Sterman's QFT - 2.7b (on functional derivatives)

[ddd123]

I've been trying to fill in my mathematical blanks of things I just took as dogma before. Especially, not having a background in functional analysis, the functional derivatives often seem to me mumbo jumbo whenever things go beyond the "definition for physicists".

In particular I tried looking everywhere to make sense of this passage from Sterman's:

I understand everything here but that last $\equiv$ . Especially what makes it look total whatever-y to me is the indices i, j on the RHS. Why does the momentum operator have j and the field i on the numerator, and the field j on the denominator? This could've been totally different and I would've equally said "mmh okay I suppose", something which worries me every time it happens.

If you think it would be beneficial, on top on the answer to this specific part, a primer of functional analysis / derivatives / ? would be appreciated.

Thanks.

 

[fzero]

The $j$ in the numerator is probably just a typo given that $f_i(x^0)$ was defined before to have $i$ on both arguments. As far as references, the wikipedia entry on "functional derivative" gives the formal mathematical definition of the derivative and develops it briefly. If you really need a longer discussion you might consult the references by Courant and Hilbert and Gel'fand that are listed there.

 

[ddd123]

The $j$ in the numerator is probably just a typo given that $f_i(x^0)$ was defined before to have $i$ on both arguments. As far as references, the wikipedia entry on "functional derivative" gives the formal mathematical definition of the derivative and develops it briefly. If you really need a longer discussion you might consult the references by Courant and Hilbert and Gel'fand that are listed there.

Apart from the supposed typo, why is that a functional derivative? I only see a delta * a derivative, and the wiki page says nothing about this form.

 

[fzero]

Apart from the supposed typo, why is that a functional derivative? I only see a delta * a derivative, and the wiki page says nothing about this form.

Yes, I should have said a bit more. The physicist version of the functional derivative is slightly different, but I think consistent with the mathematician's version. In the formal definition one introduces a test function $\phi(x)$ that defines the "direction" in which the functional derivative is taken. In the physicist's definition this test function is morally almost always a delta distribution. I can't give a rigorous derivation, so I do not want to claim too much about it.

Instead, you can derive the physicist's version as a formal process where you can compute the derivative of functionals,
$$
F[\phi_i] = \int f(\phi_i(\mathbf{x}), \nabla \phi_i(\mathbf{x}) ) d \mathbf{x},
$$
by a process in which we use the chain rule (and integration by parts) in order to reduce operations to a fundamental definition
$$ \frac{\delta \phi_i(\mathbf{x})}{\delta \phi_j(\mathbf{y})} = \delta^j_i \delta(\mathbf{x}-\mathbf{y}).$$
The presence of the delta distribution also means that, while the functional expression includes an integral over the domain of the functions $\phi_i$, in the physicist's functional derivative, the integral is gone, having been taken care of by the delta.

Sterman is trying to justify the form of the functional derivative in a slightly different way, which I suppose would be more clearly related to the mathematician's definition. You should be able to verify that the rule I suggested above is completely consistent with Sterman's treatment and leads to the Euler-Lagrange equations, etc.