LOG#213. No boundary.

Hello, again!

Yes, three consecutive days posting. A change! I missed blogging these months of nightmares. I hope these posts terminate your wait. The wait is over. And I can not wait too. Tempus fugit.

Today, we will be discussing Quantum Cosmology. The no boundary proposal and related ideas. Let me add first that Quantum Cosmology is a fuzzy subject. There is no consistent theory of Quantum Cosmology yet, beyond what we accept from the cosmological standard model (LCDM) that is accepted as the macroscopic view of the Universe in corcondance with all the data. OK, those who follow the last news, you new there is a “crisis” due to some values of the Hubble parameter. 75km/s/Mpc or 67? This is not a real crisis. Not a big one at least. I remember reading as a child the controversy about H_0 being 50 or 100, even 200. That is not the point, I believe. I think everyone in the field knows that H_0 will converge in the end. The real point is to adjust the cosmological ladder with the CMB observations. Period. Of course, we will find (or not) new physics in the path, but the times we doubted about how old is the Universe seems to be almost over. OK, perhaps if a revolution arises…But revolutions are not periodic, revolutions take time to arise.

Well, to the subject! We already know the Universe (or Multiverse, but this is speculative) had to be hotter in the past. That is the Big Bang. Going even further, to the singularity…The Universe arised from some quantum fluctuations we do not understand (due to quantum gravity whatever it is), and it tunneled from “nothing” into “everything”. This tunnel effect could even be a mere bounce from another preexistent Universe, but we can not know with electromagnetism. At about 380000 years after the “singularity phase”, the Universe created atoms, and become transparent to photons when the first hydrogen, helium or even lithium atoms were formed. That is the origin of the Cosmic Microwave Background (CMB). Before that, we don’t have the enough information to tell,…But we do know the Universe passed the electroweak phase transition, the QCD phase transition creating protons and neutrons, and we know there is likely a cosmological neutrino background (CNB) emitted at the first seconds of the Universe, …Temperature about 2 K (1.945 K) or lesser, depending on the degrees of freedom available at the neutrino decoupling. Even before, to solve why the Universe is flat (please, don’t tell to flatlanders about this), the Universe had to suffer an inflationary phase (faster than the speed of light!) and nucleating likely baby Universes like us and creating a Multiverse. This view is not completely accepted, but it is true. It could show that bubble or vacuum bubbles nucleated at the inflationary point, creating not only a vast number of (currently unobservable island of Universes) but also primordial gravitational waves and/or primordial black holes we could detect in the next decades with better and better gravitational telescopes.

The idea of tunneling Universes and bubble Universes, from inflation, had one of his master minds in the head of Alexander Vilenkin. Furthermore, the idea of treating Cosmology from the quantum viewpoint was pioneered by James Hartle and Stephen W. Hawking, to whom I would like to dedicate these lines and this post as well. The Multiverse has infinite dangers. The Ancient One knew that when talking to Stephen Strange. She was right. The Multiverse also contains many inconsistencies. And the young Quantum Cosmology tries to solve it, but we are yet far to understand the whole Universe with quantum theory because we are lacking an essential part: the degrees of freedom. A misnomer for observables. The no boundary proposal, first introduced by Hartle and Hawking, joined to the Bubble Universe picture from inflation long ago, but there are not completely consistent. Indeed, they seem to be contradictory to each other. In a paper entitled “No smooth beginning for spacetime”, it was shown:

  • The Universe, if it emerges smoothly from the void/nothingness, would be wild and fluctuating. This is in apparent contradiction with the current observations.
  • The no boundary proposal, the idea that the universe itself has no boundary in time-space at the beginning, does NOT imply a large Universe like the one we live in. However, a tiny curved Universe that would collapse almost instantaneously would be produced. That is not the case since we are already here, alive. Yet.
  • We need something else. An extra idea or framework, other picture to understand the very very early Universe OR, otherwise, we must rethink the most elementary models of quantum gravity at hand at these times.

What do we know? Let me begin by some general key facts.

GENERAL relativity. The known agenda. The known known of GR.

  • Gravity=Geometry of curved spacetime.
  • Mass-energy curves spacetime. Spacetime tells mass-energy how to move.
  • Free masses move on “straight” paths in curved spacetime. Geodesics are the cool name of “straight” paths in curved spacetime.
  • General relativity can not hold at very dense extreme situations, concentrated at the smallest scales, without conflicting with our microscopic quantum description of the Universe. Therefore, we need a UV completion of gravity. The saint grail of physics: Quantum Gravity. QG would apply to black holes and the origin and extreme ends/edges of the Universe.
  • Dark matter and dark energy. We need extra missing mass-energy to explain how galaxies move. It shows this extra matter is about 1/4 of the matter-energy or 80% of the matter content of the Universe. It can NOT be Standard Model particles.
  • Dark energy. It is now the main component of the energy budget of the Universe. About th 70%. It is accelerating (positively) the Universe at largest scales. It seems it is related to the vacuum space energy-density. Dark energy is another name for a rebooted cosmological constant. It was introduced by Einstein himself and then deprecated. It seems…He was right. Dark energy exists. It blows up your mind trying to understand why the dark energy value has the value it has today. Best theories of Quantum Field Theory give so bad prediction for it that you ponder if you are understanding something really…

There are many other “issues” with GR and the LCDM view. But these would be some of the big “known known GR”. What about the Saint Grail? Well,…Not much, but:

Quantum Gravity Agenda. The known known we are hoping to get from it.

  1. Quantum gravity=Quantum theory of gravity=Microscopic theory of gravity.
  2. Quantum gravity meas a quantum theory of space-time geometry. That is, a quantum corpuscular view of the spacetime continuum.
  3. Quantum states for gravity are unknown today. They only theories giving hints of what these states are: superstring theory/M-theory, loop quantum gravity (LQG) and minor approaches (I am one of this third road) to review what quantum relativity or spacetime is. We lack a more precise counting or identification of the microscopic degrees of freedom of what space-time is made of. A secret that makes space-time itself that likely grants a Nobel Prize if proved. What are the origins of the Black Hole entropy? Remember, atomic physics arised when trying to explain solids, liquids or gases in the 19th century. Similarly, we are now in a similar position with space-time.

Quantum Mechanics agenda/The QFT keyfacts.

  • Main objects in QM and QFT are wavefunctions or quantum fields.
  • QM/QFT dynamics depends on quantum states/quantum operators. Quantum states satisfy certain wave-equations.
  • Quantum states/wavefunctions are NOT mechanical waves. Indeed, much to the deterministic philosophical fans, quantum wave functions are probability waves. Stunned? Wave functions are just complex numbers. The describe oscillations of fields. QM/QFT gives you a rule to calculate probabilities via the Born rule P=\vert\Psi\vert^2. Amplitudes are complex, but observables are real probabilities.
  • Duality. Any quantum field has wave properties and particle properties. Complementary aspects of the same object in different regimes. \lambda=h/p. Duality is consistent with special relativity SR.
  • Heisenberg uncertainty principle (HUP). You can not observe certain observables without altering others, simultaneously. Observations are context dependent.

    \[\Delta A\Delta B\geq\hbar\vert<\left[A,B\right]>\vert/2\]

  • States evolve unitarily, and they follow Schrödinger-like equations H\Psi=E\Psi.

The GQ agenda: two main paths.

  1. Loop quantum gravity (LQG). You can non-perturbatively canonically quantize gravity via connections variables that take the form of loops and include Hilbert spaces into this approach. It can calculate some BH entropy in certain examples with the aid of spin networks. Area and volume are quantized. Singularities are likely to be replaced by quantum bounces. Issues with the introduction of matter.
  2. (Super)String theory/M-theory. Quantum particles are really vibrations of supertiny fundamental strings (or p-branes).  Every fundamental force is derived from the string excitations. You require extra dimensions to do it consistently. Dual formulations and holography are also compatible. Issues: the predict a vast number of vacuum or possible Universes. No one knows how to select our vacuum/Universe from every possible configuration. This has produced the Landscape/Swampland nightmare. In other words, nobody knows how to select the field values to tune the possible configurations to our current Universe. Bad things? Maybe not. Maybe yet. Divergent views are available at this moment of time.

Quantum gravity needs and requires experiments, tests. We are now in the dawn of neutrino and gravitational wave astronomy. Maybe, we will have dark matter and dark energy astronomy as well! However, in the past, some proposals were done to try to make accessible tests of quantum gravity. There is a problem. QG is, at least from naive expectations, hidden at energy scales of 10^{16} TeV 10^{TeV}. No forthcoming collider is going to have these energies. The LHC works at about 10 TeV. Future colliders will likely work at 100 or even 1000 TeV. We are far by thirteen or fourteen orders of magnitude in energy! Solutions? Well, maybe quantum gravity energy is really much lower than taht. That is the idea of extradimensional gravitational/gauge models. The fundamental true gravity scale is high but it gets diluted into the extra dimensions, and that is why gravity is weak. The LHC has limited these possibilities. Option two, try to get the higher energy particles/interactions of the universe, via gamma rays, neutrinos or…gravitational waves! The true power of gravitational waves is that they can in principle test everything. They are probes of strong gravity, not only of weak gravity as from observations here on Earth!

In the beginning of the Universe, the expanding Universe already reached the Planck energy: we have 14 Gyr of data to accumulate! Before the expansion make other parts of the Universe non-observable! However, the dispersion is brute. Data is dispersed on 42 Gyr of distance due to the cosmic expansion rate.

Where does Quantum Cosmology (QC) enter this game? It yields that cosmological observations of the Universe over the Universe’s classical history could drive us towards a whole quantum view of the Universe. What are the main points of QC?

  • Most of our observations of the Universe on cosmological scales are properties of its classical history. For instance: homogeneity, isotropy, evolution of fluctuations giving born to galaxies and galactic clusters or superclusters. They all made up the CMB, galaxies, planets, biota, us, and every possible configuration. At last, you find quantum things…
  • The rate of expansion of the universe, the dark matter and dark energy, the radiation to other stuff rate are surely quantum stuff too.
  • Classical behaviour is a matter of quantum probabilities! Quantum makes up everything!
  • Classical behaviour is not general but an approximation to QM. I wish you read my previous logs about the non-deterministic formulation of classical theories.
  • A quantum Universe needs a quantum description. So, should the Universe itself have a wavefunction or state? What is the quantum state of the Universe? Note that this question is hard. It is also related to gravity. The target of Quantum Cosmology is to describe the quantum state of the Universe! But it is also a target of Quantum Gravity. Thus, quantum gravity and quantum cosmology are united, or at least, complementary. Remarkly, string theory or quantum gravity require quantum states! The tryumph of string theories over the rest QG approaches is that it gives a sense for what are these states. Fuzzballs, stringy states of stuff, p-branes…But, of course, the question would be…What is string theory after all?
  • Probabilities. Basically, run! They are the square of complex amplitudes. For real, the square of the operator valued quantities applied onto the quantum states.

So, does the universe have a quantum state? Is it unique? What is that state? We have only one system to observe from our current perspective (our universe). However, can we handle for other quantum states? In fact, there is a cool variation of this. It is called third quanntization. In 3rd quantization, you are allowed to create and destroy (erase) Universes from the Multiverse (just like Zeno-same in Dragon Ball super, no joke here!). Contemporary theories and final theories have two parts (three if you allow for a lagrangian!): the hamiltonian part and the wave function part. Hamiltonians define dynamics. Wave functions or states define kinematics. Even if you manage to elucidate what quantum states are, you are driven to the next level and ask what is the evolution following the state. This is other unsolved task of unified theories. Remark: regularities in the hamiltonian H provide classical dynamics, and it could be chaotic and not predictable. Regularities in the wave function arise in the observed classical spacetime. They also should explain early homogeneity/isotropy, inflation and have certain fluctuations giving rise to anisotropies producing our galaxies and clusters. The different arrows of time remain unexplained. Why is there only a single time direction? Why do not observe the future?  The CMB, the large scale structures, the existence of certain isolated systems, the topology of spacetime could change and introduce a spacetime foam into the picture. That would imply a varying number of dimensions of spacetime, even depending on the observed scale.

The problem of finding the quantum state of the Universe gave rise long time ago to an approach named minisuperspace models. That is quite a disturbing name since it has nothing to do with superspace or miniscales! In minisuperspace models, you are forced to suppose that the universe is certain homogeneous and isotropic closed metric

    \[ds^2=-dt^2+a^2(t)d\Omega_3^2\]

Matter is just any scalar field plus a cosmological constant. a(t) models the history of the Universe, and the Universe itself is associated to certain wavefunction \Psi(a,\psi), where \psi=\psi(x,t) is the scalar field. Over the 3-space surface, \Psi evolves like a wavefunction. The problem IS:

  1. The state is NOT an initial condition for the Universe, rather a description of it.
  2. It predicts probabilities for ALL p ossible alternative 4d histories of the Universe, though. What goes on now? What went on then in the past?What is future end of the Universe? We lack a dynamics here. Conservative approaches use the so-called Wheeler-de-Wit equation for \Psi, but it could have any other dynamics!

Fascinating! Now, the no boundary idea by Hawking himself (and Hartle, who worked out hardly on it). We should search for an analogue or a cosmological analogue of the ground state in quantum physics. Thus, no H to be a lowest energy. For a closed Universe you could take H=0. Inside the all possible 3-geometry is everything, what about the boundary? Euclidean sum over every possible 4-geometry with one boundary for the arguments of the wave function should be allowed only! In other words, the boundary condition of the Universe is that it has no boundary! Technically:

    \[\Psi(b,\chi)=\int DaD\psi\exp\left(-I_E\left[a,\psi\right]/\hbar\right)\]

This integral should be regular configurations for a,\psi which match the boundary at say (b,\psi) from a,\psi. The classical spacetime is just a semiclassical approximation. This object predicts an ensemble of classical histories similar to the WKB approximation in the classical field theory setting. The main problem with this proposal turns to be that not all classical spacetimes seem to be predicted. Take, e.g., inflation (dS spacetime!). The no boundary wave function proposal plus classicality over certain scales favor low inflation scales. However, it seems that we are likely to live in a Universe that has undergone more inflation than the one the no boundary proposal predicts. there are more places for us to be. The fluctuating probabilities start in a ground stante given by a dS spacetime (something technically called Bunch-Davies vacuum). The good thing is that large fluctuations would produce large anisotropies…However, they are not observed. Different CMB maps have restricted as you know the possible degree of anisotropies in the EM sector. Therefore, we have to seek a way to live with fluctuations tending to vanish…We need symmetry here. But no idea of what symmetry is there.

What about the arrows of time? We know there is a privileged fluctuation arrow giving rise to the growth of fluctuations in the Universe past. We also know about the thermodynamics arrow of time associated to the growth os cosmic entropy, the radiation arrow fo time related to the retarded EM radiation field and the psychological arrow of time that allows us to remember the past but not the future. In fact, the no boundary idea CAN explain these arrows of time by a special quantum state selection. This is true EVEN if the dynamical laws (GR+QM) are time resal invariant! What the no boundary idea says is that these arrows are essentially the same arrow. The fluctuations of the selected special wave functions vanish only at one place on the fuzzy Universe. It is an instanton like event. A South pole in the cosmic 3-geometry. Fluctuations are small at only one of the places where the Universe is small. For instance, bouncing universes (like the ones LQG favor) have fluctuation with increased values away from the bounce on both sides of the 3-geometry. The arrows of time points in opposite directions on the opposite sides of the bounce. That is why we can not observe or remember the future. We are doomed in one side of the 3-geometry! (Note: what if we are entangled to other cosmic “clons”? Yeah, this is a crazy idea. Sorry to mention it).

Other big unsolved problem of Quantum Cosmology is named as “the topology change of spacetime problem”. Simply, the topology of the Universe could be allowed to change from a quantum viewpoint. That is not observed, of course, at macroscopic scales. But it comes as inevitable as far as you go higher and higher in energy, and you go to QG. Spacetime is manifold. Manifolds use to have metrics, isometries and symmetries. What is indeed our macroscopic manifold? What is its topology? No much is known about this from cosmological data. In fact, things get complicated whenever you make the metric or the manifold quantum stuff. There is no unique definition of quantum geometry of spacetime at this point! However, the idea of the no boundary can predict simple stuff and probabilities for the change of the topology of the spacetime. Note that the large scale of the topology of our Universe could be observable by its effect on the CMB and CNB or even other future observations. However, there is no information about how the geometries behave on topologically complex manifolds.

To summarize up:

  • We do not know the topology of spacetime.
  • We do not know how small and large dimensions are related. Evidence that the microworld is twodimensional somehow is known. The macroworld seems to be 4d.
  • Should we allow for more that one time dimension?
  • What about the coupling constants in the Universe?
  • Classical spacetime, early Universe, fluctuation, arrows of time the CMB and even isolated systems can be handled with no boundary proposal.
  • What about singularities and black holes?

GR became important in 1956 when Robert Oppenheimer highlighted that GR is important when GM/rc^2\sim 1. The Universe is about 10^{28} cm big today, but by the end on the inflation had only about 10^{-30} cm only! Primordial black holes could be evaporating today and be one of the sources of cosmic rays, fast radio bursts or even some strange emissions in the observed Universe. GR is the key to the quasiclassical realm an central to the origins and properties of current isolated subsystems. After a century of GR, GR is as central to physics as QM and the SM. GR reigns over the macroscopic scales. The SM rules over the microscale. Not good enough. we have apparently reached a limit of our comprehension of the Universe. everything is quantum and relativistic, BUT, we do NOT know how to study big heavy objects tiny enough. Spacetime is like a fluid of something else. QM says that action (energy and angular momentum as well) is quantized. What else?

Galaxies are the building blocks of the big Universe out there. The universe is indeed “a gas of galaxies” somehow. There is about 10^{12} galaxies out there, and new galaxies become visible and others disappear every second due to cosmic expansion. The Big Bang was a special moment in time, not a place in space. Space itself did not existed before the Big Bang by definition from our current conceptions of the Universe. The matter was infinite dense, infinite hot, and infinite opaque (to EM at least) at the beginning of time (but GWaves come to the rescue here!). You could see the BB light in your ancient TV devices (1% of the snow are BB photons). That the Universe had a beginning of time seems inevitable from QM+GR. This is quite a shift from Aristotle, Copernicus, Kepler, Newton and Einstein himself times. All these guys thought the Universe was STATIC, nondynamical. The hope is that quantum cosmology, the fusion of QM and cosmology, will allow a deeper insight into the creation of the Universe and its future evolution (if not chaotic!). Deterministic classical physics (newtonian physics) is an approximation to QM whose probabilities depends on wave function amplitudes and its square gives you probabilities. That is, if QC is right, you could only know what are the probabilities of your future are, not what is your future going to be or to happen. The wave function of the universe predicts probabilities for histories of the Universe and everything on it. Even you can have classically forbidden histories as well. In other words, knowing the wave function of the Universe  is not a way of knowing how to be rich or inmortal, it is a way to see what the odds are of being rich and inmortal. A SETI like question: what is the probability that there are more than 1000Earth-like planets in the galaxy hosting intelligent life that we could communicate with? In principle, that is a fair question in QC. However, it is very hard to calculate that. We do not know how to compute those probabilities for stars or galaxies or planets. However, it seems than gravity is universal everywhere (apparently the same gravitational constant everywhere out there) and newtonian gravity works well for galaxies (excepting some basic GR corrections). However, at the beginning of time, GR can not be valid. Or even in critical BH systems we do know GR is not enough. Even when its lowest energy state is reached, quantum fluctuations make quantum rest an absolute impossibility. However, the no boundary idea could provide a way to give up the initial singularity. A quantum regime (with properties to be defined) at the beginning of the time, could provide a way to define a non-singular beginning. However, the size of the smoothness at the beginning can not be arbitrary high or low. Specially if you want to get our universe.

As a final part of this essay, let me remark about the question of relativity and observers. Observers are an important part of physics:

  • We are physical systems within the universe with only a very small probability to exist in any region!
  • Probabilities for what we observe are conditioned on us being here. Observations change the odds and the states, that is a basic result of QM!
  • We won’t observe what is where can not exist.
  • The cosmological constant of the Universe is the main component of our current state.

No boundary does not predict: large dimensions, small dimensions, extra time dimensions. No boundary predicts classical features, early homogeneity and isotropy, the arrows of time, the CMB structure, isolated systems and GW from early times, plus parameters like the cosmological constant.

Unless probes of extra time dimensions are found, space travel is pointless (unless you figured out how to get a wormhole, an Alcubierre drive or any other form of faster-than-light propulsion). Stars and galaxies seem to be doomed in the future. Atoms or even protons could decay and get destroyed by dark/phantom energy. In the end, black holes will evaporate and matter as we know will vanish…Leaving what? That is an issue of QG and the quantum structure of the vacuum and spacetime itself.  We will end cold, dark, empty, lonely and simple constituents. Space communication will be more expensive in the future of the Universe due to cosmic expansion. Galaxies will be too redshifted, we will become alone in our local group (merged into a single decaying galaxy?). For the whole Universe, it seems there is no single slit experiment to try. Most of what you are is here from the seeds at the beginning of time. Frozen accidents of the Universe we are. How to calculate the state?How to evolve the state? What are the odds to survive the future?

Strings are the biggest hope for unification, even when no boundary and other LQG ideas are into the market. Gravity is derived from strings and its coupling constant G_N=(g_s/M_s)^2, cf. with the g_F=g^2/M_W^2 origin of the Fermi constant. The problem with strings is that they can say everything about nothing. They seem to imply an UNOBSERVED higher dimensional spacetime, and no adjustable dimensionless parameters arise, but vacuum expectation values at inevitable symmetry breaking events can not predict what our Universe is between zillions of possibilities. The landscape. Is just a real landscape or just the same seen from different observers? Anthropic ideas yet valid here? We are far from inflationary era, about ten to the sixty one orders of manitudes away but we know something like inflation should occur. Otherwise, the Universe could not look like this one. We would not be reading me right now. The holographic principle says that boundaries are important places. Even if no boundary at hand? Words to remember today:

  • No boundary and wave functions of the Universe.
  • Gravity, thermodynamics, entropy and arrows of time.
  • Gravity can be seen holographically as a field theory on the boundary.
  • Gauge-gravity unification is inevitable at some point.
  • Observers are important in physics.
  • GR and QM are not enough to explain the state of art of the Universe at critical points (superdense, superheavy, supertiny objects).

Epilogue: some people like A. Valentini, L. Smolin, R. Penrose, and many others have speculated about the idea of QM not being the final story but being an approximation to another theory. It can be true. I find hardly to be true that it could give up or erase probabilities. We abhor probabilities because we do not like uncertainty (do we hate free will at movies?Just curious). Valentini idea is that maybe the Born rule is just an equilibrium law. Non-equilibrium QM and extensions of QM could give rise to non QM predictions testable at the near future. For instance, we could get quantum noise from the Big Bang, and the fact non local entanglement could be caused by position dependent wave functions is a possible solution, but it has not produced any experimental departure from quantum mechanical predictions. Problems with this idea, is that non-equilibrium particles could be used to send faster than light signals to the past and to define absolute time we gave up with current theory of special relativity. Causality could be challenged by this picture as well. However, the issue of why quantum mechanics prefer a probabilistic interpretation with the squared modulus rule is a mystery than only a few scientists have tried to face with. Others, simply, in time, we become realistic in the sense we only worry about probabilities and their realizations and measurements as the real problem to understand the universe. That is, the complexity are calculating probabilities, and to know why are the states we observe as we observe, not why they arised or what selected them from all the mathematical possibilities the abstract mathematics could have realized. Why are we here? What are the odds to survive in the near future? Is that all that matters after all?

    \[\boxed{Z=\int DX\exp\left(i\int d^4x L_{TOE}\sqrt{-g}\right)}\]

    \[L_{TOE}=R-F^2-G^2-W^2+i\overline{\Psi}\Gamma\cdot D\Psi+D_\mu H^+D^\mu H-V(H)-\lambda\overline\Psi H\Psi+h.c.\]

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