Grassmann angle formulas and identities

Grassmann angles improve upon similar concepts of angle between subspaces that measure volume contraction in orthogonal projections, working for real or complex subspaces, and being more efficient when dimensions are different. Their relations with contractions, inner and exterior products of multivectors are used to obtain formulas for computing these or similar angles in terms of arbitrary bases, and various identities for the angles with certain families of subspaces. These include generalizations of the Pythagorean trigonometric identity $\cos^2\theta+\sin^2\theta=1$ for high dimensional and complex subspaces, which are connected to generalized Pythagorean theorems for volumes, quantum probabilities and Clifford geometric product.Comment: arXiv admin note: text overlap with arXiv:1910.0732

Quantum fractionalism: the Born rule as a consequence of the complex Pythagorean theorem

Everettian Quantum Mechanics, or the Many Worlds Interpretation, lacks an explanation for quantum probabilities. We show that the values given by the Born rule equal projection factors, describing the contraction of Lebesgue measures in orthogonal projections from the complex line of a quantum state to eigenspaces of an observable. Unit total probability corresponds to a complex Pythagorean theorem: the measure of a subset of the complex line is the sum of the measures of its projections on all eigenspaces. To show that projection factors can work as probabilities, we postulate the existence of a continuum infinity of identical quantum universes, all with the same quasi-classical worlds. In a measurement, these factors give the relative amounts of worlds with each result, which we associate to frequentist and Bayesian probabilities. This solves the probability problem of Everett's theory, allowing its preferred basis problem to be solved as well, and may help settle questions about the nature of probability

Blade products in Grassmann and Clifford algebras

Formulas relating dot and cross products of vectors to their angle are generalized for products of real or complex blades (simple multivectors). Their inner product and contraction are related to a Grassmann angle, the exterior product to a complementary Grassmann angle, and products of Clifford geometric algebra are also linked to such angles. We interpret geometrically the Clifford product of blades and some of its properties, relate it to an angle bivector, and give a formula for the exponential of any bivector in terms of Grassmann angles. The relations between such angles and blade products cast new light on both and reveal new properties.Comment: The manuscript was reorganized for clarity, and new results were include

Analysis of Wallace's Proof of the Born Rule in Everettian Quantum Mechanics: Formal Aspects

To solve the probability problem of the Many Worlds Interpretation of Quantum Mechanics, D.Wallace has presented a formal proof of the Born rule via decision theory, as proposed by D.Deutsch. The idea is to get subjective probabilities from rational decisions related to quantum measurements, showing the non-probabilistic parts of the quantum formalism, plus some rational constraints, ensure the squared modulus of quantum amplitudes play the role of such probabilities. We provide a new presentation of Wallace's proof, reorganized to simplify some arguments, and analyze it from a formal perspective. Similarities with classical decision theory are made explicit, to clarify its structure and main ideas. A simpler notation is used, and details are filled in, making it easier to follow and verify. Some problems have been identified, and we suggest possible corrections.Comment: In comparison with its previous version arXiv:1504.05259v2 [quant-ph], this article has been changed to focus only on the formal aspects of the proo

Everettian Decoherent Histories and Causal Histories

D. Wallace has tried to use decoherence to solve the preferred basis problem of Everettian Quantum Mechanics, and this solution lays the foundation for his proof of the Born rule. But this is a circular argument, as approximations used in decoherence usually rely on the probabilistic interpretation of the Hilbert space norm. He claims the norm can measure approximations even without probabilities, but this assumption has not been properly justified. Without it, the combination of the Everettian and decoherent histories formalisms leads to strange consequences, such as a proliferation of small amplitude histories with lots of macroscopic quantum jumps. Still, this erratic behavior may provide a way to justify the approximations, in a new histories formalism, in which macroscopic causal relations play a central role. Small histories, suffering too much interference, may lose causality, being thus discarded as invalid. The remaining branches can present some small interference, opening the possibility of experimental verification

SUB--DEGREE ANISOTROPY OBSERVATIONS: GROUND--BASED, BALLOON--BORNE AND SPACE EXPERIMENTS

Extensive, accurate imaging of the Cosmic Background Radiation temperature anisotropy at sub--degree angular resolution is widely recognized as one of the most crucial goals for cosmology and astroparticle physics in the next decade. We review the scientific case for such measurements in relation with sky coverage, attainable sensitivity, confusing foreground radiation components, and experimental strategies. Although ground--based and balloon--borne experiments will provide valuable results, only a well--designed, far--Earth orbit space mission covering a wide spectral range and a significant part of all the sky will provide decisive answers on the mechanism of structure formation.Comment: Astrophys. Lett \& Comm., in press. Tex file, 20 pages + 3 figures (file1.PS, file2.PS, file3.PS) appended

From WMAP to Planck: Exact reconstruction of 4- and 5-dimensional inflationary potential from high precision CMB measurements

We make a more general determination of the inflationary observables in the standard 4-D and 5-D single-field inflationary scenarios, by the exact reconstruction of the dynamics of the inflation potential during the observable inflation with minimal number of assumptions: the computation does not assume the slow-roll approximation and is valid in all regimes if the field is monotonically rolling down its potential. Making use of the {\em Hamilton-Jacobi} formalism developed for the 5-D single-field inflation model,we compute the scale dependence of the amplitudes of the scalarand tensor perturbations by integrating the exact mode equation. We analyze the implications of the theoretical uncertainty in the determination of the reheating temperature after inflation on the observable predictions of inflation and evaluate its impact on the degeneracy of the standard inflation consistency relation.Comment: 30 pages and 7 figures processed with LATEX macros v5.2 accepted for publication in Astrophysical Journa

Polarisation as a tracer of CMB anomalies: Planck results and future forecasts

The lack of power anomaly is an intriguing feature at the largest angular scales of the CMB anisotropy temperature pattern, whose statistical significance is not strong enough to claim any new physics beyond the standard cosmological model. We revisit the former statement by also considering polarisation data. We propose a new one-dimensional estimator which takes jointly into account the information contained in the TT, TE and EE CMB spectra. By employing this estimator on Planck 2015 low-$\ell$ data, we find that a random $\Lambda$CDM realisation is statistically accepted at the level of $3.68 \%$. Even though Planck polarisation contributes a mere $4 \%$ to the total information budget, its use pushes the lower-tail-probability down from the $7.22 \%$ obtained with only temperature data. Forecasts of future CMB polarised measurements, as e.g. the LiteBIRD satellite, can increase the polarisation contribution up to $6$ times with respect to Planck at low-$\ell$. We argue that the large-scale E-mode polarisation may play an important role in analysing CMB temperature anomalies with future mission.Comment: 24 pages, 9 figures. Figures simplified, appendix added. Final version to appear in Physics of the Dark Univers

Grassmann angles between real or complex subspaces

The Grassmann angle improves upon similar angles between subspaces that measure volume contraction in orthogonal projections. It works in real or complex spaces, with important differences, and is asymmetric, what makes it more efficient when dimensions are distinct. It can be seen as an angle in Grassmann algebra, being related to its products and those of Clifford algebra, and gives the Fubini-Study metric on Grassmannians, an asymmetric metric on the full Grassmannian, and Hausdorff distances between full sub-Grassmannians. We give formulas for computing it in arbitrary bases, and identities for angles with certain families of subspaces, some of which are linked to real and complex Pythagorean theorems for volumes and quantum probabilities. Unusual features of the angle with an orthogonal complement, or the angle in complex spaces, are examined.Comment: The manuscript has been merged with arXiv:2005.12700. It was reorganized and simplified, and some new results were include

Analysis of Wallace's Proof of the Born Rule in Everettian Quantum Mechanics II: Concepts and Axioms

Having analyzed the formal aspects of Wallace's proof of the Born rule, we now discuss the concepts and axioms upon which it is built. Justification for most axioms is shown to be problematic, and at times contradictory. Some of the problems are caused by ambiguities in the concepts used. We conclude the axioms are not reasonable enough to be taken as mandates of rationality in Everettian Quantum Mechanics. This invalidates the interpretation of Wallace's result as meaning it would be rational for Everettian agents to decide using the Born rule