Deriving the Poincare algebra in scalar field theory

[Pacopag]

Homework Statement

Find the commutators $$[P^\sigma,J^{\mu \nu}]$$

The answer is part of the Poincare algebra
$$[P^\sigma,J^{\mu \nu}]=i(g^{\mu \sigma}P^\nu-g^{\nu \sigma}P^\mu)$$

If someone can convince me that $$\partial_i T^{0\mu} = 0$$, (i.e. the energy-momentum tensor has no explicit spatial dependence) then I got it.
But I'll still post my solution below.

Homework Equations

$$P^\mu = \int d^3x T^{0\mu}$$
$$J^{\nu \sigma}=\int d^3x(x^\nu T^{0\sigma}-x^\sigma T^{0\nu})$$

The Attempt at a Solution

An unsatisfactory explanation of why $$\partial_i T^{0\mu} = 0$$ is that $$T^{\mu \nu}$$ depends on the fields, so there is no $$explicit$$ spacetime dependence, but the fields in turn depend on spacetime. You'll see where this comes in below.

Since $$J^{\nu \sigma}=-J^{\sigma \nu}$$, we only have to consider the commutators
$$[P^0,J^{0i}]$$, $$[P^0,J^{ij}]$$, $$[P^i,J^{0j}]$$, $$[P^i,J^{jk}]$$

Let's take $$[P^i,J^{0j}]$$ as an example. If I can get this one, then I can get them all.
$$[P^i,J^{0j}] = \left[P^i, \int d^3x(x^0 T^{0j}-x^j T^{00}) \right]$$
$$=x^0[P^i,P^j]-[P^i,\int d^3x x^j T^{00}]$$
The first term is zero since momenta commute. For the second term, since $$P^i$$ is the generator of translation in the i-direction, then as in quantum mechanics we get
$$[P^i,J^{0j}] = -i \partial^{'}_i \int d^3x x^j T^{00}$$
$$=-i \left( \delta_{ij} \int d^3x T^{00} + \int d^3x x^j \partial_j T^{00}\right)$$
$$=-i \left( \delta_{ij} P^0 + \int d^3x x^j \partial_j T^{00}\right)$$
If the second term vanishes, then we get the right answer. There are a couple of ways this can happen:
(1) $$\partial_i T^{0\mu} = 0$$, which is what I'd really like to show.
(2) $$T^{00}$$ is for some reason a spatially odd function (i.e. $$T^{00}(x)=-T^{00}(-x)$$.
The latter seems completely unlikely due to the whole Lorentz invariance thing. It just seems weird.
The (1) reason seems better to me. Of course, we do know that $$T^{\mu \nu}$$ is conserved, but in the sense that $$\partial_\mu T^{\mu \nu} = 0$$. But I don't see how this tells us that individual partial derivatives are zero.

 

[Avodyne]

You haven't told us what you're allowed to assume.

You assume
$$[P^i,J^{0j}] = -i \partial'_i \int d^3x \,x^j T^{00}$$
but this is wrong. I'm not sure what the prime is supposed to mean on the derivative; you can't differentiate with respect to a dummy integration variable.

What is true is that, for a field or product of fields $\varphi(x)$ at a spacetime point x,
$$[P^i,\varphi(x)] = +i \partial^i \varphi(x);$$
note the sign. So we have
$$[P^i,J^{0j}] = +i \int d^3x \,x^j \partial^i T^{00}(x)$$
But, again, it's not clear if you're allowed to assume this property of the momentum operator.

 

[Pacopag]

Well that's perfect! Thanks very much.
Integration by parts gives the answer, and I don't have to worry about the other term that I wanted to drop. About assuming the property of P_i. I don't see why not. Why do you think that I should not assume this? If I was really ambitious, I guess I could just follow the derivation in Weinberg I.

 

[Avodyne]

Well it just depends on what you what to take as the starting point. You could, for example, write the stress-energy tensor in terms of the fields, and then use canonical commutation relations for the fields. For scalar fields, this is problem 22.3 in Srednicki. Or, since you're willing to assume $[P^\mu,\varphi(x)] = i \partial^\mu \varphi(x),$ you might also be willing to assume $[J^{\mu\nu},\varphi(x)] = i (x^\mu\partial^\nu-x^\nu\partial^\mu)\varphi(x).$ Then you can consider multiple commutators like $[J^{\mu\nu},[P^\rho,\varphi]]$ and $[P^\rho,[J^{\mu\nu},\varphi]]$, and then use the Jacobi identity to get $[[P^\rho,J^{\mu\nu}],\varphi]$. This gives you $[P^\rho,J^{\mu\nu}]$, up to a possible central charge; problem 2.8 in Srednicki uses this method to get $[J^{\mu\nu},J^{\rho\sigma}].$

 

[Pacopag]

Thank you once again. My first attempt at this problem was to write the stress-energy tensor in terms of the fields and use the canonical commutation relations, but this got really messy. I was only able to get a couple of the commutators this way. Probably, I just keep making sign, or prime/unprime typos. Assuming P and J to act as derivative and angular momentum operator give a much cleaner solution. But I guess you have a point that I should PROVE that P acts as derivative, etc.

 

[Pacopag]

Ya. Srednicki was most helpful. Thanks.

 

[Avodyne]

My first attempt at this problem was to write the stress-energy tensor in terms of the fields and use the canonical commutation relations, but this got really messy.

Yes, this is a huge mess, so the more abstract arguments are useful.