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28

I think that you are referring to space-filling curves and how they can map a line segment to more than one dimension. For example the Hilbert space filling curve can be used to map the interval $[0,1]$ to $[0,1]\times[0,1]$. I am afraid while a continuous bijection is possible one-way, it is not possible to have a homeomorphism between two different ...


20

Dear asmaier, you shouldn't view $\vec L = \vec x \times \vec p$ as a primary "definition" of the quantity but rather as a nontrivial result of a calculation. The angular momentum is defined as the quantity that is conserved because of the rotational symmetry - and this definition is completely general, whether the physical laws are quantum, relativistic, ...


19

Yes, absolutely. In fact, Gauss's law is generally considered to be the fundamental law, and Coulomb's law is simply a consequence of it (and of the Lorentz force law). You can actually simulate a 2D world by using a line charge instead of a point charge, and taking a cross section perpendicular to the line. In this case, you find that the force (or ...


14

Actually, let's give this a shot. This isn't evidence for extra dimensions (the non-obesrvation of extra dimensions/supersymmetry is one of the big reasons string theory is not accepted universally as true, after all), but this is an argument as to why small extra dimensions are unobservable. Consider a particle in a box in quantum mechanics in n spatial ...


14

Lubos Motl's answer is completely right, but I'll add my perspective anyway. For many compound units, you shouldn't try to "visualize" the meaning of the unit, but you should think of it as reminding you about relationships between that quantity and others. Why are the units of Newton's constant $G$ ${\rm N\ m^2/kg^2}$? It's because $G$'s "purpose in life" ...


14

This is not paradoxical and it is not necessary for any physical phenomenon to a priori have to obey any particular law. Some phenomena do have to obey inverse-square laws (such as, particularly, the light intensity from a point source) but they are relatively limited (more on them below). Even worse, gravity and electricity don't even follow this in ...


13

I think the best answer to your question is simply "because that's all we can see when we do experiments." That is, no matter how hard anyone tries or how much energy they toss into the processes, electrons and quarks show no signs of any appendages, surfaces, hair-like structures, bumps, volume, whatever. When you model them mathematically as points, the ...


13

Specifically what that is referring to is the 'inverse-square law', nature of the gravitational force, i.e. the force of gravity is inversely proportional to the square of the distance: $F_g \propto \frac{1}{d^2}$. If you expand this concept to that of general power-law forces (e.g. when you're thinking about the virial theorem), you can write: $F ...


13

Varun's answer basically tells it all, but perhaps it's useful to also explain why we want a homeomorphism rather than just a bijection. The whole idea of modelling "real physical space" by an Euclidean $\mathbb{R}^3$ is so we can make predictions about physical processes, like how an electromagnetic wave spreads. Typically, we use differential equations ...


12

A classical1 theory of (relativistic) $p$-dimensional membranes exists for any non-negative integer $p$. Such classical membrane-like objects appears in the full non-perturbative formulation of string theory. The problem arises if one tries to quantize a membrane (in a first quantized sense) using standard perturbative quantization methods, which have ...


11

Related answer: http://math.stackexchange.com/a/532746/24293 Looking at the comments, you seem to be asking why there are chiral 'pairs' and not chiral multiplets. Looking at the tag, it looks like you want an analysis of higher dimensions as well. TL;DR Short answer: In any number of dimensions, chiral objects come in pairs. This is because numbers ...


11

Qmechanic's answer has the full details, but I figure it might be useful to break this down in a little more detail. The proper way to generalize Maxwell's equations to higher-dimensional spaces is to use the field tensor $F^{\mu\nu}$. In our normal 3+1D space, it basically looks like this: $$F = \begin{pmatrix}0 & E_1 & E_2 & E_3 \\ -E_1 & ...


10

There is no reason why you should be "imagining" a squared second. Most quantities in physics don't have any canonical "geometric" visualization and there is no reason why they should have. What matters is that you should be able to calculate with it. For example, the gravitational acceleration on Earth is $9.81\,\,{\rm m/s}^2$. This simply means that the ...


10

The notion of negative dimension has appeared in various places of modern physics. For instance: Grassmann-odd variables. Recall that the dimension ${\rm dim}(V)$ of a group representation $\rho: G \to GL(V)$ is given by the trace ${\rm dim}(V)={\rm Tr}(\rho(1))$ of the identity element. For a supergroup, one should use the supertrace, so Grassmann-odd ...


9

According to our best understanding of how Maxwell's equations would generalize to other dimensions, then yes, it is. The electromagnetic field is represented by an antisymmetric tensor, $$F^{\mu\nu} = \begin{pmatrix}0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & ...


9

This question has changed in such a way that my answer (previously here) didn't seem even related anymore. I therefore came up with something new, gladly inheriting 4 upvotes, but much less confident. In fact, I can plainly state that I'm fully incompetent in these matters. With that out of the way, another introductory remark. Science doesn't prove ...


8

One example of such an approach is Ambjorn and Loll's Causal Dynamical Triangulations, which is very similar in many ways to the very old idea of Regge calculus, whereby spacetime is discretized. At small scales, non integer dimensions can emerge. For an introductory article , see Jan Ambjørn, Jerzy Jurkiewicz and Renate Loll. The Self-Organizing ...


8

It is not so much as one single particle will be seen with different masses as it is that that one type of particle will be seen as having multiple different masses when it is detected multiple times. For example if the extra dimension is like a rolled up microscopic cylinder, the particle can have an infinite number of discrete masses starting from the ...


8

Several authors (in particular Itzhak Bars) have written papers about two-time-physics that should help build intuition for the topic. Infinitely many 'times' appear in integrable systems. F-Theory, which is a 12 dimensional theory, has been described as having extra temporal dimensions, however see Wikipedia.


8

M-theory compactified on a 2-torus is the same as M-theory compactified on a circle and then compactified on another circle because $T^2=S^1\times S^1$. M-theory compactified on a circle is type IIA string theory with $g_s$ being an increasing power of the radius of the compactified dimension. And if type IIA is compactified on a circle of a small radius, ...


7

In addition to what dmckee said, another hint at ("large") extra dimensions would be the detection of Kaluza-Klein particles at the LHC for example. Kaluza-Klein particles are in principle nothing but the known standard model particles which can propagate into the extra dimensions if these are large enough. It can be shown that the angular momentum in these ...


7

From special relativity we know that a Lorentz transformation: \begin{equation} x'^\mu = \Lambda^\mu {}_\nu x^\nu \end{equation} preserves the distance: \begin{equation} g^{\mu \nu} \Delta x_\mu \Delta x_\nu = g^{\mu \nu} \Delta x_\mu' \Delta x_\nu' \end{equation} The above two equations imply: \begin{equation} g^{\mu \nu} = g^{\rho \sigma}\Lambda_\rho ...


6

I would say yes ! Actually some theories explaining quantum gravity use also this reasoning: gravity is a very weak interaction at a quantum level because it "leaks" into other dimensions, not observable at our scale, but that are present at this scale. The mathematical tools are different, but if you just think about gauss's law you can imagine one ...


6

It's true that if you just have a set of points, with no additional mathematical structure, the notion of "dimension" is problematic as you say. But the spaces we deal with in physics usually have extra structures that make the notion well-defined. Often, the definition works by making precise a notion of different "directions" at a given point, and then ...


6

The formal statement would be that there is no one notion of time, and that one persons definition of time may be intermixed with another's definition of space. What is not correct is that time and space are wholly interchangeable, as Moshe says--there is a distinction between events separated in time and events separated in space. Causality theory is ...


6

I will here only comment on the traditional superstring theory story, say, from the first superstring revolution in the 1980s, and leave it to others to include more recent developments. Traditionally, the $10$-dimensional target space $(M^{10},g^{(10)})$ with a metric $g^{(10)}$ is viewed as a product $M^{10}=M^4 \times K^6$ with metric ...



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