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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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