Wave function of the Universe as a sum over eventually inflating universes

By using the classically solvable minimal massless scalar minisuperspace model we study two examples of wave functions of the universe in the semiclassical limit. The first one consists of a Lorentzian component and a Euclidean component and admits a clear semiclassical interpretation as a superposition of universes. The second one consists of a Lorentzian (or Euclidean) component and another oscillatory component which corresponds to (neither Lorentzian nor Euclidean) complex classical solutions. This example has some resemblance to Hawking's minimal massive scalar minisuperspace model. We suggest a possible way of recovering the classical interpretation in such a case.

We implement the no-boundary proposal for the wave function of the universe in an exactly solvable Bianchi IX minisuperspace model with two scale factors. We extend our earlier work (Phys. Rev. Lett. 121, 081302, 2018 / arXiv:1804.01102) to include the contribution from the $\mathbb{C}\text{P}^2 \setminus B^4$ topology. The resulting wave function yields normalizable probabilities and thus fits into a predictive framework for semiclassical quantum cosmology. We find that the amplitude is low for large anisotropies. In the isotropic limit the usual Hartle-Hawking wave function for the de Sitter minisuperspace model is recovered. Inhomogeneous perturbations in an extended minisuperspace are shown to be initially in their ground state. We also demonstrate that the precise mathematical implementation of the no-boundary proposal as a functional integral in minisuperspace depends on detailed aspects of the model, including the choice of gauge-fixing. This shows in particular that the choice of contour cannot be fundamental, adding weight to the recent proposal that the semiclassical no-boundary wave function should be defined solely in terms of a collection of saddle points. We adopt this approach in most of this paper. Finally we show that the semiclassical tunneling wave function of the universe is essentially equal to the no-boundary state in this particular minisuperspace model, at least in the subset of the classical domain where the former is known.

The Everett-type interpretations of quantum mechanics and quantum cosmology proposed independently by Hartle, Geroch, and Wada are discussed. They essentially involve regarding a strong peak in the wave function as a definite prediction. Wave functions in quantum cosmology are usually peaked about correlations between coordinates and momenta, and methods for identifying such correlations are introduced. The first method involves Wigner's function, a quantum-mechanical analogue of the classical phase-space distribution. The properties of this distribution are discussed and it is shown how it can be of use in describing the emergence of classical behavior from quantum systems. The second method involves a suitably chosen canonical transformation. These methods are applied to harmonic-oscillator examples, which are of relevance to scalar field fluctuations in inflationary universe models. These methods are also applied to WKB wave functions in quantum mechanics and quantum cosmology. The manner in which the wave function becomes peaked about sets of classical solutions is elucidated. This is extended to include inhomogeneous perturbations about minisuperspace in quantum cosmology, and the derivation of the semiclassical Einstein equations, ${G}_{\ensuremath{\mu}\ensuremath{\nu}}$=8\ensuremath{\pi}G〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$〉, from the Wheeler-DeWitt equation is considered. A condition under which they are valid is derived. It is essentially the requirement that the distribution of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$, as a function of the matter modes, is strongly peaked about its average value. Some situations in which this condition is satisfied are discussed.

We continue to study exponential ansatz as a candidate of the structure of the exact wave function. We divide the Hamiltonian into ND (number of divisions) parts and extend the concept of the coupled cluster (CC) theory such that the cluster operator is made of the divided Hamiltonian. This is called extended coupled cluster (ECC) including ND variables (ECCND). It is shown that the S(simplest)ECC, including only one variable (ND=1), is exact in the sense that it gives an explicit solution of the Schrödinger equation when its single variable is optimized by the variational or H-nijou method. This fact further implies that the ECCND wave function with ND⩾2 should also have a freedom of the exact wave function. Therefore, by applying either the variational equation or the H-nijou equation, ECCND would give the exact wave function. Though these two methods give different expressions, the difference between them should vanish for the exact wave function. This fact solves the noncommuting problem raised in Paper I [H. Nakatsuji, J. Chem. Phys. 113, 2949 (2000)]. Further, ECCND may give more rapidly converging solution than SECC because of its non-linear character, ECCND may give the exact wave function at the sets of variables different from SECC. Thus, ECCND is exact not only for ND=1, but also for ND⩾2. The operator of the ECC, exp(S), is an explicit expression of the wave operator that transforms a reference function into the exact wave function. The coupled cluster including general singles and doubles (CCGSD) proposed in Paper I is an important special case of the ECCND. We have summarized the method of solution for the SECC and ECCND truncated at order n. The performance of SECC and ECC2 is examined for a simple example of harmonic oscillator and the convergence to the exact wave function is confirmed for both cases. Quite a rapid convergence of ECC2 encourages an application of the ECCND to more general realistic cases.

We calculate the four point correlation function for scalar perturbations in the canonical model of slow-roll inflation. We work in the leading slow-roll approximation where the calculation can be done in de Sitter space. Our calculation uses techniques drawn from the AdS/CFT correspondence to find the wave function at late times and then calculate the four point function from it. The answer we get agrees with an earlier result in the literature, obtained using different methods. Our analysis reveals a subtlety with regard to the Ward identities for conformal invariance, which arises in de Sitter space and has no analogue in AdS space. This subtlety arises because in de Sitter space the metric at late times is a genuine degree of freedom, and hence to calculate correlation functions from the wave function of the Universe at late times, one must fix gauge completely. The resulting correlators are then invariant under a conformal transformation accompanied by a compensating coordinate transformation which restores the gauge.

Anisotropies in Kaluza-Klein quantum cosmology

In quantum cosmology the DeWitt boundary condition is a proposal to set the wave function of the universe to vanish at the classical big-bang singularity. In this Letter, we show that in many gravitational theories including general relativity, the DeWitt wave function does not take a desired form once tensor perturbations around a homogeneous and isotropic closed universe are taken into account: anisotropies and inhomogeneities due to the perturbations are not suppressed near the classical singularity. We then show that Hořava-Lifshitz gravity provides a satisfactory DeWitt wave function. In particular, in the limit of z=3 anisotropic scaling, we find an exact analytic expression for the DeWitt wave function of the universe with scale-invariant perturbations. In general cases with relevant deformations, we show that the DeWitt wave function can be systematically expanded around the classical big-bang singularity with perturbations under control.

We investigate pfaffian trial wave functions with singlet and triplet pair\norbitals by quantum Monte Carlo methods. We present mathematical identities and\nthe key algebraic properties necessary for efficient evaluation of pfaffians.\nFollowing upon our previous study \\cite{pfaffianprl}, we explore the\npossibilities of expanding the wave function in linear combinations of\npfaffians. We observe that molecular systems require much larger expansions\nthan atomic systems and linear combinations of a few pfaffians lead to rather\nsmall gains in correlation energy. We also test the wave function based on\nfully-antisymmetrized product of independent pair orbitals. Despite its\nseemingly large variational potential, we do not observe additional gains in\ncorrelation energy. We find that pfaffians lead to substantial improvements in\nfermion nodes when compared to Hartree-Fock wave functions and exhibit the\nminimal number of two nodal domains in agreement with recent results on fermion\nnodes topology. We analyze the nodal structure differences of Hartree-Fock,\npfaffian and essentially exact large-scale configuration interaction wave\nfunctions. Finally, we combine the recently proposed form of backflow\ncorrelations \\cite{drummond_bf,rios_bf} with both determinantal and pfaffian\nbased wave functions.\n

We consider the eigenfunctions of well-defined operator related to $\ensuremath{\int}{\ensuremath{\Psi}}^{*}({1}^{\ensuremath{'}},2,\dots{},N)\ifmmode\times\else\texttimes\fi{}H(1,2,\dots{},N)\ensuremath{\Psi}(1,2,\dots{},N)d(2,3,\dots{},N)$, where $\ensuremath{\Psi}$ is a configuration-interaction (CI) wave function which includes the Hartree-Fock configuration and all double excitations and $H$ is the $N$-electron Hamiltonian of an atomic or molecular system. They satisfy for the energy the same optimum convergence properties as L\"owdin's natural spin orbitals satisfy for the wave function. They are equal to the natural spin orbitals when $\ensuremath{\Psi}$ is the exact wave function or a full CI wave function; $\ensuremath{\Psi}$ may correspond to the ground state or to any excited state.

A process of idea filtration in two distinct streams of physics i.e., 1) The dimensionality perspective of spacetime, and 2) The quantum perspective leads us to an understanding of what might be a true reality of all that we perceive. The conclusions arrived at in this paper are a bit perplexing in the sense that our perceived reality could be a manifestation of a combination of 4D + n (n > 0) flat space-time, universal wave function, and cognizance. The work is based on a review and analysis of the main concepts in quantum theory, relativistic physics, and cosmology. Key ideas and conclusions are filtered and logically connected to arrive at what might be a view of the true reality. A significant part of the paper is dedicated to the concept of the “observer” and the “ability of cognizance” that should accompany the “observer”. Though the “observer” is central to modern physics, it is not known what constitutes observation, and the term observer, often open to interpretations, does not have a standard definition and hence, is lacking in clarity. In our analysis, we have argued that the environment, in which the observer-observed system is embedded, emerges as an all-knowing, cognizant, and ideal observer that has the knowledge of the observer-observed system. At a philosophical level, we link to the fundamentals of physics, “consciousness” or “ability of cognizance” as an unavoidable and key element in not only carrying out the observation but perhaps, as believed by many, having a role in shaping the reality perceived. In the review and analysis of another stream of physics, that of General Theory of Relativity (GTR) and cosmology, we examine the question of reality from a cosmological and dimensionality perspective. Research on the 4D and 5D constructs of the universe indicates that the “reality” perceived in 4D spacetime as matter, distance, time, etc., is a manifestation of a higher dimensional 4D + n (n > 0) reality. Theoretical research on this front points towards a 4D-spacetime embedded in a 5D or higher dimensional flat space with matter and energy being a manifestation of the higher dimensions. The flow of logic in this paper leans towards a view of an ultimate true reality that is flat 4D + n (n > 0) space combined with cognizance, universal wave function, and the environment.

The quantum state of a wormhole can be represented by a path integral over all asymptotically Euclidean four-geometries and all matter fields which have prescribed values, the arguments of the wave function, on a three-surface S which divides the spacetime manifold into two disconnected parts. The ground-state wave function is picked out by requiring that there be no matter excitations in the asymptotic region. Once the path integrals over the lapse and shift functions are evaluated, the requirement that the spacetime be asymptotically Euclidean can be accomplished by fixing the asymptotic gravitational momentum in the remaining path integral. It is claimed that no wave function exists which corresponds to asymptotic field configurations such that the effective gravitational constant is negative in the asymptotic region. The wormhole wave functions are worked out in minisuperspace models with massless minimal and conformal scalar fields.

Normalizability of the wave function of the universe, particle physics and supersymmetry

Given an NT×NT interaction matrix W∞(R) which describes the dissociation of a diatomic molecule into NT asymptotic atomic channel states, we can generate exact numerical solutions to the close-coupled scattering equations. At total energies E above the highest dissociation threshold we obtain an NT×NT scattering matrix S(E) which defines the asymptotic structure of the NT- fold degenerate multichannel scattering or continuum wave functions. This matrix varies rapidly with energy and is nonanalytic at thresholds. However, based on a multichannel quantum defect analysis (MCQDA) of the coupled equations we find that the numerical S(E) matrix can be made to yield a real, symmetric matrix Y(E) which is analytic in E. This matrix can then be analytically continued across threshold to provide rigorous analytic descriptions of the multichannel diatomic wave functions in the predissociating and bound-state regions of the energy spectrum. Since the extraction of Y(E) is predicated on assigning a reference potential 𝒱γ(R) to each channel, the detailed energy variation of Y(E) is dependent on the choice of potentials. Fortunately, the physics contained in W∞(R) generally dictates an obvious set of reference potentials which usually make Y(E) slowly varying. As E is reduced below threshold we can use Y(E), often well represented as a constant over a wide range of energies, to provide a description of the predissociating molecule, including such observable properties as linewidths, level shifts, and branching ratios. At still lower energies, when all channels are closed, Y(E) offers a complete, nonperturbative description of the configuration interaction between bound electronic-rotational states of the molecule. Although many insights in this paper are provided by a WKB analysis of the reference solutions associated with each reference potential, the MCQDA yields a complete quantum mechanical description of the exact close-coupled wave functions. The quality of these wave functions is limited only by the accuracy of the molecular potentials and interaction matrix elements that are used to construct W∞(R), and by the number NT and specific set of channels we choose to include in the close-coupled theory. The analysis is equally valid when applied to either adiabatic avoided crossings or diabatic curve crossings. More importantly, the formal structure of the close-coupled wave functions dictated by MCQDA yields a rigorous framework for the analysis of strongly interacting continuum, predissociating, and bound state channels without recourse to perturbation theory.

The energy of the one-dimensional hydrogen molecule with δ-function interactions between the particles is obtained using exact hydrogen molecule ion wave functions, a symmetrized product of hydrogen atomic wave functions and a linear combination of hydrogen atomic orbital wave functions. A variational calculation is carried out for each of the three cases to obtain wave functions that yield lower total molecular electronic energies.

Efficient update of determinants for many-electron wave function overlaps