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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
1
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
2
. We
just have to look at fundamentals of measurement, missed in numerous publications on the subject.
1
The objectivity is defined as independence of objective facts on observer [basis]; objective facts being synonymous
to classical information
2
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
complete
3
observation operator basis, which is the max number of mutually orthogonal
information-extracting
4
configurations of measuring device
5
. 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
6
any information.
Operator outcome is the fact
7
of the measurement
8
. 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
9
[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
10
. 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
3
The completeness of basis is manifested through group-forming commutator relations [19] between basis operators
4
With measurement defined as extraction of classical information [3]
5
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
6
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
7
A competed measurement event, e.g., a click of a particle detector, implicitly includes [29]
8
Sometimes it is incorrectly stated [19] that is “do nothing” operator
9
Also called the fundamental representation [30]
10
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
11
. 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
map
12
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
13
.
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
14
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
15
. 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
11
The space is defined as real vector space of measurement basis operators, i.e., the space of vectors in
12
is defined by
real parameters, while is by . They are
equal only for
13
The notion of observer is meaningless vis-a-vi observed world if outcome of observation is not objective
14
There is no physical basis for standard model stopping at , apart from the math becoming intractable and
results uninterpretable, for higher
15
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
16
, 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
17
. 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
18
. 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
19
as:
, 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
16
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’
17
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
18
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
19
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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