We consider the construction of gauge theories of gravity, focussing in particular on the extension of local Poincaré invariance to include invariance under local changes of scale. We work exclusively in terms of finite transformations, which allow for a more transparent interpretation of such theories in terms of gauge fields in Minkowski spacetime. Our approach therefore differs from the usual geometrical description of locally scale-invariant Poincaré gauge theory (PGT) and Weyl gauge theory (WGT) in terms of Riemann–Cartan and Weyl–Cartan spacetimes, respectively. In particular, we reconsider the interpretation of the Einstein gauge and also the equations of motion of matter fields and test particles in these theories. Inspired by the observation that the PGT and WGT matter actions for the Dirac field and electromagnetic field have more general invariance properties than those imposed by construction, we go on to present a novel alternative to WGT by considering an “extended” form for the transformation law of the rotational gauge field under local dilations, which includes its “normal” transformation law in WGT as a special case. The resulting “extended” Weyl gauge theory (eWGT) has a number of interesting features that we describe in detail. In particular, we present a new scale-invariant gauge theory of gravity that accommodates ordinary matter and is defined by the most general parity-invariant eWGT Lagrangian that is at most quadratic in the eWGT field strengths, and we derive its field equations. We also consider the construction of PGTs that are invariant under local dilations assuming either the “normal” or “extended” transformation law for the rotational gauge field, but show that they are special cases of WGT and eWGT, respectively.

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In his original paper,52 Weyl attempted to interpret the new gauge field Bμ as the electromagnetic 4-potential and thereby provide a unified description of gravity and electromagnetism. As Einstein later showed, however, the field Bμ interacts in the same manner with both particles and antiparticles, contrary to all experimental evidence about electromagnetic interactions. It was only later realised53 that electromagnetism was related to localisation of invariance under change of quantum-mechanical phase and, much later, that Bμ might be interpreted as an additional gravitational interaction.

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63.

To keep the notation simple, however, we will still indicate Lorentz transformations by Λ, rather than SL(2, C) elements.

64.

A re-expression of the Dirac Lagrangian is presented in  Appendix A, which lends itself to the modelling of classical point matter particles, which we will discuss in Section II M.

65.

It is worth noting that the form adopted for the (Λ, ρ)-covariant derivative Dμ*φ in (20) constitutes a choice about how to include the rotational and dilational gauge fields Aabμ and Bμ, respectively, which leads directly to the requirements (23) and (24) on their transformation properties.

66.

It is worth noting that, under a local dilation alone, if the field φ has weight w, then Dμ*φ also transforms covariantly with weight w, whereas Da*φ has weight w − 1.

67.

Since h is not symmetric, placing J into its first “slot” instead will, in general, lead to a new vector field that is different from 𝒥, so that, for example, hμaJμJa. We will not, however, need to consider this eventuality and may restrict our attention to the case in which the first index on the components of h (and b) is Latin and the second is Greek.

68.

Unfortunately, this presents us with a notational difficulty. For objects possessing several indices, one should, in principle, use a different kernel symbol for each possible combination of Latin and Greek indices, but such an approach would quickly become unwieldy. So, in general, we will simply retain the original kernel letter, while keeping the above observations in mind. As a compromise, however, we will adopt the notation used above, which is in keeping with the definition (29), that the kernel letter of a quantity possessing only Latin indices (and its contractions over such indices) is the calligraphic font version of the kernel letter of the quantity possessing only Greek indices (following Lasenby, Doran and Gull), with the exception of quantities having Greek or lower-case kernel letters.

69.

It is also worth pointing out that, in terms of their counterparts 0Aabc and Kabc in PGT (see  Appendix B), one has 0A*abc=0Aabc+δcaBbδcbBa and K*abc=KabcδcaBb+δcbBa, so that the decomposition (46) or (47) also holds with all asterisks removed from the RHS.

70.

It is worth noting that the PGT (and WGT) rotational gauge field strength tensor (or “curvature”) Rabcd is already covariant under local dilations, whereas the PGT translational gauge field strength (or “torsion”) Tabc is not; one must instead introduce the WGT “torsion” (44).

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One could alternatively define taμhδLG/δhaμ, sabμhδLG/δAabμ, and jμhδLG/δBμ, and similarly for the matter sector. This choice corresponds more closely to the standard “metrical” versions of such tensors, but we will not adopt this convention here.

73.

We note that, slightly unusually, in the last two definitions it is not the h-field, but its inverse, that is used to convert a Greek index into a Roman one; this should be borne in mind when dealing with these quantities.

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75.

To include gravitational effects in the action (A8) for a spinning point particle, one must, in addition, make the replacement ṘẋμμRv̇aDa*R.

76.

It is worth noting that this form for the equation of motion of a test particle appears naturally in scalar-tensor gravity theory.77,78 In the standard interpretation, the scalar field in scalar-tensor gravity in fact plays the same role of introducing a physical scale, in a similar manner to the scalar field in the WGT, although see Section II L for a further discussion of this issue.

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It is easily shown that ψ̄γa0Da*ψ=ψ̄γa0Daψ (which does not depend on the dilation gauge field Bμ), so the kinetic term in the Dirac action in reduced PGT is already invariant under local dilations, in the same way as its counterpart in “full” PGT, as discussed in Section II J.

82.
Q.
Exirifard
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M. M.
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Phys. Lett. B
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85.

Since the eμ(x) are derived from a coordinate system, they form a holonomic set of basis vectors, whereas the set eˆa(x) is, in general, non-holonomic.

86.

We need not consider the local Lorentz and GCT parts of the transformation, since the behaviour of the gravitational gauge fields haμ and Aabμ under these transformations will remain the same as in WGT.

87.

The arbitrary parameter θ could, in principle, be promoted to a field θ(x). Provided it has Weyl weight w = 0, one would obtain the same θ-independent form for the eWGT covariant derivative Dμ and the same gauge theory.

88.

The dagger on the derivative operator is simply to distinguish the eWGT covariant derivative from the PGT and WGT covariant derivatives Dμ and Dμ*, respectively, and should not be confused with the operation of Hermitian conjugation. In particular, we note that the eWGT covariant derivative Dμφ=[Dμ+12(VabbμVbbaμ)ΣabwVμ13wTμ]φ represents a far more substantial modification to Dμφ than the WGT covariant derivative Dμ*φ=(Dμ+wBμ)φ.

89.
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M.
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92.

In terms of their counterparts 0Aabc and Kabc in PGT (see  Appendix B), one has 0Aabc=0Aabcδca(Vb+13Tb)+δcb(Va+13Ta) and Kabc=Kabc+13δcaTb13δcbTa, so confirming that the decomposition (158) also holds with all daggers removed, as discussed in Section II E.

93.

An alternative and more “automated” route for constructing covariant forms of tab and τab is given in Section III J.

94.

It is easily shown that ψ̄γa0Daψ=ψ̄γa0Daψ (which does not depend on the dilation gauge field Vμ), so the kinetic term in the Dirac action in reduced PGT is already invariant under extended local dilations, in the same way as its counterpart in “full” PGT, as discussed in Section III K.

95.

It is worth noting that the semi-metricity condition (211) and that given in (124) for WGT can both be written as σgμν=12(Γλλσ0Γλλσ)gμν, where 0Γλμν is the metric connection; this also holds in PGT.

96.

One can straightforwardly construct parity-odd terms for inclusion in the free gravitational Lagrangian in an analogous manner to WGT, but we do not consider such terms here.

97.

The expression (222) can also be written in a form for which its symmetries are more manifest, namely, LR2=α1R2+α2R(ab)R(ab)+α3R[ab]R[ab]+α4Ra(bc)dRa(bc)d+α5Ra[bc]dRa[bc]d+α6RabcdRcdab, where the new coefficients are α2=α2+α3, α3=α2α3, α4=α4+α5, and α5=α4α5. In this case, the eWGT version of the Gauss–Bonnet identity allows one to set α1 or α6 to zero as before, or set α2=α3.

98.

The expression (227) can also be written in the more manifestly symmetric form LT2=β1T(ab)cT(ab)c+β2T[ab]cT[ab]c, where β1=β1+β2 and β2=β1β2.

99.

One might even consider the case in which SG and SM are themselves not individually invariant under (extended) local dilations, but the total action ST = SG + SM is so. We will not pursue this class of theories, however, since they would require the presence of (non locally scale-invariant) matter to induce local scale invariance of the full theory in the presence of gravitation (although such a possibility cannot be ruled out a priori). It seems more natural to demand the theory to be scale invariant in vacuo, in which case SG and SM must be separately locally scale invariant.

100.

One may also straightforwardly include the electromagnetic field in LM, coupled both to the Dirac field and to gravity, in an analogous manner to that discussed at the end of Section II K, but will we not consider this extension here.

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103.

Once again, one may also straightforwardly include the electromagnetic field in LM, coupled both to the Dirac field and to gravity, but will we not consider this extension here.

104.
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Doran
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105.

Moreover, after a long but straightforward calculation, one may show that 0Rabcd is itself related to the corresponding WGT quantity 0R*abcd by 0R*abcd=0Rabcd2δd[b(0DcBc)Ba]+2δc[b(0DdBd)Ba]2BeBeδc[aδdb], from which one finds that their contractions are related by 0R*ac=0Rac2(0DcBc)Ba(0De+2Be)Beδca and 0R*=0R6(0De+Be)Be. Similarly, 0Rabcd and its contractions are related to the corresponding eWGT quantities by the same relationships, but with the replacements 0R*abcd0Rabcd and Ba(Va+13Ta).

106.

More appropriately, the theory is also known as Einstein–Cartan–Kibble–Sciama (ECKS) theory, but we shall adopt the more usual, shorter naming convention (with apologies to Kibble and Sciama).

107.
A.
Lewis
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108.

It is probably fair to say that Dirac would not have considered LG in (D1) to be his entire gravitational Lagrangian and would have also included the term proportional to ϕ2R that appears in (D2). Since this term contains the scalar field ϕ, however, we have instead included it in the matter Lagrangian.

109.

In fact, Dirac’s original theory has vanishing torsion and so he proposed a matter Lagrangian of the form (D2), but with the WGT covariant derivative and Ricci scalar replaced by their “reduced” versions 0Da* and 0R*, respectively, as described in Section II E. We extend his approach here to include torsion.

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