I've heard it said before that $e^{i\pi }=-1$ has many physics applications. What are these applications?
Thanks,

Brad

It's just too good!

See ya

$e^{ix}=\cos x+i\sin x$

(put $x=\pi$ to obtain your result).

The main way in which it is used is as a mathematical trick because if you take the real part of both sides of this equation (for $x$ real), then

$\mathrm{Re}(e^{ix})=\cos x$

Now this is useful because many, many systems are based around some kind of harmonic motion, so we get $\cos x$ appearing. But it turns out that it is easier to operate with exponentials, because they have nice properties. In dealing with alternating electrical circuits in particular, everything becomes much easier if we work using exponentials rather than trigonometric functions (basically part of the idea is that you look for solutions to the equations of motion of the form $e^{iax}$ and sometimes you find that $a$ is real, in which case you get oscillatory solutions, and sometimes you find that $a$ is complex in which case you get exponentially damped or growing oscillations, the power of writing it as an exponential is that you incorporate both kinds of motion in a compact notation - the power of notation should never be underestimated!!).

Now in QM, imaginary exponentials appear all the time and here we don't take the real part because the wavefunction is a complex function. The reason they appear is that the general solution of the time-dependent Schrodinger equation

${\displaystyle i\overline{h}\frac{d\phi }{\mathrm{dt}}=H\phi }$

with

${\displaystyle H\phi =-\frac{{\overline{h}}^2}{2m}\frac{d^2\phi }{{\mathrm{dx}}^2}+v(x)\phi }$

can be written as something like

${\displaystyle \phi (x,t)=\sum _{n=0}^{\infty }{\phi }_n(x)e^{i{E}_{n}x/\overline{h}}}$

with ${\phi }_n(x)$ and $E_n$ related by the time-independent Schrodinger equation $H{\phi }_n(x)=E_n{\phi }_n(x)$

(basically we have just done a Fourier expansion of the time dependence).

The bottom line is that whenever you get a linear differential equation, you are going to get solutions that are in some sense either oscillatory or damped. And the best notation in which to incorporate these kinds of behvaiour is using complex exponentials.

Hope this is vaguely enlightening, it is of necessity a bit sketchy. Any questions welcome,

Sean

I think I remember reading a while ago that Stephen Hawking had come up with a way to use imaginary numbers (I think it was called imaginary time) to find an origin of the universe without a singularity. Can anyone explain this to me?

Thanks,
Brad

I'm not sure about that, Brad...but I know he also made use of tau and t time...using Dirac's equations!

Imaginary time is discussed in 'A Brief History of Time'; I'll try and summarise from that but I might leave some important bits out.

The American scientist Richard Feynman proposed the idea of sum over histories: an approach in which a quantum particle does not possess a single history, or path in space time, as in classical theory. Instead the particle moves from A to B by every possible path. Eath path is considered as a wave, with amplitude and period. The sum over histories describes the summation (by superposition) of these waves. Such a concept must be incorporated in any attempt to combine a theory of gravity and quantum mechanics.

However, when one tries to perform the additions, severe technical problems are encountered. To avoid these difficulties, time is measured with imaginary numbers. A space-time in which time is measured in imaginary numbers in said to be Euclidean. In Euclidean space-time there is no difference betweeen the time direction and spacial directions.

Now from those conditions it is possible to make a proposal that there is no boundary condition for the universe, that is, that space and time together form a surface that is finite in size but does not have any boundary or edge, in the same way that a sphere is finite in size but has no boundary or edge. This proposal becomes possible in the quantum theory of gravity because the time direction is on the same footing as directions in space, whereas in classical general relativity, time is distinct from spacial directions.

Now under the no boundary proposal (and Stephen Hawking emphasises that it is only a proposal) there is a particular family of histories that is much more likely than the others. In these circumstances, there will be no singularities at the beginning or end of the universe. This is argued by analogy as follows:

"These histories may be pictured as being like the surface of the earth, with the distance from the North Pole representing imaginary time and the size of a circle of constant distance from the North Pole representing the spatial size of the universe. The universe starts at the North Pole as a single point. As one moves south, the circles of latitude at constant distance from the North pole get bigger, corresponding to the universe expanding with imaginary time. The universe would reach a maximum size at the equator and would contract with increasing imaginary time to a single point at the South Pole. Even though the universe would have zero size at the North and South Poles, these points would not be singularities, any more than the North and South Poles on the earth are singular. The laws of science will hold at them, just as they do at the North and South Poles on the earth.

The history of the universe in real time, however, would look very different...The universe would expand to a very large size and eventually in would collapse again into what looks like a singularity in real time. Thus, in a sense, we are still all doomed."

I hope some of that made sense. If it didn't, I'll try explaining anything.

Tom.

The technicalities in the sum over histories idea are somewhat complicated - but it is very beautiful, Feynman was a real genius.

However, the basic idea in the motivation of imaginary time is quite easy to explain...

In the sum over histories, you get what are something like integrals of the form