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I am reading the "Quantum Field Theory lectures of Sidney Coleman". In the first chapter (subchapter 1.2), the author talks about translation invariance. In particular, he states that $U(a)=e^{iP\cdot a}$ is the translation operator, with $$O(x+a)=U(a)O(x)U^{\dagger}(a)$$ where $O$ is any other operator depending on $x^{\mu}$. Then, the author reduces the translations to space translations and therefore $$e^{-i\textbf{P}\cdot\textbf{a}}O(\textbf{x})e^{i\textbf{P}\cdot\textbf{a}}= O(\textbf{x}+\textbf{a}).$$ But then, the author states that only operators localized in space transform according to this rule, giving a counter-example the operator $\hat{\textbf{q}}$ $$e^{i\textbf{P}\cdot\textbf{a}}\hat{\textbf{q}}e^{-i\textbf{P}\cdot\textbf{a}}= \hat{\textbf{q}}+\textbf{a}.$$

I have three questions:

  1. What is the underlying reason for only operators localized in space being transformed under this rule?

  2. Why is $\hat{\textbf{q}}$ an operator that is not localized in space?

  3. How do the operators that are not localized in space transform?

Any help will be appreciated.

Qmechanic
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schris38
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1 Answers1

3

  1. Consider the following example of an operator "localized in space": The orthogonal projection operator $$\Pi_\delta(x) = \int\limits_{x-\delta}^{x+\delta} \! \! \! dq \, |q \rangle \langle q |, \quad {\rm where} \quad Q | q \rangle = q | q \rangle, $$ corresponds to the observable of finding the particle in the interval $[x-\delta, x+\delta]$ centered around $x$. I have assumed only a single spatial dimension for simplicity (the generalization to three spatial dimensions is trivial). $Q$ denotes the position operator. Using $e^{-iPa} |q \rangle = |q+a \rangle$ (c.f. second line of eq. (1.39) in the book), one obtains $$e^{-iPa} \Pi_\delta(x) e^{iPa}= \! \int\limits_{x-\delta}^{x+\delta} \! \! \! dq \, |q+a \rangle \langle q+a| =\! \! \!\! \int\limits_{x+a-\delta}^{x+a+\delta} \! \! \! \! dq \, | q\rangle \langle q | = \Pi_\delta(x+a)$$ in accordance with the third line of eq. (1.39).

  2. $Q$ itself does not single out a specific point in space. You could take e.g. the somewhat pathological "operator" $\delta(Q-x)= |x\rangle \langle x|$, corresponding to the limiting case of $\Pi_\delta(x)/2 \delta$ for $\delta \to 0$.

  3. The canonical commutation relation $[ Q, P] = i \mathbf{1}$ implies $e^{-iPa} f(Q) e^{iPa} = f(Q-a)$ and $e^{-iPa} g(P) e^{iPa}= g(P)$.

Hyperon
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