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Let's start with the quantised field operators $\phi(\mathbf{x})$ and $\pi(\mathbf{x})$, defined as usual, and the relativistic normalisation $|\mathbf{p}\rangle = \sqrt{2E_p}a_p^{\dagger}$.

I know that : $$\phi(\mathbf{x}) \left|0\right> = \int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}e^{-i\mathbf{p} \cdot \mathbf{x}} \left|\mathbf{ p}\right> , $$ which can be interpreted as $\phi(\mathbf{x})$ creating a particle at $\mathbf{x}$, according to Peskin and Schroeder, page 24.

How can we interpret $$\pi(\mathbf{x}) \left|0\right> = \int \frac{d^3p}{(2\pi)^3}\frac{-i}{2}e^{-i\mathbf{p} \cdot \mathbf{x}} \left|\mathbf{ p}\right> ?$$

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OK in the end I think I got somewhere on my own. I am just posting this here in case it is useful for someone else.

The reason $\phi(\mathbf{x}) \left|0\right>$ can be interpreted as creating a particle at position $\mathbf{x}$ is because: $$\phi(\mathbf{x}) \left|0\right> = \int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}e^{-i\mathbf{p} \cdot \mathbf{x}} \left|\mathbf{ p}\right> ,$$ and if we take $\langle \mathbf{x}|\mathbf{p} \rangle = $$\langle {0}|\phi(\mathbf{x}) \left|\mathbf{p}\right>$, we get $e^{i\mathbf{p}\cdot\mathbf{x}}$ which is analogous to the quantum mechanical momentum eignestate in the position representation.

If we look at $\langle {0}|\pi(\mathbf{x}) \left|\mathbf{p}\right>$, we get $-i\,E_{\mathbf{p}}\,e^{i\mathbf{p}\cdot\mathbf{x}}$.

Now recall that $\pi(\mathbf{x}) \equiv \frac{\partial \phi(\mathbf{x})}{\partial t} $ for the Klein-Gordon Lagrangian. We can extend $e^{i\mathbf{p}\cdot\mathbf{x}} $ to $e^{-ip_{\mu}\,x^{\mu}}$ in order to include time (using the $-+++$ signature). So taking the derivative of $\langle {0}|\phi(\mathbf{x}) \left|\mathbf{p}\right> = e^{i\mathbf{p}\cdot\mathbf{x}}$ with respect to time brings down a factor of $-i\,E_{\mathbf{p}}$.

Basically what I am saying that it all makes sense, $\pi$ is still just the derivative of $\phi$. There is nothing to interpret really.

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  • $\begingroup$ Why can we make this extension? At the end of the day, aren't they different expressions? $\endgroup$
    – Jbag1212
    Oct 6 '21 at 21:13
  • $\begingroup$ Sorry which “extension”? $\endgroup$
    – SuperCiocia
    Oct 6 '21 at 22:11
  • $\begingroup$ "We can extend $e^{ip\cdot x} to e^{-ip_{\mu}x^{\mu}}$." $\endgroup$
    – Jbag1212
    Oct 7 '21 at 2:04
  • $\begingroup$ if you didn't include the time part the time derivative would be 0. Usually the time bit is neglected cause it's just a time dependent phase factor. $\endgroup$
    – SuperCiocia
    Oct 7 '21 at 2:48
  • $\begingroup$ From what I understand, the time part isn't neglected "because it's just a time dependent phase factor." The definition of $\phi(x)$ has no time dependence in the Schrodinger interpretation. The field itself, though, has a time dependence ? ... $\endgroup$
    – Jbag1212
    Oct 7 '21 at 4:54

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