Bianchi IX and VIII Quantum Cosmology with a Cosmological Constant, Aligned Electromagnetic Field, and Scalar Field
BBianchi IX and VIII Quantum Cosmology with a Cosmological Constant, AlignedElectromagnetic Field, and Scalar Field
Daniel Berkowitz ∗ (Dated: February 5, 2021)We investigate the quantum cosmologies of the Bianchi IX and VIII models when a cosmologicalconstant, aligned electromagnetic field and free scalar field are present. The conserved quantity p φ associated with our free scalar field results in φ classically being a quantity which monotonicallyincreases with respect to time, thus allowing it to fulfill the role of an ’emergent’ internal clockfor our constrained quantum systems. We embark on this investigation to better understand howmatter sources can affect general anisotropic quantum cosmologies. To aid us we use the Euclidean-signature semi classical method to obtain our wave functions and analyze them. In addition westudy briefly the quantum Taub models when an e φ potential and aligned electromagnetic field arepresent. One of the interesting things we found was that our aligned electromagnetic field, dependingon how strong it is, can create or destroy geometric states in our ’excited’ state wave functions thatquantum Bianchi IX universes can tunnel in and out of. This creation of a state is somewhat similarto how non-commutativity in the minisuperspace variables can cause new quantum states to emergein the quantum Kantowski-Sachs and Bianchi I models. Our results further show the utility of theEuclidean-signature semi classical method towards tackling Lorentzian signature problems withouthaving to invoke a Wick rotation. This feature of not needing to apply a Wick rotation makes thismethod potentially very useful for tackling a variety of problems in bosonic relativistic field theoryand quantum gravity. I. INTRODUCTION
Even though on large scales our universe is incredi-bly isotropic and homogeneous it is highly likely thatour early universe possessed a considerable degree ofanisotropy and inhomogeneity which originated fromquantum fluctuations within particle fields whose sizewas comparable to the primordial cosmological horizon.Thus it is useful to study anistropic or inhomogeneousclassical/quantum cosmologies so we can better under-stand what our universe could have been like when it wasextremely young. To accomplish this we will study thequantum Bianchi IX, VIII and Taub models with mat-ter sources in order to obtain an idea of what possibleeffects those matter sources could have induced in ourearly universe. In the future it would be worthwhile toinclude perturbative inhomogeneities in either the elec-tric/magnetic or scalar field.The wave functions we will obtain are easiest to in-terpret in a qualitative manner when we allow Λ < .Choosing a negative cosmological constant does not nec-essarily make our solutions non-physical. Recently therehas been some interesting work[1–5] done in studying in-flation with a negative cosmological constant and theconnection between asymptotically AdS wave functionsto classical cosmological histories which exhibit phe-nomenology that one expects from universes with a pos-itive cosmological constant. From a theoretical point ofview it is worthwhile to study quantum cosmologies that ∗ Physics Department, Yale [email protected] work is in memory of my parents, Susan Orchan Berkowitz,and Jonathan Mark Berkowitz possess a negative cosmological constant.In addition we will only consider an electromagneticfield in which the electric and magnetic components of itare parallel with each other and are both non-vanishing.The case when only a primordial magnetic field is presentcan be realized in the quantum Bianchi I[6] models whichcan also be thoroughly analyzed with the method we willemploy in this paper.Despite these restrictions the wave functions we ob-tain exhibit fascinating properties. This is especially truein regards to how our aligned electromagnetic field af-fects the ’excited’ states of our quantum models whichwe computed using the Euclidean-signature semi classi-cal method. As a result they provide further incentives tostudy wave functions of the universe derived from moregeneral electric/magnetic field configurations.Beyond theoretical considerations, new evidence[7, 8]for the existence of a femto Gauss strength intergalacticmagnetic field has been uncovered by observing gammarays. This provides further reasons to continue [9–15] studying electric/magnetic fields through the lens ofquantum cosmology. Through studying the effects ofelectric/magnetic fields on quantum universes we can po-tentially better understand how seeds of anisotropy de-veloped in our early universe which we can observe[16–18]today in the CMB. Recent work[19–22] has been con-ducted in trying to determine what signatures a primor-dial magnetic field would induce on the various com-putable spectrums derived from the CMB. By studyingwhat effects aligned electromagnetic fields induce on both’ground’ state and ’excited’ state wave functions of quan-tum cosmological models we can contribute to the theo-retical portion of that task.The Bianchi IX models possess a rich history and havebeen studied in a plethora of contexts [23–30] Investiga- a r X i v : . [ g r- q c ] F e b tions into them began with the works of Misner, Ryan,and Belinskii, Khalatnikov and Lifshitz[31–34]. Whatpartly made these models so appealing was that theyshared the 3-sphere spatial topology of the k=1 FLRWmodels, which were widely believed to be a good approx-imation for our physical universe until more precise cos-mological observations [35] showed otherwise. In additionthe equations which govern the dynamics of the BianchiIX mini-superspace variables, which we will take to be theMisner variables ( α, β + , β − ) [31, 36–39] admit chaotic so-lutions. It was originally thought that the chaos presentnear its singularity, which took the form of an erraticsequence of Kasner contractions and expansions couldexplain why we observe the universe to be mostly homo-geneous, hence the origin of the name Mixmaster[31].The quantum diagonalized Bianchi IX models werefirst studied by Misner [36], and later by Moncrief-Ryan [40]; among others in a plethora of different con-texts, such as supersymmetric quantum cosmology[41,42]. They were later revisited in the 2010s by Baeand Moncrief[43, 44], when they applied the Euclidean-signature semi classical method[44, 45] to prove for thewormhole case that a smooth and globally defined asymp-totic solution exists for arbitrary Hartle-Hawking[46] or-dering parameter. They also studied the ’excited’ statesof the Bianchi IX models. The quantum cosmology ofBianchi IX models with a cosmological constant was pre-viously investigated by employing Chern-Simons solu-tions in Ashtekar’s variables [47, 48].The Bianchi VIII quantum/classical models havenot been as thoroughly studied as the Bianchi IXmodels. However, there has been some interestinginvestigations[49–51] into them and exact solutions ofthem have been obtained in both classical and quantumregimes.The classical Taub models [52] and their extension, theTaub-NUT models have a rich history of their own, whichincludes being used to model the space-time around ablack hole [53, 54]. Recently work has been conducted infinding new solutions to the symmetry reduced WheelerDeWitt equation of the LRS Bianchi IX models [55] usingKilling vectors and tensors. Their quantum cosmologyhas also been investigated within the context of the gen-eralized uncertainty principle [56] and using the WKBapproximation[57, 58]. In addition the author proved[59]the existence of a countably infinite number of ’excited’states for the quantum Taub models when a cosmologicalconstant is present and is currently working on provingtheir existence when both a cosmological constant andaligned electromagnetic field are present. We will fur-ther expand upon what was previously done by obtain-ing a leading order or closed form solution depending onthe operator ordering of the Wheeler DeWitt equationfor the quantum Taub models when an exponential e φ scalar field potential is present.This paper will have the following structure. First wewill introduce our models and discuss the problem of timeassociated with them. Afterwards we will show explicitly show how we quantize the electromagnetic field in ourmodels to form their Wheeler Dewitt equations. Thenwe will introduce the Euclidean-signature semi classicalmethod. From there we will study the Bianchi IX modelsin the semi classical limit when a cosmological constant,aligned electromagnetic field and stiff matter are present.The semi-classical wave functions we obtain in closedform for the case when a negative cosmological constantis present behaves qualitatively in a similar manner to thewave functions reported in [47]. This supports the ideathat our semi-classical wave functions reasonably cap-ture the effects of our matter sources. In addition, thefact that we were able to find closed form solutions tothe Euclidean-signature Hamilton Jacobi equation whichcan be used to construct semi-classical solutions to theLorentzian signature symmetry reduced WDW equationwithout needing to use a Wick rotation further shows thepromise of this method to be a useful alternative[44] toEuclidean path integrals[60, 61] for tackling problems inquantum gravity.Next we will study wave functions of the Bianchi IXmodels with our matter sources which have as their clas-sical analogues trajectories in minisuperspace in which β − is fixed at zero. Next we will turn our attention tothe Taub models and briefly analyze them when an ex-ponential scalar field and aligned electromagnetic fieldpotentials are present. We will interpret our wave func-tions by their atheistic characteristics. For example wewill assume, as was done in [62], that each visible peakwhich is present for our wave functions represents a geo-metric state a quantum universe can tunnel in and out of.Finally we will present a few closed form solutions to theWDW equation for the Bianchi IX and VIII models thatthe author found[63] and give some concluding remarks. II. OUR MODELS
All Bianchi cosmological models can be represented bythe following metric ds = − N dt + L π e α ( t ) (cid:16) e β ( t ) (cid:17) ij ω i ω j (cid:16) e β ( t ) (cid:17) ij = e β + ( t ) diag (cid:16) e √ β − ( t ) , e − √ β − ( t ) , e − β + ( t ) (cid:17) . (1)The ω i terms are one forms defined on the spatial hy-persurface of each Bianchi cosmology and obey dω i = C ijk ω j ∧ ω k where C ijk are the structure constants ofthe invariance Lie group associated with each particu-lar class of Bianchi models. We choose to make our oneforms ω i dimensionless by introducing L which has unitsof ’length’ and sets a length scale for the spatial size ofour cosmology. This can be seen because any shift in thescale factor e α ( t )+ δ where δ is a real number can be reab-sorbed into L . Beyond inspecting (1), it can be seen that e α ( t ) acts as the scale factor for our models in these Mis-ner variables by computing √− det g of the metric tensor(1) expressed in orthonormal coordinates, yielding e α ( t ) which measures the total volume of our Bianchi IX space-time and the relative volume in a region for our BianchiVIII space-time.For the Bianchi IX and VIII models their one formsare ω = dx − k sinh( ky ) dzω = cos( x ) dy − sin( x ) cosh( ky ) dzω = sin( x ) dy + cos( x ) cosh( ky ) dz (2)where k=1 is for the Bianchi VIII models and k= i is forthe Bianchi IX models.Upon setting c = 1 , G = π , and L = 1 we can expressthe Einstein-Hilbert action with matter as S E − H = (cid:90) (cid:112) − det g (cid:18) R − Λ + 12 g µν ∂ µ φ∂ ν φ (cid:19) d x + S matter . (3)If we only concern ourselves with the gravitation andscalar field sector of the E-H action we can obtain thefollowing action in terms of ADM variables S ADM = N √ h (cid:32) K ij K ij − K + R − Λ+ 12 g µν ∂ µ φ∂ ν φ (cid:33) d x, (4)where K ij are the components of the extrinsic curvaturewhich measures the curvature induced on the Rieman-nian manifold equipped with spatial metric h ij , from thehigher dimensional space-time g µν it is embedded in, and K is the trace of the extrinsic curvature.The kinetic term belonging to the Hamiltonian con-straint which can be derived from (4) is e − α (cid:0) − p α + p + p − + 12 p φ (cid:1) (5)and can be quantized as follows − e − α p α −→ ¯ h e (3 − B ) α ∂∂α (cid:0) e − Bα ∂∂α (cid:1) e − α p −→ − ¯ h e α ∂ ∂β e − α p − −→ − ¯ h e α ∂ ∂β − e − α p φ −→ − ¯ h e α ∂ ∂φ (6)Using these quantized canonical momenta and the wellknown[51, 64] contributions to the gravitational and mat-ter components of the potentials for the Bianchi A mod-els we can write down the Wheeler DeWitt equationsthat will aid us in studying quantum cosmologies of theBianchi VIII, IX(7), and Taub models(8) where ρ is aconstant stiff matter term and ∨ is the logical "or" sym-bol, and U + is the Bianchi VIII potential while U − is theBianchi IX potential (cid:3) Ψ − B ∂ Ψ ∂α + U Ψ = 0 U ± = ( f ) e β + (cid:16) e β + sinh (cid:16) √ β − (cid:17) ± cosh (cid:16) √ β − (cid:17)(cid:17) + 2 e a Λ9 π + U em + 14 f + ρU em = 2 b e α − β + ∨ b e ( α ±√ β − + β + ) f = 43 e α − β + , (7) ∂ Ψ ∂α − B ∂ Ψ ∂α − ∂ Ψ ∂β − ∂ Ψ ∂φ + V Ψ = 0 V = (cid:18) e α − β + (cid:0) − e β + (cid:1) + e α + φ (cid:19) + 2 b e α +2 β + . (8)In (7) (cid:3) has signature (+1 , − , − , − . Also our equa-tions are written using the Hartle-Hawking[46] operatorordering where B is the Hartle-Hawking ordering param-eter. We will derive the electromagnetic components ofthe potentials in the next section. In addition we willfind solutions to the Wheeler DeWitt equation using adifferent operator ordering that we will introduce later.The Wheeler DeWitt equations (7 and 8) are the ana-logues to the time dependent Schrödinger equations forour quantum cosmologies. Viewing the Wheeler De-Witt(WDW) equation as ˆ H ⊥ Ψ = 0 , and trying to relateit to the conventional Schrödinger equation results in theproblem of time manifesting itself as i ¯ h ∂ Ψ ∂t = N ˆ H ⊥ Ψ = 0 , (9)where ∂ Ψ ∂t = 0 . Due to the absence of the time derivativeterm of the Schrödinger equation in the WDW equation,the construction of a unitary time evolution operator isnot trivial, thus leading to the potential breakdown of asimple probabilistic interpretation of the wave functionof the universe.A Klein-Gordon current J = i ∗ ∇ Ψ − Ψ ∇ Ψ ∗ ) (10)can be defined [65, 66] which could be used to constructa probability density. It however, possesses unattractivefeatures such as it vanishing when the wave function usedto construct the current is purely real or imaginary andnot always being positive definite.Besides the issue of constructing a probability den-sity function, it appears the quantized Hamiltonian con-straint admits only zero’s as eigenvalues. This may leadone to the conclusion that all of the states which satisfythe WDW equation possess vanishing energy. This onthe surface makes it impossible to distinguish betweenground and excited states because all states seeminglyhave the same energy. This apparent obstacle to delin-eate ’ground’ and ’excited’ states can be overcome byexamining the nuanced nature of the ADM formalism[67]. When cast in the ADM formalism general relativityis a constrained theory with four Lagrange multipliers,the lapse and the three components of the shift. Theconstraint associated with the lapse is due to general rel-ativity being invariant under reparameterization of theevolution parameter. Likewise the constraint associatedwith the shift is due to diffeomorphism invariance andis called the diffeomorphism constraint. The diffeomor-phism constraint is due to the configuration space h ab being too large to the point of it being physically re-dundant. To remedy this one can define a superspace[68, 69] where an equivalence class for h ab is constructedsuch that two h ab are in the same class if they can be car-ried into one another by a diffeomorphism. This shrinksthe configuration space, allowing the diffeomorphism con-straint to be satisfied. The same cannot be done for thereparameterization constraint [70]. This explains whyit wouldn’t even make sense for a time derivative to bepresent because there is no unique "time" to use andpartially explains the origins of the "problem of time".To get a better feel for what is going on, one can exam-ine the vanishing Hamiltonian of a fully constrained sys-tem. One can formulate the Lagrangian of a free particlemoving in one dimension , and introduce another con-figuration variable by defining the function t(T) whereT is some arbitrary evolution parameter. If one were totreat both X(t(T)) and t(T) as configuration variablesand formulate the system’s Hamiltonian, they would no-tice that the Hamiltonian vanishes. Obviously the energyof a free one dimensional particle moving at a particularvelocity cannot be zero. This is resolved by realizingthat the dynamics of the system are now encoded in howX(t(T)) evolves with respect to t(T) where both are con-figuration variables. For an explicit demonstration of theabove vanishing Hamiltonian construction, we refer thereader to [71]. In other words, for these types of con-strained systems the Hamiltonian no longer correspondsto the total energy. Thus the Hamiltonian constraint wequantized does not represent the total energy of a space-time in general relativity, and its vanishing eigenvaluesdo not mean that only states which possess zero energyare physically allowed. This allows leeway in defining’ground’ and ’excited’ states in which features of ordinaryquantum mechanics manifest as will be demonstrated insection 4.A more in depth discussion in regards to how theEuclidean-signature semi classical method can be used todefine ’ground’ and ’excited’ states despite them both be-ing annihilated by the quantized Hamiltonian constraintcan be found in [44]. To deal with the problem of timewe will choose one of our variables ( α, β + , β − , φ ) to actas an evolution parameter [72]. Both α and φ are goodcandidates to be our "clock" for various reasons. Thescale factor α would be an intuitive clock to use becauseit corresponds to size of our models and thus is funda- mentally intrinsic to them. In addition if we look at (5)we can rewrite it as e − α p i G ij p j (cid:126)p = ( p α , p + , p − , p φ ) G ij = diag ( − , , , , (11)where the signature associated with α has the same signas the time component of our metric (1). A downsideto using α as an internal clock though is that classicallyBianchi IX universes experience a transition from an ex-pansionary epoch to contractionary one which causes α to "tick" backwards. An alternative is to use our freescalar field φ as our internal clock as has been done inloop quantum cosmology[73–75]. Because p φ is a con-served quantity for our Bianchi A models(excluding ourTaub model with a e φ scalar filed) classically φ alwaysincreases monotonically and thus in a quantum cosmo-logical context emerges as a candidate for time. This isnot the case though when we add our exponential poten-tial term to the Taub models (8). In this paper we willuse both α and φ as our internal clocks depending onwhat points the author wishes to elucidate. III. ELECTROMAGNETIC POTENTIALS FORTHE TAUB MODELS
In this section we will compare two methods for ob-taining the WDW equations (7). The first method will bebased on directly quantizing the class of classical Hamili-tonians for Bianchi A models that was developed in [76].This will lead to a semi-classical treatment of our elec-tromagnetic degree of freedom and will be what we usein the following sections to analyze how matter sourcesaffect our wave functions. However we will also do a fullquantum treatment of the electromagnetic degree of free-dom and compare the two approaches. We will assumeall of our electric and magnetic fields are parallel to eachother as was done in [76, 77].With this in mind our first task is to obtain solutionsfor Maxwell’s equations in the space-time (1) in termsof the Misner variables. In our calculations we will set L = √ π(cid:96) where (cid:96) is a quantity which equals unity andhas some arbitrary unit of length. Starting from A = A dt + A ω + A ω + A ω , (12)where ω = dt , and using the fact that dω i = C ijk ω j ∧ ω k to aide us in computing F = d A = F µv ω µ ∧ ω v resultsin the following expression for F µv F µv = A v,µ − A µ,v + A α C αµv . (13)In (13) differentiation is done through a vector dual toour one forms ω µ which we denote as X µ . Thus A v,µ = X µ A v . The electromagnetic portion of S matter in (3) is S matter = (cid:90) dtdx N √ h (cid:18) − π F µν F µν (cid:19) , (14)where h ab = e α ( t )+2 β + ( t ) diag (cid:16) e √ β − ( t ) , e − √ β − ( t ) , e − β + ( t ) (cid:17) . (15)Writing the action (14) in terms of its vector potential A and our structure constants results in the Lagrangiandensity which is derived in [76] L = Π s A ,s − N H L = Π s A ,s − Π s A s, − N π √ h Π s Π p h sp − N √ h π h ik h sl (cid:0) A [ i,s ] + A m C mis (cid:1) (cid:0) A [ k,l ] + A m C mkl (cid:1) , (16)where Π s = ∂ L ∂ ( X A s ) = h sj √ h N π (cid:0) − A ,j + A j, + A α C α j (cid:1) , (17)and we allow the shift N k to vanish. If we invoke thehomogeneity of (1) then we can say that A i,j = 0 , and A ,j = 0 which results in (16) simplifying to L = Π s A s, − N (cid:34) π √ h Π s Π p h ps + √ h π h ik h sl C mkl C nis A m A n (cid:35) . (18)The non zero Bianchi VIII and IX structure constantsare the following respectively C = − C = 1 C = − C = 1 C = 1 C = − . (19) C ijk = (cid:15) ijk (20)We will now set A , A , Π , and Π to zero and onlyconsider the electromagnetic field produced by A and Π ; doing so results in the following Lagrangian density L = Π A , − N (cid:34) π √ h Π Π h + √ h π h ik h sl C kl C is A A (cid:35) . (21)As the reader can easily verify h ik h sl C kl C is = h h ,which allows us to obtain the following set of Maxwell’sequations when A and Π are varied ˙ A − π √ h Π h = 0 (22) and ˙Π + 14 π √ h h A = 0 . (23)For the last equation we applied an integration by partsto the term Π A , and dropped the total derivative termwhich vanishes at the spatial boundary. The solutions for(22) and (23) are A = √ B cos ( θ ( t )) (24) Π = 12 √ π B sin ( θ ( t )) (25)where θ ( t ) is an integral which is immaterial for our pur-poses and B is an integration constant. Inserting (24)and (25) back into (21) results in L = Π A , − N B π √ h h (cid:0) sin ( θ ( t )) + cos ( θ ( t )) (cid:1) = Π A , − N B π e − α ( t )+2 β ( t ) + +2 √ β ( t ) − , (26)From (16) we can easily identify the electromagneticHamiltonian as H em = B π e − α ( t )+2 β ( t ) + +2 √ β ( t ) − (27)which can be added to the Hamiltonian constraint whichis derivable from our action (3) e − α ( t ) (cid:0) − p α + p + p − + 12 p φ (cid:1) + U g + B π e − α ( t )+2 β ( t ) + +2 √ β ( t ) − = 0 , (28)where U g is the gravitational component of the poten-tial. Quantizing (28) using the Hartle-Hawking operatorordering, multiplying each side by e α ( t ) , and rescaling B results in one of the the Wheeler DeWitt equationsof (7).As the reader can verify for the cases when A = 0 , A = 0 , Π = 0 , and Π = 0 the resulting electromag-netic term is U em = 2 b e α +2 β + − √ β − . (29)and the case when A = 0 , A = 0 , Π = 0 , and Π = 0 results in the term reported in [76] U em = 2 b e α − β + . (30)For the Taub models there are only two unique elec-tromagnetic field potentials terms, (30) and (29) when β − = 0 . If we start with (21) and directly quantize our com-ponent of the total Hamiltonian constraint which is pro-portional to the lapse N we obtain a similar, but slightlydifferent contribution to the potential. Simplifying theterm in brackets of (21) results in the following contribu-tion to the Hamiltonian constraint which can be derivedwith (3) H em = N (cid:34) e − α +2 β + +2 √ β − (cid:0) π Π Π + A (cid:1) π (cid:35) . (31)The term (cid:0) π Π Π + A (cid:1) commutes with our totalHamiltonian constraint H gravity + H em + H scalar . Thuswe can solve the following WDW equation constructed bydirectly quantizing (31) with the rest of our constraint (cid:3) Ψ − B ∂ Ψ ∂α − e α +2 β + +2 √ β − (cid:18) − π ∂ Ψ ∂A + 18 π A Ψ (cid:19) + U ± Ψ = 0 , (32)by solving this eigenvalue problem − π ∂ Ψ ∂A + 18 π A Ψ = b n Ψ . (33)This is simply the Schrödinger equation for a harmonicoscillator whose solutions are well known Ψ = ψ ( α, β + , β − , φ ) e − A π H b n − (cid:18) A √ π (cid:19) b n = 12 (1 + 2 n ) . (34)Inserting our Ψ from (34) into (32) yields (cid:3) ψ − B ∂ψ∂α − b n e α +2 β + +2 √ β − + U ± ψ = 0 . (35)This WDW equation is similar to what we had beforeexcept for the fact that the strength b n of the electro-magnetic field is now quantized thanks to (32). By firstsolving the classical A i equations (22) in terms of theMisner variables we eliminate the electromagnetic fielddegree of freedom. For now though working with just(28), (29), and (30) is sufficient for what will follow. IV. THE EUCLIDEAN-SIGNATURE SEMICLASSICAL METHOD
Our outline of this method will follow closely [44]. Themethod described in this section and its resultant equa-tions can in principle be used to find solutions and quan-tum corrections to a wide class of quantum cosmologicalmodels such as all of the Bianchi A, Kantowski Sachsmodels, and the FLRW models.The first step we will take in analyzing our WDW equa-tions is to introduce the ansatz (0) Ψ ¯ h = e − S ¯ h / ¯ h (36) where S ¯ h is a function of ( α, β + , β − , φ ) . We will rescale S ¯ h in the following way S ¯ h := Gc L S ¯ h (37)where S ¯ h is dimensionless and admits the following powerseries in terms of this dimensionless parameter X := L Planck L = G ¯ hc L . (38)The series is given by S ¯ h = S (0) + X S (1) + X S (2) + · · · + X k k ! S ( k ) + · · · , (39)and as a result our initial ansatz now takes the followingform (0) Ψ ¯ h = e − X S (0) −S (1) − X S (2) −··· . (40)Substituting this ansatz into the Wheeler-DeWitt equa-tion and requiring satisfaction, order-by-order in powersof X leads immediately to the sequence of equations (cid:18) ∂ S (0) ∂α (cid:19) − (cid:18) ∂ S (0) ∂β + (cid:19) − (cid:18) ∂ S (0) ∂β − (cid:19) − (cid:18) ∂ S (0) ∂φ (cid:19) + U ± = 0 (41) (cid:34) ∂ S (0) ∂α ∂ S (1) ∂α − ∂ S (0) ∂β + ∂ S (1) ∂β + − ∂ S (0) ∂β − ∂ S (1) ∂β − − ∂ S (0) ∂φ ∂ S (1) ∂φ (cid:35) + B ∂ S (0) ∂α − ∂ S (0) ∂α + ∂ S (0) ∂β + ∂ S (0) ∂β − + 12 ∂ S (0) ∂φ = 0 , (42), (cid:34) ∂ S (0) ∂α ∂ S (1) ∂α − ∂ S (0) ∂β + ∂ S (1) ∂β + − ∂ S (0) ∂β − ∂ S (1) ∂β − − ∂ S (0) ∂φ ∂ S (1) ∂φ (cid:35) + k (cid:34) B ∂ S ( k − ∂α − ∂ S ( k − ∂α + ∂ S ( k − ∂β + ∂ S ( k − ∂β − + 12 ∂ S ( k − ∂φ (cid:35) + k − (cid:88) (cid:96) =1 k ! (cid:96) ! ( k − (cid:96) )! (cid:32) ∂ S ( (cid:96) ) ∂α ∂ S ( k − (cid:96) ) ∂α − ∂ S ( (cid:96) ) ∂β + ∂ S ( k − (cid:96) ) ∂β + − ∂ S ( (cid:96) ) ∂β − ∂ S ( k − (cid:96) ) ∂β − − ∂ S ( (cid:96) ) ∂φ ∂ S ( k − (cid:96) ) ∂φ (cid:33) = 0 (43)We will refer to S (0) in our WDW wave functions asthe leading order term, which can be used to constructapproximate solutions to the Lorentzian signature WDWequation, and call S (1) the first order term. The S (1) termcan also be viewed as our first quantum correction, withthe other S ( k ) terms being the additional higher orderquantum corrections, assuming that they are smooth andglobally defined. This is reflected in the fact that thehigher order transport equations depend on the operatorordering used in defining the Wheeler Dewitt equation,which is an artifact of quantization. Additionally in somecases one can find a solution to the S (1) equation whichallows the S (2) equation to be satisfied by zero. Thenone can write down the following as a solution to theWDW equation for either a particular value of the Hartle-Hawking ordering parameter, or for an arbitrary orderingparameter depending on the S (1) which is found. (0) Ψ ¯ h = e − X S (0) −S (1) (44). This can be easily shown. Let’s take S (0) and S (1) asarbitrary known functions which allow the S (2) transportequation to be satisfied by zero, then the k = 3 transportequation can be expressed as (cid:34) ∂ S (0) ∂α ∂ S (3) ∂α − ∂ S (0) ∂β + ∂ S (3) ∂β + − ∂ S (0) ∂β − ∂ S (3) ∂β − − ∂ S (0) ∂φ ∂ S (3) ∂φ (cid:35) = 0 (45)which is clearly satisfied by S (3) =0. The S (4) equationcan be written in the same form as (45) and one of itssolution is 0 as well, thus resulting in the S (5) equationpossessing the same form as (45). One can easily con-vince oneself that this pattern continues for all of the k ≥ S ( k ) transport equations as long as the solution ofthe S ( k − transport equation is chosen to be 0. Thusif an S (1) exists which allows one to set the solutions toall of the higher order transport equations to zero theinfinite sequence of transport equations generated by ouransatz truncates to a finite sequence of equations whichallows us to construct a closed form wave function satis-fying the WDW equation. Not all solutions to the S (1) transport equation will allow the S (2) transport equationto be satisfied by zero; however in our case, we were ableto find S (1) ’s which cause the S (2) transport equation tobe satisfied by zero, thus allowing one to set all of thesolutions to the higher order transport equations to zeroas shown above. This will enable us to construct closedform solutions to the Lorentzian signature Bianchi VIIIand IX WDW equations. It should be noted that us-ing an alternate form of operator ordering for the WDWequation which we will introduce later one can constructsolutions to it using just the S (0) term.Under certain conditions which we cannot rigorouslyarticulate yet, wave functions that behave as ’excited’states can be calculated by introducing the following ansatz. Ψ ¯ h = Φ ¯ h e − S ¯ h / ¯ h (46)where S ¯ h = c L G S ¯ h = c L G (cid:18) S (0) + X S (1) + X S (2) + · · · (cid:19) is the same series expansion as before and Φ ¯ h can beexpressed as the following series Φ ¯ h = Φ (0) + X Φ (1) + X
2! Φ (2) + · · · + X k ( ∗ ) k ! Φ ( k ) + · · · (47)with X being the same dimensionless quantity as before.Inserting (46) with the expansions given by (39) and (47)into the Wheeler DeWitt equation (7) and by matchingequations in powers of X leads to the following sequenceof equations. − ∂ Φ (0) ∂α ∂ S (0) ∂α + ∂ Φ (0) ∂β + ∂ S (0) ∂β + + ∂ Φ (0) ∂β − ∂ S (0) ∂β − + 12 ∂ Φ (0) ∂φ ∂ S (0) ∂φ = 0 , (48), − ∂ Φ (1) ∂α ∂ S (0) ∂α + ∂ Φ (1) ∂β + ∂ S (0) ∂β + + ∂ Φ (1) ∂β − ∂ S (0) ∂β − + 12 ∂ Φ (1) ∂φ ∂ S (0) ∂φ + (cid:32) − ∂ Φ (0) ∂α ∂ S (1) ∂α + ∂ Φ (0) ∂β + ∂ S (1) ∂β + + ∂ Φ (0) ∂β − ∂ S (1) ∂β − + 12 ∂ Φ (0) ∂φ ∂ S (1) ∂φ (cid:33) + 12 (cid:32) − B ∂ Φ (0) ∂α + ∂ Φ (0) ∂α − ∂ Φ (0) ∂β − ∂ Φ (0) ∂β − − ∂ Φ (0) ∂φ (cid:33) = 0 , (49) − ∂ Φ ( k ) ∂α ∂ S (0) ∂α + ∂ Φ ( k ) ∂β + ∂ S (0) ∂β + + ∂ Φ ( k ) ∂β − ∂ S (0) ∂β − + 12 ∂ Φ ( k ) ∂φ ∂ S (0) ∂φ + k (cid:32) − ∂ Φ ( k − ∂α ∂ S (1) ∂α + ∂ S (1) ∂β + ∂ S (1) ∂β + + ∂ Φ ( k − ∂β − ∂ S (1) ∂β − + 12 ∂ Φ ( k − ∂φ ∂ S (1) ∂φ (cid:33) + k (cid:32) − B ∂ Φ ( k − ∂α + ∂ Φ ( k − ∂α − ∂ Φ ( k − ∂β − ∂ Φ ( k − ∂β − − ∂ Φ ( k − ∂φ (cid:33) − k (cid:88) (cid:96) =2 k ! (cid:96) ! ( k − (cid:96) )! (cid:32) ∂ Φ ( k − (cid:96) ) ∂α ∂ S ( (cid:96) ) ∂α − ∂ Φ ( k − (cid:96) ) ∂β + ∂ S ( (cid:96) ) ∂β + − ∂ Φ ( k − (cid:96) ) ∂β − ∂ S ( (cid:96) ) ∂β − − ∂ Φ ( k − (cid:96) ) ∂φ ∂ S ( (cid:96) ) ∂φ (cid:33) = 0 . (50)It can be seen from computing d Φ (0) ( α,β + ,β − ,φ ) dt =˙ α ∂ Φ (0) ∂α + ˙ β + ∂ Φ (0) ∂β + + ˙ β − ∂ Φ (0) ∂β − + ˙ φ ∂ Φ (0) ∂φ , and extrapolatingfrom (4 . , . − . of [44] what ˙ φ is in terms of S (0) that Φ (0) is a conserved quantity under the flow of S .This means that any function F (cid:0) Φ (0) (cid:1) is also a solutionof equation (48). Wave functions constructed from thesefunctions of Φ are only physical if they are smooth andglobally defined. If we choose our φ to have the form f ( α, β + , β − , φ ) m where f ( α, β + , β − , φ ) is some functionwhich is conserved under the flow of S (0) and vanishes forsome finite values of the Misner variables then we mustrestrict m to be either zero or a positive integer. Forbound states m can plausibly be interpreted as gravitonexcitation number [78]. This makes our ’excited’ statesakin to bound states in quantum mechanics like the har-monic oscillator whose excited states are denoted by dis-crete integers as opposed to a continuous index. Thisdiscretization of the quantity that denotes our ’excited’states is the mathematical manifestation of quantizationone would expect excited states to possess in quantumdynamics.If our conserved quantity f ( α, β + , β − , φ ) does not van-ish in minisuperspace then our ’excited’ states can be in-terpreted as ’scattering’ states akin to the quantum freeparticle and m can be any real number. Beyond the lead-ing order limit, if smooth globally defined solutions canbe proven to exist for the higher order φ transport equa-tions then one can construct asymptotic or closed form’excited’ state solutions for the quantum Taub models aswe shall do. Additional information for why we can callthe above ’excited’ states despite them being solutionsto an equation which does not have the same form asthe Schrödinger equation can be found in [44]. In whatfollows we will work in units in which X = 1 . V. BIANCHI VIII AND IX QUANTUMCOSMOLOGY WITH AN ALIGNEDELECTROMAGNETIC FIELD, SCALAR FIELD,COSMOLOGICAL CONSTANT, AND, STIFFMATTER
The author found the following solutions to theEuclidean-signature Hamilton Jacobi equations (41)when an electromagnetic field, scalar field, cosmologicalconstant, and stiff matter are present S ± ) = 16 e α − β + (cid:16) e β + cosh (cid:16) √ β − (cid:17) ± (cid:17) ∓ Λ e α +4 β + π ∓ α b ∓ β + b + φ √ ρ √ , (51) S ± ) = 16 e α − β + (cid:16) e β + cosh (cid:16) √ β − (cid:17) ± (cid:17) − Λ e α − β + − √ β − π − α b + β + b + √ β − b + φ √ ρ √ , (52) S ± ) = 16 e α − β + (cid:16) e β + cosh (cid:16) √ β − (cid:17) ± (cid:17) − Λ e α +2 β + − √ β − π − α b + β + b − √ β − b + φ √ ρ √ . (53)In our solutions the plus + sign or the top operation in ± and ∓ are for the Bianchi IX models while the bot-tom symbols/operators are for the Bianchi VIII models.It is interesting to note that in the limit of our mat-ter sources vanishing these solutions approach the wellknown ’wormhole’ [40] Bianchi IX solutions and its ana-logue for the Bianchi VIII models. The author was un-able to find elementary solutions to (41) with the afore-mentioned matters sources that exhibited this propertyfor the Bianchi IX ’no boundary’[26] or "arm" solutions[79], or for their Bianchi VIII analogues [27]. This poten-tially suggests that their is something special about the’wormhole’ solution.Using these expressions we can obtain a semi-classicalsolution to the WDW equation expressed in the Hartle-Hawking semi general operator ordering which respectsthe π symmetry in β space present in the Bianchi IXpotential when our electromagnetic field is zero( b = 0 )or a solution to the WDW expressed in an alternativeoperator ordering when b (cid:54) = 0 .As pointed out by Moncrief and Ryan in [40] and shownexplicitly in [80], using a different operator ordering thefor Wheeler DeWitt equation than the Hartle-Hawkingone[46] allows one to construct wave functions which sat-isfy it if one possesses pure imaginary solutions to its cor-responding Lorentzian signature Hamilton Jacobi equa-tion. We will review the derivation presented in [80]which allows us to construct solutions using just an S (0) .The Hamiltonian constraint for the Bianchi A models canbe expressed as H = G AB p A p B + U = 0 (54)where G AB is the DeWitt supermetric and p i are thecanonical momenta. Likewise the regular Lorentzian sig-nature Hamiliton Jacobi is expressed as G AB ∂J∂x A ∂J∂x B + U = 0 . (55)Because (55) is the Lorentzian signature Hamilton Jacobiequation the signs in its derivatives are the opposite ofthose for the Euclidean case. That means for the BianchiVIII and IX models with a cosmological constant, a pri-mordial electromagnetic field, scalar field, and stiff mat-ter (55) is satisfied by our previous solutions (51), (52),and (53) multiplied by √− such that J = √− S i (0) .This allows us to rewrite (55) as G AB p A p B + G AB ∂ S i ∂x A ∂ S i ∂x B = G AB ( x ) π ∗ A π B = 0 (56)where π A = p A − √− ∂ S i ∂x A (57)and is quantized as follows ˆ π A = −√− ∂∂x A − √− ∂ S i ∂x A . (58) Due to this quantization, wave functions of the form Ψ = e −S i (0) mathematically behave in the following way ˆ π B Ψ = 0 . (59)Thus we can satisfy the Bianchi VIII and IX WDW equa-tions when a cosmological constant, a primordial electro-magnetic field, scalar field, and stiff matter are present ifwe order the WDW as follows (cid:112) | G | (cid:104) ˆ π ∗ A (cid:16)(cid:112) | G | G AB ˆ π B (cid:17)(cid:105) Ψ = 0 . (60)We will first study a semi-classical solution Ψ = 13 (cid:16) e −S + e −S + e −S (cid:17) (61)to the WDW equation(7) when b = 0 and then turn ourattention to a closed form solution which satisfies (60)when b (cid:54) = 0 . As we previously mentioned there are twocandidates for our evolution parameter α and φ . Eventhough φ is in some respects a better variable becauseclassically it is guaranteed to increase monotonically forthe Bianchi VIII and IX models we are considering, wewill analyze (61) using α because it better facilitates theauthor’s ability to convey the physical implications of ourwave functions for this particular case. Below are threeplots of (61) for three different values of α . (a) α = − − (b) α = 1 .
65 Λ = − (c) α = 2 Λ = − FIG. 1
Three different plots of (61) for three different values of α where we suppress the the φ degree of freedom.From here one out we will interpret the peaks which0appear in our wave functions as geometric states that ourquantum universes can tunnel in and out of as was donein [62]. When α is less than 1.6 our wave function’s peakis located at the origin in β space which corresponds toisotropy. For negative values of α our wave function isroughly spread evenly around the origin as can be seenin figure (1a). Thus we can say that figure (1a) describesa universe with a "fuzzy" geometry. As α grows thoughour wave function becomes more sharply peaked at theorigin. This is expected because geometrical "fuzziness"described by a wave function which is spread out in β space is a quantum effect we intuitively expect to dimin-ish as the universe grows in size dictated by α .However around α ≈ . a dent forms where the wavefunction used to have an isotropic peak as can be seen in(1b) and our wave function begins to split apart. This isa result of the influence of our negative cosmological con-stant Λ . As α continues to grow our wave function splitsinto three parts whose peaks are not centered around theorigin. This indicates that the cosmological constant forour wave function acts as a driver of anisotropy by tear-ing our wave functions away from the origin in β space.This is reminiscent of how a positive cosmological con-stant causes distant objects which are not gravitationalbound to each other in the universe to accelerate awayfrom each other. Of course this is just an analogy becauseour wave function is in minisuperspace, not space-time.The wave function shown in (1c) is aesthetically similarto the wave functions[47] which were computed by em-ploying Chern-Simons solutions in Ashtekar’s variables. Investigating the connection between the caustics whichwere studied in [47] and these elementary closed form so-lutions to the Euclidean-signature Hamilton Jacobi equa-tion could potentially yield some interesting results.In comparison to the exotic and technical methodswhich were used to compute Bianchi IX wave functionswith a negative cosmological constant in [47], the factthat the Euclidean-signature semi classical method al-lowed us to compute similar Bianchi IX wave functions inclosed form and expressed in terms of elementary func-tions is an impressive feat. This provides further sup-port that the Euclidean-signature semi classical methodis an effective way to prove the existence of solutions toLorentzian signature equations. As is discussed in sec-tion 7 of [44], it may be easier to prove the existenceof formal solutions to the full functional Lorentzian sig-nature Wheeler DeWitt equation using the Euclidean-signature Einstein-Hamilion-Jacobi(EHJ) equation thanit is through the Lorentzian signature (EHJ). By suc-cessfully employing[43, 59, 81] this method in quantumcosmology we further support the idea that this methodcan in some circumstances be a useful alternative[44]to Euclidean path integrals[60, 61] for tackling prob-lems in quantum gravity. More information on howthe Euclidean-signature semi classical method applies toquantum cosmology, quantum gravity, and a plethora offield theories, including the Yang Mills ’mass gap’ prob-lem is provided here[44, 45, 82–84].Another way of visualizing how the cosmological con-stant is a driver of anisotropy in our wave functions isthrough the plots in figure (2). (a) Λ = − (b) Λ = − FIG. 2
Two plots of the wave functions constructed from (51), (52), and (53) which show their maximum values in β space as a function of α . The red line corresponds to (53), the green line corresponds to (52), and the purple linecorresponds to (51).Moving on to the case when an electromagnetic field ispresent we will analyze the solution to (60) constructedfrom (51). We don’t lose much from not analyzing (52)and (53) because those are just (51) rotated by ± π in β space. In figure 3 are four plots which showcase thepossible effects that an aligned electromagnetic field canhave on our particular ’ground’ state wave functions. As can be seen by comparing figures (3a) and (3b) oneof the effects of our electromagnetic field is that it causesour wave function to form a peak at a larger value of β + than it would otherwise. However owing to the primor-dial nature of our electromagnetic field for large values of α this effect is far less drastic as can be seen by comparingfigures (3c) and (3d). This suggest that an electromag-1netic field can play a decisive role in increasing the preva-lence of anisotropy in the early universe but still allow fora universe which becomes roughly isotropic as it contin-ues to grow in size. To test this assertion, in the futurewe will need to study other quantum cosmological mod-els with more general electromagnetic fields. In additionwe would need to add inhomogeneities to those models.For now this finding contributes towards a theoretical un-derstanding of how a primordial electric/magnetic fieldcould have influenced the seeds of anisotropy in the earlyuniverse and how those seeds developed as it grew in size.Another noticeable effect of our electromagnetic fieldis that it causes our wave function to be more sharplypeaked at smaller values of α than it otherwise wouldbe as can be seen by comparing figures (3a) and (3b).This suggest that an electromagnetic field could playa role in causing a universe which initially possesses a"fuzzy" geometry in the sense that its wave function ofthe universe is evenly spread over β space to transitionto one which is sharply peaked at a particular point in β space. This transition from a "fuzzy" geometry to asemi-classical one could occur if the primordial electro-magnetic field was something that emerged in an early universe at some point in its evolution or was present inthe beginning, but its effects were initially suppressed byPlanck or GUT level physics. Such a mechanism beingpresent in general anisotropic quantum cosmologies canhelp explain how a universe can transition from a statewhere it can only be accurately described using quantummechanics to one which can be adequately described byclassical mechanics.A surprising effect of our electromagnetic field is thatit can actually reduce the level of anisotropy of a quan-tum universe at certain values of α . This can be seenby comparing figures (3c) and (3d). Our cosmologicalconstant has a proclivity towards causing our wave func-tions to be centered at a negative value of β + while ouraligned electromagnetic field causes our wave functionsto shift toward the positive portion of the β + axis. Thusfor certain values of α these two competing effects to in-crease anisotropy can cancel each other out and result ina reduction of anisotropy. This effect may also be causedby an aligned electromagnetic field and other forms ofmatter. Visually this dual nature concerning the tenden-cies of our two matter sources to induce anisotropy in ourquantum cosmological can be visualized in figure (4). (a) α = − − b = 0 (b) α = − − b = 1 (c) α = 1 Λ = − b = 0 (d) α = 1 Λ = − b = 1 . FIG. 3
These are four plots of our wave functions constructed from (51) with listed values for their cosmologicalconstant and electromagnetic field.We should note that this dual nature may be a resultof us considering an aligned electromagnetic field and notone which has components in multiple directions. Whenboth a negative cosmological constant and an electro-magnetic field are present, isotropy is more likely to be reached at a smaller value of α than it is when eitherone or none of the matter sources are present. When analigned electromagnetic field and a negative cosmologi-cal constant are present the magnitude of our wave func-tions decay as α continues to grow past a point. Theseresults are in concord with those that the author ob-2 (a) α = − − b = 0 FIG. 4
This plot shows the maximum values of our wave functions in β space as a function of α . The green plot isof the wave function constructed from (51) when no matter sources are present. The purple line is when both anegative cosmological constant Λ = − and an electromagnetic field b = 2 are present. The red line is when just anelectromagnetic field b = 2 is present.served in his previous works where he studied the effectsof aligned electromagnetic fields on Taub[59], Bianchi II,and VII h [81] quantum cosmologies. VI. IX QUANTUM COSMOLOGY ON THE β + AXIS
In this section we will further explore Bianchi IX quan-tum cosmologies with matter sources by analyzing wave functions which correspond to classical trajectories inminisuperspace where β − starts at a fixed point in β space. To start we need to pick an S (0) which induces aflow in minisuperspace which possesses fixed points. Todetermine this we will write out the classical flow equa-tion for an arbitrary S (0) (a) φ = 0 b = 0 (b) φ = 0 b = 2 FIG. 5
Two plots of (70) for two different values for the strength of the electromagnetic field. p α = ∂ S (0) ∂αp + = ∂ S (0) ∂β + p − = ∂ S (0) ∂β − p φ = ∂ S (0) ∂φ (62)3 ˙ α = (6 π ) / e α N (cid:12)(cid:12)(cid:12)(cid:12) Eucl p α ˙ β + = − (6 π ) / e α N (cid:12)(cid:12)(cid:12)(cid:12) Eucl p + ˙ β − = − (6 π ) / e α N (cid:12)(cid:12)(cid:12)(cid:12) Eucl p − ˙ φ = − (6 π ) / e α N (cid:12)(cid:12)(cid:12)(cid:12) Eucl p φ (63)where N | Eucl is the lapse. The lapse N | Eucl can be anyfunction of the Misner variables and φ as long as it nevervanishes or changes sign within the range −∞ to ∞ ofall four variables. To keep our analysis straightforwardwe will set N | Eucl = e α (6 π ) / . Using this lapse and (51)results in the following classical flow equations dαdt = 13 e α − β + (cid:16) e β + cosh (cid:16) √ β − (cid:17) ± (cid:17) − Λ e α +4 β + π − b , (64) dβ + dt = − e α +2 β + cosh (cid:16) √ β − (cid:17) + bb + Λ e α +4 β + π +23 e α − β + (cid:16) e β + cosh (cid:16) √ β − (cid:17) ± (cid:17) ) , (65) dβ − dt = − e α +2 β + sinh (cid:0) √ β − (cid:1) √ , (66) dφdt = − √ ρ √ . (67)As it can be seen when β − = 0 the time derivative of β − vanishes which indicates that if β − is initially zero, then itwill remain zero indefinitely. Thus the wave functions wewill find and analyze correspond to classical trajectoriesin minisuperspace where β − is initially zero.Despite this we stress that we are not analyzing theLRS Bianchi IX models. Even though we will be eventu-ally setting β − = 0 in our calculations, the existence ofa β − axis will impact our wave functions through thederivatives of β − that will appear in our calculationswhich do not vanish when β − = 0 . As the reader canverify, (51) is the only S (0) with matter sources for theBianchi IX models which possesses this fixed point at β − = 0 in its flow equations.If we start with the case when only an aligned electro-magnetic field is present( Λ = 0 ) we can insert the entiretyof (51) into (42) which will give us a complicated lookingtransport equation. However because β − = 0 is a fixed point on the β − axis we can set β − = 0 for this com-plicated transport equation which results from inserting(51) into (42). The resultant transport equation as thereader can verify can be solved by inserting this ansatzinto it S β − =0 := x α + x φ, (68)which yields the following solution S β − =0 := 12 α ( − B − − √ b φ √ ρ . (69)This is our first quantum correction for our Bianchi IXquantum cosmologies which correspond to classical cos-mologies which are formed from a flow in minisuperspacewhich starts on a fixed point on the β − axis. The abovequantum correction takes into account the existence ofthe β − axis in the full Bianchi IX models by taking thefull derivative of S in (42). An interesting feature ofthis quantum correction is how the aligned electromag-netic field b , stiff matter ρ , and the scalar field φ arecoupled.Despite it not being the only solution to (42) and itssimple nature we claim that we are justified in employ-ing it because it generates such fascinating wave func-tions as we will show shortly. The non-trivial effects thatthese wave functions imply for the universes that they de-scribe should be chronicled as possible phenomena thata toy model of quantum gravity can induce on the evo-lution of a quantum universe(s). In addition physicallythis solution makes sense because the effects of quantumcorrections S ( k> in our wave function should becomenegligible compared to the leading order term S (0) as α increases which is the case as can be seen by comparing(69) to (51).Using this first order quantum correction (69) and S β − =0 := 16 e α − β + (cid:0) e β + ± (cid:1) + α b + β + b + φ √ ρ √ . (70)we can study the quantum cosmology of Bianchi IX’ground’ states which are restricted to the β + axis. Inaddition because our wave functions only possess threeminisuperspace variables we can use φ as our time param-eter to obtain wave functions which are easy to interpretvia their aesthetic qualities.Starting with the Bianchi IX models we form the fol-lowing wave function ψ β − =0 = e −S β − =0 −S β − =0 (71)and plot them for two different values b of the alignedelectromagnetic field as can be seen in (5). As can be seen4in figure (5a) when b = 0 our wave function is describinga quantum universe which is most likely in a geometricconfiguration in which its scale factor α ≈ . and β + = 0 .However when we turn on our aligned electromagneticfield its wave function is now peaked at larger value ofboth α and β + .Next we will examine Bianchi IX ’excited’ states whichare restricted to the β + axis when an electromagneticfield and cosmological constant are present. First we in-sert the entirety of (51) into (48); then only after tak-ing the derivative of (51) with respect to β − do we set β − = 0 which results in the following homogeneous trans-port equationb ∂ Φ (0) ∂α − b ∂ Φ (0) ∂β + − e α − β + ∂ Φ (0) ∂α − e α +2 β + ∂ Φ (0) ∂α − e α − β + f ( β + ) + 23 e α +2 β + f ( β + )+ 2 √ √ p ∂ Φ (0) ∂φ + Λ e α +4 β + π ∂ Φ (0) ∂α − Λ e α +4 β + π ∂ Φ (0) ∂β + = 0 . (72)The author found the following solutions to this transportequation Φ = (cid:16) πe α − β + − πe α + β + ) + 27 π b e α + β + ) + Λ e α + β + ) (cid:17) n (73) where in order for Φ to be globally defined and smoothwe must restrict n to be either a positive integer or zero.Using (73) we form the following wave function Ψ β − =0 = Φ e −S β − =0 −S β − =0 (74)which we plot in (6). Figure 6 emphatically shows thatwhen both a negative cosmological constant and an elec-tromagnetic field are present the effects of the electro-magnetic field are dependent upon its strength. Whenthe field is relatively weak as can be seen (6a) and (6b) itcan cause an additional highly probably geometric stateto come into existence which our quantum universe cantunnel in and out of. However when the electromag-netic field is strong it can also destroy a highly probablestate, leaving only one highly probable state that a quan-tum universe can be in. This ability to create an addi-tional state that a quantum universe can tunnel in andout of is similar to how non-commutativity in the min-isuperspace variables can cause new quantum states toemerge in the quantum Kantowski-Sachs[62] and BianchiI models[85]. Chronicling these effects of our aligned elec-tromagnetic field can help us understand what our earlyuniverse could have been like. (a) φ = 0 Λ = − b = 0 n = 1 (b) φ = 0 Λ = − b = 1 . n = 1 (c) φ = 0 Λ = − b = 3 n = 1 FIG. 6
Two plots of (74) for the first ’excited’ states of our Bianchi IX wave functions restricted to the β + axis fortwo different values of the strength of the electromagnetic field b .5 VII. QUANTUM TAUB MODELS WITH ASCALAR FIELD(a) α = 1 c = − φ = 0 (b) α = 1 c = − φ = − (c) α = 1 c = − φ = 5 FIG. 7
These are three plots of our Taub wave functions when an exponential scalar potential is present.The author found the following S (0) for the quantumTaub models when an exponential scalar potential andan aligned electromagnetic field are present S := 16 e α − β + + 13 e α +2 β + − c e α − β + + φ − α b + β + b − b φ , (75)where c can in principle be any real number. However forthe purposes of elucidating the points the author wisheswe to make we will assume c < . Utilizing (42) theauthor found the following S (1) quantum correction to(75) S := 12 α ( − B − − β + − φ . (76)The existence of (75) and (76), suggest that an analogousclosed form solution involving an exponential scalar fieldpotential may exist for the Bianchi IX models as well.First we will plot e −S −S when B = 0 and b = 0 and use both α and φ as our internal clocks.Our plots(7) indicate that as our scalar potential de-cays the universes described by these wave functionsapproache isotropy. However when our scalar poten-tial increases in magnitude these universes become more anisotropic. Similar qualitative behavior may be presentfor other types of scalar potentials such as φ n .These effects are interesting in themselves, however tofurther determine how typical or robust they are, moregeneral scalar fields need to be studied. To accomplishthis in the future various approximate techniques whichdo not rely on obtaining closed form solutions wouldneed to be employed. However for the meantime thesefindings encourage further work to be done to determinehow scalar fields effect quantum cosmological evolutionin anisotropic models.For the case when an aligned electromagnetic field ispresent we will use φ as our sole internal clock. As canbe seen in figure 8, if we use φ as our internal clock,the aligned electromagnetic field induces some dramaticeffects on our wave functions. If we compare (8a) to(8b) we see that the electromagnetic field causes the very’fuzzy’ wave function in (8a) to become sharp, or peakedat a certain value α and β + . This effect would be presenteven if we kept φ = 0 . As φ increases, our wave functionstravel in the negative α direction which indicates that theuniverses we are describing are more likely to be smallerin size as φ grows. In addition as φ grows it moves inthe positive β + direction which indicates its anisotropyin general increases as φ increases. The fact that theEuclidean-signature semi classical method was able togenerate such a wealth of information for the solutions of(8) further shows its ability to prove[43, 44] the existenceof solutions to Lorentzian signature equations.6 (a) b = 0 c = − φ = 0 (b) b = 1 . c = − φ = − (c) b = 1 . c = − φ = 3 FIG. 8
These are three plots of our Taub wave functions when an exponential scalar potential and alignedelectromagnetic are present.
VIII. CLOSED FORM SOLUTIONS TOBIANCHI IX AND VIII WDW
If we start with (51) and set
Λ = 0 , and solve theequation which results from inserting (51) into (42), weobtain the following S (1) quantum correction S ± ) := α x1 + 18 ( B + 2 x1 + 6) log (cid:16) sinh (cid:16) √ β − (cid:17)(cid:17) + β + (cid:18)
14 ( − B − − x1 (cid:19) ∓ φ (cid:32) √ b ( B + 2)8 √ ρ + √ b x1 √ ρ (cid:33) , (77)where x1 is an arbitrary constant, and B is the Hartle-Hawking ordering parameter. Technically this S (1) termis not smooth and globally defined, however we can formsmooth and globally defined wave functions from it if we restrict ( B + 2 x1 + 6) to be − y , where y is eitherzero or a positive integer. Using (43) we can obtain so-lutions to (7) of the form e −S ± ) −S ± ) , when Λ = 0 andour aligned electromagnetic field is given by b e α − β + if the constants ( x , ρ, b, B ) in (77) satisfy the followingrelations b ( B − x1 + 2) ρ − Bx1 + 2 x1 = 0 B + 2 x1 + 6 = 0 . (78)Beyond these closed form solutions we can also constructsmooth and globally defined wave functions using (51)and (77) as long as ( B +2 x1 +6) is a negative integer whichis a multiple of 8. The question of solving the higher ordertransport equations for the Bianchi IX and VIII modelswhen an aligned electromagnetic field is present in theother two perpendicular directions could be the topic ofa future investigation. For now we will be content withthis closed form solution for an aligned electromagneticfield.For the sake of completeness we will report a few other7closed form solutions that the author found for the vac-uum Bianchi IX and VIII models which were reported inearlier works. For the vacuum Bianchi IX models when ρ = 0 , Λ = 0 , and b = 0 , Joseph Bae[43] found twofamilies of quantities which are conserved under the flowproduced by (51), and thus satisfy (48). The author thenmodified those quantities so that they are conserved un-der the flow given by (51) for the Bianchi VIII models.Thus we will use the following quantities which satisfy(48) for the vacuum Bianchi IX and VIII models Φ ± := S m O m ,S = (cid:16) e α − β + (cid:16) e β + ∓ cosh (cid:16) √ β − (cid:17)(cid:17)(cid:17) ,O = (cid:16) e α − β + sinh (cid:16) √ β − (cid:17)(cid:17) , (79)where m1 is either zero or a positive integer for theBianchi IX case, or can be any number for the BianchiVIII models, m2 is always either zero or a positive inte-ger. Joseph Bae found the following solution to (42) forthe vacuum Bianchi IX models, which also happens tosatisfy (42) for the vacuum Bianchi VIII models S (1 Bae ) := − α ( B + 6) . (80)Solutions of the form Φ ± e −S (0 ± ) −S Bae ) exist for thevacuum Bianchi VIII and IX models for the followingvalues of ( B, m , m B = ± , m , m B = ± √ , m , m B = ± √ , m , m (81)More information on these closed form solutions can befound in[63]. IX. CONCLUDING REMARKS
Using the Euclidean-signature semi classical methodwe have obtained in closed form some very interestingwave functions of the universe for the Bianchi IX, VIIIand Taub models. By doing so we have shown how cer-tain matter such as a negative cosmological constant, an aligned electromagnetic field, and scalar field can effectthe evolution of a quantum universe. Most notably wehave provided additional evidence[59, 81] that an alignedelectromagnetic field can modify the states/peaks thatare present in ’excited’ states.The next step in establishing how an electric/magneticfield can effect the ’excited’ states of the WDW equa-tion for Bianchi A models would be to study the BianchiI models with a magnetic field and cosmological con-stant. Afterwards it would be instructive to investigatehow general/non-aligned electric/magnetic fields with in-homogeneous perturbations can effect the evolution ofquantum universes.Nonetheless the toy model we studied yielded inter-esting wave functions whose effects deserve to be chron-icled as possible phenomena that quantum gravity caninduce on the evolution of a universe. In addition by ob-taining these wave functions for the Lorentzian signatureWDW equation by finding closed form solutions to theEuclidean signature Hamilton-Jacobi equation we havefurther shown the utility of the Euclidean-signature semiclassical method.The author hopes that the plethora of results obtainedin this paper, in conjunction with all of the results pre-viously obtained by this method will hope promote itsuse in solving important problems in both field theory,such as the ’mass gap’ problem and in quantum cosmol-ogy. An attractive feature of the Euclidean-signaturesemi classical method as it applies to interacting bosonicfield theories is that it doesn’t require splitting the the-ory up into one part which is linear(non-interacting) andanother part which is a nonlinear(interacting) perturba-tion. This allows the full interacting nature of the fieldtheory to be present[45, 84] at every level of its analysis.
X. ACKNOWLEDGMENTS
I am grateful to Professor Vincent Moncrief for valu-able discussions at every stage of this work. I would alsolike to thank George Fleming for facilitating my ongoingresearch in quantum cosmology. Daniel Berkowitz ac-knowledges support from the United States Departmentof Energy through grant number DE-SC0019061. I alsomust thank my aforementioned parents. [1] T. Biswas and A. Mazumdar, Physical Review D ,023519 (2009).[2] J. B. Hartle, S. Hawking, and T. Hertog, arXiv preprintarXiv:1205.3807 (2012).[3] A. T. Mithani and A. Vilenkin, Journal of Cosmologyand Astroparticle Physics , 024 (2013). [4] J. B. Hartle, S. Hawking, and T. Hertog, Journal ofCosmology and Astroparticle Physics , 015 (2014).[5] L. Visinelli, S. Vagnozzi, and U. Danielsson, Symmetry , 1035 (2019).[6] M. Ryan Jr, S. Waller, and L. Shepley, The AstrophysicalJournal , 425 (1982). [7] A. Neronov and I. Vovk, Science , 73 (2010).[8] F. Tavecchio, G. Ghisellini, L. Foschini, G. Bonnoli,G. Ghirlanda, and P. Coppi, Monthly Notices of theRoyal Astronomical Society: Letters , L70 (2010).[9] A. Y. Kamenshchik and I. Mishakov, Physical Review D , 1380 (1993).[10] G. Esposito, A. Y. Kamenshchik, I. V. Mishakov, andG. Pollifrone, Physical Review D , 2183 (1995).[11] T. Kobayashi and M. S. Sloth, Physical Review D ,023524 (2019).[12] J. B. Jiménez and A. L. Maroto, Journal of Cosmologyand Astroparticle Physics , 016 (2009).[13] J. Louko, Physical Review D , 478 (1988).[14] A. Karagiorgos, T. Pailas, N. Dimakis, P. A. Terzis, andT. Christodoulakis, Journal of Cosmology and Astropar-ticle Physics , 030 (2018).[15] M. Pavšič, Physics Letters B , 441 (2012).[16] C. L. Bennett, D. Larson, J. L. Weiland, N. Jarosik,G. Hinshaw, N. Odegard, K. Smith, R. Hill, B. Gold,M. Halpern, et al. , The Astrophysical Journal Supple-ment Series , 20 (2013).[17] G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel,C. Bennett, J. Dunkley, M. Nolta, M. Halpern, R. Hill,N. Odegard, et al. , The Astrophysical Journal Supple-ment Series , 19 (2013).[18] P. A. Ade, N. Aghanim, M. Arnaud, M. Ashdown,J. Aumont, C. Baccigalupi, A. Banday, R. Barreiro,J. Bartlett, N. Bartolo, et al. , Astronomy & Astrophysics , A13 (2016).[19] T. Kahniashvili, A. Kosowsky, A. Mack, and R. Dur-rer, in AIP Conference Proceedings , Vol. 555 (AmericanInstitute of Physics, 2001) pp. 451–456.[20] D. Paoletti and F. Finelli, Physical Review D , 123533(2011).[21] K. Miyamoto, T. Sekiguchi, H. Tashiro, andS. Yokoyama, Physical Review D , 063508 (2014).[22] H. J. Hortúa and L. Castañeda, Journal of Cosmologyand Astroparticle Physics , 020 (2017).[23] M. Ryan and L. Shepley, “Relativistic homogeneous cos-mology,” (1975).[24] R. Graham, Physical review letters , 1381 (1991).[25] R. Graham, Physical Review D , 1602 (1993).[26] R. Graham and J. Bene, Physics Letters B , 183(1993).[27] J. Bene and R. Graham, Physical Review D , 799(1994).[28] R. Graham and H. Luckock, Physical Review D ,R4981 (1994).[29] P. D’eath, Physical Review D , 713 (1993).[30] A. Csordas and R. Graham, Physical review letters ,4129 (1995).[31] C. W. Misner, Physical Review Letters , 1071 (1969).[32] M. P. Ryan Jr, Annals of Physics , 301 (1972).[33] V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifshitz,Advances in Physics , 525 (1970).[34] V. Belinskii, G. Gibbons, D. N. Page, and C. Pope,Physics Letters B , 433 (1978).[35] P. De Bernardis, P. Ade, J. Bock, J. Bond, J. Borrill,A. Boscaleri, K. Coble, C. Contaldi, B. Crill, G. De Troia, et al. , The Astrophysical Journal , 559 (2002).[36] C. W. Misner, Physical Review , 1319 (1969).[37] J. D. Barrow, Physics Reports , 1 (1982).[38] D. F. Chernoff and J. D. Barrow, Physical Review Letters , 134 (1983). [39] N. J. Cornish and J. J. Levin, Physical Review Letters , 998 (1997).[40] V. Moncrief and M. P. Ryan Jr, Physical Review D ,2375 (1991).[41] R. Graham, Physics Letters B , 393 (1992).[42] A. Macias, O. Obregón, and J. Socorro, InternationalJournal of Modern Physics A , 4291 (1993).[43] J. H. Bae, Classical and Quantum Gravity , 075006(2015).[44] V. Moncrief, Surveys in Differential Geometry; commem-orating the 100th anniver ¬ sary of the development ofGeneral Relativity (S_T. Yau and L. Bieri, eds.),(InvitedSubmission) (2014).[45] A. Marini, R. Maitra, and V. Moncrief, Communicationsin Analysis and Geometry Volume 28, Number 4, 979-1056 (2020).[46] J. B. Hartle and S. W. Hawking, Physical Review D ,2960 (1983).[47] R. Paternoga and R. Graham, Physical Review D ,4805 (1996).[48] R. Graham and R. Paternoga, Physical Review D ,2589 (1996).[49] A. Banerjee and N. Santos, General Relativity and Grav-itation , 217 (1984).[50] D. Lorenz, Physical Review D , 1848 (1980).[51] O. Obregón and J. Socorro, International Journal of The-oretical Physics , 1381 (1996).[52] A. H. Taub, Ann. Math , 472 (1951).[53] E. Newman, L. Tamburino, and T. Unti, Journal ofMathematical Physics , 915 (1963).[54] R. Kerner and R. B. Mann, Physical Review D ,104010 (2006).[55] A. Karagiorgos, T. Pailas, N. Dimakis, G. Papadopoulos,P. A. Terzis, and T. Christodoulakis, Journal of Cosmol-ogy and Astroparticle Physics , 006 (2019).[56] M. V. Battisti and G. Montani, Physical Review D ,023518 (2008).[57] V. Cascioli, G. Montani, and R. Moriconi, arXiv preprintarXiv:1903.09417 (2019).[58] M. De Angelis and G. Montani, Physical Review D ,103532 (2020).[59] D. Berkowitz, arXiv preprint arXiv:2011.04229 (2020).[60] G. W. Gibbons and C. N. Pope, Communications inMathematical Physics , 267 (1979).[61] G. W. Gibbons and S. W. Hawking, Euclidean quantumgravity (World Scientific, 1993).[62] H. Garcia-Compean, O. Obregon, and C. Ramirez, Phys-ical review letters , 161301 (2002).[63] D. Berkowitz, arXiv preprint arXiv:1910.11970 (2019).[64] C. Uggla, M. Bradley, and M. Marklund, Classical andQuantum Gravity , 2525 (1995).[65] A. Vilenkin, Physical Review D , 1116 (1989).[66] A. Mostafazadeh, Annals of Physics , 1 (2004).[67] R. Arnowitt, S. Deser, and C. W. Misner, Physical Re-view , 1322 (1959).[68] A. E. Fischer, in Relativity (Springer, 1970) pp. 303–357.[69] D. Giulini, General Relativity and Gravitation , 785(2009).[70] R. M. Wald, Chicago, University of Chicago Press, 1984,504 p (1984).[71] C. Rovelli and F. Vidotto, Covariant loop quantum grav-ity: an elementary introduction to quantum gravity andspinfoam theory (Cambridge University Press, 2014).[72] B. S. DeWitt, Physical Review , 1113 (1967). [73] A. Ashtekar, T. Pawlowski, and P. Singh, Physical re-view letters , 141301 (2006).[74] A. Ashtekar, T. Pawlowski, and P. Singh, Physical Re-view D , 084003 (2006).[75] A. Ashtekar, T. Pawlowski, and P. Singh, Physical Re-view D , 124038 (2006).[76] S. Waller, Physical Review D , 176 (1984).[77] L. P. Hughston and K. C. Jacobs, The AstrophysicalJournal , 147 (1970).[78] J. H. Bae, arXiv preprint arXiv:1410.3446 (2014).[79] J. F. Barbero and M. P. Ryan Jr, Physical Review D ,5670 (1996). [80] G. Giampieri, Physics Letters B , 411 (1991).[81] D. Berkowitz, arXiv preprint arXiv:2011.05972 (2020).[82] V. Moncrief, A. Marini, and R. Maitra, Journal of math-ematical physics , 103516 (2012).[83] A. Marini, R. Maitra, and V. Moncrief, arXiv preprintarXiv:1601.01765 (2016).[84] A. Marini, R. Maitra, and V. Moncrief, Communicationsin Analysis and Geometry , 979 (2020).[85] B. Vakili, N. Khosravi, and H. Sepangi, Classical andQuantum Gravity24