Abstract
We present some interesting connections between PT symmetry and conformal symmetry. We use them to develop a metricated theory of electromagnetism in which the electromagnetic field is present in the geometric connection. However, unlike Weyl who first advanced this possibility, we do not take the connection to be real but to instead be PT symmetric, with it being \(iA_{\mu }\) rather than \(A_{\mu }\) itself that then appears in the connection. With this modification the standard minimal coupling of electromagnetism to fermions is obtained. Through the use of torsion we obtain a metricated theory of electromagnetism that treats its electric and magnetic sectors symmetrically, with a conformal invariant theory of gravity being found to emerge. An extension to the non-Abelian case is provided.
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Notes
Since charge conjugation leaves the coordinates untouched, one could equally have said that in the coordinate sector CPT is compatible with Lorentz invariance. Further discussion of T transformations in relativistic quantum theory may be found in [26].
Not only are the \(\mathbf E \) and \(\mathbf B \) fields reducible under real Lorentz transformations, they are reducible under complex Lorentz transformations as well, since if \(\exp (iw_{\mu \nu }M^{\mu \nu })\) does not mix the D(1, 0) and D(0, 1) components with each other when \(w_{\mu \nu }\) is real, it does not do so if \(w_{\mu \nu }\) is complex. This is to be contrasted with representations that contain both left- and right-handed components such as D(1 / 2, 1 / 2), since here all four components do mix under real Lorentz transformations, and thus continue to do so under complex ones, with the PT transformation that takes \(x_{\mu }\) to \(-x_{\mu }\) corresponding to a sequence of three complex Lorentz transformations \(x^{\prime }=x\cosh \xi +t\sinh \xi \), \(y^{\prime }=y\cosh \xi +t\sinh \xi \), \(z^{\prime }=z\cosh \xi +t\sinh \xi \), each with a boost angle \(\xi =i\pi \). Thus under a sequence of PT transformations and Lorentz boosts with complex boost angle one can transform \(\mathbf E (t,\mathbf x )\pm i\mathbf B (t,\mathbf x )\) first into \(-[\mathbf E (-t,-\mathbf x )\mp i\mathbf B (-t,-\mathbf x )]\) and then into \(-[\mathbf E (t,\mathbf x )\mp i\mathbf B (t,\mathbf x )]\).
Since \(P\psi (t,\mathbf x )P^{-1}=\gamma ^0\psi (t,-\mathbf x )\), \(T\psi (t,\mathbf x )T^{-1}=i\gamma ^1\gamma ^3\psi (-t,\mathbf x )\), \(PT\psi (t,\mathbf x )T^{-1}P^{-1}=-\gamma ^2\gamma ^5\psi (-t,-\mathbf x )\), it follows that \(PT(1\mp \gamma ^5)\psi (t,\mathbf x )T^{-1}P^{-1}=-(1\pm \gamma ^5)\gamma ^2\gamma ^5\psi (-t,-\mathbf x )\).
The scalar \(\bar{\psi }\psi \) and pseudoscalar \(\bar{\psi }i\gamma ^5\psi \) will be associated with the fermion condensate mass generating mechanism that is to break the conformal symmetry dynamically.
That \(I_\mathrm{D}\) would have all these invariances is due to the fact that the action \((1/2)\int d^4x(-g)^{1/2}i\bar{\psi }\gamma ^{a}\partial _a\psi +H. c.\) that we started with before we sought any local structure at all was that of a free flat space massless fermion field, viz. a field that is constrained to propagate on the light cone and thus possess its full conformal structure. However, we should note that transformations of the form \(\psi (x)\rightarrow e^{-3\alpha (x)/2}\psi (x)\) in which the argument of the field does not change are initially somewhat different than an \(x^{\mu }\rightarrow x^{\prime \mu }=\lambda ^{-1} x^{\mu }\) transformation since under the latter the argument of the field would change from \(x^{\mu }\) to \(x^{\prime \mu }\). To see that these two procedures are equivalent it is simplest to consider the free flat space massless scalar field action \(I=\int d^4x(-\eta )^{1/2}\eta ^{\mu \nu }\partial _{\mu }\phi (x)\partial _{\nu }\phi (x)\). With the scalar field having conformal weight equal to \(-1\), under a global dilatation the action transforms into \(I=\int d^4x(-\eta )^{1/2}\eta ^{\mu \nu }\partial _{\mu }\phi (x^{\prime })\partial _{\nu }\phi (x^{\prime })\lambda ^{-2}\). On changing the integration variable to \(x^{\prime \mu }\) the action takes the form \(I=\int d^4x^{\prime }(-\eta )^{1/2}\eta ^{\mu \nu }\lambda ^{2}\partial ^{\prime }_{\mu }\phi (x^{\prime })\partial ^{\prime }_{\nu }\phi (x^{\prime })\lambda ^{-2}\), to thus be invariant. However, if we define a new metric \(g_{\mu \nu }=\lambda ^2\eta _{\mu \nu }\) and a new field \(\phi ^{\prime }=\lambda ^{-1}\phi \), we can rewrite the action as \(I=\int d^4x^{\prime }(-g)^{1/2}g^{\mu \nu }\partial ^{\prime }_{\mu }\phi ^{\prime }(x^{\prime })\partial ^{\prime }_{\nu }\phi ^{\prime }(x^{\prime })\), and can thus transfer the transformation on the coordinates to a transformation on the metric.
We also note that gauging can result in a reduction in symmetry even when internal symmetries are involved. Consider for instance an action \(\int d^4x \bar{\psi }_ii\gamma ^{\mu }\partial _{\mu }\psi ^i\) where i runs from 1 to 8. As such, this action has a full global SU(8) symmetry with the eight fermions being in its fundamental representation. And since in SU(8) \(8\otimes 8^*=63\oplus 1\) one can gauge 63 SU(8) currents and obtain a full local SU(8) symmetry with 63 gauge bosons. However since the adjoint representation of SU(3) is also 8 dimensional, in the very same action one could put the eight fermions in the adjoint of SU(3), and since \(8 \otimes 8\) contains a symmetric \(1\oplus 8\oplus 27\) part and an antisymmetric \(8\oplus 10 \oplus 10^*\) part under SU(3), one could instead gauge the eight SU(3) currents to obtain a local SU(3) gauge theory and only have eight gauge bosons. Thus after a very specific local gauging, the full global SU(8) symmetry of the action is reduced to a local SU(3).
The situation here is actually even simpler. Since the axial vector current involves no spacetime derivatives, no derivatives of \(\alpha (x)\) would be generated in a local conformal transformation on it. Hence simply on global scale invariance grounds \(S_{\mu }\) must have conformal weight zero. On setting \(S_{\mu }=V_{\mu }^aS_a\) and equally \(A_{\mu }=V_{\mu }^aA_a\), with \(V_{\mu }^a\), \(S_a\), and \(A_a\) having respective conformal weights of \(+1\), \(-1\), and \(-1\), both \(S_{\mu }\) and \(A_{\mu }\) have conformal weight zero.
Extension to a local \(SU(N)\times SU(N)\) via minimal coupling is straightforward with \(A_{\mu }\) and \(S_{\mu }\) being replaced in \(\tilde{J}_\mathrm{D}\) by \(gT^iA^i_{\mu }\) and \(gT^iS^i_{\mu }\). Below we show that this same extension can be obtained via metrication.
If we introduce a vector potential \(A_{\mu }\) according to \(F^{\mu \nu }=\nabla ^{\mu }A^{\nu }-\nabla ^{\nu }A^{\mu }\) and vary with respect to \(A_{\mu }\), we would only obtain \(\nabla _{\nu }F^{\nu \mu }=J^{\mu }\) via variation with \((-g)^{-1/2}\epsilon ^{\mu \nu \sigma \tau }\nabla _{\nu }[\nabla _{\sigma }A_{\tau }-\nabla _{\tau }A_{\sigma }]=0\) being satisfied identically on every variational path. In order to only have \((-g)^{-1/2}\epsilon ^{\mu \nu \sigma \tau }\nabla _{\nu }[\nabla _{\sigma }A_{\tau }-\nabla _{\tau }A_{\sigma }]=0\) be obeyed at the stationary minimum alone, we need it to be non-zero away from the minimum. Since \((-g)^{-1/2}\epsilon ^{\mu \nu \sigma \tau }\nabla _{\nu }[\nabla _{\sigma }A_{\tau }-\nabla _{\tau }A_{\sigma }]\) vanishes identically, we would need to relate the dual of \(F_{\mu \nu }\) to some four vector other than \(A_{\mu }\). As noted in [16] \(S_{\mu }\) serves this purpose.
The utility of generating the magnetic monopole sector via \(S_{\mu }\) rather than by \(A_{\mu }\) is that it does not require \(A_{\mu }\) to have either the singularities (Dirac string) or the non-trivial topology (grand unified monopoles) that are used in order to evade the vanishing of \((-g)^{-1/2}\epsilon ^{\mu \nu \sigma \tau }\nabla _{\nu }[\nabla _{\sigma }A_{\tau }-\nabla _{\tau }A_{\sigma }]\). In the \(S_{\mu }\) case \((-g)^{-1/2}\epsilon ^{\mu \nu \sigma \tau }\nabla _{\nu }[\nabla _{\sigma }A_{\tau }-\nabla _{\tau }A_{\sigma }]\) does vanish identically, with the monopole sector not being associated with \(A_{\mu }\) at all. A second benefit to introducing \(S_{\mu }\) is that the action in (39) is renormalizable.
Since both \(A_{\mu }\) and \(S_{\mu }\) couple to the fermionic currents in \(\tilde{J}_\mathrm{D}\), through fermionic loops one could have transitions between the \(A_{\mu }\) and \(S_{\mu }\) sectors.
As noted in [14], if one sets \(q=1\) in (32) the spin connection associated with the connection \(\tilde{\Gamma }^{\lambda }_{\phantom {\alpha }\mu \nu }=\Lambda ^{\lambda }_{\phantom {\alpha }\mu \nu }+K^{\lambda }_{\phantom {\alpha }\mu \nu }\) would be locally conformal invariant too, as would then be the generalized Riemann tensor as built from this particular spin connection.
Our discussion here follows a similar discussion for theories with torsion that was given in [14].
It is actually unnecessary to show that the Weyl connection decouples from \(F_{\mu \nu }\), since in generalizing beyond standard Riemannian geometry one can only replace the Levi-Civita connection by a generalized connection in those places where the Levi-Civita connection actually appears. Since the Levi-Civita connection decouples from \(F_{\mu \nu }\) in a standard Riemannian geometry where \(\nabla _{\mu }A_{\nu }-\nabla _{\nu }A_{\mu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }\), there is no Levi-Civita connection to generalize. While this does not matter for the Weyl connection since it would decouple anyway because of its symmetry, it does matter for the torsion connection since its antisymmetry structure would permit it to couple, with \(\tilde{\nabla }_{\mu }A_{\nu }-\tilde{\nabla }_{\nu }A_{\mu }\) then being given by \(\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu } +Q^{\lambda }_{\phantom {\lambda }\mu \nu }A_{\lambda }\). However, this is not the correct definition of \(F_{\mu \nu }\) in the torsion case, and indeed it could not be since it would not be gauge invariant, so even in the torsion case one has to set \(F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }\). Moreover, if one then takes the action to be of the form \(-(1/4)\int d^4x(-g)^{1/2}(\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu })(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu })\), the Maxwell equations that are then produced by variation with respect to \(A_{\mu }\) will only depend on the Levi-Civita connection derivative and be of the form \(\nabla _{\nu }(\partial ^{\nu }A^{\mu }-\partial ^{\mu }A^{\nu })=\partial _{\nu }(\partial ^{\nu }A^{\mu }-\partial ^{\mu }A^{\nu })+ (-g)^{-1/2}\partial _{\nu }(-g)^{1/2}(\partial ^{\nu }A^{\mu }-\partial ^{\mu }A^{\nu })=0\), to thus be independent of the generalized connection altogether.
While it is intriguing to give electromagnetism such a chiral structure, we need to explain why there is no sign of any axial massless photon, and why its presence would not impair the great success achieved by a quantum electrodynamics that is based purely on \(A_{\mu }\) alone. To this end it was suggested in [16] that the axial symmetry is spontaneously broken with \(S_{\mu }\) acquiring a Higgs mechanism type mass. On noting that \(\bar{\psi }\gamma ^{\mu }\psi A_{\mu }+\bar{\psi }\gamma ^{\mu }\gamma ^5\psi S_{\mu }=(1/2)\bar{\psi }(\gamma ^{\mu }-\gamma ^{\mu }\gamma ^5)\psi (A_{\mu }-S_{\mu })+(1/2)\bar{\psi }(\gamma ^{\mu }+\gamma ^{\mu }\gamma ^5)\psi (A_{\mu }+S_{\mu })\), we see that a straightforward way to implement a Higgs mechanism for \(S_{\mu }\) is to embed not just \(A_{\mu }\) but also \(S_{\mu }\) into a non-Abelian chiral weak interaction such as the \(SU(2)_\mathrm{L}\times SU(2)_\mathrm{R}\times U(1)\) type theory discussed in [30] and references therein. An advantage of doing this is that if the theory is broken down to \(SU(2)_\mathrm{L}\times U(1)\) by making right-handed gauge bosons very heavy, this would explain the lack of detection to date of the right-handed neutrinos that are required by the conformal symmetry. (If the chiral symmetry breaking is achieved by giving a right-handed neutrino Majorana mass \(\psi ^\mathrm{Tr}(1+\gamma ^5)i\gamma ^2\gamma ^0(1+\gamma _5)\psi \) a non-zero vacuum expectation value, then since its PT transform is \(\psi ^\mathrm{Tr}(1-\gamma ^5)i\gamma ^2\gamma ^0(1-\gamma _5)\psi \), PT would be spontaneously broken too.) Thus, rather than being some arcane geometrical curiosity, because of its association with a metrication of \(S_{\mu }\), torsion would actually be manifest as a perfectly normal and even quite mundane gauge boson that gets its mass via the Higgs mechanism. Thus if we seek a metrication of the fundamental forces through the Weyl and torsion connections, we are led to a quite far reaching conclusion, namely that not only must the fundamental forces be described by local gauge theories, they must be described by spontaneously broken ones. (It was also suggested in [16] that intrinsically antisymmetric torsion might instead have escaped detection by being based on hard to detect anticommuting Grassmann numbers).
From the perspective of minimal coupling there is however a distinction in principle, since one is not actually obliged to couple electromagnetism minimally at all as one could introduce a fundamental \((e/m)\bar{\psi }F^{\mu \nu }i[\gamma _{\mu },\gamma _{\nu }]\psi \) type coupling into electromagnetism as well. However such a coupling is not generated geometrically via the Weyl connection, and could anyway not be generated in a conformal invariant theory since given its m dependence, the coupling is not conformal invariant.
While the effective \(I_\mathrm{EFF}\) action given in (53) and (55) is motivated by local conformal invariance and the generalized Weyl and torsion connections, we note that it is actually more general than that. Specifically, while this \(I_\mathrm{EFF}\) arises as the one fermion loop radiative correction to the \(\tilde{J}_\mathrm{D}\) action given in (52), actions such as the \(\tilde{J}_\mathrm{D}\) action itself will arise in any local non-Abelian gauge theory even if the connection is just the Levi-Civita one. In other words this action is not just a standard action, but with the appropriate non-Abelian gauge group, it is the one that is expressly used for the fundamental forces. Hence, regardless of what explicit form the gravitational sector action might take, the \(I_\mathrm{EFF}\) action given in (53) will always be generated in any gravitational theory. Thus no matter what the gravity theory, one will always have to deal with a log divergent radiatively-induced conformal gravity action. Moreover, as noted in [10] radiative loops due to other standard fields such as scalars and gauge bosons yield a log divergence of the same sign, and thus the fermionically generated \(I_\mathrm{EFF}\) could not be cancelled by other fundamental fields. To cancel this divergence one must therefore introduce a counter term of exactly the same form as \(I_\mathrm{EFF}\), and thus one must introduce the \(I_\mathrm{W}\) Weyl action given in (49) into the theory. If that is all that one introduces, one then has a fully renormalizable quantum gravitational theory.
If we replace \(g_{\mu \nu }\) by \(ig_{\mu \nu }\), and thus \(g^{\mu \nu }\) by \(-ig^{\mu \nu }\) (since \(g^{\mu \lambda }g_{\lambda \nu }=\delta ^{\mu }_{\nu }\)), then neither the connection nor the Riemann tensor undergo any change. Standard gravitational measurements are thus insensitive as to whether the overall phase of the gravitational field is real or purely imaginary, with the phase only being measurable via interference with another field such as the electromagnetic one.
Because the conformal gravity theory potential obeys a fourth-order rather than a second-order Poisson equation, the theory departs from the standard Newton-Einstein theory on large distance scales, and it is through these specific departures that the theory is actually able to fit galactic rotation curves without any need for dark matter. Specifically, in [35] it was shown by Mannheim and Kazanas that in a static, spherically symmetric geometry the equation of motion that follows from (48) is given exactly and without approximation by \(\nabla ^4B(r)=f(r)\) where \(B(r)=-g_{00}(r)\) and \(f(r)=3(T^0_{\phantom {0}0}-T^r_{\phantom {r}r})/4\alpha _gB(r)\). The general solution to the equation of motion is given by \(B(r)=-(1/6)\int _0^r dr^{\prime }f(r^{\prime })(3r^{\prime 2}r+r^{\prime 4}/r) -(1/6)\int _r^{\infty } dr^{\prime }f(r^{\prime })(3r^{\prime 3}+r^{\prime }r^2)+B_0(r)\) where \(B_0(r)\) obeys \(\nabla ^4B_0(r)=0\). Consequently exterior to a localized source of radius R the source puts out a potential \(B(r)=1-2\beta /r+\gamma r\) where \(2\beta =(1/6)\int _0^R dr^{\prime }f(r^{\prime })r^{\prime 4}\), \(\gamma =-(1/2)\int _0^R dr^{\prime }f(r^{\prime })r^{\prime 2}\). Thus in the fourth-order theory the Newtonian potential is modified at large distances, and gravity is sensitive to more moments of the source than in the familiar second-order case. If one applies \(\nabla ^4B(r)\) to \(1-2\beta /r+\gamma r\) one obtains \(-8\pi \gamma \delta ^3(\mathbf{x})+8\pi \beta \nabla ^2 \delta ^3(\mathbf{x})\), and thus, as noted by Mannheim and Kazanas, in the conformal theory sources must be extended or singular, with an extended f(r) having both a height (cf. the \(\gamma \) term) and a width (cf. the \(\beta \) term), terms that are in principle of independent strength since they correspond to different moments of f(r). While such a structure for sources is not forbidden in standard second-order gravity, in the second-order case such a structure cannot be encountered since there one is only able to measure one moment of the source (its second order one). However, since all mass has to be dynamical in a conformal theory, when the conformal symmetry is spontaneously broken (via scalar bilinear fermion condensates or via fundamental scalar Higgs fields) one expressly encounters extended sources (such as the bag models of elementary particle theory or the vortices of superconductors). Furthermore, since the conformal theory energy-momentum tensor is traceless, one cannot dominate f(r) by \(T_{00}\), and not only that, as also noted by Mannheim and Kazanas, in dynamical theories of mass generation, gravity is sensitive to the contribution to the energy-momentum tensor of the Higgs field that dynamically gives mass to fermions. Specifically, in the Higgs field case the full energy-momentum tensor is given by \(T_{\mu \nu }=T^\mathrm{kin}_{\mu \nu }\,+\,T^\mathrm{Higgs}_{\mu \nu }\) where \(T^\mathrm{kin}_{\mu \nu }\) is given by the standard kinematical perfect fluid \((\rho +p)U_{\mu }U_{\nu }+pg_{\mu \nu }\) and where \(T^\mathrm{Higgs}_{\mu \nu }\) is given by the Higgs field contribution \(2\nabla _{\mu } \nabla _{\nu }S /3 -g_{\mu \nu }\nabla ^{\alpha }S\nabla _{\alpha }S/6 -S\nabla _{\mu }\nabla _{\nu }S/3+g_{\mu \nu }S\nabla ^{\alpha }\nabla _{\alpha }S/3 -S^2(R_{\mu \nu }-g_{\mu \nu }R^\alpha _{\phantom {\alpha }\alpha }/2)/6-g_{\mu \nu }\lambda S^4\) (so that in a static, spherically symmetric geometry \(f(r)=(1/4\alpha _g)[-3(\rho ^\mathrm{kin}+p^\mathrm{kin})/B(r)+SS^{\prime \prime }-2S^{\prime 2}]\)). It is only through the interplay of the \(T^\mathrm{kin}_{\mu \nu }\) and \(T^\mathrm{Higgs}_{\mu \nu }\) terms that the tracelessness of the full \(T_{\mu \nu }\) can be maintained, since the purely kinematic \(T^\mathrm{kin}_{\mu \nu }\) (with \(T^\mathrm{kin}_{00}=mc^2/(4\pi R^3/3),~T^\mathrm{kin}_{rr}\sim 0,~T^\mathrm{kin}_{\theta \theta }\sim 0,~T^\mathrm{kin}_{\phi \phi }\sim 0\)) is not itself traceless. Because of all of these very specific aspects of the conformal theory, experience based on studies of sources in standard gravity is not a good guide to the structure of sources in the conformal case. Because this is not fully appreciated in the literature, it has led to claims appearing in the literature from time to time that conformal gravity is not viable (a recent example of this may be found in [36], as responded to in [37]). However, all that is shown in papers like Yoon’s that exclude extended or singular sources, that dominate the full \(T_{\mu \nu }\) by \(T^\mathrm{kin}_{00}=mc^2/(4\pi R^3/3)\), that ignore the contribution of the Higgs field to \(T_{\mu \nu }\), and that ignore the fact that \(T_{\mu \nu }\) is traceless, is that there would in fact be difficulties for conformal gravity if one were to couple it to the sources that are commonly used in standard gravity rather than to sources that are constrained by the very conformal invariance of the conformal theory.
References
Hehl, F.W., von der Heyde, P., Kerlick, G.D., Nester, J.M.: General relativity with spin and torsion: foundations and prospects. Rev. Mod. Phys. 48, 393–416 (1976)
Shapiro, I.L.: Physical aspects of the space-time torsion. Phys. Rep. 357, 113–213 (2002)
Hammond, R.T.: Torsion gravity. Rep. Prog. Phys. 65, 599–649 (2002)
Scholz, E.: Weyl geometry in late 20th century physics (2011). arXiv:1111.3220 [math.HO]
Yang, C.N.: The conceptual origins of Maxwell’s equations and gauge theory. Phys. Today 67, 45–51 (2014)
Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007)
Mannheim, P.D.: Alternatives to dark matter and dark energy. Prog. Part. Nucl. Phys. 56, 340–445 (2006)
Mannheim, P.D.: Comprehensive solution to the cosmological constant, zero-point energy, and quantum gravity problems. Gen. Relativ. Gravit. 43, 703–750 (2011)
Mannheim, P.D.: Making the case for conformal gravity. Found. Phys. 42, 388–420 (2012)
’t Hooft, G.: Probing the small distance structure of canonical quantum gravity using the conformal group (2010). arXiv:1009.0669 [gr-qc]
’t Hooft, G.: The conformal constraint in canonical quantum gravity (2010). arXiv:1011.0061 [gr-qc]
’t Hooft, G.: A class of elementary particle models without any adjustable real parameters. Found. Phys. 41, 1829–1856 (2011)
’t Hooft, G.: Local conformal symmetry: the missing symmetry component for space and time. Int. J. Mod. Phys. D 24, 1543001 (2015)
Fabbri, L., Mannheim, P.D.: Continuity of the torsionless limit as a selection rule for gravity theories with torsion. Phys. Rev. D 90, 024042 (2014)
Mannheim, P.D., Poveromo, J.J.: Gravitational analog of Faraday’s law via torsion and a metric with an antisymmetric part. Gen. Relativ. Gravit. 46, 1795 (2014)
Mannheim, P.D.: Torsion, magnetic monopoles and Faraday’s law via a variational principle. J. Phys. Conf. Ser. 615, 012004 (2015)
Hayashi, K., Kasuya, M., Shirafuji, T.: Elementary particles and Weyl’s gauge field. Prog. Theor. Phys. 57, 431–440 (1977)
Kibble, T.W.B.: Canonical variables for the interacting gravitational and Dirac fields. J. Math. Phys. 4, 1433–1437 (1963)
Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)
Bender, C.M., Berry, M.V., Mandilara, A.: Generalized PT symmetry and real spectra. J. Phys. A: Math. Gen. 35, L467–L471 (2002)
Bender, C.M., Mannheim, P.D.: PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues. Phys. Lett. A 374, 1616–1620 (2010)
Mannheim, P.D.: PT symmetry as a necessary and sufficient condition for unitary time evolution. Phil. Trans. R. Soc. A 371, 20120060 (2013)
Bender, C.M., Brandt, S.F., Chen, J.-H., Wang, Q.: Ghost busting: PT-symmetric interpretation of the Lee model. Phys. Rev. D 71, 025014 (2005)
Bender, C.M., Mannheim, P.D.: No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model. Phys. Rev. Lett. 100, 110402 (2008)
Bender, C.M., Mannheim, P.D.: Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart. Phys. Rev. D 78, 025022 (2008)
Bender, C.M., Mannheim, P.D.: PT symmetry in relativistic quantum mechanics. Phys. Rev. D 84, 105038 (2011)
Buchbinder, I.L., Shapiro, I.L.: On the renormalization of models of quantum field theory in an external gravitational field with torsion. Phys. Lett. B 151, 263–266 (1985)
Shanmugadhasan, S.: The dynamical theory of magnetic monopoles. Can. J. Phys. 30, 218–225 (1952)
Cabibbo, N., Ferrari, E.: Quantum electrodynamics with Dirac monopoles. Nuovo Cim. 23, 1147–1154 (1962)
Mannheim, P.D.: Neutrino pairing as the origin of parity violation in a chiral flavor theory of weak interactions. Phys. Rev. D 22, 1729–1752 (1980)
Mannheim, P.D., O’Brien, J.G.: Impact of a global quadratic potential on galactic rotation curves. Phys. Rev. Lett. 106, 121101 (2011)
Mannheim, P.D., O’Brien, J.G.: Fitting galactic rotation curves with conformal gravity and a global quadratic potential. Phys. Rev. D 85, 124020 (2012)
O’Brien, J.G., Mannheim, P.D.: Fitting dwarf galaxy rotation curves with conformal gravity. Mon. Not. R. Astron. Soc. 421, 1273–1282 (2012)
Mannheim, P.D., O’Brien, J.G.: Galactic rotation curves in conformal gravity. J. Phys. Conf. Ser. 437, 012002 (2013)
Mannheim, P.D., Kazanas, D.: Newtonian limit of conformal gravity and the lack of necessity of the second order Poisson equation. Gen. Relativ. Gravit. 26, 337–361 (1994)
Yoon, Y.: Problems with Mannheim’s conformal gravity program. Phys. Rev. D 88, 027504 (2013)
Mannheim, P.D.: Comment on “Problems with Mannheim’s conformal gravity program”. Phys. Rev. D 93, 068501 (2016)
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Mannheim, P.D. PT Symmetry, Conformal Symmetry, and the Metrication of Electromagnetism. Found Phys 47, 1229–1257 (2017). https://doi.org/10.1007/s10701-016-0017-8
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DOI: https://doi.org/10.1007/s10701-016-0017-8