# Derive the representation of the momentum acting on a field

1. Apr 25, 2016

### D_Cross

1. The problem statement, all variables and given/known data
consider the space-time transformation of translation

xμ → x'μ = xμ + aμ

where xμ is a point in space-time and aμis a constant 4-vector. Assuming translations are generated by the operator U=e-iPμaμ acting on fields Φ(x), derive the representation of Pμ on the field Φ(x).

2. Relevant equations

3. The attempt at a solution
From looking at it I would assume it would look something like i∂μΦ
Though if this is the case I'm not entirely sure why.

2. Apr 26, 2016

### CAF123

An abstract element of the translation group looks like $\exp(-ia^{\mu}P_{\mu})$. When this acts on fields, $P_{\mu}$ indeed manifests itself as a derivative and we say the representation of the generator of translations on space time fields is a differential operator. To see this, start by shifting the space time point $x'^{\mu} \rightarrow x^{\mu} + a^{\mu}$, where $a^{\mu}$ is an infinitesimal shift and look at the corresponding transformation of the field from $\phi(x)$ to $\phi'(x)$.

3. Apr 27, 2016

### D_Cross

Thank you for the reply. I think I almost have it, so under the infinitesimal transformation x'μ=xμ+aμ, we have δΦ(x)=Φ'(x)-Φ(x). Rearranging the first equation gives xμ=x'μ-aμ. so we can rewrite δΦ=Φ'(x-a)-Φ(x), expanding this out gives δΦ=-aμμΦ(x).

Now we also have Φ'(x)=e-iPμaμΦ(x), but I don't know what to do next. I've seen in a book that it should be Φ'(x'-a)=e-iPμ(-aμ)Φ'(x'), but I don't see how to get to this.

I'm sorry if I'm actually going in completely the wrong direction.

4. Apr 28, 2016

### CAF123

It's correct - given $\phi'(x) = \exp(-i P_{\mu} a^{\mu}) \phi(x)$, expand the rhs for an infinitesimal shift $a^{\mu}$ and compare with your derivation of $\delta \phi = \phi'(x)-\phi(x)$.