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Register a new userExplicit Baker-Campbell-Hausdorff-Dynkin formula for Spacetime via Geometric Algebra
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Joseph Wilson
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We present a compact Baker-Campbell-Hausdorff-Dynkin formula for the composition of Lorentz transformations $e^{sigma_i}$ in the spin representation (a.k.a. Lorentz rotors) in terms of their generators $sigma_i$: $$ ln(e^{sigma_1}e^{sigma_2}) = tanh^{-1}left(frac{ tanh sigma_1 + tanh sigma_2 + frac12[tanh sigma_1, tanh sigma_2] }{ 1 + frac12{tanh sigma_1, tanh sigma_2} }right) $$ This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension $leq 4$, naturally generalising Rodrigues formula for rotations in $mathbb{R}^3$. In particular, it applies to Lorentz rotors within the framework of Hestenes spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex $2times2$ matrix representation realised by the Pauli spin matrices. The formula is applied to the composition of relativistic $3$-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.
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The well-known Baker-Campbell-Hausdorff theorem in Lie theory says that the logarithm of a noncommutative product e X e Y can be expressed in terms of iterated commutators of X and Y. This paper provides a gentle introduction t{o} Ecalles mould calculus and shows how it allows for a short proof of the above result, together with the classical Dynkin explicit formula [Dy47] for the logarithm, as well as another formula recently obtained by T. Kimura [Ki17] for the product of exponentials itself. We also analyse the relation between the two formulas and indicate their mould calculus generalization to a product of more exponentials.
There are two classes of topologies most often placed on the space of Lorentz metrics on a fixed manifold. As I interpret a complaint of R. Geroch [Relativity, 259 (1970); Gen. Rel. Grav., 2, 61 (1971)], however, neither of these standard classes correctly captures a notion of global spacetime similarity. In particular, Geroch presents examples to illustrate that one, the compact-open topologies, in general seems to be too coarse, while another, the open (Whitney) topologies, in general seems to be too fine. After elaborating further the mathematical and physical reasons for these failures, I then construct a topology that succeeds in capturing a notion of global spacetime similarity and investigate some of its mathematical and physical properties.
A general formula is calculated for the connection of a central metric w.r.t. a noncommutative spacetime of Lie-algebraic type. This is done by using the framework of linear connections on central bi-modules. The general formula is further on used to calculate the corresponding Riemann tensor and prove the corresponding Bianchi identities and certain symmetries that are essential to obtain a symmetric and divergenceless Einstein Tensor. In particular, the obtained Einstein Tensor is not equivalent to the sum of the noncommutative Riemann tensor and scalar, as in the commutative case, but in addition a traceless term appears.
We give an upper bound of the relative entanglement entropy of the ground state of a massive Dirac-Majorana field across two widely separated regions $A$ and $B$ in a static slice of an ultrastatic Lorentzian spacetime. Our bound decays exponentially in $dist (A, B)$, at a rate set by the Compton wavelength and the spatial scalar curvature. The physical interpretation our result is that, on a manifold with positive spatial scalar curvature, one cannot use the entanglement of the vacuum state to teleport one classical bit from $A$ to $B$ if their distance is of the order of the maximum of the curvature radius and the Compton wave length or greater.
We build a family of explicit one-forms on $S^3$ which are shown to form a complete set of eigenmodes for the Laplace-de Rahm operator.
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