- #1
user1139
- 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?