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Riemannian Geometry Framed as a Non-Commutative Algebra of Observables

Joseph E. Johnson1

1. Distinguished Professor Emeritus, Department of Physics & Astronomy, University of South Carolina, Columbia SC, 29208

Abstract: This paper reframes Riemannian geometry (RG) as a generalized Lie algebra allowing the

equations of both RG and then General Relativity (GR) to be expressed as commutation relations among

fundamental operators. We begin with an Abelian Lie algebra of n operators, X whose simultaneous

eigenvalues, y, define a real n-dimensional space R(n). Then with n new operators defined as independent

functions, X’(X), we define contravariant and covariant tensors in terms of their eigenvalues, y’, on a

Hilbert space representation. We then define n additional operators, D,whose exponential map is to

translate Xas defined by a noncommutative algebra of operators (observables) where the “structure

constants” are shown to be the metric functions of the X operators thus allowing for spatial curvature

resulting in a noncommutativity among the D operators. The Doperators then have a Hilbert space

position-diagonal representation as generalized differential operators which, with the metric, written as a

commutator, can express the Christoffel symbols, and the Riemann, Ricci and other tensors as

commutators in this representation. Traditional RG and GR are obtained in a position diagonal

representation of this noncommutative algebra of operators. Our motivation was suggested by the fact that

Quantum Theory (QT), Special Relativity (SR), and the Standard Model (SM) are framed and well-

established in terms of Lie algebras. But GR, while also well-established, is framed in terms of nonlinear

differential equations for the space-time metric and space-time variables. We seek to provide a more

general framework for RG to support an integration of GR, QT, and the SM by generalizing Lie algebras

as described.

Keywords: Riemannian Geometry, Lie algebra, Quantum Theory, metric, Heisenberg algebra >November 18, 2019 version

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1. Introduction

Lie algebras and the Lie groups which they generate have played a central role in both

mathematics and theoretical physics since their introduction by Sophius Lie in 1888 [1]. Both relativistic

quantum theory (QT) and the gauge algebras of the phenomenological standard model (SM) of particles

and their interactions are framed in terms of observables which form Lie algebras and are firmly

established [2,3,4,5]. The Heisenberg Lie algebra (HA) among (generalized) momentum and position

operators, [D, X], in QT also has applications in mathematics in studies related to Fourier transforms and

harmonic analysis [6,7,8,9]. Likewise, in QT one has the Poincare symmetry Lie algebra (PA) of space-time

observables whose representations define free particles.

But the theory of gravitation as expressed in Einstein’s general theory of relativity (GR), although

also firmly established, is formulated in terms of a Riemannian geometry (RG) of a curved space-time

where the metric is determined by nonlinear differential equations from the distribution of matter and

energy [10,11]. In GR there are no operators representing observables, and thus no commutation rules to

define Lie algebras, and thus no representations of such algebras. The observables in GR are (a) the

positions of events in space-time, and (b) the metric function of position in space-time (and its

derivatives) which define the distance between events, and thereby define the curvature of space-time.

Thus QT and GR are expressed in totally different mathematical frameworks and their merger into a

single theory has been a central problem in physics for over a century. However, the space-time events in

QT are the eigenvalues of the space-time operators which are an essential part of the HA which also

contains the Minkowsky metric which defines the associated translated distance when space-time is not

curved. If the associated space were curved, one would have a metric that was a function of the position

in space-time. Such a generalized HA would no longer allow closure as a traditional Lie algebra but rather

closure in the enveloping algebra of analytic functions of the basis elements of the Lie algebra.

This led us to consider a noncommutative algebra of operators (NAO), generalizing the form of a

Lie algebra, with n operators, X, and n corresponding operators D, which by definition are to execute

infinitesimal translations in the associated representation space of the X eigenvalues, y ( = 0, 1, .n-1).

The X are to form an Abelian algebra whose eigenvalues represent a “space-time” manifold of four or a

larger number of dimensions as the associated X eigenvalues are simultaneously measureable. But we

will allow the space to be curved so the corresponding D operators will not in general commute as the

translations in such a curved space can interfere with each other. We found that this approach generalized

the HA “structure constants” to be the Riemann metric thus allowing the metric to be a function of the

position operators, X, in the algebra [12]. This generalizes the concept of a Lie algebra to allow for

“structure constants” that are functions of the operators in the algebra and thus are no longer constants

except approximately in small neighborhoods. This paper formally reframes RG [13] in terms of a NAO or

generalized HA. We show that the fundamental concepts in RG such as the coordinate transformations,

contravariant and covariant tensors, Christoffel symbols, Riemann and Ricci tensors, and the Riemann

covariant derivative can all be expressed in terms of commutation relations among the fundamental

operators. This framework is reminiscent of contractions of Lie algebras where the structure constants as

functions are modified to vary smoothly among different algebras based upon certain external parameters [14, 15, 16, 17, 18, 19] but not in the algebra itself. In a similar way, our algebra allows the structure constants to

be dependent upon the X operators in the algebra so that RG is retrieved as a position-diagonal

representation of the algebra as one moves over the Riemann manifold of X eigenvalues.

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2. Riemannian Geometry Framed as a Noncommutative Algebraic Geometry of Observables

Consider a set of n independent linear self adjoint operators, X, which form an Abelian Lie

algebra of order n, where

[XXand where = 0, 1, 2, … (n-1). (2.1)

Consider a Hilbert space of square integrable complex functions |> as a representation space for this

algebra where a scalar product is used to normalize the vectors to unity: . The simultaneous

eigenvectors of the Abelian Lie algebra can be written as the outer product of the Xeigenvectors with the

Dirac notation

|y0> |y1>|y2> …|yn-1> = | y0, y1, y2, …yn-1 > =|y> (2.2)

where the eigenvalues y label the associated eigenvectors |y> of the X operators and where we use the

notation

X|y> = y|y> where the y are real numbers. (2.3)

These independent real variables y can be thought of as the coordinates of an n-dimensional space Rn

since each set of values defines a point in Rn. Let the eigenvectors be normalized to be orthonormal with

the scalar product

<ya|yb> = (y0a-y0

b) (y1a-y1

b)… (yn-1a-yn-1

b). (2.4)

Let the decomposition of unity

1 = dy|y><y| (2.5)

project the entire space onto the basis vectors |y> where <y|, using Dirac notation, is the dual vector to

|y>. A general vector in the representation (Hilbert) space of this Lie algebra can then be written as

|> = dy|y><y|> = dyy) |y>, (2.6)

where the functiony) gives the “components” of the abstract vector |> on the basis vectors |y>. Thus

<|> = 1 = dy <|y><y|> = dy y y

Now consider another set of n linear operators, X’, which are independent analytic functions, X’X),

of the Xoperators also forming an Abelian Lie algebra on the same representation space for this algebra

where it follows that

[X’X’

Let the X’ have eigenvectors |y’> and eigenvalues y’ given by

X’|y’> = y’|y’> where y’ are real numbers (2.9)

The same orthonormality and decomposition of unity also obtain for the |y’> vectors which are also a

complete basis for the space. Then we can let the X’X act to the left on the dual vector y’| and also act

to the right on the vector |y> as

<y’| X’X|y> = <y’| X’X|y> to give (2.10)

y’y’|y> = y’y)y’|y>. (2.11)

Thus the eigenvalues y’ = y’y)give the transformation from the y coordinates to the y’ coordinates if the

Jacobian does not vanish i.e. | y’ y| ≠ 0 which we require to be the case. Thus the operators X’X)

define a coordinate transformation in Rn between the eigenvalues (coordinates) y and the eigenvalues y’

(transformed coordinates) define Rn’. Then the set of n real variables y and the alternative variables y’

both can be interpreted as specifying the coordinates of points in this n-dimensional real space Rn with

coordinate transformations given by the functions

y’y’ y

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It now follows that

dy’ y’ y) dy

and any set of n functions V(y) that transform as the coordinates,

V’(y’) = y’ y) V(y) is to be called a contravariant vector. (2.14)

We use the summation convention for repeated identical indices. The derivatives y transform as

y' = y y’) y (2.15)

and any such vector V(y) which transforms in this manner as

V’(y’) = y y’) V(y) is defined as a covariant vector. (2.16)

Upper indices are defined as contravariant indices while lower indices are covariant indices. Functions

with multiple upper and lower indices that transform as the contravariant and covariant indices just shown

are defined as tensors of the rank of the associated indices.

One would like to have transformations that translate one in the Rn space of the operators X (and

thus their eigenvalues y). We define a new additional set of n operators, D, that by definition translates a

point an infinitesimal distance, ds, in the Rn respectively in each corresponding direction y by using the

transformation generated by the D elements of the algebra via the exponential map with transformations:

G(ds, ) =exp(ds D/b) where b is a constant (2.17)

In this transformation is to be a unit vector in the y space, b is an unspecified constant, and ds is

defined to be the distance moved in the direction as defined below. Then

X’G XG

By taking ds to be infinitesimal, then one gets

X’Xsds)exp( ds D/b) Xsexp(- ds D

/b)

= (1 + ds D/b) Xs- ds D/b)

= X s ds D, X/bhigher order in ds., (2.19)

Thus the commutator D, Xdefines the way in which the transformations commute (interact)

with each other in executing the translations in keeping with the theory of Lie algebras and Lie groups

although in general the D & X do not close as a standard Lie algebra. If the space is flat then there is no

dependence of the commutator upon location, and thus there is no interference among the D. Then D

Xcan be normalized to ±(since D

is defined to translate X ) thus

D, XI± b ±

(2.20)

where ± is the diagonal n x n matrix with ±1 on the diagonal with off-diagonal terms zero. This is the

customary Heisenberg Lie algebra with structure constants ± and with D, D 0 for ≠. The

operator I is to commute with all elements and by definition has a single eigenvalue b, and is needed to

close the basis of the Lie algebra which is of dimension 2n+1. Thus confirming that the distance is ds:

dX s ds b/b± ds + higher order terms in ds. (2.21)

We now wish to allow for curvature in the space Rn of the X eigenvalues. Thus the [D, X]

commutator is now allowed to be dependent upon the operators X and can vary from point to point in the

space. We define the functions g(X) as generalized structure functions (no longer constants):

D, X bg(X) (with the requirement that |g|≠ 0 ) (2.22)

wherehas the single eigenvalue b with the commutators

[D , ] = 0 = [X , ], and [X , X] = 0 . (2.23)

These generalized structure functions can also be written as

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g(X) = D, X/b

where g(X) are assumed to be analytic with g(X) defined by

g(X) g(X) = in the representation space. (2.25)

Then using (2.21) one gets

Xsds) - Xs) = dXds g(X) = ds . (2.26)

Then

g(X) dXdXds2 g(X)

= ds2 since is a unit vector on this metric, or (2.27)

ds2 g(X) dXdX proving that g(X) is the metric for the space. (2.28)

One notes that equation [2.22] would be identical to the Heisenberg algebra if one were to set

b = iħ to get D, X iħ g(X) (2.29)

since D is the translation operator on X, and then invoke the operator interpretation of quantum theory.

But for now we retain the purely mathematical framework where b is simply an undefined constant.

One notes that g(X) can have an antisymmetric component as well as a symmetric. But only

the symmetric portion of g(X) contributes to the distance metric for the space since it is contracted

with the symmetric form dXdX. The antisymmetric component of g(X) can however support a torsion

(twisting) for the transformation although not contributing to the distance function ds. This results in a

2n+1 dimensional noncommutative algebraic geometry with D, X, and I as the algebras basis elements

(observables).

One notes that the commutator [D , D ] has not yet been defined. D can be represented on the

basis vectors of the Hilbert representation space where X is diagonal as

<y|[D , X ] = <y| b g(X) as (2. 30)

<y| Db g(y) /yA(y) )<y| = b A(y) )where ∂ = g(y) (∂/∂y) (2.31)

as the representation of Don the space of eigenvectors <y| and where A(y) is an arbitrary collection of

vector functions of y. Note that this arbitrary vector function A(y) can include other terms such as

b g(y) y/y . So one could write

D’ = DA(X) (2.32)

in the commutators with X as this would not alter the commutation rules of D with X (but which does alter

the commutators [D,D]. This is the most general representation of the commutation rules with the

operators available using the scalar, vector and second rank tensor representations. Both the vector

function A(y) and the scalar function (y) could consist of multiple higher order tensor components

including g(X), arbitrary scalar functions, arbitrary contravariant vector function A(X) and derivatives of

such objects because any contravariant vector function of the X will commute with the X in the defining

commutator of D and X. The A(y) can thus support a Yang Mills gauge transformation group, acting both

on the representation space and on the A(y) vector functions. In that case the A(y) will have the

commutation rules of that algebra with additional indices supporting Yang Mills gauge transformations.

If that gauge algebra were to be extended to include g(X) then the commutators are more complex.

Since [D, X] = I g(X), this is a generalization of the normal definition of a Lie algebra since

g(X) is now a function of the position operators X which, in the position representation |y>, become the

eigenvalues which determine the position in the n dimensional space. Consequently, this “generalized Lie

algebra” has “structure functions” (not constants), g(y), which vary from point to point in the space.

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From now on we assume the general case where g = g (y) is to be understood in the position

representation.

In the position representation one now has

<y| D|> =(b g(y) (∂/∂y) + A (y) ) (y) = ( b ∂ + A (y) ) (y) where (2.33)

(y) = <y|> and (2.34)

∂ = g(y) (∂/∂y) (2.35)

and A(y) is an yet undetermined vector function of X. In the position representation, one can write

g (∂ /∂y) (y) <y| = ∂ (y) <y| = <y| ( 1/b) [D, (X)]) (2.36)

for any function (X) allowing one to convert differential operators into commutators with D. It follows

that [D, [D, X]] ≠ 0 so that this Heisenberg type algebra is no longer nilpotent. But instead one gets

<y| [D, [D, X]] = b2 g (∂g/∂y) <y| since (2.37)

[A, g] = 0 (2.38)

as they both are only functions of X.

We have not specified the commutators [D, D ] (often written as F ) yet as they are no longer

zero but which in the position representation with b=1 give

<y| [D, D ] = [(g(y) (∂/∂y) + A (y)), (g(y) (∂/∂y)+ A (y))] <y| or (2.39)

= -(( g(y)(∂g(y)/∂y)– (g(y) (∂g(y)/∂y)) (∂/∂y)

-(( g(y)(∂A(y)/∂y)– (g(y) (∂A(y)/∂y)) + [A, A ] ) <y|. (2.40)

If the vector fields are generalized to Yang-Mills gauge fields such as in the standard model in physics

one gets

[D, D ] = -(∂g- ∂g ) ∂ (∂AC ∂AC )CABC AA, AB (2.41)

where [AA, AB ] = CABC AA ABwhere the A, B, & C indices are those of the SM. ()

3. Reframing of Riemannian Geometry in the Non-Commutative Algebra

The Christoffel symbols are given by

= (½) (∂, g + ∂, g- ∂, g ) (3.1)

and can be written in the position diagonal representation, in terms of the commutators of D with the

metric as

= (½) (1/b) ( [D, g] + [D, g] - [ D, g ] ). (3.2)

Then using

g (X) = (1/b) [D, X] one obtains (3.3)

= (½) (b-2) ( [D, [D, X]] + [D, [D, X]] - [ D, [D, X] ] ). (3.4)

The Riemann tensor then becomes

R = (1/b) ( [D, ] - [D, ] ) + ( -

) (3.5)

where is to be inserted for the Christoffel symbols using [3.4] giving only commutators. One then

defines the Ricci tensor using [3.5] for the Riemann tensor as

R = g R = (1/b) [D, X] R and also defines (3.6)

R = g R = (1/b) [D, X] R . (3.7)

It is well known that the ordinary derivative of a scalar function, V = /y , in Riemann

geometry will transform under arbitrary coordinate transformations as a covariant vector. But such a

derivative of a vector function of the coordinates will not transform as a tensor. The covariant derivative

with respect to y of a contravariant vector is

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/y

and the covariant derivative of a covariant vector is given by

/y

where both and transform as tensors with respect to the metric g.

One recalls for Riemannian geometry that there is a Christoffel symbol on the right hand side for

each index of the tensor being differentiated. In this algebraic framework one can write the covariant

differentiation of a contravariant vector as:

i [D ](½) ( [D, [D, X]] + [D, [D, X]] - [D, [D, X] ] ) (3.10)

assuming that A is at most a function of the X operators. Thus we are able to write both the regular

derivative (first term) and complete it with the index contraction with the Christoffel symbol (second

term). It is important to distinguish this covariant differentiation from the regular differentiation that

occurs as a representation of the operator D in the position representation. It follows that we can write the

covariant derivative of any tensor in the same way but with a contraction of the Christoffel symbol with

each of the tensor indices as is well known in Riemannian geometry. When the SM is incorporated then

the [D, A] commutator must not only include the Riemann covariant correction given by (3.10) but must

also include the covariant correction due to the [B, C from the SM.

4. General Relativity Framed as a Noncommutative Algebraic Geometry of Operators:

Up to now, the framework is for an abstract space. To connect this with GR and the foundations

of physics, we first connect this framework to special relativity with the assignment of the four-position

operators X and their eigenvalues as X0 = ct, and Xi = (x, y, z) with i = 1, 2, & 3 where c is the speed of

light with SI units of meters and seconds and D as the effective four momentum with eigenvalues (E/c,

Pi). In a theory of additional dimensions, subsequent values of Xi would refer to hidden dimensions.

In general relativity the Einstein equations for the metric:

R - ½ g R + g = (8 π G/c4) T can now be written (4.1)

R + (b [D, X]) ( ½ R - ) = (8 π G/c4) T

where R and R are now given in terms of commutators as shown above in (3.6) and (3.7) while Tis the

energy-momentum tensor. Thus all terms on the LHS consist only of commutators and (4.2) is an exact

reproduction of the Einstein equations in GR expressed totally as NAO commutators thus framing GR as

a non-commutative algebra of the X, D, I, and g(X) observables and their eigenvalues. The expression of

Einstein’s equations in the form of commutators of operators does not lead to new solutions, but one can

utilize the solutions that are already known. In a strong gravitational field near a star, such as a non-

rotating white dwarf, one can treat the metric as constant using the Schwarzschild solution over a region

that is small relative to the size of the star. The radial direction can be taken as the y1 direction as the

distance to the center of the star, with

g00 = (1 – rs/y1 ) and g11 = -1/(1 – rs/y1) (4.3)

where rs = 2GM/c2 with g22 = g33 = -1 (4.4)

and where G is the gravitational constant, M is the mass of the star, c is the speed of light, and y1 is the

distance to the center of the star giving g(X) as the Schwarzschild solution. Equation 4.2 is still exactly

the classical equation for the metric but recast in NAO commutation relations.

5. On Integrating this NAO framework of GR into a Quantum Theory Framework:

At this point we have reframed RG and GR both as a NAO thus providing a classical framework

that is similar to the Lie algebra framework of the HA and PA of relativistic quantum theory as well as the

LA structure of the SM. At this point, both RG and GR are fundamentally the same exact theory as

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before. But one can see suggestive parallels between the NAO as a generalization of the HA where the

four momentum D is not only the translation generator for space time, X, but where that commutator not

only fixes the constant b as the value iћ, but also naturally supports arbitrary vector fields as a

component of D and thus support the Yang Mills gauge basis for the standard model with D interpreted

physically as the effective four momentum. From these parallels, we invoke quantization with Din the

reframed GR (1) be taken as the four momentum with (b) the general formalism as in standard quantum

field theory with (c) the metric and vector fields Aas those of the SM, and (d) quantized with b = iћ.

This is a very difficult program which we have just begun to investigate. However, there are

certain results that follow immediately that could test the general concept of this approach: The position

and momentum operators are now to have the interpretational structure given by quantum theory with free

particles as representations of the position extended Poincare algebra. The essentially new feature is that

by virtue of the presence of a particle in a gravitational field such as near a star, the commutation rules

with the rationalized Planks constant, ћ, are effectively modified by the metric in the radial (X1) and time

(X0 ) directions with the specific prediction that in a small region, with the Schwarzschild metric, one gets

the altered uncertainty principles:

Xr Pr ≥ (ћ/2)(1/(1-rs/r)) and (5.1)

t Er ≥ (ћ/2) (1-rs/r) (5.2)

where rs = 2GM/c2 and where r = the distance to the center of the spherical mass. This is because the

generalized algebra effectively alters the value of Planks constant in both the X1 and X0 directions as well

as the wave nature of particles in to the altered local Fourier transform. This would in turn alter the

creation rate of virtual pairs in the vacuum in a strong gravitational field. What is maintained is a more

general form of the Heisenberg uncertainty principle obtained by multiplying (5.1) and (5.2) together to

obtain

t Er Xr Pr ≥ (ћ/2)2 (5.3)

while the other two uncertainty relations remain unchanged. Because the metric would be quantized along

with the vector fields in D, it follows that distance and angle in space-time would now be observationally

“granular” or “quantized” while leaving the underlying space itself as continuous.

One would now use traditional quantum field theory with the standard model intact, with all free

fields quantized as creation and annihilation operators as representations of the Poincare algebra. Then

the gravitational field g now included in D, would be quantized as the spin 2, (b0 = 0, b1 = 3 symmetric

tensor Poincare representation) massless field determined by equation (4.2) with other spin and helicity

components gauged away. Then the RHS would be expressed in terms of the symmetrized energy-

momentum density operator, T where D not only contains the vector fields of the standard model but

now also contains the gravitational tensor field g in parallel with the mediating forces of the vector

fields. The Twould be formed in the traditional way containing all fields for the energy-momentum

tensor in the Lagrangian as the source of the gravitational field as well as source terms for vector fields.

The generalization of the Fourier transform follows from <y|D|k> = <y|D|k> where the D acts

first to the left on the bra vector and then to the right on the ket vector, which is to be an eigenstate of D

with eigenvalue k giving the differential equation

(iħ g(y) (∂/∂y) + A(y) )<y|k> = (k + A(y)) <y|k>. (5.4)

When there is no vector field present and when g is constant (no y dependence & using a Minkowski

metric), then this can be solved (with normalization for a four dimensional space-time) with:

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<y|k> = (2 )-2 exp (g yk ) as in standard quantum theory. (5.5)

But in the general case with g(y) as a function of y this is no longer a solution and in the general case

one cannot solve this equation except formally. In fact, since the D do not commute among themselves,

one does not generally have a complete set of simultaneous Deigenvectors. However, one can consider

very small regions of space where the metric is effectively a constant as giving by a modified Fourier

transform. Then the general solution would be approximately the smoothing of these local traditional

solutions into a global solution maintaining functional and derivative continuity.

6. Lie Algebra Limit with no Gravitation as the Extended Poincare Algebra.

As quantum theory is founded upon the relationship between momentum and position operations

as defined in the HA, with [X, P] = iħ, and [E, t] = iħ, then a full Lie algebra of space-time observables

must also include a four-position operator X in order to formally include the foundations of quantum

theory as well as, M, the generators of the Lorentz group. This led us previously to extend the Poincare

algebra [20,21] (P & M) by adding a four-vector position operator, X ( … = 0, 1, 2, 3) whose components

are to be considered as fundamental observables using a manifestly covariant form of the HA. We briefly

review that algebra whose representations give free particles (fields). As the X generate translations in

momentum, they do not generate symmetry transformations or represent conserved quantities but do

provide the critical observables of space-time. We choose the Minkowski metric g = (+1, -1, -1, -1) and

write the HA in the covariant form as [P, X] = iħ I g where I is an operator that commutes with all

elements and has the unique eigenvalue “1” with P = E/c, P= Px ..., X = ct, X.= x … . Here “I” is needed to

make the fifteen (15) fundamental observables in this extended Poincare algebra (X, P, M, I) close into a

Lie algebra with the structure constants as follows:

[I, P] = [I, X ] = [I, M ] = 0 (6.1)

thus I commutes with all operators and has “1” as the only eigenvalue.

[P, X] = iħ g where the I eigenvalue is iħ (6.2)

which is the covariant Heisenberg Lie algebra – the foundation of quantum theory,

[P, P] = 0 (6.3)

which insures noninterference of energy momentum measurements in all four dimensions.

[X, X] = 0 (6.4)

which insures noninterference of time and position measurements in all four dimensions.

[M, P] = iħ (gPgP

which guarantees that P transforms as a vector under Mand thus the Lorentz group M

[M, X] = iħ (gX gX

which guarantees that Xalso transforms as a vector under Mand thus the Lorentz group.

M, M] = iħ (gM + gM - gM - gM). (6.6)

which guarantees that Mtransforms as a tensor under the Lorentz group generated by M.

The representations of the Lorentz algebra are well-known [8, 9 ] and are straight forward but the

extension to include the four-momentum with the Poincare algebra make determination of the Poincare

representations are somewhat complicated. But with our extension of the Poincare algebra to include a

four-position operator, X, the representations are clearer. Because X is now in the algebra, one can now

define the orbital angular momentum four-tensor, operator L as:

LX P – X P

From this it follows that

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[L, P] = iħ (gPgP

[L, X] = iħ (gX gX (6.9)

L, L] = iħ (g L + g L - g L - g L).

One can then define an intrinsic spin four-tensor as:

S = M - L (6.11)

with the result that

[S, P] = 0 ; [S, X] = 0 ; S, L] = 0; and (6.12)

S, S] = iħ (g S + g S - g S - g S) (6.13)

Now one can separate this EP algebra into the product of two Lie algebras, the nine parameter

HA (consisting of X, P, I) and the six parameter homogeneous Lorentz algebra (consisting of the S & I).

Thus one can write all representations as products of the representations of the two algebras. For the HA

one can choose the position representation:

X y > = y y > or the momentum representation (6.14)

P k > = k k > (6.15)

or equivalently diagonalize the mass and the sign of the energy and three momenta as

P P= m2 c2; ( Pkiwith eigenstates written as |m, ( Pk > (6.16)

The transformations between the position diagonal and momentum diagonal basis vectors are given by the

Fourier transform as is well known.

All representations of the homogeneous Lorentz group have been found by Bergmann and by

Gelfand, Neimark, and Shapiro [2,9] to be given by the two Casimir operators b0 and b1 defined as:

b02 + b1

2 – 1 = ½ ggS S (6.17)

where b0 = 0, ½, 1, 3/2, …(|b1|-1) and where b1 is a complex number defined by

b0b1 = - ¼ S S (6.18)

with the rotation Casimir operator as S2 which has the spectrum s(s+1) with the total spin

s = b0 , b0+1, …, (|b1| - 1) (6.19)

and the z component of spin:

= -s, -s+1, ….s-1, s (6.20)

Thus the homogeneous Lorentz algebra representation can be written as | b0 , b1 , s, > which joined with

the Heisenberg algebra gives the full representation space as either

|kb0 , b1 , s, > = a+ kb0, b1, s,|0> for the momentum representation or (6.21)

| y b0 , b1 , s, > = a+ y b0,b1,s,0> for the position representation. (6.22)

To obtain the effective extended Poincare algebra, one must generalize the equations above by

replacing Pwith Dand the appropriately altered commutators which contain both the gravitational

metric and the vector bosons A as in equation (2.30) and likewise change (5.7) to

LX D–X D

7. Conclusions and Discussion:

1. The primary objective and result of this work is to reframe RG as a noncommutative algebraic

geometry following the customary expression of differential equations in terms of commutators of

operators in Lie algebras thus casting differential equations in a more powerful and formal

mathematical context. A RG can now be seen in the form of commutation rules with the differential

equations for the Christoffel connections and the Ricci, Riemann, and other tensor expressions

11

expressed as commutators of operators generalizing the framework of a Lie algebra. The potential

applicability to geometry is then of interest mathematically.

2. The second result is the subsequent reframing of Einstein’s equations and GR as a noncommutative

algebra of operators corresponding to observables thus providing a framework for GR that is more

compatible with the foundation of QT and the SM as a Lie algebras of observables expressed as

commutator products with the associated interpretation of interference of measurements. This

reframing does not in itself change GR but reexpressed it as commutators as described.

3. Our next objective will be to replace the effective momentum of QT and the SM with this more

general form for D described above that includes the metric as a function of operators that is to be

quantized as a massless spin 2 graviton as well as the vector particles that mediate the electroweak

and strong forces as described in the SM. Our first result in that direction is that the generalized HA,

[D, X] = iћ g(X), in the framework of QT implies that the interference of the simultaneous

measurement of momentum and position is to be generalized from the constant ћ to the operator ћ

g(X). This in turn suggests that the metric of space-time curvature represents the degree of

momentum-position interference of measurement and conversely. It also has the potential observable

and testable result that the uncertainty relations, between position and momentum (in the direction of

the gravitational field) and between energy and time, are altered. This could be possibly tested for a

Schwarzschild or Kerr metric near a gravitating body both with the altered Fourier transform, atomic

energy states, and virtual pair production.

4. The fourth result is the suggested replacement of the momentum operator in the Poincare,

Heisenberg, and Lorentz algebras with the D operator resulting in a generalization of the Lie algebra

framework as a NAO with the inclusion of the metric operator. The resulting commutation rules for

rotation and Lorentz transformations in a Riemannian geometry are very complex expressions but are

well defined and can be investigated in small regions of space-time where the metric is known and is

approximately constant. The Casimir operators are g(X) DD = m2c2, ½ ggL L = b0

2 + b12 – 1,

and - ¼ L L = b0b1 and as well as with the L operator for orbital angular momentum replaced

by S and also by M. We are currently investigating these equations.

5. The fifth result is that this framework exactly reduces to Einstein’s equations in an astronomical

environment with no quantum effects and where the energy-momentum tensor is taken with its

classical value. Likewise, if there is no gravitational force then the metric is Minkowski and one has

the standard QT and SM systems. But when all forces and scales are present, then this formulation is

formidable but can yet can be viewed in approximation when the metric is known and for small

domains where it can be taken as constant.

6. It is also of interest to note that the functions g(y) can contain an antisymmetric component related

to torsion although this component does not contribute to the metric for distance [22, 23,24], but does

allow more freedom in the connection as explored by Einstein and Cartan. And most important,

this framework can be extended to higher dimensions as in string theory as there is no restriction of

the space-time to four dimensions.

Acknowledgements: I am deeply appreciative of discussions and suggestions from Dr. James Knight.

12

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