- #1

Haorong Wu

- 415

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- TL;DR Summary
- Why the coefficients in expansions of a vector potential are different in different papers?

I have seen two expansions of a vector potential,

$$\mathbf A=\sum_\sigma \int \frac{d^3k}{(16 \pi^3 |\mathbf k|)^{1/2}} [\epsilon_\sigma(\mathbf k) \alpha_\sigma (\mathbf k) e^{i \mathbf k \cdot \mathbf x}+c.c.],$$

and

$$\mathbf A=\sum_\sigma \int \frac{d^3k}{ (2 \pi)^3(2 |\mathbf k|)^{1/2}} [\epsilon_\sigma(\mathbf k) \alpha_\sigma (\mathbf k) e^{i \mathbf k \cdot \mathbf x}+c.c.],$$

where ##\epsilon_\sigma## are polarizations, ##\alpha_\sigma (\mathbf k)## are amplitudes.

My problem is the two coefficients ## (2 \pi)^{3/2}## and ## (2 \pi)^3## in these two expressions, respectively. Why they are different?

I do not remember where I have read something about it, that it is related to the definition of Fourier transform.

Thanks for any hints.

$$\mathbf A=\sum_\sigma \int \frac{d^3k}{(16 \pi^3 |\mathbf k|)^{1/2}} [\epsilon_\sigma(\mathbf k) \alpha_\sigma (\mathbf k) e^{i \mathbf k \cdot \mathbf x}+c.c.],$$

and

$$\mathbf A=\sum_\sigma \int \frac{d^3k}{ (2 \pi)^3(2 |\mathbf k|)^{1/2}} [\epsilon_\sigma(\mathbf k) \alpha_\sigma (\mathbf k) e^{i \mathbf k \cdot \mathbf x}+c.c.],$$

where ##\epsilon_\sigma## are polarizations, ##\alpha_\sigma (\mathbf k)## are amplitudes.

My problem is the two coefficients ## (2 \pi)^{3/2}## and ## (2 \pi)^3## in these two expressions, respectively. Why they are different?

I do not remember where I have read something about it, that it is related to the definition of Fourier transform.

Thanks for any hints.