arXiv:1407.1820v3 [hep-th] 28 Jan 2015 an natatv betv,i so neett e if see to interest of is it objective, attractive an mains invariant. conformal locally dimensionless is via constants interacting coupling bosons gauge fermions massless massless of of level and consisting the universe a at attractive Lagrangian since potentially the theory now fundamental invariance a for scale desideratum introduced, electromag- local metricate had the Weyl loses to one that able Also longer way. no this is netism one is and electromagnetism lost, metricate to attempt Weyl’s nection n with and tal oee,b taking by However, all. at A ∇ ope ag rnfrainta cson acts that transformation by gauge replaced complex was a transformation given scale any real Weyl’s at chanics, system history. a prior its of on state dependent de- being the a path moment than with being rather and transport geometry pendent, parallel Weyl with a metric one, has the Riemannian one this of and of derivative non-zero presence covariant is the the in connection since Weyl successful not was attempt rnfrainudrwihbt Λ both which under transformation a h moiino el oa cl rcnomltrans- conformal or scale local was both real, second that a formation the of And imposition the potential. vector electromagnetic rclycagdfilsbtde o c on act not does but fields charged trically to W ns h rtwsagnrlzto fteLevi-Civita the of ingredi- generalization key a two Λ was involved grav- first connection approach of The Weyl’s formulation ents. geometric itself. a Ein- metricate ity developed after to long first tried that stein who not Weyl ap- electromagnetism by an (geometrize) Such pioneered seek geometric. was to intrinsically proach appealing is very that is approach it an forces fundamental other the oicuean include to µ µ ic osbemtiaino lcrmgeimre- electromagnetism of metrication possible a Since olwn h usqetdvlpeto unu me- quantum of development subsequent the Following ncnepaigapsil nfiaino rvt with gravity of unification possible a contemplating In λ A g µν en elcdby replaced being µν T P ν = ( ∇ − x − ) ymty ofra ymty n h ercto fElect of Metrication the and Symmetry, Conformal Symmetry, A g ν → µ λα A nain hoyo rvt en on oeeg.A extens An s emerge. magnetic to and found being electric gravity its of theory o treats invariant use that the modific Through electromagnetism this With of obtained. is connection. fermions the to in electromagnetism appears then that itself o aetecneto ob elbtt nta be instead to but real be to w Weyl connection electromagnetism unlike the However, take of not connection. theory geometric metricated the a in present develop to them use λ o pern ntegoercconnection geometric the in appearing not µ ( A e µν epeetsm neetn oncin between connections interesting some present We g 2 r etivrat oee,a uh the such, as However, invariant. left are µ να .INTRODUCTION I. α dpnetgoercWy connection Weyl geometric -dependent ( (1 = x A ) µ g g iA µν + µν / ( µ g x 2) and µα nteculn ocagdfields charged to coupling the in ), g A A λα A µ A ν µ ( − eateto hsc,Uiest fCnetct Storrs, Connecticut, of University Physics, of Department µ u ftegoerccon- geometric the of out ∂ ( µ x atcpt naccording in participate g T029 S.eal [email protected] email: USA. 06269, CT g ) νµ να → A + α where ) λ A ∂ µν µ g ν ( µν g x + µα A + ) tal with all, at µ W Dtd aur 8 2015) 28, January (Dated: − A hlpD Mannheim D. Philip n elec- and λ ∂ µ µν ∂ µ α sthe is α g ( and νµ x ), ) T P ht f ncmlt aallt h a n rasthe treats one way the to parallel of complete this coupling show In in we Specifically, acceptable. if, possibility. be that, a would such then it present it we modify that paper and so way program some original in Weyl’s revisit could one opoueaflyacpal ercto felectromag- of which able metrication in then acceptable netism is fully one a itself, produce connection to geometric Weyl the in ants ntefis lc.Dsieti,w aefound replace have we we this, if Despite that place. Weyl’s first Thus electro- the in described all. have magnetism at could never fermions connection charged geometric to couple not thus h ia cinWy’ ueyra eeaie connec- generalized real into purely inserted in- ( Weyl’s when tion fields that action fermion note, Dirac on to the but particular fields in metric and stead, and gauge the on forces. lead fundamental thus the to all fields, of gauge metrication axial to a torsion fields to via gauge this and vector Moreover, well, as non-Abelian geometry. to Riemannian as generalizes just standard approach given in be to is out well, it turns as then solved transport being parallel even since problem with transport way, coupling parallel minimal the standard the than other none to ic h tnadReana eiCvt connection Levi-Civita Riemannian requires. Λ coupling standard minimal as the do just then couple Since it to does found only is not it potential and so, vector fermions, to the couple complex, does it then make thereby to nection replace the i opigt emosi h tnadlclminimal local standard the an in via or fermions way either to electromagnetism, coupling of via a description obtain dual we connection completely geometric complex now this With httasom h aewyas way same the transforms that inteimdaecneni that is concern immediate the tion nusbigteslsm n ntetocss n with and cases, two the metricated. in being one electromagnetism selfsame the being ensues rgnlWy rsrpin.(ne h iceeantilin- discrete the (Under ear prescription). Weyl original T P λ h e tpta stkni hsppri ofcsnot focus to is paper this in taken is that step key The o,wt uhan such with Now, ∂ µν ymti,wt tbeing it with symmetric, R T P µ A ymtyadcnomlsmer.We symmetry. conformal and symmetry − d sbsdon based is 4 µ cossmercly ihaconformal a with symmetrically, ectors o otennAeincs sprovided. is case non-Abelian the to ion ∂ 2 to h tnadmnmlculn of coupling minimal standard the ation x oso eoti erctdtheory metricated a obtain we torsion f ofis dacdti osblt,w do we possibility, this advanced first ho rnfrainta ecnie eo tis it below consider we that transformation iA µ en el un u odo u dnial and identically out drop to out turns real) being ( − nwihteeetoantcfil is field electromagnetic the which in by µ A g ) ntegoerccneto,rte hnto than rather connection, geometric the in µ 1 ∂ iA / µ ocagdfils n replaces one fields, charged to 2 µ i − A ψγ ∂ dpnetgoerccneto,with connection, geometric -dependent ¯ µ µ A 2 h rsrpini ogeneralize to is prescription the , A a µ ste ope ocagdfilsin fields charged to coupled then is V µ iA a by µ ntegoerccneto (the connection geometric the in ( µ ∂ iA dpnetgoercconnec- geometric -dependent iA µ +Σ µ µ ahrthan rather nWy’ emti con- geometric Weyl’s in bc ω ∂ romagnetism µ bc µ g − µν n not and iA ol hncou- then would A µ ) µ ψ cinthat action A A µ µ by itself.) iA iA ∂ µ µ µ 2 ple through a then generalized, and even complex, Rie- ometric connection and a standard purely Riemannian mann tensor and lead to a gravity theory that does not Levi-Civita-based spin connection, then, as we discuss in look anything like the gravity that is observed. How- detail below, on doing a path integration on the fermions ever, because of Weyl’s very same conformal invariance (equivalent to a fermion one loop Feynman diagram) one this does not in fact occur. Specifically, for the standard obtains an effective action for gravity and electromag- µνστ µν minimal coupling of fermions to Aµ to be able to pos- netism that is precisely of the Cµνστ C plus Fµν F µνστ sess Weyl’s local conformal invariance, Aµ would have to form, where the Cµνστ C term is based on the stan- have conformal weight zero and not transform under a dard Levi-Civita connection alone. With the generalized µν conformal transformation at all, viz. Aµ → Aµ, (just as Weyl connection only appearing as the Fµν F term, µνστ gµν does not transform under an electromagnetic gauge and with the Cµνστ C term being based on the stan- transformation). Because of this, the only geometric ac- dard Levi-Civita connection alone, the geometry is then tion one could write down that would be locally con- strictly Riemannian and the iAµ dependence does not ap- formal invariant would be the one based on the square pear in the coupling of the metric to the geometry. Thus µνστ (viz. Cµνστ C ) of the Weyl tensor Cµνστ (cf. (47) the path integration on the fermions serves to produce an and (48) below) as constructed via the Levi-Civita con- effective action for gravity and electromagnetism in which nection alone, with a generalized Weyl tensor built out of the Levi-Civita and Weyl connections are completely de- the Levi-Civita plus iAµ-dependent Weyl connection not coupled. Moreover this decoupling persists even if we i being locally conformal invariant for an Aµ that does extend the theory to non-Abelian Aµ and even if we add not transform under a conformal transformation. Now in torsion as well, and even if we generalize torsion to such a generalized Weyl-connection-dependent Weyl ten- the non-Abelian case. That this decoupling occurs is be- sor would have been locally conformal invariant had Aµ cause the Dirac action for a fermion coupled to the vari- transformed non-trivially (Aµ → Aµ + ∂µα(x)) under ous Weyl, torsion, and Levi-Civita connections turns out a conformal transformation just as Weyl had originally to be locally conformally invariant (up to fermion mass proposed. However, with an Aµ that does not trans- terms), so that a path integration over the fermions will form at all under a local conformal transformation, this necessarily produce an effective action for the various gµν very same conformal invariance then forces the geometry and Aµ fields whose leading term is locally conformal in- to depend on the Levi-Civita connection alone, with the variant too (the effect of mass is non-leading since the Weyl connection contributing solely to and being buried mass term is a soft operator). The fermion path integra- in the coupling of Aµ to the fermions. With the pure tion thus does the separation of the gµν and Aµ sectors metric sector of the theory only depending on the Levi- for us without our needing to impose it in advance. Civita connection, parallel transport is thus strictly Rie- To underscore the point we note that had we started mannian, and thus through the replacement of Aµ by iAµ with a completely conventional Dirac action in which the in the Weyl connection we convert Weyl geometry into fermion is coupled to the geometry through a Levi-Civita- Riemannian geometry. based spin connection and coupled to Aµ through con- Thus by making two changes in Weyl’s approach, ventional minimal coupling (viz. (33) below), fermion namely by replacing Aµ by iAµ in the Weyl geometric path integration would generate an effective action con- connection and by giving Aµ the zero conformal weight taining a purely Riemannian geometry in the gµν sector µν that the Dirac action requires, one can then construct a and a purely conventional Fµν F term in the Aµ sector metrication of electromagnetism. Moreover, the exten- (viz. (43)). We would not at all expect to get an effective sion to the strong and weak gauge theories is immediate, action that would involve Weyl geometry, and of course since if one also gives the non-Abelian gauge fields confor- we do not. With the metrication that we present here mal weight zero and couples them to the geometry via an leading to a complete duality between minimal coupling analogous non-Abelian iAµ-type geometric connection, and the iAµ-based Weyl connection approach, the Weyl µνστ they also do not couple in the geometric Cµνστ C but connection approach must generate the selfsame Dirac µν only in non-Abelian generalizations of Fµν F . Since the action, and thus it too must lead to an effective action in strong, electromagnetic and weak interactions are based which there is a complete separation between the grav- on the non-Abelian SU(3) × SU(2) × U(1) local gauge ity and electromagnetic sectors, with the gravity sector theory, our approach thus permits a metrication of all being based on the Levi-Civita connection alone. the fundamental forces, with their chiral aspects being To develop the results presented in this paper we need accommodated through the introduction a further geo- to explore the interplay of geometric connections with metric connection, namely one with torsion. PT symmetry, CPT symmetry, and conformal symme- Now the reader might be concerned that our results are try. We present the various geometric connections of in- somewhat restrictive since they appear to require the a terest to us in Sec. II, and in Sec. III we present the vari- priori imposition of local conformal invariance. However, ous PT , CPT , conformal and Lorentz symmetry aspects this turns out not to be the case, since one can obtain of interest to us here. In Sec. IV we discuss metrication our results via a completely different procedure. Specif- associated with torsion, and in Sec. V we discuss metri- ically, if one starts with the Dirac action for a fermion cation associated with the Weyl connection. Finally, we coupled to the geometry via both an iAµ-dependent ge- comment on the fact that our approach leads us to con- 3

λ formal gravity rather than to standard Newton-Einstein a form which follows since δΓ µν is a true tensor. gravity. Reviews of torsion may be found in [1–3], and λ Each different choice of δΓ µν defines its own geome- recent reviews of Weyl geometry may be found in [4] and try, each with its own R˜λ . Our interest here is two [5]. A review of PT symmetry (P is parity, T is time µνκ particular connections: the previously introduced Weyl reversal) may be found in [6]. Some recent discussion of connection conformal gravity may be found in [7–9] and [10–13]. λ λα W µν = −g (gναAµ + gµαAν − gνµAα), (9) II. THE VARIOUS SPACETIME CONNECTIONS AND THE DIRAC ACTION and the contorsion connection 1 Kλ = gλα(Q + Q − Q ), (10) A. The Spacetime Connections µν 2 µνα νµα ανµ

In order to construct covariant derivatives in any where curvature-based theory of gravity one must introduce a λ λ λ λ Q µν =Γ µν − Γ νµ (11) three-index connection Γ µν , with the only requirement on it being that it transform under a coordinate trans- formation xµ → x′µ as is the Cartan torsion tensor associated with a connec- tion that has an antisymmetric part. With the Weyl dx′λ dxβ dxγ d2xρ dx′λ Γ′λ (x′)= Γα (x)+ . (1) connection being symmetric on its two lower indices and µν dxα dx′µ dx′ν βγ dx′µdx′ν dxρ the contorsion connection being antisymmetric on them, With this condition covariant derivatives such as we can anticipate that these two connections will respec- tively have some relation to vector and axial vector fields. λν λν λ αν ν λα ∇µg = ∂µg +Γ αµg +Γ αµg (2) Of the two connections the metric obeys a metricity λ λ transform as true general coordinate tensors, i.e. as condition when δΓ µν = K µν . However it does not do λ λ ˜ µν ′ ′ so when δΓ µν = W µν , since for it one has ∇σg = dx λ dx ν dxγ µν ∇′ gλν (x′)= ∇ gαβ(x). (3) −2g Aσ. While this is actually a quite intriguing rela- µ α β ′µ γ 2α(x) dx dx dx tion since it is left invariant under gµν (x) → e gµν (x), Moreover, given only that the connection transforms as Aµ(x) → Aµ(x)+ ∂µα(x), it nonetheless leads to a path in Eq. (1), the four-index object dependence to parallel transport, thereby rendering Weyl geometry untenable as is. λ λ λ η λ η λ R µνκ = ∂κΓ µν − ∂ν Γ µκ +Γ µν Γ ηκ − Γ µκΓ ην (4) Nothing that we know of requires us to consider either λ transforms as a true rank four tensor and is known as the of these two choices for δΓ µν , and nothing would ap- Riemann curvature tensor. pear to go wrong if they are not considered. However, For pure Riemannian geometry the connection is given they do have certain advantages. Use of the torsion con- by the Levi-Civita connection nection provides insights into spin and axial gauge sym- metry, and use of the Weyl connection provides insights λ 1 λα into vector gauge invariance and conformal invariance. Λ µν = g (∂µgνα + ∂ν gµα − ∂αgνµ), (5) 2 Recently, we have shown [14–16] that torsion provides in- and with it the metric obeys the metricity (or metric sights into both gravitation and electromagnetism. And λν compatible) condition ∇µg = 0. in this paper we show that these developments are inter- λ related with PT symmetry and Weyl geometry in a way However, one is free to add on to Λ µν any additional rank three tensor δΓλ since Γ˜λ =Λλ + δΓλ will that will enable us to both metricate electromagnetism µν µν µν µν and convert Weyl geometry into standard Riemannian λ ˜λ still obey (1) if δΓ µν is itself a tensor. In terms of Γ µν geometry, and thereby dispose of its parallel transport one defines covariant derivatives such as problem. ˜ λν λν ˜λ αν ˜ν λα ∇µg = ∂µg + Γ αµg + Γ αµg , (6) and whether or not the metric obeys the generalized B. The Spin Connection λν metricity condition ∇˜ µg = 0 depends on the choice λ of δΓ µν . Additionally, the four index object λ While one uses the connection Γ µν to implement local ˜λ ˜λ ˜λ ˜η ˜λ ˜η ˜λ R µνκ = ∂κΓ µν − ∂ν Γ µκ + Γ µν Γ ηκ − Γ µκΓ ην (7) translation invariance, to implement local Lorentz invari- a ance one introduces a set of vierbeins Vµ where the co- is also a true tensor. In terms of the Levi-Civita-based ordinate a refers to a fixed, special-relativistic reference derivative ∇ the generalized R˜λ can be rewritten as µ µνκ coordinate system with metric ηab, with the Riemannian metric then being writable as g = η V aV b. With the R˜λ = Rλ + ∇ δΓλ − ∇ δΓλ µν ab µ ν µνκ µνκ κ µν ν µκ vierbein carrying a fixed basis index its covariant deriva- η λ η λ λ + δΓ µν δΓ ηκ − δΓ µκδΓ ην , (8) tives are not given by Γ µν alone. Rather, one introduces 4

ab a second connection known as the spin connection Ωµ , this is no longer the case. When one has a more general with it being the derivative connection the Dirac action is given by

aλ aλ λ aν ab λ 1 4 1/2 a µ bc DµV = ∂µV +Λ V +Ω V (12) I˜D = d x(−g) iψγ¯ V (∂µ +Σbcω˜ )ψ + H.c. (18) νµ µ b 2 Z a µ that will transform as a tensor under both local trans- Following a few algebraic steps I˜D is found to take the lations and local Lorentz transformations provided the form spin connection transforms as 1 I˜ = I + d4x(−g)1/2iψV¯ aµV bλV cν ′ab a b cd bc a D D 16 Ωµ =Λ c(x)Λ d(x)Ωµ − Λ (x)∂µΛ c(x) (13) Z † × (δΓλνµγa[γb,γc] + (δΓλνµ) [γb,γc]γa)ψ (19) under V a(xλ) → Λa (x)V c(Λλ xτ ). For a standard Rie- µ c µ τ with some additional terms now being generated. It is mannian geometry the spin connection is given by these explicit additional terms that will enable us to met- − ωab = V b∂ V aν + V bΛλ V aν , (14) ricate the fundamental forces. µ ν µ λ νµ With a view to what is to follow below, in (19) we λ and with this connection the vierbein obeys metricity have expressly not taken δΓ νµ to be real or Hermitian. aλ Recalling that in the form DµV = 0. Finally, when one uses the ˜λ λ λ a b c b c a ab c ac b generalized connection Γ µν = Λ µν + δΓ µν , one must γ [γ ,γ ] − [γ ,γ ]γ = 4η γ − 4η γ ab ab ab a b c b c a abcd 5 use the generalized spin connectionω ˜µ = ωµ + δωµ , γ [γ ,γ ] + [γ ,γ ]γ = 4iǫ γdγ , where γ5 = iγ0γ1γ2γ3, ab ab b λ aν ǫabcdV µV ν V σV τ = (−g)−1/2ǫµνστ , (20) − ω˜µ = −ωµ + Vλ δΓ νµV , (15) a b c d

we can rewrite I˜D as withω ˜ab obeying (13) if Γ˜λ obeys (1). Given the gen- µ µν 1 eralized spin connection the metric will only obey the I˜ = I + d4x(−g)1/2iψγ¯ δΓdψ (21) λν D D d generalized metricity condition ∇˜ µg = 0 if the vier- 4 Z ˜ aλ bein obeys the generalized DµV = 0. where 1 δΓd = [δΓ + (δΓ )†](−g)−1/2ǫµλντ iγ5V d 2 µλν µλν τ C. Connections and the Dirac Equation 1 + [δΓ − (δΓ )†][gµλV dν − gµν V dλ]. (22) 2 µλν µλν To introduce spinors one starts with the free mass- λ less Dirac action in flat space, viz. the Poincare in- As we see, if δΓ νµ is in fact real, the only connection 4 a ˜ variant (1/2) d xiψγ¯ ∂aψ plus its Hermitian conjugate that could couple in ID would be that part of it that is 4 ¯ a (or equivalentlyR (1/2) d xiψγ ∂aψ plus its CPT conju- antisymmetric on all three of its indices. Thus of the two gate), where the fixedR basis Dirac gamma matrices obey connections of interest to us only the torsion-dependent λ λ γaγb + γbγa = 2ηab (with diag[ηab] = (1, −1, −1, −1) K νµ as evaluated with a real Q νµ could possibly couple λ here). To make this action invariant under local trans- to the fermion, with W νµ as evaluated with a real Aµ lations one introduces a (−g)1/2 factor in the measure not being able to couple to the fermion at all, a result first a a µ and replaces γ ∂a by γ Va ∂µ, and to make the action noted in [17]. Thus the Weyl connection as introduced locally Lorentz invariant one introduces the spin connec- by Weyl (viz. one with a real Aµ) could not serve to met- tion. Thus, in a standard curved Riemannian space with ricate electromagnetism, and such an Aµ could not serve λ ab connections Λ µν and ωµ , the Dirac action is given by as the electromagnetic vector potential. As we will show below, we will rectify this by taking the Weyl connection 1 4 1/2 ¯ a µ bc not to be Hermitian at all but to be PT symmetric in- ID = d x(−g) iψγ Va (∂µ +Σbcωµ )ψ + H.c.,(16) 2 Z stead, in consequence of which Aµ will be replaced by iAµ in it. where Σ = (1/8)(γ γ −γ γ ). Following an integration ˜ λ ab a b b a For the torsion contribution to ID with a real Q νµ by parts and some algebraic steps ID can be written as evaluation is straightforward and yields (see e.g. [18],[2])

4 1/2 a µ bc ˜ 4 1/2 ¯ a µ bc 5 ID = d x(−g) iψγ¯ V (∂µ +Σbcω )ψ. (17) ID = d x(−g) iψγ Va (∂µ +Σbcωµ − iγ Sµ)ψ, (23) Z a µ Z where As is familiar from experience with flat space actions, µ 1 −1/2 µαβγ we see that the inclusion of the Hermitian conjugate did S = (−g) ǫ Qαβγ, not generate any new terms in the action. However, for 8 −1/2 µ connections more general than the Levi-Civita-based one, −4(−g) ǫµαβγS = Qαβγ + Qγαβ + Qβγα. (24) 5

In the action I˜D we note that even though the torsion is path is. Since path integral quantization is a completely only antisymmetric on two of its indices, just as required c-number approach to quantization, it makes no reference the only components of the torsion that appear in its to any Hilbert space at all and thus makes no reference torsion-dependent Sµ term are the four that constitute to any quantum Hamiltonian at all. Rather, the path that part of the torsion that is antisymmetric on all three integral generates the Green’s functions of the quantum of its indices. These four torsion components couple to theory, i.e. it generates matrix elements of quantum op- the fermion via an axial vector current, and thus couple erators. It is only after constructing the Hilbert space not to the electric current but to a magnetic current in- in which those operators act could one then determine stead. A possible role for Sµ in electromagnetism as an whether or not the quantum Hamiltonian might be Her- axial vector potential was discussed in [16], and we will mitian. With PT symmetry on the other hand one knows return to the issue below. However before we do this, a lot about the quantum theory before even starting to we need to discuss the relation between PT symmetry, evaluate the path integral. In the same way as working conformal symmetry, and Lorentz symmetry. not with the Hamiltonian but with the action integral of the Lagrangian has always been beneficial for establish- ing the symmetry structure of a quantum theory, it is III. PT , LORENTZ, CONFORMAL, AND CPT equally the case for PT symmetry. SYMMETRIES When a Hamiltonian is not Hermitian it is not appropriate to use the Dirac norm, since if |R(t)i A. PT Symmetry is a right eigenstate of H then hR(t)|R(t)i = † hR(0)|eiH te−iHt|R(0)i is not equal to hR(0)|R(0)i, with A PT transformation differs from either a conformal the norm not being time independent. However, if in- transformation or a Lorentz transformation in two signif- stead of being Hermitian the Hamiltonian is PT sym- icant ways. First it is not a continuous transformation metric, then one should use a norm involving not the but a discrete one, and second it is not a linear transfor- Dirac conjugate of |R(t)i but its PT conjugate instead mation but through time reversal is an antilinear one. Its [6]. If we introduce a left eigenstate hL(t)| of H, then the utility for physics was developed by Bender and collabo- appropriate PT theory norm can be written [22] as the iHt −iHt rators [6] following the discovery [19] that the eigenvalues time independent hL(t)|R(t)i = hL(0)|e e |R(0)i = of the non-Hermitian Hamiltonian H = p2 + ix3 were all hL(0)|R(0)i. In this way one can obtain unitary time real. As we thus see, while Hermiticity is sufficient to evolution in theories with non-Hermitian Hamiltonians, yield real eigenvalues it is not necessary. With the Hamil- with it being shown in [22] that PT symmetry of a Hamil- tonian H = p2 +ix3 being PT symmetric (P xP −1 = −x, tonian is a both necessary and sufficient condition for TiT −1 = −i), and with E∗ being an eigenvalue of unitary time evolution, with Hermiticity only being suf- any PT -symmetric Hamiltonian H if E is an eigenvalue ficient one. (HPT |ψi = PTH|ψi = P T E|ψi = E∗PT |ψi), it was A further benefit of the PT theory norm is that in recognized that one could also get real eigenvalues via PT cases where the Dirac norm hR(t)|R(t)i is found to be of symmetry. Subsequently it was recognized that the key negative ghost state form, a cause for this can be that issue was not the reality of the eigenvalues themselves but the Hamiltonian is not Hermitian, with one then not be- of the secular equation f(λ)= |H − λI| that determines ing permitted to use the Dirac norm. Thus rather than them, with it being shown first that if H is PT symmet- signaling that a theory is not unitary, the presence of a ric then f(λ) is a real function of λ [20], and second that negative Dirac norm could be signaling that one is not if f(λ) is a real function of λ, then H must possess a in a Hermitian theory and that one should not be using PT symmetry [21]. Since a complex f(λ) would require the Dirac norm at all, and in such a situation the prop- ′ that at least one eigenvalue be complex, PT symmetry agator would be given not by hΩR|T (φ(x)φ(x ))|ΩRi but ′ was thus identified as being the necessary condition for by hΩL|T (φ(x)φ(x ))|ΩRi instead. There are two cases reality of eigenvalues. with negative Dirac norms that have been identified in A benefit of PT symmetry is that with it one can make the literature as being PT theories, with both of their statements about the eigenvalues of a Hamiltonian just hL(t)|R(t)i norms then being found to be positive defi- by checking its symmetry structure, not only without any nite. The PT norm has been found to be relevant [23] need to determine whether or not the Hamiltonian is Her- to the Lee model, and [24, 25], [8, 9] to the conformal mitian (which requires studying its behavior at asymp- gravity theory that we shall encounter below. totic spatial infinity to check whether one can drop sur- face terms in integrations by parts), but without even needing to solve for the eigenvalues at all. Moreover, with B. PT Symmetry and the Lorentz Group PT being a symmetry, one can study the symmetry of every path in a path integral quantization, and thus with- While PT symmetry is thus seen to be more general out actually doing the integration one can know ahead of than Hermiticity, as stressed in [6] it is also a physical time that the Hamiltonian of the quantum theory that requirement on a theory rather than the mathematical will result will be PT symmetric if every path integral requirement that H = H†. Indeed, both parity and time 6 reversal symmetries are physical ones that many theories generators transform xµ and x2 according to possess, and in relativistic field theory properties of PT µ µ µ µ µ ν invariance carry over to CPT invariance in those cases x → x + ǫ , x → Λ ν x where PT is not a symmetry but CPT is. As regards xµ + cµx2 Poincare invariance, we note that the Hamiltonian is the xµ → λxµ, xµ → , 1+2c · x + c2x2 generator of time translations regardless of whether or x2 not it might be Hermitian. And as regards Lorentz in- x2 → λ2x2, x2 → . (25) variance, we note that the Lorentz group has a PT ex- 1+2c · x + x2 tension. Specifically, under the combined PT transfor- With the 15 infinitesimal generators acting on the co- mation xµ transforms as xµ →−xµ, with PT thus being µ µ ordinates x according to (∂µ = (∂/∂t, ∂/∂x), ∂ = compatible with Lorentz invariance as PT (but not P or (∂/∂t, −∂/∂x) here) T separately) treats all four components of xµ equiva- lently [26]. P µ = i∂µ,M µν = i(xµ∂ν − xν ∂µ), Moreover, there is an intimate connection between PT D = ixµ∂ Cµ = i(x2ηµν − 2xµxν )∂ , (26) symmetry and the structure of the irreducible represen- µ ν tations of the Lorentz group. Consider for instance the E B E together they form the 15-parameter SO(4, 2) conformal standard and fields of electromagnetism. The group, with algebra field is P odd and T even, to thus be PT odd, while the B field is P even and T odd, to thus be PT odd also. [Mµν ,Mρσ]= i(−ηµρMνσ + ηνρMµσ Lorentz transformations that mix the E and B fields thus mix fields with the same PT . Now the E and B fields − ηµσ Mρν + ηνσ Mρµ), transform according to the D(1, 0) ⊕ D(0, 1) representa- [Mµν , Pσ]= i(ηνσPµ − ηµσPν ), [Pµ, Pν ]=0, tion of the Lorentz group. However, this representation [Mµν , Cσ]= i(ηνσCµ − ηµσCν ), [Mµν ,D]=0, is reducible, with the irreducible components being the [Cµ, Cν ]=0, [Cµ, Pν ]=2i(ηµν D − Mµν ), left- and right-handed E − iB and E + iB. While irre- ducible under the Lorentz group, as we see under a PT [D, Pµ]= −iPµ, [D, Cµ]= iCµ. (27) transformation E − iB →−(E + iB) [27]. The six fields The utility of the conformal group is that while time- E and B while reducible under SO(3, 1) alone are thus ir- like, lightlike or spacelike distances are preserved by reducible under SO(3, 1) × PT . Exactly the same is true the 10 Poincare transformations, lightlike distances are of the left- and right-handed fermions, which respectively preserved by all 15 conformal group transformations, transform as D(1/2, 0) and D(0, 1/2). They are reducible with the light cone thus having a symmetry larger than under SO(3, 1) but irreducible under SO(3, 1) × PT [28]. Poincare. With the flat space free massless particle prop- An analogous pattern occurs for the vector and axial agator also depending only on the distance (cf. 1/x2 for vector currents. For the vector current J µ = ψγ¯ µψ we spin zero scalars and γ xµ/x4 for spin one half fermions), note that J 0 is P even and T even, to thus be PT even, µ free flat space massless particles possess all 15 conformal while J i is P odd and T odd, to thus be PT even also. group invariances. Theories in which all particles are Since the vector current couples to A , A is PT even. µ µ massless at the level of the Lagrangian and all coupling For the axial vector current Kµ = ψγ¯ µγ5ψ we note that constants are dimensionless thus have an underlying con- K0 is P odd and T even, to thus be PT odd, while Ki formal structure. With conformal invariance being tied is P even and T odd, to thus be PT odd also. Since in with masslessness at the level of the Lagrangian, to Sµ couples to the axial current in the generalized Dirac ˜ generate masses we would thus have to break the con- action ID given in (23), it follows that Sµ is PT odd. formal symmetry via vacuum dynamics. Moreover, this is precisely the standard SU(3) × SU(2) × U(1) picture of strong, electromagnetic and weak interactions, where C. Global Conformal Symmetry all fermions and gauge bosons have no mass at the level of the Lagrangian and all couplings in the pure fermion As well as being able to relate left- and right-handed gauge boson sector are dimensionless. When we make irreducible representations of the Lorentz group via a dis- the conformal transformations local, which we do below, crete PT symmetry, it is also possible to relate them via this will lead us to a theory of gravity, conformal gravity a set of continuous transformations instead, with the req- (a strictly Riemnannian variant of the Weyl geometry of uisite transformations being conformal transformations, interest to us in this paper), in which its coupling con- viz. precisely those transformations that are relevant to stants are dimensionless too. the Weyl geometry of interest to us in this paper. In flat The conformal algebra admits of a 4-dimensional space the conformal group enlarges the 10 parameter flat spinor representation since the 15 Dirac matrices γ5, space Poincare group with its P µ and M µν generators γµ, γµγ5,[γµ,γν ] also close on the SO(4, 2) algebra. to include five more flat space generators, a dilatation The group SU(2, 2) is the covering group of SO(4, 2) operator D and four conformal generators Cµ. With re- with the 4-dimensional spinor being its fundamental rep- µ µ µ spective constant parameters ǫ ,Λ ν , λ and c the 15 resentation. Thus unlike the Lorentz group SO(3, 1) 7 where a 4-component spinor transforms according to the obey D(1/2, 0)⊕D(0, 1/2) representation, under the conformal group all four components are irreducible, with the con- ∂µKν + ∂ν Kµ = 2(λ − 2c · x)ηµν . (30) formal transformations mixing the left- and right-handed If we now allow a , b , λ, and c to become space- spinors, doing so via transformations that are continuous. µ µν µ time dependent, we note that the λ − 2c · x factor in Since this holds for all spinors no matter what their inter- (30) becomes just one general spacetime-dependent func- nal quantum numbers might be, in a conformal invariant tion. We thus anticipate having only 11 local symmetries theory neutrinos would have to have four components rather than the initial 15 global ones. (This is to be ex- too, with right-handed neutrinos being needed to accom- pected since under the global D and C transformations pany the observed left-handed ones. µ given in (25) x2 transforms as x2 → λ2x2, x2 → x2/(1 + The fact that 4-component fermions are irreducible un- 2c · x + x2), with a global C being a particular local D. der the conformal group means that conformal transfor- µ Also, as can be seen from (27), M , P and D close on mations mix components with opposite PT . In [28] we µν µ an algebra all on their own.) Referring now to the Dirac had noted that under a PT transformation a Dirac spinor action I = d4x(−g)1/2iψγ¯ aV µ(∂ + Σ ωbc)ψ given transforms as P T ψ(t, x)T −1P −1 = −γ2γ5ψ(−t, −x), D a µ bc µ in (17), we see that it possesses four local translation in- with its conjugate transforming as PT ψ¯(t, x)T −1P −1 = R variances and six local Lorentz invariances (as it of course −ψ¯(−t, −x)γ2γ5. We now recognize this transformation must since the V µ vierbeins and the ωbc spin connection as being none other than a conformal transformation a µ were expressly introduced for this purpose). However, since γ2γ5 is one of the 15 generators of the conformal ID also possesses one local conformal invariance as well, group. PT symmetry is thus integrally connected with 2α(x) since it is left invariant under gµν (x) → e gµν (x), conformal symmetry. And because of this, conformal a α(x) a −3α(x)/2 transformations will thus mix Lorentz group represen- Vµ (x) → e Vµ (x), ψ(x) → e ψ(x) with arbi- tations such as E − iB and E + iB. trary spacetime dependent α(x). With this local confor- Given the fundamental 4-dimensional representation mal invariance we find that ID does indeed have 11 local of the conformal group, by constructing the 4 ⊗ 4∗ direct invariances [30], just as anticipated [31]. product we can make both a 15-dimensional adjoint rep- As had been noted above, we could generalize ID to the ˜ λ resentation of the conformal group and a singlet. With general ID given in (19) provided δΓ νµ is itself a true λ the 15 Dirac gamma matrices and the identity matrix rank-three tensor. For any δΓ νµ that is a rank-three spanning a general 4 × 4 matrix space, we see that in tensor the general I˜D will still be both locally translation the irreducible decomposition of 4 ⊗ 4∗ we have precisely invariant and locally Lorentz invariant. However, requir- λ ˜ the needed number of independent Dirac gamma matri- ing that the contribution of δΓ νµ to ID also be locally ces. We can thus anticipate that the associated fermion conformal invariant will constrain how the fields in δΓλ ¯ 5 µ µ 5 µ ν νµ bilinear currents ψΓψ with Γ = 1,iγ ,γ ,γ γ ,i[γ ,γ ] are to transform under a local conformal transformation. µ could play a central role in physics, with Γ = γ and To therefore identify the conformal properties needed µ 5 ˜ ˜ Γ= γ γ being seen to appear in ID (cf. (21)) or JD, its for the Sµ term in I˜D, we note that since the Levi-Civita Aµ extension given in (33) below [29]. connection transforms as

λ λ λ λ λ Λ µν → Λ µν + (δµ∂ν + δν ∂µ − gµν ∂ )α(x), (31) D. Local Conformal Symmetry a straightforward transformation for the torsion that In order to extend the above global conformal symme- takes into account its antisymmetry structure is [2, 32] try to a local symmetry, we note that while the confor- Qλ → Qλ + q(δλ∂ − δλ∂ )α(x), (32) mal group has 15 generators no 4-dimensional space can µν µν µ ν ν µ have more than 10 Killing vectors, viz. vectors that obey where q is the conformal weight of the torsion tensor. ∇µKν + ∇ν Kµ = 0. Since flat spacetime is maximally 4- While the specific value taken by q is not known, there symmetric it has 10 vectors that obey ∂µKν + ∂ν Kµ = 0, appear to be two natural choices for it. One of course viz. the 10 Kµ that are embodied in is simply q = 0. And since the torsion tensor has to ν Kµ = aµ + bµν x , (28) have the same engineering dimension as the Levi-Civita connection, it must have engineering dimension equal to where aµ is a constant four-vector and bµν is a constant one, with q = 1 thus being the other. However regardless 6-component antisymmetric rank two tensor. To account of this, it was noted in [14] that in fact no matter what for the remaining five generators of the conformal group the value of q, the α(x)-dependent term in (32) actually we introduce conformal Killing vectors, viz. vectors that drops out identically in Sµ, with Sµ thus having confor- obey ∇µKν + ∇ν Kµ = f(x)gµν where f(x) is an appro- mal weight equal to zero. Since the term that Sµ couples priate scalar function. For flat spacetime we find that to in I˜ , viz. (−g)1/2ψγ¯ aV µγ5ψ, has conformal weight with λ being a constant scalar and c being a constant D a µ zero itself (4 − 3/2 − 1 − 3/2 = 0), we thus establish that four-vector the five Kµ that are embodied in the Sµ-dependent term in I˜D term is locally conformal 2 Kµ = λxµ + cµx − 2xµc · x (29) invariant, just as required. 8

There is, however, a completely different local way to E. PT Symmetry and CPT Symmetry view the Sµ-dependent term in I˜D. Suppose we start with the torsion independent I and instead of changing D Beyond all these continuous symmetries, J˜ has the connection at all require that the action be invari- D two further symmetries, as it is invariant under a ant under a local chiral transformation on the fermion of 5 discrete PT symmetry and a discrete CT P symme- the form ψ(x) → eiγ β(x)ψ(x) with spacetime-dependent try. As regards first PT symmetry, we note that β(x). To maintain the chiral symmetry we would need to given the PT transformation properties of the fermion minimally couple in an axial vector field S (x) that trans- µ fields, the generic ψ¯(xµ)Γψ(xµ) will transform into forms as S (x) → S (x)+ ∂ β(x), and the resulting ac- µ µ µ ψ¯(−xµ)γ2γ5Γ∗γ2γ5ψ(−xµ). Thus ψψ¯ , ψiγ¯ 5ψ, and ψγ¯ µψ tion that we would obtain would be precisely none other are PT even, while ψγ¯ µγ5ψ and ψi¯ [γµ,γν ]ψ are PT odd. than I˜ as given in (23). In such a case we would have to D Now we had noted earlier that A is PT even and S is appeal to the zero conformal weight of (−g)1/2ψγ¯ aV µγ5ψ µ µ a PT odd. With i∂ and iwbc both being PT even [in (14) to establish that S (x) should have conformal weight zero µ µ µ the [∂/∂xµ]V aν (−xµ) = −[∂/∂(−xµ)]V aν (−xµ) term in itself [33]. While inspection of I˜D alone could thus not b aν λ bc V ∂µV , and analogously for Λ , makes w act as an tell us whether S (x) is associated with a torsionless ge- ν νµ µ µ odd PT operator in the d4x = d4(−x) integration], ometry or with one with torsion, the geometry would still ˜ λ ˜λ we see that every term inRJD is PTR even. PT symmetry know, since one would have to use either R µνκ or R µνκ. However, as we will see below, even this distinction will is thus again seen to accompany conformal symmetry. disappear; and even if there were to be a distinction, As regards CPT symmetry, we recall that, with it itself would not involve any parallel transport prob- Cψ(t, x)C−1 = iγ2ψ†(t, x), Cψ¯(t, x)C−1 = ψ(t, x)iγ2γ0, λν x −1 −1 −1 lem since the covariant derivative ∇˜ µg of the metric as under CPT we obtain CPTψ(t, )T P C = λ † x 5 ¯ x −1 −1 −1 constructed with the contorsion tensor K µν is zero. iψ (−t, − )γ , CPT ψ(t, )T P C = −iγ0γ5ψ(−t, −x). Following an antisymmetric in- In the same way that we could introduce Sµ via a terchange of ψ and ψ¯ both ψγ¯ µψ and ψγ¯ µγ5ψ are found local axial symmetry, we could equally of course intro- to be CPT odd, while ψψ¯ , ψiγ¯ 5ψ, and ψi¯ [γµ,γν ]ψ duce the vector potential Aµ via a local vector symmetry, are CPT even. Under the same antisymmetric in- since on requiring invariance under ψ(x) → eiα(x)ψ(x) 4 1/2 ¯ a µ terchange we obtain (1/2) d x(−g) iψγ Va (∂µ + with spacetime-dependent α(x) we would need to min- Σ ωbc)ψ = −(1/2) d4x(−R g)1/2i(∂ ψ)V µ(γa)Trψ¯ − imally couple in a vector field A that transforms as bc µ µ a µ (1/2) d4x(−g)1/2iψV µR(Σ )Trωbc(γa)Trψ¯ (Tr denotes A (x) → A (x)+∂ α(x). With such a coupling I˜ would a bc µ µ µ µ D transpose), to find that the CPT and Hermitian con- be replaced by R jugates of this expression are equal. We thus establish that ID as given in (16) is CPT symmetric. With both ψV¯ µγaψ and ψV¯ µγaγ5ψ being CPT odd, the full J˜ is 4 1/2 a µ bc a a D J˜D = d x(−g) iψγ¯ V (∂µ +Σbcω µ µ Z a µ CPT invariant since A and S are both CPT odd also 5 (Aµ is PT even and C odd, and Sµ is PT odd and C −iAµ − iγ Sµ)ψ. (33) even). Minimal coupling is thus fullly CPT symmetric. bc bc bc For the contribution of δωµ =ω ˜µ − ωµ , Given that (−g)1/2ψγ¯ aV µψ has conformal weight zero, we note that the Hermitian and CPT conju- a 4 1/2 ¯ a µ bc gates of (1/2) d x(−g) iψγ Va Σbcδωµ )ψ are J˜D will be locally conformal invariant if, just like the axial R 4 1/2 ¯ µ bc † a Sµ, the vector Aµ has conformal weight zero too. In fact given by (1/2) d x(−g) iψVa (δωµ ) Σbcγ ψ and 4 1/2 ¯ µ bc CP T a just as had been discussed in [33] in regard to Sµ, the con- −(1/2) d x(−g)R iψVa (δωµ ) Σbcγ ψ. These two bc † bc CP T formal weight of Aµ can also be determined from global conjugatesR will thus coincide if (δωµ ) = −(δωµ ) , scale invariance considerations alone. The fact that Aµ is but not otherwise. With a metricated Sµ (and thus not to transform under a local conformal transformation Qαβγ) being Hermitian and CPT odd, and with the is of significant import since it constitutes a quite major iAµ-based connection that we actually use below being departure from Weyl’s original intent that it is to trans- anti-Hermitian and CPT even, the metrication of both form non-trivially under a conformal transformation, a Aµ and Sµ studied in this paper is thus fully compatible point that will prove crucial below. with both PT symmetry and CPT symmetry. Regardless of how it may or may not have been de- Given all of these remarks, we see that in general if we rived, as an action J˜D is quite remarkable as it has a wish to consider any specific contribution to the generic λ very rich local invariance structure. J˜D is invariant un- connection δΓ νµ, each such contribution is constrained λ der local translations, local Lorentz transformations, lo- in three distinct ways. The contribution to δΓ νµ would cal gauge transformations, local axial gauge transforma- need to be a true rank-three tensor, it would need to keep tions, and local conformal transformations. Moreover, I˜D locally conformal invariant, and it would need to keep J˜d is not just invariant under any arbitrary set of local I˜D PT (and also CPT ) even. Since we have seen that we transformations, it is invariant under some of the key can introduce Sµ either by a local gauge invariance or by local transformations in physics [34]. a metrication that meets these three requirements, it is 9 natural to ask whether we could do the same for Aµ and with the Aµ and Sµ sectors thus being decoupled in the introduce it by a metrication procedure that meets these action. Inspection of (39) shows it to be both locally three requirements as well. However in order to do so for conformal invariant and PT symmetric, again just as we the electromagnetic vector potential Aµ, we first need to would want [38]. discuss the relation of the axial S to electromagnetism. 01 12 µ With the usual F = −Ex, F = −Bz etc. identifi- cation of the field strengths, we can give physical signifi- cance to the Sµ sector by introducing a second set of field IV. THE RELATION OF TO Sµ strengths S01 = −B′ , S12 =+E′ , Sˆ01 = E′ , Sˆ12 = B′ , ELECTROMAGNETISM x z x z etc. In terms of the field strengths, we find that in flat µ µ space with J = (ρe, Je) and K = (ρm, −Jm), the gen- A. An Axial Vector Potential eralized Maxwell equations given in (37) decompose into the standard sector While we have related Sµ to torsion in the above, Sµ can also be related to electromagnetism. If we consider E ∇ B ∂ J ∇ E the standard Maxwell equations as coupled to an elec- × − = e, · = ρe, µ ∂t tric vector current J in a standard curved Riemannian ∂B background geometry, viz. ∇ × E + =0, ∇ · B =0, (40) ∂t νµ µ −1/2 µνστ ∇ν F = J , (−g) ǫ ∇ν Fστ =0, (34) we count a total of eight equations. If we wish to obtain and a primed sector all eight of these equations via a variational principle we would need to vary with respect to eight different ∂E′ ∇ × B′ − =0, ∇ · E′ =0, quantities [35]. As noted in [16], given the structure of ∂t (34) these eight would need to be a vector Aµ and an axial ∂B′ ∇ × E′ + = J , ∇ · B′ = ρ . (41) vector Sµ. In fact one should use these eight potentials ∂t m m if magnetic currents are present. Indeed, recalling the study [36, 37] of the magnetic monopole problem, it is E E E′ B B B′ very convenient to introduce Finally, if we define TOT = + , TOT = + , we can combine (40) and (41) into Xµν = ∇µAν − ∇ν Aµ 1 −1/2 µνστ ∂E − (−g) ǫ (∇σSτ − ∇τ Sσ) (35) ∇ × B − TOT = J , ∇ · E = ρ , 2 TOT ∂t e TOT e µν µν µ ν ν µ as a generalized F . On setting S = ∇ S −∇ S , we ∂BTOT µν µν µ ν ∇ × E + = J , ∇ · B = ρ . (42) can rewrite X in terms of the standard F = ∇ A − TOT ∂t m TOT m ν µ µν −1/2 µνστ µν ∇ A and the dual Sˆ = (1/2)(−g) ǫ Sστ of S according to: Thus even if Jm and ρm can be neglected, it is ETOT and µν µν µν µν µν µν X = F − Sˆ , Xˆ = Fˆ + S . (36) BTOT that are measured in electromagnetic experiments. 0123 µν (If ǫ = +1, ǫ0123 = −1.) Given this X , (34) is replaced by νµ νµ µ ˆ νµ νµ µ ∇ν X = ∇ν F = J , ∇ν X = ∇ν S = K , B. PT Structure of Chiral Electromagnetism νµ νµ ∇ν Fˆ = 0, ∇ν Sˆ =0, (37) µ νµ 0 where K is a magnetic current, with it being ∇ν Xˆ = In terms of P , T assignments, K = ρm is P odd and i i Kµ that is to describe the magnetic monopole sector. T even, to thus be PT odd, while K = −Jm is P even On introducing the action and T odd, to thus be PT odd also. Consequently, the E′ field is P odd and T odd, to thus be PT even, while 4 1/2 1 µν µ µ the B′ field is P even and T even, to thus be PT even I = d x(−g) − Xµν X − AµJ − SµK , (38) Z  4  also. With E and B both being PT odd, we see that ETOT and BTOT contain components with opposite PT . we find that stationary variation with respect to Aµ and However, no transition between them could be generated Sµ then immediately leads to (37), just as we would want. Moreover, up to surface terms this action decomposes by the action given in (39) since in it the Aµ and Sµ into two sectors according to sectors are decoupled. To obtain any such transitions we could introduce the conformal invariant, CPT invariant µ µ 4 1/2 1 µν µ couplings AµK and SµJ , though PT symmetry would I = d x(−g) − Fµν F − AµJ µ ν Z  4 then be lost. The higher order coupling AµK Sν J is 1 both PT and CPT invariant. − S Sµν − S Kµ . (39) 4 µν µ  We summarize the discrete transformation properties 10 of the fields and currents of interest to us in a table invariance, with standard QED being set up this way. However, starting from the same action it is just as nat- PTPTCPT PTPTCPT ural to equally introduce Sµ via a local axial gauge in- ′ E − + − + E − − + + variance, with a chiral QED then being set up. That B + − − + B′ +++ + this option is not ordinarily followed is because QED is ordinarily discussed without consideration either of set- ρe ++ + − ρm − + − − J − − + − J + −− − ting up a variational procedure for Faraday’s Law or of e m magnetic monopoles. However, one of the arguments in A0 ++ + − S0 − + − − favor of monopoles is to be symmetric between the elec- A − − + − S + −− − tric and magnetic currents. But then, if one wants to ∇ · B − − + − ∇ · B′ − + − − consider such symmetry one should extend it to poten- ∇ · E ++ + − ∇ · E′ + −− − tials that couple to these currents. A second reason not to consider an axial potential is that in QED the chiral in which we have also listed the properties of ∇ · B, symmetry is broken since fermions have mass. Since it is ∇ · B′, ∇ · E, and ∇ · E′. As we see, only ∇ · B′ could now understood that mass can be induced by dynamics, couple to ρm, and only ∇ · E could couple to ρe. The that objection is no longer valid. Moreover, not only can primed sector B′ is thus needed to provide a coupling to mass be induced dynamically, in a conformal invariant a magnetic monopole ρm that B itself could not provide. theory mass must be induced dynamically since there can As introduced above Sµ is just an axial vector potential be no mass scales at the level of the Lagrangian. Since to be used in Maxwell theory, and does not need to pos- the tachyonic mass term associated with a fundamental sess any relation to the Sµ that appears in the fermionic Higgs scalar field would violate the conformal symmetry, J˜D given in (33). To establish a relation we recall that there should be no such tachyonic term present in the Lagrangian, with all mass scales having to come from when one does a J˜D path integration DψDψ¯ exp(iJ˜D) over the fermions (equivalent to a one fermionR loop Feyn- quantum fluctuations. Finally, if one does want symme- man graph) one generates [10], [2] an effective action of try between the electric and magnetic sectors, with Sµ the form being able to have a geometric origin, it is thus natural to seek a geometric origin for Aµ too. In fact not only 4 1/2 1 µν 1 α 2 is it natural, that is what led Weyl to Weyl geometry in IEFF = d x(−g) C Rµν R − (R ) Z 20  3 α  first place. 1 + (∂ A − ∂ A )(∂µAν − ∂ν Aµ) 3 µ ν ν µ V. METRICATION OF ELECTROMAGNETISM 1 µ ν ν µ + (∂µSν − ∂ν Sµ)(∂ S − ∂ S ) , 3  A. Implementing Conformal Invariance 4 1/2 1 µν 1 α 2 = d x(−g) C Rµν R − (R α) Z 20  3  In developing Weyl geometry Weyl generalized the 1 Levi-Civita connection by augmenting it with the Weyl + X Xµν (43) 3 µν  connection to give a full connection of the form ˜λ λ λ where C is a log divergent constant, Rµν is the standard Γ µν = Λ µν + W µν Levi-Civita-based, torsionless, Ricci tensor, and Xµν is 1 = gλα(∂ g + ∂ g − ∂ g ) as given in (35). The action IEFF possesses all the local 2 µ να ν µα α νµ symmetries possessed by J˜D, with the appearance of the λα µν α 2 − g (gναAµ + gµαAν − gνµAα). (44) strictly Riemannian Rµν R − (1/3)(R α) term being characteristic of a gravity theory that is locally conformal Weyl introduced this particular connection since under a invariant (see e.g. [7–9]). Comparing now with (39), we local conformal transformation of the form see that, up to renormalization constants, the action IEFF 2α(x) is precisely of the form needed for Maxwell theory, with gµν (x) → e gµν (x), Aµ(x) → Aµ(x)+ ∂µα(x) (45) torsion thus providing a natural origin for the second ˜λ potential that Maxwell theory needs [39]. Γ µν transforms into itself, to thus be locally conformal invariant. In consequence, the generalized Riemann ten- ˜λ ˜λ sor R µνκ built as per (7) with this Γ µν would be locally C. The Key Role of the Fermion conformal invariant too [40]. However, if one uses this ˜λ Γ µν connection the metric would obey In our work the fermionic action plays a central role. If µν µν we start with the free massless Dirac action in flat space, ∇˜ σg (x)= −2g Aσ(x), (46) 4 a viz. the Poincare invariant (1/2) d xiψγ¯ ∂aψ + H.c., then it is natural to introduce Aµ viaR a local vector gauge with parallel transport then being path independent. 11

As well as develop Weyl geometry, Weyl made a par- under a local conformal transformation. ticularly useful discovery for Riemann geometry itself. If we now introduce the Weyl connection, we need Specifically, he found a purely Riemannian, Levi-Civita- to ask whether it is possible to construct an action based tensor, the Weyl conformal tensor, viz. for the gravitational sector that would contain it and 1 still be invariant under (50). Since on dimensional Cλµνκ = Rλµνκ − (gλν Rµκ − gλκRµν grounds such an action would have to be quadratic, 2 1 the most general one possible would be the combina- −g R + g R )+ Rα (g g − g g ), (47) tion d4x(−g)1/2[aR˜ R˜λµνκ + bR˜ R˜µκ + c(R˜α )2] µν λκ µκ λν 6 α λν µκ λκ µν λµνκ µκ α for someR choice of the coefficients a, b, c. Now under in which, remarkably, all derivatives of α(x) drop out (45) this combination is invariant for any choice of a, b identically under a local conformal transformation on and c. However, if Aµ is not to transform under a confor- 2α(x) the metric of the form gµν (x) → e gµν (x). The mal transformation, this combination would need to be Weyl tensor thus bears the same relation to a lo- invariant order by order in Aµ. For the zeroth order term cal conformal transformation as the Maxwell tensor we noted above that the needed combination is the one does to a local gauge transformation, with the IW = that appears in IW, to thus have a = 1, b = −2, c =1/3, 4 1/2 µνστ −αg d x(−g) Cµνστ C Weyl action with dimen- and thus to have b = 1, c = −1/3 following the use of the sionlessR gravitational coupling constant αg being the con- Gauss-Bonnet theorem. The term that is linear in Aµ in 4 1/2 µν formal analog of the d x(−g) Fµν F Maxwell ac- the combination involves a cross term between the term tion. R that is zeroth order in Aµ and a first order term in Aµ When written in terms of the Riemann tensor IW takes that according to (8) is a total divergence in the Levi- the form Civita-based ∇µ. Recalling that the Riemann tensor ραβγ β αγ γ αβ obeys ∇ρR = ∇ R − ∇ R , up to a total diver- 4 1/2 λµνκ IW = −αg d x(−g) CλµνκC gence the net linear term for the combination is found to Z 4 1/2 λ µν be of the form d x(−g) [(8a +2b)Wλµν ∇ R −(b + 4 1/2 λµνκ λ µ α = −αg d x(−g) RλµνκR 4c)W µλ∇ R αR] [41]. Using the Bianchi identity and the Z  λ explicit form for W µν given in (9), we can write the net µκ 1 α 2 linear term as d4x(−g)1/24(b + 4c)A ∇µRα . Since − 2RµκR + (R α) . (48) µ α 3  this term is notR left invariant under (50) (the Ricci scalar not being a conformal invariant), conformal invariance With (−g)1/2 R Rλµνκ − 4R Rµκ + (Rα )2 being λµνκ µκ α requires b +4c be zero, to thus require b = 1, c = −1/4. a total divergence (the Gauss-Bonnet theorem), the Weyl   Consequently, there is no choice for the coefficients a, b action can be written more compactly as and c for which both the zeroth and first order terms in 4 1/2 ˜ ˜λµνκ ˜ ˜µκ ˜α 2 4 1/2 µκ 1 α 2 d x(−g) [aRλµνκR + bRµκR + c(R α) ] could IW = −2αg d x(−g) RµκR − (R ) , Z  3 α  Rsimultaneously obey (50). Thus if we introduce the (49) Weyl connection and wish to write down an action that to give the form presented in (43). obeys local conformal invariance as realized via (50), the Weyl thus provides us with two specific ways to im- only choice is the Aµ-independent, strictly Riemannian 4 1/2 µκ α 2 plement conformal invariance. To determine which one, IW = −2 d x(−g) RµκR − (1/3)(R α) . Thus if either, might be the relevant one for physics we need even in theR presence of the Weyl connection, the only to determine how Aµ is to transform under a confor- allowed conformal action in the metric sector is the one mal transformation. To this end we look to the cou- that is completely independent of the Weyl connection pling of Aµ not to the geometry but to fermions in- term. Thus if we are able to generate a Dirac action in stead. And as noted above, without any reference to which the Aµ term is associated with the Weyl connec- Weyl geometry, if we construct the Dirac action by cou- tion in some way, the path integration over the fermions pling the fermion to the geometry in a strictly Rieman- would still have to lead to the separation between the µνστ µν nian way while coupling the fermion to Aµ in a standard Cµνστ C and Fµν F terms that is exhibited in (43). minimally coupled local electromagnetic gauge invariant 4 1/2 ¯ a µ bc way, the d x(−g) iψγ Va (∂µ +Σbcωµ − iAµ)ψ ac- tion that resultsR will be locally conformal invariant under 2α(x) a α(x) a B. Complex Weyl Connection gµν (x) → e gµν (x), Vµ (x) → e Vµ (x), ψ(x) → e−3α(x)/2ψ(x) only if A (x) undergoes no transforma- µ In order to be able to actually obtain an tion at all. Hence immediately we see that if we want to 4 1/2 ¯ a µ bc metricate electromagnetism and recover this same Dirac d x(−g) iψγ Va (∂µ + Σbcωµ − iAµ)ψ action action via a generalized connection, we must do so with Rin which the Aµ term is generated geometrically, we recall, as noted above, that this cannot be done with the an Aµ that has conformal weight zero, with gµν and Aµ then respectively transforming as Weyl connection with its real Aµ as is, since the Weyl connection drops out of the Dirac action identically. 2α(x) gµν (x) → e gµν (x), Aµ(x) → Aµ(x) (50) Now one would of course initially want to take Aµ to 12

4 1/2 λµνκ be real, since, first, it is to describe the electromagnetic d x(−g) CλµνκC . Hence with an Aµ with con- λ field, and, second, Aµ plays the same role in W µν as Rformal weight zero (and analogously for Sµ) the theory λ is strictly Riemannian and no parallel transport path de- ∂µ does in Λ µν . However, from the perspective of a complex phase invariance on the fermion field, minimal pendence problem can be encountered. Thus by making electromagnetic coupling is not of the form ∂µ − Aµ but two key changes in Weyl’s metrication program, namely of the form ∂µ − iAµ instead, with Aµ being Hermitian replacing Aµ by iAµ in the Weyl connection and by tak- and iAµ being anti-Hermitian. Moreover, minimal ing Aµ to have conformal weight zero, we are not only coupling must be of this latter form since if Aµ is PT able to metricate electromagnetism in principle, but are even and ∂µ is PT odd, one needs the extra i factor in actually able to obtain the precise structure that any such order to to enforce PT symmetry. electromagnetic metrication must possess. We thus see Now precisely the same reasoning has to apply to the a dual description of electromagnetism. We can induce connection, since we had noted above that the connection it by a local phase transformation on the fermion field ˜λ ˜ has to be PT odd (i.e. iΓ µν has to be PT even if ID in a standard Riemannian background geometry or can λ obtain it by enlarging the connection to include the Weyl is to be PT even). Thus, with Λ µν being PT odd we λ connection. There are no operative distinctions between would need W µν to be PT odd too. To achieve this with λ the two cases [44], and for either one fermion path in- a PT even and Hermitian Aµ we thus replace W µν by tegration yields a purely Riemannian Weyl-tensor-based 2 locally conformal invariant theory of gravity. V λ = − igλα (g A + g A − g A ) , (51) µν 3 να µ µα ν νµ α

λ C. Non-Abelian Generalizations with V µν being PT odd and anti-Hermitian. Insertion λ of V µν with its convenient −2/3 charge normalization † Even though the Weyl and contorsion connections in- into the δΓµλν − (δΓµλν ) term in (22) is then found to lead to none other than the A -dependent contribution to volve fermionic electric and magnetic charge quantum µ numbers, the pure gravitational sector only involves the J˜ precisely as given and normalized in (33), to thereby D Levi-Civita connection. Consequently, the approach oblige A to have conformal weight zero and not trans- µ we have developed here can naturally be extended to form under the conformal group at all. Thus with V λ µν the non-Abelian case, with the metric not being forced we can indeed metricate electromagnetism in the fermion to acquire any internal quantum number. Thus, on sector after all. Finally, with Γ˜λ =Λλ +Kλ +V λ µν µν µν µν putting the fermions into the fundamental representation ˜ we can obtain the entire JD by metrication. Thus while of SU(N)×SU(N) with SU(N) generators T i that obey λ Weyl sidelined W once his scale transformation on Aµ i j ijk k i i µν [T ,T ] = if T , we replace Aµ by gT Aµ and Qαβγ was reinterpreted as a minimal coupling phase transfor- i i i i by gT Q , and thus replace Sµ by gT S in the con- mation with a factor i, we see that this same procedure αβγ µ nections, to obtain a locally SU(N) × SU(N) invariant applied in V λ enables us to reinstate Weyl’s metrication µν Dirac action of the form of electromagnetism after all. λ With the connection V µν not coupling in ∇µAν − ˜ 4 1/2 ¯ a µ bc JD = d x(−g) iψγ Va (∂µ +Σbcωµ ∇ν Aµ [42], and with it acting in J˜D just like conven- Z tional electromagnetic vector potential in the fermionic i i 5 i i −igT Aµ − igγ T Sµ)ψ. (52) sector, in a universe consisting solely of fermions, gauge bosons and gravitons (with mass generation by fermion On doing the path integral on the fermions the previous λ bilinear condensates), the only place where V µν could effective action given in (43) is replaced by [10] ˜λ still be manifest would be in R µνκ, i.e. in the gravi- 4 1/2 1 µν 1 α 2 tational equations of motion should they depend on the IEFF = d x(−g) C Rµν R − (R ) λ 20  3 α  generalized connection. Since the only role of V µν in Z the fermion sector is to act as a standard electromag- 1 1 + Gi Gµν + Si Sµν , (53) netic potential, parallel transport of fermions with a dy- 3 µν i 3 µν i  namics described by J˜D would be just the same as the µν α 2 conventional parallel transport of fermions in a standard with an unmodified Rµν R − (1/3)(R α) term and the Riemannian geometry in the presence of a background same log divergent constant C as before, but with ∂µAν − i i i ijk j k electromagnetic field (and its axial analog [43]). Like- ∂ν Aµ being replaced by Gµν = ∂µAν −∂ν Aµ+gf AµAν , i i i wise, parallel transport of gauge bosons would be the and ∂µSν −∂ν Sµ being replaced by S = ∂µS −∂νS + λ µν ν µ same as in standard Riemannian geometry, since V µν gf ijkSj Sk, just as one would want. In the same vein ˜ ˜ µ ν drops out of ∇µAν − ∇ν Aµ. The only problematic case Xµν of (35) generalizes to would be parallel transport of the gravitational field it- µν µ ν ν µ self. However, we have just seen that this not a prob- Xi = ∇ Ai − ∇ Ai lem either since the only allowed locally conformal in- 1 − (−g)−1/2ǫµνστ (∇ Si − ∇ Si ), (54) variant action is the purely Riemannian-geometry-based 2 σ τ τ σ 13

α 2 with (53) then being written as (1/3)(R α) ] involves fourth-order derivative equations of motion, the theory had long been thought to pos- 4 1/2 1 µν 1 α 2 sess negative norm states or negative energies. However, IEFF = d x(−g) C Rµν R − (R ) Z 20  3 α  detailed examination of the quantization procedure re- vealed [24, 25], [8, 9] that the quantum Hamiltonian is 1 µν + Xi X , (55) not in fact Hermitian but is instead PT symmetric, and 3 µν i  that when one uses the requisite hL(t)|R(t)i norm and a very compact form. hΩL|T (φ(x)φ(y))|ΩRi type Green’s functions there are then neither negative norm states nor negative energies. Consequently, conformal gravity is a fully consistent and D. Final Comments unitary quantum theory of gravity. Interestingly for our purposes here, the key step needed to avoid negative ener- gies was to recognize that the gravitational field gµν had As an action the effective IEFF contains all the symme- tries of J˜ , both all its local ones and its PT and CPT to be an anti-Hermitian rather than a Hermitian field D and be a PT eigenstate [46]. Intriguingly, to be able to symmetries. However, while IEFF contains the Maxwell λ λ action, we note that it does not actually contain the go from W µν to V µν Weyl’s electromagnetic field Aµ Einstein-Hilbert action, and indeed it could not since the had to be reinterpreted in precisely the same way. Einstein-Hilbert action is not locally conformal invari- Moreover, not only is conformal gravity a consistent ant. The gravitational action that IEFF does contain is quantum gravity theory, there is even some encouraging locally conformal invariant, as it of course would have to observational support for it. Specifically, in [47–50] fits be given the local conformal invariance of the underly- were provided to the rotation curves of 141 spiral galax- ing J˜D. Thus we see that local conformal invariance is ies using a universal formula provided by the conformal to gravity what local gauge invariance is to electromag- theory with only one free parameter per galaxy (the stan- netism, and the two are naturally linked to each other dard mass to light ratio of the luminous matter, a param- since photons and gravitons both propagate on the con- eter that is common to all rotation curve studies). In the formal invariant light cone. With a fermion generically fits no need was found for any of the copious amounts transforming as eαRE ψ under a conformal transformation of dark matter required of the standard Newton-Einstein and as eiαIM ψ under an electromagnetic gauge transfor- gravity treatment of rotation curves. With current dark mation, we thus unify gravitation and electromagnetism matter halo studies requiring two free parameters for the by gauging both the real and imaginary parts of the phase halo of each galaxy, to fit the same 141 galaxies dark of the fermion. matter fits require 282 more free parameters than con- The other unifying feature of the conformal gravity formal gravity, with the fitting thus currently favoring sector and the Maxwell sector actions given in IEFF is the conformal theory. that both sectors involve dimensionless couplings alone, To conclude we note that Weyl’s ideas on conformal so that as quantum theories both are renormalizable invariance and unification can still be of relevance today, [45]. However, because a conformal gravity theory based and could be much closer to conventional fundamental 4 1/2 λµνκ 4 1/2 µν on d x(−g) CλµνκC ≡ 2 d x(−g) [Rµν R − physics than had previously been thought to be the case. R R

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[22] P. D. Mannheim, Phil. Trans. Roy. Soc. A 371, 20120060 [31] We also note that gauging can result in a reduction in (2013). symmetry even when internal symmetries are involved. 4 µ i [23] C. M. Bender, S. F. Brandt, J.-H. Chen, and Q. Wang, Consider for instance an action R d xψ¯iiγ ∂µψ where Phys. Rev. D 71, 025014 (2005). i runs from 1 to 8. As such, this action has a full [24] C. M. Bender and P. D. Mannheim, Phys. Rev. Lett. 100, global SU(8) symmetry with the eight fermions being 110402 (2008). in its fundamental representation. And since in SU(8) ∗ [25] C. M. Bender and P. D. Mannheim, Phys. Rev. D 78, 8 ⊗ 8 = 63 ⊕ 1 one can gauge 63 SU(8) currents and ob- 025022 (2008). tain a full local SU(8) symmetry with 63 gauge bosons. [26] Since charge conjugation leaves the coordinates un- However since the adjoint representation of SU(3) is also touched, one could equally have said that in the coor- 8 dimensional, in the very same action one could put the dinate sector CPT is compatible with Lorentz invari- eight fermions in the adjoint of SU(3), and since 8 ⊗ 8 ance. Further discussion of T transformations in relativis- contains a symmetric 1 ⊕ 8 ⊕ 27 part and an antisym- ∗ tic quantum theory may be found in C. M. Bender and metric 8 ⊕ 10 ⊕ 10 part under SU(3), one could instead P. D. Mannheim, Phys. Rev. D 84, 105038 (2011). gauge the eight SU(3) currents to obtain a local SU(3) [27] Not only are the E and B fields reducible under gauge theory and only have eight gauge bosons. Thus af- real Lorentz transformations, they are reducible un- ter a very specific local gauging, the full global SU(8) der complex Lorentz transformations as well, since if symmetry of the action is reduced to a local SU(3). µν exp(iwµν M ) does not mix the D(1, 0) and D(0, 1) com- [32] I. L. Buchbinder and I. L. Shapiro, Phys. Lett. B 151, ponents with each other when wµν is real, it does not 263 (1985). do so if wµν is complex. This is to be contrasted with [33] The situation here is actually even simpler. Since the ax- representations that contain both left- and right-handed ial vector current involves no spacetime derivatives, no components such as D(1/2, 1/2), since here all four com- derivatives of α(x) would be generated in a local confor- ponents do mix under real Lorentz transformations, and mal transformation on it. Hence simply on global scale thus continue to do so under complex ones, with the invariance grounds Sµ must have conformal weight zero. PT transformation that takes xµ to −xµ correspond- [34] Extension to a local SU(N) × SU(N) via minimal cou- ing to a sequence of three complex Lorentz transforma- pling is straightforward with Aµ and Sµ being replaced in ′ ′ ′ ˜ i i i i tions x = x cosh ξ + t sinh ξ, y = y cosh ξ + t sinh ξ, z = JD by gT Aµ and gT Sµ. Below we show that this same z cosh ξ + t sinh ξ, each with a boost angle ξ = iπ. Thus extension can be obtained via metrication. under a sequence of PT transformations and Lorentz [35] If we introduce a vector potential Aµ according to boosts with complex boost angle one can transform F µν = ∇µAν − ∇ν Aµ and vary with respect to νµ µ E(t, x) ± iB(t, x) first into −[E(−t, −x) ∓ iB(−t, −x)] Aµ, we would only obtain ∇ν F = J via varia- −1/2 µνστ and then into −[E(t, x) ∓ iB(t, x)]. tion with (−g) ǫ ∇ν [∇σAτ − ∇τ Aσ] = 0 being −1 −1 [28] Since P ψ(t, x)P = γ0ψ(t, −x), T ψ(t, x)T = satisfied identically on every variational path. In or- 1 3 −1 −1 2 5 −1/2 µνστ iγ γ ψ(−t, x), P T ψ(t, x)T P = −γ γ ψ(−t, −x), der to only have (−g) ǫ ∇ν [∇σAτ − ∇τ Aσ] = − − it follows that PT (1 ∓ γ5)ψ(t, x)T 1P 1 = −(1 ± 0 be obeyed at the stationary minimum alone, we γ5)γ2γ5ψ(−t, −x). need it to be non-zero away from the minimum. Since 5 −1/2 µνστ [29] The scalar ψψ¯ and pseudoscalar ψiγ¯ ψ will be associated (−g) ǫ ∇ν [∇σAτ −∇τ Aσ] vanishes identically, we with the fermion condensate mass generating mechanism would need to relate the dual of Fµν to some four vector that is to break the conformal symmetry dynamically. other than Aµ. As noted in [16] Sµ serves this purpose. [30] That ID would have all these invariances is due to the [36] S. Shanmugadhasan, Can. Jour. Phys. 30, 218 (1952). 4 1/2 a fact that the action (1/2) R d x(−g) iψγ¯ ∂aψ + H.c. [37] N. Cabibbo and E. Ferrari, Nuovo Cim. 23, 1147 (1962). that we started with before we sought any local struc- [38] The utility of generating the magnetic monopole sec- ture at all was that of a free flat space massless fermion tor via Sµ rather than by Aµ is that it does not field, viz. a field that is constrained to propagate on require Aµ to have either the singularities (Dirac the light cone and thus possess its full conformal struc- string) or the non-trivial topology (grand unified ture. However, we should note that transformations of monopoles) that are used in order to evade the vanish- −3α(x)/2 −1/2 µνστ the form ψ(x) → e ψ(x) in which the argument ing of (−g) ǫ ∇ν [∇σAτ −∇τ Aσ]. In the Sµ case −1/2 µνστ of the field does not change are initially somewhat differ- (−g) ǫ ∇ν [∇σAτ −∇τ Aσ] does vanish identically, µ ′µ µ ent than an x → x = λx transformation since under with the monopole sector not being associated with Aµ at the latter the argument of the field would change from all. A second benefit to introducing Sµ is that the action ′ xµ to x µ. To see that these two procedures are equiva- in (39) is renormalizable. lent it is simplest to consider the free flat space massless [39] Since both Aµ and Sµ couple to the fermionic currents in 4 1/2 µν scalar field action I = R d x(−η) η ∂µφ(x)∂νφ(x). J˜D, through fermionic loops one could have transitions With the scalar field having conformal weight equal to between the Aµ and Sµ sectors. −1, under a global dilatation the action transforms into [40] As noted in [14], if one sets q = 1 in (32) the spin connec- 4 1/2 µν ′ ′ −2 I = R d x(−η) η ∂µφ(x )∂ν φ(x )λ . On changing ˜λ λ λ ′ tion associated with the connection Γ µν = Λ µν +K µν the integration variable to x µ the action takes the form 4 ′ 1/2 µν 2 ′ ′ ′ ′ −2 would be locally conformal invariant too, as would then I = R d x (−η) η λ ∂µφ(x )∂ν φ(x )λ , to thus be 2 be the generalized Riemann tensor as built from this par- invariant. However, if we define a new metric gµν = λ ηµν ′ − ticular spin connection. and a new field φ = λ 1φ, we can rewrite the action 4 ′ 1/2 µν ′ ′ ′ ′ ′ ′ [41] Our discussion here follows a similar discussion for theo- as I = R d x (−g) g ∂µφ (x )∂ν φ (x ), and can thus ries with torsion that was given in [14]. transfer the transformation on the coordinates to a trans- [42] It is actually unnecessary to show that the Weyl con- formation on the metric. nection decouples from Fµν , since in generalizing be- 15

yond standard Riemanian geometry one can only replace be described by spontaneously broken ones. (It was also the Levi-Civita connection by a generalized connection suggested in [16] that intrinsically antisymmetric torsion in those places where the Levi-Civita connection actu- might instead have escaped detection by being based on ally appears. Since the Levi-Civita connection decou- hard to detect anticommuting Grassmann numbers.) ples from Fµν in a standard Riemannian geometry where [44] From the perspective of minimal coupling there is how- ∇µAν −∇ν Aµ = ∂µAν − ∂ν Aµ, there is no Levi-Civita ever a distinction in principle, since one is not actually connection to generalize. While this does not matter for obliged to couple electromagnetism minimally at all as µν the Weyl connection since it would decouple anyway be- one could introduce a fundamental (e/m)ψF¯ i[γµ,γν ]ψ cause of its symmetry, it does matter for the torsion con- type coupling into electromagnetism as well. However nection since its antisymmetry structure would permit such a coupling is not generated geometrically via the it to couple, with ∇˜ µAν − ∇˜ ν Aµ then being given by Weyl connection, and could anyway not be generated in a λ ∂µAν − ∂ν Aµ + Q µν Aλ. However, this is not the correct conformal invariant theory since given its m dependence, definition of Fµν in the torsion case, and indeed it could the coupling is not conformal invariant. not be since it would not be gauge invariant, so even [45] While the effective IEFF action given in (53) and (55) is in the torsion case one has to set Fµν = ∂µAν − ∂ν Aµ. motivated by local conformal invariance and the gener- Moreover, if one then takes the action to be of the form alized Weyl and torsion connections, we note that it is 4 1/2 µ ν ν µ −(1/4) R d x(−g) (∂µAν − ∂ν Aµ)(∂ A − ∂ A ), the actually more general than that. Specifically, while this Maxwell equations that are then produced by variation IEFF arises as the one fermion loop radiative correction with respect to Aµ will only depend on the Levi-Civita to the J˜D action given in (52), actions such as the J˜D ac- ν µ connection derivative and be of the form ∇ν (∂ A − tion itself will arise in any local non-Abelian gauge theory µ ν ν µ µ ν −1/2 1/2 ν µ ∂ A )= ∂ν (∂ A −∂ A )+(−g) ∂ν (−g) (∂ A − even if the connection is just the Levi-Civita one. In other ∂µAν ) = 0, to thus be independent of the generalized words this action is not just a standard action, but with connection altogether. the appropriate non-Abelian gauge group, it is the one [43] While it is intriguing to give electromagnetism such a that is expressly used for the fundamental forces. Hence, chiral structure, we need to explain why there is no regardless of what explicit form the gravitational sector sign of any axial massless photon, and why its pres- action might take, the IEFF action given in (53) will al- ence would not impair the great success achieved by a ways be generated in any gravitational theory. Thus no quantum electrodynamics that is based purely on Aµ matter what the gravity theory, one will always have to alone. To this end it was suggested in [16] that the deal with a log divergent radiatively-induced conformal axial symmetry is spontaneously broken with Sµ ac- gravity action. Moreover, as noted in [10] radiative loops quiring a Higgs mechanism type mass. On noting that due to other standard fields such as scalars and gauge µ µ 5 µ µ 5 ψγ¯ ψAµ +ψγ¯ γ ψSµ = (1/2)ψ¯(γ −γ γ )ψ(Aµ −Sµ)+ bosons yield a log divergence of the same sign, and thus µ µ 5 (1/2)ψ¯(γ + γ γ )ψ(Aµ + Sµ), we see that a straightfor- the fermionically generated IEFF could not be cancelled ward way to implement a Higgs mechanism for Sµ is to by other fundamental fields. To cancel this divergence embed not just Aµ but also Sµ into a non-Abelian chiral one must therefore introduce a counter term of exactly weak interaction such as the SU(2)L × SU(2)R × U(1) the same form as IEFF, and thus one must introduce the type theory discussed in P. D. Mannheim, Phys. Rev. D IW Weyl action given in (49) into the theory. If that is all 22, 1729 (1980) and references therein. An advantage that one introduces, one then has a fully renormalizable of doing this is that if the theory is broken down to quantum gravitational theory. µν µν SU(2)L × U(1) by making right-handed gauge bosons [46] If we replace gµν by igµν , and thus g by −ig (since µλ µ very heavy, this would explain the lack of detection to g gλν = δν ), then neither the connection nor the Rie- date of the right-handed neutrinos that are required by mann tensor undergo any change. Standard gravitational the conformal symmetry. (If the chiral symmetry break- measurements are thus insensitive as to whether the over- ing is achieved by giving a right-handed neutrino Ma- all phase of the gravitational field is real or purely imag- Tr 5 2 0 jorana mass ψ (1 + γ )iγ γ (1 + γ5)ψ a non-zero vac- inary, with the phase only being measurable via interfer- uum expectation value, then since its PT transform is ence with another field such as the electromagnetic one. Tr 5 2 0 ψ (1 − γ )iγ γ (1 − γ5)ψ, PT would be spontaneously [47] P. D. Mannheim and J. G. O’Brien, Phys. Rev. Lett. broken too.) Thus, rather than being some arcane geo- 106, 121101 (2011). metrical curiosity, because of its association with a metri- [48] P. D. Mannheim and J. G. O’Brien, Phys. Rev. D 85, cation of Sµ, torsion would actually be manifest as a per- 124020 (2012). fectly normal and even quite mundane gauge boson that [49] J. G. O’Brien and P. D. Mannheim, Mon. Not. R. As- gets its mass via the Higgs mechanism. Thus if we seek a tron. Soc. 421, 1273 (2012). metrication of the fundamental forces through the Weyl [50] P. D. Mannheim and J. G. O’Brien, J. Phys. Conf. Ser. and torsion connections, we are led to a quite far reaching 437, 012002 (2013). conclusion, namely that not only must the fundamental forces be described by local gauge theories, they must