# Unit issues with commutator relations with two dependent variables

Suppose I have the following commutator relation for an operator $$a[x,\omega]$$ which depends on position $$x$$ and frequency $$\omega$$

$$\left[a[x,\omega],a^{\dagger}[x^{\prime},\omega^{\prime}]\right]=\delta(x-x^{\prime})\delta(\omega-\omega^{\prime}).$$

This is an unequal position and unequal frequency commutator. Based on the two Dirac delta functions, the unit for $$a[x,\omega]$$ must be $$\text{Dim}[a[x,\omega]] = \frac{1}{\sqrt{\text{length}\times\text{frequency}}}.$$

Suppose now that the position is fixed, namely, I'm looking at an equal-space commutator. Hence $$x=x^{\prime}$$ and the commutator simplifies to $$\left[a[x,\omega],a^{\dagger}[x,\omega^{\prime}]\right]=\delta(x-x)\delta(\omega-\omega^{\prime})=\delta(0)\delta(\omega-\omega^{\prime}) =\delta(\omega-\omega^{\prime}),$$ where now, the unit of $$a[x,\omega]$$ becomes $$1/\sqrt{\text{frequency}}$$.

My question is, how is this possible? Surely fixing the position can't possibly change the dimension of the operator? What am I missing here and what are some good resources to read up on something like this?

Edit: Another analogy that I can think of are simply the raising and lowering operators of the harmonic oscillator. In general, they are time-dependent (but dimensionless) such that $$[a(t),a^{\dagger}(t)]=1.$$ But it is possible for them to be evaluated at unequal times. In which case $$[a(t),a^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})$$ However, the Dirac delta here implies that the unit of $$a(t)$$ must be $$1/\sqrt{\text{time}}$$. How can this be?

Edit 2: I noticed I wrote $$\delta(0) = 1$$ which is clearly not true. I think I meant to say $$\int\delta(0)dx = 1$$. This suggests that some integration might be involved in making the dimensions consistent when fixing one of the variables.

• What is $\delta(0)$...? Mar 22 at 20:03
• @JasonFunderberker I see your point. $\delta (0)$ should be infinity. I think I just confused Kronecker delta with Dirac delta. In either case, what is the correct way of approaching the collapse of the delta function of one variable? Mar 22 at 20:11
• The correct way is to not approach it at all. The delta function is not a true function, and evaluating it at specific points will lead to trouble. Mar 22 at 21:23