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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[ 29818, 752]*) (*NotebookOutlinePosition[ 59816, 1810]*) (* CellTagsIndexPosition[ 59772, 1806]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Bracket notation and quaternions: quaternions as a complete inner-product space\ \>", "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["doug "]], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "Introductory definitions\nThe positive definite norm of a quaternion\n\ Completeness\nIdentities and inequalities (triangle, Schwarz)\n\ Implications"]], "Text"], Cell[CellGroupData[{ Cell[" Introductory definitions", "Subsection"], Cell["\<\ A mathematical connection between the bracket notation of quantum \ mechanics and quaternions is detailed. It will be argued that quaternions \ have the properties of a complete inner-product space (a Banach space for the \ field of quaternions). A central issue is the definition of the square of \ the norm. In quantum mechanics:\ \>", "Text"], Cell[BoxData[ \(\( || \[CurlyPhi]\( || \^2\)\)\ = \( < \[CurlyPhi]\) | \(\[CurlyPhi] > \)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\( || \[CurlyPhi]\( || \^2\)\)\ = \( < \[CurlyPhi]\) | \(\[CurlyPhi] > \)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ In this notebook, the following assertion will be examined (* is the \ conjugate, so the vector flips signs):\ \>", "Text"], Cell[BoxData[ \(\( || q[t, x, y, z]\( || \^2\)\)\ = \ \(q[t, x, y, z]\^*\).\ q[t, x, y, z]\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\( || \((t, X\&\[RightVector])\)\( || \^2\)\)\ = \ \(\((t, X\&\[RightVector])\)\^*\).\ \((t, X\&\[RightVector])\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ The inner-product of two quaternions is defined here as the transpose (or \ conjugate) of the first quaternion multiplied by the second. The inner \ product of a function with itself is the norm.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ " The positive definite norm of a quaternion"]], "Subsection", CellMargins->{{-1, Inherited}, {Inherited, Inherited}}], Cell["\<\ The square of the norm of a quaternion can only be zero if every \ element is zero, otherwise it must have a positive value.\ \>", "Text"], Cell[BoxData[{ \(inner[q[t, x, y, z]]\ .\ {1, 0, 0, 0}\), \({t\^2 + x\^2 + y\^2 + z\^2, 0, 0, 0}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\(\((t, X\&\[RightVector])\)\^*\) \((t, X\&\[RightVector])\) = \((t\^2 + X\&\[RightVector].X\&\[RightVector], \(0\& \[RightVector] \)) \)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ This is the standard Euclidean norm for a real 4-dimensional vector \ space.\ \>", "Text"], Cell["\<\ The Euclidean inner-product of two quaternions can take on any \ value, as is the case in quantum mechanics for . The adjective \ \"Euclidean\" is used to distinguish this product from the Grassman \ inner-product which plays a central role in special relativity (see \ alternative algebra for boosts).\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Completeness"]], "Subsection", CellMargins->{{-1, Inherited}, {Inherited, Inherited}}], Cell["\<\ With the topology of a Euclidean norm for a real 4-dimensional \ vector space, quaternions are complete.\ \>", "Text"], Cell["\<\ Quaternions are complete in a manor required to form a Banach space \ if there exists a neighborhood of any quaternion x such that there is a set \ of quaternions y\ \>", "Text"], Cell[BoxData[ \(\( || x\ - \ y\)\( || \^2\)\( < \[Epsilon]\)\^2\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\( || x\ - \ y\)\( || \^2\)\( < \[Epsilon]\)\^2\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["for some fixed value of epsilon.", "Text"], Cell["Construct such a neighborhood.", "Text"], Cell[BoxData[{ \(Simplify[ \((\n\t\tinner[ q[t\_1, x\_1, y\_1, z\_1]\ - \n\t\t\t\t\t q[t\_1\ + \[Epsilon]\/4, x\_1 + \[Epsilon]\/4, y\_1 + \[Epsilon]\/4, z\_1 + \[Epsilon]\/4]])\).{1, 0, 0, 0}] \), \({\[Epsilon]\^2\/4, 0, 0, 0}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\(\(( \((t, X\&\[RightVector])\) - \(\[Epsilon]\/4\) \((t, X\&\[RightVector])\))\)\^*\) \((\((t, X\&\[RightVector])\) - \(\[Epsilon]\/4\) \((t, X\&\[RightVector])\))\)\n\ = \((\[Epsilon]\^2\/4, 0, 0, 0)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["An infinite number of quaternions exist in the neighborhood.", "Text"], Cell["\<\ Any polynomial equation with quaternion coefficients has a \ quaternion solution in x (a proof done by Eilenberg and Niven in 1944, cited \ in Birkhoff and Mac Lane's \"A Survey of Modern Algebra.\")\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Identities and inequalities"]], "Subsection", CellMargins->{{-1, Inherited}, {Inherited, Inherited}}], Cell["\<\ The following identities and inequalities eminate from the \ properties of a Euclidean norm. They are worked out for quaternions here in \ detail to solidify the connection between the machinery of quantum mechanics \ and quaternions.\ \>", "Text"], Cell["\<\ The conjugate of the square of the norm equals the square of the \ norm of the two terms reversed.\ \>", "Text"], Cell[BoxData[ \(\( < \[Phi]\) | \(\[CurlyPhi]\( > \^*\)\)\ = \ \( < \[CurlyPhi]\) | \(\[Phi] > \)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\( < \[Phi]\) | \(\[CurlyPhi]\( > \^*\)\)\ = \ \( < \[CurlyPhi]\) | \(\[Phi] > \)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["For quaternions,", "Text"], Cell[BoxData[{ \(Simplify[\n\t Transpose[inner[q[t\_1, x\_1, y\_1, z\_1], q[t\_2, x\_2, y\_2, z\_2]]]. \n\t\t{1, 0, 0, 0}]\), \({t\_2\ t\_1 + x\_2\ x\_1 + y\_2\ y\_1 + z\_2\ z\_1, \(-x\_2\)\ t\_1 + x\_1\ t\_2 + z\_2\ y\_1 - z\_1\ y\_2, \n \(-y\_2\)\ t\_1 + y\_1\ t\_2 - z\_2\ x\_1 + z\_1\ x\_2, \(-z\_2\)\ t\_1 + z\_1\ t\_2 + y\_2\ x\_1 - y\_1\ x\_2}\), \(Simplify[\n\t inner[q[t\_2, x\_2, y\_2, z\_2], q[t\_1, x\_1, y\_1, z\_1]].\n \t\t{1, 0, 0, 0}]\), \({t\_2\ t\_1 + x\_2\ x\_1 + y\_2\ y\_1 + z\_2\ z\_1, \(-x\_2\)\ t\_1 + x\_1\ t\_2 + z\_2\ y\_1 - z\_1\ y\_2, \n \(-y\_2\)\ t\_1 + y\_1\ t\_2 - z\_2\ x\_1 + z\_1\ x\_2, \(-z\_2\)\ t\_1 + z\_1\ t\_2 + y\_2\ x\_1 - y\_1\ x\_2}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ \(\(\(( \(\((t, X\&\[RightVector])\)\^*\) \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\))\)\^*\)\ = \ \((t\ t\^\[Prime] + \ X\&\[RightVector].\(X\^\[Prime]\)\&\[RightVector], \(-t\) \(X\^\[Prime]\)\&\[RightVector]\ + \ \(X\&\[RightVector]\) t\^\[Prime]\ + \ \(X\&\[RightVector]\) x \( X\^\[Prime]\)\&\[RightVector])\)\), \(\(\((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\^*\) \((t, X\&\[RightVector])\)\ = \ \((\(t\^\[Prime]\) t\ + \ \(X\^\[Prime]\)\&\[RightVector].X\&\[RightVector], \(t\^\[Prime]\) X\&\[RightVector] - \(X\^\[Prime]\)\&\[RightVector]\ t\ - \(\(X\^\[Prime]\)\&\[RightVector]\) x X\&\[RightVector])\)\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["These are identical.", "Text"], Cell["For products of squares of norms in quantum mechanics,", "Text"], Cell[BoxData[ \(\(\( < \[CurlyPhi]\[Phi]\) | \(\[CurlyPhi]\[Phi] > \)\ = \ \( < \[CurlyPhi]\) | \[CurlyPhi] > < \[Phi] | \(\[Phi] > \)\ \)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\(\( < \[CurlyPhi]\[Phi]\) | \(\[CurlyPhi]\[Phi] > \)\ = \ \( < \[CurlyPhi]\) | \[CurlyPhi] > < \[Phi] | \(\[Phi] > \)\ \)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["This is also the case for quaternions.", "Text"], Cell[BoxData[{ \(Simplify[\n\t inner[q[t\_1, x\_1, y\_1, z\_1].q[t\_2, x\_2, y\_2, z\_2]].\n \t\t{1, 0, 0, 0}]\), \({\((t\_1\%2 + x\_1\%2 + y\_1\%2 + z\_1\%2)\)\ \((t\_2\%2 + x\_2\%2 + y\_2\%2 + z\_2\%2)\), 0, 0, 0}\), \(Simplify[\n\t inner[q[t\_1, x\_1, y\_1, z\_1]].inner[q[t\_2, x\_2, y\_2, z\_2]].\n \t\t{1, 0, 0, 0}]\), \({\((t\_2\%2 + x\_2\%2 + y\_2\%2 + z\_2\%2)\)\ \((t\_1\%2 + x\_1\%2 + y\_1\%2 + z\_1\%2)\), 0, 0, 0}\), \(Simplify[\n\t inner[q[t\_1, x\_1, y\_1, z\_1].q[t\_2, x\_2, y\_2, z\_2]].\n \t\t{1, 0, 0, 0}]\), \({\((t\_1\%2 + x\_1\%2 + y\_1\%2 + z\_1\%2)\)\ \((t\_2\%2 + x\_2\%2 + y\_2\%2 + z\_2\%2)\), 0, 0, 0}\), \(Simplify[\n\t inner[q[t\_1, x\_1, y\_1, z\_1]].inner[q[t\_2, x\_2, y\_2, z\_2]].\n \t\t{1, 0, 0, 0}]\), \({\((t\_2\%2 + x\_2\%2 + y\_2\%2 + z\_2\%2)\)\ \((t\_1\%2 + x\_1\%2 + y\_1\%2 + z\_1\%2)\), 0, 0, 0}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\( < \((t, X\&\[RightVector])\) \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\) | \(\((t, X\&\[RightVector])\) \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\) > \)\n\t\t\n = \(\(\((\((t, X\&\[RightVector])\) \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\))\)\^*\)\ \((t, X\&\[RightVector])\) \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\n = \ \(\(\((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\^*\) \(\((t, X\&\[RightVector])\)\^*\)\ \((t, X\&\[RightVector])\) \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\n = \(\(\((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\^*\) \((t\^2 + x\^2 + y\^2 + z\^2, 0, 0, 0)\) \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\n = \(\((t\^2 + x\^2 + y\^2 + z\^2, 0, 0, 0)\) \(\((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\^*\) \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\n = \(\(\((t, X\&\[RightVector])\)\^*\)\ \((t, X\&\[RightVector])\) \(\((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\^*\) \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\n \t\t\t\t\t\t\n = \( < \((t, X\&\[RightVector])\)\) | \((t, X\&\[RightVector])\) > < \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\) | \(\((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\) > \)\)\)\)\)\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["The triangle inequality in quantum mechanics:", "Text"], Cell[BoxData[ \(\( < \[CurlyPhi]\ + \ \[Phi]\) | \[Phi]\ + \ \[CurlyPhi]\( > \^2\) \[LessEqual] \((\( < \[CurlyPhi]\) | \[CurlyPhi] > \(+\ \( < \[Phi]\)\) | \(\[Phi] > \))\)\^2\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\( < \[CurlyPhi]\ + \ \[Phi]\) | \[Phi]\ + \ \[CurlyPhi]\( > \^2\) \[LessEqual] \((\( < \[CurlyPhi]\) | \[CurlyPhi] > \(+\ \( < \[Phi]\)\) | \(\[Phi] > \))\)\^2\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["For quaternions,", "Text"], Cell[BoxData[{ \(Simplify[\n\t \((inner[q[t\_1, x\_1, y\_1, z\_1] + \ q[t\_2, x\_2, y\_2, z\_2]])\).{ 1, 0, 0, 0}\n\t \[LessEqual] \ \ \n\t \((inner[q[t\_1, x\_1, y\_1, z\_1]]\ + \ inner[q[t\_2, x\_2, y\_2, z\_2]] + \n\t\t 2 \@\(\ inner[q[t\_1, x\_1, y\_1, z\_1]].\ inner[q[t\_2, x\_2, y\_2, z\_2]]\))\).\n \t\t\t{1, 0, 0, 0}]\), \({\((t\_1 + t\_2)\)\^2 + \((x\_1 + x\_2)\)\^2 + \((y\_1 + y\_2)\)\^2 + \((z\_1 + z\_2)\)\^2, 0, 0, 0}\n \[LessEqual] \n{t\_1\%2 + t\_2\%2 + x\_1\%2 + x\_2\%2 + y\_1\%2 + y\_2\%2 + z\_1\%2 + z\_2\%2 + \n 2\ \@\(\((t\_2\%2 + x\_2\%2 + y\_2\%2 + z\_2\%2)\)\ \((t\_1\%2 + x\_1\%2 + y\_1\%2 + z\_1\%2)\)\), 0, 0, 0}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ \(\( < \((t, X\&\[RightVector])\) + \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\) | \(\((t, X\&\[RightVector])\) + \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\( > \^2\)\)\n \t\t\n = \(\(( \(\((t + t\^\[Prime], X\&\[RightVector] + \(X\^\[Prime]\)\&\[RightVector])\)\^* \) \((t + t\^\[Prime], X\&\[RightVector] + \(X\^\[Prime]\)\&\[RightVector])\))\)\^2\n = \((t\^2\ + \ \(t\^\[Prime]\)\^2 + X\&\[RightVector]\^2 + \(X\^\[Prime]\)\&\[RightVector]\^2 + \ 2 t\ t\^\[Prime]\ + 2 X\&\[RightVector].\(X\^\[Prime]\)\&\[RightVector], 0)\)\^2 \n\t\t \[PrecedesEqual] \n \((t\^2\ + X\&\[RightVector]\^2 + \ \(t\^\[Prime]\)\^2 + \(X\^\[Prime]\)\&\[RightVector]\^2 + \ 2 \@\(\(\((t, X\&\[RightVector]\ )\)\^*\) \((t, X\&\[RightVector]\ )\) \(\((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\^* \) \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector]) \)\), 0)\)\^2\)\n\t\t\), \(\((\( < \((t, X\&\[RightVector])\)\) | \((t, X\&\[RightVector])\) > \(+\ \( < \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\)\)\) | \(\((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\) > \))\)\^2\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ If the signs of each pair of component are the same, the two sides \ will be equal. If the signs are different (a t and a -t for example), then \ the cross terms will cancel on the left hand side of the inequality, making \ it smaller than the right hand side where terms never cancel because there \ are only squared terms.\ \>", "Text"], Cell[TextData[StyleBox[ "The Schwarz inequality in quantum mechanics is analogous to dot products and \ cosines in Euclidean space."]], "Text"], Cell[BoxData[ \(\(| < \[CurlyPhi]\) | \(\[Phi] > \)\(|\^2\)\ \( \[LessEqual] \ < \[CurlyPhi]\) | \[CurlyPhi] > < \[Phi] | \(\[Phi] > \)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(\(| < \[CurlyPhi]\) | \(\[Phi] > \)\(|\^2\)\ \( \[LessEqual] \ < \[CurlyPhi]\) | \[CurlyPhi] > < \[Phi] | \(\[Phi] > \)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ Look a the norm of the difference between two quaternions, which is \ necessarily equal to or greater than zero. \ \>", "Text"], Cell[BoxData[{ \(0\ \[LessEqual] \ Simplify[\n\t inner[q[\[Alpha], 0, 0, 0].q[t\_1, x\_1, y\_1, z\_1]\ - \ \n \t\t\t\t\t\t\tq[\[Beta], 0, 0, 0].q[t\_2, x\_2, y\_2, z\_2]].{ 1, 0, 0, 0}]\), \(0 \[LessEqual] \n{\((\[Alpha]\ t\_1 - \[Beta]\ t\_2)\)\^2 + \((\[Alpha]\ x\_1 - \[Beta]\ x\_2)\)\^2 + \((\[Alpha]\ y\_1 - \[Beta]\ y\_2)\)\^2 + \((\[Alpha]\ z\_1 - \[Beta]\ z\_2)\)\^2, 0, 0, 0}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(0\ \[LessEqual] \ < \[Alpha] \((t, X\&\[RightVector])\) - \[Beta] \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\) | \(\[Alpha] \((t, X\&\[RightVector])\) - \[Beta] \((t\^\[Prime], \(X\^\[Prime]\)\&\[RightVector])\) > \)\n \t\t\n = \ \((\((\[Alpha]\ t\ - \ \[Beta]\ t\^\[Prime])\)\^2\ + \ \((\[Alpha]\ X\&\[RightVector] - \[Beta] \( X\^\[Prime]\)\&\[RightVector])\). \((\[Alpha]\ X\&\[RightVector] - \[Beta] \( X\^\[Prime]\)\&\[RightVector])\), 0\&\[RightVector])\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["There is an efficient way to represent the cross terms.", "Text"], Cell[BoxData[{ \(Simplify[ \((\n\t\tinner[ q[\[Alpha]\ t\_1, \[Alpha]\ x\_1, \[Alpha]\ y\_1, \[Alpha]\ z\_1], q[\[Beta]\ t\_2, \[Beta]\ x\_2, \[Beta]\ y\_2, \[Beta]\ z\_2]]\ + \n\t\tinner[ q[\[Beta]\ t\_2, \[Beta]\ x\_2, \[Beta]\ y\_2, \[Beta]\ z\_2], q[\[Alpha]\ t\_1, \[Alpha]\ x\_1, \[Alpha]\ y\_1, \[Alpha]\ z\_1]])\).\n\t\t{1, 0, 0, 0}]\), \({2\ \[Beta]\ \[Alpha]\ \((t\_2\ t\_1 + x\_2\ x\_1 + y\_2\ y\_1 + z\_2\ z\_1)\), 0, 0, 0}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(cross\ terms\ = \ \((\(-2\)\ \[Alpha]\ \[Beta] \((t\ \ t\^\[Prime]\ + \ X\&\[RightVector].\(X\^\[Prime]\)\&\[RightVector])\), 0\&\[RightVector])\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ Note that the above operation carries out the operation of twice dot product \ between two 4-vectors. Place the cross terms on the other side of the \ inequality. Divide by alpha beta, assigning the norm of the second \ quaternion to alpha and the norm of the first to beta. \ \>", "Text"], Cell[BoxData[{ \(Simplify[\n\t \(\((inner[q[\[Alpha]\ t\_1, \[Alpha]\ x\_1, \[Alpha]\ y\_1, \[Alpha]\ z\_1], q[\[Beta]\ t\_2, \[Beta]\ x\_2, \[Beta]\ y\_2, \[Beta]\ z\_2]]\ + \n\t inner[q[\[Beta]\ t\_2, \[Beta]\ x\_2, \[Beta]\ y\_2, \[Beta]\ z\_2], q[\[Alpha]\ t\_1, \[Alpha]\ x\_1, \[Alpha]\ y\_1, \[Alpha]\ z\_1]])\)\/\(2\ \[Alpha]\ \[Beta]\)\).\n \t\t{1, 0, 0, 0}\n \[LessEqual] \ \n\t\t\t \(\((inner[q[\[Alpha]\ t\_1, \[Alpha]\ x\_1, \[Alpha]\ y\_1, \[Alpha]\ z\_1]]\ + \n\t inner[q[\[Beta]\ t\_2, \[Beta]\ x\_2, \[Beta]\ y\_2, \[Beta]\ z\_2]])\)\/\(2\ \[Alpha]\ \[Beta]\)\)\ .{1, 0, 0, 0} /. \n \t\t{\[Alpha]\ -> \@\((t\_2\%2 + x\_2\%2 + y\_2\%2 + z\_2\%2)\), \[Beta]\ -> \@\((t\_1\%2 + x\_1\%2 + y\_1\%2 + z\_1\%2)\)}]\), \({t\_2\ t\_1 + x\_2\ x\_1 + y\_2\ y\_1 + z\_2\ z\_1, 0, 0, 0}\n \[LessEqual] \n{\@\(t\_2\%2 + x\_2\%2 + y\_2\%2 + z\_2\%2\)\ \@\(t\_1\%2 + x\_1\%2 + y\_1\%2 + z\_1\%2\), 0, 0, 0}\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[{ \(\((2\ \[Alpha]\ \[Beta] \((t\ \ t\^\[Prime]\ + \ X\&\[RightVector].\(X\^\[Prime]\)\&\[RightVector])\), 0\&\[RightVector])\) \[LessEqual] \((\[Alpha]\ t\^2 + \ \[Alpha]\ X\&\[RightVector]\^2 + \ \[Beta]\ \(t\^\[Prime]\)\^2\ + \[Beta] \( X\^\[Prime]\)\&\[RightVector]\^2, 0\&\[RightVector])\)\), \(Divide\ by\ \[Alpha]\ \[Beta], \ and\ let\ \ \[Alpha] -> \((\(t\^\[Prime]\)\^2\ + \(X\^\[Prime]\)\&\[RightVector]\^2)\), \ \[Beta] -> \((t\^2\ + X\&\[RightVector]\^2)\)\), \(\((t\ \ t\^\[Prime]\ + \ X\&\[RightVector].\(X\^\[Prime]\)\&\[RightVector], 0\&\[RightVector])\) \[LessEqual] \((\((\ t\^2 + \ X\&\[RightVector]\^2)\) \((\ \(t\^\[Prime]\)\^2 + \(X\^\[Prime]\)\&\[RightVector]\^2)\), 0\&\[RightVector])\)\)}], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["This is now the Schwarz inequality.", "Text"], Cell["Another inequality:", "Text"], Cell[BoxData[ \(2\ Re\ < \[CurlyPhi] | \[Phi] > \ \[LessEqual] \ < \[CurlyPhi] | \[CurlyPhi] > \(+\( < \[Phi]\)\) | \(\[Phi] > \)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell[BoxData[ \(2\ Re\ < \[CurlyPhi] | \[Phi] > \ \[LessEqual] \ < \[CurlyPhi] | \[CurlyPhi] > \(+\( < \[Phi]\)\) | \(\[Phi] > \)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ Examine the square of the norm of the difference between two \ quaternions which is necessarily equal to or greater than zero.\ \>", "Text"], Cell[BoxData[{ \(0\ \[LessEqual] \ Simplify[\((\n\t\t inner[q[t\_1, x\_1, y\_1, z\_1]\ - 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From that solitary observation, the power of a mathematical field \ was harnessed to solve a wide range of problems in special relativity.\ \>", "Text"], Cell["\<\ In a similar fashion, it is hoped that because the product of a \ transpose of a quaternion with a quaternion has the properties of a complete \ inner product space, the power of the mathematical field of quaternions can \ be used to solve a wide range of problems in quantum mechanics. This is an \ important area for further research.\ \>", "Text"], Cell["\<\ Note: this goal is different from the one Stephen Adler sets out in \ \"Quaternionic Quantum Mechanics and Quantum Fields.\" He tries to \ substitute quaternions in the place of complex numbers in the standard \ Hilbert space formulation of quantum mechanics. The analytical properties of \ quaternions do not play a critical role. It is the properties of the Hilbert \ space over the field of quaternions that is harnessed to solve problems. 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