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tensor approach to EM, the current and potential are contracted. To \ look at the issue of spin, one usually take the Fourier transform of the \ potential. We do this for both the Hamilton and Even representation of \ quaterrnions:\ \>", "Text", CellChangeTimes->{{3.410791188991247*^9, 3.410791219777223*^9}, { 3.410791462220625*^9, 3.410791468960992*^9}, {3.410791533436656*^9, 3.410791533772889*^9}, {3.410791572273374*^9, 3.410791656527569*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"-", RowBox[{"J", ".", "A"}]}], ")"}], ".", RowBox[{"{", RowBox[{"1", ",", "0", ",", "0", ",", "0"}], "}"}]}], "/.", RowBox[{"{", RowBox[{ RowBox[{"\[Phi]", "\[Rule]", RowBox[{"\[Rho]", "'"}]}], ",", RowBox[{"Ax", "\[Rule]", RowBox[{"Jx", "'"}]}], ",", RowBox[{"Ay", "\[Rule]", RowBox[{"Jy", "'"}]}], ",", RowBox[{"Az", "\[Rule]", RowBox[{"Jz", "'"}]}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"-", RowBox[{"J2", ".", RowBox[{"Conjugate2", "[", "A2", "]"}]}]}], ")"}], ".", RowBox[{"{", RowBox[{"1", ",", "0", ",", "0", ",", "0"}], "}"}]}], "/.", RowBox[{"{", RowBox[{ RowBox[{"\[Phi]", "\[Rule]", RowBox[{"\[Rho]", "'"}]}], ",", RowBox[{"Ax", "\[Rule]", RowBox[{"Jx", "'"}]}], ",", RowBox[{"Ay", "\[Rule]", RowBox[{"Jy", "'"}]}], ",", RowBox[{"Az", "\[Rule]", RowBox[{"Jz", "'"}]}]}], "}"}]}]}], "Input", CellChangeTimes->{{3.410767989814432*^9, 3.4107680929273233`*^9}, { 3.410768271669338*^9, 3.41076827598094*^9}, {3.410791259944934*^9, 3.410791293619334*^9}, {3.410791330334771*^9, 3.410791448940608*^9}, { 3.410791558691531*^9, 3.410791563298567*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Jx", " ", RowBox[{"Jx", "'"}]}], "+", RowBox[{"Jy", " ", RowBox[{"Jy", "'"}]}], "+", RowBox[{"Jz", " ", RowBox[{"Jz", "'"}]}], "-", RowBox[{"\[Rho]", " ", RowBox[{"\[Rho]", "'"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{"-", "\[Rho]"}], " ", RowBox[{"Jx", "'"}]}], "+", RowBox[{"Jz", " ", RowBox[{"Jy", "'"}]}], "-", RowBox[{"Jy", " ", RowBox[{"Jz", "'"}]}], "-", RowBox[{"Jx", " ", RowBox[{"\[Rho]", "'"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{"-", "Jz"}], " ", RowBox[{"Jx", "'"}]}], "-", RowBox[{"\[Rho]", " ", RowBox[{"Jy", "'"}]}], "+", RowBox[{"Jx", " ", RowBox[{"Jz", "'"}]}], "-", RowBox[{"Jy", " ", RowBox[{"\[Rho]", "'"}]}]}], ",", RowBox[{ RowBox[{"Jy", " ", RowBox[{"Jx", "'"}]}], "-", RowBox[{"Jx", " ", RowBox[{"Jy", "'"}]}], "-", RowBox[{"\[Rho]", " ", RowBox[{"Jz", "'"}]}], "-", RowBox[{"Jz", " ", RowBox[{"\[Rho]", "'"}]}]}]}], "}"}]], "Output", CellChangeTimes->{{3.4107912793247004`*^9, 3.410791294181714*^9}, 3.410791353197638*^9, {3.41079142086224*^9, 3.410791450484851*^9}, 3.410791566681622*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Jx", " ", RowBox[{"Jx", "'"}]}], "+", RowBox[{"Jy", " ", RowBox[{"Jy", "'"}]}], "+", RowBox[{"Jz", " ", RowBox[{"Jz", "'"}]}], "-", RowBox[{"\[Rho]", " ", RowBox[{"\[Rho]", "'"}]}]}], ",", RowBox[{ RowBox[{"\[Rho]", " ", RowBox[{"Jx", "'"}]}], "+", RowBox[{"Jz", " ", RowBox[{"Jy", "'"}]}], "+", RowBox[{"Jy", " ", RowBox[{"Jz", "'"}]}], "-", RowBox[{"Jx", " ", RowBox[{"\[Rho]", "'"}]}]}], ",", RowBox[{ RowBox[{"Jz", " ", RowBox[{"Jx", "'"}]}], "+", RowBox[{"\[Rho]", " ", RowBox[{"Jy", "'"}]}], "+", RowBox[{"Jx", " ", RowBox[{"Jz", "'"}]}], "-", RowBox[{"Jy", " ", RowBox[{"\[Rho]", "'"}]}]}], ",", RowBox[{ RowBox[{"Jy", " ", RowBox[{"Jx", "'"}]}], "+", RowBox[{"Jx", " ", RowBox[{"Jy", "'"}]}], "+", RowBox[{"\[Rho]", " ", RowBox[{"Jz", "'"}]}], "-", RowBox[{"Jz", " ", RowBox[{"\[Rho]", "'"}]}]}]}], "}"}]], "Output", CellChangeTimes->{{3.4107912793247004`*^9, 3.410791294181714*^9}, 3.410791353197638*^9, {3.41079142086224*^9, 3.410791450484851*^9}, 3.410791566693112*^9}] }, Open ]], Cell["\<\ The scalars are identical as they must be to be identical to the Lorentz \ invariant contraction found in EM. What requires closer study is the signs \ of terms in the phase. There are pairs of terms which have the same sign. \ if the J current points in the directin of the J' current, these will add \ together and only take \[Pi] radians to return, a sign of spin 2 symmetry. \ There are other pairs where the signs are different. They will require \ 2\[Pi] radians to return, a sign of spin 1 symmetry. The Hamilton product \ has spin 1 for the cross terms, spin 2 for the \[Rho] source terms. The even \ product does the opposite: spin 2 for the cross terms and spin 1 for the \ \[Rho] source terms. Sounds like a complete story to me.\ \>", "Text", CellChangeTimes->{{3.410791671900751*^9, 3.4107918508506804`*^9}, { 3.4107924279669933`*^9, 3.4107924538573523`*^9}, {3.410792512934688*^9, 3.41079267562821*^9}}], Cell["Look at the sum:", "Text", CellChangeTimes->{{3.4108116938609943`*^9, 3.41081170046769*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Simplify", "[", RowBox[{ FractionBox["1", "2"], RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", RowBox[{"J", ".", "A"}]}], "-", RowBox[{"J2", ".", RowBox[{"Conjugate2", "[", "A2", "]"}]}]}], ")"}], ".", RowBox[{"{", RowBox[{"1", ",", "0", ",", "0", ",", "0"}], "}"}]}]}], "]"}], "/.", RowBox[{"{", RowBox[{ RowBox[{"\[Phi]", "\[Rule]", RowBox[{"\[Rho]", "'"}]}], ",", RowBox[{"Ax", "\[Rule]", RowBox[{"Jx", "'"}]}], ",", RowBox[{"Ay", "\[Rule]", RowBox[{"Jy", "'"}]}], ",", RowBox[{"Az", "\[Rule]", RowBox[{"Jz", "'"}]}]}], "}"}]}]], "Input", CellChangeTimes->{{3.410811599902356*^9, 3.410811618700058*^9}, 3.410811653342597*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Jx", " ", RowBox[{"Jx", "'"}]}], "+", RowBox[{"Jy", " ", RowBox[{"Jy", "'"}]}], "+", RowBox[{"Jz", " ", RowBox[{"Jz", "'"}]}], "-", RowBox[{"\[Rho]", " ", RowBox[{"\[Rho]", "'"}]}]}], ",", RowBox[{ RowBox[{"Jz", " ", RowBox[{"Jy", "'"}]}], "-", RowBox[{"Jx", " ", RowBox[{"\[Rho]", "'"}]}]}], ",", RowBox[{ RowBox[{"Jx", " ", RowBox[{"Jz", "'"}]}], "-", RowBox[{"Jy", " ", RowBox[{"\[Rho]", "'"}]}]}], ",", RowBox[{ RowBox[{"Jy", " ", RowBox[{"Jx", "'"}]}], "-", RowBox[{"Jz", " ", RowBox[{"\[Rho]", "'"}]}]}]}], "}"}]], "Output", CellChangeTimes->{3.410811619368978*^9, 3.410811654759468*^9}] }, Open ]], Cell["\<\ Recall the phase does not play a direct roll in electromagnetic theory. \ Instead it is used to confirm what particles particoipate in a \ current-current interaction.\ \>", "Text", CellChangeTimes->{{3.410811720441248*^9, 3.410811771927885*^9}, { 3.4108119132187223`*^9, 3.410811940234295*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Fields", "Subsection", CellChangeTimes->{{3.4107679781251574`*^9, 3.410767984719227*^9}, { 3.4107927640188227`*^9, 3.4107927683517847`*^9}}], Cell["\<\ There are 5 fields in the GEM poposal. Two are familiar: the electric field \ E and the magnetic field B. For gravity, there are two symmetric analogues \ to these fields I call e and b, which have the same differential terms but \ different signs. The final field is composed of terms down the diagonal, \ which arise in both expressions.\ \>", "Text", CellChangeTimes->{{3.4104225655656967`*^9, 3.410422794978692*^9}, { 3.410422859767229*^9, 3.410422926929956*^9}}], Cell["The field strength tensor is a combination of E and B", "Text", CellChangeTimes->{{3.40794548229352*^9, 3.4079455128142548`*^9}, { 3.41042375996949*^9, 3.4104238129470387`*^9}, 3.410423866735525*^9, 3.410424303221326*^9, {3.410427503106442*^9, 3.4104275364811172`*^9}, 3.410427664676282*^9, 3.410429323067041*^9}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"-", "g"}], "+", "E", "+", "B"}], "=", RowBox[{ RowBox[{"-", " ", "A"}], "\[Del]"}]}]], "Input", CellChangeTimes->{{3.410811957918127*^9, 3.410811994631748*^9}, { 3.410812038483235*^9, 3.410812106728026*^9}, {3.410812566157362*^9, 3.410812585010706*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Simplify", "[", RowBox[{"-", RowBox[{"qd", "[", "A", "]"}]}], "]"}], ".", RowBox[{"{", RowBox[{"1", ",", "0", 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