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AN INTEGRATION OF GENERAL
RELATIVITY AND RELATIVISTIC
QUANTUM THEORY
Joseph E. Johnson, PhD
jjohnson@sc.edu
May 5, 2016
Physics Department
University of South Carolina
I. INTRODUCTION
A Linear Vector Space (LVS):
1.
1.
2.
3.
Allows two operations: |A>+|B> = |C> and a|A> = |B>
A basis |i> for the space gives any vector as |A> = ai |i>
A LVS becomes more powerful with two products:
A Metric Space (MS) is a LVS with the scalar
product:
2.
1.
2.
3.
4.
5.
A*B = SAiBi = a number = |A| |B| cos q = <A | B >
So we get the ‘metrics of length and angle.
Examples: regular space and the unitary scalar product
In infinite dimensions this is called a Hilbert space
Or in a Riemannian geometry <a|b> = gmn am bn where g is the “metric”
A Lie Algebra (LA) is a LVS with the product
1.
1.
2.
3.
2.
[Li, Lj] = cijk Lk & the Jacobi Identity Scyc per [[L1, L2],L3]=0
Lie Group : G(t) = etL = 1 +tL + t2L2/2!+…
LA Examples: Rotation Sxi2, Translations xi=xi+ai, Lorentz c2t2-r2, Poincare
(Lorentz with translations), Unitary SYa*Ya=1, General Linear xi = Gij
xj, Markov Sxi = Sxi, Scaling xi=elixi & Heisenberg X, P, I.
LA Representations: LA can be represented by
matrices Li acting upon vectors |Y> in a metric
space: |Y’> = L |Y>.
1.
Vectors in a LA representation space are
distinguished by a list of internal values that
normally represent the eigenvalues of a complete set
of commuting operators: |a1, a2, …. >
2.
One can also use creation, a+, and annihilation
operators, a, that create a given vector from the
vacuum: |a1, a2, …. > = a+ a1, a2, …. |0>
3.
In this framework, the original LA (Li i = 1, 2….)
and the “creation operators” that constitute the
representation, a+ a1, a2, …. , both have well defined
commutation rules and constitute a larger LA !!!
NOTATION
1.
We use Greek indices for time & space with
values 0, 1, 2, 3 with the metric gmn = (+1, -1, -1,
-1), and English indices for space alone.
2.
Repeated indices indicate contraction by
Einstein convention.
3.
Generally X0 = ct, X1 = x, X2 = y …
1. And P0 = E/c, P1 = Px , P2 = Py …
FUNDAMENTAL FRAMEWORK
1.
The fundamental observables (measurements) in nature L1,
L2, … can be represented by an algebraic structure (usually
a LA) whose commutators show how these observations
(measurements) may interfere with each other.
2.
The exponential maps from a LA, M(a) = exp (ai Li), form Lie
Groups (LG) where the associated Li are conserved
quantities if the transformation group leaves a system
invariant.
3.
The possible states of nature | > are the vectors in a metric
space that form a representation space of the LA of possible
observables.
4.
That which can exist is a representation space for the
algebra of actions (measurements / observations) which can
be performed.
OPERATORS (OBSERVABLES OR ACTIONS)
1.
Operators can generate:
1.
2.
3.
2.
Operators can be classified as:
1.
2.
3.
Symmetries & associated conserved quantities
(momentum)
Approximate symmetries (isospin)
No associated symmetry but are spectrum generating
(position)
Global like momentum, angular momentum, …
Local like gauge transformations
Operators can create LA representations
from |0>
1.
a+a1,a2,… |0> = |a1,a2,… >
FOUR DOMAINS ACCEPTED:
1.
RQT: Quantum Theory, Special Relativity, &
Field Theory
2.
SM: The Standard LA Gauge Model of
Particles & Dynamics
3.
GR: Einstein’s Theory of General Relativity
4.
Inversions: Space, Time, & Particle Conjugation
5.
Dark Matter (DM) and Dark Energy (DE) are still not understood and
are only peripherally considered.
THE PROBLEM:
Is, It, Ic spin/stat
RQT  Well Integrated  SM
?
?
GR
DM
DE
1.
Quantum theory follows from the Heisenberg
algebra:
1.
2.
and [E, Time] = iħ I
Intrinsic spin follows from the rotation algebra:
1.
3.
[Pi, Xj] = -iħ dij I
[Si, Sj] = eijk Sk
Special Relativity follows from the Lorentz
algebra:
1.
2.
[Mmn, Mrs] = iħ (gmsMnr + gurMms - gmrMns - gusMmr)
and
[Mmn, Pl] = iħ (gln Pm - glm Pn) for any vector Pl
II. THE EXTENDED POINCARE (EP) ALGEBRA
(POINCARE LIE ALGEBRA EXTENDED WITH A FOUR-VECTOR POSITION XN OPERATOR)
THE POINCARE ALGEBRA IS IN BLACK. THE EXTENSION WITH XN IS TAN.
6.
[I, Pm] = [I, Xn ] = [I, Mmn ] = 0
[Pm, Xn] = iħ gmn I (covarient form of [P , X ] = -iħ d
[Pm, Pn] = 0
[Xm, Xn] = 0
[Mmn, Pl] = iħ (gln Pm - glm Pn)
[Mmn, Xl] = iħ (gln Xm - glm Xn)
7.
[Mmn, Mrs] = iħ (gmsMnr + gurMms - gmrMns -
1.
2.
3.
4.
5.
i
j
ij I
Heisenberg LA)
gusMmr)
THE EP REPRESENTATIONS
1.
Define Lmn = Xm Pn – Xn Pm (orbital angular momentum
tensor)
1.
From which it follows that
2.
[Lmn, Pl] = iħ (gln Pm - glm Pn)
3.
[Lmn, Xl] = iħ (gln Xm - glm Xn)
4.
[Lmn, Lrs] = iħ (gms Lnr + gur Lms - gmr Lns - gus Lmr)
One can then define an intrinsic spin four-tensor as
1.
Smn = Mmn - Lmn
EP DECOMPOSED INTO HEISENBERG (XPI)
& HOMOGENEOUS LORENTZ (S)
1.
Then the commutators for the intrinsic spin Smn are
2.
[Smn, Pl] = 0
3.
[Smn, Xl] = 0
4.
[Smn, Lrs] = 0
5.
[Smn, Srs] = iħ (gms Snr + gur Sms - gmr Sns - gus Smr)
6.
Thus the EP algebra becomes the product of the two
algebras: EP = XPI * S (Heisenberg * Lorentz) and the
full representations are the products of the separate
representations.
XPI HEISENBERG LA REPRESENTATIONS
1.
One can use the position representation
1.
Xm | y > = y m | y >
or the momentum representation
2.
Pm | k > = k m | k >
or equivalently diagonalize the mass Pm Pm = m2,
the sign of the energy, e( P0), and Pi as
3.
|m, e( P0), k >
THE LORENTZ GROUP REPRESENTATIONS
1.
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:
2.
b02 + b12 – 1 = ½ gmrgnsSmn Srs where b0 = 0, ½, 1, 3/2,
…(|b1|-1) and where b1 is a complex number
3.
b0 b1 = - ¼ e mnrs Smn Srs and with the Casimir operator for
the rotation subalgebra as
4.
S2 which has the spectrum s(s+1) with the total spin s = b0 ,
b0+1, …, (|b1| - 1) and
5.
s = -s, -s+1, ….s-1, s which is the z component of spin.
THE STANDARD MODEL (SM) ADMITS
ONLY THREE REPRESENTATIONS:
1.
b0 = ½, b1= ± 3/2: A Dirac spin s=½ particle
where the two signs of b1 are conjugate
representations giving four components and the
g matrices (quarks & leptons).
2.
b0 = 0, b1=1:
3.
b0 = 0, b1 = 2: A vector particle with spin
s = 0, 1 which can be written as a four-vector Am
(like the electromagnetic field potential for
photons). (photon, W, Z)
A spin s = 0 particle (Higgs)
THE EP REPRESENTATION SPACE
1.
2.
Thus one had the representation|b0 ,b1 , s, s >
Which gives the EP representations as:
|k m , b0 , b1 , s, s > = a+k , b0, b1, s, s |0>
for the momentum or position representation or
3.
| y m , b0 , b1 , s, s > = a+y, b0, b1, s, s |0>
for the position representation.
III. THE STANDARD MODEL OF PARTICLES
& DYNAMICS (SM)
1.
A gauge LA model using the unitary groups:
U(1) x SU(2) x SU(3) on spin ½ (Dirac) spinors
generating the associated 4-vector and scalar
bosons responsible for the strong, weak, and
electromagnetic forces.
2.
This phenomenological (non space-time) algebra
contains 1+3+8 = 12 observables (operators)
including electrical charge.
1.
The fields (particles) that support this group are
EP representations (a+….) mixed with EP
operators (Pm…).
2.
Although the SM is still a work in progress, it
has had remarkable success in predicting forms
of the interactions, masses, and the dynamical
evolution of interacting systems using a
Lagrangian for the system.
IV. EINSTEIN’S THEORY OF GENERAL
RELATIVITY (GR)
In GR the metric of space time is curved by the
presence of matter and energy as given by the
equation
1.
Rab - ½ gab R + gabL
= (8pG/c4) Tab
where Rab is the Ricci tensor, R is its contraction, L
is the cosmological constant and Tab is the energy
momentum tensor expressed in terms of the
particles in the SM theory.
1.
The Ricci and Riemann tensor terms on the
LHS are defined in terms of the derivatives of
the Christoffel symbols which are in turn
defined in terms of derivatives of the metric
tensor.
2.
Terms on the RHS are defined in terms of the
energy momentum tensor, Tab, expressed in
terms of the operators in the SM theory acting
on the state of the system.
V.
A PROPOSED INTEGRATION
THE INTEGRATION ARGUMENT
1.
The EP is the Lie algebra of all space time
observables with structure constants defined in
terms of the Minkowski flat metric giving RQT.
2.
As the Minkowski metric is altered by GR, it
seems reasonable that this metric, the
associated space-time operators, and the
defining EP structure constants be generalized
to the Riemannian curved space time metric
required by GR thus capturing all space-time
operators in an Extended Poincare Einstein
(EPE) algebra.
THE INTEGRATION ARGUMENT
1.
This merger with quantum theory can now be done
by allowing the metric to be a function of the new
four-position operators as gmn -> gmn(X).
2.
The energy momentum tensor on the RHS of the
Einstein equations cannot come from the RQT but
rather from the SM operators acting on the state of
the current system. (In practice, it comes from large
masses such as black holes or stars and thus the
classical expression is valid).
3.
Thus GR becomes a bridge between the 15
parameter EP global algebra of space-time
observables and the 12 parameter SM gauge algebra
of phenomenological observables.
THE EPE ALGEBRA
1.
Is a ‘kinematical’ foundational algebra that
encompasses the global operators of space and
time but which does not explain forces or the
states of physical systems.
2.
The EPE representations must also support the
gauge algebras and all interactions.
3.
Encompasses the foundations of quantum
theory (Heisenberg algebra), special relativity
(Poincare algebra), and general relativity (gmn)
THE INTEGRATION POSTULATE
1.
The gmn in all EP structure constants is
now to be taken as a function of the fourposition operators, gmn(X), which in the
position representation becomes a
function of space-time variables to be
determined by the Einstein equations
using the energy-momentum tensor
density Tmn from SM operators acting on
the state of the system.
2.
We call this generalized EP LA the
Extended Poincare Einstein (EPE) LA
RQT – SM – GR INTERDEPENDENCE
1.
Thus the “space-time” EPE LA gives the
representations (particles) which also must
serve as the representation space for the SM
“gauge algebra”.
2.
The energy momentum tensor density operator
defined by the representations in the SM then
are to determine by GR the metric that defines
the EPE algebra which in turn defines the
possible SM representations.
THE PROPOSED SOLUTION:
Is, It, Ic spin/stat
RQT (EPE)
gmn (X)
L
SM
Tmn
GR
new DM
particle
DE
conjectured links in tan
DM
Dynamics is given by Feynman Paths with the SM Lagrangian
DETAILS OF THE PROPOSED INTEGRATION 1
1.
The basic commutation rules for EP remain the
same for EPE (gmn now= gmn (X) & I has the single eigenvalue
“1”.)
2.
[Pm, Xn] = iħI gmn(X) so we can now write
3.
gmn (X) = (-i/ ħ) [Pm, Xn]
4.
In the position representation one now has
5.
<y| Pm | Y> = iħ gmn(y) ∂/∂yn) Y(y)
1.
= iħ ∂m Y(y)
where Y(y) = <y| Y >.
DETAILS OF THE PROPOSED INTEGRATION 2
1.
Although one still has
1.
2.
3.
4.
2.
where from now on
1.
3.
[Pm, Pn ] = 0 but since
[Pm, gab ] ≠ 0 it follows that
[Pm, [Pn, Xr]] ≠ 0 so that the Heisenberg algebra is
no longer nilpotent. It also follows that
[Pm, gab] = iħ gmn (∂gab /∂yn) = iħ ∂mgab = (-i/ħ )[Pm, [Pa, Xb]]
gab = gab (y) is to be understood.
Thus in the position representation all ∂m f(x)
are converted into (–i/ħ) [Pm, f(X)]
DETAILS OF THE PROPOSED INTEGRATION 3
Christoffel symbols are defined as:
1.
Gcab = (½) ( [∂b, gca] + [∂ a, gcb] - [∂ c, gab])
2.
Gcab = (½) (-i/ħ) ( [Pb, gca] + [Pa, gcb] - [ Pg, gab])
Then using
3.
4.
gab (X) = (-i/ ħ) [Pa, Xb]
one obtains
Ggab = (-½) (1/ħ2) ( [Pb, [Pg, Xa]] + [Pa, [Pg, Xb]] - [ Pg, [Pa, Xb] ] )
DETAILS OF THE PROPOSED INTEGRATION 4
Then the Riemann tensor becomes:
1.
Rlijk = (-i/ħ) ( [Pj, Glik ] - [Pk, Glij ] ) + (Gljs Gsik - Glks Gsij )
2.
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 only:
DETAILS OF THE PROPOSED INTEGRATION 5
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] ]))
DETAILS OF THE PROPOSED INTEGRATION 6
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 to be inserted into the Einstein equations,

Rlj - ½ glj R + gljL

Rlj + ((i/ ħ) [Pl, Xj]) ( ½ R + L ) = (8 π G/c4) Tlj
= (8 π G/c4) Tlj or
DETAILS OF THE PROPOSED INTEGRATION 7
1.
This expresses the LHS of the GR equations in
terms of the fundamental EP operators and
their commutators.
2.
The energy momentum tensor Tab must now be
expressed in terms of the operators in the
standard model:
3.
Tab = Y (ga Pb)Y where the (ga Pb ) term is both
symmetrized and also acts in both directions on
all Dirac fields and similar expressions obtain
for the gauge bosons.
1.
Since the EPE LA describes the structure of all
space-time observables, it follows that the
generalization of gab to gab(X) incorporates the
curved nature of space time generalizing both
special relativity and quantum theory.
2.
It captures the dynamics of geodesic motion of
entities as dictated by the metric.
3.
It does not provide information on the allowable
particle mass and spin states, nor the interactions of
the strong and electroweak force nor the dynamical
evolution under those forces all of which is still done
in the SM.
SYSTEM SOLUTIONS - 1
One might begin with a static environment
where particles are in a large gravitational field such as
near a black hole or neutron star.
Here we can compute the metric as the
Schwarzschild solution. Having the metric then we can
use the representation of P in the position
representation as:
 <y| Pm | Y> = iħ gmn(y) ∂/∂yn) Y(y)
The internal spin tensor is not altered via the
metric. Then one can write the representations as
before such as for the spin ½ fields for the SM.
SYSTEM SOLUTIONS - 2
The Lagrangian can be constructed as before but
now with the new representations for Pb and ga .
The dynamics would proceed by path-integral
solutions and the energy momentum tensor would
be computed from the dominant fields (which might
just consist of the contribution from the massive
sphere or black hole.
The particle paths would then follow geodesics in
the curved space time in order to minimize the
Lagrangian phase interference.
VI A: DARK ENERGY (DE)
1.
It is currently known that the cosmological
constant L in the GR equations is the simplest
account for the DE expansion of the universe.
2.
And since the accelerated expansion of the
universe is a space time transformation, it is
natural to assume that DE arises from this GR
parameter (until otherwise informed by
experiments).
VI B: DARK MATTER (DM)
1.
It is known that DM is not contained in the current SM as
it only interacts gravitationally while the SM represents
only the phenomenological LA structure of the strong,
electromagnetic, and weak interactions.
2.
Thus there is no expectation for it to be an existing known
particle.
3.
It would be reasonable for DM to be a new particle in the
SM like the Higgs, but stable, that naturally would not be
predicted from its existing phenomenological basis in
other interactions.
4.
This assumption then will require an associated Tmn term
to account for this unknown particle whose mass and spin
would have to be determined from gravitational
interactions or theoretical arguments.
POSSIBLE TEST AREAS:
1.
2.
3.
Study & simplify the 30 parameter algebra of
space-time and gauge operations along with the
a, a+ representation operations and analyze for
patterns.
Use the Schwarzschild solution for the metric
near a massive object where a test atom,
nucleus, or molecule is placed to study possible
alterations in the photon emissions.
Using the altered uncertainty relations
between X and P, seek effects that could be
detectable in states of systems.
PROPOSED PROGRAM
1.
I propose to study the combined 30 parameter
algebra composed of the 15 parameter EPE and
the 12 parameter gauge LA and adjoining the 3
parameter Inversion algebra.
2.
To study this algebra seeking simplifications in
the tensor expressions with the Bianchi, Jacobi,
and other means with the hope that some
‘beautiful structure’ might emerge as it did with
Maxwell’s equations.
SPECIFICALLY NEEDED:
1.
2.
3.
4.
5.
Formulate the 12 parameter gauge algebra
along with the three discrete inversions as
algebraic commutators including expressions of
all mixing angles and related SM parameters.
Study and simplify the EPE and general 30
parameter LA using symbol manipulating
software (Mathematica, Python/Sage, MatLab)
Study the mathematical properties of the 30
parameter algebra and the associated group
manifolds.
Seek new predictions involving quantum effects
in strong gravitational files
Seek new implications in the SM framework
THANK YOU
THE EPE ALGEBRA
(EXTENDED POINCARE EINSTEIN LIE ALGEBRA)


[I, Pm] = [I, Xn ] = [I, Mmn ] = 0
[Pm, Xn] = iħ gmn I


which is the Heisenberg Lie algebra – the foundation of quantum theory
[Pm, Pn] = 0
insuring the noninterference of energy momentum measurements all four dimensions
 [Xm, Xn] = 0

insuring the noninterference of time and position measurements all four dimensions
 [Mmn, Pl] = iħ (gln Pm - glm Pn)




which guarantees that Pl transforms as a vector under
Mmn
[Mmn, Xl] = iħ (gln Xm - glm Xn)
 which guarantees that Xl transforms as a vector under Mmn
[Mmn, Mrs] = iħ (gmsMnr + gurMms - gmrMns - gusMmr)

which guarantees that Mrs transforms as a tensor
TO BEGIN:
1.
2.
3.
4.
Assume that any physical system (components of the
universe) can be represented by vectors in a linear vector
space |a> which is also a metric space: <a|b> = a
number.
Assume that any action on |a> can be represented by an
operator L in linear vector space of fundamental
operators, L = aiLi , i = 1, 2, … with closure L|a> = |b>
Assume that the Li form a non-commutative algebra of
fundamental actions [Li , Lj] = cijk Lk. and which is
(normally) a Lie algebra.
Thus |a> must be a representation space of this algebra
where the order of actions define the interference among
the fundamental actions as defined by the structure
constants: cijk.
THE FUNDAMENTAL ALGEBRAS OF
RELATIVITY AND QUANTUM THEORY
1.
Quantum theory follows from the Heisenberg
algebra:
1.
2.
Intrinsic spin follows from the rotation algebra:
1.
3.
[Pi, Xj] = -iħ dij I
[Si, Sj] = eijk Sk
Special Relativity follows from the Lorentz
algebra:
1.
2.
[Mmn, Mrs] = iħ (gmsMnr + gurMms - gmrMns - gusMmr)
and
[Mmn, Pl] = iħ (gln Pm - glm Pn) for any vector Pl
EXTENSION OF THE POINCARE LIE ALGEBRA TO
INCLUDE A POSITION FOUR VECTOR XM



When a position operator X1 is viewed by a
moving observer, it must transform into a “time
operator” X0.
Thus we extended the Heisenberg Lie algebra to
include a four-vector space-time position operator
which obeys [Pm, Xn] = iħ gmu I where “I”
commutes with all operators.
Thus we are extending the Poincare algebra by
five new operators Xn and I forming a 15
parameter Lie algebra of M, P, X, and I.
THE EP OR EPE ALGEBRA
(EXTENDED POINCARE LIE ALGEBRA)


[I, Pm] = [I, Xn ] = [I, Mmn ] = 0
[Pm, Xn] = iħ gmn I


which is the Heisenberg Lie algebra – the foundation of quantum theory
[Pm, Pn] = 0
insuring the noninterference of energy momentum measurements all four dimensions
 [Xm, Xn] = 0

insuring the noninterference of time and position measurements all four dimensions
 [Mmn, Pl] = iħ (gln Pm - glm Pn)




which guarantees that Pl transforms as a vector under
Mmn
[Mmn, Xl] = iħ (gln Xm - glm Xn)
 which guarantees that Xl transforms as a vector under Mmn
[Mmn, Mrs] = iħ (gmsMnr + gurMms - gmrMns - gusMmr)

which guarantees that Mrs transforms as a tensor
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