Four vectors in complex notation

[Nick89]

Hi,

I recently completed a course on Electrodynamics, with a short introduction to special relativity (and E/B fields in special relativity).

Both in the book we were asked to study, and the final exam, the four vectors we used used real components, with the time coordinate being the zeroth component. For example:
$$(x^\mu) = (ct, x, y, z)$$
$$(A^\mu) = (\phi / c, A_x, A_y, A_z)$$
(A = vectorpotential, $\phi$ = scalar potential)
etc...

Also, what the book calls the 'four dimensional scalar product' has a, in my opinion, rather strange minus sign in the zeroth component:
$$\sum_{\mu = 0}^3 a_\mu b^\mu = -a^0b^0 + a^1b^1+a^2b^2+a^3b^3$$

In the classes we were given however, we used a different convention (well, I guess it's a convention), where the time coordinate as the fourth component (counting from 1 to 4), carrying a factor of $i$:
$$(x^\mu) = (x, y, z, ict)$$
$$(A^\mu) = (A_x, A_y, A_z, i\phi / c)$$
etc...

Now, the product doesn't need the pesky minus sign, because it comes from the factor $i^2$...

I could only find the first convention (real vectors) in practice, I couldn't find the complex vector convention anywhere I looked, except in our classes. But, I think it's a much neater definition, because everything generalizes from 3D, without the pesky minus sign...

I was just wondering, is there any particular reason for using the real vectors, instead of the complex ones? If not, is the real vector definition really used more often, or did I just not look hard enough?

I just find it a bit strange that everything seems to be about generalization (putting time and space into a single vector, and putting vector potential and scalar potential into one vector, etc), yet an, in my opinion, beautiful use of complex vectors is not used, so that an awkward definition of the scalar product has to be used..?

 

[bcrowell]

It's just a matter of taste. Some people do special relativity one way, some do it the other. In general relativity, the coefficients occurring in the inner product are varying rather than constant, so it becomes pointless to worry about trying to make them all equal +1.

 

[Fredrik]

As bcrowell said, it's a matter of taste. I personally find the "i" weird and awkward, mainly because we wouldn't use complex numbers anywhere else in the entire theory. I have never seen that convention used in a book on differential geometry or general relativity. Also note that any bilinear form g on $\mathbb R^4$ can be defined as $g(x,y)=x^T M y$, where M is a 4×4 matrix and x and y are 4×1 matrices. We have g(x,y)=g(y,x) if and only if M^T=M. When M=I (the identity matrix), g is the standard inner product, and when $M=\eta$, it's the Minkowski bilinear form. (I don't want to call it an inner product or a scalar product, because it's not positive definite). I think this is a much more attractive way to deal with these things.

In case you don't know already, $\eta$ is defined by

$$\eta=\begin{pmatrix}-1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}$$

 

[Meir Achuz]

Use of the i that way was dropped about 30 years ago, because it avoided the fact that there really is a minus sign in the metric, which is called an 'indefinite metric'. I think Panofsky and Phillips used it in their first edition, but went to the indefinite metric for their second edition. The i leads to confusion in relativistic QM, and can't be extended to GR.
The metric used by most physicists now is +1,-1,-1,-1, because this leads to
p.p=m^2 for the length of the momentum 4-vector.

 

[utesfan100]

As bcrowell said, it's a matter of taste. I personally find the "i" weird and awkward, mainly because we wouldn't use complex numbers anywhere else in the entire theory.

I agree with this strongly. If we are going to introduce complex variables we should not limit ourselves to pure complex numbers. What would 3+2i seconds look like? 4+3i meters would be a 3-4-5 triangle, but what significance is that?

Fringe warning, proceed with caution.
I like to consider from time to time that these imaginary values might have meaning in a phase space frequency relationship. I also like to think that the metric looks too similar to quaternion arithmetic to be a coincidence, but we still need the pesky pure i's there (which then generates a signature of (+---), rather than the (++++) of the underlying quaternions ).

Complex numbers, quaternions and split complex numbers are part of a family of algebras related by Cayley-Dickson doubling. One can observe that split doublings naturally generate algebras with Lorentz boosts.

http://en.wikipedia.org/wiki/Cayley–Dickson_construction
http://en.wikipedia.org/wiki/Split-octonion

If one uses Cayley-Dickson doubling three times from the real numbers using split values for each additional step one obtains the three split primary components, three generated proper roots value and an additional split root with a combined signature of (++++----).

If the real value is taken as time and the three original split roots as real space, we get a metric of (+---), but we have ignored the phase space.

The split algebra generated after the third cycle have a signature (+++-). Using the phase-space analogy, the three proper roots correspond to some type of spatial frequency and the split root a temporal frequency. If we are basing a theory on observed peaks and troughs of waves this is all of the algebra we are using.

Thus if we limit our investigations to space-time or phase space-frequency we will make observations consistent with SR. Only when we attempt to combine the two will complications arise.

This algebra obeys the property that a metric is preserved under multiplication but breaks the associative property, requiring some type of bra-ket-like notation to be employed. It also contains zero divisors that are similar to the light cones of SR. (A spatial direction with an equal spatial frequency form a new class of cones, as does a temporal frequency locked to the real time. The latter obviously represents a dynamic system appearing static, the former is not so clear.)

Then I through up my hands and say "whatever."