Showing that a certain summation is equal to a Dirac delta?

  • #1
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Homework Statement
Quantum Field Theory for the Gifted Amateur Exercise 3.1 (reworded a bit): Suppose we are working with a system defined in a volume of space ##V'##. For boson operators satisfying ##[\hat{a}_p, \hat{a}^{\dagger}_q] = \delta_{pq}## show that $$\frac{1}{V}\sum\limits_{pq}e^{i(p \cdot x - q \cdot y)}[\hat{a}_p, \hat{a}^{\dagger}_q] = \delta^{(3)}(x - y)$$, where ##V## is the magnitude of ##V'##. Show the same with fermion operators.
Relevant Equations
Seen above.
I'm studying Quantum Field Theory for the Gifted Amateur and feel like my math background for it is a bit shaky. This was my attempt at a derivation of the above. I know it's not rigorous, but is it at least conceptually right? I'll only show it for bosons since it's pretty much the same for fermions except the commutator is replaced with the anticommutator.

First note that
$$\frac{1}{V}\sum\limits_{pq}e^{i(p \cdot x - q \cdot y)}[\hat{a}_p, \hat{a}^{\dagger}_q]$$
$$= \frac{1}{V}\sum\limits_{pq}e^{i(p \cdot x - q \cdot y)}\delta_{pq}$$
$$= \frac{1}{V}\sum\limits_p e^{ip \cdot (x - y)}$$

To prove that the sum above is equivalent to ##\delta^{(3)}(x - y)##, we must show 2 things:

1. ##\frac{1}{V}\sum\limits_p e^{ip \cdot (x - y)} = 0## whenever ##x \neq y##
2. $$\int\limits_{V'} d^3x \frac{1}{V}\sum\limits_p e^{ip \cdot (x - y)} = 1$$

The first follows from the periodicity of ##e^{ip \cdot (x - y)}## [EDIT: the periodicity and the fact that for each value, the negative of that value will also appear within one period]. When ##x - y## is nonzero, each ##e^{ip \cdot (x - y)}## term will be cancelled out by some other term ##e^{ip' \cdot (x - y)}##.

For the second,
$$\int\limits_{V'} d^3x \frac{1}{V}\sum\limits_p e^{ip \cdot (x - y)}$$
$$= \frac{1}{V}\sum\limits_p \int\limits_{V'} d^3x e^{ip \cdot (x - y)}$$

Note that due to the same periodicity mentioned above, ##\int\limits_{V'} d^3x e^{ip \cdot (x - y)} = 0## when ##p \neq 0##. So

$$\frac{1}{V}\sum\limits_p \int\limits_{V'} d^3x e^{ip \cdot (x - y)}$$
$$= \frac{1}{V} \int\limits_{V'} d^3x e^{i(0) \cdot (x - y)}$$
$$= \frac{1}{V} \int\limits_{V'} d^3x$$
$$= \frac{1}{V}(V)$$
$$= 1$$

Therefore we have shown that

$$\frac{1}{V}\sum\limits_{pq}e^{i(p \cdot x - q \cdot y)}[\hat{a}_p, \hat{a}^{\dagger}_q] = \delta^{(3)}(x - y)$$

Does this make sense? Is there anything that should be corrected or made more rigorous? Thanks for your help!
 
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  • #2
This is just the completeness of the functions ##\exp(\mathrm{i} \vec{p} \cdot \vec{x})## with ##\vec{p} \in \frac{2 \pi}{L} \mathbb{Z}^3## on the square-integrable functions on a cube with periodic boundary conditions. It's proven in many analysis textbooks dealing with Fourier Series.
 

FAQ: Showing that a certain summation is equal to a Dirac delta?

What is the Dirac delta function?

The Dirac delta function, often represented as δ(x), is a generalized function or distribution that is zero everywhere except at x = 0, where it is infinitely high in such a way that its integral over the entire real line is equal to one. It is used in various fields of science and engineering to model an idealized point mass or point charge.

Why is the Dirac delta function important in summation problems?

The Dirac delta function is important in summation problems because it can be used to represent discrete distributions and to simplify the analysis of systems with impulses or point sources. It allows the transformation of discrete sums into continuous integrals, making it easier to work with certain types of problems in physics and engineering.

How can a summation be shown to equal a Dirac delta function?

To show that a summation equals a Dirac delta function, one typically uses properties of the delta function and Fourier transforms. By expressing the summation in terms of an integral and demonstrating that it satisfies the sifting property of the delta function (i.e., it picks out a specific value of a function at a point), one can show that the summation converges to a delta function in the limit.

What are common examples of summations that equal a Dirac delta function?

Common examples include the summation of complex exponentials over integer multiples of a fundamental frequency, such as the Dirac comb or Shah function. Another example is the summation of sinc functions, which in the limit of an infinite number of terms, approximates the Dirac delta function.

What mathematical tools are used to prove that a summation equals a Dirac delta function?

Mathematical tools commonly used include Fourier series and Fourier transforms, properties of orthogonal functions, and the Poisson summation formula. These tools help to convert the summation into a form that can be analyzed and shown to exhibit the characteristics of the Dirac delta function.

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