- #1

- 72

- 8

- Homework Statement
- The statement is in title

- Relevant Equations
- The equations are given below

Given the commutation relation

$$\left[\phi\left(t,\vec{x}\right),\pi\left(t,\vec{x}'\right)\right]=i\delta^{n-1}\left(\vec{x}-\vec{x}'\right)$$

and define the Fourier transform as

$$\tilde{\phi}(t,\vec{k})=\frac{1}{\sqrt{L^{n-1}}}\int_{\,0}^{\,L}\phi(t,\vec{x})e^{-i\vec{k}\cdot\vec{x}}\,\mathrm{d}^{n-1}\vec{x}$$

$$\tilde{\pi}(t,\vec{k})=\frac{1}{\sqrt{L^{n-1}}}\int_{\,0}^{\,L}\pi(t,\vec{x})e^{-i\vec{k}\cdot\vec{x}}\,\mathrm{d}^{n-1}\vec{x}$$

Is it then correct to say the following?

$$\left[\tilde{\phi}(\vec{k}),\tilde{\pi}(\vec{k}')\right]=i\delta_{\vec{k},-\vec{k}'}=i\delta_{-\vec{k},\vec{k}'}$$

i.e. can I use both Kronecker deltas interchangeably?

$$\left[\phi\left(t,\vec{x}\right),\pi\left(t,\vec{x}'\right)\right]=i\delta^{n-1}\left(\vec{x}-\vec{x}'\right)$$

and define the Fourier transform as

$$\tilde{\phi}(t,\vec{k})=\frac{1}{\sqrt{L^{n-1}}}\int_{\,0}^{\,L}\phi(t,\vec{x})e^{-i\vec{k}\cdot\vec{x}}\,\mathrm{d}^{n-1}\vec{x}$$

$$\tilde{\pi}(t,\vec{k})=\frac{1}{\sqrt{L^{n-1}}}\int_{\,0}^{\,L}\pi(t,\vec{x})e^{-i\vec{k}\cdot\vec{x}}\,\mathrm{d}^{n-1}\vec{x}$$

Is it then correct to say the following?

$$\left[\tilde{\phi}(\vec{k}),\tilde{\pi}(\vec{k}')\right]=i\delta_{\vec{k},-\vec{k}'}=i\delta_{-\vec{k},\vec{k}'}$$

i.e. can I use both Kronecker deltas interchangeably?