## Abstract

Weyl’s original scale geometry of 1918 (“purely infinitesimal geometry”) was withdrawn by its author from physical theorizing in the early 1920s. It made a surprising comeback, however, in the last third of the 20th century in several different contexts: scalar-tensor theories of gravity, foundations of space-time theories, foundations of quantum mechanics, elementary particle physics, and cosmology. It seems that Weyl geometry continues to offer an open window for research on the foundations of physics even after the turn into the new millenium.

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## Notes

- 1.
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 was of restricted range.

- 2.
SM fields are here called

*pseudo-classical*if they are considered before, or better abstracting from (so-called “second”) quantization. Mathematically they are classical fields (spinor fields or gauge connections), but the field components do not correspond to physically measurable quantities. Observationally relevant information can be extracted only after applying perturbative quantization methods or, in semiclassical approximations, from their wave-function-like probability currents. - 3.
This distinguishes Euclidean geometry from the other classical geometries of constant curvature. The consideration of material systems, like hydrogen or caesium atoms etc., may be used to introduce units and reduce the allowed automorphisms of the congruences.

- 4.
In mathematical terminology, the implementation of a similarity structure happens at the

*infinitesimal*level. In the following the physicists’ use of the terminology “local” will be used. A discussion, given by Weyl later in his life, of the role of mathematical and physical autormorphisms can be found in (Weyl 1949/2016), some aspects of this also appear in (Weyl 1949, chap. III, sec. 14). - 5.
- 6.
- 7.
Presentations of Weyl geometry can be found, among others, in (Blagojević 2002; Israelit 1999b), (Tonnelat 1965, chap. IX), (Drechsler/Tann 1999, appendix A) and (Perlick 1989) (difficult to access). For selected aspects see (Codello et al. 2013) and (Ohanian 2016, sec. 4). Integrable Weyl geometry is presented in (Dahia et al. 2008; Romero et al. 2011; Almeida et al. 2014; Quiros 2014b), (Scholz 2011a, sec. 2.1). Be aware of different conventions for the scale connection. Expressions for Weyl-geometric derivatives and curvature quantities are derived in (Gilkey et al. 2011; Yuan/Huang 2013) and (Miritzis 2004, App.). For a more mathematical perspective consult (Folland 1970; Calderbank/Pedersen 1998; Gauduchon 1995; Higa 1993; Ornea 2001; Gilkey et al. 2011).

- 8.
- 9.
In a way, this may be called a geometrical “tensor-vector scalar” theory

*sui generis*, in which all components have geometrical meaning. - 10.
- 11.
Obviously the Einstein gauge exists also in the non-integrable case.

- 12.
- 13.
- 14.
Warning: One has to check carefully the sign convention used in the definition of Riemann and scalar curvature. Jordan, e.g., used sign inverted definitions of the curvature terms with respect to those used here and in much of the present literature (Jordan 1952, 40). In Fujii/Maeda’s notation (see below) this would correspond to

*𝜖*= −1 and thus to a “ghost” field. - 15.
- 16.
The conformal factor

*Ω*was (unnecessarily) restricted by the condition*Ω*^{2}=*χ*^{γ}for some constant*γ*∈**R**. - 17.
In the 1950s Sciama had considered the possibility that the gravitational “constant” was related to the mass and the “radius” of the visible universe.

- 18.
Also in the English edition of

*Philosophy of Mathematics and Natural Sciences*Weyl expressed this disassociation quite clearly, appealing to the constants of atomic physics which regulate the frequencies of spectral lines (Weyl 1949, 83). But this was only one part of his perspective. In the appendix he argued that for a deeper insight it would be necessary to understand how the “adaptation” of the mass of the electron to the local field constellation is achieved (Weyl 1949, 288f). This was close to the intentions of his 1918 approach, although no longer a claim that the goal had been achieved. Einstein, in his later papers, agreed (Einstein 1949, 555f); see (Lehmkuhl 2014). - 19.
*𝜖*= 1 corresponds to a normal field having a positive energy, in other words, not a “ghost”. Fuji/Maeda add that*𝜖*= −1 looks unacceptable because it seems to indicate negative energy, but “this need not be an immediate difficulty owing to the presence of the nonminimal coupling” (ibid.). - 20.
See, e.g., (Capozziello/Faraoni 2011, chap. 3.6).

- 21.
See, e.g., (Gilkey et al. 2011).

- 22.
- 23.
(Weyl 1920)

- 24.
Weyl’s note (Weyl 1921) became better known by his calculation and discussion of projective and conformal curvature tensors, which followed.

- 25.
See (Trautman 2012).

- 26.
I thank S. Walter for the hint.

- 27.
- 28.
- 29.
The flow could be characterized by the 0-th order probability current \(j^{\mu }_o\) of the pseudo-classical field. In their argument that the current is geodesic the authors neglected terms proportional to \(j^{\mu }_o\) (Audretsch/Gähler/Straumann 1984, eqn.(5.14), (5.16)). The repercussions of this generousness remains unclear to me (ES).

- 30.
Dirac presented his proposal for a retake of Weyl geometry at the occasion of the symposium honouring his 70th birthday, 1972 at Trieste. This talk remained unpublished. According to (Charap/Tait 1974, p. 249 footnote) the talk was close to his 1973 publication. For the broader historical context of this enterprise, the background in Dirac’s reflection on large numbers in the 1920s, and a surprising link to geophysics see (Kragh 2016).

- 31.
The qualifications “sign inverted” and “generally accepted” refer to the sign convention which agrees with the coordinate-free definition

*Riem*(*Y*,*Z*)*X*= ∇_{ Y }∇_{ Z }*X*−∇_{ Z }∇_{ Y }*X*−∇_{[Y,Z]}*X*. It is preferred in the mathematical literature, including (Weyl 1918b, 5th ed., 131), and also used in the majority of the more recent physics books. The “sign inverted” convention in some of the physics literature goes back to Einstein, e.g. (Einstein 1916, 801), who in turn may have followed Ricci and Levi-Civita. It is found in much of the physics literature from the first half of the 20th century, (Eddington 1923, § 37), (Pauli 1921) up to the influential (Weinberg 1972, eqn. (2.1.3)). Weyl, on the other hand, used the above convention long before the coordinate-free definition of the Riemann tensor was available. It seems to be dominant in the more recent literature on GRT, although Rindler speaks of a 50 % distribution among the two conventions (Rindler 2006, 219). - 32.
In his discussion with Einstein Weyl had argued that atomic clocks somehow adapt to the local field constellation via the Weylian scalar curvature (letter to Einstein April 28, 1918 etc (Einstein 1998, vol. 8B, doc. 526, 619)), and similarly in the fourth edition of

*Raum - Zeit - Materie*(Weyl 1922, 308f). In the fifth edition Weyl pondered the possibility that the Bohr theory of the atom might give a clue for such an adaptation (Weyl 1918b, 5-th ed., p. 298); compare (Scholz 2017, sec. 5.4). - 33.
On the work of Canuto and Maeder, see (Kragh 2006, pp. 126ff)

- 34.
For an illuminating historical report on the rise of dark matter, see (Sanders 2010). From a methodological point of view, Maeder’s hypothesis was not so far from the later, more pragmatic and more successful approach of modified Newtonian dynamics, MOND, as introduced by

*Mordechai Milgrom*. - 35.
Rosen’s “meson” was a hypothetical massive fundamental boson, not a bound state of quarks like the ones in the SM.

- 36.
This was more than a year before the Trieste symposium at which Dirac talked about his ideas. Apparently the paper remained unknown to Dirac. A second paper by Omote followed after Dirac’s publication and after Utiyama had jumped in (Omote 1974).

- 37.
- 38.
Dirac included a similar scale curvature term in his Lagrangian, but did not study its consequences.

- 39.
Apparently Hayashi/Kugo used different signature conventions from Utiyama, which resulted in another sign flip in

*ε*. - 40.
In fact,

*ϕ*and*φ*were not even coupled to the electromagnetic field, as Hayashi and Shirafuri showed in another paper that same year. - 41.
- 42.
Notations have been slightly adapted.

- 43.
For \(g_{\mu \nu } = h_{\mu }^{\;\;a} h_{\nu a}\) the convention (11.21) boils down to \(g_{\mu \nu } \mapsto g^{\prime }_{\mu \nu } = \varOmega ^2 g_{\mu \nu }\).

- 44.
Bregman, like many other authors, used a sign inverted convention for the scale connection form.

- 45.
Cf. (Sharpe 1997).

- 46.
The torsion term \(\frac {2}{3} T^{\alpha }_{\mu \alpha } \partial ^{\mu }\sigma ^2\) in Bregman’s equation is not scale invariant by itself, but his entire Lagrangian density is scale invariant.

- 47.
- 48.
\( \mathfrak {T}^k_{\mu }= \frac {\partial \mathfrak {L}}{\partial h^{\mu }_k}\, , \quad \mathfrak {S}^{\mu }_{ij}= -2 \frac {\partial \mathfrak {L}}{\partial A^{i j}_{\mu }}\, , \quad \mathfrak {D}^{\mu }= \frac {\partial \mathfrak {L}}{\partial A_{\mu }}\) (Charap/Tait 1974, p. 256, notation slightly changed). Compare the

*dynamical currents*by Hehl et al. below (11.29). For the dynamical matter energy current variational and partial derivatives usually coincide. Charap and Tait may have (wrongly) generalized this property to the other currents. - 49.
(Charap/Tait 1974, eqn. (3.22)).

- 50.
All of them are mentioned in (Blagojević/Hehl 2013, chap. 8).

- 51.
- 52.
- 53.
According to later estimates the torsion-spin coupling of Einstein-Cartan gravity becomes important only close to 10

^{38}times the density of a neutron star, which signifies energy densities at the hypothetical grand unification scale of elementary particle interactions (Trautman 2006, p. 194), (Blagojević/Hehl 2013, p. 108). Moreover, there seems to be a problem in defining the spin density of gauge particles, in particular of gluons. This would lead to leaving a large part of the nucleon spin underdetermined (Kleinert 2008, sec. 20.2) (I thank H. Ohanian for this information). - 54.
- 55.
Bohm realized the kinship of his approach to the earlier proposals of de Broglie only after he had finished his manuscript (Bohm 1952a, p. 167). In a footnote added in proof he also referred to Madelung’s “similar” approach of 1926, adding the remark “…but like de Broglie he did not carry this interpretation to a logical conclusion” (ibid.).

- 56.
“Die hydrodynamischen Gleichungen sind also gleichwertig mit denen von Schrödinger und liefern alles, was jene geben, d. h. sie sind hinreichend, um die wesentlichen Momente der Quantentheorie der Atome modellmäßig darzustellen” (Madelung 1926, p. 325).

- 57.
- 58.
Cf. (Nicolic 2005, p. 554 ) for vanishing

*em*potential. - 59.
According to de Broglie it was J.-P. Vigier who made him aware of a parallel between his hypothesis and the work of Einstein and Grommer (de Broglie 1960, p. 92).

- 60.
For E. Nelson’s program to re-derive the quantum dynamics from classical stochastic processes and classical probability see (Bacciagaluppi 2005).

- 61.
See, e.g., (Woodhouse 1991) or (Hall 2001, chaps. 22/23). A classical monograph on the symplectic approach to

*classical*mechanics is (Abraham/Marsden 1978); but this does not discuss spinning particles. In the 1980s the symplectic approach was already used as a starting platform for (pre-)quantization to which proper quantization procedures could then hook up, see e.g. (Śniatycki 1980). Souriau was an early advocate of this program. In his book he discussed relativistic particles with spin (Souriau 1970, §14). - 62.
Carlos Castro later added his mother’s name Perelman to his second name. Under this name he is mentioned in the acknowledgements of (Santamato 1984b). At that time he was a research assistant at the University of Texas, Austin, where he acquired his Ph.D. in 1991.

- 63.
Santamato and De Martini promoted the geometrical side of their research under different headings: at first they talked about “affine quantum mechanics (AQM)” in “conformal differential geometry” (Santamato/DeMartini 2013), then they shifted to “conformal quantum geometrodynamics (CQG)” (De Martini/Santamato 2014a,b). Physical problems for their approach, like those of negative probability densities for relativistic particles, remain outside the scope of this article.

- 64.
That is,

*ω*∈*Ω*, the sample space of a probability triple \((\varOmega ,\mathfrak {F}, P)\), where \(\mathfrak {F}\subset \mathfrak {P}(\varOmega )\) are the random events and*P*is a probability measure on*Ω*. - 65.
*L*^{∗}has the same Euler-Lagrange equations as the original*L*. - 66.
Cf. fn 31.

- 67.
Compare Section 11.6.4.1.

- 68.
A sign error in the formula of the Weyl-geometric affine connection (De Martini/Santamato 2014a, eqn. (4)) notwithstanding.

- 69.
(De Martini/Santamato 2014a, p. 3310)

- 70.
For van der Waerdens spinor symbolism see (Schneider 2011).

- 71.
This term, \( \frac {e^2 a^2}{c^2}(H^2-E^2)\), is comparable with the linear term in the field strengths only if the latter are very large,

*E*∼ 10^{18}*V**m*^{−1},*H*∼ 10^{9}*T*. “To have an idea how large is this field, an electron at rest is accelerated by such field up to 10^{9}*GeV*in a linear accelerator 1 m long” (De Martini/Santamato 2014a, p. 3315). - 72.
It remains unclear to me (E.S.) how the representation matrices of the “Euler angles” of configuration space are suppressed, while the change of coordinate frames in Minkowski space gets represented on the spinor fields.

- 73.
Among others, this had led to the measurement problem for atomic clocks.

- 74.
In the physics literature, and also in the paper by Shojai and Golshani,

*Ω*is often referred to as the “conformal” degree of freedom of the metric, or even the “conformal structure”.The latter is clearly mistaken, whereas the first is, at best, misleading. Therefore I avoid this terminology in favour of*scaling degree of freedom*. - 75.
- 76.
The authors differentiated between “scale transformations” and “conformal transformations”. In their terminology the first operated only on the metric, while the latter rescaled all physical fields according to their weights.

- 77.
The Dirac Lagrangian was stripped of the Yang-Mills term of the scale connection (Shojai/Shojai 2004, eqn. 6).

- 78.
The claim that one can identify a Dirac-type scalar field with a “quantum mass” field remains, in my view, an unfounded speculation; E.S.

- 79.
- 80.
*Not*“S. Carroll”, as listed on occasion in the bibliography of later papers. - 81.
See, e.g. Karaca (2013).

- 82.
The motivation for considering

*n*≥ 4 was the method of dimensional regularization for the quantization of the theory. - 83.
- 84.
Signs have to be taken with caution. They may depend on conventions for defining the Riemann curvature, the Ricci contraction, and the signature. Smolin, e.g., used a different sign convention for

*Riem*to the one used in this survey. Signs given here are adapted to*signature**g*= (3, 1). The Riemann tensor and Ricci contraction are those usually adopted in the mathematical literature, see fn. 31. - 85.
This seems to have been widely known. For an explicit statement see, e.g., (Hehl et al. 1996).

- 86.
(Englert et al. 1975) was not quoted by Smolin.

- 87.
In scalar-field gauge with

*ϕ*≐*ϕ*_{ o }=*F*, his reduced Lagrangian (square gravitational terms dropped) was (Smolin 1979, eqn. (3.17)) - 88.
It is possible to choose the scale gauge such that

*ϕ*becomes constant (scalar-field gauge, see Section 11.2.1). - 89.
For a survey of the status of investigations in 1981 see (Adler 1982); but note in particular (Zee 1982, 1983). In fact, Zee’s first publication on the subject preceded Smolin’s. (Zee 1979) was submitted in December 1978 and published in February 1979; (Smolin 1979) was submitted in June 1979. The topic of “origin of spontaneous symmetry breaking” by radiative correction was much older (Borrelli 2015; Karaca 2013). A famous paper was (Coleman/Weinberg 1973).

- 90.
Flato/Ra̧cka’s paper appeared as a preprint of the

*Scuola Internazionale Superiore di Studi Avanzati*, Trieste, in 1987; the paper itself was submitted in December 1987 to*Physics Letters B*and published in July 1988. Cheng’s paper was submitted in February 1988, published in November. Only a decade later, in March 2009, Drechsler and Tann got acquainted with the other two papers. This indicates that the Weyl-geometric approach in field theory had not yet acquired the coherence of a research program with a stable communication network. - 91.
Personal communication to ES, April 04, 2017.

- 92.
In the sequel the isospin extended scalar field will be denoted by

*Φ*. - 93.
Cheng added another coupling coefficient for the scale connection, which is here suppressed.

- 94.
See footnote 31.

- 95.
The second term in (11.82) is missing in Cheng’s publication. That is probably not intended, but a misprint. Moreover he did not discuss scale weights for Dirac matrices in the tetrad approach.

- 96.
Remember that the

*φ*terms of scale-covariant derivatives in the Lagrangian of spinor fields cancel. - 97.
More than a decade earlier Flato had sketched a covariant (“curved space”) generalization of the Wightman axioms (Flato/Simon 1972), different from the one discussed by R. Wald in this volume.

- 98.
- 99.
For example (Drechsler/Mayer 1977).

- 100.
(Tann 1998, eqn. (372)), (Drechsler 1999, eqn. (2.29)). Both authors used coefficients as in the case of conformal coupling in Riemannian geometry, \(\beta =\frac {1}{6}\). In the Weyl-geometric framework this was an unnecessary restriction but it suppressed the mass factor at the Planck scale for the Weyl field

*φ*. Tann wrote the modified Hilbert term with a negative sign, because he used the sign inverted convention for the Riemann tensor, see footnote 31. - 101.
- 102.
In the appendix Drechsler and Tann showed that the squared Weyl-geometric conformal curvature

*C*^{2}=*C*_{ λμνρ }*C*^{λμνρ}arises from the conformal curvature of the Riemannian component by adding a scale curvature term: (Drechsler/Tann 1999, (A 54)). So one may wonder, why they did not replace the square term \( \mathcal {L}_{R^2}\) by the Weyl-geometric conformal curvature term \(\mathcal {L}_{\text{conf}} = \tilde {\alpha } C^2 \sqrt {|det\, g|}\). - 103.
(11.82) can equivalently be written with a Weylianized scale-covariant derivative \(\overline {D}_{\mu } = \left ( \partial _{\mu } + i \tilde {\varGamma } _{\mu } + w(\psi ) \varphi _{\mu } + \frac {i q}{\hbar c} A_{\mu } \right )\). Because

*φ*_{ μ }is real, the scale connection terms*w*(*ψ*)*φ*_{ μ }in the Lagrangian cancel. - 104.
One could then just as well consider a complex valued connection

*z*= (*z*_{ μ }) with values \(z_{\mu }= \varphi _{\mu } + \frac {i}{\hbar c} A_{\mu }\) in \(\mathbb C= \mathfrak {Lie}(\mathbb C^{\ast })\) and weight \(W(\psi )= (-\frac {3}{2}, q)\). Then \(D_{\mu } \psi = (\partial _{\mu } + \tilde {\varGamma }_{\mu } + W(\psi )z_{\mu })\psi \), presupposing an obvious convention for applying*W*(*ψ*)*z*. - 105.
This argument is possible, but not compelling:

*γ*|*Φ*| has the correct scaling weight of mass and may be considered as such. - 106.
A similar approach is already used in Tann’s PhD dissertation.

- 107.
Note that one could just as well do without (11.84) and proceed with fully scale-covariant masses – compare last footnote.

- 108.
In the literature it is often also called “unitary gauge”.

- 109.
Mathematically speaking, it amounts to a change of trivialization of the

*SU*(2) ×*U*(1)-bundle. - 110.
\(W^{\pm }_{\mu } = \frac {1}{\sqrt {2}} (W^1_{\mu } \mp i W^2 _{\mu })\), \(Z_{\mu }= \cos \varTheta \, W^3_{\mu } - \sin \varTheta \, B_ {\mu } \).

- 111.
One has to be careful, however. Things become more complicated if one considers the trace. In fact,

*tr**T*_{ Φ }contains a mass terms of the Dirac field of form \(\gamma | \varPhi _o | \hat {\psi }^{\ast } \hat {\psi }\), with*γ*coupling constant of the Yukawa term (\(\hat {\psi }\;\) indicating electromagnetic gauge). One of the obstacles for making quantum matter fields compatible with classical gravity is the vanishing of*tr**T*_{ ψ }, in contrast to the (nonvanishing) trace of the energy momentum tensor of classical matter. Might Drechsler’s analysis indicate a way out of this impasse? – Warning: The mass-like expressions for*W*and*Z*in (11.87) cancel in*tr**T*_{ Φ }(Drechsler 1999, eqn. (3.55)) like in the energy-momentum tensor of the*W*and*Z*fields themselves. - 112.
See, e.g., (Franklin 2017).

- 113.
For an exception, still standing under the spell of Drechsler/Tann though with a consistently scale-covariant approach and no explicit scale symmetry breaking term, see (Scholz 2011a).

- 114.
In the high energy physical context, and accordingly in our section 3, signature of

*g*= (+ −−−). For sign conventions regarding curvature see footnote 31. - 115.
See also (Blagojević 2002, p. 81).

- 116.
For an explicit form of Dirac kinetic terms and Yukawa mass terms see, e.g., (Nishino/Rajpoot 2009, eqn. (1.2)).

- 117.
The expression for the scalar curvature is given in the paper (and also in the later papers by the same authors) as , where a coupling constant

*f*introduced by the authors is here set to*f*= 1 and transcribed into our notation, (Nishino/Rajpoot 2004, eqn. (14)). The Weyl-geometric value (11.2) in our sign conventions would be , cf. (Weyl 1918c, p. 21), (Drechsler/Tann 1999, eqn. (A 31)) and others. Nishino and Rajpoot apparently used inverted sign conventions for the curvature and the scale connection. - 118.
The non-broken

*U*(1) symmetry is important for the BRST relations, the quantum analogue of the Noether relations. See (Ruegg et al. 2003, 75ff). - 119.
The factor \(\zeta _1^{-\frac {1}{2}}\) in (Nishino/Rajpoot 2009, eqn. (2.1)) was set by them to

*ζ*_{1}= 1 while transforming the Lagrangian into their eqn. (2.3). The follow up paper (Nishino/Rajpoot 2011, second paragraph of section 2) shows that this reduction was intended. Of course, a different factor*ζ*_{1}would heavily influence the mass calculation in (11.101). - 120.
Warning: Nishino/Rajpoot used the notation

*φ*for the Stueckelberg “compensator”, i.e. our*β*, and*S*_{ μ }for the scale connection (the potential of the “Weyl field”), our*φ*_{ μ }. In order to avoid confusion the notation in the present paper has been homogenized for the authors discussed here. - 121.
Nishino/Rajpoot did not consider the contribution of the modified Hilbert term, in contrast to Smolin and Cheng (see Section 11.5.1).

- 122.
Compare (Stoeltzner 2014).

- 123.
Strictly speaking, their framework does not contain any meaningful “Jordan frame” because their Weyl structure is not integrable, and thus the purely Riemannian representation of the affine connection presupposed for an ordinary Jordan frame does not exist. The Einstein frame, on the other hand, is meaningful in any Weyl-geometric gravity approach with a scale-covariant scalar field and corresponds to scalar-field gauge (11.92).

- 124.
“Note that the Weyl rescaling we made is a field re-definition, but it is not a part of any local scale transformation which is defined to act not only on

*g*_{ μν }but also on*Φ*and*φ*as in (2.2) [the equation for the full gauge transformation, E.S.]” (Nishino/Rajpoot 2011, p. 4). - 125.
(Cai/Wei 2007, Acknowledgments)

- 126.
Note that the differential form

*b*=*b*_{ μ }*dx*^{μ}is sign inverted in comparison with our conventions of Section 11.2.1. - 127.
Ohanian preserved the label “scale transformation” for a global usage in Minkowski space, where, in addition to the rescaling of the fields \(X\mapsto \tilde {X} = \varOmega ^k X\), a space dilation \(x\mapsto \tilde {x}= \varOmega x\) is applied (Ohanian 2016, p. 25).

- 128.
\(\partial _{\nu }\left (\sqrt {|g|}f^{\mu \nu } \right ) =\mathfrak {J}^{\mu }\). In Ohanian’s Lagrangian

*φ*couples only to the “dilaton” scalar field*χ*. This leads to a form for the variation of the Lagrangian under scale transformations such that the dynamical current coincides with the Noether current (Ohanian 2016, eqn. (13)). - 129.
See C.Will’s contribution to this volume.

- 130.
- 131.
For a historical discussion of oscillating models see (Kragh 2009) and, in an even wider perspective, H. Kragh’s contribution to this book.

- 132.
The authors declared geodesic incompleteness as “an artifact of an unsuitable frame choice: geodesically incomplete solutions in Einstein frame may be completed in other frames, even though the theories are entirely equivalent away from the singularity” (Bars et al. 2014, p. 13).

- 133.
- 134.
Because of scale invariance there is, in fact, only one true scalar field degree of freedom (compare Section 11.2.1).

- 135.
- 136.
Compare on this point (Capozziello/Faraoni 2011, pp. 86ff).

- 137.
J.E. Madriz Aguilar was a student of C. Romero of the Brazilian school, see Section 11.6.4.

- 138.
A few years later high precision numerical modelling showed that thermal effects can completely account for the observations known as the flyby anomaly of the Pioneer spacecrafts (Rievers/Lämmerzahl 2011).

- 139.
- 140.
It was the topic of the author’s talk at the Mainz conference.

- 141.
Bekenstein, as late as 2004, took it as “evident” that measurements with “ clocks and rods” are expressed by the Jordan metric. Moreover, its Levi-Civita connection was taken to govern the free fall of test particles, although the dynamics in the Jordan frame does not satisfy the “usual Einstein equation” (because of explicit terms in the scalar field). The Einstein frame represented for him the “primitive metric” because here the gravitational action reduces to the classical form of the Hilbert term, and the dynamics is given by the Einstein equation (Bekenstein 2004, p. 5f). In their common paper, Milgrom and Bekenstein used the terminology of “dual descriptions” working in “gravitational units” (Einstein frame) respectively “atomic units” (Jordan frame), which sounded a bit like Dirac’s distinction (Bekenstein/Milgrom 1984, p. 14).

- 142.
In his later review paper Bekenstein qualified this point by stating that only so long as the scalar field “…contributes comparatively little to the energy-momentum tensor, it cannot affect light deflection, which will thus be due to the visible matter alone” (Bekenstein 2004, p. 6). One can read this observation the other way round: If the scalar field carries a considerable contribution to the energy-momentum this influences light deflection.

- 143.
Cf. the preceding footnote.

- 144.
The data for 2 clusters are outliers, already from the phenomenological point of view.

- 145.
The famous Coma cluster which led Zwicky to introduce the hypothesis of dark matter is among those for which the Weyl-geometric model gives results consistent with most recent empirical data on mass distributions and accelerations.

- 146.
Unpublished calculations indicate scenarios of a cosmic evolution in agreement with many features of standard cosmology. These would support such features as an initial singularity, large parts of the cosmological redshift due to the expansion of spatial folia in Einstein gauge, accelerated “late time” expansion etc.

- 147.
In this paper Novello still thought in terms of Weyl’s first interpretation of the scale connection, the

*em dogma*in the terminology above. - 148.
In this paper

*ω*(*x*) was not yet introduced as a scalar field of its own, but via the square of the electromagnetic potential*A*_{ μ }, i.e., \(\omega =\log {A_{\mu }A^{\mu }}\). - 149.
In many papers of the Brazilian tradition (Novello et al. 1992) is quoted as a starting point: (Salim/Sautú 1996; de Oliveira 1997; Romero et al. 2011, 2012), to cite just a few. Sometimes it is even called the “first approach to scalar-tensor theory in WIST” [Weyl integrable space-time] Pucheu et al. (2016).

- 150.
- 151.
WIST was (and is) the abbreviation, preferred by the Brazilian authors, for “Weyl integrable spacetime”.

- 152.
This adds flavour to the hybrid approach mentioned above.

- 153.
The fluid Lagrangian was taken from (Ray 1972).

- 154.
For the Hawking-Penrose singularity theorem the geometrical energy condition is the crucial point. It has to be distinguished from the

*strong energy condition*in the*physical sense*which is given by \(T_{\mu \nu } - \frac {1}{2} tr\, T\, g_{\mu \nu }V^{\mu } V^{\nu }\). The geometrical and physical conditions are equivalent in Einstein gravity; see, e.g., (Curiel 2017, p. 49). Later investigations of the Brazilian school would show that the geometrical energy condition could be violated, while the physical energy condition might still be satisfied (see end of section 5.4.3). - 155.
- 156.
“An important conclusion (…) is that general relativity can perfectly ‘survive’ in a non-Riemannian environment” (Romero et al. 2011). In a personal communication with ES (April 3, 2017) C. Romero added “Philosophically, this could perhaps be interpreted as an illustration of the tenets put forward by Poincaré in his conventionalist ideas”.

- 157.
- 158.
One only needed to put the JBD Lagrangians in a Weyl-geometric framework. Alternatively, if one wants to start from the Brazilian point of view, one may read the constant coefficient of the Hilbert term in (11.119) as the value of a scale-covariant scalar field

*χ*in (Einstein-) scalar-field gauge,*χ*_{ o }≐ 1, and*∂*_{ μ }*ϕ∂*^{μ}*ϕ*as the scalar field gauged expression of the scale-covariant kinetic term*D*_{ μ }*χD*^{μ}*χ*with scale-covariant derivative (11.3). - 159.
- 160.
Cf. C. Will’s contribution to this volume.

- 161.
E.g. (Drechsler/Hartley 1994).

- 162.
Studies of QFT on Weylian manifolds, comparable to the corresponding researches for Lorentzian manifolds, as discussed in R. Wald’s contribution to this volume, are still a desideratum.

- 163.
These two authors referred neither to the Dirac tradition in Weyl-geometric gravity nor to Utiyama’s; their Weyl-geometric starting point was “self-made” (Drechsler/Hartley 1994) aside from Weyl’s original papers.

- 164.
(Bars et al. 2014, p. 2)

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## Acknowledgements

This paper owes its existence to *David Rowe*’s initiative in several respects. He encouraged me to present heterodox ideas on Weyl-geometric methods in cosmology at the conference and invited me to rethink the case after a cool reception of the talk by the other participants. That gave me the chance to place my views in the wider range of recent attempts to use Weyl-geometric methods in physics. After an interruption of several years, an earlier first draft of this paper (Scholz 2011b) had be to be rewritten completely for the final version of this book. The new version overlaps nicely with the wider ambit of the investigations of the interdisciplinary group *Epistemology of the LHC* with center at Wuppertal and supported generously by the DFG/FWF. This group offers the chance for a close communication between historians and philosophers of science and collegues from the elementary particle community. H. Cheng, F. Hehl, J. Miritzis, C. Romero, D. Rowe, A. Trautman, S. Walter gave helpful hints for the final version of the paper.

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Scholz, E. (2018). The Unexpected Resurgence of Weyl Geometry in late 20th-Century Physics. In: Rowe, D., Sauer, T., Walter, S. (eds) Beyond Einstein. Einstein Studies, vol 14. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-7708-6_11

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