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From a real world perspective each dimension in the 3-D Cartesian System can be represented by an axis that is perpendicular to 2 other axes. I read somewhere else that the effect of ${i}$ is to reorient data 90 degrees on the "imaginary" axis. I guess my question is this: What role ( if any) does $i$ serve in everyday or even quantum physics?

${Edit}$

To clarify, I am more concerned with the dimensional aspects of $i$. If for instance you were to consider a 3-D location to be a "point" with length width and height in a complex graph representation with three additional mutually perpendicular axes?

PS. I know this thread is getting a bit hair brained. But in my defense, I did look for a "specualation" tag before I posted this thread here.

$EDIT$

I changed $\sqrt{-1}$ back to $i$, because I am concerned with the hypothetical real world effect of the entity. Not the mathematical representation.

user33995
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    possible duplicate of Can one do the maths of physics without using $\sqrt{-1}$? – jinawee Jan 27 '14 at 17:38
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    To see the importance of the imaginary unit, hence complex numbers, see: http://physics.stackexchange.com/search?q=complex+numbers. – jinawee Jan 27 '14 at 17:40
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    @user33995: flagging a question as a duplicate doesn't mean we are accusing you of plagiarism. On a site of this size only the long time members are likely to remember duplicates so new members frequently and accidentally create duplicate questions. I too have voted to close your question as a duplicate. If you think there are aspects of your question not covered by the previous one you can edit your question to highlight the differences and I will withdraw my VTC. – John Rennie Jan 27 '14 at 18:00
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    Don't worry about it... guess the flag threw me off a bit. Perhaps I should peruse before asking, but I often the other question is not exactly what I am looking for. – user33995 Jan 27 '14 at 18:01
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    I personally think the two questions are a bit different, but I'll let the community decide here. – Manishearth Jan 27 '14 at 20:21
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    Comment to the question (v3): OP might want to put his complexification $\mathbb{R}^3\to\mathbb{C}^3$ in physical context to motivate his concern about dimensional aspects. Else the answers will likely reduce to duplicates of previous posts. – Qmechanic Jan 27 '14 at 20:47
  • Is that a better change I title? – user33995 Jan 27 '14 at 21:29
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    @Manishearth the questions are not exactly the same and there have indeed been some people saying leave open. – Dilaton Jan 27 '14 at 22:30
  • You know, I read those other posts. They don't even come close to what I was asking even before the edit. Do you "flag" happy folks even read the OP v the NP or did you just see ${\sqrt {-1}}$ and assume "Hey this must be a duplicate"! – user33995 Jan 28 '14 at 15:12
  • Could elaborate your question a bit more? I see at least to questions: What role ( if any) does $\sqrt {-1} $serve in everyday or even quantum physics? $\implies $ How are complex numbers used in Physics? How are complex numbers used in quantum physics? The other question: you were to consider a 3-D location to be a "point" with length width and height in a complex graph representation with three additional mutually perpendicular axes? is not finised. I think you are hypothesising that we could associate a complex space to each point. – jinawee Jan 30 '14 at 20:54
  • In that case, how do you define it? The complex plane has a real and imaginary axis, so the complex space should have another real or imaginary axis, like $\mathbb{C}\times \mathbb{R}$. Is there any reason why you think this should be possible. Note that in particle physics maps like $\mathbb{R}^3\to\mathbb{C}$ are widely used. – jinawee Jan 30 '14 at 21:00
  • I don't know. but if we were to be (for the sake of argument ) a 3d point riding on an imaginary axis that is perpendicular to two other axes. And 3-space behaved similarly on the other two? Is there any reason to think someone riding either of the other axes would measure their 3-space as any less real than we consider ours to be> – user33995 Jan 31 '14 at 00:42
  • This question (v6) was flagged by an anonymous user as unclear what you're asking. However I'm closing it as a duplicate even if it is not an exact duplicate, in order to point in the right direction. More Phys.SE posts on complex numbers: http://physics.stackexchange.com/questions/tagged/complex-numbers – Qmechanic Jan 31 '14 at 20:50
  • You are the moderator but I really don't see what the heck folks are talking about. – user33995 Feb 01 '14 at 02:34

2 Answers2

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What you're talking about seems to be (or at least lead to) Wick rotation which leads to all sort of crazy dualities between, for example, quantum and thermal physics or Minkowski and Euclidean geometries.

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Firstly, you're referring to geometry on the 2D plane. Represent every point on your 2-plane by a position vector from an origin. The 2D (Real) plane is (in some very useful ways) equivalent to the Complex plane, which can be seen from the formula that any complex number can be written in two equivalent forms $$x + \mathbf{i} y \equiv r e^{i \phi}$$ The format on the left is like using cartesian coordinates on the plane, while the format on the right is like using polar coordinates on the plane -- with $r = \sqrt{x^2 + y^2}$ and $\phi = \arctan{\frac{y}{x}}$, as usual.

One can show that multiplying any position vector by a Real number is akin to "scaling" the size of this position vector, but leaving invariant the direction in which it is pointing. One can also show that multiplying by some kinds of imaginary numbers (unimodular numbers of the form $e^{\mathbf{i} \phi}$) is like rotating them.

On the other hand, for 3D space, there is no such simple analogy to complex numbers. However, one could use quaternions (generalization of complex numbers) but that's more involved.

Siva
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  • No, I am not but perhaps my terminology is not sophisticated enough for the topic. I am trying to describe what I am getting at in words, more difficult in words than in mathematical terms. Lacking the latter I am stuck with the former. Thus, I am referring to a normal 3 space traveling through a constructed space that likewise possesses 3 mutually perpendicular IN addition to the 3 we experience in daily life. – user33995 Jan 29 '14 at 03:28
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    So you're talking about some kind of 6-dimensional space? – Siva Jan 30 '14 at 03:39
  • Actually I thought about it more like 3 dimensions in each dimension So A nine dimensional space.. I thought some value of ${\sqrt{-1}}$ might be indicative of a 90 degree shift in this imaginary space. – user33995 Jan 30 '14 at 17:25