Abstract
Yang-Mills theories are a very fruitful concept in quantum field theory. Fundamental interactions and its unifications can be described with Yang-Mills theory. However, gravity is still not modeled in the framework of Yang-Mills theory. It is modeled in terms of the Einstein-Hilbert action in the case of semiclassical field theory, but ordinary quantization of the spacetime field fails due to UV divergences. A possible approach to quantum gravity called E-gravity theory avoids UV-divergences. Primary, this theory is based on a spacetime discretization and the assignment of a curvature measure to discretized spacetime. This paper shows that this approach is also a special case of Yang-Mills theory.Introduction
The most plausible approach for making a
model for fundamental elementary particle interaction is the Yang-Mills theory.
In this theoretical framework, a local invariance of a physical action under a
gauge group action is assumed. This invariance imposes the existence of a gauge
connection that is also called “gauge boson field”. Electromagnetic,
electroweak and strong interaction can be described very well in terms of
Yang-Mills theory. Electroweak interaction is generated by imposing a gauge group invariance
and strong interaction can be obtained if the action functional is invariant
under
group
transformations. The Standard model of particle physics is based on the gauge
group
. May be
a gauge connection
in spacetime direction
with generator
index
,
the structure
constants of the gauge group,
the coupling
constant and
the associated
field strength. Then the Yang-Mills Lagrangian density
has the form:
The additional Lagrangian density is arising from the
measure factor of the quantum path integral and contains the Faddeev-Popov
ghost fields
that are
Grassmannian variables. It holds:
However, gravity, also another fundamental
force, cannot be described in terms of Yang-Mills theory. The only well-known
consistent way for describing gravity is General Relativity, which is a
classical theory of gravity. An open question is whether gravity can be
formulated as a Yang-Mills theory. Several research papers show that there is
existing a relationship between gravity and Yang-Mills theory (Bern, Z. et al.
1998) (Hsu, J.P. 2006).
There are existing theories that quantize gravity, e.g. Loop Quantum Gravity.
The derivation of Loop Quantum Gravity is based on a reformulation of General
Relativity by introducing gauge fields
(Ashtekar,A. 1985). However, Loop Quantum Gravity does not coincide with
Yang-Mills theory, e.g. due to the use of constraint terms and the quantization
in terms of Wilson loops. Another recent approach to quantum gravity is
E-gravity theory (Linker, P. 2016). This theory provides a discretization of
spacetime with simplices. Simplices can have two or more simplex corners that
are identified (in mathematical terms: there exists a nontrivial equaliser).
Because of this additional property, a curvature measure can be assigned to
each simplex. General Relativity is rewritten in this discrete form and a path
integral over all possible spacetime geometries is performed.
In this research paper, an alternative derivation of E-gravity theory is shown.
At first, spacetime is discretized by finite elements which are the simplices.
Finite element methods are typically used for engineering problems, but are also
applied to quantum field theories (Sopuerta, Carlos F. et al. 2005). Instead of
using assumptions about the discretization of the Einstein-Hilbert action, a
Yang-Mills theory with the diffeomorphism operator as the gauge group is worked
out. Also, the field strength tensor, which is different from the smooth fiber
bundle curvature form, is derived. With proper projection of the Yang-Mills
theory to physical states, E-gravity theory is recovered.
The Derivation of E-gravity theory by Yang-Mills theory
The theory of gravity is assumed to be a
gauge theory with respect to the Poincaré group. If General Relativity is
transformed locally by the Poincaré group, the action remains the same.
Generators of the Poincaré group are local spacetime translations and local
Lorentz transformations. Since the translation generators corresponding to a
4-vector
are operators and
not finite-dimensional matrices, Yang-Mills theory in the form (1) cannot be
applied. The goal is to find a matrix-like gauge group for gravity. The first
step to find a proper gauge group is to use a finite element discretization of
spacetime. In general, a physical field
(can be scalar,
spinor, vector or tensor field) on a spacetime point
is discretized as
follows:
Here, are the field
values at simplex corner (or node)
and
the corresponding
form functions that have the value 1 at simplex point
and the value 0
elsewhere. These form functions generate a discretized spacetime without self-intersection
of simplices. Additionally, if two simplex nodes are identified (a key feature
of E-gravity theory), the sum in equation (3) runs from
to
with the number of
identified nodes
. Denotes
the set of all
possible form functions that discretize the spacetime and
the set of all
nodes in the spacetime, one obtains the path integral:
Clearly, the set is quite different
from the simplex node equaliser set. However, the path integral (4) can be
reduced to E-gravity theory by an appropiate gauge group. May the gauge
operator
acting on a
function
with the entire
spacetime region
be defined as follows:
Because the operator (5) modifies the
physical fields globally, the field is a global gauge
transformation. The operator (5) can be turned into a local gauge
transformation by replacing
.
For a plausible model of gravity, the invariance of the action under the
modified Poincaré group with group elements
acting on physical
states in 4-dimensional spacetime is imposed. This group is called “modified”,
because all elements of the group acting like the convolution operator (5);
these groups are “infinite-dimensional matrices”. Moreover it holds
, i.e.
is an unitary
transformation. Due to linearity of
and the invariance
of the action
under group
transformations in
, the choice of form
functions is arbitrary. If the deformation of simplices by
is diffeomorphic,
one can set without loss of generality
, i.e. local and
global gauge transformations are the same. This can be easily seen if a differential
operator
with respect to
is performed: By
chain rule,
and equation (6) is only true if and only
if . Hence, the gauge
connection vanishes in case of any diffeomorphic deformations.
If two or more simplex nodes are identified, one performs a non-diffeomorphic
deformation. The left hand side of equation (6) does not exist in general. By
switching the gauge transformation from a global to a local transformation, one
can generalize the gauge group to a
non-diffeomorphic gauge group. This implies the existence of a non-smooth (!)
gauge connection
.
Gauge group ensures that
physical laws are unchanged not only if local diffeomorphic deformations are
present as it is the case in General Relativity. Local non-diffeomorphic deformations also let physical laws be the same.
Diffeomorphic spacetime deformations of General Relativity can be assumed as a very
dilute collection of non-diffeomorphic deformations on Planck scales in average.
An important condition for this gauge group is that given the map
between form
functions
for arbitrary
, the resulting form
function
generates also a
simplex element.
Clearly, a gauge connection which is not
differentiable, makes not sense in a physical field theory. If on a certain
spacetime point
, then
is singular, since
is necessary only
for non-diffeomorphic deformations. As a consequence, physical states can only
be observed in regions with
for all spacetime
directions
. Field strengths
are not vanishing
necessarily, because derivatives of
are not zero in
general even if
.
The path integral (4) can be reduced with the assumptions made above.
Introducing a parameter , where
denotes the index
of the topological simplex structure (i.e. it denotes which nodes are
identified). This parameter runs over all different states of a simplex with
same topology and therefore it is called the “diffeomorphic deformation
parameter”; in diffeomorphic deformation this parameter changes. For
non-diffeomorphic deformations, the gauge connection
is changing.
Therefore:
The set is the set of all
possible simplex topologies, i.e. the set of all identified simplex nodes.
Physical action does not change under diffeomorphic deformations (including
Lorentz transformations) and hence, the integral over
for each
is constant. The
integration over all gauge connections makes only sense for
. All contributions
with
can be neglected;
only physical states are taken into account, while the number of unphysical
states is vanishing small. Hence, the Faddeev-Popov factor in the path integral
factorizes in the path integral and can be omitted. One is left with the
following path integral:
The path integral (8) is the path integral
of E-gravity theory if the field strength term corresponds to the
Einstein-Hilbert action in vacuum. It holds
for a 1-form gauge
connection
and a 2-form field
strength tensor
for
. Using theorem of
Stokes over a 2-dimensional area
, one obtains
. If
is sufficiently small,
i.e.
(
denotes the measure
of a set), one can define the field strength corresponding to the area
:
From (9) it follows that if and the path
integration is not performed around a singularity, it holds
. This is the case
if there are no identified simplex nodes in a simplex element. Every node which is
identified with another node causes a nonvanishing value of
due to the change of
spacetime structure by making topological "shortcuts" in it. The statement
with coboundary map
, E-semigroup
elements
and indicator
function
(that is 1 if there
are no identified simplex nodes and 0 otherwise) where the
are pairwise different is proven. If there are 2 or more identified
simplex nodes (these induce singularities), one can choose closed paths in
arbitrary spacetime directions such that all 3 or more singularities are
contained in the path. Since
is small, one has
everywhere on
and in this case
.
It has to be shown that if exactly one simplex node is identified with
, it holds
if
is even and
is odd or
is even and
is odd,
if
and
are odd and
if
and
are even; the value
is a specific
gravitational constant. At first, the following decomposition of the gauge
field action (which is known from electromagnetic theory) can be performed:
The indices are running from 1
to 3 (all space directions). The term
is the
“gravito-electric” field strength and the term
is the
“gravito-magnetic” field strength. By Lorentz transformations, the time
direction can be replaced by a direction pointing anywhere in spacetime, where
all other indices denote all other directions perpendicular to the direction
with index 0; action functional does not change after any Lorentz
transformation.
Simplex elements have to be oriented
properly while discretizing spacetime. Non-diffeomorphic deformations have
singular values of the connection across the 0-direction, if the simplex node
generators or
are identified with
another simplex node (by convention of the coboundary map, all nodes with odd
index are negatively oriented), and have singular values of the connection
perpendicular to this 0-direction, if all other simplex node generators
and
are identified with
another simplex node (by convention of the coboundary map, all nodes with even
index are positively oriented). Every singularity has the same magnitude of the
value
given by
. This assumption
relies on the choice of infinite hypercomplex numbers
with ordinary
finite complex numbers
for the gauge
connection field; indeed, it holds
by nonstandard
analysis. If
is even and
is odd or
is even and
is odd, the path
integration passing through 0-direction encounters one singularity and the path
integration passing through directions only perpendicular to 0-direction
encounters also one singularity; hence
and
.
If and
are odd, the path
integration in arbitrary 0-direction encounters two singularities of
with equal
magnitude, while path integration in directions perpendicular to the
0-direction have no singularities; hence:
,
and
with a constant
. Finally, if
and
are even, the path
integration in directions perpendicular to 0-direction encounters two
singularities of
with equal
magnitude, while path integration in directions in 0-direction have no
singularities; hence:
,
and
with a constant
. The values of
are also infinite
values, but with renormalization of the Yang-Mills coupling constant
this infinity can
be removed.
Discussion
In four spacetime dimensions, the Ricci scalar defined by (averaged over a
volume) can be also recovered by a Yang-Mills theory. Some idealizations like
the equal values of a topological singularity are assumed, which might lead to
small inaccuracies if the field strength is computed by more advanced concepts
(not treated in this research paper). The equivalence between Yang-Mills theory
and discretized Einstein-Hilbert action with curvature measure
is only valid for
3+1-dimensional spacetime (one direction generated by 2 simplex nodes makes the
pure “electric” field strength and all three other directions generated by all
other 3 simplex nodes make the pure “magnetic” field strength). Extra
dimensions can play a role if E-gravity theory is embed in a certain plausible
framework of superstring theory or M-theory. However, there is still not found
a way to do this. However, there is evidence that the Standard model of
particle physics with gravity is recovered if superstring theories are compactified
on the Calabi-Yau manifolds (Candelas et al. 1985). Getting the Standard model
combined with E-gravity theory from superstring theory or M-theory by suitable
compactification is still not clear even since the gauge group
has different
structure compared with gauge groups that impose all other fundamental
interactions.
Inhomogenity in spacetime, which is also a
feature of E-gravity theory, is caused exactly by the field which is
nonvanishing between the simplex elements, where physical states can occur.
This gauge connection separates simplex elements which are different. E-gravity
theory in terms of a Yang-Mills theory would conserve energy, momentum and
spin-angular momentum, but if the 1-form field
is “cut out”,
energy and momentum has to be created and annihilated to include effects
performed by the gauge connection.
Gravitational interactions are able to materialize multiple particles. From the
Yang-Mills theory, gravity couples on the currents (here, the fermionic case
with fermion fields and coupling
constant
)
where is the generator of
diffeomorphic deformations. It is a convolution operator and from (11) it holds
that gravity is induced by correlations of particles in the entire spacetime. A
simple example is the correlation of an apple that correlates with the earth;
the apple falls towards earth because gravity is triggered by the apple-earth
correlation. In systems where quantum entanglement can occur, physical states
can correlate; therefore gravity is also produced in entangled systems.
Assuming that the generators
of the simplex
modification are acting only on a small, bounded region, the correlation
current (11) involves only correlations of states that are very close to each
other. Moreover, due to the very small support of these generators, one can
assume that the commutators of these are neglectible; an abelian gauge theory
arises. Therefore, gravity has an “electromagnetic” character and the force is
transmitted with speed of light.
The fact that entangled states produce
gravitational energy can be understood as follows: If two particles are
interacting with each other, the structure of quantum uncertainty between these
particles is modified by gravity. Two different independent random motions that
are interacting with each other, will combine to one correlated random motion.
But such a process would lead to decreasing entropy . Gravity that is caused by
correlation avoids a violation of the second law of Thermodynamics, it increases
entropy by and therefore, the
energy that is produced by gravity has to be of order
for a given
temperature
. Energy has to be
bounded by Planck energy which implies that the entropy change due to
gravitational interaction
has to be finite.
Quantum Gravity theories are still not proven by experiments. Some other research was performed to discuss effects predicted by quantum gravity (Takeuchi et al. 2016). However, experiments for effective gravity theories (e.g. General Relativity) were performed. A recent experimental verification of General Relativity (Abbott 2016) is the detection of gravitational waves from binary Black Holes. In terms of E-gravity theory, two Black Holes are correlating with each other and therefore these would induce a gravitational current given by (11). By using the Yang-Mills form of E-gravity theory one would obtain equations similar to Maxwell’s equations to electromagnetism with a correlation current instead of a electric current. The fields of the gauge connection have large values on small length and time scales. If the action of E-gravity theory is treated as a mean-field theory for large length and time scales, General Relativity can be recovered; moreover, since the regions of spacetime where large values of the nonvanishing gauge connection occur are very little, one can treat these gauge connections also as small values on large length and time scales. This implies that E-gravity theory predicts also waves induced by gravitational correlations. The largest amplitudes of gravitational waves occurring when the two Black Holes are collided; the Black Holes have the largest correlation in this case.
Another experimental result, where predictions of E-gravity theory agree, is the entropy of a Black Hole. Black Hole entropy computed with E-gravity theory is proportional to the event horizon area of the Black Hole (Linker 2016).
Gravity is a very weak force in comparison with all other fundamental forces of nature. In E-gravity theories the value of the curvature measure vanishes at the most of the possible simplex configurations; making gravitational effects nearly irrelevant for particle scattering at low energies and low particle densities. Therefore, E-gravity provides an explaination of the weakness of gravity in comparison with all other fundamental forces.
Conclusions
An interesting correspondence between
Yang-Mills theory and General Relativity is shown in this research paper.
Yang-Mills theory with another version of the Poincaré group, called with special
assumptions about simplex topology turns out to coincide with the
Einstein-Hilbert action that uses a generalized curvature measure based on
chain complexes of simplices. Formulating fundamental interactions in terms of
Yang-Mills theory makes it easier to find unified descriptions of fundamental
forces. Indeed, the gauge group of Standard model of particle physics combined
with E-gravity theory can be regarded as the gauge group
that looks like a
unified description of fundamental forces.
Unfortunately, a plausible Grand Unified Theory is still not found; only
experimentally unverified models are proposed. However, E-gravity theory also
applies to effects, where not all other fundamental forces are unified.
References
Ashtekar,A. 1985. “New variables for classical and quantum gravity.” Physical Review Letters 57(18): 2244. doi:http://dx.doi.org/10.1103/PhysRevLett.57.2244.
Bern, Z. et al. 1998. “On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences.” Nuclear Physics B 530(1): 401–56. doi:10.1016/S0550-3213(98)00420-9.
Hsu, J.P. 2006. “ YANG–MILLS GRAVITY IN FLAT SPACE–TIME I: CLASSICAL GRAVITY WITH TRANSLATION GAUGE SYMMETRY.” International Journal of Modern Physics A 21(25): 5119–39. doi:http://dx.doi.org/10.1142/S0217751X06034082.
Linker, P. 2016. “E-gravity theory.” The Winnower 3: e145441.18359. doi:10.15200/winn.145441.18359.
Sopuerta, Carlos F. et al. 2005. “A toy model for testing finite element methods to simulate extreme-mass-ratio binary systems.” Classical and Quantum Gravity 23(1): 251. doi:http://dx.doi.org/10.1088/0264-9381/23/1/013.
B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration) 2016. "Observation of Gravitational Waves from a Binary Black Hole Merger". Physical Review Letters 116 (6). doi: 10.1103/PhysRevLett.116.061102.
Candelas, P. et al. 1985: “Vacuum configurations for superstrings.” Nuclear Physics B. 258: 46–74, doi: 10.1016/0550-3213(85)90602-9.
Linker, P. 2016. “Black Hole Entropy and dynamics of quantum fluctuations predicted by E-gravity theory.” The Winnower 3:e145934.43470. doi: 10.15200/winn.145934.43470.
Takeuchi, T. et al. 2016. “Observable Effects of Quantum Gravity.” arXiv Preprint. arXiv: 1605.04361
Showing 7 Reviews
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1
Attempts to describe properties of a theory due to the author that does not seem to have any real connection to known physics or measurable effects. It is my opinion that this would be dismissed by most journal editors without even being sent to a reviewer.
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Well I find it very interesting Patrick but here is my question and that is Did extra dimensions have any role to play ? Thanks :)
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Is your theory agree with expermental results ?
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Thanks for the review! I have updated and improved the paper (fixed also an error).
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It is an interesting and inspiring article about the topic and it is clearing the view onto the methodology of Quantum Gravity in its here discussed form.
The mechanism between quantum correlation, i.e. entanglement, and gravitation should be extended to a more physically based understanding and not only derived from the mathematical framework.
The way how the access energy out of correlation between particles converts into gravity should be in the future appended under consideration of Planck Energy. A treatment of the entropy argument for the in this article proposed view is recommended.
Equation (11) claims to express a particle correlation over the hole space time. May be it should be combined with some considerations how quantum correlations over such large distances can be established especially under consideration of the time necessary to propagate through the universe.
From EPR experiments we know about the instantaneous propagation of influence on super positioned systems. So it could be interesting to make some propositions about the propagation velocity of the this way described gravity effects. Recent measurement of gravitational waves in coincidence with synchronized data from e-m-based telescopes are indicating that gravitational waves are traveling at the speed of light. So it is worth to show how the instantaneous propagated dynamics of the change in superpositioned systems is to be in accordance with gravity on the speed of c.-
Very good review!
Added some things in the paper (see "Discussion" section).
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0
Is your theory agree with experimental data ?
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I have added a reference where agreement of E-gravity theory with experimental data occurs (Black Hole entropy).
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Technically good,but i don't understand its proper physical meaning, can you explain me ?
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I have added some clarification to the gauge group MP(1,3).
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Promising approach with the derivative Yang Mills and Superstring correspondence, looking forward more development of this project Partick, you might be interested to look into the Z2 gauge,lattice theories and modular-quasi modular gravity.
License
This article and its reviews are distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and redistribution in any medium, provided that the original author and source are credited.
I have updated the paper now. E-gravity from its original point of view has nothing to do with extra dimensions, but some string theorists can try to find a theory, where the compactification of it leads to Standard model with E-gravity theory.
If string theory is to be relevant at all for physics,it is because it provides evidence for the existence of a more fundamental theory.
Extra dimensions role not needed for this concept.