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- Homework Statement
- Show that the commutator for field operators is zero for spacelike separations

- Relevant Equations
- $$\hat \phi(x) = \int \frac{d^3p}{(2\pi)^{3/2}(2E_p)^{1/2}}(\hat{a}_{\vec p}exp(-ip\cdot x)+\hat{a}^{\dagger}_{\vec p}exp(ip\cdot x))$$

I got as far as:

$$[\hat \phi(x), \hat \phi(y) ] = \int \frac{d^3p}{(2\pi)^{3}(2E_p)}(\exp(-ip.(x-y) - \exp(-ip.(y-x))$$

Then I simplified the problem by taking one of the four-vectors to be the origin:

$$[\hat \phi(0), \hat \phi(y) ] = \int \frac{d^3p}{(2\pi)^{3}(2E_p)}(\exp(ip.y) - \exp(-ip.y))$$

We need to show that this is zero for spacelike ##y##. But, I can't see how the spacelike condition is relevant. The hint in the book is that we can change ##-y## to ##y## in the second term. If ##p## was a three-vector I can see this, but the zeroth component of ##p## is the positive ##E_p##. And, if it's a simple ##p## substitution, then that should work for any ##y##, spacelike or otherwise.

Any help would be appreciated.

$$[\hat \phi(x), \hat \phi(y) ] = \int \frac{d^3p}{(2\pi)^{3}(2E_p)}(\exp(-ip.(x-y) - \exp(-ip.(y-x))$$

Then I simplified the problem by taking one of the four-vectors to be the origin:

$$[\hat \phi(0), \hat \phi(y) ] = \int \frac{d^3p}{(2\pi)^{3}(2E_p)}(\exp(ip.y) - \exp(-ip.y))$$

We need to show that this is zero for spacelike ##y##. But, I can't see how the spacelike condition is relevant. The hint in the book is that we can change ##-y## to ##y## in the second term. If ##p## was a three-vector I can see this, but the zeroth component of ##p## is the positive ##E_p##. And, if it's a simple ##p## substitution, then that should work for any ##y##, spacelike or otherwise.

Any help would be appreciated.

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