Why Gravity Cannot Be Quantized Canonically, and What We Can We Do About It

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WHY GRAVITY CANNOT BE QUANTIZED CANONICALLY, AND WHAT WE CAN WE DO ABOUT IT

Philip D. Mannheim Department of Physics University of Connecticut

Presentation at Miami 2013, Fort Lauderdale December 2013

1 GHOST PROBLEMS, UNITARITY OF FOURTH-ORDER THEORIES AND PT QUANTUM MECHANICS 1. P. D. Mannheim and A. Davidson, Fourth order theories without ghosts, January 2000 (arXiv:0001115 [hep-th]). 2. P. D. Mannheim and A. Davidson, Dirac quantization of the Pais-Uhlenbeck fourth order oscillator, Phys. Rev. A 71, 042110 (2005). (0408104 [hep-th]). 3. P. D. Mannheim, Solution to the ghost problem in fourth order derivative theories, Found. Phys. 37, 532 (2007). (arXiv:0608154 [hep-th]). 4. C. M. Bender and P. D. Mannheim, No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model, Phys. Rev. Lett. 100, 110402 (2008). (arXiv:0706.0207 [hep-th]). 5. C. M. Bender and P. D. Mannheim, Giving up the ghost, Jour. Phys. A 41, 304018 (2008). (arXiv:0807.2607 [hep-th]) 6. C. M. Bender and P. D. Mannheim, Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart, Phys. Rev. D 78, 025022 (2008). (arXiv:0804.4190 [hep-th]) 7. C. M. Bender and P. D. Mannheim, PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues, Phys. Lett. A 374, 1616 (2010). (arXiv:0902.1365 [hep-th]) 8. P. D. Mannheim, PT symmetry as a necessary and sufficient condition for unitary time evolution, Phil. Trans. Roy. Soc. A. 371, 20120060 (2013). (arXiv:0912.2635 [hep-th]) 9. C. M. Bender and P. D. Mannheim, PT symmetry in relativistic quantum mechanics, Phys. Rev. D 84, 105038 (2011). (arXiv:1107.0501 [hep-th]) CONFORMAL GRAVITY AND THE COSMOLOGICAL CONSTANT AND DARK MATTER PROBLEMS 10. P. D. Mannheim, Alternatives to dark matter and dark energy, Prog. Part. Nuc. Phys. 56, 340 (2006). (arXiv:0505266 [astro-ph]). 11. P. D. Mannheim, Why do we believe in dark matter and dark energy – and do we have to?, in ”Questions of Modern Cosmology – Galileo’s Legacy”, M. D’Onofrio and C. Burigana (Eds.), Springer Publishing Company, Heidelberg (2009). 12. P. D. Mannheim, Conformal Gravity Challenges String Theory, Proceedings of the Second Crisis in Cosmology Conference, CCC-2, Astronomical Society of the Pacific Conference Series Vol. 413. (F. Potter, Ed.), San Francisco (2009). (arXiv:0707.2283 [hep-th]) 13. P. D. Mannheim, Dynamical symmetry breaking and the cosmological constant problem, Proceedings of ICHEP08, eConf C080730. (0809.1200 [hep-th]). 14. P. D. Mannheim, Comprehensive solution to the cosmological constant, zero-point energy, and quantum gravity problems, Gen. Rel. Gravit. 43, 703 (2011). (arXiv:0909.0212 [hep-th]) 15. P. D. Mannheim, Intrinsically quantum-mechanical gravity and the cosmological constant problem, Mod. Phys. Lett. A 26, 2375 (2011). (arXiv:1005.5108 [hep-th]). 16. P. D. Mannheim and J. G. O’Brien, Impact of a global quadratic potential on galactic rotation curves, Phys. Rev. Lett. 106, 121101 (2011). (arXiv:1007.0970 [astro-ph.CO]). 17. P. D. Mannheim and J. G. O’Brien, Fitting galactic rotation curves with conformal gravity and a global quadratic potential, Phys. Rev. D 85, 124020(2012). (arXiv:1011.3495 [astro-ph.CO]) 18. P. D. Mannheim, Making the case for conformal gravity, Found. Phys. 42, 388 (2012). (arXiv:1101.2186 [hep-th]). 19. J. G. O’Brien and P. D. Mannheim, Fitting dwarf galaxy rotation curves with conformal gravity. MNRAS 421, 1273 (2012). (arXiv:1107.5229 [astro-ph.CO]) 20. P. D. Mannheim, Cosmological perturbations in conformal gravity, Phys. Rev. D. 85, 124008 (2012). (arXiv:1109.4119 [gr-qc]) 21. P. D.Mannheim Astrophysical Evidence for the Non-Hermitian but PT -symmetric Hamiltonian of Conformal Gravity, Fortschritte der Physik - Progress of Physics 61, 140 (2013). (arXiv:1205.5717 [hep-th]) 22. P. D. Mannheim and J. G. O’Brien, Galactic rotation curves in conformal gravity, Journal of Physics: Conference Series 437, 012002 (2013). (arXiv:1211.0188 [astro-ph.CO])

2 1 Does gravity actually know about quantum mechanics?

1.1 Experimental considerations

Before discussing how one might quantize gravity we need to discuss whether we need to. Macroscopically, there are two established sources of gravity that are intrinsically quantum-mechanical: 3/2 2 (1) Pauli pressure of white dwarf stars, with MCH ∼ (¯hc/G) /mp. 2 4 4 3 3 (2) The energy density ρ = π kBT /15c h¯ and pressure p = ρ/3 of black-body radiation in cosmology. Microscopically, the Colella-Overhauser-Werner (Phys. Rev. Lett. 34, 1472 (1975)) experiment shows that the quantum-mechanical phase of the wave function of a neutron of mass m and velocity v is modiﬁed as it traverses 2 the gravitational ﬁeld g of the earth, viz. ∆φCOW = −mgH /vh¯ where AB = BD = DC = CA = H (see e.g. P. D. Mannheim (Phys. Rev. A 57, 1260 (1998)), yielding an observable fringe shift at D even though H is of the order of centimeters — it is just that mgH2/v is not small on the scale of Planck’s constant.

C D3 D b C D 1 1 2b

x ¡

A2 A B S 2b b B1 A1

Figure 1

3 1.2 Theoretical considerations and concerns

In equations such as the Einstein equations:   1 µν 1 µν α µν − R − g R  = T , (1) 8πG 2 α M if the Einstein tensor is equal to the matter field energy-momentum tensor, then either both sides are classical c-numbers or both sides are quantum-mechanical q-numbers. Otherwise, if the gravity side were to be classical µν while the matter side were to be quantum mechanical, then the quantum TM would have to be equal to a c-number in every single field configuration imaginable. (This is actually not impossible for a specific field configuration since a quantum-mechanical quantity such as the equal time field commutator [φ(t, x¯), π(t, y¯)] = ihZδ¯ 3(x¯ − y¯) is a c-number - it is just impossible if it has to be true for every field configuration.) However, from gravitational experiments described above we know that the source of gravity is quantum-mechanical, and we know that gravity knows it. Hence gravity must be quantized. However, since these gravitational experiments do not involve quantum gravity effects themselves (graviton loops), for phenomenological purposes it is conventional to use a hybrid in which we keep gravity classical but take c-number matrix elements of its source, to give   1 µν 1 µν α µν − R − g R  = hT i. (2) 8πG 2 α M µν But hTM i involves products of fields at the same point, so it is not finite. Thus we additionally subtract off the µν µν µν infinite zero-point part and take the source to be the normal-ordered hTM iFIN = hTM i − hTM iDIV instead, to thereby yield:

4   1 µν 1 µν α µν − R − g R  = hT i . (3) 8πG 2 α M FIN It is in this subtracted form that the standard applications of gravity are made. Thus in Σ(a†a + 1/2)¯hω we keep the Σa†ahω¯ term but ignore the Σ(1/2)¯hω zero-point energy density term 00 in hTM i. ij Also ignore the zero-point pressure in the spatial hTM i. As of today there is no known justification for doing this, with this subtracted hybrid not having been derived from a fundamental theory or having been shown to be able to survive quantum gravitational corrections. There is no apparent reason why the zero point energy density of the matter sector should not gravitate. While one only needs to consider energy differences in flat space physics, in gravity one has to consider energy itself, with the hallmark of Einstein gravity being that gravity couple to everything. Hence for gravity one cannot ignore zero-point contributions.

5 Moreover, even if one does start with the subtracted hybrid, then the order G contribution to gravity is given µν µν by the flat spacetime hΩ|TM |ΩiFIN, with Lorentz invariance allowing a finite flat spacetime hΩ|TM |ΩiFIN to be of the generic form −Λgµν. Thus even if one ignores the matter sector zero-point energy contributions one still has a vacuum energy problem; with the standard strong, electromagnetic, and weak interactions typically then generating a huge such Λ. We thus recognize two types of vacuum problem, zero-point and cosmological constant problems. Moreover, if we take gravity to be quantum-mechanical (as we must), it too will have zero-point contributions. As I have shown in my papers and discussed at Miami 2011, it is the interplay of the gravity and matter field zero-point contributions that leads to a solution to the cosmological constant µν problem, with the problem having arisen entirely due to ignoring how hTM iFIN got to be finite in the first µν µν α µν place, and then using R − (1/2)g R α = −8πGhTM iFIN as the starting point. However, in order to be able to discuss gravitational zero-point contributions, we need to to quantize gravity. As we now show, such quantization cannot be done canonically, with gravity actually being quantized by the source that it is coupled to, rather than by being quantized on its own.

6 2 Canonical quantization of matter ﬁelds

2.1 Massless fermions

R 4 ¯ µ ¯ µ For massless flat space fermion fields with IM = d xih¯ψγ ∂µψ, stationary fields obey δIM/δψ = ihγ¯ ∂µψ = 0. We introduce creation and annihilation operators according to 3 Z d k ψ(x, t) = X b(k, s)u(k, s)e−ik·x + d†(k, s)v(k, s)eik·x , s (2π)3/2 3 Z d k ψ†(x, t) = X b†(k, s)u†(k, s)eik·x + d(k, s)v†(k, s)eik·x . (4) s (2π)3/2 † 0 3 0 Quantizing the fermion field according to {ψα(x, t), ψβ(x , t)} = δ (x − x )δα,β then requires that its creation and annihilation operators obey † 0 0 3 0 † 0 0 3 0 {b(k, s), b (k , s )} = δ (k − k )δs,s0, {d(k, s), d (k , s )} = δ (k − k )δs,s0. (5) In the Fock space vacuum the vacuum matrix element of the energy-momentum tensor is given by   3 −1/2 δIM M 2¯h Z d k   ¯ hΩ| 2g µν |Ωi = hΩ|Tµν|Ωi = hΩ|ψihγ¯ µ∂νψ|Ωi = − 3 kµkν. (6) δg (2π) k0 and we can identify zero point energy density and pressure terms of the form 3 2 M 2¯h Z 3 M M M 2¯h Z d k k ρM ρM = hΩ|T00 |Ωi = − 3 d kωk, pM = hΩ|T11 |Ωi = hΩ|T22 |Ωi = hΩ|T33 |Ωi = − 3 = . (2π) (2π) ωk 3 3 (7) M M R 3 M Thus even though hΩ|T0i |Ωi and the total matter vacuum momentum Pi = d xhΩ|T0i |Ωi are both zero, M M M hΩ|T11 |Ωi = hΩ|T22 |Ωi = hΩ|T33 |Ωi are not, to thus yield a vacuum pressure. Thus have two zero-point infinities to deal with, viz. ρM and pM, and neither behaves like a cosmological constant (for which p = −ρ), since pM and ρM have same sign.

7 2.2 Massive fermions R 4 ¯ µ ¯ ¯ µ For massive flat space free fermions with IM = d x[ih¯ψγ ∂µψ − mψψ], stationary fields obey δIM/δψ = ihγ¯ ∂µψ − mψ = 0. µν ¯ µ ν The matter field energy-momentum tensor is again of the form TM = ih¯ψγ ∂ ψ and again its vacuum expectation value is of the perfect fluid form µ ν µν 2¯h Z ∞ 3 k k µ ν µν hΩ|T |Ωi = − d k = (ρM + pM)U U + pMη , (8) M (2π)3 −∞ k0 where now  2 2 4  2 2  4  h¯ 4 m K m 4¯h K m ρ = − K + − ln   +  , M 4π2 h¯2 4¯h4 m2 8¯h4  2 2 4  2 2  4  h¯ 4 m K 3m 4¯h K 7m p = − K − + ln   −  . (9) M 12π2 h¯2 4¯h4 m2 8¯h4

µν We note that hΩ|TM |Ωi is still not in the form of cosmological constant where one would have pM = −ρM, and indeed could µν not be since in the m = 0 limit TM is traceless and pM = ρM/3. Also we note three separate divergences: quartic, quadratic, and logarithmic.

If we introduce a set of covariant Pauli-Villars regulator masses Mi with each such regulator contributing an analogous term P 2 P 2 4 P 4 as multiplied by some overall factor ηi, the choice 1 + ηi = 0, m + ηiMi = 0, m + ηiMi = 0 will not only then lead to finite regulated ρREG and pREG, it will give them the values h¯ hΩ|T µν|Ωi = p ηµν, ρ = −p = − m4lnm2 + X η M 4lnM 2 . (10) M REG REG REG REG 16π2 i i i µν The regulated and thus now finite hΩ|TM |Ωi is indeed in the form of a cosmological constant, but it is completely uncontrollable, being as large as the regulator masses. M Now the fact that hΩ|Tµν|Ωi is non-zero is not of concern in flat space field theory, and one can ignore it. Hence canonical quantization is based on stationarity with respect to the matter fields, and it is not of M concern if it forces products of fields in the energy-momentum tensor to lead to a non-zero hΩ|Tµν|Ωi since it can be normal-ordered away. But, what happens if gravity is present, and does this same canonical prescription work for gravity?

8 3 Quantization of the gravitational ﬁeld

3.1 Quantization of the gravitational ﬁeld cannot be canonical

For gravity let us introduce some general coordinate scalar action function of the metric IGRAV, and let us identify the energy-momentum tensor of gravity itself according to 2 δI GRAV = T µν . (11) g1/2 δgµν GRAV (See e.g. Weinberg’s gravitation book and P. D.Mannheim, Phys. Rev. D 74, 024019 (2006)). For the Einstein Hilbert action R 4 1/2 α µν µν µν µν α IEH = (−1/16πG) d x(−g) R α for instance we identify TGRAV as the Einstein tensor G = (1/8πG)[R −(1/2)g R α], though the prescription is generic.

For the general IGRAV the gravitational equations of motion take the form: µν TGRAV = 0, (12) and in a typical linearization around ﬂat spacetime of the form gµν = ηµν + hµν would lead to a ﬁrst-order wave equation of the form µν TGRAV(1) = 0. (13) α For the Einstein Hilbert action for instance we would obtain ∇α∇ hµν = 0 (in the harmonic gauge).

If we now were to quantize hµν canonically by identifying the coefficients of its negative and positive frequency modes as µν creation and annihilation operators, we would find that the term that is second order in hµν, viz. TGRAV(2), would not be zero (its vacuum expectation value would actually possess zero-point infinities). However, since the stationarity condition µν TGRAV = 0 is to hold to all orders in hµν, stationarity with respect to the metric requires that µν TGRAV(2) = 0. (14) Now for gravity we cannot normal order away its zero-point contributions since gravity couples to gravity, and thus we have µν 00 a contradiction. (We anyway cannot get rid of TGRAV(2) since the one-graviton matrix element of TGRAV(2) is the energy carried by a gravity wave.) Hence unlike all other fields for which the zero-point contribution does not conflict with canonical quantization, for gravity there is a conflict, since its stationarity condition involves a product of its fields at the same point. Hence, gravity cannot be quantized canonically, and in any quantization prescription for it one would have to take into account its zero-point infinities.

9 3.2 Quantization of gravity via coupling to a quantized matter ﬁeld

Consider a generic action for the universe consisting of gravitational plus matter actions of the form:

IUNIV = IGRAV + IM. (15) Stationary variation with respect to the metric yields µν µν µν TUNIV = TGRAV + TM = 0. (16) µν µν µν If both TGRAV and TM are well-defined quantum-mechanically, we can treat the condition TUNIV = 0 as a quantum-mechanical operator identity, with its finite and divergent parts separately having to obey µν µν (TGRAV)DIV + (TM )DIV = 0, (17) µν µν (TGRAV)FIN + (TM )FIN = 0. (18) µν µν µν Moreover, even if TGRAV is not well-defined quantum-mechanically, we can still apply TGRAV + TM = 0 if we µν ignore radiative loop corrections. Now for massless fermions we noted that hΩ|TM |Ωi is given by 3 M 2¯h Z d k hΩ|Tµν|Ωi = − 3 kµkν, (19) (2π) k0 with quartically divergent 4 3 2 4 2¯h Z 3 hK¯ 2¯h Z d k k hK¯ ρM = − 3 d kωk = − 2 , pM = − 3 = − 2 . (20) (2π) 4π (2π) ωk 3 12π Thus for 4-component fermions we get a zero-point energy of −2¯hω for each momentum state k¯, i.e. −hω¯ for each of two two-components fermions.

10 In order to cancel the −2¯hω fermion zero-point contribution we need to generate +2¯hω in the gravity sector µν hΩ|TGRAV|Ωi. 1/2 Since the graviton zero-point contribution is quadratic in hµν, we need hµν to be orderh ¯ . µν Since the ﬁrst-order equation of motion for hµν, viz. hΩ|TGRAV(1)|Ωi = 0 is homogeneous in hµν, in and of µν itself it does not force us to have to choose a nontrivial solution to hΩ|TGRAV(1)|Ωi = 0, and we can satisfy µν hΩ|TGRAV(1)|Ωi = 0 with hµν = 0.

Moreover, we would have to set hµν = 0 if we tried to quantize gravity on its own, since we would then need µν to also satisfy the second-order hΩ|TGRAV(2)|Ωi = 0. µν However, if gravity is coupled to a matter source, then the non-vanishing of the matter hΩ|TM (2)|Ωi requires µν that the gravitational hΩ|TGRAV(2)|Ωi be non-vanishing too. Hence coupling to a matter source forces gravity to have to choose the non-trivial solution to the ﬁrst-order µν 1/2 hΩ|TGRAV(1)|Ωi = 0, with hµν then being forced to non-trivially be orderh ¯ . Gravity is not quantized on its own. It is quantized by being coupled to a quantum- mechanical source.

11 To how the quantization works in detail, for Einstein gravity for instance, we have one massless transverse traceless graviton with two helicity components, but a zero point of only +¯hω/2 for each momentum state, to give a total of +¯hω. Thus we need to generically quantize the graviton ﬁeld according to [φ(t, x¯), π(t, y¯)] = ihZδ¯ 3(¯x − y¯), (21) and thus need to set Z = 2, (22) and not the canonical Z = 1. Moreover, if the source of gravity has N two-component fermions (i.e. N/2 four-component fermions) and M massless gauge bosons (with +¯hω/2 for each of two helicity components), the net matter sector zero-point energy would be −(N − M)¯hω, and so for the gravitational ﬁeld we would have to set Z = N − M (23)

The normalization of canonical commutators or anti-commutators in the matter sector and the matter content in the matter sector thus ﬁxes the normalization in the gravity sector, with the gravity sector adjusting to whatever its source might be. Still to be addressed is what is to happen when we do take radiative corrections into consideration, and what is to happen when we generate masses dynamically via a change in vacuum to a spontaneously broken vacuum |Si. To address these issues we need a theory of gravity in which radiative corrections are under control, and so we turn now to conformal gravity, as it is a renormalizable theory of gravity, one for which mass generation would have to be spontaneous since unbroken conformal symmetry requires that all masses be zero. However, such an unbroken symmetry also requires that any fundamental cosmological constant be zero, to thus provide a very good starting point to attack the cosmological constant problem.

12 4 Conformal gravity

2α(x) Conformal gravity is based on the requirement of local conformal invariance of the action under gµν → e gµν. The Weyl conformal tensor 1 1 C = R + Rα [g g − g g ] − [g R − g R − g R + g R ] , (24) λµνκ λµνκ 6 α λν µκ λκ µν 2 λν µκ λκ µν µν λκ µκ λν possesses a remarkable geometric property, namely that under a local conformal transformation it transforms as Cλµνκ → 2α(x) e Cλµνκ with all derivatives of α(x) dropping out identically. Given this fact, a locally conformal invariant gravitational action can then uniquely be prescribed to be of the form:

Z 4 1/2 λµνκ IGRAV = −αg d x(−g) CλµνκC Z " 1 # ≡ −2α d4x(−g)1/2 RµνR − (Rα )2 . (25) g µν 3 α

Then, since the coupling constant αg is dimensionless, conformal gravity is power-counting renormalizable. However, since the action contains four derivatives of the metric, the equations of motion are fourth-order derivative equations, and for a long time were thought to lead to states of negative norm (ghost states) and violations of unitarity. However, recently Bender and Mannheim have reanalyzed fourth-order theories and shown that they have to be reinterpreted as PT theories, and that they can then be formulated in a Hilbert space with no states of negative norm and no states of negative energy. Conformal gravity is thus recognized as a consistent quantum theory of gravity, one expressly formulated in four spacetime dimensions, the only spacetime dimensions for which we have any evidence. Stationary variation of the conformal gravity action with respect to the metric leads to gravitational equations of motion of the form 1 δIW µν µν 1/2 = 4αgW = TM , (26) (−g) δgµν where 1 1 W µν = gµν(Rα );β + Rµν;β − Rµβ;ν − Rνβ;µ − 2RµβRν + gµνR Rαβ 2 α ;β ;β ;β ;β β 2 αβ 2 2 2 1 − gµν(Rα );β + (Rα );µ;ν + Rα Rµν − gµν(Rα )2, (27) 3 α ;β 3 α 3 α 6 α µν µν with both W and a conformal TM being both covariantly conserved and traceless.

13 5 Quantizing conformal gravity

5.1 Linearizing around ﬂat 1 gµν = ηµν + hµν,T µν (1) = −4α W µν(1) = −2α (∂ ∂α)2Kµν,Kµν = hµν − ηµνη hαβ, ∂ Kµν = 0. (28) GRAV g g α 4 αβ µ µν TGRAV(1) = 0 has generic solution

Kµν = Aµνeik·x + (n · x)Bµνeik·x, nµ = (1, 0, 0, 0), (29) and full solution

1/2 2 3 h¯ X Z d k (i) ¯ (i) ¯ ik·x (i) ¯ (i) ¯ ik·x Kµν(x) = 1/2 3/2 3/2 A (k)µν(k)e + iωkB (k)µν(k)(n · x)e 2(−αg) i=1 (2π) (ωk) ˆ(i) ¯ (i) ¯ −ik·x ˆ(i) ¯ (i) ¯ −ik·x + A (k)µν(k)e − iωkB (k)µν(k)(n · x)e , (30)

(i) ¯ (j) ¯0 (i) ¯ (j) ¯0 3 ¯ ¯0 [A (k), Bˆ (k )] = [B (k), Aˆ (k )] = Z(k)δi,jδ (k − k ), [A(i)(k¯), Aˆ(j)(k¯0)] = 0, [B(i)(k¯), Bˆ(j)(k¯0)] = 0, [A(i)(k¯),B(j)(k¯0)] = 0, [Aˆ(i)(k¯), Bˆ(j)(k¯0)] = 0. (31) α Z I (2) = − g d4x∂ ∂αK ∂ ∂βKµν, (32) W 2 α µν β 3 µν µν 2¯h Z d k µ ν hΩ|TGRAV(2)|Ωi = −4αghΩ|W (2)|Ωi = 3 Z(k)k k . (33) (2π) ωk Unlike Einstein gravity this time there are two graviton-type modes to thus give a zero-point of +2¯hω. As previously µν µν µν TUNIV = TGRAV + TM = 0. (34) With one four-component massless fermion matter source we obtain 3 3 µν 2¯h Z d k µ ν 2¯h Z d k µ ν TUNIV = 3 Z(k)k k − 3 k k = 0, (35) (2π) ωk (2π) ωk to yield Z(k) = 1. (36)

14 5.2 Some phenomenology

If M massless gauge bosons and N massless two-component spinors, need

(N − M) Z(k) = ,Z(k) > 0 −→ N > M. (37) 2 For SU(3) × SU(2) × U(1): M=12, N=16 per generation, so N > M generation by generation. For SO(10): M=45, N=16, so need 3 generations. For all generations in a common multiplet need SO(2n) where 2n ≥ 16. (No solution for SU(N) if all generations in a single multiplet, so triangle-anomaly-free grand-unifying groups preferred.) For SO(16): M=120, N=128, so 8 generations. Have asymptotic freedom up to SO(20) if all fermions in same multiplet (Wilczek and Zee (1982)). So just SO(16), SO(18) and SO(20). In general µν µν (TGRAV)DIV + (TM )DIV = 0, (38) µν µν (TGRAV)FIN + (TM )FIN = 0. (39) So zero-point ﬂuctuations of gravity and matter take care of each other, and conformal trace anomalies take care µν of each other also since the condition gµνTUNIV = 0 is not a conformal Ward identity, but is instead an equation µν µν of motion. Ability to solve trace-anomaly problem is because we do not start with (−1/8πG)G = hTM i and µν try to show that hTM i is conformal-anomaly-free all on its own. µν µν µν Since TUNIV = TGRAV + TM = 0 is an identity, it holds in any vacuum. Thus we now examine how things change when mass is generated dynamically through a change of vacuum, something that has to happen in an otherwise massless conformal invariant theory.

15 6 Spontaneously broken conformal gravity

6.1 Particle physics motivation for conformal invariance

Old idea that we can have conformal invariance at energies much greater than particle masses. Violated by radiative corrections, but can be restored at a renormalization group ﬁxed point where we get scaling with anomalous dimensions. Johnson, Baker and Willey (1964, 1967): if QED at Gell-Mann-Low ﬁxed point then

˜ 2 2 γψψ¯ /2 2 2 γψψ¯ /2 Γψψ¯ (p, p, 0) = (−p /m ) , dψψ¯ = 3 + γψψ¯ , m0 = m(Λ /m ) . (40)

Thus if γψψ¯ < 0, bare mass is zero, and all mass is dynamical, with physical mass obeying a homogeneous equation. However, why should physical mass not be zero too, since zero is a solution to a homogeneous equation. So could non-zero solution be chosen by spontaneous breakdown. So we need to adapt Nambu-Jona-Lasinio to theories with a ﬁxed point. In Nambu-Jona-Lasinio Z 4 ¯ µ ¯ 2 IM(4F) = − d x[ih¯ψγ ∂µψ − (g/2)[ψψ] ] (41)

4  2 2 4  2 2  4  i Z d p µ h¯ 4 m K m 4¯h K m ρ = (m, 4F) = Tr Ln [γ p − m + i] = − K + − ln   +  . (43) MF h¯ (2π)4 µ 4π2 h¯2 4¯h4 m2 8¯h4 Deﬁne 0(M, 4F) = M/g.

2  4  2 2  4 2 2  2 2  2 2  m h¯ 4 m 4¯h K m m M 4¯h K m M ρ + Λ = ˜(m, 4F) = (m, 4F) − = − K − ln   + + ln   −  , (44) MF MF 2g 4π2 4¯h4 m2 8¯h4 2¯h4 M 2 2¯h4 ˜(M, 4F) − ˜(M = 0, 4F) < 0. So massive vacuum lies lower than massless vacuum (Nambu and Jona-Lasinio (1961)), 2 and mass-dependent quadratic divergence cancelled by mass generation mechanism. Induced ΛMF = −M /2g 2 behaves as K , and quadratic divergence cancelled identically in ρMF + ΛMF (with no need for supersymmetry).

16 Apply to Johnson-Baker-Willey QED. Find (Mannheim (1974, 1975, 1978)) that massive vacuum lies lower then massless one if γψψ¯ = −1, dψψ¯ = 2. (As γψψ¯ gets more negative, theory gets softer in ultraviolet, and thus more divergent in infrared, and get long range order phase transition at γψψ¯ = −1.)   4  2 −1/2 4 2 2   2 2   i Z d p µ −p hK¯ m M m M ρ = (m) = Tr Ln γ p − m   + i = − + ln   − 1 . (45) MF h¯ (2π)4  µ M 2  4π2 16π2h¯3 16¯h4K4

2 2 4 2 2   2   m m 0 hK¯ m M m ρ + Λ = ˜(m) = (m) − = (m) − (M) = − + ln   − 1 . (46) MF MF 2g 2M 4π2 16π2h¯3 M 2 Produce induced double-well shaped potential dynamically, with ˜(M) − ˜(M = 0) < 0. Also now have cancelled mass- dependent logarithmic divergence, and with dψψ¯ = 2 the 4-Fermi interaction is non-perturbatively scale invariant and 2 renormalizable. Note that in dynamical models the induced cosmological constant term −M /2g = ΛMF is log divergent, i.e. not ﬁnite, but is cancelled. (Induced cosmological constant of double-well Higgs potential is ﬁnite and not cancelled.).

Thus to summarize:

Initially there are quartic, quadratic and log divergences in the vacuum energy. In Nambu-Jona-Lasinio the mass generation mechanism produces a self-consistent m2/2g mean-ﬁeld term that cancels the quadratic divergence, leaving only quartic and log divergences. (If Higgs boson is dynamical bound state then quadratically divergent self–energy problem naturally solved.)

In Nambu-Jona-Lasinio with γψψ¯ = −1, dψψ¯ = 2 the quadratic divergence is reduced to a log divergence, and this log divergence is then cancelled by the m2/2g term, to thus leave only a quartically divergent K4 term

But what about this quartic divergence? We cancel it by gravity, and in so doing we solve the cosmological constant problem at the same time.

17 7 Gravity solves the cosmological constant problem

µν µ ν µν µν µν hS|TMF|Si = (ρMF + pMF)U U − pMFη + ΛMF η , ηµνhS|TMFSi = 0. (47)

1 hS|T µν |Si = (ρ + p ) [4U µU ν − ηµν] (48) MF 4 MF MF

  4  2 −1/2 4 2 2   2 2   i Z d p µ −p hK¯ m M m M ρ = (m) = TrLn γ p − m   + i = − + ln   − 1 . (49) MF h¯ (2π)4  µ M 2  4π2 16π2h¯3 16¯h4K4

2 4 2 2   2   00 m hK¯ m M m ρ + Λ = hS|T |Si = ˜(m) = (m) − = + ln   − 1 , MF MF MF 2g 4π2 16π2h¯3 M 2  4  4  2 3  4 4 1 1 M M 8π h¯ 0 0 M hK¯ M = ln   , = exp   , ˜(M) = (M) − = 0, ˜(M) = − − . (50) g 8π2h¯3 16¯h4K4 16¯h4K4 g g 4π2 16π2h¯3

   2 1/2 2  2 1/2 2 2¯h Z 3 2 M M 2 M M ˜(M) = − d k k +  − + k −  +  . (51) (2π)3  h¯2 4¯h2(k2 + M 2/h¯2)1/2 h¯2 4¯h2(k2 − M 2/h¯2)1/2 

µν µν µν µν (TGRAV)DIV + (TM )DIV = 0, (TGRAV)FIN + (TM )FIN = 0. (52)

 2 1/2 2  2 1/2 2 2 M M 2 M M kZ(k) = k +  − + k −  + , (53) h¯2 4¯h2(k2 + M 2/h¯2)1/2 h¯2 4¯h2(k2 − M 2/h¯2)1/2

Z(k) of graviton is ﬁxed by whatever quantized source gravity is coupled to, and cannot be assigned a priori. When there is fermion mass generation, even though graviton stays massless, Z(k) of graviton depends on induced mass of fermion, just as needed to cancel induced cosmological constant. Just as Z(k) cancels zero- µν 2 2 2 2 2 point hΩ|T |Ωi when ωk = k , as modes readjust to ωk = k + M , Z(k) readjusts to cancel broken symmetry µν hS|TMF|Si. Such behavior for Z(k) is completely foreign to canonical quantization.

18 8 Summary

In a renormalizable theory of gravity the equations of motion can be treated as well-deﬁned operator identities of the form:

µν µν (TGRAV)DIV + (TM )DIV = 0, (54) µν µν (TGRAV)FIN + (TM )FIN = 0. (55)

They couple the gravity sector infinities to those in the matter sector, with the matter sector quantization forcing the gravity sector to be quantized so as to cause all these infinities to mutually cancel each other while fixing the renormalization constant Z in the gravity sector. These equations hold in no matter what states one takes matrix elements, but in a mass generating spontaneously broken vacuum the gravitational Z will readjust so as to then cancel the cosmological constant that is induced by the breaking. If all mass and length scales come from symmetry breaking, then all scales come from quantum mechanics. I.e. without length scales there cannot be any curvature. SPACETIME CURVATURE IS INTRINSICALLY QUANTUM-MECHANICAL, AND GRAVITY IS QUANTIZED BY ITS SOURCE.

19 Following are notes in case there are questions.

20 9 The Zero-Point Energy Density and Zero-Point Pressure Problems in More Detail

Consider a free fermion of mass m in flat, four-dimensional spacetime. With kµ = ((k2 + m2/h¯2)1/2, k¯) the µν ¯ µ ν vacuum expectation value of the matter field energy-momentum tensor TM = ih¯ψγ ∂ ψ evaluates to µ ν 2¯h Z ∞ k k hΩ|T µν|Ωi = − d3k . (56) M (2π)3 −∞ k0 00 11 We recognize a zero-point energy density ρM = hΩ|TM |Ωi and a zero point pressure pM = hΩ|TM |Ωi = 22 33 µν µν hΩ|TM |Ωi = hΩ|TM |Ωi with ηµνhΩ|TM |Ωi = 3pM − ρM, and, with momentum cutoff K, can write hΩ|TM |Ωi in the form of a perfect fluid µν µ ν µν hΩ|TM |Ωi = (ρM + pM)U U + pη , (57) h¯  m2K2 m4 4¯h2K2  m4  ρ = − K4 + − ln   +  , M 4π2  h¯2 4¯h4  m2  8¯h4  h¯  m2K2 3m4 4¯h2K2  7m4  p = − K4 − + ln   −  . (58) M 12π2  h¯2 4¯h4  m2  8¯h4  µν We note that hΩ|TM |Ωi is not in the form of cosmological constant where one would have pM = −ρM, and µν indeed could not be since in the m = 0 limit TM is traceless and pM = ρM/3. Also we note three separate divergences: quartic, quadratic, and logarithmic. If we introduce a set of covariant Pauli-Villars regulator masses Mi with each such regulator contributing an analogous term as multiplied by some overall factor ηi, P 2 P 2 4 P 4 the choice 1 + ηi = 0, m + ηiMi = 0, m + ηiMi = 0 will not only then lead to finite regulated ρREG and pREG, it will give them the values h¯ hΩ|T µν|Ωi = p ηµν, ρ = −p = − m4lnm2 + X η M 4lnM 2 . (59) M REG REG REG REG 16π2 i i i µν The regulated and thus now finite hΩ|TM |Ωi is indeed in the form of a cosmological constant, but it is completely uncontrollable, being as large as the regulator masses.

21 10 The Additional Problem due to Symmetry Breaking: Scalar Field Case

¯ In flat space in a normal vacuum |Ni where φN = hN|φ|Ni = 0 or hN|ψψ|Ni = 0 we already have divergent zero-point energy density and pressure contributions, and regulating these divergences leads to a cosmological constant term, one as large as the scale associated with the regulator cut-off scale, being possibly as large as the Planck scale. Moreover, if for the strong, electromagnetic and weak interactions we change the vacuum ¯ by spontaneous breakdown to a vacuum |Si where φS = hS|φ|Si 6= 0 or hS|ψψ|Si 6= 0, we then induce an additional contribution to the cosmological constant. For the scalar field case we note that if we have a double-well potential

4 2 2 V (φ) = λφ − µ φ + V0,

2 1/2 with a local maximum in which φN = 0 and a global minimum where φS = (µ /2λ) , then not only do we induce a contribution 4 V (φS) − V (φN) = −µ /4λ due to the change in vacuum, in addition we do not know where the actual of energy of the global minimum is since V0 is unconstrained. Thus in flat space we only measure the energy difference V (φS) − V (φN), but gravity couples to energy itself, i.e. to 4 V (φS) = −µ /4λ + V0. 4 Even though both hS|φ|Si and V (φS) = −µ /4λ+V0 are finite (though large), since φ is a fundamental scalar field it has an uncontrollable quadratically divergent self-energy, and should thus have a mass at the Planck cut-off scale rather than at the now observed 126 GeV.

22 11 The Additional Problem due to Symmetry Breaking: Fermion Condensate Case

If in flat space the symmetry breaking is instead done by a fermion bilinear condensate, then for a massive fermion we obtain  2 3  2 2  3  dρM h¯ 2mK m 4¯h K m hS|ψψ¯ |Si = = −  − ln   +  dm 4π2  h¯2 h¯4  m2  h¯4  and it is not finite, but is instead quadratically divergent. We thus appear to have traded one quadratic divergence for another. Moreover, with dynamical symmetry breaking a cosmological constant term is also induced, and it would be quadratically divergent too. However, when we introduce the fermion condensate through a Nambu-Jona-Lasinio (1961) type mean field theory the vacuum energy divergence will automatically be reduced to logarithmic. Specifically, in Nambu-Jona-Lasinio the action is of the 4-Fermi form Z 4 ¯ µ ¯ 2 IM(4F) = − d x[ih¯ψγ ∂µψ − (g/2)[ψψ] ], (60) and one looks for self-consistent mean field solutions in which hS|[ψψ¯ − m/g]2|Si = hS|[ψψ¯ − m/g]|Si2 = 0, hS|ψψ¯ |Si = m/g, with the mean field action then taking the form Z 4 ¯ µ ¯ 2 IMF(4F) = − d x[ih¯ψγ ∂µψ − mψψ + m /2g]. In the mean field the vacuum energy density is given as 4 i Z d p (m, 4F) = Tr Ln [γµp − m + i] . h¯ (2π)4 µ 0 ¯ 2 2 0 Then on defining (M, 4F) = hS|ψψ|Sim=M = M/g and identifying ΛMF = −m /2g = −m (M)/2M, for the mean-field action we obtain a mean-field energy-momentum tensor of the form

23 µν ¯ µ ν 2 µν µ ν µν µν hS|TMF|Si = hS|ih¯ψγ ∂ ψ|Si − (m /2g)η = (ρMF + pMF)U U − pMFη + ΛMFη , where, with momentum cut-off K, h¯  m2K2 m4 4¯h2K2  m4  ρ = (m, 4F) = − K4 + − ln   +  MF 4π2  h¯2 4¯h4  m2  8¯h4  h¯  m2K2 m2M 2 4¯h2K2  m2M 2  Λ = − − + ln   −  . (61) MF 4π2  h¯2 2¯h4  M 2  2¯h4  00 Consequently we obtain a mean-field potential VMF = hS|TMF|Si = ρMF + ΛMF of the form h¯  m4 4¯h2K2  m4 m2M 2 4¯h2K2  m2M 2  V = − K4 − ln   + + ln   −  , MF 4π2  4¯h4  m2  8¯h4 2¯h4  M 2  2¯h4   3  2 2  2  2 2  3 2  dVMF h¯ m 4¯h K mM 4¯h K m mM = − − ln   + ln   + −  . (62) dm 4π2  h¯4  m2  h¯4  M 2  h¯4 h¯4  This potential has a dynamically induced double-well form with a maximum at m = 0 and a minimum at m = M (VMF(m = M) − VMF(m = 0) < 0), so M is the physical mass. Moreover, even though the induced ΛMF is quadratically divergent, this divergence is cancelled identically in VMF, leaving VMF only logarithmically divergent, with the zero-point energy thus readjusting itself automatically. Thus by starting µν µν α µν with R − (1/2)g R α = −8πGhTM iFIN we have omitted a zero-point term that can cancel at least some part of the cosmological constant. Moreover, we note that we have been able to remove the quadratic divergence in VMF without the need to introduce any supersymmetric scalar partner of the fermion. Finally, when one extends Nambu-Jona-Lasinio to a conformal invariant theory with a renormalization group fixed point, one finds (Mannheim 1974, 1975, 1978) that dynamical mass generation occurs when the dimension of ψψ¯ is ¯ 2 reduced from 3 to 2 (with (ψψ) then being renormalizable), with the logarithmic divergence in VMF then being cancelled too. All that remains is the quartic divergence, and it will be cancelled by gravity.

24 12 The Moral of the Story

Since symmetry breaking through either a fundamental scalar field or a fermion bilinear condensate can occur in flat space in the absence of gravity, gravity is unable to control any of it. Gravity must this live with whatever flat space symmetry breaking bequeaths it, with the lowest order in G contribution to standard gravity being µν due to the flat space contributions of the strong, electromagnetic and weak interactions to hΩ|TM |Ωi. To avoid this, we should not try to treat gravity via a power series expansion in the gravitational coupling constant.

25 13 The Solution

To enable gravity to have some control over the cosmological constant then, we need to put gravity on an equal footing with the other three fundamental interactions. Moreover, we need that footing to be one that does control the vacuum contributions of particle physics. So immediately we need to get rid of the uncontrollable fundamental Higgs potential with its intrinsic −µ2 tachyonic mass scale. Thus we need to generate symmetry breaking mass scales dynamically via hN|ψψ¯ |Ni fermion condensates. In such a case there are no fundamental Higgs ﬁelds, no longer any need to have to deal with any quadratically divergent self-energy term, AND THERE IS NO MEXICAN HAT. Then, if all particle physics mass scales are dynamical, there are no mass scales in particle physics at the level of the Lagrangian at all, with all SU(3) × SU(2) × U(1) coupling constants being dimensionless. Moreover, the massless matter action Z 4 1/2 ¯ µ IM = − d x(−g) ψ(x)γ (x)[i∂µ + iΓµ(x) + Aµ(x)]ψ(x) (63) with fermion spin connection 1 Γ (x) = ([γν(x), ∂ γ (x)] − [γν(x), γ (x)]Γσ ), γµ(x) = V µ(x)γˆa (64) µ 8 µ ν σ µν a is not just locally Lorentz invariant (the reason for introducing the spin connection) in the presence of gravity, it turns out to be locally conformal invariant under

−3α(x)/2 µ −α(x) µ ψ(x) → e ψ(x),Va (x) → e Va (x),Aµ(x) → Aµ(x) (65) as well. Thus, even though the spin connection was introduced purely to maintain local Lorentz invariance, it also serves to maintain local conformal invariance as well. We thus get local conformal invariance for free, with the spin connection serving as a gauge ﬁeld for conformal invariance.

26 Thus to put gravity on the same footing as the other fundamental interactions, we need gravity to be locally 2α(x) conformal invariant too, i.e. invariant under gµν(x) → e gµν(x). To this end we note that the Weyl conformal tensor 1 1 C = R + Rα [g g − g g ] − [g R − g R − g R + g R ] , (66) λµνκ λµνκ 6 α λν µκ λκ µν 2 λν µκ λκ µν µν λκ µκ λν 2α(x) transforms as Cλµνκ → e Cλµνκ under a local conformal transformation, a quite remarkable geometric property in which all derivatives of α(x) drop out identically. Given this fact, a locally conformal invariant gravitational action is then uniquely prescribed to be of the form:

Z 4 1/2 λµνκ IGRAV = −αg d x(−g) CλµνκC   Z 4 1/2 µν 1 α 2 ≡ −2α d x(−g) R R − (R )  , (67) g µν 3 α As we see, the underlying conformal symmetry forbids the presence of either any fundamental Planck scale or any fundamental cosmological constant in the gravitational action. Moreover, the gravitational coupling constant αg must be dimensionless, with a conformal invariant theory of gravity thus being renormalizable. And now one can write a meaningful gravitational equation of motion in which quantum equals quantum, one 4 2 2 in which the cosmological constant now is under control. (For the V (φ) = λφ − µ φ + V0 potential for instance only the λφ4 term is allowed by the conformal symmetry, with the conformal symmetry forcing both 2 µ and V0 to be zero.)

27 Now if all particle physics mass and length scales are dynamical, then all such scales are intrinsically quantum- mechanical. However, it is mass and length scales that characterize spacetime curvature. HENCE IN THE ABSENCE OF QUANTUM MECHANICS THERE CAN BE NO CURVATURE, AND GRAVITY MUST THUS BE INTRINSICALLY QUANTUM-MECHANICAL. We thus need to expand the gravitational equations not as power series in the gravitational coupling constant but as a power series inh ¯. If we define the action of the Universe as IUNIV = IGRAV + IM, then on defining the functional variation of each one of these terms with respect to the metric to be its associated energy-momentum tensor, stationarity with respect to the metric yields µν µν µν TUNIV = TGRAV + TM = 0. (68) µν µν µν If both TGRAV and TM are well-defined quantum-mechanically, we can treat the condition TUNIV = 0 as a quantum-mechanical operator identity, with its finite and divergent parts separately having to obey µν µν µν µν (TGRAV)DIV + (TM )DIV = 0, (TGRAV)FIN + (TM )FIN = 0. (69) µν µν Since TUNIV is zero identically, gravity now knows where the zero of energy is, and matrix elements of TUNIV in any state must be zero. Hence stationarity in a renormalizable theory of gravity guarantees that the zero-point energy density and pressure contributions of the matter fields and gravitational fields in the vacuum |Ni must cancel each other identically (the gravity sector also has a zero-point pressure and energy density, and they take care of the matter sector quartic divergence). And moreover, each piece must readjust after we change to the vacuum |Si, with the finite or infinite symmetry breaking Λ that is induced in the matter sector being cancelled by a readjustment of the gravity sector. For this to happen, gravity cannot be independently quantized. Rather, the value of the wave function renormalization constant Z in the (generic) canonical equal time commutator [φ, π] = ihZδ¯ 3(x) must be determined via the coupling to the matter sector, so that Z can readjust as needed. THUS GRAVITY IS QUANTIZED VIA ITS COUPLING TO A QUANTIZED MATTER SOURCE, AND THERE IS NO NEED TO QUANTIZE IT ON ITS OWN.

28 14 Why Conformal Gravity Action is Unavoidable

R 4 1/2 ¯ µ With IM = − d x(−g) ψ(x)γ (x)[i∂µ +iΓµ(x)+Aµ(x)]ψ(x), either evaluate the one fermion loop Feynman diagram in the gravitational and gauge ﬁeld background, or do the fermionic path integral R D[ψ¯]D[ψ]eiIM = eiIEFF (’t Hooft 2011) in the same background, to obtain the dimensionally regularized

  µ Z 4 1/2 1 µν 1 α 2 1 µν I = Tr ln[γ (x)[i∂ + iΓ (x) + A (x)]] → d x(−g) C  [R R − (R ) ] + F F  ,(70) EFF µ µ µ 20 µν 3 α 3 µν where 1 C = . 8π2(4 − D) R 4 1/2 λµνκ The IW = −αg d x(−g) CλµνκC action is thus unavoidable – and must appear in ANY theory of gravity, with it being the Weyl action rather than the Einstein-Hilbert action that is on same footing as the Maxwell action. Hence uniﬁcation of gauge theories with gravity.

Now add spacetime-dependent fermion mass term and do path integral again to generate additional IMF.

Z 4 1/2 ¯ µ IM = − d x(−g) ψ(x)γ (x)[i∂µ + iΓµ(x) + Aµ(x) + M(x)]ψ(x). (71)

  Z 4 1/2 4 1 2 α µ ν I = d x(−g) C −M (x) + M (x)R − g ∂ M(x)∂ M(x) . (72) MF 6 α µν Now gauge M(x) and do path integral again (Eguchi and Sugawara 1974, Mannheim 1976).

Z 1 I = d4x(−g)1/2C − M 4(x) + M 2(x)Rα − g (∂µ + iAµ(x))M(x)(∂ν − iAν(x))M(x) . (73) MF 6 α µν

29 Obtain Standard Model. Either induced M(x) ∼ ψ(¯x)ψ(x), or conformally coupled fundamental scalar field φ(x) ∼ M(x), with no input tachyonic mass, but with mass scale and α double-well potential coming from non-vanishing R α when conformal symmetry spontaneously broken. Now the gravitational sector of the conformal theory is based on fourth-order equations of motion, and they are widely thought to possess negative norm ghost states, with the theory not then being unitary. And if fourth- order theories do indeed possess them, then everybody has to live with them. However, when one quantizes the conformal gravity theory it turns out to be a PT theory rather than a Hermitian one, and one is then able to construct (Bender and Mannheim 2008) a Hilbert space in which there are no states of negative norm and no states of negative energy. Conformal gravity is thus offered as a bona fide, fully consistent quantum theory of gravity, being so in the four spacetime dimensions for which there is evidence.

30 15 Einstein gravity: what must be kept

1 Γµ = gµλ [∂ g + ∂ g − ∂ g ] , (74) νσ 2 ν λσ σ λν λ νσ

2 µ ν ds = gµν(x)dx dx (75)

d2xµ dxν dxσ + Γµ = 0. (76) ds2 νσ ds ds

∂Γλ ∂Γλ Rλ = µν + Γλ Γη − µκ − Γλ Γη . (77) µνκ ∂xκ κη µν ∂xν νη µκ

ds2 = −B(r)dt2 + A(r)dr2 + r2dθ2 + r2 sin2 θdφ2, (78)

On solar distance scales outside the sun have Rµν = 0, i.e.

B(r) = A−1(r) = 1 − 2β/r, β = MG/c2, (79)

31 16 Einstein gravity: what could be changed

1 Z I = I + I = − d4x(−g)1/2Rα + I . (80) UNIV EH M 16πG α M

1 1 C = R + Rα [g g − g g ] − [g R − g R − g R + g R ] . λµνκ λµνκ 6 α λν µκ λκ µν 2 λν µκ λκ µν µν λκ µκ λν 2α(x) 2α(x) gµν(x) → e gµν(x),Cλµνκ → e Cλµνκ remarkable geometric property (81)

Z 4 1/2 λµνκ IUNIV = IW + IM = −αg d x(−g) CλµνκC + IM   Z 4 1/2 λµνκ µκ 1 α 2 = −α d x(−g) R R − 2R R + (R )  + I g λµνκ µκ 3 α M   Z 4 1/2 µν 1 α 2 ≡ −2α d x(−g) R R − (R )  + I . (82) g µν 3 α M

1/2 λµνκ µκ α 2 Lanczos : LL = (−g) RλµνκR − 4RµκR + (R α) = total divergence (83)

αg is dimensionless, so conformal theory is renormalizable

1 δIW µν 1 µν α ;β µν;β µβ;ν νβ;µ µβ ν 1 µν αβ 1/2 = W = g (R α) ;β + R ;β − R ;β − R ;β − 2R R β + g RαβR (−g) δgµν 2 2 2 µν α ;β 2 α ;µ;ν 2 α µν 1 µν α 2 1 µν − g (R α) ;β + (R α) + R αR − g (R α) = TM . (84) 3 3 3 6 4αg

Wµν = 0 if Rµν = 0. Einstein suﬃcient to give Schwarzschild and Newton, but not necessary.

32 The Einstein equations were ﬁrst derived for solar system physics. If continue Einstein equations to galaxies get dark matter problem. If continue Einstein equations to cosmology get cosmological constant problem. If continue Einstein equations to strong gravity get singularity problem. If quantize Einstein equations get renormalizability and zero-point problems. Conclusion: The standard extrapolation of solar system wisdom is not reliable. If all problems have a common origin, then maybe they all have a common solution. So seek an alternate extrapolation. Hence conformal gravity. Already know that it can also give Schwarzschild, that it is renormalizable, and that if conformal symmetry is exact at the level of the Lagrangian, there can be no fundamental cosmological constant.

33 17 Conformal gravity: an ab initio approach

iβ(x) Z 4 ¯ µ ψ(x) → e ψ(x),Aµ(x) → Aµ(x) + ∂µβ(x),IM = − d xψ(x)γ [i∂µ + Aµ(x)]ψ(x). (85)

µν Z ωµν (x)M 4 1/2 ¯ µ ψ(x) → e ψ(x),IM = − d x(−g) ψ(x)γ (x)[i∂µ + iΓµ(x)]ψ(x),

µν X µ ν µ µ a ν ν σ g (x) = Va (x)Va (x), γ (x) = Va (x)ˆγ , Γµ(x) = ([γ (x), ∂µγν(x)] − [γ (x), γσ(x)]Γµν)/8, a 2α(x) −3α(x)/2 gµν(x) → e gµν(x), ψ(x) → e ψ(x). (86)

Z 4 1/2 ¯ µ IM = − d x(−g) ψ(x)γ (x)[i∂µ + iΓµ(x) + Aµ(x)]ψ(x), −3α(x)/2+iβ(x) µ −α(x) µ ψ(x) → e ψ(x),Va (x) → e Va (x),Aµ(x) → Aµ(x). Get local conformal invariance for free.(87)

2 µ ν 17.1 Global invariances of the ﬂat spacetime light cone ds = ηµνdx dx = 0 Symmetry of gauge/fermion sector of Yang-Mills since fermions and gauge bosons have no kinematical mass. 15 invariances: xµ → xµ + µ 4 translations P µ = −i∂µ µ µ ν µν µ ν ν µ x → Λν x 6 rotations M = −i(x ∂ − x ∂ ) µ ν µ x → λx 1 dilatation D = ix ∂µ xµ + cµx2 xµ → 4 conformal Cµ = −i(xµxν∂ − x2∂µ). (88) 1 + 2c · x + c2x2 ν

[Mµν,Mρσ] = −i(ηµσMνρ − ηµρMνσ + ηνρMµσ − ηνσMµρ)

[Mµν,Pσ] = i(ηµσPν − ηνσPµ), [Mµν,Cσ] = i(ηµσCν − ηνσCµ), [Mµν,D] = 0,

[Pµ,Pν] = 0, [Cµ,Cν] = 0, [Cµ,Pν] = 2i(Mµν − ηµνD)

[D,Pµ] = −iPµ, [D,Cµ] = iCµ (89)

Close on SO(4,2). However equivalent to SU(2,2) invariance since 4+6+1+4=15 Dirac matrices: γµ, i[γµ, γν], γ5, γµγ5 close on same algebra. 4-component fermions reducible under Lorentz, but irreducible under conformal group. Thus all fermions 4-component, and right-handed neutrinos must exist. Hence both strong and weak interactions are chiral.

34 17.2 10 Killing vectors and 5 conformal Killing vectors in ﬂat spacetime

ν Kµ = aµ + bµνx , bµν = −bνµ, ∂µKν + ∂νKµ = 0, 4 + 6 = 10. (90) 2 Kµ = λxµ + cµx − 2xµc · x, ∂µKν + ∂νKµ = 2(λ − 2c · x)ηµν, 1 + 4 = 5. (91)

17.3 Connecting Conformal Gravity and Einstein Gravity

2 Under gµν(x) = ω (x)ˆgµν(x)

1 Z 1 Z I = − d4x(−g)1/2Rα = − d4x(−gˆ)1/2 ω2Rˆα − 6ˆgµν∂ ω∂ ω , (92) EH 16πG α 16πG α µ ν

’t Hooft (2011): Do path integral over conformal factor ω(x)

1 C Z 1 I = Tr ln[ˆgµν∇ˆ ∇ˆ + Rˆα ] −→ d4x(−gˆ)1/2[RˆµνRˆ − (Rˆα )2], (93) EFF µ ν 6 α 120 µν 3 α

’t Hooft (2011): go from Einstein to conformal. Maldacena (2011): go from conformal to Einstein.

35 17.4 Particle physics motivation for conformal invariance

˜ 2 2 γψψ¯ /2 2 2 γψψ¯ /2 Γψψ¯ (p, p, 0) = (−p /m ) , dψψ¯ = 3 + γψψ¯ , m0 = m(Λ /m ) . (94)

4  2 2 4  2 2  4  i Z d p µ h¯ 4 m K m 4¯h K m ρ = (m, 4F) = Tr Ln [γ p − m + i] = − K + − ln   +  . (97) MF h¯ (2π)4 µ 4π2 h¯2 4¯h4 m2 8¯h4 Deﬁne 0(M, 4F) = M/g.

2  4  2 2  4 2 2  2 2  2 2  m h¯ 4 m 4¯h K m m M 4¯h K m M ρ + Λ = ˜(m, 4F) = (m, 4F) − = − K − ln   + + ln   −  , (98) MF MF 2g 4π2 4¯h4 m2 8¯h4 2¯h4 M 2 2¯h4 ˜(M, 4F) − ˜(M = 0, 4F) < 0. So massive vacuum lies lower than massless vacuum (Nambu and Jona-Lasinio (1961)), 2 and mass-dependent quadratic divergence cancelled by mass generation mechanism. Induced ΛMF = −M /2g 2 behaves as K , and quadratic divergence cancelled identically in ρMF + ΛMF (with no need for supersymmetry).

36 Apply to Johnson-Baker-Willey QED. Find (Mannheim (1974, 1975, 1978)) that massive vacuum lies lower then massless one if γψψ¯ = −1, dψψ¯ = 2. (As γψψ¯ gets more negative, theory gets softer in ultraviolet, and thus more divergent in infrared, and get long range order phase transition at γψψ¯ = −1.)   4  2 −1/2 4 2 2   2 2   i Z d p µ −p hK¯ m M m M ρ = (m) = Tr Ln γ p − m   + i = − + ln   − 1 . (99) MF h¯ (2π)4  µ M 2  4π2 16π2h¯3 16¯h4K4

2 2 4 2 2   2   m m 0 hK¯ m M m ρ + Λ = ˜(m) = (m) − = (m) − (M) = − + ln   − 1 . (100) MF MF 2g 2M 4π2 16π2h¯3 M 2 Produce induced double-well shaped potential dynamically, with ˜(M) − ˜(M = 0) < 0. Also now have cancelled mass-dependent logarithmic divergence, and with dψψ¯ = 2 the 4-Fermi interaction is non-perturbatively scale invariant and 2 renormalizable. Note that in dynamical models the induced cosmological constant term −M /2g = ΛMF is log divergent, i.e. not ﬁnite, but is cancelled. (Induced cosmological constant of double-well Higgs potential is ﬁnite and not cancelled.). But what about K4 term? Can we cancel it by gravity? Particle physics says that inertial mass is dynamical (Higgs or hS|ψψ¯ |Si.) Equivalence principle says that inertial mass is equal to gravitational mass. Hence gravitational mass must be R 4 1/2 λµνσ dynamical too. But Planck mass not dynamical. So forbid. Hence conformal gravity with IW = αg d x(−g) CλµνσC as unique action, with no Einstein-Hilbert term and no fundamental cosmological constant term.

37 17.5 The cosmological constant problem in D = 4

µν µ ν µν µν µν hS|TMF|Si = (ρMF + pMF)U U − pMFη + ΛMF η , ηµνhS|TMFSi = 0. (101)

1 hS|T µν |Si = (ρ + p ) [4U µU ν − ηµν] (102) MF 4 MF MF

  4  2 −1/2 4 2 2   2 2   i Z d p µ −p hK¯ m M m M ρ = (m) = TrLn γ p − m   + i = − + ln   − 1 . (103) MF h¯ (2π)4  µ M 2  4π2 16π2h¯3 16¯h4K4

m2 M ρ + Λ = hS|T 00 |Si = ˜(m) = (m) − , ˜0(M) = 0(M) − = 0, MF MF MF 2g g 1 1  M 4  M 4 8π2h¯3  hK¯ 4 M 4 = ln   , = exp   , ˜(M) = − − . (104) g 8π2h¯3 16¯h4K4 16¯h4K4 g 4π2 16π2h¯3

2¯h Z M 2 M 2 ˜(M) = − d3k (k2 + M 2/h¯2)1/2 − + (k2 − M 2/h¯2)1/2 + . (105) (2π)3 4¯h2(k2 + M 2/h¯2)1/2 4¯h2(k2 − M 2/h¯2)1/2

µν µν µν µν (TGRAV)DIV + (TM )DIV = 0, (TGRAV)FIN + (TM )FIN = 0. (106)

M 2 M 2 kZ(k) = (k2 + M 2/h¯2)1/2 − + (k2 − M 2/h¯2)1/2 + , (107) 4¯h2(k2 + M 2/h¯2)1/2 4¯h2(k2 − M 2/h¯2)1/2

38 18 The dark energy problem

18.1 Standard cosmology

 dr2  ds2 = −c2dt2 + R2(t)  + r2dθ2 + r2sin2θdφ2 (108) 1 − kr2 

R˙ RR¨ H = , q = − (109) R R˙ 2

A T = (ρ + p )U U + p g − Λg , ρ = (110) µν M M µ ν M µν µν M Rn

2 2 2 R˙ + kc = R˙ [ΩM + ΩΛ] (111)

RR¨ n ! q = − = − 1 Ω − Ω (112) R˙ 2 2 M Λ

8πGρ 8πGΛ Ω = M , Ω = , (113) M 3c2H2 Λ 3cH2

4 ΩΛ cΛ TV = = 4 (114) ΩM ρM T

39 18.2 Conformal cosmology

Z 1 1 I = − d4x(−g)1/2 S;µS − S2Rµ + λS4 + iψγ¯ µ[∂ + Γ ]ψ − hSψψ¯ (115) M 2 ;µ 12 µ µ µ   1 2 1 α 4 T = (ρ + p )U U + p g − S R − g R  − g λS (116) µν M M µ ν M µν 6 0 µν 2 µν α µν 0 Cλµνκ = 0,Tµν = 0 (117)

  1 2 1 α 4 S R − g R  = (ρ + p )U U + p g − g λS (118) 6 0 µν 2 µν α M M µ ν M µν µν 0 3 Geﬀ = − 2 (119) 4πS0

RR¨ n ! R˙ 2 + kc2 = R˙ 2 hΩ¯ + Ω¯ i , q = − = − 1 Ω¯ − Ω¯ (120) M Λ R˙ 2 2 M Λ ¯ 4 ¯ 8πGeﬀρM ¯ 8πGeﬀΛ ΩΛ cΛ TV ΩM = 2 2 , ΩΛ = 2 , = = − 4 (121) 3c H 3cH Ω¯ M ρM T  T 2 −1  T 2T 2 −1 T 4 ¯    max  ¯ ¯ ΩΛ = 1 − 2  1 + 4  , ΩM = − 4 ΩΛ (122) Tmax TV TV

0 ≤ Ω¯ Λ ≤ 1, − 1 ≤ q ≤ 0 (123)

2   1/2 c (1 + z)  q0  d = − 1 − 1 + q −   (124) L   0 2   H0 q0 (1 + z )

(ΩΜ,ΩΛ) = 26 ( 0, 1 ) (0.5,0.5) (0, 0) ( 1, 0 ) (1, 0) 24 (1.5,–0.5) (2, 0) Supernova Cosmology Flat 22 Project Λ = 0 B m 20 Calan/Tololo (Hamuy et al, A.J. 1996) effective 18

14 0.02 0.05 0.1 0.2 0.5 1.0

redshift z 41

Figure 1: Supernovae data as log plot 3 No Big Bang 99% 95% 90% 2 68%

1 Λ Ω

expands forever 0 lly recollapses eventua

closed Flat Λ = 0 flat -1 Universe open

0 1 2 3

Ω42Μ

Figure 2: Hubble plot conﬁdence contours in ΩM (t0), ΩΛ(t0) plane. 24

20 apparent magnitude

14 0.0 0.2 0.4 0.6 0.8 1.0 redshift

Figure 3: The q0 = −0.37 conformal gravity ﬁt (upper curve) and the ΩM (t0) = 0.3, ΩΛ(t0) = 0.7 standard model ﬁt (lower curve) to the z < 1 supernovae Hubble plot data.

43 30

22 apparent magnitude

14 0 1 2 3 4 5 redshift

Figure 4: Hubble plot expectations for q0 = −0.37 (highest curve) and q0 = 0 (middle curve) conformal gravity and for ΩM (t0) = 0.3, ΩΛ(t0) = 0.7 standard gravity (lowest curve).

44 19 The dark matter problem

Keplerian expectation for planetary orbital velocities

45 Face on view of a spiral galaxy

46 Edge on view of a spiral galaxy

47 19.1 Dark matter – fact or fantasy? dr2 ds2 = −B(r)dt2 + + r2dΩ , (125) B(r) 2

000 0000 3 0 r 4 0000 4B (rB) 3 0 r (W 0 − W r) = ∇ B = B + = = (T 0 − T r) ≡ f(r) (126) B(r) r r 4αgB(r)

2β B(r > r ) = 1 − + γr, (127) 0 r

1 Z r0 1 Z r0 2β = dr0r04f(r0), γ = − dr0r02f(r0). (128) 6 0 2 0

β∗c2 γ∗c2r V ∗(r) = − + (129) r 2

∗ ∗ 2 2 " ! ! ! !# ∗ ∗ 2 2 ! ! 2 N β c R R R R R N γ c R R R vLOC = 3 I0 K0 − I1 K1 + I1 K1 . (130) 2R0 2R0 2R0 2R0 2R0 2R0 2R0 2R0

1 Z r Z ∞ dφ(r) 1 Z r ∇2φ = g(r), φ(r) = − dr0r02g(r0) − dr0r0g(r0), = dr0r02g(r0). (131) r 0 r dr r2 0 Newtonian Gravity is LOCAL

r Z r 1 Z r 1 Z ∞ r2 Z ∞ ∇4φ = h(r), φ(r) = − dr0r02h(r0) − dr0r04h(r0) − dr0r03h(r0) − dr0r0h(r0) (132) 2 0 6r 0 2 r 6 r dφ(r) 1 Z r 1 Z r r Z ∞ = − dr0r02h(r0) + dr0r04h(r0) − dr0r0h(r0). (133) dr 2 0 6r2 0 3 r ∇4φ = h(r) never reduces to ∇2φ = g(r), but solutions ∇4φ = h(r) do reduce to solutions to ∇2φ = g(r) when r is small. Large r solutions to ∇4φ = h(r) involve matter distribution at large distances from a local source.

48 Conformal Gravity is GLOBAL. So cannot ignore the rest of the universe. Rest of universe has homogeneous Hubble ﬂow and inhomogeneous clusters of galaxies.

4r Z ρ = 1/2 , τ = dtR(t) (134) 2(1 + γ0r) + 2 + γ0r

2  2  2  2 2 dr 2 1 1 + γ0ρ/4 2 2 R (τ) 2 2 −(1 + γ0r)c dt + + r dΩ2 = 2   −c dτ + 2 2 2 dρ + ρ dΩ2  (135) (1 + γ0r) R (τ) 1 − γ0ρ/4 [1 − γ0ρ /16]

γ c2R v2 = v2 + 0 . (136) TOT LOC 2

∗ −41 −1 −30 −1 γ = 5.42 × 10 cm , γ0 = 3.06 × 10 cm . (137)

γ c2R N ∗β∗c2 N ∗γ∗c2R γ c2R v2 = v2 + 0 − κc2R2, v2 → + + 0 − κc2R2, (138) TOT LOC 2 TOT R 2 2

∗ ∗ 2 2 " ! ! ! !# ∗ ∗ 2 2 ! ! 2 N β c R R R R R N γ c R R R vLOC = 3 I0 K0 − I1 K1 + I1 K1 . (139) 2R0 2R0 2R0 2R0 2R0 2R0 2R0 2R0

κ = 9.54 × 10−54 cm−2 ≈ (100 Mpc)−2. (140)

Fit 138 galaxies with VISIBLE N ∗ of each galaxy (i.e. galactic mass to light ratio M/L) as only variable, ∗ ∗ ∗ with β , γ , γ0 and κ all universal, and with NO DARK MATTER, and with 276 fewer free parameters than 2 2 −30 −1 in dark matter calculations. Works since (v /c R)last ∼ 10 cm for every galaxy.

49 70 100 140 150 60 120 80 50 100 100 60 40 80

30 60 40 50 20 40 20 10 20 DDO 154 IC 2574 NGC 925 NGC 2403 0 0 0 0 0 2 4 6 8 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 5 10 15 20

100 300 250 200

250 80 200 150 200 60 150

150 100 40 100 100 50 20 50 50 NGC 2841 NGC 2903 NGC 2976 NGC 3031 0 0 0 0 0 10 20 30 40 50 0 5 10 15 20 25 30 0.0 0.5 1.0 1.5 2.0 2.5 0 2 4 6 8 10 12 14

250

200 150 150 200

150 150 100 100

100 100

50 50 50 50

NGC 3198 NGC 3521 NGC 3621 NGC 3627 0 0 0 0 0 10 20 30 0 5 10 15 20 25 30 35 0 5 10 15 20 25 0 2 4 6 8 200 250 200 200 150 200 150 150 150 100 100 100 100

50 50 50 50

NGC 4736 NGC 4826 NGC 5055 NGC 6946 0 0 0 0 0 2 4 6 8 10 0 5 10 15 0 10 20 30 40 0 5 10 15 20

140

250 120

200 100

80 150 60 100 40

50 20 NGC 7331 NGC 7793 0 0 0 5 10 15 20 0 2 4 6 8 10 FIG. 1: Fitting to the rotational velocities (in km sec−1) of the THINGS 18 galaxy sample

50 140 200 120 150

150 100

80 100 100 60

40 50 50 20 NGC 3726 NGC 3769 NGC 3877 0 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 2 4 6 8 10

200 150

150 150 100 100 100

50 50 50

NGC 3893 NGC 3917 NGC 3949 0 0 0 0 5 10 15 20 0 2 4 6 8 10 12 14 0 1 2 3 4 5 6 7

250 140 250 200 120 200 100 150 80 150

100 60 100 40 50 50 20 NGC 3953 NGC 3972 NGC 3992 0 0 0 0 5 10 15 0 2 4 6 8 0 10 20 30 40 50

140 200 120 150

100 150

80 100

60 100

40 50 50 20 NGC 4010 NGC 4013 NGC 4051 0 0 0 0 2 4 6 8 10 0 5 10 15 20 25 30 0 2 4 6 8 10

200 150 200

150 150 100

100 100

50 50 50

NGC 4085 NGC 4088 NGC 4100 0 0 0 0 1 2 3 4 5 6 0 5 10 15 0 5 10 15 20 25 FIG. 2: Fitting to the rotational velocities of the Ursa Major 30 galaxy sample – Part 1

51 140 200 200 120

150 150 100

100 100 60

40 50 50 20 NGC 4138 NGC 4157 NGC 4183 0 0 0 0 5 10 15 0 5 10 15 20 25 30 0 5 10 15 200 120 100

100 150 80 80 60 100 60 40 40 50 20 20 NGC 4217 NGC 4389 UGC 6399 0 0 0 0 5 10 15 0 1 2 3 4 0 2 4 6 8

100 100 80

80 80 60 60 60 40 40 40

20 20 20

UGC 6446 UGC 6667 UGC 6818 0 0 0 0 2 4 6 8 10 12 0 2 4 6 8 0 2 4 6 8

100 120 120

100 80 100

80 80 60

60 60 40 40 40

20 20 20 UGC 6917 UGC 6923 UGC 6930 0 0 0 0 2 4 6 8 10 0 1 2 3 4 5 0 5 10 15

100 120 200 100 80

150 80 60

60 100 40 40

50 20 20 UGC 6973 UGC 6983 UGC 7089 0 0 0 0 2 4 6 8 10 0 5 10 15 0 1 2 3 4 5 6 7 FIG. 3: Fitting to the rotational velocities of the Ursa Major 30 galaxy sample – Part 2

52 120 80 120 120

100 100 100 60 80 80 80

60 60 40 60

40 40 40 20 20 20 20 DDO 64 F 563-1 F 563-V2 F 568-3 0 0 0 0 0.0 0.5 1.0 1.5 2.0 0 5 10 15 0 1 2 3 4 5 6 0 2 4 6 8 10 50 100 100 80 40 80 80 60 60 30 60 40 40 20 40

20 20 20 10

F 583-1 F 583-4 NGC 959 NGC 4395 0 0 0 0 0 2 4 6 8 10 12 14 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8

120 140 120 80 120 100 100

60 100 80 80 80 60 40 60 60 40 40 40 20 20 20 20 NGC 7137 UGC 128 UGC 191 UGC 477 0 0 0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 10 20 30 40 50 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10 120 50 100 120 100 40 80 100 80 30 60 80 60 60 20 40 40 40

10 20 20 20 UGC 1230 UGC 1281 UGC 1551 UGC 4325 0 0 0 0 0 5 10 15 20 25 30 35 0.0 0.5 1.0 1.5 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 3.0

120 120 80 100 150 100

80 60 80 100 60 60 40

40 40 50 20 20 20 UGC 5005 UGC 5750 UGC 5999 UGC 11820 0 0 0 0 0 5 10 15 20 25 0 2 4 6 8 0 2 4 6 8 10 12 14 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 FIG. 4: Fitting to the rotational velocities of the LSB 20 galaxy sample

53 50 60

80 300 50 40 250 60 40 200 30 30 150 40 20 20 100 20 10 50 10 ESO 140040 ESO 840411 ESO 1200211 ESO 1870510 0 0 0 0 0 5 10 15 20 25 30 0 2 4 6 8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 70 140 100 60 120 150

80 50 100

40 80 60 100

60 30 40 40 20 50 20 20 10 ESO 2060140 ESO 3020120 ESO 3050090 ESO 4250180 0 0 0 0 0 2 4 6 8 10 0 2 4 6 8 10 0 1 2 3 4 5 0 2 4 6 8 10 12 14

120 150 140 150 100 120

100 80 100 100 80 60 60 40 50 50 40

20 20 ESO 4880490 F 571 - 08 F 579 -V1 F 730- V1 0 0 0 0 0 2 4 6 8 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 60 250 100 150 50 200 80 40 150 100 60 30

100 40 20 50

10 50 20 UGC 4115 UGC 6614 UGC 11454 UGC 11557 0 0 0 0 0.0 0.5 1.0 1.5 0 10 20 30 40 50 60 0 2 4 6 8 10 12 0 1 2 3 4 5 6 60 150 150 250 50

200 40 100 100 150 30

100 20 50 50

10 50 UGC 11583 UGC 11616 UGC 11648 UGC 11748 0 0 0 0 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10 0 2 4 6 8 10 12 0 5 10 15 20

150

100

UGC 11819 0 0 2 4 6 8 10 12 FIG. 5: Fitting to the rotational velocities of the LSB 21 galaxy sample 54 80 80 120 100

100 60 80 60 80 60 40 40 60 40 40 20 20 20 20 DDO 168 DDO 170 M 33 NGC 55 0 0 0 0 0 1 2 3 4 0 2 4 6 8 10 12 0 2 4 6 8 0 2 4 6 8 10 12 120 140 120 250 100 120 100 200 100 80 80 80 150 60 60 60 100 40 40 40

50 20 20 20 NGC 247 NGC 300 NGC 801 NGC 1003 0 0 0 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 0 10 20 30 40 0 5 10 15 20 25 30 100 250 250 80 80 200 200 60 60 150 150 40 40 100 100

20 20 50 50

NGC 1560 NGC 2683 NGC 2998 NGC 3109 0 0 0 0 0 2 4 6 8 10 0 5 10 15 20 25 30 35 0 10 20 30 40 0 1 2 3 4 5 6 7

300 350 100 250 250 300 80 200 250 200 60 150 200 150 150 40 100 100 100

50 50 20 50 NGC 5033 NGC 5371 NGC 5533 NGC 5585 0 0 0 0 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 50 0 2 4 6 8 10 12 14

300 350 140 100

250 120 300 80 250 200 100

80 200 60 150 60 150 40 100 40 100

50 20 20 50 NGC 5907 NGC 6503 NGC 6674 UGC 2259 0 0 0 0 0 10 20 30 40 0 5 10 15 20 0 10 20 30 40 50 60 0 2 4 6

350 300 300 250 250 200 200 150 150

100 100

50 50 UGC 2885 Malin 1 0 0 0 10 20 30 40 50 60 70 0 20 40 60 80 100 FIG. 6: Fitting to the rotational velocities of the Miscellaneous 22 galaxy sample 55 140 120 80 120 100

100 80 60 80 60 40 60 40 40 20 20 20 F 568-V1 F 574-1 UGC 731 0 0 0 0 5 10 15 0 2 4 6 8 10 12 14 0 2 4 6 8 10 120 100 80 100 80 60 80

60 60 40 40 40

20 20 20 UGC 3371 UGC 4173 UGC 4325 0 0 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7

80 100 80

80 60 60 60

40 40 40

20 20 20

UGC 4499 UGC 5414 UGC 5721 0 0 0 0 2 4 6 8 0 1 2 3 4 0 2 4 6 8

100 60 100 50 80 80 40 60 60 30 40 40 20

20 10 20 UGC 5750 UGC 7232 UGC 7323 0 0 0 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 0 1 2 3 4 5

FIG. 7: Fitting to the rotational velocities (in km sec−1) of the 24 dwarf galaxy sample – Part 1 56 100 50 120

100 80 40

80 60 30

60 40 20 40

20 10 20 UGC 7399 UGC 7524 UGC 7559 0 0 0 0 5 10 15 20 25 30 0 2 4 6 8 0.0 0.5 1.0 1.5 2.0 2.5 40 80 100

80 30 60

20 40 40

10 20 20

UGC 7577 UGC 7603 UGC 8490 0 0 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 2 4 6 8 0 2 4 6 8 10

120 80 100 150

60 80 100 60 40

40 50 20 20 UGC 9211 UGC 11707 UGC 11861 0 0 0 0 2 4 6 8 0 5 10 15 20 0 2 4 6 8 10 12

100 120 80 100 80

60 80 60 60 40 40 40

20 20 20 UGC 12060 UGC 12632 UGC 12732 0 0 0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 12 14

FIG. 8: Fitting to the rotational velocities of the 24 dwarf galaxy sample – Part 2 57 60 50 80 50 40

60 40 30 30 40 20 20

20 10 10 UGC 3851 UGC 4305 UGC 4459 0 0 0 0 2 4 6 8 10 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 60 60 100

50 50 80

40 40 60

30 30 40 20 20

20 10 10 UGC 5139 UGC 5423 UGC 5666 0 0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10

FIG. 9 Fitting to the rotational velocities of the 6 dwarf galaxy sample

100 140 300 80 120 250 100 60 200 80 150 40 60 100 40 20 20 50 DDO 154 UGC 128 Malin 1 0 0 0 0 10 20 30 40 50 0 20 40 60 80 0 50 100 150 FIG. 10: Extended distance predictions for DDO 154, UGC 128, and Malin 1

58 Properties of the THINGS 18 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI DDO 0154 LSB 4.2 0.007 0.8 8.1 0.03 0.003 0.45 1.12 [?][?][?][?] IC 2574 LSB 4.5 0.345 4.2 13.1 0.19 0.098 0.28 1.69 [?][?][?][?] NGC 0925 LSB 8.7 1.444 3.9 12.4 0.41 1.372 0.95 4.17 [?][?][?][?] NGC 2403 HSB 4.3 1.647 2.7 23.9 0.46 2.370 1.44 2.89 [?][?][?][?] NGC 2841 HSB 14.1 4.742 3.5 51.6 0.86 19.552 4.12 5.83 [?][?][?][?] NGC 2903 HSB 9.4 4.088 3.0 30.9 0.49 7.155 1.75 3.75 [?][?][?][?] NGC 2976 LSB 3.6 0.201 1.2 2.6 0.01 0.322 1.60 10.43 [?][?][?][?] NGC 3031 HSB 3.7 3.187 2.6 15.0 0.38 8.662 2.72 9.31 [?][?][?][?] NGC 3198 HSB 14.1 3.241 4.0 38.6 1.06 3.644 1.12 2.09 [?][?][?][?] NGC 3521 HSB 12.2 4.769 3.3 35.3 1.03 9.245 1.94 4.21 [?][?][?][?] NGC 3621 HSB 7.4 2.048 2.9 28.7 0.89 2.891 1.41 3.18 [?][?][?][?] NGC 3627 HSB 10.2 3.700 3.1 8.2 0.10 6.622 1.79 15.64 [?][?][?][?] NGC 4736 HSB 5.0 1.460 2.1 10.3 0.05 1.630 1.60 4.66 [?][?][?][?] NGC 4826 HSB 5.4 1.441 2.6 15.8 0.03 3.640 2.53 5.46 [?][?][?][?] NGC 5055 HSB 9.2 3.622 2.9 44.4 0.76 6.035 1.87 2.36 [?][?][?][?] NGC 6946 HSB 6.9 3.732 2.9 22.4 0.57 6.272 1.68 6.39 [?][?][?][?] NGC 7331 HSB 14.2 6.773 3.2 24.4 0.85 12.086 1.78 9.61 [?][?][?][?] NGC 7793 HSB 5.2 0.910 1.7 10.3 0.16 0.793 0.87 3.61 [?][?][?][?]

59 Properties of the Ursa Major 30 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI NGC 3726 HSB 17.4 3.340 3.2 31.5 0.60 3.82 1.15 3.19 [?][?][?][?] NGC 3769 HSB 15.5 0.684 1.5 32.2 0.41 1.36 1.99 1.43 [?][?][?][?] NGC 3877 HSB 15.5 1.948 2.4 9.8 0.11 3.44 1.76 10.51 [?][?][?][?] NGC 3893 HSB 18.1 2.928 2.4 20.5 0.59 5.00 1.71 3.85 [?][?][?][?] NGC 3917 LSB 16.9 1.334 2.8 13.9 0.17 2.23 1.67 4.85 [?][?][?][?] NGC 3949 HSB 18.4 2.327 1.7 7.2 0.35 2.37 1.02 14.23 [?][?][?][?] NGC 3953 HSB 18.7 4.236 3.9 16.3 0.31 9.79 2.31 10.20 [?][?][?][?] NGC 3972 HSB 18.6 0.978 2.0 9.0 0.13 1.49 1.53 7.18 [?][?][?][?] NGC 3992 HSB 25.6 8.456 5.7 49.6 1.94 13.94 1.65 4.08 [?][?][?][?] NGC 4010 LSB 18.4 0.883 3.4 10.6 0.29 2.03 2.30 5.03 [?][?][?][?] NGC 4013 HSB 18.6 2.088 2.1 33.1 0.32 5.58 2.67 3.14 [?][?][?][?] NGC 4051 HSB 14.6 2.281 2.3 9.9 0.18 3.17 1.39 8.52 [?][?][?][?] NGC 4085 HSB 19.0 1.212 1.6 6.5 0.15 1.34 1.11 10.21 [?][?][?][?] NGC 4088 HSB 15.8 2.957 2.8 18.8 0.64 4.67 1.58 5.79 [?][?][?][?] NGC 4100 HSB 21.4 3.388 2.9 27.1 0.44 5.74 1.69 3.35 [?][?][?][?] NGC 4138 LSB 15.6 0.827 1.2 16.1 0.11 2.97 3.59 5.04 [?][?][?][?] NGC 4157 HSB 18.7 2.901 2.6 30.9 0.88 5.83 2.01 3.99 [?][?][?][?] NGC 4183 HSB 16.7 1.042 2.9 19.5 0.30 1.43 1.38 2.36 [?][?][?][?] NGC 4217 HSB 19.6 3.031 3.1 18.2 0.30 5.53 1.83 6.28 [?][?][?][?] NGC 4389 HSB 15.5 0.610 1.2 4.6 0.04 0.42 0.68 9.49 [?][?][?][?] UGC 6399 LSB 18.7 0.291 2.4 8.1 0.07 0.59 2.04 3.42 [?][?][?][?] UGC 6446 LSB 15.9 0.263 1.9 13.6 0.24 0.36 1.36 1.70 [?][?][?][?] UGC 6667 LSB 19.8 0.422 3.1 8.6 0.10 0.71 1.67 3.09 [?][?][?][?] UGC 6818 LSB 21.7 0.352 2.1 8.4 0.16 0.11 0.33 2.35 [?][?][?][?] UGC 6917 LSB 18.9 0.563 2.9 10.9 0.22 1.24 2.20 4.05 [?][?][?][?] UGC 6923 LSB 18.0 0.297 1.5 5.3 0.08 0.35 1.18 4.43 [?][?][?][?] UGC 6930 LSB 17.0 0.601 2.2 15.7 0.29 1.02 1.69 2.68 [?][?][?][?] UGC 6973 HSB 25.3 1.647 2.2 11.0 0.35 3.99 2.42 10.58 [?][?][?][?] UGC 6983 LSB 20.2 0.577 2.9 17.6 0.37 1.28 2.22 2.43 [?][?][?][?] UGC 7089 LSB 13.9 0.352 2.3 7.1 0.07 0.35 0.98 3.18 [?][?][?][?]

60 Properties of the LSB 20 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI DDO 0064 LSB 6.8 0.015 1.3 2.1 0.02 0.04 2.87 6.05 [?][?][?][?] F563-1 LSB 46.8 0.140 2.9 18.2 0.29 1.35 9.65 2.44 [?][?][?][?] F563-V2 LSB 57.8 0.266 2.0 6.3 0.20 0.60 2.26 6.15 [?][?][?][?] F568-3 LSB 80.0 0.351 4.2 11.6 0.30 1.20 3.43 3.16 [?][?][?][?] F583-1 LSB 32.4 0.064 1.6 14.1 0.18 0.15 2.32 1.92 [?][?][?][?] F583-4 LSB 50.8 0.096 2.8 7.0 0.06 0.31 3.25 2.52 [?][?][?][?] NGC 0959 LSB 13.5 0.333 1.3 2.9 0.05 0.37 1.11 7.43 [?][?][?][?] NGC 4395 LSB 4.1 0.374 2.7 0.9 0.13 0.83 2.21 2.29 [?][?][?][?] NGC 7137 LSB 25.0 0.959 1.7 3.6 0.10 0.27 0.28 3.91 [?][?] ES [?] UGC 0128 LSB 64.6 0.597 6.9 54.8 0.73 2.75 4.60 1.03 [?][?][?][?] UGC 0191 LSB 15.9 0.129 1.7 2.2 0.26 0.49 3.81 15.48 [?][?][?][?] UGC 0477 LSB 35.8 0.871 3.5 10.2 1.02 1.00 1.14 4.42 [?][?] ES [?] UGC 1230 LSB 54.1 0.366 4.7 37.1 0.65 0.67 1.82 0.97 [?][?][?][?] UGC 1281 LSB 5.1 0.017 1.6 1.7 0.03 0.01 0.53 3.02 [?][?][?][?] UGC 1551 LSB 35.6 0.780 4.2 6.6 0.44 0.16 0.20 3.69 [?][?][?][?] UGC 4325 LSB 11.9 0.373 1.9 3.4 0.10 0.40 1.08 7.39 [?][?][?][?] UGC 5005 LSB 51.4 0.200 4.6 27.7 0.28 1.02 5.11 1.30 [?][?][?][?] UGC 5750 LSB 56.1 0.472 3.3 8.6 0.10 0.10 0.21 1.58 [?][?][?][?] UGC 5999 LSB 44.9 0.170 4.4 15.0 0.18 3.36 19.81 5.79 [?][?][?][?] UGC 11820 LSB 17.1 0.169 3.6 3.7 0.40 1.68 9.95 8.44 [?][?][?][?]

61 Properties of the LSB 21 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI ESO 0140040 LSB 217.8 7.169 10.1 30.0 20.70 3.38 8.29 [?][?][?] NA ESO 0840411 LSB 82.4 0.287 3.5 9.1 0.06 0.21 1.49 [?][?] ES NA ESO 1200211 LSB 15.2 0.028 2.0 3.5 0.01 0.20 0.66 [?][?] ES NA ESO 1870510 LSB 16.8 0.054 2.1 2.8 0.09 1.62 2.02 [?][?][?] NA ESO 2060140 LSB 59.6 0.735 5.1 11.6 3.51 4.78 4.34 [?][?][?] NA ESO 3020120 LSB 70.9 0.717 3.4 11.2 0.77 1.07 2.37 [?][?] ES NA ESO 3050090 LSB 13.2 0.186 1.3 5.6 0.06 0.32 1.87 [?][?] ES NA ESO 4250180 LSB 88.3 2.600 7.3 14.6 4.79 1.84 5.17 [?][?][?] NA ESO 4880490 LSB 28.7 0.139 1.6 7.8 0.43 3.07 4.34 [?][?] ES NA F571-8 LSB 50.3 0.191 5.4 14.6 0.16 4.48 23.49 5.10 [?][?][?][?] F579-V1 LSB 86.9 0.557 5.2 14.7 0.21 3.33 5.98 3.18 [?][?][?][?] F730-V1 LSB 148.3 0.756 5.8 12.2 5.95 7.87 6.22 [?][?][?] NA UGC 04115 LSB 5.5 0.004 0.3 1.7 0.01 0.97 3.42 [?][?][?] NA UGC 06614 LSB 86.2 2.109 8.2 62.7 2.07 9.70 4.60 2.39 [?][?][?][?] UGC 11454 LSB 93.9 0.456 3.4 12.3 3.15 6.90 6.79 [?][?][?] NA UGC 11557 LSB 23.7 1.806 3.0 6.7 0.25 0.37 0.20 3.49 [?][?][?][?] UGC 11583 LSB 7.1 0.012 0.7 2.1 0.01 0.96 2.15 [?][?][?] NA UGC 11616 LSB 74.9 2.159 3.1 9.8 2.43 1.13 7.49 [?][?][?] NA UGC 11648 LSB 49.0 4.073 4.0 13.0 2.57 0.63 5.79 [?][?][?] NA UGC 11748 LSB 75.3 23.930 2.6 21.6 9.67 0.40 1.01 [?][?][?] NA UGC 11819 LSB 61.5 2.155 4.7 11.9 4.83 2.24 7.03 [?][?][?] NA

62 Properties of the Miscellaneous 22 Galaxy Sample 2 2 Galaxy Type Distance LB R0 Rlast MHI Mdisk (M/L)stars (v /c R)last Data Sources 10 10 10 −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) v L R0 HI DDO 0168 LSB 4.5 0.032 1.2 4.4 0.03 0.06 2.03 2.22 [?][?][?][?] DDO 0170 LSB 16.6 0.023 1.9 13.3 0.09 0.05 1.97 1.18 [?][?][?][?] M 0033 HSB 0.9 0.850 2.5 8.9 0.11 1.13 1.33 4.62 [?][?][?][?] NGC 0055 LSB 1.9 0.588 1.9 12.2 0.13 0.30 0.50 2.22 [?][?][?][?] NGC 0247 LSB 3.6 0.512 4.2 14.3 0.16 1.25 2.43 2.94 [?][?][?][?] NGC 0300 LSB 2.0 0.271 2.1 11.7 0.08 0.65 2.41 2.69 [?][?][?][?] NGC 0801 HSB 63.0 4.746 9.5 46.7 1.39 6.93 2.37 3.59 [?][?][?][?] NGC 1003 LSB 11.8 1.480 1.9 31.2 0.63 0.66 0.45 1.53 [?][?][?][?] NGC 1560 LSB 3.7 0.053 1.6 10.3 0.12 0.17 3.16 2.16 [?][?][?][?] NGC 2683 HSB 10.2 1.882 2.4 36.0 0.15 6.03 3.20 2.28 [?][?][?][?] NGC 2998 HSB 59.3 5.186 4.8 41.1 1.78 7.16 1.75 3.43 [?][?][?][?] NGC 3109 LSB 1.5 0.064 1.3 7.1 0.06 0.02 0.35 2.29 [?][?][?][?] NGC 5033 HSB 15.3 3.058 7.5 45.6 1.07 0.27 3.28 3.16 [?][?][?][?] NGC 5371 HSB 35.3 7.593 4.4 41.0 0.89 8.52 1.44 3.98 [?][?][?][?] NGC 5533 HSB 42.0 3.173 7.4 56.0 1.39 2.00 4.14 3.31 [?][?][?][?] NGC 5585 HSB 9.0 0.333 2.0 14.0 0.28 0.36 1.09 2.06 [?][?][?][?] NGC 5907 HSB 16.5 5.400 5.5 48.0 1.90 2.49 1.89 3.44 [?][?][?][?] NGC 6503 HSB 5.5 0.417 1.6 20.7 0.14 1.53 3.66 2.30 [?][?][?][?] NGC 6674 HSB 42.0 4.935 7.1 59.1 2.18 2.00 2.52 3.57 [?][?][?][?] UGC 2259 LSB 10.0 0.110 1.4 7.8 0.04 0.47 4.23 3.76 [?][?][?][?] UGC 2885 HSB 80.4 23.955 13.3 74.1 3.98 8.47 0.72 4.31 [?][?][?][?] Malin 1 LSB 338.5 7.912 84.2 98.0 5.40 1.00 1.32 1.77 [?][?][?][?]

63 Properties of the 30 Dwarf Galaxy Sample 2 2 Galaxy Distance LB i (R0)disk Rlast MHI Mdisk (M/LB)disk (v /c R)last 9 B ◦ 9 9 B −30 −1 (Mpc) (10 L ) (kpc) (kpc) (10 M ) (10 M )(M /L ) (10 cm ) F568-V1 78.20 2.15 40 3.11 17.07 2.32 16.00 7.45 2.95 F574-1 94.10 3.42 65 4.20 13.69 3.31 14.90 4.35 2.77 UGC 731 11.80 0.69 57 2.43 10.30 1.61 3.21 4.63 1.91 UGC 3371 18.75 1.54 49 4.53 15.00 2.62 4.49 2.91 1.78 UGC 4173 16.70 0.33 –5 4.43 12.14 2.24 0.07 0.20 1.21 UGC 4325 11.87 1.71 41 1.92 6.91 1.04 6.51 3.82 4.37 UGC 4499 12.80 1.01 50 1.46 8.38 1.15 1.80 1.79 2.37 UGC 5414 9.40 0.49 55 1.40 4.10 0.57 1.13 2.29 3.31 UGC 5721 7.60 0.48 +10 0.76 8.41 0.57 1.90 3.96 2.27 UGC 5750 56.10 4.72 64 5.60 21.77 1.00 3.68 0.78 1.03 UGC 7232 3.14 0.08 59 0.30 0.91 0.06 0.14 1.76 7.64 UGC 7323 7.90 2.39 47 2.13 5.75 0.70 4.19 1.75 4.59 UGC 7399 24.66 4.61 2.32 32.30 6.38 5.42 1.18 1.32 UGC 7524 4.12 1.37 46 3.02 9.29 1.34 5.29 3.86 2.67 UGC 7559 4.20 0.04 61 0.87 2.75 0.12 0.05 1.32 1.43 UGC 7577 2.13 0.05 –15 0.51 1.39 0.04 0.01 0.20 1.18 UGC 7603 9.45 0.80 78 1.24 8.24 1.04 0.41 0.52 1.81 UGC 8490 5.28 0.95 +10 0.71 11.51 0.72 1.53 1.61 1.47 UGC 9211 14.70 0.33 44 1.54 9.62 1.43 1.23 3.69 1.55 UGC 11707 21.46 1.13 68 5.82 20.30 6.78 9.89 8.76 1.77 UGC 11861 19.55 9.44 50 4.69 12.80 4.33 45.84 4.86 6.55 UGC 12060 15.10 0.39 40 1.70 9.89 1.67 3.45 8.94 2.00 UGC 12632 9.20 0.86 46 3.43 11.38 1.55 4.26 4.97 1.82 UGC 12732 12.40 0.71 39 2.10 14.43 3.23 4.11 5.76 2.40 UGC 3851 4.85 2.33 2.07 11.70 1.32 0.47 0.20 1.37 UGC 4305 2.34 0.41 –5 0.68 4.75 0.31 0.08 0.20 1.18 UGC 4459 3.06 0.03 27 0.60 2.47 0.04 0.01 0.20 0.97 UGC 5139 4.69 0.20 14 0.96 3.58 0.21 0.05 0.24 1.19 UGC 5423 7.14 0.14 45 0.61 1.97 0.05 0.28 2.01 1.82 UGC 5666 3.85 2.53 56 3.56 11.25 1.37 1.96 0.77 1.88

64 20 Summary

To conclude, we note that at the beginning of the 20th century studies of black-body radiation on microscopic scales led to a paradigm shift in physics. Thus it could that at the beginning of the 21st century studies of black-body radiation, this time on macroscopic cosmological scales, might be presaging a paradigm shift all over again.