Recollection

# Commutation Relations

Commutation relation in quantum mechanics has triggered my interests ever since i discovered the existence and equivalence of two commutations. The story is, one day while i was reading a book on relativity, i started to wonder if quantum mechanics should be better formulated in a four-vector (space-time) representation. Here is my thought process. The two commutation relations are

$[\hat{x}, \hat{p}] = i\hbar$

$[\hat{H}, \hat{t}] = i \hbar$

In theory of relativity we have,

$x^a = (ct, x^i)$

$p^a = m\frac{dx^a}{d\tau} = \gamma(mc, p^i) = \gamma(E/c, p^i)$

The quest is, we know $[\hat{x^i}, \hat{p^i}] = i\hbar$, does this relation holds even up to four dimensions? i.e. $[\hat{x^a}, \hat{p^a}]$? Before that let’s mention a simple fact about commutation relations,

$[a\hat{A}, b\hat{B}] = ab[\hat{A}, \hat{B}]$

Imagining we are on a rest frame, i.e. $\gamma = 1$, replacing all variables with operators of correspondence and replacing energy term $E$ with $H$, I got the following

$[\hat{x^0}, \hat{p^0}] = [c\hat{t}, \hat{H}/c] = [\hat{t}, \hat{H}] = -i\hbar$

However,

$[\hat{x^i}, \hat{p^i}] = i\hbar$

They have different signs! It troubles me for some time because it doesn’t seem to be as elegant, until i realized something is wrong with my calculation! Time and space are treated completely equal footing, this shouldn’t happen because time is uni-directional, unlike space! This inspires me to add this extra term to make the three coordinate equal footing

$x^a = (ict, x^i)$

It follows then,

$[\hat{x^0}, \hat{p^0}] = [ic\hat{t}, i\hat{H}/c] = -[\hat{t}, \hat{H}] = i\hbar$

$[\hat{x^a}, \hat{p^a}] = i\hbar$

The commutation relation indeed holds for four dimensions which is no only remarkable but also mysterious that an addition of imaginary number makes the picture complete.

After i discovered this, i was so excited that i called my friend to tell him about it. He, who has taken a graduate course on quantum field theory, told me that this is indeed what people have found, but instead of solving the problem by introducing imaginary number, people uses the metric tensor to solve the inconsistency.

We had a big argument which i think is worth mentioning here. My friend insisted that using metric tensor approach is more fundamental because apparently some famous guy said so and all textbooks are consistent with that, my solution is just a mathematical trick which bears no fundamental truth beneath. I didn’t agree very well with this statement.

There is no authorities in physics. Uncountable cases we have, where orthodox belief was completely proven wrong. If there is only one thing to be called fundamental, it is the experimental truth, unchangeable, unprejudiced, upon which all theoretical architectures build up.

Metric tensor is really an invention of notation that encapsulates the distinction of time and space in a geometric interpretation. I see no justification, that this point of view is more fundamental than introducing imaginary number.

I am personally fond of my own notation for the following reasons. It gives rise to a nice equivalence of space and time, which is elegant in its own right. If whichever law applies to one dimension, it applies to any other dimensions.

I believe the fact that time travels along an imaginary axis has profound implications. Anything oscillating in space and time could only behave in two ways, it either oscillates in space but decaying in time, or oscillates in time but decaying in space. That’s how nature works, nothing could happen at all time over all places . Some conservation law seems to be at work, in a mysterious but elegant way.