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I think that it in this case is a bit of semantics. In an ideal beam-splitter there is really no "true" reflection. Rather the beam is split into two output directions. Which one of the two directions you call transmitted and deflected is a matter of taste. Naively I would say the example with transition $t$ on the diagonal looks more natural. In this case ...


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The contravariant metric tensor is the inverse metric tensor. If you have a general $g_{ab}$ you can find $g^{ab}$ by matrix inversion (which can usually be done in Mathematica or any other program of the kind). In the special case of a diagonal metric tensor you can verify that $g^{ii} = 1/g_{ii}$.


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If it's diagonal, you can just focus on the diagonal elements, that is : $g_{aa} g^{aa} = 1$


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Our rigorous definition of "dimension" comes from linear algebra. This is going to be a quick run-through of the mathematical way of describing dimensions, and then its physical significance. The first concept needed is a vector space. A vector space can effectively be thought of as a collection of points that satisfy a few particular (and very useful) ...


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In a geometrical context, the dimension (at a point) roughly speaking is the number of coordinates you need to identify any point in a fixed neighbourhood. Note that in a geometrical context, the least you need is a topology, in order to be able to speak of continuity, neigbourhoods, etc. This intuitive definition used to work quite well, but at a certain ...


6

I think this depends a lot on what you are doing and how you look at whatever you are looking at. Speaking of which, how many dimensions does the content displayed by your computer monitor have? Two, I guess, could be one answer. It's not three dimensional and it certainly is not one strip of pixels. Let me quote from Carl's great answer that I like to ...


28

In this context, I usually explain it (non-mathematically) by saying that the number of dimensions is the number of values you need to specify where an event occurs. For most people this involves space and time (but for particle physicists it might involve more values ;). Anyway, certainly even people before Einstein would need to specify the time as well ...


35

Coming from a math perspective, I would define a dimension as "any property which is orthogonal to all other properties." "Orthogonal" here means you cannot get to one property by applying scalar operations on another. For example, the x-axis dimension can never become a y-axis value, and similarly for time vs. spatial dimensions. For that matter, ...


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I agree with Phoenix87 that the answer must contain a little bit of maths. Basic (nonrelativistic) quantum systems are well described by the Schrödinger equation, $H \psi = E \psi$, which is a linear (partial differential) equation: It contains the wave function $\psi$ only to the first power (no $\psi^2$ etc.). Thus, if two wave functions, say $\psi_1$ ...


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There is not a direct link between the linearity of some physical laws and the superposition of quantum mechanics. The latter is more of a special kind of linear superposition which requires some restrictions on the coefficients. The existence of the phenomenon of superposition of states is a characteristic of quantum mechanics. In classical mechanics such ...



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