(*********************************************************************** 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[ 9894, 288]*) (*NotebookOutlinePosition[ 38989, 1333]*) (* CellTagsIndexPosition[ 38945, 1329]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[ "The time reversal transformation for intervals represented as quaternions"], "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["doug "]], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ The time reversal quaternion Classical time reversal Relativistic time reversal Implications\ \>", "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox[" The time reversal quaternion"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}], Cell["\<\ The following transformation R for quaternions reverses time:\ \>", "Text"], Cell[BoxData[ \(q[t, \ x, \ y, \ z]\ \ \ -> \ \ \(q'\)[\(-t\), \ x, \ y, \ z]\ = \ R\ q[t, \ x, \ y, \ z]\)], "Input", FontSize->16], Cell[BoxData[ \(\((t, \ \(X\& \[RightVector] \))\)\ -> \ \ \((\(-t\), \ \(X\& \[RightVector] \))\)\ = \ R\ \((t, \ \(X\& \[RightVector] \))\)\)], "Input", FontSize->16], Cell["The quaternion R exist because quaternions are a field. ", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ R will equal (-t, X )(t, X)^-1. The inverse of quaternion is the transpose \ divided by the square of the norm, which is the scalar term of the transpose \ of a quaternion times itself.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(Simplify[ \((\n\t\t\t q[\(-t\), x, y, z]\ .\ Transpose[q[t, x, y, z]]\/\(( Transpose[q[t, x, y, z]].q[t, x, y, z])\)[[1, 1]])\)\n \ \ \ \ \ \ .{1, 0, 0, 0}\ ]\ \), \({\(\(-t\^2\) + x\^2 + y\^2 + z\^2\)\/\(t\^2 + x\^2 + y\^2 + z\^2\), \(2\ t\ x\)\/\(t\^2 + x\^2 + y\^2 + z\^2\), \n\t \(2\ t\ y\)\/\(t\^2 + x\^2 + y\^2 + z\^2\), \(2\ t\ z\)\/\(t\^2 + x\^2 + y\^2 + z\^2\)}\)}], "Input", CellMargins->{{42, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\(\ R\ \ = \(\((\(-t\), \ \(X\& \[RightVector] \))\) \((t, \ \(X\& \[RightVector] \))\)\^\(-1\) = \((\(-t\^2\) + \ \(X\& \[RightVector] \).\(X\& \[RightVector] \), 2\ t\ \ \(X\& \[RightVector] \))\)/ \((t\^2 + \ \(X\& \[RightVector] \).\(X\& \[RightVector] \)) \)\)\)\)], "Input", CellMargins->{{42, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["For any given time, R can be defined based on the above.", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Classical time reversal"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}], Cell["\<\ Examine the form of the quaternion which reverses time under two conditions. \ A interval normalized to the interval takes the form (1, beta), a scalar one \ and a 3-vector relativistic velocity beta . In the classical region, \ beta<<<1. Calculate R in this limit to one order of magnitued in beta.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(R\_time[q[1, \[Beta]\_x, \[Beta]\_y, \[Beta]\_z]]\ .\ {1, 0, 0, 0}\), \({\(\(-1\) + \[Beta]\_x\%2 + \[Beta]\_y\%2 + \[Beta]\_z\%2\)\/\(1 + \[Beta]\_x\%2 + \[Beta]\_y\%2 + \[Beta]\_z\%2\), \(2\ \[Beta]\_x\)\/\(1 + \[Beta]\_x\%2 + \[Beta]\_y\%2 + \[Beta]\_z\%2\), \n\t \(2\ \[Beta]\_y\)\/\(1 + \[Beta]\_x\%2 + \[Beta]\_y\%2 + \[Beta]\_z\%2\), \(2\ \[Beta]\_z\)\/\(1 + \[Beta]\_x\%2 + \[Beta]\_y\%2 + \[Beta]\_z\%2\)}\)}], "Input", FontSize->16], Cell[BoxData[{ \(\ R\ \ = \(\((\(-t\), \ \(\[Beta]\& \[RightVector] \))\) \((t, \ \(\[Beta]\& \[RightVector] \))\)\^\(-1\) = \((\(-t\^2\) + \ \(\[Beta]\& \[RightVector] \).\(\[Beta]\& \[RightVector] \), 2\ t\ \ \(\[Beta]\& \[RightVector] \))\)/ \((t\^2 + \ \(\[Beta]\& \[RightVector] \).\(\[Beta]\& \[RightVector] \)) \)\)\n\t\t\t\), \(if\ \ \ \[Beta] << <1, \ then\ \ R \[TildeTilde] \((\(-1\), 2\ t\ \ \(\[Beta]\& \[RightVector] \))\)\)}], "Input", FontSize->16], Cell["\<\ The operator R is almost the negative identity, but the vector is non-zero, \ so it would not commute.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Relativistic time reversal"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}], Cell["\<\ For a relativistic interval involving one axis, the interval could \ be characterized by the following: \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(q[T\ + \ \[Epsilon], T, 0, 0]\)], "Input", FontSize->16], Cell[BoxData[ \(\((T\ + \ \[Epsilon], T, 0, 0)\)\)], "Input", FontSize->16], Cell["\<\ Find out what quaternion is required to reverse time for this \ relativistic interval to first order in epsilon.\ \>", "Text"], Cell[BoxData[{\(R\_time[q[T\ + \ \[Epsilon], T, 0, 0]]\ .\ {1, 0, 0, 0}\), \({\(T\^2 - \((T + \[Epsilon])\)\^2\)\/\(T\^2 + \((T + \[Epsilon])\)\^2\), \(2\ T\ \((T + \[Epsilon])\)\)\/\(T\^2 + \((T + \[Epsilon])\)\^2\), 0, 0}\), \(R\ == \ Series[%, {\[Epsilon], \ 0, \ 1}]\), RowBox[{"R", "==", RowBox[{"{", RowBox[{ InterpretationBox[ RowBox[{\(-\(\[Epsilon]\/T\)\), "+", InterpretationBox[\(O[\[Epsilon]]\^2\), SeriesData[ \[Epsilon], 0, {}, 1, 2, 1]]}], SeriesData[ \[Epsilon], 0, { Times[ -1, Power[ T, -1]]}, 1, 2, 1]], ",", InterpretationBox[ RowBox[{"1", "+", InterpretationBox[\(O[\[Epsilon]]\^2\), SeriesData[ \[Epsilon], 0, {}, 0, 2, 1]]}], SeriesData[ \[Epsilon], 0, {1}, 0, 2, 1]], ",", "0", ",", "0"}], "}"}]}]}], "Input", FontSize->16], Cell[BoxData[{ RowBox[{ \(R = \((\(T\^2 - \((T + \[Epsilon])\)\^2\)\/\(T\^2 + \((T + \[Epsilon])\)\^2\), \(2\ T\ \((T + \[Epsilon])\)\)\/\(T\^2 + \((T + \[Epsilon])\)\^2\), 0, 0)\)\), "\n"}], RowBox[{"R", "\[TildeTilde]", RowBox[{"(", RowBox[{ InterpretationBox[ RowBox[{\(-\(\[Epsilon]\/T\)\), "+", InterpretationBox[\(O[\[Epsilon]]\^2\), SeriesData[ \[Epsilon], 0, {}, 1, 2, 1]]}], SeriesData[ \[Epsilon], 0, { Times[ -1, Power[ T, -1]]}, 1, 2, 1]], ",", InterpretationBox[ RowBox[{"1", "+", InterpretationBox[\(O[\[Epsilon]]\^2\), SeriesData[ \[Epsilon], 0, {}, 0, 2, 1]]}], SeriesData[ \[Epsilon], 0, {1}, 0, 2, 1]], ",", "0", ",", "0"}], ")"}]}]}], "Input", FontSize->16], Cell["\<\ This approaches q[-e/T, 1, 0, 0], almost a pure vector, a result \ distinct from the classical case.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Implications"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}], Cell["\<\ In special relativity, the interval between events is considered to \ be 4 vector are operated on by elements of the Lorentz group. The element of \ this group that reverses time has along its diagonal {-1, 1, 1, 1}, zeroes elsewhere. There is no dependence on relative \ velocity. Therefore special relativity predicts the operation of time \ reversal should be indistinguishable for classical and relativistic \ intervals. Yet classically, time reversal appears to involve entropy, and \ relativistically, time reversal involves antiparticles.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ In this notebook, a time reversal quaternion has been derived and \ shown to work. 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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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