Answering prima facie questions in quantum gravity using quaternions

doug  <sweetser@world.com>

Introduction
What is quantum gravity?
Why do we study quantum gravity?
What is a prima dacie question?
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)

Introduction

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.

What is quantum gravity?

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.

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.

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.

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.

Why do we study quantum gravity?

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 :-)

What are prima facie questions?

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.

((((t,  (Overscript[X, ⇀])) - 
            ((t^′),  ((Overscript[X, ⇀])^′))))^2) = 
      ((dt^2) - d  (Overscript[X, ⇀]).d  (Overscript[X, ⇀]), 
        2 dt d  (Overscript[X, ⇀]))

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!

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

\(\(\((
              \((t, \ X\&⇀)\) - 
                \((t\^′, \ X\&⇀\^′)\))\)\^*\) 
        \((\((t, \ X\&⇀)\) - 
            \((t\^′, \ X\&⇀\^′)\))\)
 = 
      \((dt\^2 + d  X\&⇀.d  X\&⇀, 0\&⇀)
        \)\)

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.

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.

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.

Isham points out the difficulty of clearly marking a boundary between theories and fact.  He writes:

"...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."

My structure is this:  the description of events in spacetime using the topological algebraic field of quaternions is physics.

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.

Prima facie questions in quantum gravity

Isham is concerned with the form of these approaches.  He writes:

"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."

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:

"In using real or complex numbers in quantum theory we are arguably making a prior assumption about the continuum nature of space."

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.

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.

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.

"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."

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.

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.

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.

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.

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.

The role of the spacetime diffeomorphism group Diff(M)

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.

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.

The problem of time

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.

Isham raises three questions:

"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?"

Three solutions are noted: fix the background causal structure, locate events within functionals of fields, or make no reference to time.

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.

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.

Approaches to quantum gravity

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.

What is the shortest distance between two points A and B in Euclidean 3-space?

A  =  (0,  ax,  ay,  az)    B  =  
          (0,  bx,  by,  bz)

What is the shortest distance between two worldlines A(t) and B(t) in spacetime?

\(A \((t)\)\  = \ 
        \((t, \ ax \((t)\), \ ay \((t)\), \ az \((t)\))\)\ \ \ \ 

      B \((t)\)\  = \ \((t, \ bx \((t)\), \ by \((t)\), \ bz \((t)\))\)\)

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).

\(N \((k)\)\  = \ 
        \(\(\((A\  + \ k\ B)\)\^*\)\ \((A\  + \ k\ B)\)
\ \ \ \ \ \ \  = \ 
          \(A\^*\) A\  + \ k \((\(A\^*\)\ B\  + \ \(B\^*\)\ A)\)\  + \ 
            k^2\ \(B\^*\)\ B\)\)

Find the extremum of the parameterized norm.

(dN/dk) = 0 = \(\(A\^*\)\ B\) + \(\(B\^*\)\ A\) + 2 k \(B\^*\) B

The extremum is a minimum

(((d^2) N)/(dk^2)) = 2 \(B\^*\) B ≥ 0

The minimum of a quaternion norm is zero.  Plug the extremum back into the first equation.

\(0\  ≤ \ 
      \(A\^*\) A - \ 
        \((\(A\^*\)\ B\  + \ \(B\^*\)\ A)\)\^2\/\(2\ \(B\^*\)\ B\)\  + \ 
        \((\(A\^*\)\ B\  + \ \(B\^*\)\ A)\)\^2\/\(4\ \(B\^*\)\ B\)\)

Rearrange.

\(\((\(A\^*\)\ B\  + \ \(B\^*\)\ A)\)\^2 ≤ 
      4 \( A\^*\) A\ \(B\^*\)\ B\)

Take the square root.

\(\(A\^*\)\ B\  + \ \(B\^*\)\ A ≤ 
      2 \@\(\( A\^*\) A\ \(B\^*\)\ B\)\)

Add the norm of A and B to both sides.

\(A*\ A\  + \ \(A\^*\)\ B\  + \ \(B\^*\)\ A\  + \ \(B\^*\)\ B\  
      ≤ \ 
      \(A\^*\) A\  + \ 2\ \@\(\(A\^*\) A\ \(B\^*\)\ B\) + \ \(B\^*\)\ B\)

Factor.

N (A  +  B)=(A  +  B\()\^*\) (A  +  B) ≤ (\(\@\(\(A\^*\) A\)\) + \(\(\@\(\(B\^*\)\ B\))\)\^2\)

The norm of the worldline of A plus B is less than the norm of A plus the norm of B.

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.

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).

Certainty is seven for seven

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.