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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 10221, 272]*) (*NotebookOutlinePosition[ 39590, 1339]*) (* CellTagsIndexPosition[ 39546, 1335]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[StyleBox[ "Rules for multiplying quaternion exponentials"]], "Subtitle", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times New Roman"], Cell[TextData[StyleBox["doug "]], "Subsubtitle", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->16], Cell["\<\ Introduction New method for multiplying quaternion exponentials Testing the proposal Implications\ \>", "Text", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}], Cell[TextData[StyleBox["Introduction"]], "Subsection", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}], Cell[TextData[StyleBox[ "Multiplying two exponentials is at the heart of modern analysis, whether one \ works with Fourier transforms or Lie groups. Given a Lie algebra of a Lie \ group in a sufficiently small area the identity, the product of two \ exponentials can be defined using the Campbell-Hausdorff formula:"]], "Text", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ \(Exp[X] Exp[y]\ = \ \((X\ + \ Y)\)\ + \ \(1\/2\)[X, \ Y] \((X\ + \ Y)\)\ \n + \ \(1\/12\) \((\([\([X, \ Y]\), Y]\)\ - \ \([\([X, \ Y]\), X]\))\) \((X\ + \ Y)\)\ + \ ...\)], "Input", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, Evaluatable->False, FontSize->16], Cell[TextData[StyleBox[ "This formula is not easy to use, and is only applicable in a small area \ around unity. Quaternion analysis that relies on this formula would be very \ limited."]], "Text", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}], Cell[TextData[StyleBox[ "New method for multiplying quaternion exponentials"]], "Subsection", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}], Cell[TextData[StyleBox[ "I have discovered (perhaps for the first time) a simpler and general way to \ express the product of two quaternion exponentials as the sum of two \ components. The product of two quaternions splits into a commuting and an \ anti-commuting part. The rules for multiplying commuting quaternions are \ identical to those for complex numbers. The anticommuting part needs to be \ purely imaginary. The Grassman product (q q') of two quaternion exponentials \ and the Euclidean product (q* q') should both have these properties. \ Together these define the needs for the product of two quaternion \ exponentials."]], "Text", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}], Cell[BoxData[{ \(Let\ q = \ \(Exp[X]\ \ \ q'\ = \ Exp[Y]\)\n\t\), \(q\ q'\ = \ {q, \ q'}\/2\ + \ \(Abs[q, \ q']\/2\) Exp[\[Pi]\/2\ \([q, \ q']\)\/Abs[q, \ q']]\ \), \(\twhere\ \ \ {q, \ q'}\ = \ \(q\ q'\ + \ q\^\('*\)\ \ \(q\^*\)\ \ \ and\ \ [q, \ q']\ = \ q\ q'\ - \ q\^\('*\)\ \ \(q\^*\)\)\ \n\), \(\(q\^*\) q'\ = \ same\ as\ above\), \(\twhere\ \ \ {q, \ q'}\ = \ \(\(q\^*\)\ q'\ + \ q\^\('*\)\ \ q\ \ \ and\ \ [q, \ q']\ = \ \(q\^*\)\ q'\ - \ q\^\('*\)\ \ q\)\ \ \)}], "Input", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, Evaluatable->False, FontSize->16], Cell[TextData[StyleBox[ "Changing the commutator changes the type of product of quaternion \ exponentials. Andrew Millard made the suggestion for the Grassman product \ that unifies these approaches nicely. This may have an interesting \ philosophical impact, but for now, let's just check the math :-)"]], "Text", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}], Cell[TextData[StyleBox["Testing the proposal"]], "Subsection", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}], Cell[TextData[{ StyleBox["Define a few "], StyleBox["Mathematica", FontSlant->"Italic"], StyleBox[" functions to test these proposals."] }], "Text", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}], Cell[BoxData[ \(expProduct[comm_, \ anti_]\ := \n\t anti/2\ + \ abs[comm]\ qexp[\[Pi]\/2\ comm\ /\ abs[comm]]/2\)], "Input",\ CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(commGrassman[q1_, \ q2_]\ := \n\t q1\ .q2\ - \ Transpose[q2].Transpose[q1]\)], "Input", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(antiGrassman[q1_, \ q2_]\ := \n\t q1.q2 + Transpose[q2].Transpose[q1]\)], "Input", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(commEuclid[q1_, \ q2_]\ := \n\t Transpose[q1]\ .q2\ - \ Transpose[q2].q1\)], "Input", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(antiEuclid[q1_, \ q2_]\ := \n\t Transpose[q1]\ .q2\ + \ Transpose[q2].q1\)], "Input", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(A\ = \ q[2, 1, 3, 4]; \nB\ = \ q[1, 2, 3, 5]; \)], "Input", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16], Cell[TextData[StyleBox["Without the new tools, find the products."]], "Text", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(N[\((qexp[A].qexp[B])\).{1, 0, 0, 0}]\)], "Input", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \({5.34631928617794294`, \(-3.70340669789332554`\), \(-11.0942475386486072`\), \(-15.4288079242748015`\)}\)], "Output", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[\((Transpose[qexp[A]].qexp[B])\).{1, 0, 0, 0}]\)], "Input", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \({9.69426691278217056`, 3.12108445764449982`, 10.2207641782753677`, 13.9730023236527367`}\)], "Output", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16] }, Open ]], Cell[TextData[StyleBox["Test the proposal."]], "Text", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(N[expProduct[commGrassman[qexp[A], qexp[B]], \n \t\t\t\t\t\t\t\t\t\t\t\t\tantiGrassman[qexp[A], qexp[B]]]].{1, 0, 0, 0}\)], "Input", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \({5.34631928617794383`, \(-3.70340669789332554`\), \(-11.0942475386486072`\), \(-15.4288079242748015`\)}\)], "Output", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[expProduct[commEuclid[qexp[A], qexp[B]], \n\t\t\t\t\t\t\t\t\t\t\t\t\t antiEuclid[qexp[A], qexp[B]]]].{1, 0, 0, 0}\)], "Input", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \({9.69426691278217234`, 3.12108445764449982`, 10.2207641782753677`, 13.9730023236527367`}\)], "Output", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}, FontSize->16] }, Open ]], Cell[TextData[StyleBox["It works!"]], "Text", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}], Cell[TextData[StyleBox["Implications"]], "Subsection", CellMargins->{{Inherited, 42}, {Inherited, Inherited}}], Cell[TextData[StyleBox[ "In terms of mathematics, a few new identities for multiplying quaternion \ exponentials were established. These might be part of an effort to do \ Fourier analysis with quaternions. That analysis would be applicable to both \ Grassman and Euclidean products, since work with the former can be \ transformed into the latter by changing the commutator relations. This may \ have the greatest impact for physics. Quantum mechanics has grown to very \ high levels of sophistication due in part to the norm being definite in a \ Hilbert space. General relativity relies on the indefinite Minkowski metric, \ and that leads to technical problems with non-compact groups. Doing physics \ with quaternions does not even use metrics! It may become transparent to \ bounce between Hilbert spaces where the scalar of squaring a quaternion is a \ norm, and spacetime where the scalar of squaring a quaternion is an interval. \ Prove something in quantum mechanics, then apply it to spacetime. 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If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", Background->GrayLevel[1]], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{1, -5}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }, Closed]] }, Open ]] }] ] (*********************************************************************** Cached data follows. 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