Abstract
Differential geometry is a powerful tool in various branches of science, especially in theoretical physics. Ordinary differential geometry requires differentiable manifolds. This research paper shows how concepts of differential geometry can also be applied to pure topological spaces. Such a theory is based on concepts like cohomology theory. It allows to define a curvature operator also on pure topological spaces without connection. The main advantage of this theory is that the only required information about the topological spaces is the structure of these spaces. A formulation of quantum gravity is also possible with this theory.1. Introduction
Since differential geometry is applied very
successful in sciences like theoretical physics (e.g. general relativity, gauge
theory), computer graphics and many more, there are no significant doubts about
the logical structure of this branch of mathematics. However, differential
geometry is based mainly on the requirement of the smooth -manifold.
Also the theory of Lie groups which is used very frequently in theoretical
physics bases on smooth manifold. Despite the great success of this concepts
there is remaining a disadvantage which is the failure of derivatives at
certain regions of the manifold. Such a problem is avoided in discrete
differential geometry due to the fact that only finitely many objects (e.g.
simplices) are used. However, many structures cannot be modeled sufficiently
accurate with discrete differential geometry. Another possibility of
introducing a differential even if the manifold is not differentiable in
ordinary analysis is Nonstandard analysis (Schmieden 1958).
This research paper focuses also on the
unification of quantum theory with general relativity. There is still not found
a way how to quantize general relativity (Hamber 2009). The reason is because
of the singularity at the Big Bang that is predicted by the classical general
relativity theory which is based on the geometry of smooth manifolds.
Heisenberg’s incertainty principle with
momentum uncertainty
and
length resolution
states
that for the continuum limit, i.e.
,
the incertainty of momentum becomes infinite which is clearly unphysical. To
solve this problem, various theories of quantum gravity were proposed. An
example of such a theory is String theory (Schwarz 2007) that assumes that
elementary particles are not pointlike such that
.
Other approaches of quantum gravity are Loop Quantum Gravity (Ashtekar 1987)
and Causal Dynamical Triangulation (Loll 1998). These theories provide a
discretization of spacetime, where Causal Dynamical Triangulation relies on
discrete differential geometry. There are still no experimental validations of
these theories and therefore the plausibility of these theories is still an
open question in theoretical physics.
Also an open question is how to define a proper directional derivative in general manifolds or even in topological spaces. Formal descriptions of manifolds are existing in mathematics literature; one of these are pseudogroups (Golab 1939). Another formal description of manifolds is synthetic differential geometry (Katz 1970). This research paper shows how topological spaces can be formalized with the use of semigroups and category theoretical elements. With the definition of the E-semigroups calculus on general topological spaces can be performed. The groups are called E groups because the letter “E” is an acronym of the word “Equalizer”; one can also call this theory “Equalizer-Theory”. After the basic definitions and theorems about such topological spaces the application to quantum gravity is shown. It will be shown the following
Theorem 1.1: A quantum gravity theory is possible without singularities.
The proof of this theorem is given in section 2 of this research paper.
2. Theoretical Concept
The
E-Theory is based only on topological spaces that
have finite cardinality. Here, the main ingredient of the E-Theory are the
E-semigroups (groups without the inverse and identity property).
Definition 2.1: May be a
semigroup, where operations between elements
are
only multiplications. If the maximum number of indices that are attached on an
element of
is
,
this semigroup has the characteristic
.
The semigroup
is
called an E-semigroup if the following properties are satisfied:
(i)
or more general
; here the elements
are called generators of
(ii)
The group contains the empty element with
and
arbitrary
.
(iii)
If there is an equality in indices, i.e. then
(iv)
The semigroup is commutative.
Property (iii) of the definition 2.1
contains the equalizer property: The set is
clearly not empty for
and
if the element
has
two or more factors
that
are equal, it holds
.
E-semigroups are strongly related to relations between objects.
Example 2.2: The set of all possible (generalized) relations between objects
which also includes the objects is an E-semigroup of characteristic
.
A very important fact is that E-semigroups can formalize topological spaces.
Lemma 2.3: Let be
a topological space which can be covered by minimal closed subsets
, i.e.
and
cannot be subdivided into smaller subsets. Then it exists a functor
between the category of topological spaces
and the category of E-semigroups
.
Proof: A
closed subset has
a boundary
that
can be determined by computing the following map:
.
Since
is
a minimal subset, it can be regarded as an element of
.
An E-semigroup of characteristic n has only one element
with
due
to property (iii) of definition 2.1. Therefore, one can define an isomorphism
where
is
the E-semigroup associated with subset
.
The set of all elements
can
be obtained by removing one generator from the factorization of
;
this map is denoted by
.
Finally, one can construct a commutative diagram:
and therefore the functor exists.
From Lemma 2.3 a calculus on topological spaces can be defined that is similar to the calculus on manifolds (exterior calculus).
Lemma 2.4: For every closed subset that covers a topological space
a
long exact sequence
with
can
be constructed if
for
arbitrary
.
Proof: If for
arbitrary
,
there is no empty element contained in the sets
for
due
to the equalizer property. Hence, every element of
is
well-defined. Defining the map
as
where the superscript
denotes
that this index is omitted. Then it is easy to show that it holds the exactness
condition
.
Clearly one can apply a functor with
a group
to
the exact sequence constructed in Lemma 2.4. This leads to an exact sequence in
functions on a topological space (by respecting Lemma 2.3). Such an operation
is very similar to the conversion of the simplicial complex to the deRham
complex; a Hodge dual can be defined analogously.
From ordinary differential geometry it is
known that for a scalar it
follows
with
the 2-form torsion tensor
and
for a vector
it
follows
with
the 2-form curvature tensor
if
is
the exterior covariant derivative. Both quantities are based on the loss of
exactness in a chain complex. Since the E-Theory related to topological spaces
is based on a chain complex, one can define a curvature in topological spaces.
Theorem 2.5: A curvature value (the analogous quantity is the curvature 2-form in differential
geometry) can be assigned to every closed subset
.
Proof: The
exact sequence of Lemma 2.4 requires that the E-semigroup has no nonempty equalizers.
If there are nonempty equalizers, the chain complex loses
the exactness. Now one can define an indicator function
that
is applied on a E-semigroup element. This indicator function has always the
value 1 with exception if it is applied on an empty element; this indicator
function has the value 0 when applied on an empty element. therefore, this
indicator function measures the presence of equalizers. The curvature value can
be obtained by computing the inexactness function
.
Here, the indicator function is evaluated on the element
,
because it represents (by applying the functor
used
in Lemma 2.3) the whole closed subset
.
Equation (*) can also be written as
since
the inexactness function lies also on
.
Comparing this equation with the equation
with
curvature operator
of
differential geometry leads to the choice that it can be set
;
it holds
,
because the element
has
to be well-defined (in other words: this element is not the empty element).
Theoretical frameworks given in this
section of this research papers can be used to rewrite General relativity in a
form such that it can be quantized without UV-divergences. It is clear that the
action functional of General relativity has the form with
a coupling constant
and
tetrads
.
Original General relativity has well-defined distance and angle values, whereas
general topological spaces have not such values. To assign distance measures to
a general topological space governed by E-semigroups (as described in Lemma 2.3)
it is assumed that the two neighboring closed subsets
are
separated exactly one Planck length (or one Planck time). With this assumption
one can get rid of pure geometrical quantities like the tetrads.
Proof of Theorem 1.1: The integration over spacetime is replaced by summation over all closed
subsets of the topological space. Also the tetrads and connections are deformed
in a manner such that the physical spacetime coincides with a topological space
governed by E-semigroups. Hence, the Lagrangian density can be rewritten as ,
where
is
the curvature tensor after the deformation process and
is
a modified coupling constant. The sum
can
be interpreted as the average curvature value times 16 and hence it makes sense
to redefine the action of General relativity in the following way:
.
Here,
is
a new coupling constant. Finally, the Feynman path integral for quantum gravity
has the form:
The set denotes
the set of all possible E-semigroup generators and has the cardinality
.
For compact topological spaces,
is
finite if
is
also finite. For the standard E-Theory it is assumed that the sum over
can
be omitted, because
(since
the Minkowski spacetime has 4 topological dimensions).
The E-Theory applied to gravity is a quantum field theory that is based on the non-homogeny of spacetime; by Noether’s theorem it is a theory that does not conserve energy and momentum.
3. Conclusions
With the use of commutative semigroup theory, a quantization of gravity is possible due to the introduction of a curvature measure. Such a theory has only equalizers as a degree of freedom that are simply Boolean variables (is there an equalizer in semigroup element or not?). This makes the theory easy to implement in computer simulations. A disadvantage of this theory is that the new action which is linked to the loss of exactness in a chain complex has a small deviation from the original General relativity.
References
Schmieden, C. et al. “Eine Erweiterung der Infinitesimalrechnung.” Mathematische Zeitschrift, 1958, 69: 1-39
Hamber, H. W. “Quantum Gravitation - The Feynman Path Integral Approach.” Springer Publishing, 2009. ISBN 978-3-540-85292-6
Schwarz, J. H. "String Theory: Progress and Problems." Progress of Theoretical Physics Supplement, 2007, 170: 214–226. arXiv:hep-th/0702219. Bibcode:2007PThPS.170..214S. doi:10.1143/PTPS.170.214.
Ashtekar, A. "New Hamiltonian formulation of general relativity." Physical Review D, 1987, 36 (6): 1587–1602. Bibcode:1987PhRvD..36.1587A. doi:10.1103/PhysRevD.36.1587
Loll, R ."Discrete Approaches to Quantum Gravity in Four Dimensions". Living Reviews in Relativity, 1998, 1: 13. arXiv:gr-qc/9805049. Bibcode:1998LRR.....1...13L. doi:10.12942/lrr-1998-13
St. Golab. "Über den Begriff der "Pseudogruppe von Transformationen"." Mathematische Annalen, 1939, 116: 768–780. doi: 10.1007/BF01597390
Katz, N. "Nilpotent connections and the monodromy theorem." IHES Publ. Math., 1970, 39: 175-232.
Showing 19 Reviews
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2
I think you are in the right way by this theory. but may be this theory just in four dimension spacetime, so can you expand the theory in 10 or 11 dimension as string theory ? and how may you think that the energy-mass can be written by this theory because the theory should has expectations.
your work is in the right way and I hope you more great ideas like this.
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The theory is a mathematically consistent theory (a recent Review comment of "Patrick Linker", the author of the paper, Shows this). The data does Support author's conclusions. Wish you all the best.
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It's not clear what is going on in this paper, except that it is clear that it doesn't do what the author suggests it does. That is, there is no chance that a legitimate quantum theory of gravity is obtained. Would not be considered seriously by any qualified mathematicians or physicists. The author should work with mathematicians and physicists to help himself better understand the deficiencies in these claims.
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A brilliant paper. Nice discussion on the Big Bang and String Theory. Good work on Differential geometry with introduction to E theory.
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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.
The theory is a quite general theory; of course one can formulate quantum gravity in 10 or 11 (or any other) dimension with E-theory. Applying this E-theory to strings, branes and its background space would reformulate string theory in a new way.
It is no problem to compute expectational values of energy-mass; for this also the matter Lagrangian is required. From this the energy- momentum-tensor can be obtained (at first one has to take ordinary qmatter lagrangian in continuum and reformulate this in terms of E-semigroups). Simply using definition of path integral expectation value to this energy-momentum tensor leads to the expectation value of energy-momentum tensor.