by: Roger Penrose, read in 2005

xx URL for solutions: www.roadsolutions.ox.ac.uk

14 Penrose doubts the axiom of choice

18 Three worlds: The Physical, the Mental, and the Platonic Mathematical. Three mysteries: Each seems to "cause" the next in a rock-paper-scissors relationship.

22 My "transfer of omniscience" explains this mystery

60 "It is as though Nature herself is guided by the same kind of criteria of consistency and elegance as those that guide human mathematical thought."

62 Erwin Schroedinger: "The idea of a *continuous range*, so familiar to mathematicians in our days, is something quite exorbitant, an enormous extrapolation of what is accessible to us."

62 Einstein: "One can give good reasons why reality cannot be represented as a continuous field...Quantum phenomena...must lead to an attempt to find a purely algebraic theory for the description of reality. But nobody knows how to obtain the basis of such a theory." I wonder if my Practical Numbers could be such a basis

62 "real numbers still form a fundamental ingredient of our understanding of the physical world"

95 Five fundamental numbers in one equation - the Euler Formula

97 If you take C = angle of intersection, then I think this graph depicts our observations of the cosmos where the steeper curve is the path of light and the other is the path of the observer

103 What calculus (analysis) is all about: differentiation (local behavior) is the inverse of integration (general behavior).

120 Dirac Delta Function

138 Manifolds and Riemann surfaces

145 Genus of complex Reimann surfaces; moduli; m = 3g - 3 for g > 1

165 Momentum, p, in QM

165 Switching from discrete to continuous variables using the limit.

190 Partial differential operators are "arrows" pointing along coordinate axes, i.e. a Killing vector field

208 Spinors and Clifford Algebras

214 "Grassman algebra provides a powerful means of describing the basic geometrical linear elements of arbitrary (finite) dimension."

222 Hausdorff condition

224 Vector spaces: vectors taken at only a single point of a manifold.

224 Vector space = tangent space at a point

233 Fundamental Theorem of Exterior Calculus

225 Covectors are generalized contour lines.

239 Physicists' vs. Mathematicians' views of math: Physicists want coordinates!

239 Coordinate systems facilitate construction of new operators.

239 Pure mathematicians are "embarrassed" to resort to calculation with coordinates.

240 A vector can be thought of as a single thing or as a set of components.

240 Abstract index notation and abstract markers.

240 Definition of a Tensor.

244 Newlander-Nirenberg Theorem allows free movement between the philosophical notions of real manifolds vs. complex manifolds.

262 "An eigenvector of a linear transformation T is a non-zero complex vector v which T sends to a multiple of itself. That is to say, there is a complex number ?, the corresponding eigenvalue, for which Tv = ?v

265 "the conceptual difference between a linear transformation and a matrix is that the latter refers to some basis-dependent presentation, whereas the former is abstract, not depending upon a basis."

266 "quantum mechanics is all to do with linear transformations."

271 Abstract definition of a tensor.

278 Definition of a metric.

291 Definition of a tensor.

326 Kaluza-Klein

326 The reason for "curling up" extra dimensions: so we are not aware of them.

328 The notion of a "bundle" seems to exactly describe the relationship between a TV screen (the manifold) and the set of paths taken by electrons from the electron gun to the individual points on the screen.

332 Bundle cross sections generalizes the notion of a graph of a function.

357 Penrose: It appears to be believed that the universe is infinite.

357 A finite universe is not inherently inconsistent. Y. Ahmavaara pursued such an idea.

358 Penrose considers infinite sets as rigorously defined.

359 "To my mind, a physical theory which depends fundamentally upon some absurdly enormous prime number would be a far more complicated (and improbable) theory than one that is able to depend upon a simple notion of infinity."

363 "awesome mathematical richness...inherent in the infinite" - in his brief consideration of finite cases, he didn't consider "Practical Numbers".

363 Penrose did consider discrete systems early in his career.

365 Frege: antinomies arise from the notion of "all sets". Penrose "evades...this disturbing issue."

365 Finite cardinals makes more intuitive sense than finite ordinals.

366 Axiom of Choice ==> non-comparable cardinals do not exist.

366 Penrose is "cautious" about accepting the Axiom of Choice.

366 Axiom of Choice ==> Banach-Tarske Theorem = "an alarming consequence".

369 Cantor's proof that the cardinality of the reals > the cardinality of the rationals is equivalent to Turing's and Goedel's theorems.

371 The Continuum Hypothesis and the Axiom of Choice.

372 Russell's Paradox and the "class of all classes".

372 The "set of all sets" is a forbidden concept; there is no such thing.

373 Mathematicians distinguish sets from classes to avoid Russell's Paradox.

373 Penrose's dissatisfaction with this ploy.

373 Penrose: Constructivists are not immune from Cantor's diagonal slash.

374 His notion of "construct" presupposes the "infinite" set of natural numbers and the "infinite" capacity of the Turing Machine. I think this begs the question of how those infinities got constructed.

376 Penrose argues against "extreme conservatism" because "We are always driven to consider "additional sets". I say simply, resist the temptation.

377 Penrose's interpretation of Goedel's Theorem.

378 "It is perhaps remarkable, in view of the close relationship between mathematics and physics, that issues of such basic importance in mathematics as transfinite set theory and computability have as yet had a very limited impact on our description of the physical world. It is my own personal opinion that we shall find that computability issues will eventually be found to have a deep relevance to future physical theory..." It is my opinion that reality is finite and thus transfinite considerations will be irrelevant.

379 The different notions of "size" of a space as given by the number of dimensions vs. the cardinality of its points.

386 Earth's position in the hypergalactic structure of the universe: Milky Way galaxy < local group < Virgo cluster < Coma supercluster < the Great Attractor.

388 I don't get the 6D family of "Aristotelian" uniform motions.

389 "Galilean spacetime, considered as a manifold, possesses a connection which has both vanishing curvature and vanishing torsion (which is quite different from it possessing a bundle connection..."

390 Newton's Theory is accurate to 1 in 10^7 while his data was accurate to 1 in 10^3 !

407 The signature of the spacetime metric.

409 Spacetime metric, inertial motion, proper time, and the 'clock paradox'

413 The signs in the Euclidean metric vs. the Minkowski metric

415 How complexifying 4D Euclidean space leads to 10 dimensions - symmetry groups

416 10D Poincare symmetry group

418 Orthogonal complement

420 Time as the arc length of a world line.

428 Noether's Theorem

442 Maxwell's euations described

445 Maxwell's equations

449 Poincare Lemma

449 Electromagnetic potential

450 Electromagnetic potential is not locally measurable.

450 QM wavefunction = cross-section of a bundle describing charged fields

451 Gauge invariance

451 Weyl's conformal rescaling of the metric

451 Weyl: no absolute 'ideal clocks' - rate depends on history.

453 QM requires a "non-observable" 'gauge transformation' complex multiplier outside of spacetime. Isn't this hyperspace??

454 Aharonov-Bohm effect: electrons affected by magnetic field they don't encounter.

455 Einstein's energy-momentum tensor - the source of gravity.

457 Could the ten independent Killing vectors of Minkowski space form a basis for 10D hyper space-time?

457 Noether's Theorem = Lagrangian formalism

457 How each Killing vector yields a conservation law

457 A hint of a possible mechanism for cosmological introduction of new dimensions.

459 Principle of general covariance

462 Einstein's Field Equation and the Vacuum Equation

462 The cosmological constant.

473 The Euler-Lagrange equations encode the entire Newtonian behavior.

477 Hamiltonian flow representing Newtonian time evolution

478 Penrose's awkwardness with some formalisms are reminiscent of Dr. Dick's

482 General description of how classical systems can vibrate

482 The analysis of vibrating systems here applies to the finite case and is then extended to the case of infinite degrees of freedom.

483 Ask Dick if this isn't a description of part of what he has done.

484 Liouville's Theorem: Phase space volume is preserved by the dynamics.

486 Penrose: Physical fields are continuous thus their configuration space will be infinite-dimensional in some regions.

487 Functionals and functional derivatives

489 Noether's Theorem in Lagrangians

489 Gauge invariance and the failure of Noether's Theorem for General Relativity

490 There is an open issue with respect to angular momentum in General Relativity.

490 Hilbert derived Einstein's theory from a Lagrangian approach.

490 Mie's Theory: a 1915 vintage TOE

491 Modern fundamental physics theories are usually in the form of Lagrangian functionals.

491 The Lagrangian for free Maxwell Theory: "1/8th of the difference between the squared lengths of the electric and magnetic field vectors, in 3-dimensional terms."

491 The value of the potential of a Maxwell Lagrangian is not directly observable. Could it be because it exists in an inaccessible hyperspace?

491 "In most situations, the Lagrangian density does not itself seem to have clear physical meaning." Could these densities have meaning in hyperspace?

491 Penrose is uneasy about depending on Lagrangians too much in formulating fundamental theories.

495 Partial differential operators describe infinitesimal translations in the direction of the axes of an affine space, each expressing an independent symmetry.

496 In QM, momentum is actually identified with the differential operator - crazy!

496 Quantum momentum and canonical quantization: put QM momentum in the classical Hamiltonian.

499 The Schroedinger Equation

505 "momentum is indeed identified with 'differentiation with respect to position', and energy with 'differentiation with respect to time'."

506 Eigenfunctions, eigenstates, eigenvectors, eigenvalues, and state vectors.

519 Penrose: The wavefunction shouldn't be thought of as a probability wave.

520 Measurement, eigenstates, eigenvalues, and the Dirac Delta Function.

521 Position states are products of Dirac Delta Functions.

521 The wavefunction, expressed as a function of position, is a continuous linear combination of Dirac Delta Functions.

522 The wavefunction expressed as a function of momentum.

523 Pictures of wavefunctions and Heisenberg's Uncertainty Principle.

528 Quantum jump

528 The choice of eigenstate is by pure chance.

533 Wavefunction, normalization and probability density

533 Unitary evolution means that the norm is preserved.

534 Hilbert space

534 'bra' and 'ket' vectors and Hermitian scalar products. Bra is the Hermitian conjugate of ket and vice versa.

538 History of Heisenberg's vs. Schroedinger's pictures. Born, Jordan, Dirac.

540 The orthogonality of the eigenstates of a quantum measurement jump

541 "Passing from the position representation to the momentum representation amounts to a change of basis in the Hilbert space."

541 "Rigour mortis"

547 Photon spin axis is always the direction of travel.

547 Spin numbers are helicities.

550 I think the choice of O(3) is based on an assumption we should doubt.

556 Spin 1/2 is immediately associated with 3D spatial directions.

562 Spherical polar coordinates = Latitude and longitude

570 QM description of an isolated quantum object: the answer to my question to Prof. Roothan.

574 Compactness implies discreteness; I don't see the irony.

577 "The term 'quantum number' usually refers to the possible discrete eigenvalues of some significant quantum observable, such as angular momentum, charge, baryon number, etc. which is used to classify a particle or simple quantum system."

579 In QM there is only one time for all particles.

579 "the 'path-integral' approach to relativistic quantum theory,...is based on a relativistic Lagrangian rather than on a Hamiltonian formalism"

580 QM treatment of many-particle systems; the answer to my question to Prof. Roothan.

592 Measurement cuts an entangled particle free of its entanglement.

593 "It is my own view that Nature herself is continually enacting R-process effects, without any deliberate intentions on the part of an experimenter or any intervention by a 'conscious observer'."

594 Examples of bosons and fermions.

595 The Pauli Exclusion Principle for fermions.

596 Bose-Einstein condensates, superfluidity, superconductivity, and Cooper pairs

608 Bell's Theorem; Bell's inequality

609 In QM time is treated differently than space. Cosmic time??

610 Quantum Field Theory is mathematically inconsistent!

612 Basic form of Schroedinger's Equation

613 Wonderful example of the deep subtle relationship between mathematics and physical reality

620 The Dirac Equation

621 "A spinor may be thought of as an object upon which the elements of the Clifford algebra act as operators."

623 "Pauli matrices (which are basically quaternions, with a factor i) are also Clifford algebra elements, but for the 3-diomensional rotation group."

627 Knowledge of electrons, positrons, and photons in 1928

628 The particles of the Standard Model

638 CPT Theorem

641 It seems to me that this suggests existence in real hyperspace.

642 "Nature chooses"?? Do you suppose the choice was a conscious one?

643 Mass as a consequence of symmetry breaking

645 Families of particles and their relationships

648 It seems to me that this suggests existence in real hyperspace.

649 Yang-Mills theory - generalization of electromagnetism and the basis of QCD

651 "The basic symmetry group of the entire standard model is taken to be SU(3)XSU(2)XU(1)/Z6

652 The Chan-Tsou extension to the standard model

655 "The predictive power of the theory indeed depends crucially upon the mathematical consistency of such theoretical underpinnings."

660 The spin-statistics theorem has to do with energy positivity (fermions) and particle-number positivity (bosons).

675 Feynman diagrams without closed loops reproduce classical theory.

679 If 10^-40 is "almost unimaginable", how much less imaginable is "infinity"?

680 Feynman graphs represent a perturbation expansion of a Lagrangian.

681 "Roughly speaking, the symmetry is used in order to ensure that certain divergent terms cancel each other out..."

689 Simplest form of 2nd Law of Thermodynamics; I agree with it in this form.

690 Temperature = energy per degree of freedom

690 Boltzmann's notion of entropy

691 My concerns re entropy:

...1. Consciousness is required for a meaningful interpretation of 'indistinguishability',

...2. What makes V constant over all phase space given a particular discriminating observer?

...3. The boundaries of the boxes seem to be arbitrary and not unique even for a single discriminating observer.

...4. I am not convinced that S is constant for different degrees of coarseness (since the Vs would be different).

692 Penrose: "My own position concerning the physical status of entropy is that I do not see it as an 'absolute' notion in present-day physical theory, although it is certainly a very useful one." and "In my view, entropy has the status of a 'convenience', in present-day theory, rather than being 'fundamental'"

693 The answer to problem [27.5] might clear up doubts I have about this assertion

700 How about conscious attention traversing phase space rather than individual world lines in space-time?

718 I don't think the FLRW model is completely "spatially homogeneous and isotropic". If you consider 4D space-time, the time-like directions are qualitatively different from the space-like directions. It seems to me that it might be fruitful to posit additional anisotropy even in spatial directions where the directions on our 3D spatial manifold are qualitatively different from directions outside the manifold, to wit, they are inaccessible to observation by 3D instruments. The same goes for spatial homogeneity. See note 416.

719 The various cosmological models

720 "...decoupling [is] the time that we effectively 'look back to' when we observe the microwave background."

722 This seems to imply that there actually are spatial points outside the universe, to wit: the center of the "balloon". I would think we can conclude that large, extra, nearly flat spatial dimensions exist.

738 "...superconductivity [is] a phase transition which accompanies the symmetry reduction that breaks the ordinary U(1) symmetry of electromagnetism."

744 "The issue I wish to raise is whether there can ever be 'enough time' in the case of eletroweak spontaneous symmetry breaking." Hmmmmm. "enough time"?

746 Penrose doubts the fundamental role of symmetry.

753 Penrose doubts inflationary cosmology.

754 The inflationary model claims to obviate the "fine tuning" that the standard model requires to solve the "horizon problem", the "flatness problem", and the "smoothness problem".

754 Inflation is fashionable.

755 Entropy increases the difficulty of explaining the specialness of the universe in the inflationary model.

757 Good explanation of "the real problem" of inflationary cosmology explaining the specialness of the early universe.

757 There is an apparently coincidental relation between the age of the universe and the ratio of strengths of electro-magnetism and gravity.

759 Hoyle's prediction of a specific energy level of carbon.

760 Penrose doesn't like the use of the Strong Anthropic Principle.

761 Lee Smolin's idea of cosmic evolution with competing universes born from parent's black holes.

764 Odds of random collisions producing our solar system including life = 1/10^10^60!

769 "...despite over fifty years of determined efforts to bring general relativity and quantum mechanics together, there is still nothing that even approaches a consensus as to the correct approach to the subject."

769 Wick Rotation and Euclideanization. Is this anywhere close to Dr. Dick's approach?

777 "...the cosmological problem that, in my opinion, overshadows all others, [is] the extraordinarily 'special' Big Bang -- to at least the degree of a part in 10^10^123 -- which underlies the Second Law."

782 Mysteries solved by QM

783 Copenhagen interpretation: state vector = experimenter's knowledge, quantum jump = a "jump" or increase in that knowledge.

784 Everett's many-world's interpretation is consistent with my view that PC (now CC) deliberately makes all choices determining the outcomes of reality.

786 Six ontologies of quantum theory

787 A projector is an operator that squares to itself and is self-adjoint. Orthogonality and completeness of a projector set.

788 Penrose's musings that the role of the insertion of a projector set in the 'consistent histories' ontology could be to "refine" the history rather than change the world. In my view, these are the points at which PC (now CC) observes the evolution in order to know what is going on so it can make deliberate choices.

788 Penrose's guess at the ontology of 'consistent histories'.

789 The Bohmian (pilot wave) ontology seems to fit with my idea of PC (now CC) constructing and playing "virtual reality games". "Position" would be an address in a memory substrate at some level.

790 Unitary quantum evolution and quantum jumps are mathematically inconsistent!

791 Definition of 'density matrix'.

793 "...in quantum mechanics, a measurement [is] the form of question posed to a quantum system -- and let us restrict attention to a YES/NO question -- is phrased in terms of the action of some projector E applied to the (normalized) state vector..."

793 "...we do not need to know the complete information of the distribution of probabilities for the alternative states ... in order to be able to calculate probabilities for a standard YES/NO question in quantum mechanics (or, indeed, for the expectation value of any other quantum-mechanical observable)"

793 The density matrix combines classical probabilities with quantum probabilities without directly distinguishing between them. John von Neumann figured this out.

793 "simple" example of a density matrix

794 The ontological problem of probability mixtures of states is typically confused by quantum physicists.

795 If the density matrix best describes quantum reality, then it seems physical reality may indeed by implemented as a VR game!

795 Different usages of the term 'state'.

796 Bloch sphere and ambiguous ontologies

800 "Orthogonality is not a requirement for the probability mixture of states composing a density matrix"

800 "there is no unique ontology of 'probability-weighted alternative states' whatever density matrix is used."

802 "the environmental-decoherence viewpoint...maintains that state reduction **R** can be understood as coming about because the quantum system under consideration becomes inextricably entangled with its environment."

802 "any density matrix is diagonal in *some* basis!"

803 "any density matrix has a host of ontological interpretations."

806 "if we take the stand that **U** fails for conscious beings, then we are driven to a version of (f) [new theory with objective **R**] according to which some new type of behaviour, outside the ordinary predictions of quantum mechanics, comes into play with beings who possess consciousness."

808 "what [is it] about a quantum state that allows it to be considered as a 'perception'[?]"

809 "we still need a theory of perception,...and such a theory is lacking."

809 My "PC" idea would explain why macroscopic superpositions are not perceived, but not why the squared modulus rule is so precise.

809 The "ontological stuff" he talks about could be explained as a shift or a focusing of PC's attention.

810 "The requirement of 'consistency' for (maximally refined) coarse-grained histories appears to be a long way from what is needed in order to provide a model for observed physical reality."

811 The DeBroglie-Bohm 'pilot wave' viewpoint has two levels of reality: the firmer "particle" level and the less firm "wave" level. Might there not be several more levels all the way to PC at the ultimate level?

811 The solution to Penrose's "difficulty" with "big" and "small" might be analogous to the difference in driving a car with the "big" decisions regarding navigation and the "small" nearly reflexive decisions regarding keeping the car on the road.

811 "ordinary (deterministic) dynamics alone can never achieve [the derivation of the apparent occurrence of **R** from the dynamics of (say) **U**, -- as is clear, if only for the reason that there are no probabilities in such a dynamical equation as the Schroedinger equation."

812 A list of difficulties with theories for an "objective **R**"

813 The notion that consistency (decoherence) is equivalent to obeying the laws of probability reminds me of Dick's work.

818 "It seems to me to be clear that the mystery of the extraordinarily special nature of the Big Bang cannot be resolved within the standard framework of quantum field theory."

818 The CPT Theorem applies to flat Minkowski space, not to the curved spacetime of GR.

825 Could the problem of finding an appropriate notion of 'positive frequency' in a curved background and a naturally defined 'time parameter' be the consequence of PC operating in a separate hyper-temporal dimension?

825 It seems to me that PC's attention might track along the integral curves of a Killing vector.

826 Thermal vacuum and 'acceleration temperature'

828 Complexification and imaginary time

829 Periodicity in imaginary time.

829 By looping and closing the axes in (b), (see my drawing in the margin) you get two antipodal (b)s and four (a)s, a closed finite structure that suggests a memory mechanism for PC.

844 Penrose suspects that the loss of degrees of freedom in black hole formation is required in the quantum **R** process.

849 Different Killing vectors lead to QM/GR clash.

851 The "problem of time" in quantum cosmology is the inability to specify Schroedinger's partial derivative operator wrt time!

860 Penrose is convinced that QM has no credible ontology.

861 John Wheeler suggested quantum foam in the 1950s.

861 "Heisenberg uncertainty restricts the precision whereby two non-commuting measurements can be carried out."

862 **U**-evolution can't give rise to 'quantum fluctuations so it must be **R**.

863 How can we be sure there were no experimenters around at the Big Bang?

868 "On Wheeler's own variant, the 'participatory universe', ... it would be the ultimate presence of conscious observers who somehow (teleologically) determine the particular selection of spacetime geometry that occurred in the early universe."

869 A list of general approaches to 21st Century physics.

870 Good summary of the renormalizability of the Standard Model and of the problem of constants like mass and charge values.

870 The fine structure constant -- Eddington

872 There is some hope for a finite theory

873 Penrose is unconvinced of the relevance of supergravity.

873 "bosons satisfy commutation laws, whereas fermions satisfy anticommutation laws."

877 See if a super-algebra can be defined for Practical Numbers.

879 "body" and "soul" referring to a manifold and a hypermanifold.

880 Eleven dimensional space-time

880 "How is it that physicists could take seriously the possibility that the dimensionality of spacetime might be other than the four that we directly experience (one time and three space)?" I think the answer is straightforward!

881 String theory has considered 26 and 10 dimensions.

881 Kaluza/Klein small dimensions. Penrose seems to buy into the idea that extra dimensions should be detectable!

881 Fallacious argument: If the "being" inhabited the hosepipe, it would know about the extra dimension. If it is "at a great distance" then it is not inhabiting the hosepipe.

882 Erroneous (IMHO) opinion that large extra dimensions would change inverse square laws (my "sheep" analogy counters this.)

882 Penrose: extra dimensions is a 'cute idea'.

882 Penrose's dismissal of Kaluza-Klein theory, "elegant as it is".

888 String theory has a history of diminishing observational support.

893 Renormalizability = finiteness is crucial for QFT theorists; Deep QM/GR conflicts are crucial for theorists like Penrose coming from a GR perspective.

893 "Most people in the relativity community have the expectation that the true 'quantum geometry' should take on some elements of discreteness, or should at least differ profoundly from the classical smooth-manifold picture."

894 "the condition for a *vacuum* in ordinary Einsteinian gravity is Ricci flatness"

895 "the string Lagrangian is defined by 1/2(alpha prime) multiplied by the surface area of the 2-surface history -- the *world sheet* -- that the string traces out in spacetime.'

896 Why Ed Witten thinks string theory predicts gravity.

897 Why don't we see the extra degrees of freedom in 10 dimensions? I say it is structural like shadows or sheep in a pasture. The extra freedom on the higher, inaccessible, dimensions provides the possibility for hyper complex structures that could be responsible for such things as consciousness, free will, ID, etc.

902 Penrose is puzzled at why our 3 dimensions behave so very differently from the other six. I say it's for the same reason the two dimensions of a movie screen behave so differently from the third.

905 The expected instability of extra dimensions depends on the assumption that the extra dimensions are of Planck size. I suggest waiving this and assume the extra dimensions are large, compact, and Ricci flat.

906 Penrose-Hawking Singularity Theorem.

906 "singularity theorems...[establish] that there is an obstruction of some sort to timelike or null geodesics being extendable within the spacetime to infinite length (or to infinite affine length, in the case of null geodesics)"

906 "...then we must expect the singularities...to occur in a comparable timescale..." If we don't assume that the extra dimensions are "small" then the timescale of the singularity could be as long as we like. I think these "obstructions" to the time evolution are points at which the conscious intervention of deliberate choice of one quantum outcome is necessary in order for the time evolution to proceed. As when an artist pauses to look at the unfinished painting in order to decide how and where to apply the next brush stroke. A submanifold of the entire hyper spacetime evolves in this manner.

907 I don't think that the perturbations necessarily need spill out into ordinary space. Perturbations in a shadow don't spill out into the room,. I see no "gross conflict with observation".

910 Calabi-Yau spaces

912 Heterotic strings don't make sense to Penrose, although they make formal sense. They make sense to me. I think they could explain why some dimensions are accessible to us while others are not.

912 Deep symmetries among various string theories are analogous to the duality between electricity and magnetism. I think this idea is on the right track.

913 Counting rational curves; mirror symmetry: complex structure <--> symplectic structure; Ellingstrud and Stromme

914 Ed Witten's M-Theory. I think this is exactly on track. I think that which dimensions are noticeable and accessible varies depending on the observer's manifold (frame) or reference. Thus from different hypothetical perspectives, you get different string theories.

915 M-Theory is 11-dimensional; F-theory is 12-dimensional with two time dimensions. I think that whether a dimension is temporal or spatial depends on the observer. If it is traversing, it seems temporal; if just observing, it seems spatial.

916 Penrose sees a problem with extra dimensions having to do with "vast differences in functional freedom". I say consider shadows.

916 Strominger and Vafa's calculation used 5D black holes according to Greene.

920` The holographic 'principle' - or conjecture

920 Maldacenna (ADS/CFT) conjecture is holographic.

922 Penrose seems to disagree with Witten by expecting that there should be symmetry among all dimensions. I agree with Witten.

922 To explain this "discrepancy" why is it not reasonable to suppose that the path integrals are to be taken in some but not all dimensions? Just as paths to the post office are confined to the earth's surface and don't include those that are underground.

923 Penrose's "enormous increase in functional freedom" objection to extra dimensions. I say, acknowledge that extra freedom. It implies enormously complex structures existing and acting outside our 4D manifold. The asymmetry among dimensions allows only some manifestations of those actions to be detectable in our manifold ~ shadows.

924 D-branes explain the inaccessibility of extra dimensions the same way I do.

925 Penrose says that a restriction of freedom within the D-brane is in general "hardly likely", but he doesn't say why. I think it is likely.

925 The "hierarchy problem" - huge discrepancies in masses and forces.

927 A discussion of Ed Witten

928 Penrose suspects that the successes of string theory are nothing more than mathematical formalism and may have nothing to do with physics.

931 A hypertorus avoids some of the problems Penrose raises. I think a hypertorus might indeed be the topology of greater reality.

938 Penrose: "...we may take the position that there is something deeply left/right-asymmetrical about Nature..." I suspect that the asymmetry occurs only in higher dimensions but not in our 4D manifold (except for electro-weak interactions). Of course, there is the asymmetry between the temporal and spatial dimensions, although it has nothing to do with chirality.

939 Ashtekar's 'new variables' approach seems to be the same idea I have as that of 'real' structures in extra dimensions manifesting noticeable (to us) effects in our 4D spacetime.

946 "Most of known physics depends on the non-triviality of [field] equations in order that fields propagate into the future in a controlled manner."

947 Penrose studied discrete systems - spin networks - in the 1950s. He favored discreteness over continuity then.

947 Mach's principle: "Everything was to be expressed in terms of the *relation* between objects, and not between an object and some background space."

950 "notions of ordinary Euclidean geometry are seen to arise merely from the quantum combinatorics of spin networks."

950 LQG

952 Rovelli, Ashtekar, Smolin; LQG; Syracuse and Penn State

954 The 'problem of time' or of 'frozen time' in QG. Penrose suspects it requires state reduction to resolve - I suspect it requires conscious involvement.

954 "although the spin-network basis states individually have a pleasing coordinate-independent geometric description, it is most unclear how to interpret quantum *superpositions* of such basis states." I suggest superpositions are alternative states not yet chosen by consciousness.

954 "How are we expected to understand how an almost classical world is to emerge out of all this?" I say, similar to the way in which we understand how a painting emerges.

960 'Regge calculus' looks like CDT (Causal Dynamical Triangulation)

960 Category Theory: 'arrows' connecting 'objects'

963 Spin networks are indifferent to time flow.

974 Relation between the number of space dimensions and the number of time dimensions.

974 The matrix relation describing flat-space twister geometry

978 In twistor space, "Ordinary spacetime points are represented as Riemann spheres in [the space of null projective twistors]. Points of [this space] are represented as light rays in spacetime."

980 Penrose's question suggests a conflation of 'real' in the mathematical real/complex sense with 'real' in the sense of being physically accessible.

1030 "We must bear in mind that each 'world' possesses its own distinctive kind of existence, different from that of the other two."

1030 "Any universe that can 'be observed' must, as a logical necessity, be capable of supporting conscious mentality, since consciousness is precisely what plays the ultimate role of 'observer'."

1031 "the behaviour of the seemingly objective world that is actually perceived depends upon how one's consciousness threads its way through the myriads of quantum-superposed alternatives."

1032 "I envisage that the phenomenon of consciousness -- which I take to be a *real* physical process, arising 'out there' in the physical world -- fundamentally *makes use* of the actual **OR** process."

1033 "it seems to me that a 'fundamental' physical theory that lays claim to any kind of completeness at the deepest levels of physical phenomena must also have the potential to accommodate conscious mentality."

1034 A list of math subjects useful in describing the world.

1036 Penrose assumes that "actual spacetime" has a "real-number structure".

1039 "In my opinion, [Cantor's theory] is one of the most profoundly beautiful mathematical contributions in the whole of mathematical history."

1039 "extraordinarily little of [Cantor's theory] seems to have relevance to the workings of the physical world as we know it."

1040 The "miraculous" cancellation of divergent terms in supersymmetry failed at higher orders. Penrose hints that higher dimensions might be indicated, but that the work "stagnated". I think this is an avenue worth pursuing.

1044 "it is the weak force that is, through radioactive decay in the Earth's interior, largely responsible for the heating of the Earth's magma."

1052 Link to 4D visualization website

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