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Dimensionality of observable space

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I show the dimensionality of observable space is conditioned on objectivity. I explain distinction between measuring device and observer
Dimensionality of observable space
Sergei Viznyuk
I show the dimensionality of observable space is conditioned on
objectivity. I explain distinction between measuring device and observer
Commonly, the implications of objectivity
are either overlooked or misappropriated. E.g., the
Schrödinger equation is postulated [1, 2] like primordial law, instead of being derived from
objectivity-imposed unitarity [3]. Here I show the objectivity also effectuates the dimensionality
of observable space.
The question of why the observable space is three-dimensional, and if there are extra
dimensions, is listed as one of the major unresolved problems of physics [4]. The currently existing
propositions largely fall under three categories:
1. As exemplified by quote from [5]:
Quantum (and classical) binding energy considerations in n-dimensional space indicate
that atoms (and planets) can only exist in three-dimensional space. This is why observable
space is solely 3-dimensional
2. Extra dimensions are compactified microscopically in a form of so-called "Calabi-Yau
Spaces" [6, 7], stipulated by assortment of string theories [8, 9]
3. Immediately after the moment of creation, popularly named Big Bang, the things went bad
for all but    dimensions [10, 11, 12]
A brief excursion into each category suffices to illustrate why the question is still open:
1. There is ample argument [13, 14, 15, 16, 17], that familiar  objects can’t exist in   
space. That would include virtually anything, from atoms to planets to any known form of
matter. Yet, it does not prove in any way that other forms of matter and objects, including
intelligent life, may not exist, governed by vastly different laws of [classical] physics
2. No one has managed to extract any sort of experimental prediction out of the [string]
theory other than that the cosmological constant should probably be at least 55 orders of
magnitude larger than experimental bounds [18]. In string theories, we deal with complete
absence of factual basis. Below I point to the core reason for the failure of string theories
to comply with empirical evidence. As it stands, any proposition based on a string theory
is an unsubstantiated speculation
3. Even more speculative are propositions [10, 11] referring to the immediate aftermath of
the moment of creation, Big Bang. No such hypothesis has any chance of experimental
confirmation. Any theory not rendering itself to empirical validation falls out of scientific
methodology into the realm of religion
And yet, the answer to a basic question must not be convoluted and impossible to attest to
. We
just have to look at fundamentals of measurement, missed in numerous publications on the subject.
The objectivity is defined as independence of objective facts on observer [basis]; objective facts being synonymous
to classical information
Transformation of question into answer effectively is a measurement on input state (question). A question, expressed
in terms of fundamental notions, has an answer expressed in terms of no more than fundamental notions
It appears to be a common practice to use the term space[time] dimensionality [16, 15, 17]
with no definition. Authors think the notion is so obvious that providing definition is superfluous.
However, operating with undefined notion is even less credible endeavor than theorizing off a false
assumption. Therefore, I start with definition. The space dimensionality is the cardinality of
observation operator basis, which is the max number of mutually orthogonal
configurations of measuring device
. The device configurations and
are orthogonal if output does not convey any information about output . Formally, the
orthogonality condition is:
 
, where and are traceless Hermitian operators. The device configuration has to be represented
by a traceless operator, because traced part of a Hermitian operator, up to an invariant factor, is
identity operator
. Since output of is same for any input, does not extract
any information.
Operator outcome is the fact
of the measurement
. Only the traceless part of Hermitian operator
extracts information [about input].
The space dimensionality, defined above, is related to cardinality of measurement outcomes
in defining representation
[19] of measurement operator. There are real parameters defining
measurement operator. The traceless condition reduces the number of parameters by , i.e., the
number of real parameters specifying configuration of measuring device is . If space
dimensionality is , then, by definition, there are traceless Hermitian operators ; each being
orthogonal to other    operators:
      
For a given the above represents    real linear equations
. The non-trivial solution of ,
for real parameters defining , only exists if
  
From , the max value of is . It corresponds to the complete measurement operator
basis, commonly represented by generalized Gell-Mann operators [20, 21]:
  
, where
The completeness of basis is manifested through group-forming commutator relations [19] between basis operators
With measurement defined as extraction of classical information [3]
For now, I consider measurement synonymous to observation, to avoid introducing any awkward terminology, such
as, e.g., “observation device”. Few paragraphs down I shall draw a distinction between measurement and observation
Operator is perfectly able to distinguish orthogonal inputs , since  . It’s just not able to encode its
output as classical information, because output of is the same for any input:  . The information-
producing measurement can only be performed by operator whose output is different for different inputs
A competed measurement event, e.g., a click of a particle detector, implicitly includes [29]
Sometimes it is incorrectly stated [19] that is “do nothing” operator
Also called the fundamental representation [30]
Trace of a product of two Hermitian operators is always real
  
 
     
 
Thus, the space dimensionality is:   
It immediately follows, can only take values    [22]. Other numbers of spatial
dimensions, as proposed by now largely defunct theories, such as Kaluza-Klein [23]   or
M-theory [6, 18, 12] (   do not correspond to a complete measurement basis, and, therefore,
do not make a space
. E.g., there is no  space, since two information-extracting operators do
not make a complete basis. Any  or  model implies projection from   .
Everyday experience points to the fact that we live in   space, corresponding to   ,
i.e., qubit measurement basis. The unambiguous relations:
  ;   ;
  ;   , where minus sign means objectively opposite, are only
possible in    basis. Had we lived in    world, there would be    spatial dimensions,
mandating different laws of classical physics [16]. None of the known states of matter or objects
would exist [17]. Furthermore, in    world nothing could be objectively ascertained, as one
cannot unambiguously assign  
to   mutually exclusive measurement outcomes,
similar to how we routinely assign    and    in    basis. Unlike 
 map, the  transformations, which include spatial rotations, do not homomorphically
onto , for
  . It means, the extracted by measurement information, expected to
be objective, for observer living in    space, depends on orientation of observer basis.
Consequently, there is no objectivity in    world, and therefore, could not be an observer
Having inferred that observer, as a notion, can only exist in    space, I thus arrived at
anthropic principle [24, 25, 13]. Yet it does not mean the measurement basis is limited to   .
There is no reason
to pick any particular over  , given  transformation group is
a subgroup of a bigger . Measurements in all    bases are legitimate and,
therefore, omnipresent. What happens to dimensions other than those corresponding to  
basis? For    the answer is obvious: measurement in    basis results in the same outcome
for any input, and, therefore, produces no information to observe, i.e., no dimension,   . For
   bases I could resort to a primitive interpretation of anthropic principle, by stating that the
known world is not feasible in   , and thus claim “explanation” why we only see   
dimensions. This is how many authors [17, 5, 15, 13, 16, 14] approach the subject
. However, the
anthropic principle is not an exercise in tautology or circular reasoning. At the base of anthropic
principle is the presence of observer, which implies objectivity. The objectivity signifies
The space is defined as real vector space of measurement basis operators, i.e., the space of vectors in
 is defined by
real parameters, while  is by . They are
equal only for   
The notion of observer is meaningless vis-a-vi observed world if outcome of observation is not objective
There is no physical basis for standard model stopping at , apart from the math becoming intractable and
results uninterpretable, for higher
From [14]: In the absence of a truly convincing argument [about space dimensionality] however we may rely instead
on anthropic reasoning
invariance of extracted by measurement classical information, formally enforced through unitarity
[3, 26]. This is where comes the difference between measurement and observation. The
measurement transforms quantum information into classical. Quantum and classical information
are not invariants of measurement, only their sum is [26]. Contrary to some statements
, the
observer is not the measuring device. The observer only deals with extracted by measurement
classical information, in a form of various classical objects, which make up the objective reality.
The representation of classical information in observer basis (-basis) is the observation. Unlike
measurement, the observation conserves both quantum and classical information
. The
information extracted by measurement in any -basis, is represented, i.e., observed, in -basis.
Therefore, the dimensionality of observable space is the cardinality of observation operator basis,
not of measurement operator basis. I prove below, the common -basis for information extracted
by measurements in all   bases can only be of cardinality  , and corresponding space
dimensionality can only be      .
All basis operators in only have distinct non-zero eigenvalues for any    basis.
Among are    commuting, i.e., simultaneously diagonalizable. These diagonal operators
are encoding operators. They unambiguously assign input eigenstate a classical outcome - an
eigenvalue (encoding symbol), whenever there is an output, i.e., non-zero eigenvalue. A
measurement, whose output (a classical information) is encoded as one of distinct symbols is
effectively the measurement on [generalized] qubit
. The outcome of a measurement in -basis
is thus encoded as tensor product of outcomes of measurement on    qubits. E.g., the possible
outcomes  of a measurement in    basis are encoded
                  
, where  are qubit measurement outcomes. Effectively, the cardinality of measurement
outcomes is the number of ways to distribute identical inputs into distinct observation bins.
Therefore, the common -basis for measurements in all    bases, has to satisfy:
          
  
The above holds only if   . Q.E.D.
I have proved   is the only dimensionality of observable space wherein the existence of
observer, and observation of outcomes of a measurement in all cardinality bases is possible.
The  dimensionality is effectuated by [objective] representation of classical information
extracted by measurements in all cardinality bases. I have shown the outcome of a measurement
in any -basis is a tensor product of outcomes of a measurement on    generalized qubits. I
have explained the distinction between measurement and observation. The objectivity-imposed
From [13]: Even quantum mechanics, which supposedly brought the observer into physics, makes no use of
intellectual properties; a photographic plate would serve equally well as an ‘observer’
That answers the question, if, by observing Moon, we make its wave function collapse; or if there was a sound of a
tree falling, if no one listened
In    case, the definition of generalized qubit measurement operator effectively merges definitions of isospin
and hypercharge operators, since both have 2 distinct non-zero eigenvalues
The order of qubits in tensor products is irrelevant
unitarity [3] may only pertain to transformation of observation basis, not necessarily to
transformation of measurement basis (configuration of measuring device). In accordance with
everyday experience, the observation basis is expected to transform under -dimensional
representation of  group [27], [locally] isomorphic to  [28]. The configuration of
measuring device is not under such restriction, and, generally, is expected to only abide the
intertwist relation [26]:
 
, where and are device configurations measuring the same input, and is transformation from
to . The unitarity of follows from  only if    and   [26].
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Full-text available
Kaluza was the first to realize that the four-dimensional gravitational field of general relativity and the classical electromagnetic field behave as if they were components of a five-dimensional gravitational field. We present a novel experimental test of the macroscopic classical interpretation of the Kaluza fifth dimension. Our experiment design probes a key feature of Kaluza unification—that electric charge is identified with motion in the fifth dimension. Therefore, we tested for a time dilation effect on an electrically charged clock. This test can also be understood as a constraint on time dilation from a constant electric potential of any origin. This is only the second such test of time dilation under electric charge reported in the literature, and a null result was obtained here. We introduce the concept of a charged clock in the Kaluza context, and discuss some ambiguities in its interpretation. We conclude that a classical, macroscopic interpretation of the Kaluza fifth dimension may require a timelike signature in the five-dimensional metric, and the associated absence of a rest frame along the fifth coordinate.
Properties of the Lie algebra gl(n,C) are described for a basis which is a generalization of the 2×2 Pauli matrices. The 3×3 case is described in detail. The remarkable properties of that basis are the grading of the Lie algebra it offers (each grading subspace is one dimensional) and the matrix group it generates [it is a finite group with the center of SL(n,C) as its commutator group].
The anthropic principle is very important in many aspects including determining of extraterrestrial life. In this presentation the discussion of the theory of quantum cosmology of Hawking, Linde and Vilenkin will be done. The application of the anthropic principle to determine the parameters of the Universe will be discussed as well.
Copenhagen Interpretation of Quantum Theory and the Measurement Problem
  • M Kober
M. Kober, "Copenhagen Interpretation of Quantum Theory and the Measurement Problem," arXiv:0905.0408 [physics.hist-ph], 2009.
The origin of unitary dynamics
  • S Viznyuk
S. Viznyuk, "The origin of unitary dynamics," 2020. [Online]. Available:
Why Observable Space Is Solely Three Dimensional
  • M Rabinowitz
M. Rabinowitz, "Why Observable Space Is Solely Three Dimensional," arXiv:1502.04989 [physics.gen-ph], 2014.
  • M Duff
M. Duff, "M-Theory (the Theory Formerly Known as Strings)," arXiv:hep-th/9608117, 1996.
  • B Greene
B. Greene, "String Theory on Calabi-Yau Manifolds," arXiv:hep-th/9702155, 1997.
String Theory -From Physics to Metaphysics
  • R Hedrich
R. Hedrich, "String Theory -From Physics to Metaphysics," arXiv:physics/0604171 [physics.hist-ph], 2006.
Introduction to String Theory
  • T Weigand
T. Weigand, "Introduction to String Theory,", 2011.