(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 12638, 326]*) (*NotebookOutlinePosition[ 42317, 1395]*) (* CellTagsIndexPosition[ 42246, 1389]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Inner and outer products of quaternions", "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["doug "]], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Introduction\nThe Grassman inner and outer products\nThe Euclidean inner and \ outer products\nImplications"]], "Text"], Cell[CellGroupData[{ Cell[" Introduction", "Subsection"], Cell["\<\ A good friend of mine has wondered what is means to multiply two \ quaternions together (this question was a hot topic in the nineteenth \ century). I care more about what multiplying two quaternions together can \ do. There are two basic ways to do this: just multiply one quaternion by \ another, or first take the transpose of one then multiply it with the other. \ Each of these products can be separated into two parts: a symmetric (inner \ product) and an antisymmetric (outer product) components. The symmetric \ component will remain unchanged by exchanging the places of the quaternions, \ while the antisymmetric component will change its sign. Together they add up \ to the product. In this notebooks, both types of inner and outer products \ will be formed and then related to physics.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[" The Grassman inner and outer products", "Subsection"], Cell["\<\ There are two basic ways to multiply quaternions together. There \ is the direct approach.\ \>", "Text"], Cell[BoxData[{ \(\((q[t\_1, \ x\_1, y\_1, z\_1].q[t\_2, \ x\_2, y\_2, z\_2])\).{1, 0, 0, 0}\), \({t\_1\ t\_2 - x\_1\ x\_2 - y\_1\ y\_2 - z\_1\ z\_2, \ \ t\_2\ x\_1 + t\_1\ x\_2 - y\_2\ z\_1 + y\_1\ z\_2, \n\t t\_2\ y\_1 + t\_1\ y\_2 + x\_2\ z\_1 - x\_1\ z\_2, \(-x\_2\)\ y\_1 + x\_1\ y\_2 + t\_2\ z\_1 + t\_1\ z\_2}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\((t, \(X\& \[RightVector] \))\) \((t\^\[Prime], \(X\& \[RightVector] \)\^\[Prime])\) = \((t\ t\^\[Prime]\ - \(X\& \[RightVector] \).\(X\& \[RightVector] \)\^\[Prime], t \( X\& \[RightVector] \)\^\[Prime] + \(X\& \[RightVector] \) t\^\[Prime] + \(X\& \[RightVector] \) x \( X\& \[RightVector] \)\^\[Prime])\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ I call this the Grassman product (I don't know if anyone else does, but I \ need a label). The inner product can also be called the symmetric product, \ because it does not change signs if the terms are reversed.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(Simplify[\n \((\(q[t\_1, \ x\_1, y\_1, z\_1]\ .\ q[t\_2, \ x\_2, y\_2, z\_2]\ + \ \nq[t\_2, \ x\_2, y\_2, z\_2]\ .\ q[t\_1, \ x\_1, y\_1, z\_1]\)\/2)\).{1, 0, 0, 0}]\), \({t\_1\ t\_2 - x\_1\ x\_2 - y\_1\ y\_2 - z\_1\ z\_2, \n\t t\_2\ x\_1 + t\_1\ x\_2, t\_2\ y\_1 + t\_1\ y\_2, t\_2\ z\_1 + t\_1\ z\_2}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ StyleBox["{", FontWeight->"Bold"], \(\((t, \(X\& \[RightVector] \))\), \((t\^\[Prime], \(X\& \[RightVector] \)\^\[Prime])\)\), StyleBox["}", FontWeight->"Bold"]}], StyleBox[" ", FontWeight->"Bold"], "\[Congruent]", \(\(\((t, \(X\& \[RightVector] \))\) \((t\^\[Prime], \(X\& \[RightVector] \)\^\[Prime])\) + \ \((t\^\[Prime], \(X\& \[RightVector] \)\^\[Prime])\) \((t, \(X\& \[RightVector] \))\)\)\/2\)}], "\n", "=", \((t\ t\^\[Prime]\ - \(X\& \[RightVector] \).\(X\& \[RightVector] \)\^\[Prime], t \( X\& \[RightVector] \)\^\[Prime] + \(X\& \[RightVector] \) t\^\[Prime])\)}]], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ I have defined the anticommutator (the bold curly braces) in a non-standard \ way, including a factor of two so I do not have to keep remembering to write \ it. The first term would be the Lorentz invariant interval if the two \ quaternions represented the same difference between two events in spacetime \ (i.e. t1=t2=delta t,...). The invariant interval plays a central role in \ special relativity. The vector terms are a frame-dependent, symmetric \ product of space with time and does not appear appear on the stage of \ physics, but is still a valid measurement.\ \>", "Text"], Cell["\<\ The Grassman outer product is antisymmetric and is formed with a \ commutator.\ \>", "Text"], Cell[BoxData[{ \(Simplify[\n \((\(q[t\_1, \ x\_1, y\_1, z\_1]\ .\ q[t\_2, \ x\_2, y\_2, z\_2]\ - \ \nq[t\_2, \ x\_2, y\_2, z\_2]\ .\ q[t\_1, \ x\_1, y\_1, z\_1]\)\/2)\).{1, 0, 0, 0}]\), \({0, \(-y\_2\)\ z\_1 + y\_1\ z\_2, x\_2\ z\_1 - x\_1\ z\_2, \(-x\_2\)\ y\_1 + x\_1\ y\_2}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ StyleBox["[", FontWeight->"Bold"], \(\((t, \(X\& \[RightVector] \))\), \((t\^\[Prime], \(X\& \[RightVector] \)\^\[Prime])\)\), StyleBox["]", FontWeight->"Bold"]}], StyleBox[" ", FontWeight->"Bold"], "\[Congruent]", \(\(\((t, \(X\& \[RightVector] \))\) \((t\^\[Prime], \(X\& \[RightVector] \)\^\[Prime])\) - \ \((t\^\[Prime], \(X\& \[RightVector] \)\^\[Prime])\) \((t, \(X\& \[RightVector] \))\)\)\/2\)}], "\n", "=", \((0, \(X\& \[RightVector] \) x \( X\& \[RightVector] \)\^\[Prime]) \)}]], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ This is the cross product defined for two 3-vectors. It is unchanged for \ quaternions.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[" The Euclidean inner and outer products", "Subsection"], Cell["\<\ Another important way to multiply a pair of quaternions involves first taking \ the transpose of one of the quaternions. For a real-valued matrix \ representation, this is equivalent to multiplication by the conjugate which \ involves flipping the sign of the 3-vector.\ \>", "Text"], Cell[BoxData[{ \(\((Transpose[q[t\_1, \ x\_1, y\_1, z\_1]]\ .\ q[t\_2, \ x\_2, y\_2, z\_2])\).\n\t\t{1, 0, 0, 0}\), \({\ \ t\_1\ t\_2 + x\_1\ x\_2 + y\_1\ y\_2 + z\_1\ z\_2, \(-t\_2\)\ x\_1 + t\_1\ x\_2 + y\_2\ z\_1 - y\_1\ z\_2, \n\t \(-t\_2\)\ y\_1 + t\_1\ y\_2 - x\_2\ z\_1 + x\_1\ z\_2, \ \ x\_2\ y\_1 - x\_1\ y\_2 - t\_2\ z\_1 + t\_1\ z\_2}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\(\((t, \(X\& \[RightVector] \))\)\^*\) \((t\^\[Prime], \(X\& \[RightVector] \)\^\[Prime])\) = \(\((t, \(-\(X\& \[RightVector] \)\))\) \((t\^\[Prime], \(X\& \[RightVector] \)\^\[Prime])\)\n\t\t = \((t\ t\^\[Prime]\ + \(X\& \[RightVector] \).\(X\& \[RightVector] \)\^\[Prime], t \( X\& \[RightVector] \)\^\[Prime] - \(X\& \[RightVector] \) t\^\[Prime] - \(X\& \[RightVector] \) x \( X\& \[RightVector] \)\^\[Prime]) \)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["Form the Euclidean inner product.", "Text"], Cell[BoxData[{ \(Simplify[\n \((\(Transpose[q[t\_1, \ x\_1, y\_1, z\_1]]\ .\ q[t\_2, \ x\_2, y\_2, z\_2]\ + \ \n Transpose[q[t\_2, \ x\_2, y\_2, z\_2]]\ .\ q[t\_1, \ x\_1, y\_1, z\_1]\)\/2)\).\n\t\t\t{1, 0, 0, 0}]\), \({t\_1\ t\_2 + x\_1\ x\_2 + y\_1\ y\_2 + z\_1\ z\_2, 0, 0, 0}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\(\(\((t, \(X\& \[RightVector] \))\)\^*\) \((t\^\[Prime], \(X\& \[RightVector] \)\^\[Prime])\) + \(\((t\^\[Prime], \(X\& \[RightVector] \)\^\[Prime])\)\^*\) \((t, \(X\& \[RightVector] \))\)\)\/2 = \((t\ t\^\[Prime]\ + \(X\& \[RightVector] \).\(X\& \[RightVector] \)\^\[Prime], \(0\& \[RightVector] \))\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ The first term is the Euclidean norm if the two quaternions are the \ same (this was the reason for using the adjective \"Euclidean\"). The \ Euclidean inner product is also the standard definition of a dot \ product.\ \>", "Text"], Cell["Form the Euclidean outer product.", "Text"], Cell[BoxData[{ \(Simplify[\n \((\(Transpose[q[t\_1, \ x\_1, y\_1, z\_1]]\ .\ q[t\_2, \ x\_2, y\_2, z\_2]\ - \ \n Transpose[q[t\_2, \ x\_2, y\_2, z\_2]]\ .\ q[t\_1, \ x\_1, y\_1, z\_1]\)\/2)\).\n\t\t\t{1, 0, 0, 0}]\), \({0, \(-t\_2\)\ x\_1 + t\_1\ x\_2 + y\_2\ z\_1 - y\_1\ z\_2, \n\t\t\t\ \(-t\_2\)\ y\_1 + t\_1\ y\_2 - x\_2\ z\_1 + x\_1\ z\_2, x\_2\ y\_1 - x\_1\ y\_2 - t\_2\ z\_1 + t\_1\ z\_2}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\(\(\((t, \(X\& \[RightVector] \))\)\^*\) \((t\^\[Prime], \(X\& \[RightVector] \)\^\[Prime])\) - \(\((t\^\[Prime], \(X\& \[RightVector] \)\^\[Prime])\)\^*\) \((t, \(X\& \[RightVector] \))\)\)\/2 = \((0, t \( X\& \[RightVector] \)\^\[Prime] - \(X\& \[RightVector] \) t\^\[Prime] - \(X\& \[RightVector] \) x \( X\& \[RightVector] \)\^\[Prime])\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ The first term is zero. The vector terms are an antisymmetric \ product of space with time and the negative of the cross product. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[" Implications", "Subsection", Evaluatable->False, AspectRatioFixed->True, CellTags->"post"], Cell["\<\ When multiplying vectors in physics, one normally only considers \ the Euclidean inner product, or dot product, and the Grassman outer product, \ or cross product. Yet, the Grassman inner product, because it naturally \ generates the invariant interval, appears to play a role in special \ relativity. What is interesting to speculate about is the role of the \ Euclidean outer product. It is possible that the antisymmetric, vector \ nature of the space/time product could be related to spin. Whatever the \ interpretation, the Grassman and Euclidean inner and outer products seem \ destine to do useful work in physics.\ \>", "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 640}, {0, 451}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Automatic, WindowSize->{594, 313}, WindowMargins->{{3, Automatic}, {Automatic, 17}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], Inherited, None}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PageHeaderLines->{False, Inherited}, PrintingOptions->{"PrintingMargins"->{{72, 57.5625}, {57.5625, 72}}, "PrintCellBrackets"->False, "PrintRegistrationMarks"->False, "PrintMultipleHorizontalPages"->False, "FirstPageHeader"->False}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of all cells in \ a given style. 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{Inherited, Inherited}}, FontFamily->"Courier", FontSize->14, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], PageWidth->Infinity, CellMargins->{{42, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, LineSpacing->{1, 0}, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}, FontFamily->"Courier", 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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. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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