In the canonical quantisation of a free scalar field ##\phi## one typical constructs a mode expansion of the corresponding field operator ##\hat{\phi}## as a solution to the Klein-Gordon equation, $$\hat{\phi}(t,\mathbf{x})=\int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{\sqrt{\omega(\mathbf{k})}}\left(\hat{a}(\mathbf{k})e^{-ik\cdot x}+\hat{a}^{\dagger}(\mathbf{k})e^{ik\cdot x}\right)$$ where ##k\cdot x:=k_{\mu}x^{\mu}=k_{0}t-\mathbf{k}\cdot\mathbf{x}##.(adsbygoogle = window.adsbygoogle || []).push({});

What I'm unsure about is what exactly is the term "mode" referring to? Is it the whole component ##\hat{a}(\mathbf{k})e^{-ik\cdot x}##, or is it simply ##\hat{a}(\mathbf{k})##?

Furthermore, one has that the frequency ##\omega(\mathbf{k})## satisfies the equation $$\omega^{2}(\mathbf{k})=\mathbf{k}^{2}+m^{2}$$ but is ##\omega(\mathbf{k})## the frequency of each mode or of the field ##\hat{\phi}## itself? I think it's the frequency of each mode (with momentum ##\mathbf{k}##, but then, given my first question, I am unsure whether this is the frequency corresponding to ##\hat{a}(\mathbf{k})e^{-ik\cdot x}##, or is it simply ##\hat{a}(\mathbf{k})##?!

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A A question about the mode expansion of a free scalar field

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads - question mode expansion | Date |
---|---|

I Question about charge | Mar 14, 2018 |

A Are vacuum EM modes circularly polarized according to QED? | Mar 14, 2018 |

B Questions about Identical Particles | Mar 12, 2018 |

I Some (unrelated) questions about the measurement problem | Mar 9, 2018 |

B Questions about parity | Mar 8, 2018 |

**Physics Forums - The Fusion of Science and Community**