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## Main Question or Discussion Point

Hi,

I am working through Maggiore's QFT book and have a small problem that is really bothering me.

It involves finding the representation of the Poincare algebra generators on fields. I always end up with a minus sign for my representation of a translation on fields compared to Maggiore (and everyone else). Here's what I do that gets the wrong answer:

For the Poincare transformation [itex] x^\mu \rightarrow x'^\mu = x^\mu + a^\mu [/itex], Maggiore says that any field transforms as [itex] \phi'(x') = \phi(x) [/itex] (Eq. 2.106).

The representation of a Poincare group element is [itex] \exp(-i a_\mu P^\mu) [/itex] (Eq. 2.96). I assume this is an operator that acts on a field and gives you the transformed (primed) field.

For the infinitesimal translation by [itex] a^\mu [/itex] and at fixed [itex]x[/itex] (Eq. 2.107):

[tex] \delta_0\phi \equiv \phi'(x) - \phi(x) = \phi(x-a) -\phi(x) = (-a^\mu \partial_\mu \phi) (x) ,[/tex]

and, by definition of what a generator is, this must also be equal to

[tex] \delta_0\phi = \left(\exp(-i a_\mu P^\mu) \phi - \phi \right)(x) = (-i a_\mu P^\mu \phi)(x) .[/tex]

By equating these two expressions for [itex] \delta_0\phi [/itex] I find that the representation of the generator on fields is

[tex] P^\mu = -i \partial^\mu .[/tex]

But this disagrees with the usual result (Eq. 2.110) by a minus sign.

Can anyone tell me what I'm doing wrong? My definition still obeys all the required commutation relations but it gives the wrong answer for the momentum and energy operators (i.e. an extra minus sign).

Any thoughts are much appreciated.

Thanks!

I am working through Maggiore's QFT book and have a small problem that is really bothering me.

It involves finding the representation of the Poincare algebra generators on fields. I always end up with a minus sign for my representation of a translation on fields compared to Maggiore (and everyone else). Here's what I do that gets the wrong answer:

For the Poincare transformation [itex] x^\mu \rightarrow x'^\mu = x^\mu + a^\mu [/itex], Maggiore says that any field transforms as [itex] \phi'(x') = \phi(x) [/itex] (Eq. 2.106).

The representation of a Poincare group element is [itex] \exp(-i a_\mu P^\mu) [/itex] (Eq. 2.96). I assume this is an operator that acts on a field and gives you the transformed (primed) field.

For the infinitesimal translation by [itex] a^\mu [/itex] and at fixed [itex]x[/itex] (Eq. 2.107):

[tex] \delta_0\phi \equiv \phi'(x) - \phi(x) = \phi(x-a) -\phi(x) = (-a^\mu \partial_\mu \phi) (x) ,[/tex]

and, by definition of what a generator is, this must also be equal to

[tex] \delta_0\phi = \left(\exp(-i a_\mu P^\mu) \phi - \phi \right)(x) = (-i a_\mu P^\mu \phi)(x) .[/tex]

By equating these two expressions for [itex] \delta_0\phi [/itex] I find that the representation of the generator on fields is

[tex] P^\mu = -i \partial^\mu .[/tex]

But this disagrees with the usual result (Eq. 2.110) by a minus sign.

Can anyone tell me what I'm doing wrong? My definition still obeys all the required commutation relations but it gives the wrong answer for the momentum and energy operators (i.e. an extra minus sign).

Any thoughts are much appreciated.

Thanks!