Download In order to integrate general relativity with quantum

An Integration of General Relativity and Relativistic Quantum Theory
Joseph E. Johnson, PhD
April 6, 2016 - Draft
Abstract
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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 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 work2, the author extended the Poincare Lie algebra to include a four-vector position operator
as a natural covariant extension of the Poincare algebra to a larger Lie algebra of observables. This “Extended Poincare”
(EP) Lie algebra also was shown to provide a more transparent foundation for representations of the Lorentz, Poincare,
and Heisenberg algebras 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 algebra be generalized to be a function of these position operators
which then are to be determined from Einstein’s general relativity equations3 using the energy-momentum tensor from
the standard model4. Thus one side of Einstein’s equations determine the structure constants of the extended Poincare
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 altered extended Poincare 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 objective is to uncover the set of all fundamental
observables from each domain: general relativity, relativistic quantum theory, and the standard model, 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.
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Introduction
Prior to 1900, classical mechanics provided a foundational kinematic theory for the motion of masses and
charges in space and time along with the observables and descriptions of the state of a system of particles and the
electromagnetic field as formulated by Newton and Maxwell. But this kinematic theory did not provide the forces
among the particles that determine the accelerations in Newton’s equations of motion with the exception of centrifugal
and Coriolis forces. That dynamical theory of interactions describing the fundamental forces on these charges and
masses came from the Lorentz equation for electromagnetic forces and Newton’s law of gravitation for gravitational
forces. The forces given by these theories allowed one to predict the time evolution of a dynamical system. One notes
the parallel between that space time kinematic infrastructure and the Poincare symmetry group and position operators
that replaced it thus giving our current “kinematic theory”, Relativistic Quantum Theory (RQT). Likewise one notes the
parallel between the earlier phenomenological forces of gravity and electromagnetism and the current Standard Model
(SM) that has replaced it with the strong and electroweak forces along with the allowed observed charge and mass
spectrum. This deeper understanding of the nature of the forces and their origin, along with the observed mass and
charge spectra, had to await the beginning of the twentieth century with formulation of Einstein’s special relativity,
quantum theory, and their merger providing the RQT for the kinematical infrastructure of particle physics. This is most
elegantly expressed as an algebra of space-time based observables whose algebraic representations on a Hilbert space
provide the states of physical systems. Also, the (SM) has evolved over the last century to provide the dynamical theory
for the strong, weak, and electromagnetic forces, the particle spectra & properties, and interactions using internal gauge
operators that do not currently have a space-time basis such as electrical charge and hypercharge. The combination of
the kinematic RQT with the dynamic/phenomenological SM currently provides an extremely accurate theory of physical
phenomena. The three currently missing components in that framework are Dark Matter (DM), Dark Energy (DE), and
Einstein’s theory of gravitation as formulated in General Relativity (GR). DM and DE are both only relatively recent
discoveries and each is still far from being understood and are not the primary concern of this paper.
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 kinematical 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and P. 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 of the
creation and annihilation operators as follows from the connection between spin and statistics forming an algebraic
structure under the permutation group that combine the observables and their algebra as described above. But this RQT
framework does not include space and time as fundamental observables or operators in a covariant way.
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, it is inconsistent to have only spatial positon operators,
Xi, as these are mixed by the Lorentz group with time making time an operator to another observer. This led us to
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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. As they generate translations in momentum, they do not generate
symmetry transformations or represent conserved quantities but just additional observables. 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 commutes
with all elements and has the unique eigenvalue 1 and where P =E/c, P= Px ..., X = ct, X.= x … and where “I” is needed
to make the fifteen fundamental observables in this Extended Poincare algebra (EP) into a Lie algebra with the structure
constants as follows:
[I, P] = [I, X ] = [I, M ] = 0
[P, X] = iħ gI which is the covariant Heisenberg Lie algebra – the foundation of quantum theory
[P, P] = 0
insuring the noninterference of energy momentum measurements all four dimensions


[X , X ] = 0
insuring the noninterference of time and position measurements all four dimensions


 
[M , P ] = iħ (g P gPwhich guarantees that Ptransforms as a vector under M
[M, X] = iħ (gX gXwhich guarantees that Xtransforms as a vector under M
M, M] = iħ (gM + gM - gM - gM) which guarantees that Mtransforms as a tensorunder the
Lorentz group generated by  M.
The representations of the Lorentz algebra are well known and straight forward but the extension to the
Poincare algebra representations are rather messy. But with our extension of the Poincare algebra to include a four
position operator (EP), one can now define the orbital angular momentum four-tensor, operator L as:
LX P – X P
From this it follows that
[L, P] = iħ (gPgP
[L, X] = iħ (gX gX
L, L] = iħ (g L + g L - g L - g L). 
One can then define an intrinsic spin four-tensor as
S = M - L
with the result that
[S, P] = 0
[S, X] = 0
S, L] = 0
S, S] = iħ (g S + g S - g S - g S) 
Now one can separate the 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
P k > = k k >
or equivalently diagonalize the mass P P= m2 , the sign of the energy, ( Pand Pi as |m, ( Pk >
All representations of the homogeneous Lorentz group have been found by Bergmann and by Gelfand, Neimark, and
Shapiro to be given by the two Casimir operators b0 and b1 defined as:
b02 + b12 – 1 = ½ ggS S where b0 = 0, ½, 1, 3/2, …(|b1|-1) and where b1 is a complex number
b0 b1 = - ¼  S S and with the Casimir operator for the rotation subalgebra as
S2 which has the spectrum s(s+1) with the total spin s = b0 , b0+1, …, (|b1| - 1) and
 = -s, -s+1, ….s-1, s which is the z component of spin.
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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> for the momentum or
| y b0 , b1 , s,  > = a+ y b0,b1,s,0> for the position representation.
These simultaneous eigenvalues represent a maximal set of commuting observables for the Lorentz, Poincare, and EP Lie
algebras 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 for 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 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 spin s = ħ/2 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 one does not have (b1). There are two such
pertinent cases for bosons: (b) a unique spin s = 0 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)xSU(2)xU(1) gauge group which mixes the observables
contained in the EP algebra (X ,P ,  ) with new 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 with 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 the
equation
R - ½ g R + g = (8  G/c4) T
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 explain 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
massless spin two particle (graviton) have not been successful nor does the graviton fit naturally into the RQT.
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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 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
extended Poincare Lie algebra (the EP algebra of RQT) 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 the energy momentum tensor of particles (fields). The energy momentum tensor in turn is to be
obtained from the SM (plus that from DM), or when appropriate, expressed classically from the mass energy distribution
of 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 structure “constants” but with ggX. This EPE algebra
thus generalizes RQT. 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 (a) the discrete transformations of space inversion, (b) IS, 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 RQT observables
commute with all SM observables and where the inversion operations have their normal commutator and
anticommutator rules. 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. This results in the same basic EP Lie
algebra that constitute the extended Poincare algebra but with structure constants which are now functions of the fourposition operators with g = g(X) and now which are to be determined by the equations of GR from the energy
momentum tensor as dictated by the SM. The RQT space-time observables that constitute the EPE algebra are now to
obey:
[P, X] = iħI g(X) so we can now write
g (X) = (-i/ ħ) [P, X]
In the position representation one now has
<y| P | > = iħ g(y) (∂/∂y) (y) = iħ ∂ (y) where (y) = <y|>.
It can also be shown also that although one still has
[P, P ] = 0 but since
[P, g ] ≠ 0 it follows that [P, [P, X]] ≠ 0 so that the Heisenberg algebra is no longer nilpotent. It follows that
[P, g] = ih g (∂g /∂y) = ih ∂g 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)]
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:
cab = (½) (∂b, gca + ∂a, gcb - ∂c, gab ) thus
cab = (½) (-i/ħ) ( [Pb, gca] + [Pa, gcb] - [ Pc, gab ] )
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Then using gab (X) = (-i/ ħ) [Pa, Xb] one obtains
cab = (-½) (1/ħ2) ( [Pb, [Pc, Xa]] + [Pa, [Pc, Xb]] - [ Pc, [Pa, Xb] ] )
Then the Riemann tensor becomes:
(***indices below need checking for typos)
s
s
Rlijk = (-i/ħ) ( [Pj, lik ] - [Pk, lij ] ) + (ljs  ik - lks  ij )
Rlijk = (-ħ-2) { ½ ( [Pj, ( [Pk, gli] + [Pi, glk] - [ Pl, gik ] )]
- [Pk, ( [Pj, gli] + [Pi, glj] - [ Pl, gij ] )] )
+ (( [Ps, glj] + [Pj, gls] - [ Pl, gjs ] ) gsr ( [Pk, gri] + [Pi, grk] - [ Pr, gik ])
- (( [Ps, glk] + [Pk, gls] - [ Pl, gks ] ) gsr ( [Pj, gri] + [Pi, grj] - [ Pr, gij ] )} or in terms of commutators
Rlijk = (-i/ħ) (-½) (1/ħ2)
( [Pj, ( [Pk, [Pl, Xi]] + [Pi, [Pl, Xk]] - [ Pl, [Pi, Xk] ] )]
- [Pk, ( [Pj, [Pl, Xi]] + [Pi, [Pl, Xj]] - [ Pl, [Pi, Xj] ] ) ] )
+ ((-i/ ħ) [Pr, Xs]) (( [Ps, [Pl, Xj]] + [Pj, [Pl, Xs]] - [ Pl, [Pj, Xs] ] ) ( [Pk, [Pr, Xj]] + [Pj, [Pr, Xk]] - [ Pr, [Pj, Xk] ])
- ((-i/ ħ) [Pr, Xs]) ( [Ps, [Pl, Xk]] + [Pk, [Pl, Xs]] - [ Pl, [Pk, Xs] ] ) ( [Pj, [Pr, Xj]] + [Pj, [Pr, Xj]] - [ Pr, [Pj, Xj] ]))
One then defines the Ricci tensor as:
Rlj = gik Rlijk = (-i/ ħ) [Pi, Xk] Rlijk
and also defines
R = glj Rlj or (-i/ ħ) [Pl, Xj] Rlj
all of which must be inserted into the LHS of Einstein equations,
Rlj - ½ glj R + glj = (8 π G/c4) Tlj
Then finally we have the LHS of Einstein equations in terms of just commutators:
Rlj + ((i/ ħ) [Pl, Xj]) ( ½ R -  ) = (8 π G/c4) Tlj
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 acting in both directions giving four terms for all
fermions in the SM along with operator contributions from the boson fields and that of DM where | represents the
state of the system.
One notes that there are a very large number of terms on the left hand side of this 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 and Bianchi 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 and
RQT 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. Even then, both the diagonal and off-diagonal terms in g make the system very
complex and it is not clear how to find all representations of even locally constant states 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 utilizing contractions of the structure constants. Also for
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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 RQT, 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-GR-SM “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 iħg(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 relativity) whose representations are to
support the allowable particles in the SM which gives 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 (not unlike the “fictitious” centripetal and Coriolis “forces” which also emerge from the
kinematical space-time framework). (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 energymomentum 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 EP algebra and the
SM algebra.
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 Lie algebras for the study of differential
equations.
It is (temporarily) reasonable 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 EP structure constants and not a separate kind of particle that issues
from the SM as it is currently the simplest explanation. By contrast, it appears that dark matter (DM) is new type of
particle that has no strong, electromagnetic, or weak interactions but only gravitational. Thus it would not necessarily
emerge in any natural way from the current standard model 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. In our view, there is no need or expectation that DM is a particle that is
currently represented in the SM because 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 similar to what a stable Higgs particle would have. 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 would be another gauge group just for particles that only interacted 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 spin (if any).
Discussion:
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 EPE and seek representations of that algebra in neighborhoods where the
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metric can be considered locally constant. Thus to solve a physical system, 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. 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 EPE+SM+I Lie 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 is
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+I 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 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 EP algebra, it is
necessary to cast that framework into an algebraic structure where derivatives are replaced by commutators and
gamma matrices by spin representations. 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.
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 grr = -1/(1- rs /r) where the Schwarzschild radius
is given by rs = 2GM/c2 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. Therefore we are first specifically seeking
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. Perhaps one of these effects can be detected in the atomic or
nuclear spectra from domains with intense gravity. The integration of the EP-GR and the SM Lie algebras necessarily
bring the non-Euclidian g into the equations of the SM.
We also would expect that a particle would now move along 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 FoldyWouthuysen type formulation that generalizes the motion of a charged Dirac particle in an electromagnetic field where
the free particle would evolve primarily along the Feynman paths close to the geodesic that would create the least
phase interference. The SM formulation would then cause deviations from that geodesic as required by the strong and
electroweak 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 Stage of Research
(A) The next problem is to determine the representations of the EPE Lie algebra that can also support the SM
gauge Lie algebra. This involves a merger of the SM Lie algebras and the EP Lie algebras so that the EP representations
support the SM Lie algebras in a more transparent way than is currently done and where all equations are expressed as
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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 EP
and the SM algebras, we need to study their collective commutation relations. The representations must also support
the three discrete inversions.
(B) We then need to form Tab in terms of the energy-momentum tensor of the existing particles as operators
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 EP Lie algebra
along with GR as a single EPE-GR-SM-I algebraic system.
(C) 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.
References:
1. Poincare Lie algebra references
2. JEJ EP extension of the Poincare Lie algebra
3. Einstein’s equations in GR
4. Standard Model
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