry break-

ing due to some dynamical process (Nambu, Goldstone, Englert, Higgs,

Kibble e.a.). The increasingly successful standard model worked with

conformal invariant interaction ﬁelds, mathematically spoken connec-

tions with values in the Lie algebras of “internal” symmetry groups

(i.e., unrelated to the spacetime), SU(2) × U(1) Y for the electroweak

(ew) ﬁelds, SU(3) for the chromodynamic ﬁeld modelling strong inter-

actions, and U(1) em for the electromagnetic (em) ﬁeld, inherited from

the 1920s. In the SMEP electromagnetism appears as a residual phe-

nomenon, after breaking the isospin SU(2) symmetry of the ew group

to the isotropy group U(1) em of a hypothetical vacuum state. The lat-

ter is usually characterized by a Higgs ﬁeld Φ, a “scalar” ﬁeld (i.e. not

transforming under spacetime coordinate changes) with values in an

isospin

2 representation of the weak SU(2) group. If Φ characterizes

dynamical symmetry breaking, it should have a massive quantum state,

the Higgs boson (Higgs 1964, Weinberg 1967). The whole procedure

became known under the name “Higgs mechanism”. 3

Three interrelated questions arose naturally if one wanted to bring

gravity closer to the physics of the standard model:

(i) Is it possible to bring conformal, or at least scale covariant gen-

eralizations of classical (Einsteinian) relativity into a coherent

common frame with the standard model SMEP? 4

(ii) Is it possible to embed classical relativity in a quantized theory

of gravity?

2 It is complemented by the standard model of cosmology, SMC. Both, SMC

and SMEP, developed a peculiar symbiosis since the 1970s (Kaiser 2007). Strictly

speaking, the standard model without further speciﬁcations consists of the two

closely related complementary parts SMEP and SMC.

3 Sometimes called in more length and greater historical justness “Englert-Brout-

Higgs-Guralnik-Hagen-Kibble” mechanism.

4 Such an attempt seemed to be supported experimentally by the phenomenon

of (Bjorken) scaling in deep inelastic electron-proton scattering experiments. The

latter indicated, at ﬁrst glance, an active scaling symmetry of mass/energy in high

energy physics; but it turned out to hold only approximatively and of restricted

range.

WEYL GEOMETRY IN LATE 20TH CENTURY PHYSICS

(iii) Or just the other way round, can “gravity do something like

the Higgs”? 5 That would be the case if the mass acquirement

of electroweak bosons could be understood by a Brans-Dicke

like extension of gravitational structures.

These questions were posed and attacked with diﬀering degrees of suc-

cess since the 1970s to the present. Some of these contributions, mostly

referring to questions (i) and (iii), were closely related to Weylian scale

geometry or even openly formulated in this framework. The literature

on these questions is immense. Obviously we can only scratch on the

surface of it in our survey, with strong selection according to the crite-

rion given by the title of this paper. So we exclude discussion of topic

(ii), although it was historically closer related to the other ones than

it appears here (section 3).

In the last three decades of the 20th century a dense cooperation

between particle physics, astrophysics and cosmology was formed. The

emergence of this intellecual and disciplinary symbiosis had many causes;

some of them are discussed in (Kaiser 2007) and by C. Smeenk (this vol-

ume). Both papers share a common interest in inﬂationary theories of

the very early universe. But in the background of this reorganization

more empirically driven changes, like the accumulating evidence for

“dark matter” by astronomical observations in the 1970s, were surely

of great importance (Rubin 2003, Trimble 1990). That had again strong

theoretical repercussions. In the course of the 1980/90s it forced as-

tronomers and astrophysicists to assume a large amount of non-visible,

non-baryonic matter with rather peculiar properties. In the late 1990s

increasing and diﬀerent evidence spoke strongly in favour of a non-

vanishing cosmological constant Λ. It was now interpreted as a “dark

energy” contribution to the dynamics of the universe (Earman 2001).

The second part of the 1990s led to a relatively coherent picture of

the standard model of cosmology SMC with a precise speciﬁcation of

the values of the energy densities Ω m , Ω of (mostly “dark”) matter and

of “dark” energy as the central parameters of the model. This speci-

ﬁcation depended, of course, on the choice of the Friedman-Lemaitre

spacetimes as theoretical reference frame. Ω m and Ω together deter-

mine the adaptable parameters of this model class (with cosmologi-

cal constant). The result was the now favoured ΛCDM model. 6 In

this sense, the geometry of the physical universe, at least its empiri-

cally accessible part, seems to be well determined, in distinction to the

quantitative underdetermination of many of the earlier cosmological

world pictures of extra-modern or early modern cultures (Kragh 2007).

But the new questions related to “dark matter” and “dark energy”

also induced attempts for widening the frame of classical GRT. Scale

5 Formulation due to (Pawlowski 1990).

6 CDM stands for for cold dark matter and Λ for a non-vanishing cosmological

constant.

E. SCHOLZ

covariant scalar ﬁelds in the framework of conformal geometry, Weyl

geometry, or Jordan-Brans-Dicke theory formed an important cluster

of such alternative attempts. We shall have a look at them in section

(4). But before we enter this discussion, or pose the question of the

role of scale covariance in particle physics, we give a short review of

the central features of Weyl geometry, and its relation to Brans-Dicke

theory, from a systematic point of view. Readers with a background

in these topics might like to skip the next section and pass directly to

section (3). The paper is concluded by a short evaluation of our survey

(section 5).

2. PRELIMINARIES ON WEYL GEOMETRY AND JBD THEORY

Weylian metric, Weyl structure. A Weylian metric on a diﬀeren-

tiable manifold M (in the following mostly dim M = 4) can be given

by pairs (g,ϕ) of a non-degenerate symmetric diﬀerential two form g,

here of Lorentzian signature (3, 1) = (−, +, +, +, ), and a diﬀerential

1-form ϕ. The Weylian metric consists of the equivalence class of such

pairs, with (g, ϕ) ∼ (g,ϕ) iﬀ

(1) (i) g = Ω 2 g , (ii) ϕ = ϕ − d log Ω

for a strictly positive real function Ω > 0 on M. Chosing a represen-

tative means to gauge the Weylian metric; g is then the Riemannian

component and ϕ the scale connection of the gauge. A change of rep-

resentative (1) is called a Weyl or scale transformation; it consists of a

conformal rescaling (i) and a scale gauge transformation (ii). A mani-

fold with a Weylian metric (M, [g,ϕ]) will be called a Weylian manifold.

For more detailed introductions to Weyl geometry in the theoretical

physics literature see (Weyl 1918b, Bergmann 1942, Dirac 1973), for

mathematical introductions (Folland 1970, Higa 1993).

In the recent mathematical literature a Weyl structure on a diﬀer-

entiable manifold M is speciﬁed by a pair (c,∇) consisting of a con-

formal structure c = [g] and an a ne, i.e. torsion free, connection Γ,

respectively its covariant derivative ∇. The latter is constrained by the

property that for any g ∈ c there is a diﬀerential 1-form ϕ g such that

(2) ∇g + 2ϕ g ⊗ g = 0 ,

(Calderbank 2000, Gauduchon 1995, Higa 1993, Ornea 2001). We shall

call this weak compatibility of the a ne connection with the metric. 7

One could also formulate the compatibility by

(3) Γ − g Γ = 1 ⊗ ϕ g + ϕ g ⊗ 1 − g ⊗ ϕ g ,

where 1 denotes the identiy in Hom(V,V ) for every V = T x M, ϕ g is

the dual of ϕ g with respect to g, and g Γ is the Levi-Civita conection of

7 Physicists usually prefer to speak of a “semimetric connection”’ (Hayashi/Kugo

1977) or even of a “nonmetricity” of the connection (Hehl e.a. 1995) etc.

WEYL GEOMETRY IN LATE 20TH CENTURY PHYSICS

g. Written in coordinates that means

(4) Γ νλ = g Γ νλ + δ ν ϕ λ + δ λ ϕ ν − g νλ ϕ µ ,

if g Γ νλ denote the coe cients of the a ne connection with respect to

the Riemannian component g only.

This is just another way to specify the structure of a Weylian mani-

fold, because [(g,ϕ)] is compatible with exactly one a ne connection.

(8) is the condition that the scale covariant derivative of g vanishes in

every gauge (see below). The Weyl structure is called closed, respec-

tively exact, iﬀ the diﬀerential 1-form ϕ g is so (for any g). In agreement

with large parts of the physics literature on Weyl geometry, we shall

use the terminology integrable in the sense of closed, i.e., in a local

sense.

In some part of the physics literature a change of scale like in (1 (i))

is considered without explicitly mentioning the accompanying gauge

transformation (ii). Then a scale transformation is identiﬁed with a

conformal transformation of the metric. That may be misleading but

need not, if the second part of (1) is respected indirectly. In any case

we have to distinguish between a strictly conformal point of view and

a Weyl geometric one. In the ﬁrst case we deal with c = [g] only, in

the second case we refer to the whole Weyl metric [(g,ϕ)], respectively

Weyl structure (c,∇). 8

Covariant derivative(s), curvature, Weyl ﬁelds. The covariant

derivative with respect to Γ will be denoted (like above) by ∇. The

covariant derivative with respect to the Riemannian component of the

metric only will be indicated by g ∇. ∇ is an invariant operation for

vector and tensor ﬁelds on M, which are themselves invariant under

gauge transformations. The same can be said for geodesics γ W of

Weylian geometry, deﬁned by ∇, and for the Riemann curvature tensor

Riem = (R βγδ ) and its contraction, the Ricci tensor Ric = (R µν ). The

contraction is deﬁned with respect to the 2nd and 3rd component

(5)

R µν := R µαν

Functions or (vector, tensor, spinor . . . ) ﬁelds F on M, which trans-

form under gauge transformations like

(6) F −→ F = Ω k F .

will be called Weyl functions or Weyl ﬁelds on M of (scale or Weyl)

weight w(f) := k. Examples are: w(g µν ) = 2, w(g µν ) = −2 etc. As the

8 Both approaches work with the “localized” (physicists’ language) scale extended

Poincare group W = R 4 ⋉SO + (3, 1) ×R + as gauge automorphisms. The transition

from a strictly conformal approach to a Weyl geometric one has nothing to do with

a group reduction (or even with “breaking” of some symmetry); it rather consists

of an enrichment of the structure while upholding the automorphism group.

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