# Complex numbers in quantum mechanics and in special relativity

Is there a physical relation between the use of complex numbers for the wavefunction in (non-relativistic) quantum mechanics and in special relativity (as formulated in the setting of Minkowski space)?

Or is it just two different theories that happen to use the same mathematical trick?

• It's not really a mathematical trick but the deep mathematical connection between rotations and complex numbers that does it. We are, by the way, not using complex numbers in relativity any longer. That approach has been superseded by the use of metrics and differential geometry. – CuriousOne Dec 30 '14 at 14:49
• Complex numbers tend to be useful when one of the degrees of freedom can be treated as a phase, because a complex number in polar form naturally encodes a phase. This is useful in quantum mechanics, optics and electrical circuits, but less immediately applicable in GR. – John Rennie Dec 30 '14 at 14:57
• @CuriousOne could you please elaborate that mathematical connection? I would really like to know about it(yep! I always thought it was just a fancy trick devised by the mathematicians):) – GRrocks Dec 30 '14 at 15:07
• Did you read en.wikipedia.org/wiki/Complex_number? The phasor representation should be pretty commonly known. – CuriousOne Dec 30 '14 at 15:10
• @Bob: the rapidity is an angle, but I'm not sure it's useful to regard it as a phase. If you know of such an application then please shout. – John Rennie Dec 30 '14 at 15:59

If you are referring to using complex numbers in the sense that the spacetime metric can be written by using the Euclidean metric but with an $i$ in the time component to produce the required minus sign, then this is an antiquated way of doing things, and has been essentially abandoned by the physics community in favor of the more powerful framework of semi-Riemannian geometry.

On the other hand, it's not entirely the case that complex numbers cannot be useful in relativity, and especially in elucidating connections which exists between relativity and quantum mechanics.

The following is tangential, but it's sufficiently beautiful that I'm willing to risk down votes so that people will be exposed to certain ideas herein.

The punch line (see the end) will be that there is a deep mathematical connection between relativity and the concept of "spin" in quantum mechanics, and this connection has something to do with using complex numbers appropriately in relativity, namely by considering certain mathematical objects called groups certain of which are naturally described in terms of matrices with complex entries.

Symmetries in physics and relativity.

In physics and mathematics, symmetries of a system are encapsulated in certain objects called groups which basically tell us what sorts of things we can do to the system without changing its relevant structure.

For example, the symmetries of Minkowski space (not including parity and time-reversal) consist of the elements of the so-called Poincare group $$P = \mathbb R^{3,1}\rtimes \mathrm{SO}(3,1)^+.$$ Basically, this group consists of boosts, rotations, and spacetime translations.

Now, for reasons that I won't explain here (see Idea of Covering Group), when we want to start considering how to apply these symmeries of physical system to quantum systems, we should consider "representations" of "universal covering groups" of symmetry groups instead of the groups themselves. When we compute the universal covering group of the Poincare group, we get a group that is most naturally described using complex matrices: $$\mathbb R^{3,1}\rtimes \mathrm{SL}(2,\mathbb C).$$ Furthermore, finite-dimensional representations of the subgroup $\mathrm{SL}(2,\mathbb C)$ give us representations of dimension $2s+1$ where $$s=0,\tfrac{1}{2},1,\tfrac{3}{2},\dots.$$ If you've studied quantum mechanics (even non-relativistic), then you should recognize these numbers. These are the possible spins of particles!

In other words, the quantum mechanical concept of spin arises in a mathematically natural way when you consider symmetries in special relativity.

There is a huge amount more to say here, but I'll let the interested reader investigate for herself.

• Maybe mention Prof. Wigner, for historical completeness? – Selene Routley Mar 9 '17 at 22:33

There isn't too much of a connection, because the use of complex numbers in special relativity is itself just a physically meaningless hack that is often no longer used.1

In special relativity, we have this annoying (to some) sign difference between time and space. The modern interpretation is that the (pseudo-)metric is simply not positive definite, and that's okay. We have $$ds^2 = -(c\,\mathrm{d}t)^2 + (\mathrm{d}x)^2 + (\mathrm{d}y)^2 + (\mathrm{d}z)^2,$$ so there are real combinations $\mathrm{d}t,\mathrm{d}x,\mathrm{d}y,\mathrm{d}z$ that lead to negative values of $ds^2$.

However someone, somewhere, noticed that the sign flip in time can be hidden away by burying an extra factor of $i$ in the definition of the time direction, since $(cit)^2 = -(ct)^2$. Now we have an honest, positive-definite metric acting on "imaginary time" $\tau=it$ and real space $x,y,z$: $$ds^2 = (c\,\mathrm{d}(it))^2 + (\mathrm{d}x)^2 + (\mathrm{d}y)^2 + (\mathrm{d}z)^2 = (c\,\mathrm{d}\tau)^2 + (\mathrm{d}x)^2 + (\mathrm{d}y)^2 + (\mathrm{d}z)^2,$$ so any collection of real infinitesimal differences $\mathrm{d}\tau,\mathrm{d}x,\mathrm{d}y,\mathrm{d}z$ will lead to a nonnegative $ds^2$.

That's cute, but note we never even used the vast majority of the structure of $\mathbb{C}$. All we needed was the concept of something whose square is $-1$. And positive definiteness only works for real $\tau$, i.e. for pure imaginary $t$. You can't simply get rid of the fact that real positive distances and real positive time intervals can and should lead to spacetime intervals of both signs. Special relativity (unlike perhaps quantum mechanics) is not intimately tied to complex structure.

Moreover, general relativity puts an end to any notion that complex numbers make things simpler. One might have a metric $$ds^2 = -(\mathrm{d}t)^2 + \mathrm{d}t\,\mathrm{d}x + (\mathrm{d}x)^2 + (\mathrm{d}y)^2 + (\mathrm{d}z)^2.$$ The $t \to -i\tau$ transformation might make the first coefficient positive, but the second coefficient will become complex. Employing complex numbers just to avoid a negative sign is somehow both overkill and self defeating.

1At least when it comes to the metric. See joshphysics's answer for a different use of complex numbers.

One shouldn't be hasty in discounting complex numbers as useful for understanding the geometry of special relativity. As Penrose has pointed out, an observer of the night sky (considered as a Riemann sphere in correspondence with the extended complex plane via stereographic projection) who undergoes a lorentz transformation will see stellar positions shifted by a mobius transformation (aka linear fractional transformation) of the Riemann sphere. This is really a deep connection between lorentz transformations and complex geometry, and corresponds to the covering of SO(1,3) by SL(2,C) mentioned in the last post. See for example http://www.mathpages.com/rr/s2-06/2-06.htm and http://www.math.wustl.edu/~feres/Math496F15/Math496F15HW02Sol.pdf