Download In order to integrate general relativity with quantum theory, we

An Integration of General Relativity and Relativistic Quantum Theory
Joseph E. Johnson, PhD
[email protected]
May 21, 2016 - Draft
Abstract
1,2
The Poincare Lie algebra , and the Lie group of symmetries which follow from its exponential map, have
representations on a Hilbert space that provide the possible states of the fundamental relativistic quantum particles
(fields). But quantum theory also requires observables such as position in space-time which, while they do not generate
symmetries, do extend the set of observables via the Heisenberg Lie algebra which forms the foundation of quantum
mechanics. In previous work3,4, the author extended the Poincare Lie algebra to include a four-vector position operator
as a natural covariant extension of this algebra to a larger Lie algebra of observables. This 15 parameter “Extended
Poincare” (EP) Lie algebra also was shown to provide a more transparent foundation for representations of the Lorentz,
Poincare, and Heisenberg subalgebras and the groups which they generate.
In order to integrate general relativity with quantum theory, we propose that the Minkowsky metric
contained in the structure constants of this Extended Poincare (EP) algebra be generalized to be a function of these
position operators which then are to be determined from Einstein’s general relativity equations5,6, 7 using the energymomentum tensor from the standard model8,9. We call this new algebra the Extended Poincare Einstein (ELE) Lie
algebra. It guarantees that all space time observables conform to both relativistic quantum theory and Einstein’s
equations for the metric of space time. Thus one side of Einstein’s equations determine the structure constants of the
EPE algebra while the other side, containing the energy momentum tensor, is determined from the state of the system
using observables (operators) from the standard model. The representations of this EPE algebra are to be operators that
support the standard model’s gauge transformations as well as provide the energy momentum tensor operator for the
system along with the system Lagrangian. Our initial results give an altered uncertainty principles in the presence of a
strong spherical gravitational field: XxPx ≥ (ħ/2) (1/(1-rs/r)) and tE ≥ (ħ/2)(1-rs/r) where the Schwarzschild radius is
given by rs = 2GM/c2. In the same environment the Dirac equation for the hydrogen atom is modified also in the X0 and
X1 derivatives by the same factors thus altering the hydrogen spectra near a black hole or neutron star.
Our objective is to uncover the full set of all fundamental observables from each fundamental domain, including
the standard model (SM), along with their algebraic structure constants. Perhaps when viewed in its entirety as the
algebra of all observables, one can uncover regularities and new underlying patterns as Maxwell did with the equations
for electromagnetism. Our approach thus differs from the customary approach where gravitation is incorporated into
the SM as are the strong and electroweak forces.
1
Introduction
The twentieth century began with multiple revolutions in our previous view of the universe which was seen
obeying the equations of Newton and Maxwell for point masses, charges, and electromagnetic fields. An understanding
of the nature of the fundamental forces and their origin, along with the observed mass and charge spectra, had to await
the formulation of special relativity, quantum theory, and their merger providing the Relativistic Quantum Theory (RQT)
for the kinematical infrastructure of particle physics. This is now most elegantly expressed as a Lie algebra (LA) of spacetime based observables whose algebraic representations on a Hilbert space provide the states of physical systems.
Gradually then over the next hundred years, the phenomenological Standard Model (SM) evolved to provide the
dynamical theory for the strong, weak, and electromagnetic forces, the particle spectra & properties, and interactions
using Yang Mills internal gauge transformations given by operators that do not currently have a space-time basis such as
electrical charge, isospin, and hypercharge. The combination of the space-time based RQT observables with the
phenomenological SM currently now provides an extremely accurate theory of physical phenomena. The three currently
missing components in that framework are the theory of gravitation as formulated in Einstein’s General Relativity (GR)
and the more recently discovered Dark Matter (DM), and Dark Energy (DE) with the latter two still poorly understood.
However, the merger of GR with RQT and the SM has frustrated diverse attempts over the last hundred years
although each of these three theoretical structures has proved their separate validity beyond question in their
respective domains of applicability. More precisely, RQT describes a foundational structure for elementary particles as
representations of the ten parameter Poincare Lie algebra (M, the Lorentz algebra of three rotations and three Lorentz
transformations, along with the four-momentum P, generating translations in space-time). The Heisenberg algebra of
Xi, Pi, E, and t, likewise forms the foundation of quantum mechanics. The additional discrete group of transformations
of space inversion IS, time inversion, IT, and particle conjugation, IC, are also well defined on M, P, and X. The
representations of the maximum set of commuting elements of this algebra are used to index the representation space,
|1, 2, …> of particle states and then is used to index operators a and a+ that annihilate and create these particles
(representations) when acting on the vacuum so that |1, 2, … …> = a+1, 2, … a+1, 2, … |0>. Likewise the
transformations that interchange identical particles, a and a+ , in symmetric (for integer spin) and antisymmetric (for half
integer spin) states are incorporated in the commutation or anticommutation relations that follow from the connection
between spin and statistics thus forming an algebraic structure under the permutation group. But the Poincare Lie
algebra framework does not include space-time operators as fundamental observables or operators.
The Extended Poincare Algebra
As quantum theory is founded upon the relationship between momentum and position operations as defined in
the Heisenberg nilpotent Lie algebra, [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. This led us to
extend the Poincare Lie algebra by adjoining 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 Heisenberg algebra. As they
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
Heisenberg algebra in the covariant form as [P, X] = iħ g I 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
fundamental observables in this Extended Poincare algebra (EP) close into a Lie algebra with the structure constants as
follows:
[I, P] = [I, X ] = [I, M ] = 0
(1a)



[P , X ] = iħ g I which is the covariant Heisenberg Lie algebra – the foundation of quantum theory.
(1b)


[P , P ] = 0
insuring the noninterference of energy momentum measurements all four dimensions. (1c)
2
[X, X] = 0
insuring the noninterference of time and position measurements all four dimensions. (1d)


 
[M , P ] = iħ (g P gPwhich guarantees that Ptransforms as a vector under M

e)


 
 


[M , X ] = iħ (g X  g X which guarantees that X transforms as a vector under M 

f)


 
 
 
 
M , M ] = iħ (g M + g M - g M - g M ).
(1g)


which guarantees that M transforms as a tensorunder the Lorentz group generated by  M .
The representations of the Lorentz algebra are well-known 10 and are straight forward but the extension to
include the four momentum thus creating the Poincare algebra representations are rather messy. But with our
extension of the Poincare algebra to include a four position operator, one can now define the orbital angular
momentum four-tensor, operator L as:
LX P – X P










a)
From this it follows that
[L, P] = iħ (gPgP 








b)


 
 
[L , X ] = iħ (g X  g X 
(2c)
 
 
 
 
 
L , L ] = iħ (g L + g L - g L - g L ). 





(2d)
One can then define an intrinsic spin four-tensor as:
S = M - L
(3a)
with the result that
[S, P] = 0
(3b)


[S , X ] = 0
(3c)
 
S , L ] = 0
(3d)
 
 
 
 
 
S , S ] = iħ (g S + g S - g S - g S ) 
(3e)
Now one can separate this Extended Poincare (EP) algebra into the product of two Lie algebras, the nine
parameter Heisenberg Lie algebra (X,P,I) and the six parameter homogeneous Lorentz algebra (S). Thus one can write all
EP representations as products of the representations of the two algebras. For the Heisenberg algebra one can choose
the position representation:
X y > = y y > or the momentum representation
(4a)


P k> =k k>
(4b)
or equivalently diagonalize the mass and the sign of the energy and three momentum as
P P= m2 , ( Pwith eigenstates written as |m, ( Pk >
(4c)
All representations of the homogeneous Lorentz group have been found by Bergmann and by Gelfand, Neimark, and
Shapiro10 to be given by the two Casimir operators b0 and b1 defined as:
b02 + b12 – 1 = ½ ggS S
(5a)
where b0 = 0, ½, 1, 3/2, …(|b1|-1) and where b1 is a complex number
b0 b1 = - ¼  S S
(5b)
2
with the rotation Casimir operator as S which has the spectrum s(s+1) with the total spin
s = b0 , b0+1, …, (|b1| - 1)
(5c)
and the z component of spin:
 = -s, -s+1, ….s-1, s
(5d)
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
|kb0 , b1 , s,  > = a+ kb0, b1, s,|0>
(6a)
for the momentum representation or

| y  b0 , b1 , s,  > = a+ y b0,b1,s,0>
(6b)
for the position representation.
3
These simultaneous eigenvalues represent a maximal set of commuting observables for the EP Lie algebra that can be
used to index creation and annihilation operators representing particles (fields) with the quantum numbers described.
The fundamental entities must be in the representation space of these operators and their Lie algebra.
But while the known elementary particle states can easily be fit into this infinite array of spins and continuous
masses, one has a vast overabundance of states as well as a lack of a dynamical theory of their interactions. One would
like to have an algebraic structure that gave all possible particles and only those particles as representations. It is here
that one imposes the additional requirements of the phenomenological Standard Model (SM) which only allows three
Lorentz representations (specified by b0 and b1 ) which are the pairs of values (½ , ±3/2), (0,1) and (0,2). Specifically
these representations of the Lorentz algebra for particles in the SM are as follows: (a) a unique Dirac spin s = ½ for
fermions which is given by b0 = ½ and, b1 = ±3/2 where the sign of b1, (b1), distinguishes the representation from the
conjugate representation and thus where the four states of |b0 , b1 , s, can be abbreviated as |(b1),  =± ½These
four (spinor) states support the definition of the  matrices which result from the requirement that both representation
and conjugate representation be used in order for the state to be invariant under a spatial reflection which takes one
from the representation to the conjugate representation having the opposite sign of b1 thus giving the standard Dirac
theory. When b0 = 0 then the representation is equivalent to its own conjugate and thus one does not have (b1). There
are two such pertinent cases for bosons in the SM: (b) a unique spin s = 0 (e.g. the Higgs) which is given by b0 = 0 and, b1
= 1, and (c) the four-vector vector representation given by b0 = 0 and, b1 = 2, which gives both s = 0 and s = 1. Linear
combinations of these four spin states ( s=0, and s = 1, can be used to form a four vector
representation which is needed for the photon (electromagnetic potential A ) and the W and Z vector fields.
The dynamical theory is introduced via the phenomenological standard model (SM) which imposes the
requirement that these representations also support the SU(3) x SU(2) x U(1) gauge group which mixes the observables
contained in the EP algebra (X ,P ,  ) with new gauge observables (electric charge, hypercharge, isospin, color, and
flavor which currently lack a space-time origin) to account for the masses and spins of physical particles along with their
strong and electroweak interactions. This is similar to classical mechanics where the descriptive kinematical
infrastructure supports the separate dynamical theory which contains the interactions and a specification of what
physical states (masses, charges, magnetic moments…) are allowable.
Separate from these systems, one is confronted with Einstein’s theory of general relativity (GR) which can
normally be ignored in the quantum physics at small scales as described by the SM but which dominates the large scale
structure of the universe. In GR the metric of space-time is curved by the presence of matter and energy as given by :
R - ½ g R + g = (8  G/c4) T







(7)
where R is the Ricci tensor and T is the energy momentum tensor expressed in terms of the particles in the
SM theory (along with DM).  represents a constant value that could possibly represent the expansion of the universe
as envisioned with DE. The Ricci and Riemann tensors are defined in terms of the derivatives of the Christoffel symbols
which are in turn defined in terms of derivatives of the metric tensor. Attempts to integrate GR with the SM using a
gauge group with a massless spin two particle (graviton) have not been successful. But recalling Einstein’s basic
concepts, gravitation is not to be thought of as a force but rather the consequence of a curvature of space-time induced
by the energy momentum tensor.
Proposed Method of Integrating GR with RQT:
We propose a very different approach that incorporates GR directly into both the foundational relativistic
quantum theory (RQT) via the structure constants in the space-time based EP algebra and also incorporates the SM via
the expression for the energy momentum tensor. Specifically we postulate that the Minkowsky metric gin the
structure constants of the EP algebra is to be generalized to be a function of the X position operators in order to reflect
the space-time dependence of g in GR as expressed by the equations of Einstein that determine the metric in terms of
4
the energy momentum tensor of particles (fields). This approach makes the full algebraic structure of space-time, as
expressed in the EP Lie algebra, manifestly adherent to GR. The energy momentum tensor in turn is to be obtained from
the SM energy momentum tensor (plus that from DM), or when appropriate, expressed classically from the mass and
energy distribution of large scale interstellar matter, stars, black holes, and dark matter. Thus the LHS of Einstein’s
equations are to determine the metric tensor and thus determine the EP Lie algebra’s structure constants. We call this
modified extended Poincare Lie algebra the “Extended Poincare Einstein (EPE) “Lie” algebra which has the same
structural form as EP but with greplaced by gX. This EPE algebra thus generalizes the EP algebra which generalized
the Poincare algebra. The RHS of Einstein’s equations provide the explicit form of the energy momentum tensor T
which is to come formally from the SM. But in practice, one can define T using classical concepts when dealing with
astronomical scales of black holes and other large masses. This then forms a closed system with (1) the 15 parameter
EPE Lie algebra, (2) the 12 parameter SU(3) * SU(2) * U(1) Lie algebra of the SM, (3) the discrete 3 parameter Abelian
algebra of inversions that contains the discrete transformations of (a) space inversion IS, (b) time inversion, IT, and (c)
particle conjugation, IC and (4) Einstein’s GR equations that determine gfrom T as re-expressed in term of
commutators of the fundamental observables as shown below. Then one can express this system as an algebra of
commutation relations among these (15 + 12 +3 = 30) fundamental observables where all of the EPE observables are
normally assumed to commute with all SM observables (although we will not make that assumption). The inversion
operations have their normal commutator and anticommutator rules but permeate both the EPE and SM algebras in
ways that are not yet clearly understood. The operations of identical particle transpositions are to be the symmetric and
antisymmetric representations of the permutation group of symmetries of identical particles. One notes that this
system is incomplete to the extent that the SM is still incomplete and is to be modified in the future as appropriate. Also
we note that DM and DE are not fully understood and changes will be made there also in the future.
The space-time observables that constitute the EPE algebra are now to obey:
[P, X] = iħ I g(X)
(8a)
so we can now write

g (X) = (-i/ ħ) [P, X]
(8b)
In the position representation one now has
<y| P | > = iħ g(y) (∂/∂y) (y) = iħ ∂ (y) where (y) = <y|>.
(8c)
It can also be shown also that although one still has
[P, P ] = 0
(8d)
but since
[P, g ] ≠ 0
(8e)



it follows that [P , [P , X ]] ≠ 0 so that the Heisenberg algebra is no longer nilpotent. It follows that
<y| [P, g] = iħ g (∂g /∂y) <y| = iħ ∂g(y) <y|
(8f)


where from now on g = g (y) is to be understood. Thus in the position representation one can write
g (∂ /∂y) f(y) = ∂ f(y) = -(i/ ħ) [P, f(y)]
(8g)

for any function f(y) thus converting all differential operations into commutators with P .
We now seek to cast the LHS of the Einstein equations into the form of commutators of algebraic observables.
Although this looks complex, it is no more so than the differential equations for the Christoffel, Riemann, and Ricci
tensors. One can define the Christoffel symbols in terms of the commutators of the four-momentum with the metric as:
 = (½) (∂, g + ∂, g- ∂, g ) (understood to be in the position representation) thus
(9a)
 = (½) (-i/ħ) ( [P, g] + [P, g] - [ P, g ] )
(9b)
Then using g (X) = (-i/ ħ) [P, X] one obtains
 = (-½) (1/ħ2) ( [P, [P, X]] + [P, [P, X]] - [ P, [P, X] ] )
(9c)
Then the Riemann tensor becomes:
5
R = (-i/ħ) ( [P,  ] - [P,  ] ) + (  -  )
(9d)
where  is to be inserted for the Christoffel symbols thus giving an expression involving only commutators.
One then defines the Ricci tensor as:
R = g R = (-i/ ħ) [P, X] R
(9e)
and also defines
R = g R or (-i/ ħ) [P, X] R
(9f)
all of which must be inserted into the LHS of Einstein equations,
R - ½ g R + g = (8 π G/c4) T
(10a)
Then finally we have the LHS of Einstein equations in terms of just commutators:
R + ((i/ ħ) [P, X]) ( ½ R -  ) = (8 π G/c4) T
b






where Rlj and R are given above in terms of commutators and where
T = <  P +…|
with the  P term being symmetrized over  and  and which is to act in both directions giving four terms for all
fermions in the SM along with similar operator contributions from the boson fields and that of DM where |
represents the state of the system.
With these substitutions, one obtains a very large number of terms (commutators) on the left hand side of the
Einstein equation when the expressions are expanded in terms of the commutators of P. We do not need to expressly
write out the SU(3) x SU(2) x U(1) gauge group as this is well developed in the literature as well as the commutators of all
of these observables with the inversions (Is, It, and Ic). Our hope is that there will be simplifications and cancelations
among some of these terms and that a pattern would emerge in the GR equations expressed as commutators using the
Jacobi, Bianchi, and other identities along with other symmetry properties. When ħ is infinitesimally small compared to
the parameters of the problem, then one obtains the traditional GR equations in the Heisenberg representation for the
operators and particle trajectories are geodesics. Likewise when masses and their associated gravitational fields are
small compared to the other parameters in the problem, then gravitation can be ignored and one obtains the standard
Minkowsky metric of EP with the SM. Thus the current formulation smoothly contains RQT, the SM, and GR.
But the resulting system is very unusual since the structure constants are no longer constants and could be
considered so only in a local domain. The gauge groups of the SM are based on space-time dependent phase
transformations of the first and second kind but the SM Lie algebra has fixed unchanging structure constants. The EPE
Lie algebra we are postulating has the fundamental structure constants dependent upon the metric tensor which varies
in space-time. Thus the EPE Lie algebra is an algebra which changes its algebraic structure throughout space and time in
accordance with Einstein’s equations for the metric. This is thus a generalization of the concept of a Lie algebra as the
structure constants are not constants but constitute a smoothly changing flow from one Lie algebra to another. Both
the diagonal and off-diagonal terms in g make the system very complex and it is not clear how to find all
representations or even local representations of this algebra. For cases where the GR equations can be solved for the
metric, then this metric gives the structure constants of the EPE algebra and thus from these one can create the regular
representations. Also for such EPE solutions one can form the Cartan-Killing inner product to study the Lie algebra
structure of the EPE.
We note four aspects of this proposal: (1) Our design accepts the currently standard versions of Poincare and
Heisenberg LA, the SM, GR, and the algebra of the three discrete inversions. (2) c, ħ, and G are all in the structure
constants of this EPE “Lie algebra” on an equal footing and as such define the “natural” scale for mass, length and time
commonly known as the Plank scale. (3) Specifically all nonzero EPE structure constants contain iħ g(X). And (4)
general relativity as formulated here is not a part of the phenomenological SM of interactions but rather is a rich
extension (EPE) of the kinematic space-time infrastructure of RQT (quantum theory, special relativity, and now general
6
relativity) whose representations are to support the allowable particles in the SM which give the complementary
formulation of the strong and electroweak forces. Our approach is in keeping with the principles of Mach and Einstein
that gravitation is a result of the Riemann curvature of space time and not a separate force to be incorporated like other
forces in the SM. (5) The uncertainty relation: x p ≥ ħ g/2 is modified so that the metric now modifies the effective
value of ħ both for the position-momentum and the energy-time inequalities. (6) The energy-momentum tensor
operators in Einstein’s equations are for the particles/fields (energy and momentum) found in nature and thus this Tlj
operator must originate in the SM. So GR becomes a “bridge” equation between the EPE and the SM gauge algebras.
The most unusual aspect of this proposal is that the new EPE Lie algebra is really not a Lie algebra since the
structure constants vary from one location to another in space-time but are more like a self-consistent dynamical Lie
Algebra. Thus we must envision that there are an infinite number of “local” algebras with constant metric values that
smoothly mediate their associated representations. Having the structure constants change and thus interconnect
multiple Lie algebras presents a much more demanding mathematical environment to be investigated as we can now
look at the possible influence of simultaneously having intense gravitational fields in an environment with a fully
operable quantum mechanics supporting the standard model. In particular, the Fourier transform can be shown to still
be the projection of the position eigenstates onto the momentum eigenstates but with the presence of the space time
dependent metric. One also notes that this design we are proposing in terms of just commutator rules, structure
constants and algebras is in exact accord with Sophius Lie’s original concept of inventing Lie algebras for the study of
differential equations.
Currently the simplest explanation is to take the  term in GR as being responsible for dark energy (DE) and thus
as a pure manifestation of the GR formulation of the EPE structure constants and not a separate kind of particle that
issues from the SM as this is. Of course the question remains of how this term arises. It appears that dark matter (DM) is
new type of particle that has no strong, electromagnetic, or weak interactions but only gravitational. If this is true, then
it would not necessarily emerge in any natural way from the current SM which was built on the other interactions but
could be a particle that carried mass but had only gravitational interactions. Thus it would be just another
representation of the EPE algebra that needed to be adjoined to the SM framework. Current experiments do not require
that DM is a particle that is currently represented in the SM. The SM is totally built from strong and electroweak gauge
transformations which show no current evidence of interacting with DM. Thus DM can be just another particle that only
has a gravitational interaction. Thus the energy momentum tensor from the SM would have to contain an additional
term for DM. Following the current successful methodology of the gauge transformations with the SM, there could be
another gauge group just for particles that only interact gravitationally thus extending the 12 parameter SM gauge
group. If there is a stable DM particle similar to the Higgs, that is part of the SM but which only has gravitational
interactions, it will be very difficult to measure its mass and possible spin.
Discussion:
Thus our fundamental premise is that the energy-momentum tensor of particles that exist within the SM
framework are, by Einstein’s equations for GR, to define the structure constants through the metric and that metric in
turn alters the 30 parameter EPE+SM+I (Inversions) algebra’s structure constants and thus their representations. Yet
these representations of the 30 observables are tied into a closed system of equations that also generate the SM. Now
that one has the state of the system fixed, the system dynamics would be determined by using the method of Feynman
path integrals using the SM Lagrangian. The procedure would be: (a) the T from the SM is to determine gin the Cijk
of the combined EPE+SM algebra. (b) Having these Cijk one determines the EPE representations that also support the
actions of the SM gauge groups operators and then ( c) the SM must give a T consistent with (a). Then (d) the system
is to dynamically evolve via a Feynman’s path integral dynamical solution in terms of the SM Lagrangian. While the
fundamental EPE space time observables and the equations of GR that link the determination of the structure constants
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g, are on very solid ground separately, this proposed integration opens complex mathematical questions. Although the
SM Lie algebra of internal quantum observables and interactions is known, it is still a work in progress with a large
number of arbitrary parameters. Also since the SM contains observables from the EPE algebra, it is necessary to cast the
SM framework into an algebraic structure where derivatives are replaced by commutators and gamma matrices by spin
representations. One notes that our approach fully integrates GR with spin and angular momentum as well as spacetime. Then one can ask the most fundamental question: Is the resulting algebraic system self-consistent when the GR
equations are invoked between T and g.
However, by using those cases where Einstein’s equations have been solved, one can determine g(X) and thus
can explicitly write the structure constants for the EPE and seek representations of that algebra in neighborhoods where
the metric can be considered locally constant. Thus one begins with the state of a physical system (such as a black hole
with some particle state exterior to it such as a nucleus, atom, or molecule) some distance outside the event horizon.
Then the energy momentum tensor can be classically determined at the location of the object. Next, Einstein’s
equations for the metric tensor can be solved for this case. Then that metric tensor is to be inserted into the structure
constants for the EPE algebra. One must then find the representations of that Lie algebra which give the allowable
states in nature which are then required to support the SM gauge groups. The immediate objective is to seek new
predictions that issue from this EPE algebraic framework.
In particular for a spherical mass, the Schwarzschild metric has
g00 = (1- rs /r) and
(11a)
grr = -1/(1- rs /r)
(11b)
where the Schwarzschild radius is given by
rs = 2GM/c2
(11c)
and r = the radius of the mass M located at r = 0 as given in spherical coordinates. This implies that X P and
t E have effectively different values than are expected in traditional quantum theory if the gravitational field is very
large such as near a black hole or neutron star. For the case of a non-rotating black hole without charge we get
XrPr ≥ (ħ/2) (1/(1-rs/r)) and
(12a)
tE ≥ (ħ/2)(1-rs/r)
(12b)
which modify the uncertainty principle in a strong gravitational field. It follows that taking the product
of these two equations that one gets:
XrPr tE ≥ (ħ/2)2
(12c)
It is not obvious how to measure these altered uncertainty relations so one needs to seek effects that would be
sensitive to these altered values of “effective ħ” due to the metric, as might be observable in either energy transitions or
angular momentum values. For example, the Dirac equation for hydrogen in a strong gravitational field has the time and
X1 component modified by these Schwarzschild metric factors so the energy levels would be altered with modified
emission spectra, possibly observable. Here one can take the radial direction from the center of the black hole to be the
X1 direction since the black hole field appears as perpendicular to a plane from a position near the event horizon while
the x and y directions are not affected as the black hole is so effectively large at this proximity. The Dirac equation for
the states of the hydrogen atom thus need to be solved with the altered Schwarzschild term for P1 thus giving an
asymmetric Dirac equation to be solved for the energy levels and transitions.
A particle would now move along Feynman paths near a geodesic and that this geodesic would be altered by the
strong and electroweak forces implied by the SM. This suggests the development of a Foldy- Wouthuysen type
formulation that generalizes the motion of a charged Dirac particle in an electromagnetic field where the free particle
would evolve primarily close to the geodesic along the Feynman paths that would create the least phase interference.
The SM formulation would then cause deviations from that geodesic as required by the strong and electroweak
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interactions. The dynamical evolution of the entire system would follow current methodology using Feynman paths as
this provides the required covariant treatment of time.
The next problem is to determine merger of the SM and the EPE Lie algebras along with the three inversions
where all equations are expressed as commutators of operators. Thus for example the  matrices can be re-expressed
in terms of the spin tensor operators and the derivatives re-expressed in terms of commutators with P. Although we
know the structure constants for the EPE and the SM algebras, we need to study their collective commutation relations
and must also support the three discrete inversions and express the multiple parameters and angles of the SM within
such an algebra structure. This is not a simple problem. Next one then needs to form Tab in terms of the energymomentum tensor of the existing particles as operators or commutators from the SM so that the right hand side of the
Einstein equations with Tab are also written in the form of commutators of fundamental operators for the fields. Then
we will have a tight explicit incorporation of the SM into the EPE Lie algebra along with GR as a single EPE-GR-SM-I
algebraic system. The hope would be that if we could construct this system of algebraic operator equations for the
combined observables then one might uncover an “inherently beautiful” algebraic system such as Maxwell did when he
combined the four “pre-Maxwell” equations and discovered the correction that was needed for consistency. One would
hope that such a system would have far more constraints and fewer degrees of freedom (arbitrary parameters) than is
currently the case with the SM.
Conclusions:
An alternative approach to integrating General Relativity (GR) with quantum theory was here proposed that
began with an extended Poincare algebra, that included a four vector position operator X , to create a 15 parameter
Extended Poincare (EP) Lie algebra of space time observables thus including the Heisenberg algebra, and thus explicitly
including the foundations of quantum theory. This previous work by the author then suggested our integration postulate
that the resulting EP structure constants containing the Minkowsky metric g be generalized to be functions of those
position operators whose values were to be determined by Einstein’s GR equations for the metric (in the position
representation) from the energy momentum tensor. This “Extended Poincare Einstein” (EPE) algebra is like a Lie algebra
but with structure constants that vary over space and time as determined by Einstein’s equations for general relativity
(GR). As such, it represents a new mathematical structure where unlike a gauge transformation based algebra one here
has different LA at different points in space time. Two results were derived for strong gravitational fields such as near a
black hole: (1) an altered expression of the Heisenberg uncertainty principle, and (2) an altered Dirac equation for the
energy levels of a hydrogen atom thus predicting altered emission spectra. The latter prediction might be observable.
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