Quantum Mechanics of a Quasi-Euclidean Space with Planck Length, Rotational Symmetry and Translational Symmetry ()
1. Introduction
The fine structure constant (FSC, ~1/137.035999), which is a dimensionless quantity characterizing the strength of the electromagnetic interaction, has fascinated innumerable scientists since it appeared in 1916 [1]. Its value has been measured more and more precisely in the cosmos explored by humans [2]-[5], whereas its theoretical origin remains unknown till now. Although some interesting formulae
have been proposed, such as
[6],
[7], etc., there is never a convincing solution that has both numerical consistency and sufficient theory for FSC. It’s even unknown whether FSC is calculable in principle or is a non-calculable one determined by historical or quantum mechanical accident [8].
Nevertheless, there are still a considerable number of scientists who insist that FSC must have theoretical derivations [9]-[11]. If FSC is really a dimensionless constant with calculability, like another fundamental constant π defined by a radius and the semicircle it determines, then two prerequisites, including a natural object and its characteristic path relatively measured ~137.036, must be both present. As for the former, some mathematical models, including the point-like one, the lattice-like one, the string-like one, etc., have been assumed to be the natural objects [12] [13]. As for the latter, Euclidean spaces and manifolds have been involved to be the background for the research [14]. Although no precise solutions had been obtained directly, some attempts illuminate that the higher dimensional spaces beyond 3D/4D may be required to study some fundamental interactions [15] [16].
For the further pursuit of a calculable FSC, the present work chooses a generalized nD Euclidean space with Planck length as the background. Here the limit of Planck length turns the Euclidean space to a quasi-Euclidean space, resulting in an extremely small space bubble, inside which no distance is reasonably allowed. After taking such a space bubble as a natural object, its moving path in nD space then will be searched for. Definitely, this small bubble would obey quantum mechanics and some other basic principles of physics, such as conservation of mass, the minimum energy, etc. Besides, we should not forget the most important thing that they satisfy rotational symmetry and translational symmetry, since they are the parts of the vacuum, which is proven by all experiments to be of absolute symmetries, no matter when and where. This means geometrical limits will be strongly involved when considering the motion of such a space bubble. Therefore, we set up a quasi-Euclidean space with Planck length and symmetries, trying to quantize it via the space bubble and explore its quantum behaviors, not only physically but also geometrically.
2. Planck Units: Quantization of Quasi-Euclidean Space
2.1. Quasi-Euclidean Space
Compared to a general nD Euclidean space
(dimension symbols are marked in the upper left corner of a certain space
in this work), a special Euclidean space noted as
is established here to be a Euclidean space carrying the Planck length
, which determines the minimum distance in vacuum and comes from
, where
is the reduced Planck constant, G
is the gravitational constant and c is the speed of light in vacuum [17]. Therefore, space
can be described as follows. On the one hand, it exists
(
), showing that
behaves as same as its corresponding Euclidean space
on the relatively macroscopic scale
. On the other hand, it exists (
), showing that any 1D measure along a certain dimension of
is never less than
on the microscopic scale
, although
remains a Euclidean space at the same time. E.g., for two adjacent points on a certain dimension of
, a blank interval measured
occurs between them even they are originally infinitely close to each other on the macroscopic scale.
Based on the above commonalities and difference, relationship between quasi-Euclidean space
and its corresponding Euclidean space
can be mathematically demonstrated as follows.
Commonalities (linearity):
(1)
(same position with
)(2)
(same measure with
)(3)
Difference (Planck length):
() (4)
Figure 1. Difference between Euclidean space P and quasi-Euclidean space
at Planck scale: (
) when
and
share the same position (A) or the same measure (B).
As two equivalent spaces at macro scale,
and
always satisfy k = 1 and a = 0 simultaneously in Equation (1). However, the exclusive character of Planck length for
, as shown by Euler’s formula in Equation (4), brings
(Figure 1) and leads to the alternatively satisfied a = 0 and k = 1 in Equation (2) and Equation (3), respectively. Briefly, it exists when
(Figure 1). Besides,
when
describes the special case where the minimum distance of
is invalid since the two adjacent points coincide and act as one point. This special case demonstrates that it exists
when a single point is involved.
So quasi-Euclidean space
can be mathematically defined in the space of non-negative part as
(
) (5)
or (
)(6)
(
)(7)
Obviously, unless considering the minimum distance at the extremely small scale, a quasi-Euclidean space
is equivalent to its corresponding Euclidean space
. So,
is basically taken as a space that not only applies the axioms and definitions of Euclid space, but also applies all the physical principles in a general Euclidean space, being a normal background for experimental or theoretical physical objects, including a particle, a field, and so on.
2.2. Relationship between
and P
Based on the mathematical definition of quasi-Euclidean space
in Equations (1)-(4), the relationship between
and its corresponding Euclidean space P is obtained.
Non-commutativity in 1D. Equations (2)-(4) result in
; (8)
; (9)
; (10)
where Equation (10) demonstrates the non-commutativity between
and P in any one dimension (Part 1 in Supplementary information).
Commutativity in ≥2D. For any a pair of local ≥2D spaces
and P expanded respectively by orthogonal
and
, it exists
(11)
since the orthogonality results in
, which guarantees Equation (11). Here
(
) in Equation (11) ensures the commutativity between
and P in any ≥2D spaces.
Reciprocity. Results in Equations (8)-(10) also determine a special relationship of reciprocity between
and
for it always exists
(12)
vice versa. This reciprocity leads to the interchangeability between
and P, demonstrating the equivalence for a certain object under two conditions, one is to measure it with reference to P when it lies in
, the other is to do so with reference to
when it lies in P.
Thus, the relationship between the quasi-Euclidean space
and its corresponding Euclidean space P can be summarized into 3 points, including non-commutativity in 1D, commutativity in ≥2D, and reciprocity.
2.3. Properties of Quasi-Euclidean Space
Uncertainty and UV cutoff . According to the conclusions about non-commutativity [18],
and
, as a pair of non-commutative quantities represented in Equation (10), are relatively uncertain for they can’t be determined simultaneously. Besides, Equation (9) results in the same minimum 1D measure
of (reduced Planck length ; Part 2 in Supplementary information) for both
and
, since
; (13)
; (14)
As a visual understanding of the above results, the minimum measurements may always be the same for either of the two scenarios, one is to measure a math space with infinite scales by a physical ruler with the minimum scale , the other is to measure a physical space with the smallest length by a math ruler with infinite scales. Here, the smallest 1D measure is named UV cutoff for both
and P, when they are measured by each other.
Briefly, of the UV cutoff results in a 1D blank interval, directly leading to a quantized 1D for
.
Planck units
. The minimum blank interval along any a dimension will naturally result in an nD minimum blank interval
or
. Algebraically, these two minimum spaces
and
share the same minimum measure because UV cutoff requires any local space
or
to satisfy
; (
)(15)
Geometrically, the boundary of an nD minimum interval
is determined by two factors, including the 1D condition and the nD condition. Regarding the 1D condition about UV cutoff , any distance inside the boundary is forbidden. Let the boundary be determined by ,
which defines a boundary on a generalized nD sphere with center located at (, 0,
,
, ...) on the polar axis of a polar coordinates (Figure 2(A)). Obviously,
any distance inside the boundary will result in and the violation of the 1D condition. Regarding the nD condition that the minimum interval is of nD measure , then
should be of another boundary on the surface of a generalized nD cube with side length , and any distance outside the cube is forbidden since it will result in an nD measure and the violation of the nD condition (Figure 2(B)). These two boundaries, including the spherical one and the cubic one, determine the inner and outer boundary for the minimum interval
, respectively. Moreover, any intermediate between the outer one and the inner one is also a valid boundary for
because of the linearity of
. The above results indicate that it exists a series of boundaries for
, and
has an uncertain boundary in nature (Figure 2(C)).
![]()
Figure 2. The uncertain boundary of a Planck unit: it can be a generalized circle without any distance shorter than inside (A), a generalized cube without any distances longer than outside (B), or an intermediate between them (C). The uncertain structure of a Planck unit and the uncertain relationship between
and p (D). Space bubbles of Planck units
located in subspaces of quasi-Euclidean space
(E).
Consequently, the uncertain boundary results in the uncertain structure for
. E.g., to represent a 2D minimum local space
by
(ρ, θ), it always exists certain when θ is completely uncertain (
), or uncertain ρ () when θ is of certainty (Figure 2(D)).
Here the minimum nD blank interval with uncertain boundary and uncertain structure is named a Planck unit. In quasi-Euclidean space
, a Planck unit
generally defines a local space with the following characteristics. Firstly, it is a blank space with constant nD generalized volume but uncertain structure varying from a sphere with a diameter of to a cube with a side length of . Secondly, the longest 1D distance of the boundary varies in interval of [, ] (Figure 2(D)). Obviously, the Planck unit is a natural extension of the concept of the UV cutoff. Mathematical derivation indicates that
can also be quantized by , just like its 1D subspace can be quantized by UV cutoff . Considering the special case about
(Section 2.1), an iD Planck unit
(
) is also allowed when
and
(
), demonstrating that
can also be quantized in subspaces (Figure 2(E)). Therefore, quasi-Euclidean space
can be redefined as such a special Euclidean space, which behaves as same as Euclidean space nP at scale
on the one hand, but does differently from nP for its iD measure always satisfies at scale
on the other hand, although it remains a Euclidean space at the same time. This redefinition takes the original definition about 1D blank interval as a special case of 1D and results in a phenomenon that a serials of space bubbles exist in
at the extremely small scale (Figure 2(E)).
IR cutoff L. For a generalized nD local space
(
), its nD generalized volume is of a constant measure of , according to Equation (15). Besides, linearity of
results in
(,
) (16)
when
is taken as the linear summation of innumerous UV cutoffs between any adjacent point pairs. Obviously,
satisfies uncertainty of
, when only one dimension along
is involved. Whereas the nD condition about the commutative
and
, as shown in Equation (11), should be involved when
is included in a certain local space
(
). Considering the requirement aroused by the UV cutoff in Equation (17), the possible maximum and the minimum for 1D condition can be obtained as +∞ and (the mathematically reasonable solution about
is prohibited to ensure compliance with the physical principle of conservation of mass in the current 1D space), respectively, as shown in Equation (18). When the nD condition about the commutative volume is involved in Equation (19) as
or (17)
, (18)
(19)
the allowed maximum of a certain
included in a local space
is
IR cutoff: (when ) (20)
demonstrating the certain IR cutoff L for a general local space
. To transform
Equation (20) into , the physical or geometrical meaning for IR cutoff
can be discovered to be the longest path for a Planck unit when its motion path covers the entire local space
uniformly and without overlap (Part 3 in Supplementary information).
Thus, properties of the quasi-Euclidean space
can be summarized into 4 points, including uncertainty, UV cutoff , Planck unit
with certain measure and uncertain structure varying from a generalized cube to a generalized
sphere, and IR cutoff as the longest path for a Planck unit
confined to a local space
.
Section 2 defines a generalized quasi-Euclidean space
with Planck length. Mathematical study discovers its uncertainty and clarifies the minimum structures of Plank units in it. As a micro-object with measure , a Planck unit might play the role of a quantum object when its motion inside
is investigated at scale , where
behaves as a normal Euclidean background space. Consequently, a series of physical principles, including quantum mechanics, the minimum energy, conservation of mass, conservation of energy, etc., should be obeyed by a moving Planck unit. Besides, rotational symmetry and translational symmetry should be strictly satisfied by a Planck unit, since it is also part of the background space. Next, motion for such a Planck unit should be pursued, assuming that it can distinguish itself from the quasi-Euclidean space
of the macro background. And IR cutoff L is expected to help in determining the ground state when the object is confined to a certain local space.
3. Planck Units under Control of Rotational Symmetry
3.1. A General Planck Unit Controlled by Rotational Symmetry
As part of space, a Planck unit
should satisfy rotational symmetry (abbreviated as RS).
RS states (I, R and T) and RS spaces (PI, PR and PT). Strict RS bans Planck unit
from any radial displacement (dr = 0) by infinite potential barrier
, requiring r = c mathematically. The solution results in two types of states, one is in-situ (c = 0), named state I (Figure 3(A)), the other is revolving around the in-situ position in the surface of a certain sphere (c ≠ 0), named state R. Geometrically, the nD surface of an (n + 1)D sphere provides the simplest spherical space for an nD Planck unit
(Figure 3(B)). Considering the UV cutoff of quasi-Euclidean space
, the nearest sphere to state I is determined by , since a closer sphere is prohibited by the definition of UV cutoff . After normalizing the system by , the simplest and nearest space for the revolving state R, symbolized as PR, should be
(21)
here, the curved nD surface PR is embedded in the (n + 1)D flat space, so state R actually takes the (n + 1)D space as its background, breaks away from the original nD background space of state I, and violates space conservation directly. Space conservation requires that
always moves in the same flat nD background space, and this then requires a flattened PR, which is a finite plane PT tangent to PR (to show it more clearly, PT’s tangent point is set up to be at the bottom of PR to distinguish PT from PI in Figure 3(C), which represents the original nD background by a parallel space). Let nm be the generalized nD volume of an nD sphere with radium r = 1 (Part 5 in Supporting information), its generalized surface should be (n − 1)D space measured (nm)', and the flattened surface should be of measurement
.
Logically, RS states I, R and T for
should be of spaces measured
(22)
(23)
(24)
According to Equation (20), their IR cutoffs are
,
and
, respectively. It should be noted that these spaces are of nD measurement while their IR cutoffs are of 1D measurement, and they share the same algebra expression only because of the normalization .
So rotational symmetry requires Planck unit
exist at in situ state I, revolving state R or flattened revolving state T at the extreme condition, and these states are confined to certain spaces PI, PR and PT with certain IR cutoff LI, LR and LT, respectively. Generally, there exists infinite potential
to separate state I from the topologically connected PR/PT, resulting in localized Planck units with no transformability (Figure 3(C)). And such a background space
without any non-local Planck units is taken to be a trivial one (a general case is shown in Part 6, Supplementary information).
Figure 3. RS spaces of
at RS states of I (in situ state, A), R (revolving state, B) and T (tangent state, C). Transformable
with topologically connected I and PR/PT (D).
3.2. Non-Triviality: Transformable Planck Units in High-Dimensional Spaces
If state I and space PR/PT are topologically connected,
will be invalidated, transformation between I and R/T will be possible, and the background space
will become a non-trivial one since non-local Planck units in it are allowed (Figure 3(D)).
The topology of the RS spaces was investigated here. Assuming that a Planck unit at state I takes a generalized cube as its uncertain structure and takes diagonal
as its maximum 1D measurement, the increasing
will make itself possible to reach PR/PT in higher dimensional spaces. The possible points connecting I and PR/PT should belong to the intersecting space P∩ determined by PI (
) and PR (
) as
(25)
which is an (n − 1)D spherical surface enclosing an nD sphere (Figure 3(D)). Logically, the longest 1D measurement enclosed by P∩ is nm, since the enclosed sphere measured nm is of IR cutoff
. Thus, state I will share points with P∩ when
, meaning that the cube or some other intermediate reaches PR/PT topologically (Figure 3(D)).
Calculation demonstrates the relationship between
and nm in nD space (Figure 4), showing topology for I and PR/PT as follows. Firstly, 1 - 9D spaces are all trivial ones for
, resulting in that their state I is always separated from PR/PT by
and state transformation is always forbidden. Secondly, 10D, 11D and 12D spaces are three non-trivial spaces with
, where topological connection between I and PR/PT exists, potential
can be invalidated, and non-local Planck units are allowed. Lastly,
in ≥13D spaces makes these solutions meaningless for
violates the minimum local space
(Figure 4).
Thus, Planck units
,
and
are proven to be of non-trivial topology connections between their state I and PR/PT, discovering that transformation of a Planck unit is allowed in 10D, 11D and 12D spaces. Planck units
inside quasi-Euclidean spaces >12D involve threshold space
, e.g.,
, which violates the definition of a Planck unit, resulting in the loss of reasonability for the current case. According to the monotonic decrease trend of the generalized formulae for nm, spaces with higher dimensions are reasonably ignored (Part 5 in Supplementary information).
3.3.
: RS Space Structures and Two Transforming Paths
is the first non-trivial Planck unit with transformability, since its in-situ state I has the possibility to reach the space of its revolving state R, resulting in the probability for itself to transform into the corresponding tangent state T. This possible transformation is determined by the geometrical characters of these different states, heavily depending on that I shares same point(s) with PR. Investigation in 3.2 discovers that there is no possibility for such a sharedness in a quasi-Euclidean space
when
. Although any a Planck unit has the possibility to behave as a spherical one with completely uncertain θ but certain , which keeps itself always isolated from PR, a possible sharedness still exists when it satisfies
, where
is the possible maximum 1D distance for a Planck unit and nm equals to the maximum 1D distance inside the threshold space P∩. Essentially, algebraic relationship between
and 10m determines that
is a transformable Planck unit with the lowest dimensional structure (Figure 4). Its non-triviality, however, changes motion of
in turn, since the topological connection changes its RS space structures and the transformation changes its movements.
![]()
Figure 4. Topological relationship for state I and PR/PT in nD space.
Changed PI and PT. Topological connection invalidates the infinitely potential
, which is originally separating in-situ state I from the revolving space PR, and then the disappeared
makes state I exist actually in a space enclosed by P∩, which is the 10D sphere measured 10m (Figure 3(D)). Besides, the topological connection makes PR be flattened from a certain point on 9D space of P∩ and the corresponding PT becomes
. The above changes result in
(26)
(27)
Two paths for I⇌R/T: the highest rotational symmetry (RS↾) and the minimum energy (E⇂). Accordingly, spaces PI and PT of 10D Planck unit
are of
IR cutoff
and
, respectively. And these two IR cutoffs
might provide quantum paths for
at ground states. But as an RS state, movement of state I should also be measured equally along each dimension, meaning
that PI should be an equally expanded space of
where
and its observable IR cutoff should be
. As
for state R, dimensional equivalence would not work here since curved PR is of no dimensional equivalence intrinsically. Consequently, 10PR remains the same with
LR = (11m)', and so does 10PT with
.
Thus,
at in-situ state I has two ways for its movement, one takes
as its half wavelength when it satisfies the minimum energy, the other takes
as its half wavelength when it satisfies the highest RS. The two principles, one is the minimum energy noted as
, the other is the highest rotational symmetry noted as
, then determine the two different transforming paths for
.
I⇌T: space structures for a transforming
. Considering that
should remain in 10D space to satisfy conservation, its R state in PR, which is a 10D generalized surface embedded in 11D background space (Figure 3(B)), is unobservable in
for it violates the conservation. In fact, only state I and T for
are RS states with physical legality inside
, whereas state R is a virtual one just bridging I and T mathematically. Accordingly, the 9D space P∩ (
) in
actually determines the topological connection for I and T, instead of PR (
) embedded in
(Figure 3(B)).
Taking 10D quasi-Euclidean space
as the simplest background space with non-locality, critical space P∩ plays the role of threshold not only for
, but also for its substructures in each subspace of
. Based on the maximum length
allowed by P∩, transformability for each substructure in 1-9D proper
subspaces had been also investigated. Results show that the 1D maximum
allows Planck units
,
and
to be transformable
ones besides
, since it exists
. This means that any ≥7D substructure is of transformability for its state I can exceed the threshold and connect to the corresponding state T. In other words, any ≤6D substructure never reaches the threshold space of P∩, being prohibited from transformation and maintaining triviality.
Let triviality be of priority to nontriviality and let xi expand the ith dimension of the iD subspace (
), the maximum space with triviality, noted as tP, should be 6D space expanded by x1, x2, …, x6, and its complement space involving non-triviality, noted as ntP, should be 4D space expanded by x7, x8, x9 and x10. Then a map of space structures for
can be obtained for both state I and state T in each of the 1 - 10D subspaces (Figure 5).
Regarding the principle of the highest rotational symmetry (
), iD substructure
(
) at state I should be of PI expanded by
and
, which meets
(
) or
(
),
satisfies the dimensional equivalence as much as possible, and ensures the priority of the triviality in ≤6D. Correspondingly, PT of
includes a trivial part and a
non-trivial part measured (2π)6 and
, respectively. Taking 8D Planck
unit or 8D substructure of
as an example, its PI should be expanded by
and
, and its PT should be of IR cutoffs (2π)6 and
in 6D trivial subspace and the other 2D non-trivial subspace, respectively. Similarly, PI for
should be expanded by
and
, and its corresponding PT should be of IR cutoffs (2π)6 and
in tP and ntP, respectively (Figure 5(B)).
Regarding the principle of the minimum energy (
),
, as an nD Planck unit or nD substructure, would take IR cutoff L as its motion spaces in each of its own subspaces, acquiring
be its largest motion space. Taking 2D Planck unit or 2D substructure of
as an example, its proper substructure
takes
and
as its longest paths for state I and T, respectively, and there must be
and
. Besides,
itself takes
and
as its longest paths for state I and T, respectively, and there must be
. Obviously, 2PI and 2PT have the maximum values of
and
, respectively, if and only if
and
. Similarly, it
exists
and
for
, when
is dominated by Principle
.
Figure 5. Non-transformability or transformability for a Planck unit
inside 10D (A). 10D space structure and two transforming paths for I⇌T: Path 1 with the highest rotational symmetry (
) and half-wavelength change ~137.036 times, while Path 2 with the minimum energy (
) and half-wavelength change ~1.628 × 1038 times (B).
2 dimensionless constants. For
satisfying Principle
, it conserves triviality in ≤6D space of tP, meaning that the non-locality aroused by transformation only occurs in the 4D complement space of ntP. According to the map of
the space structure (Figure 5(B)),
would take
and
as its half wavelengths for its ground states, and
would be observed to be with wavelength change of
times during its transformation in 4D space ntP.
Similarly,
satisfying Principle
would be observed to be with wavelength change of
times during its transformation in global 10D space.
Section 3 investigated on a Planck unit dominated by rotational symmetry (RS). RS strictly defines the spaces for a Planck unit in-situ (state I) or in flattened spherical surface (state T), and these two states are generally isolated from each other, meaning that a Planck unit is always localized. Geometry study discovered that a 10D Planck unit acquired its transformability and non-locality when its two states I and T were topologically connected. Following the two principles, one was the highest rotational symmetry (
), the other was the minimum energy (
),
presented two dimensionless constants of ~1/137.036 and ~1/(1.628 × 1038) for its two transforming paths, respectively.
4. 10D Planck Unit under Control of Translational Symmetry
As part of 10D quasi-Euclidean space, Planck unit
should satisfy translational symmetry (TS) besides rotational symmetry (RS).
RS State T'. Besides PR, RS also defines other concentric surfaces with and results in innumerous state R' in infinite space. However, transformation between anyone of these R' states and the in situ sate I is forbidden because R' is not so close to I, which causes them to be separated by potential barrier
. Consequently, it also exists innumerous state T' without transformability into state I (Figure 6). To satisfy principle of least action [19], a Planck unit at state T' should be of action S = h for its ground state.
Constant velocity. Because of the translational symmetry, Planck units
in 10D background space should be identical objects, and their T states should be of constant velocity v.
Sharing-coupling effect. A system containing two centers A and B had been built up to explore the interaction between any two
. Because of the constant velocity v of state T, one center of state IA shares TB as its state
at distance r (
), since TB and
are indistinguishable for they are identical objects sharing the same velocity v. So are TA and
for another center of IB. The two shared states T and T' spontaneously bring about energy difference of
(28)
between them when they both have the least action h and take IR cutoffs as their half wavelengths at ground states. This effect caused by the shared T and T' states is named sharing-coupling effect (Figure 6).
Figure 6. Shared state T/T': sharing-coupling effect for a 2-center system.
Two long range interaction fields. The velocity v for state T or T' leads to
their constant mass of 0 since
requires
when dv = 0. So kinetic energy of T or T' should be protected by their constant velocity and mass, which results a kinetic energy difference of
. Considering that
the energy for each point in a vacuum should be always equal, there must be a potential energy different
between IA and IB to balance the energy of the system, resulting in two long-range potential fields
(29)
when the two dimensionless constants of α1 = 1/137.036082 for
process or α2 = 1/(1.628 008 × 1038) for
process had been brought in and state T took the light speed c as its constant velocity (Part 6 in Supplementary information).
Section 4 discovered the sharing-coupling effect determined by translational symmetry, obtaining a spontaneous potential field between any two shared T/T' states in 10D space. According to the two transforming paths with dimensionless constants α1~1/137.036 and α2~1/(1.628 × 1038), two long range interacting fields had been obtained accurately.
5. Conclusion and Discussion
Based on its three initial settings, including Planck length, rotational symmetry and translational symmetry, a quasi-Euclidean space is quantized by the extremely small structures of Planck units, and a 10D Planck unit is discovered to be the simplest Planck unit with two different transforming paths, resulting in two long-range interactions in a 4D subspace and in 10D global space, respectively.
The two constants, 1/137.036 for the transformation dominated by the highest rotational symmetry (
) and 1/(1.628 × 1038) for the transformation dominated by the minimum energy (
), are exactly equal to the fine structure constant (FSC = 1/137.035999) and approximately equal to the dimensionless gravitational
constant (
) [17], with deviations of ~10−6 and ~4%, respectively.
But the two corresponding forces between any 2 Planck units are neither the electromagnetic interaction between any two static electrons nor the gravitation between any two protons, and the difference between them is (2LI/2π) times. Obviously, it suggests another pseudo space with clear geometrics. Besides the pseudo space, the 4D non-trivial space expanded by the 7th, 8th, 9th and 10thD, the positive or negative nature of the two long range interactions, the obvious deviation between α2~1/(1.628 × 1038) and the dimensionless gravitational constant, etc., are also puzzling. This work distinguishes the 10th dimension x10 from the other 3 ones of x7, x8 and x9 since
,
and
all have their own subspaces to conserve their triviality but
has not. The relationship between the non-trivial 4D subspace and the current 4D space-time need to be explored further. Although two constants for the two long range interactions have been obtained, no other useful information, such as that about their negativity or positivity, has been obtained.
In fact,
had ever been discovered before, and the 10D space with 6D + 4D structure had been investigated in other fields. Wyler’s constant of
, which is algebraically equal to α1, was published in 1969 to try to
solve FSC, but all the relevant works stopped abruptly here almost at the same time [20]. And string theory assumes that a micro string with six-dimensional curved structure moves in a 10D space in some cases [13]. The current work obtained a bubble moving in 10D spaces with its 6D substructure being locked into a 6D subspace for it behaves trivially and loses transformability here. Whereas the 10D → 6D + 4D in this work is obtained via a geometrical method for a bubble-like object.
As for the outlook for this work, we look forward to a future where a particle and a vacuum can be unified to prove the unity of the physical world.
Additionally, the newly defined symbols in this work have been listed in Part 7 of the Supplementary information.
Acknowledgements
Yan Zhou would like to express the gratitude for support from Special Funding for Talents of West and Northeast China, Chinese Academy of Sciences. Special tribute is made to the mathematician Armand Wyler, who was pioneering this work.
Supplementary Information
1) Non-commutativity between
and
Based on the mathematical definition of quasi-Euclidean space
in Equation (2)-(4), relationship between
and its corresponding
in Euclidean space could be obtained.
To substitute Equation (2) into Equation (4) and to take derivative with respect to
on the both sides, it obtains
(S1)
(S2)
And Equation (S2) leads to
(S3)
Then to substitute Equation (2) and (3) into Equation (4) and to take derivative with respect to
on the both sides, it obtains
(S4)
(S5)
And Equation (S5) leads to
(S6)
Furthermore, Equation (S3) results in
(S7)
which then leads to the non-commutativity of
(S8)
Similarly, Equation (S6) results in another non-commutativity of
(S9)
Based on Equation (S8) and (S9), relationship between
and
can be obtained as
(S10)
(S11)
demonstrating the non-commutativity between
and
along any a dimension.
2) On for
For in Equation (S8), let
be Gaussian for
, then it should exist
(S12)
with
satisfying normalization condition of
(S13)
Let expansion coefficient cp be
(S14)
And let , Equation (S14) leads to
(S15)
showing that cp is also Gaussian distribution with
. So it exists
(S16)
Based on
(S17)
it could be obtained that
(S18)
Equation (S18) leads to , which then results in
given the even symmetry for Gaussian. Considering Gaussian with the narrowest contribution, it generally exists .
3) On physical meaning of IR cutoff L: the longest length of a closed or open path (Figure S1).
4) A general cases for Planck unit with or without transformability.
Generalized surface of an (n + 1)D sphere is also an nD space, since it is expanded by orthogonal X1, X2, X3, …, Xn, Xn+1, where
and
, according to the Frenet frame shown in Figure S2(A). Accordingly, an nD Planck unit
can be located in both the curve surface
s and the flat space
(Figure S2(B)).
Figure S1. Physical meaning for IR cutoff L: the theoretically longest path for a Planck unit
confined to a local space
on a surface of a 3D sphere.
Figure S2. A general case for a Planck unit
at state I and state R (without transformability).
5) Calculation on generalized volume nm for a generalized sphere
For a general sphere determined by
(i = 1, Λ, n) in nD Euclidean space, its nD measurement nm should be
;
;
;
…
And the general formulae for nm
(
)
(
)
showing obviously the monotonic decrease trend aroused by (n+2m/nm) < 1 for cases of n > 12.
Table S1 shows the longest 1D measurement
for an nD Planck sphere and diagonal
for an nD Planck unit in Figure 4.
Table S1. IR cutoffs L = xmax for an nD Planck sphere (L = nm) and the maximum length for an nD Planck unit at state I (
).
Dim. (n) |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
… |
nm |
2.00 |
3.14 |
4.19 |
4.93 |
5.26 |
5.17 |
4.72 |
4.06 |
3.30 |
2.55 |
1.88 |
… |
|
1 |
|
|
|
|
|
|
|
|
|
|
… |
6) Detailed solutions about the coupling fields
The maximum wavelengths for the two T states should be
(in 4D ntP) (S19)
(in 10D 10P)(S20)
The wavelength
for
in 10P should be of
in ntP, with the maximum wavelength being
(in 4D ntP) (S21)
For a 2-body system, TS requires parity between the two symcenters of I1 and I2 (Figure 6), resulting in the shared
,
and
at the tangent point(s) and the interacting point(s) (T1 and
are identical when mT = 0 and dvT = 0, and so do
and
), respectively (Figure S3).
Figure S3. A potential field aroused by quantum coupled state T/T'.
So it should be
and
for potential U of state T. Based on
, kinetic energy for state T satisfies
(S22)
In the flat background space for state I and state T, the conserved energy requires
and
, resulting in
(S23)
between I1 and I2 since
(
) and
. Consequently, it exists
()(S24)
Then a coupling field should be
(
)(S25)
for the two
satisfying principle
in ntP.
And another coupling field should be
(
)(S26)
for two
satisfying principle
in XP.
7) Symbol list
Symbols |
Explanations |
RS |
rotational symmetry |
TS |
translational symmetry |
|
the minimum nD local space in quasi-Euclidean space, nD Planck unit with uncertainty, where the uncertainty forbids
to be with certain measure and certain position simultaneously (
and
can’t be of determinacy simultaneously) |
I |
in-situ state for a Planck unit
with RS, with uncertain structures, such as a generalized sphere with completely uncertain azimuth but certain 1D measure
, a generalized cube with certain azimuth
but completely uncertain xi varying within [, ], or any an intermediate between the sphere and the cube |
R |
revolving state for a RS unit in Planck surface determined by |
T/T' |
state for an RS unit in tangent plane of spherical surface determined by |
nP |
nD orthogonal background space, also the super set of 1P, Λ, nP |
nP |
a local nD space |
tP |
≤6D subspace in 10D background, without non-triviality for any a Planck unit or a substructure expanded by x1, Λ, x6 |
ntP |
4D non-trivial subspace in 10D background, expanded by x7, Λ, x10 |
|
infinite barrier to forbid an object from any radial displacement (demanding dr = 0) |
L |
IR cutoff of for a local space measured nP |
λ |
wavelength of a Planck unit limited in a certain local space, where the unit always takes IR cutoff L as its motion path; when the path is not a closed loop, it exists λ = 2L for the unit in ground state |
nm |
generalized volume for an nD sphere with normalized radium r = 1 |
|
the minimum energy principle, results in a state with motion space as large as possible. |
|
the highest RS principle, results in a state with RS as high as possible |
α |
ratio
during transformation I → T for a Planck unit
in 10D background space, with value ~1/137.036 in ntP when it satisfies Principle
, or ~1/(1.628 × 1038) when it satisfies Principle
|
NOTES
*Authors contributed equally to this work.
#Corresponding authors.