CC BY
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Mathematical Structures and Modeling 2015. N. 2(34). PP. 24-28
UDC 514.8
why we need extra physical dimensions: a simple geometric explanation
O. Kosheleva
Ph.D. (Phys.-Math.), Associate Professor, e-mail: [email protected]
V. Kreinovieh
Ph.D. (Phys.-Math.), Professor, e-mail: [email protected] University of Texas at El Paso, El Paso, TX 79968, USA
Abstract. It is known that a consistent description of point-wise particles requires that we add extra physical dimensions to the usual four dimensions of space-time. The need for such dimensions is based on not-very-intuitive complex mathematics. It is therefore desirable to try to come up with a simpler geometric explanation for this phenomenon. In this paper, we provide a simple geometric explanation of why extra physical dimensions are needed.
Keywords: extra physical dimensions, discreteness of physical space-time, geometric explanation, Meyer’s Theorem on quadratic forms.
1. Need for Extra Physical Dimensions: Reminder
Problems with the usual 4-dimensional space-time models. In relativistic physics, elementary particles are points in space; see, e.g., [1]. Point-wise character of elementary particles makes many physical quantities infinite. For example, the energy density p(x) of the electric field E(x) is known to be proportional to |E(x)|2, and the electric field of a point-wise particle decreases with the distance
r to the particle according to the Coulomb law |E(x)| ~ —. Thus, the energy
density p(x) is proportional to |E(x)|2 ~ 1.
The overall energy is equal to the integral / p(x) dx and is, thus, proportional
to the integral / — dx. In polar coordinates, after integrating over angular coordinates, we get
2n • r2
dr = 2n ■
Ю
Ж 1
dr.
This integral is equal to
I = — 2n • — r
.
r
0
0
Similar physically meaningless infinities appear when we compute other quantities related to a point particle [1].
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Comment, The above computations use a non-quantum approximation, but similar infinities appear when we take into account quantum effects as well.
Current solution. It turns out that infinities can be avoided if we assume that the space-time has extra dimensions beyond the four usual ones. For example, string theory shows that we can get a consistent physical theory if we assume that the space-time is 10-dimensional; see, e.g., [4].
Remaining challenge. A problem with this solution is that it is heavily mathematical, there is no simple intuitive geometric explanation of why extra dimensions are needed.
Comment, It should be mentioned that:
• while there is no clear geometric explanation of why extra dimensions are needed,
• there are simple geometric explanations of why namely 10 is a good dimension; see, e.g., [5].
What we do in this paper. In this paper, we provide a possible geometric explanation of why extra space-time dimensions are needed.
2. Analysis of the Problem and the Resulting Explanation of Extra Physical Dimension(s)
Natural idea: discrete space-time. The infinities are caused by integration to r = 0. Thus, one possible way to avoid infinities is to assume that spatial coordinates - and other quantities - are discrete. This idea is ubiquitous in physics [1]:
• an electric charge cannot take any possible value, it must be proportional to some constant;
• quantum physics started with Planck’s hypothesis that energy of light of a given wavelength cannot take any possible value, it must be proportional to some constant (dependent on this frequency), etc.
Resulting description of space-time. Let us apply the discreteness idea to variables that describe space-time geometry, namely,
• to the space-time coordinates x1,... ,xn, and
• to the components gij of the metric tensor that describes the proper time s(x,x') between two points x = (xb ... ,xn) and x' = (x[,... ,x'n) as follows:
n n
S2(x,x') = gj ■ (xi - xi) ' (xj - xj). (1)
i=1 j=1
26 О. Kosheleva, V. Kreinovich. Why We Need Extra Physical Dimensions...
For space-time coordinates, discreteness means that all the coordinates must be integer multiples of some fixed quantum qx, i.e., that for every point x and for each coordinate i, we must have x, = X ■ qx for some integer X,,. Similarly, for the components of the metric tensor , discreteness means that there exists some fixed quantum qg for which, for each component g,j, we have g,j = Gj ■ qg for some integer Gij.
Under these two discreteness assumptions, the formula (1) that describes the square s2(x,x') of the proper time between the points x = (X1 ■ qx,... ,Xn ■ qx) and x' = (X[ ■ qx,..., ХП ■ qx) takes the form
s2(x, x') = 52(X, X') ■ qx ■ qg, (2)
where we denoted
S2(X, X') d= £ E Gij ■ (Xi - Xi) ■ (Xj - Xj). (3)
i=1 j=1
Empirical fact: there are light-like particles. It is a known physical fact that:
• in addition to usual particles like electrons and protons that travel with speeds smaller than the speed of light, and for which, therefore, s2(x,x') > 0 for every two points x = x' on the particle’s trajectory,
• there also exist “light-like” particles like photons that always travel with the speed of light and for which s2(x,x') = 0 for every two points x = x' on the particle’s trajectory.
In the continuous space-time, the possibility of light-like particles is mathematically trivial. In the continuous space-time, when each coordinate x, can take any real value, it is always possible to find pairs of points x = x' for which s2(x,x') = 0 - provided, of course, that the matrix gij is not positive or negative definite, i.e., provided that:
• there exist pairs (x,x') with s2(x,x') > 0, and
• there exist pairs (x,x') with s2(x,x') < 0.
In discrete space-time, the existence of light-like particles is automatically guaranteed only if we have extra physical dimensions. In the discrete spacetime model (2)-(3), however, it is not always true that if a quadratic form (3) with integer coefficients Gij- attains both positive and negative values, there exist integer values X, — Xi for which this form is equal to 0.
Such a general statement is true if and only if we have at least five variables,
i.e., if and only if n > 5. This result was proven by A. Meyer in 1884 [6] and is known as Meyer’s Theorem; see, e.g., [1,7,8].
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Resulting explanation. Thus, to make sure that a discrete space-time is always consistent with the existence of light-like particles, we must assume that the dimension of space-time is at least five.
This explain the need for at least one extra physical dimension - in addition to the usual four dimensions of space-time.
Acknowledgments.
This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721.
References
1. A. Borel, “Valued of indefinite quadratic forms at integral points and flows on spaces of lattices”, Bulletin of the American Mathematical Society, 1995, Vol. 32, No. 2, pp. 184-204.
2. J. W. S. Cassels, Rational Quadratic Forms, London Mathematical Society, London, 1978.
3. R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics, Addison Wesley, Boston, Massachusetts, 2005.
4. M. Green, J. H. Schwarz, and E. Witten, Superstring Theory, Vols. 1 and 2, Cambridge University Press. Cambridge, Massachusetts, 1987.
5. A. B. Korlyukov and V. Kreinovich, “Equations of physics become consistent if we take measurement uncertainty into consideration", Reliable Computing, 1995, Supplement (Extended Abstracts of APIC’95: International Workshop on Applications of Interval Computations, El Paso, TX, Febr. 23-25, 1995), pp. 111-112.
6. A. Meyer, “Mathematische Mittheilungen", Vierteljahrschrift der Naturforschenden Gesellschaft in Zurich, 1884, Vol. 29, pp. 209-222.
7. J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Springer Verlag, Berlin, Hidelberg, 1973.
8. J.-P. Serre, A Course in Arithmetic, Springer Verlag, Berlin, Hidelberg, 1973.
28 О. Kosheleva, V. Kreinovich. Why We Need Extra Physical Dimensions...
почему необходимы дополнительные физические размерности: простое геометрическое объяснение
О. Кошелева
к.Ф.-м.н., доцент, e-mail: [email protected]
В. Крейнович
к.Ф.-м.н., профессор, e-mail: [email protected] Техасский университет в Эль Пасо, США
Аннотация. Известно, что последовательное описание точечных частиц требует, чтобы мы добавили дополнительные физические размерности к обычным четырём измерениям пространства-времени. Необходимость добавления таких размерностей основана на не очень интуитивной сложной математике. Поэтому желательно придумать более простые геометрические объяснения этому феномену. В данной работе мы предоставляем простое геометрическое объяснение того, почему дополнительные физические размерности необходимы.
Ключевые слова: дополнительные физические размерности, дискретность физического пространства-времени, геометрическое объяснение, теорема Мейера о квадратичных формах.
