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The role of the spacetime diffeomorphism group Diff(M) The problem of time Approaches to quantum gravity Certainty is seven for seven (Note: this was a post sent to the newsgroup sci.physics.research June 28, \ 1998)\ \>", "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Introduction"]], "Subsection"], Cell[TextData[StyleBox[ "Chris Isham's paper \"Prima Facie Questions in Quantum Gravity\" \ (gr-qc/9310031, October, 1993) details the structure required of any \ approach to quantum gravity. I will use that paper as a template for this \ post, noting the highlights (but please refer to this well-written paper for \ details). Wherever appropriate, I will point out how using quaternions in \ quantum gravity fits within this superstructure. I will argue that all the \ technical parts required are all ready part of quaternion mathematics. These \ tools are required to calculate the smallest norm between two worldlines, \ which may form a new road to quantum gravity."]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["What is quantum gravity?"]], "Subsection"], Cell["\<\ Isham sorts the approaches to quantum gravity into four groups. First, there \ is the classical approach. This begins with Einstein's general relativity. \ Systematically substitute self-adjoint operators for classical terms like \ energy and momentum. This gets further subdivided into the 'canonical' \ scheme where spacetime is split into time and space--Ashtekar's work--and a \ covariant formulation, which is believed to be perturbatively \ non-renormalizable. \ \>", "Text"], Cell["\<\ The second approach takes quantum mechanics and transforms it into general \ relativity. Much less effort has gone in this direction, but there has been \ work done by Haag.\ \>", "Text"], Cell["\<\ The third angle has general relativity as the low energy limit of ideas based \ in conventional quantum mechanics. Quantum gravity dominates the world on \ the scale of Plank time, length, or energy, a place where only calculations can go. This is where superstring theory \ lives.\ \>", "Text"], Cell["\<\ The fourth possibility involves a radical new perspective, where general \ relativity and quantum mechanics are only different applications of the same \ mathematical structure. This would require a major \"retooling\". People \ with the patience to have read many of my post (even if not followed :-) know \ this is the task facing work with quaternions. Replace the tools for doing \ special relativity--4-vectors, metrics, tensors, and groups--with quaternions \ that preserve the scalar of a squared quaternion. Replace the tools for \ deriving the Maxwell equations--4-potentials, metrics, tensors, and \ groups--by quaternion operators acting on quaternion potentials using \ combinations of commutators and anticommutators. It remains to be shown in \ this post whether quaternions also have the structure required for a quantum \ gravity theory.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Why do we study quantum gravity?"]], "Subsection"], Cell["\<\ Isham gives six reasons: the inability to calculate using perturbation theory \ a correction for general relativity, singularities, quantum cosmology \ (particularly the Big Bang), Hawking radiation, unification of particles, and \ the possibility of radical change. This last reason could be a lot of fun, \ and it is the reason to read this post :-)\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["What are prima facie questions?"]], "Subsection"], Cell["\<\ The first question raised by Isham is the relation between classical and \ quantum physics. Physics with quaternions has a general guide. Consider two \ arbitrary quaternions, q and q'. The classical distance between them is the \ interval. \ \>", "Text"], Cell[BoxData[ \(\((\((t, \ X\&\[RightVector])\) - \((t\^\[Prime], \ X\&\[RightVector]\^\[Prime])\))\)\^2 = \((dt\^2 - d X\&\[RightVector].d X\&\[RightVector], 2\ dt\ d X\&\[RightVector])\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ This involves retooling, because the distance also includes a 3-vector. \ There is nothing inherently wrong with this vector, and it certainly could be \ computed with standard tools. To be complete, measure the difference between \ two quaternions with a quaternion containing the usual invariant scalar \ interval and a covariant 3-vector. To distinguishing collections of events \ that are lightlike separated where the interval is zero, use the 3-vector \ which can be unique. Never discard useful information!\ \>", "Text"], Cell["\<\ Quantum mechanics involves a Hilbert space. Quaternions can be used to form \ an inner-product space. The norm of the difference between q and q' is\ \>", "Text"], Cell[BoxData[ \(\(\(( \((t, \ X\&\[RightVector])\) - \((t\^\[Prime], \ X\&\[RightVector]\^\[Prime])\))\)\^*\) \((\((t, \ X\&\[RightVector])\) - \((t\^\[Prime], \ X\&\[RightVector]\^\[Prime])\))\)\n = \((dt\^2 + d X\&\[RightVector].d X\&\[RightVector], 0\&\[RightVector]) \)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ The norm can be used to build all the equipment expected of a Hilbert space, \ including the Schwarz and triangle inequalities. The uncertainty principle \ can be derived in the same way as is done with the complex-valued wave \ function.\ \>", "Text"], Cell["\<\ I call q q' a Grassman product (it has the cross product in it) and q* q' the \ Euclidean product (it is a Euclidean norm if q = q'). In general, classical \ physics involves Grassman products and quantum mechanics involves Euclidean \ products of quaternions.\ \>", "Text"], Cell["\<\ Isham moves from big questions to ones focused on quantum gravity.Which \ classical spacetime concepts are needed? Which standard parts of quantum \ mechanics are needed? Should particles be united? With quaternions, all \ these concepts are required, but the tools used to build them morph and \ become unified under one algebraic umbrella.\ \>", "Text"], Cell["\<\ Isham points out the difficulty of clearly marking a boundary between \ theories and fact. He writes:\ \>", "Text"], Cell[TextData[StyleBox[ "\"...what we call a 'fact' does not exist without some theoretical schema \ for organizing experimental and experiential data; and, conversely, in \ constructing a theory we inevitably impose some prior idea of what we mean by \ a fact.\""]], "Text", CellMargins->{{36, 75}, {Inherited, Inherited}}], Cell["\<\ My structure is this: the description of events in spacetime using the \ topological algebraic field of quaternions is physics.\ \>", "Text"], Cell["\<\ Current research programs in quantum gravityThere is a list of current \ approaches to quantum gravity. This is solid a description of the family of \ approaches being used, circa 1993. See the text for details.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Prima facie questions in quantum gravity", "Subsection"], Cell["\<\ Isham is concerned with the form of these approaches. He writes:\ \>", "Text"], Cell["\<\ \"I mean (by background structure) the entire conceptual and structural \ framework within whose language any particular approach is couched. Different \ approaches to quantum gravity differ significantly in the frameworks they \ adopt, which causes no harm--indeed the selection of such a framework is an \ essential pre-requisite for theoretical research--provided the choice is made \ consciously.\"\ \>", "Text", CellMargins->{{36, 75}, {Inherited, Inherited}}], Cell["\<\ My framework was stated explicitly above, but it literally does not appear on \ the radar screen of this discussion of quantum gravity. Moments later comes \ this comment:\ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell["\<\ \"In using real or complex numbers in quantum theory we are arguably making a \ prior assumption about the continuum nature of space.\"\ \>", "Text", CellMargins->{{36, 75}, {Inherited, Inherited}}], Cell["\<\ This statement makes a hidden assumption, that quaternions do not belong on a \ list that includes real and complex numbers. Quaternions have the same \ continuum properties as the real and complex numbers. The important \ distinction is that quaternions do not commute. This property is shared by \ quantum mechanics so it should not banish quaternions from the list. The \ omission reflects the history of work in the field, not the logic of the \ mathematical statement.\ \>", "Text", CellMargins->{{7, Inherited}, {Inherited, Inherited}}], Cell["\<\ General relativity may force non-linearity into quantum theory, which require \ a change in the formalism. It is easy to write non-linear quaternion \ functions. Near the end of this post I will do that in an attempt to find \ the shortest norm in spacetime which happens to be non-linear.\ \>", "Text", CellMargins->{{7, Inherited}, {Inherited, Inherited}}], Cell["\<\ Now we come to the part of the paper that got me really excited! Isham \ described all the machinery needed for classical general relativity. The \ properties of quaternions dovetail the needs perfectly. I will quote at \ length, since this is helpful for anyone trying to get a handle on the nature \ of general relativity.\ \>", "Text", CellMargins->{{7, Inherited}, {Inherited, Inherited}}], Cell["\<\ \"The mathematical model of spacetime used in classical general relativity is \ a differentiable manifold equipped with a Lorentzian metric. Some of the \ most important pieces of substructure underlying this picture are illustrated \ in Figure 1. The bottom level is a set M whose elements are to be identified with \ spacetime 'points' or 'events'. This set is formless with its only general \ mathematical property being the cardinal number. In particular, there are no \ relations between the elements of M and no special way of labeling any such \ element. The next step is to impose a topology on M so that each point acquires a \ family of neighborhoods. It now becomes possible to talk about relationships \ between point, albeit in a rather non-physical way. This defect is overcome \ by adding the key of all standard views of spacetime: the topology of M must \ be compatible with that of a differentiable manifold. A point can then be \ labeled uniquely in M (at least locally) by giving the values of four real \ numbers. Such a coordinate system also provides a more specific way of \ describing relationships between points of M, albeit not intrinsically in so \ far as these depend on which coordinate systems are chosen to cover M. In the final step a Lorentzian metric g is placed on M, thereby introducing \ the ideas of the length of a path joining two spacetime points, parallel \ transport with respect to a Riemannian connection, causal relations between \ pairs of points etc. There are also a variety of possible intermediate steps \ between the manifold and Lorentzian pictures; for example, as signified in \ Figure 1, the idea of causal structure is more primitive than that of a \ Lorentzian metric.\"\ \>", "Text", CellMargins->{{36, 75}, {Inherited, Inherited}}], Cell["\<\ My hypothesis to treat events as quaternions lends more structure than is \ found in the set M. Specifically, Pontryagin proved that quaternions are a \ topological algebraic field. Each point has a neighborhood, and limit \ processes required for a differentiable manifold make sense. Label every \ quaternion event with four real numbers, using whichever coordinate system \ one chooses. Earlier in this post I showed how to calculate the Lorentz \ interval, so the notion of length of a path joining two events is always \ there. As described by Isham, spacetime structure is built up with care from \ four unrelated real numbers. With quaternions as events, spacetime structure \ is the observed properties of the mathematics, inherited by all quaternion \ functions.\ \>", "Text", CellMargins->{{7, Inherited}, {Inherited, Inherited}}], Cell["\<\ Much work in quantum gravity has gone into viewing how flexible the spacetime \ structure might be. The most common example involves how quantum \ fluctuations might effect the Lorentzian metric. Physicists have tried to \ investigate how such fluctuation would effect every level of spacetime \ structure, from causality, to the manifold to the topology, even the set M \ somehow.\ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell["\<\ None of these avenues are open for quaternion work. Every quaternion \ equation inherits this wealth of spacetime structure. It is the family \ quaternion functions are born in. There is nothing to stop combining \ Grassman and Euclidean products, which at an abstract level, is the way to \ merge classical and quantum descriptions of collections of events. If a \ non-linear quaternion function can be defined that is related to the shortest \ path through spacetime, the cast required for quantum gravity would be \ complete.\ \>", "Text", CellMargins->{{7, Inherited}, {Inherited, Inherited}}], Cell["\<\ According to Isham, causal structure is particularly important. With \ quaternions, that issue is particularly straightforward. Could event q have \ caused q'? Take the difference and square it. If the scalar is positive, \ then the relationship is timelike, so it is possible. Is it probable? That \ might depend on the 3-vector, which could be more likely if the vector is \ small (I don't understand the details of this suggestion yet). If the scalar \ is zero, the two have a lightlike relationship. If the scalar is negative, \ then it is spacelike, and one could not have caused the other.\ \>", "Text", CellMargins->{{7, Inherited}, {Inherited, Inherited}}], Cell["\<\ This causal structure also applies to quaternion potential functions. For \ concreteness, let q(t) = cos(pi t (2i + 3j + 4k)) and q'(t) = sin( pi t (5i - \ .1j + 2k). Calculate the square of the difference between q and q'. \ Depending on the particular value of t, this will be positive, negative or \ zero. The distance vectors could be anywhere on the map. Even though I \ don't know what these particular potential functions represent, the causal \ relationship is easy to calculate, but is complex and not trivial.\ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "The role of the spacetime diffeomorphism group Diff(M)"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}], Cell["\<\ Isham lets me off the hook, saying \"...[for type 3 and 4 theories] there is \ no strong reason to suppose that Diff(M) will play any fundamental role in \ [such] quantum theory.\" He is right and wrong. My simple tool collection \ does not include this group. Yet the concept that requires this idea is \ essential. This group is part of the machinery that makes possible causal \ measurements of lengths in various topologies. Metrics change due to local \ conditions. The concept of a flexible, causal metric must be preserved.\ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell["\<\ With quaternions, causality is always found in the scalar of the square of \ the difference. For two events in flat spacetime, that is the interval. In \ curved spacetime, the scalar of the square is different, but it still is \ either positive, negative or zero.\ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["The problem of time", "Subsection"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}], Cell["\<\ Time plays a different role in quantum theory and in general relativity. In \ quantum, time is treated as a background parameter since it is not \ represented by an operator. Measurements are made at a particular time. In \ classical general relativity in curved spacetime, there are many possible \ metrics which might work, but no way to pick the appropriate one. Without a \ clear definition of measurement, the definition is non-physical. Fixing the \ metric cannot be done if the metric is subject to quantum fluctuations.\ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell["Isham raises three questions:", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell["\<\ \"How is the notion of time to be incorporated in a quantum theory of \ gravity? Does it play a fundamental role in the construction of the theory or is it a \ 'phenomenological' concept that applies, for example, only in some \ coarse-grained, semi-classical sense? In the latter case, how reliable is the use at a basic level of techniques \ drawn from standard quantum theory?\"\ \>", "Text", CellMargins->{{36, 75}, {Inherited, Inherited}}], Cell["\<\ Three solutions are noted: fix the background causal structure, locate events \ within functionals of fields, or make no reference to time.\ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell["\<\ With quaternions, time plays a central role, and is in fact the center of the \ matrix representation. Time is isomorphic to the real numbers, so it forms a \ totally ordered sub-field of the quaternions. It is not time per se, but the \ location of time within the event quaternion (t, x i, y j, z k) that gives \ time its significance. The scalar slot can be held by energy (E, px i, py j, \ pz k), the tangent of spacetime, by the interval of classical physics (t^2 - \ x^2 - y^2 - z^2, 2 tx i, 2 ty j, 2 tz k) or the norm of quantum mechanics \ (t^2 + x^2 + y^2 + z^2, 0, 0, 0). Time, energy, intervals, norms,...they all \ can take the same throne isomorphic to the real numbers, taking on the \ properties of a totally ordered set within a larger, unordered framework. \ Events are not totally ordered, but time is. Energy/momenta are not totally \ ordered, but energy is. Squares of events are not totally ordered, but \ intervals are. Norms are totally ordered and bounded below by zero. \ \>", "Text", CellMargins->{{7, Inherited}, {Inherited, Inherited}}], Cell["\<\ Time is the only element in the scalar of an event. Time appears in \ different guises for the scalars of energy, intervals and norms. The \ richness of time is in the way it weaves through these other scalars, sharing \ the center in different ways with space.\ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Approaches to quantum gravity"]], "Subsection", CellMargins->{{0, Inherited}, {Inherited, Inherited}}], Cell["\<\ Isham surveys the field. At this point I think I'll just explain my \ approach. It is based on a concept from general relativity. A painter \ falling from a ladder travels along the shortest path through spacetime. How \ does one go about finding the shortest path? In Euclidean 3-space, that \ involves the triangle inequality. A proof can be done using quaternions if \ the scalar is set to zero. That proof can be repeated with the scalar set \ free. The result is the shortest distance through spacetime, or gravity, \ according to general relativity.\ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell["\<\ What is the shortest distance between two points A and B in Euclidean \ 3-space?\ \>", "Text", CellMargins->{{7, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ StyleBox[ \(A\ = \ \(\((0, \ ax, \ ay, \ az)\)\ \ \ \ B\ = \ \((0, \ bx, \ by, \ bz)\)\)\)]], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ What is the shortest distance between two worldlines A(t) and B(t) in \ spacetime?\ \>", "Text", CellMargins->{{7, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ StyleBox[ \(A \((t)\)\ = \ \((t, \ ax \((t)\), \ ay \((t)\), \ az \((t)\))\)\ \ \ \ \n B \((t)\)\ = \ \((t, \ bx \((t)\), \ by \((t)\), \ bz \((t)\))\)\)]], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ The Euclidean 3-space question is a special case of the worldline question. \ The same proof of the triangle inequality answers both questions. \ Parameterize the norm N(k) of the sum of A(t) and B(t).\ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ StyleBox[ \(N \((k)\)\ = \ \(\(\((A\ + \ k\ B)\)\^*\)\ \((A\ + \ k\ B)\)\n\ \ \ \ \ \ \ = \ \(A\^*\) A\ + \ k \((\(A\^*\)\ B\ + \ \(B\^*\)\ A)\)\ + \ k^2\ \(B\^*\)\ B\)\)]], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["Find the extremum of the parameterized norm.", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ RowBox[{ StyleBox[" "], RowBox[{ StyleBox[\(dN\/dk\)], StyleBox[" "], StyleBox["="], StyleBox[" "], RowBox[{ StyleBox["0"], StyleBox[" "], StyleBox["="], StyleBox[" "], RowBox[{\(\(A\^*\)\ B\), " ", "+", " ", \(\(B\^*\)\ A\), StyleBox[" "], StyleBox["+"], StyleBox[" "], RowBox[{ StyleBox["2"], StyleBox[" "], StyleBox["k"], StyleBox[" "], \(B\^*\), " ", "B"}]}]}]}]}]], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["The extremum is a minimum", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ RowBox[{ StyleBox[" "], RowBox[{ FractionBox[ RowBox[{ SuperscriptBox[ StyleBox["d"], "2"], StyleBox[" "], StyleBox["N"]}], SuperscriptBox[ StyleBox["dk"], "2"]], StyleBox[" "], StyleBox["="], StyleBox[" "], RowBox[{ RowBox[{ StyleBox["2"], StyleBox[" "], \(B\^*\), " ", "B"}], StyleBox[" "], "\[GreaterEqual]", StyleBox[" "], StyleBox["0"]}]}]}]], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["\<\ The minimum of a quaternion norm is zero. Plug the extremum back into the \ first equation.\ \>", "Text", CellMargins->{{7, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(0\ \[LessEqual] \ \(A\^*\) A - \ \((\(A\^*\)\ B\ + \ \(B\^*\)\ A)\)\^2\/\(2\ \(B\^*\)\ B\)\ + \ \((\(A\^*\)\ B\ + \ \(B\^*\)\ A)\)\^2\/\(4\ \(B\^*\)\ B\)\)], "Input",\ CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["Rearrange.", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\((\(A\^*\)\ B\ + \ \(B\^*\)\ A)\)\^2 \[LessEqual] 4 \( A\^*\) A\ \(B\^*\)\ B\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["Take the square root.", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(\(A\^*\)\ B\ + \ \(B\^*\)\ A \[LessEqual] 2 \@\(\( A\^*\) A\ \(B\^*\)\ B\)\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["Add the norm of A and B to both sides.", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ \(A*\ A\ + \ \(A\^*\)\ B\ + \ \(B\^*\)\ A\ + \ \(B\^*\)\ B\ \[LessEqual] \ \(A\^*\) A\ + \ 2\ \@\(\(A\^*\) A\ \(B\^*\)\ B\) + \ \(B\^*\)\ B\)], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->16], Cell["Factor.", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ RowBox[{ StyleBox[\(N \((A\ + \ B)\)\), FontSize->16], StyleBox["=", FontSize->16], RowBox[{ RowBox[{ RowBox[{ StyleBox["(", FontSize->16], StyleBox[\(A\ + \ B\), FontSize->16], StyleBox[\()\^*\), FontSize->16]}], StyleBox[" ", FontSize->16], StyleBox[\((A\ + \ B)\), FontSize->16]}], StyleBox[" ", FontSize->16], StyleBox["\[LessEqual]", FontSize->16], StyleBox[" ", FontSize->16], StyleBox[ RowBox[{"(", RowBox[{\(\@\(\(A\^*\) A\)\), " ", "+", " ", FormBox[\(\(\@\(\(B\^*\)\ B\))\)\^2\), "TraditionalForm"]}]}], FontSize->16]}]}]], "Input", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell["\<\ The norm of the worldline of A plus B is less than the norm of A plus the \ norm of B.\ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell["\<\ List the mathematical structures required. To move the triangle inequality \ from Euclidean 3-space to worldlines required the inclusion of the scalar \ time component of quaternions. The proof required differentiation to find \ the minimum. The norm is a Euclidean product, which plays a central role in \ quaternion quantum mechanics. Doubling A or B does not double the norm of \ the sum due to cross terms, so the minimal function is not linear. \ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell["\<\ To address a question raised by general relativity with quaternions required \ all the structure Isham suggested except causality using the Grassman \ product. The above proof could be repeated using Grassman products. The \ only difference would be that the extremum would be an interval which can be \ positive, negative or zero (a minimum, a maximum or an inflection point). \ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}] }, Open ]], Cell[CellGroupData[{ Cell["Certainty is seven for seven ", "Subsection", CellMargins->{{8, Inherited}, {Inherited, Inherited}}], Cell["\<\ I thought I'd end this long post with a personal story. At the end of my \ college days, I started drinking heavily. Not alcohol, soda. I'd buy a \ Mellow Yellow and suck it down in under ten seconds. See, I was thirsty. \ Guzzle that much soda, and, well, I also had to go to the bathroom, even in \ the middle of the night. I was trapped in a strange cycle. Then I noticed \ my tongue was kind of foamy. Bizarre. I asked a friend with diabetes what \ the symptoms of that disease were. She rattled off six: excessive thirst, \ excessive urination, foamy tongue, bad breath, weight loss, and low energy. \ I concluded on the spot I had diabetes. She said that I couldn't be certain. \ Six for six is too stringent a match, and I felt very confident I had this \ chronic illness. I got the seventh later when she tested my blood glucose on \ her meter and it was off-scale. She gave me sympathy, but I didn't feel at \ all sorry for myself. I wanted facts: how does this disease work and how do \ I cope? Nothing was made official until I visited the doctor and he ran some tests. \ The doctor's prescription got me access to the insulin I could no longer \ produce. It was, and still is today, a lot of work to manage the disease. When I look at Isham's paper, I see six constraints on the structure of any \ approach to quantum gravity: events are sets of 4 numbers, events have \ topological neighborhoods, they live on differential manifolds, there is one \ of the three types of causal relationships between all events, the distance \ between events is the interval whose form can vary and a Hilbert space is \ required for quantum mechanics. Quaternions are six for six. The seventh \ match is the non-linear shortest norm of spacetime. I have no doubt in the \ diagnosis that the questions in quantum gravity will be answered with \ quaternions. Nothing here is official. There are many test that must be \ passed. I don't know when the doctor will show up and make it official. It \ will take a lot of work to manage this solution.\ \>", "Text", CellMargins->{{8, Inherited}, {Inherited, Inherited}}] }, Open ]] }, Open ]] }, FrontEndVersion->"Microsoft Windows 3.0", ScreenRectangle->{{0, 640}, {0, 451}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Automatic, WindowSize->{509, 244}, WindowMargins->{{5, Automatic}, {Automatic, 6}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], Inherited, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], Inherited, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, 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[TextData[StyleBox["Style Definitions"]], "Subtitle"], Cell[TextData[StyleBox[ "Modify the definitions below to change the default appearance of all cells \ in a given style. 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Cell[StyleData["Text", "Printout"], CellMargins->{{2, 2}, {6, 6}}, FontFamily->"Times New Roman"] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SmallText"], CellMargins->{{7, 10}, {6, 6}}, Evaluatable->False, CellHorizontalScrolling->False, PageBreakWithin->Automatic, LineSpacing->{1, 3}, CounterIncrements->"SmallText", AspectRatioFixed->True, FontFamily->"Times New Roman", FontSize->10], Cell[StyleData["SmallText", "Presentation"], CellMargins->{{24, 10}, {8, 8}}, LineSpacing->{1, 5}, FontFamily->"Times New Roman", FontSize->12], Cell[StyleData["SmallText", "Condensed"], CellMargins->{{8, 10}, {5, 5}}, LineSpacing->{1, 2}, FontFamily->"Times New Roman", FontSize->9], Cell[StyleData["SmallText", "Printout"], CellMargins->{{2, 2}, {5, 5}}, FontFamily->"Times New Roman", FontSize->7] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Styles for Input/Output"]], "Section", FontSize->14, FontWeight->"Plain"], Cell[TextData[StyleBox[ "The cells in this section define styles used for input and output to the \ kernel. 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Cell[StyleData["Input", "Condensed"], CellMargins->{{40, 10}, {2, 3}}, FontSize->14, FontWeight->"Plain"], Cell[StyleData["Input", "Printout"], CellMargins->{{39, 0}, {4, 6}}, FontSize->14, FontWeight->"Plain"] }, Closed]], Cell[StyleData["InputOnly"], Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", StyleMenuListing->None, FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[CellGroupData[{ Cell[StyleData["Output"], PageWidth->Infinity, CellMargins->{{42, 10}, {7, 5}}, CellEditDuplicate->True, Evaluatable->False, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, 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Cell[CellGroupData[{ Cell[StyleData["Graphics"], PageWidth->Infinity, CellMargins->{{7, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", AspectRatioFixed->True, ImageSize->{387, 393}, ImageMargins->{{34, Inherited}, {Inherited, 0}}, StyleMenuListing->None, FontFamily->"Courier", FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[StyleData["Graphics", "Presentation"], ImageMargins->{{62, Inherited}, {Inherited, 0}}, FontSize->14], Cell[StyleData["Graphics", "Condensed"], ImageSize->{175, 175}, ImageMargins->{{38, Inherited}, {Inherited, 0}}, FontSize->14], Cell[StyleData["Graphics", 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Cell[StyleData["InlineFormula", "Condensed"], CellMargins->{{8, 10}, {6, 6}}, LineSpacing->{1, 1}], Cell[StyleData["InlineFormula", "Printout"], CellMargins->{{2, 0}, {6, 6}}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}], Cell[StyleData["DisplayFormula", "Condensed"], LineSpacing->{1, 1}], Cell[StyleData["DisplayFormula", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Styles for Headers and Footers"]], "Section"], Cell[StyleData["Header"], CellMargins->{{7, 0}, {4, 1}}, Evaluatable->False, PageBreakWithin->Automatic, AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Times", FontSize->12, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{7, 0}, {0, 4}}, Evaluatable->False, PageBreakWithin->Automatic, TextAlignment->Center, AspectRatioFixed->True, StyleMenuListing->None, FontFamily->"Times", FontSize->12, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Palette Styles"]], "Section"], Cell[TextData[StyleBox[ "The cells below define styles that define standard ButtonFunctions, for \ use in palette buttons."]], "Text"], Cell[StyleData["Paste"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, After]}]&)}], Cell[StyleData["Evaluate"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookApply[ FrontEnd`InputNotebook[ ], #, All], 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on-line help \ system."]], "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"]], Cell[StyleData["MainBookLink", "Condensed"]], Cell[StyleData["MainBookLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"]], Cell[StyleData["AddOnsLink", "Condensed"]], Cell[StyleData["AddOnLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Condensed"]], Cell[StyleData["RefGuideLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"]], Cell[StyleData["GettingStartedLink", "Condensed"]], Cell[StyleData["GettingStartedLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"]], Cell[StyleData["OtherInformationLink", "Condensed"]], Cell[StyleData["OtherInformationLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Placeholder Styles"]], "Section"], Cell[TextData[StyleBox[ "The cells below define styles useful for making placeholder objects in \ palette templates."]], "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->True, StyleMenuListing->None], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SelectionPlaceholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->Primary, StyleMenuListing->None, DrawHighlighted->True], Cell[StyleData["SelectionPlaceholder", "Presentation"]], Cell[StyleData["SelectionPlaceholder", "Condensed"]], Cell[StyleData["SelectionPlaceholder", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["FormatType Styles"]], "Section"], Cell[TextData[StyleBox[ "The cells below define styles that are mixed in with the styles of most \ cells. 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