# Dimensional aspects of the imaginary unit $i$ in physics [duplicate]

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.

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## marked as duplicate by Qmechanic♦Jan 31 '14 at 20:51

possible duplicate of Can one do the maths of physics without using $\sqrt{-1}$? –  jinawee Jan 27 '14 at 17:38
To see the importance of the imaginary unit, hence complex numbers, see: physics.stackexchange.com/search?q=complex+numbers. –  jinawee Jan 27 '14 at 17:40
@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
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
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

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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Is this where some get the idea that we may live in a 9+1 universe? –  user33995 Jan 27 '14 at 18:29
@user33995 not through Wick rotation no, but there is a certain sense in which this moves us from 3+1 physics to "4+0" physics (consider the change in the metric $-\mathrm dt^2+\mathrm dx^2+\mathrm dy^2+\mathrm dz^2 \longrightarrow \mathrm d\tau^2+\mathrm dx^2+\mathrm dy^2+\mathrm dz^2$ where $\tau = it$ is the Wick time) –  Sean D Jan 27 '14 at 19:12
... not to mention the Black-Scholes option pricing equation (for which I am told Wick rotation is used to implement path integral formulation of the BS pricing equation) whose indiscriminate use led to the collapse of the ironically named "Long Term Capital Management" hedge fund. –  WetSavannaAnimal aka Rod Vance Jan 28 '14 at 23:36

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.

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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 3:28
So you're talking about some kind of 6-dimensional space? –  Siva Jan 30 '14 at 3:39
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