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It is important to understand how experiments work. With very few very very basic exceptions, all experiments and their measurements involve a theoretical framework. Fact is, we almost never measure things explicitly. For crude examples consider: Temperature: a mercury thermometer measures length (that of the column of mercury). An electric thermometer ...


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First, no evidence for other dimensions has been found. However, there are ways for particle colliders to detect other dimensions. One of the main ones is to see if any energy "disappears" under certain circumstances...then it could've gone into another dimension. Another way is to look for particles that can only exist if there are other dimensions. ...


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One classic example of the $d=1,2,3,$ NLSE is in nonlinear optics. Given a paraxial laser beam propogating in Kerr medium, a classical descriptive PDE is the following version of the NLSE: $$i\psi _z ({\bf x},z) + \Delta _{\perp} \psi + |\psi |^2\psi = 0 \,, \quad {\bf x} \in \mathbb{R} ^d, \quad z\geq0 \, , $$ $$ \psi ({\bf x},z=0) = \psi_0 ({\bf x}) \in H^...


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I would normaly understand an Integral $\int d^3\mathbf{r}$ as a volume integral over the whole space $ \mathbb{R}_3$, where I would understand the bold r there as $\vec{r}$ . I have also seen $\int d^3 \vec{r}$ meaning the same volume integral over $ \mathbb{R}_3$. Or even $\int d~ \vec{r}$ with the superscript dropped (I do not like this one but I have ...


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The notation $\mathrm d^3r$, often also $\mathrm d^3\mathbf r$, is generally understood to indicate a three-dimensional volume integral, as you correctly surmise. If $\mathbf r=(x,y,z)$ then you could also denote that as $\mathrm dx\,\mathrm dy\,\mathrm dz$, or as $\mathrm dV$ if it is clear what the integration variable is. The notation $\mathrm d^3\...


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So actually they are compatible. It says that a string theory in 3D for AdS (anti deSitter) spacetime which describes a universe with quantum gravity can be mapped (i.e., there is some correspondence) to a conformal field theory (CFT) in 2D, in the spatial boundary of that universe. This gave rise to the holographic principle, unproven but also not disproven,...


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I believe the answer to your question lies in the Gauss theorem itself $$ \oint \textbf{E}d\textbf{S} \sim Q $$ and the symmetry of the system, which defines the shape of equipotential surfaces. In case of a point charge there is a rotational symmetry about any axis going through the charge, so the equipotential surfaces are spheres whose area is ...


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I don't like the "you can't get away" explanation. There is a simple explanation with field lines: In all three cases, the field lines are straight lines from the point charge to infinity. You can easily calculate the density of the field lines for each object. For a point charge, the "number" of field lines through any sphere around the point charge is ...


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Distance measurements in $n$ dimensional flat space follows the same pattern for $n$ equal 1,2,3, or higher values. I'm going to assume a straight line, change in position to simplify the math (that is we're measuring what a introductory book would call the "displacement" $s$ rather than distance. But then distance is just an accumulation of many magnitudes ...


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One meter is a unit defined in the "real world" around us – places we can actually visit. Or it is used for the lengths and dimensions of objects we can touch. It only makes sense to use the same "meter" for other worlds if we can actually get to those worlds. If two worlds are completely separated from each other, it makes no sense to apply the units of ...


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Whatever unit you're using for distance in 1D is still good in any number of dimensions. Kilometers in manifold of dimension n is fine (assuming non-compactified dimensions).


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Let me take parts 2. and 3. of the question first: The 10 dimensions of string theory are, a priori, not "coiled up" or anything else. They are derived for a string theory where the classical version of the string propagates in d-1 spatial dimensions and 1 temporal dimension, i.e. Minkowski space $\mathbb{R}^{1,d-1}$. "Dimension" here is dimension of a ...


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The similarity of time and space is limited to Lorentz symmetry. Beyond Lorentz symmetry, the time dimension cannot be assimilated to space dimensions. Within spacetime, time is not intrinsically curved: Any observed time corresponds to the proper time of a clock, and the clock is always counting straightforward, even if according to our coordinates we may ...


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Two or more timelike dimensions is a situation that is difficult if not impossible to reconcile with the notion of causality. Suppose you want to think of a five dimensional universe with three spatial and two time dimensions. What you mean then is the metric has a $(2,\,3)$ signature, which means that at each point Riemann normal co-ordinates centered at ...


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Time is the perception of the order of change in 3 dimensions. Time does not exists, only 3 dimensions of space exists in reality. Time is not a 4th dimension. I think your question highlights the damage done to innocent brains in high schools. Small wonder we do not have enough smart physics students, their brains get destroyed with false assumptions ...


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I can define it very easy from a math point of view (the hypercube construction example), even we can apply real models based on the mathematical principles, but there is no practical way to measure/test that, not with current capabilities. As a note, a Black Hole may be the clue we're looking for. All that compressed and stored energy could manifest itself ...


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The reason the whole idea of 11 spacetime dimensions came about is because the equations of string theory without these extra dimensions have "quantum anomalies"...namely, the creation and destruction of energy, which is obviously kind of a problem. But, with 11 spacetime dimensions...voila! Problem solved. Then, of course, people were like, "but why don't ...


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In principle this could be tested in the near future, but it could be perhaps a million years ahead. The problems is that these dimensions seem to be rolled very tight, with dimensions much smaller than an atomic nucleus. However there is another twist in the opposite direction: Holographic principle. The holographic principle is a property of string ...


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The critical point of the Ising model in two dimensions is about $$ T= 2.269 \frac{J}{k_B},$$ where $J$ is a free parameter that can be interpred as the coupling between spins and has energy units. So by introducing $J$ (experimentally or by other calculation) and dividing by $k_B$ (Boltzmann's constant) you would get the temperature in Kelvin.



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