- #1
- 3,971
- 328
Hi guys, I'm not sure where to put this question, so I'll just put it here. If a mod knows of a better place, just point me to it, thanks.
I'm looking at the functional differentiation equation:
$$\left.\frac{dF[f+\tau h]}{d\tau}\right|_{\tau=0}\equiv \int\frac{\delta F[f]}{\delta f(x)}h(x)dx$$
I understand how this equation works (relatively well) for functionals of the form:
$$F[f]=\int \mathscr{L}(f,f',f'',...)dx$$
But there is another kind of functional that we often see in quantum field theory, which looks more like:
$$F[f]=\mathcal{N}\int \left([e^{i\int \mathcal{L}(\phi(x),\phi'(x),...)f(x)dx}\right)\mathscr{D}\phi$$
Here, we are integrating over functions ##\phi## rather than over the real line. Let's assume for the purposes of the discussion that this latter functional, taken as a whole (with the normalization constant), is well defined.
Superficially at least, this second functional seems to be a different beast than the first, since we are integrating over functions rather than a real variable. However, this functional is a functional in that it takes functions and maps them to real numbers. In this second case of functionals, I'm not sure how to apply the rule for functional differentiation. Does it still work?
Looking at the simplest functional of the second type, my analysis is as follows. Let's consider:
$$F[f]=\mathcal{N}\int \left(e^{i\int \phi(x)f(x)dx}\right)\mathscr{D}\phi$$
Now, let's construct ##F[f+\tau h]##:
$$F[f+\tau h]=\mathcal{N}\int \left(e^{i\int[f(x)+\tau h(x)]\phi(x)dx}\right)\mathscr{D}\phi$$
Taking the derivative ##d/d\tau## and the limit ##\tau\rightarrow 0##:
$$\left.\frac{dF[f+\tau h]}{d\tau}\right|_{\tau=0}=\mathcal{N}\int\left(e^{i\int \phi(x)f(x)dx}i\int h(t)\phi(t)dt\right)\mathscr{D}\phi$$
The problem I am left with here is that we are not left with an expression of the form in the first equation. We have a extra ##\mathcal{N}\int\mathscr{D}\phi## outside of everything. Taking the assumption that we can switch orders of integrals over the dummy variable t and the functions ##\phi## (which I am very dubious about), and looking at the first expression above, we might guess:
$$\frac{\delta F[f]}{\delta f(x)}=i\mathcal{N}\int\left(\phi(x)e^{i\int \phi(t)f(t)dt}\right)\mathscr{D}\phi$$
Even if we took this expression at face value, something seems to be very amiss since usually ##\delta F/\delta f## is a function, whereas here we just have a number (due to the integration over all ##\phi##)...
Some help would be appreciated @_@
I'm looking at the functional differentiation equation:
$$\left.\frac{dF[f+\tau h]}{d\tau}\right|_{\tau=0}\equiv \int\frac{\delta F[f]}{\delta f(x)}h(x)dx$$
I understand how this equation works (relatively well) for functionals of the form:
$$F[f]=\int \mathscr{L}(f,f',f'',...)dx$$
But there is another kind of functional that we often see in quantum field theory, which looks more like:
$$F[f]=\mathcal{N}\int \left([e^{i\int \mathcal{L}(\phi(x),\phi'(x),...)f(x)dx}\right)\mathscr{D}\phi$$
Here, we are integrating over functions ##\phi## rather than over the real line. Let's assume for the purposes of the discussion that this latter functional, taken as a whole (with the normalization constant), is well defined.
Superficially at least, this second functional seems to be a different beast than the first, since we are integrating over functions rather than a real variable. However, this functional is a functional in that it takes functions and maps them to real numbers. In this second case of functionals, I'm not sure how to apply the rule for functional differentiation. Does it still work?
Looking at the simplest functional of the second type, my analysis is as follows. Let's consider:
$$F[f]=\mathcal{N}\int \left(e^{i\int \phi(x)f(x)dx}\right)\mathscr{D}\phi$$
Now, let's construct ##F[f+\tau h]##:
$$F[f+\tau h]=\mathcal{N}\int \left(e^{i\int[f(x)+\tau h(x)]\phi(x)dx}\right)\mathscr{D}\phi$$
Taking the derivative ##d/d\tau## and the limit ##\tau\rightarrow 0##:
$$\left.\frac{dF[f+\tau h]}{d\tau}\right|_{\tau=0}=\mathcal{N}\int\left(e^{i\int \phi(x)f(x)dx}i\int h(t)\phi(t)dt\right)\mathscr{D}\phi$$
The problem I am left with here is that we are not left with an expression of the form in the first equation. We have a extra ##\mathcal{N}\int\mathscr{D}\phi## outside of everything. Taking the assumption that we can switch orders of integrals over the dummy variable t and the functions ##\phi## (which I am very dubious about), and looking at the first expression above, we might guess:
$$\frac{\delta F[f]}{\delta f(x)}=i\mathcal{N}\int\left(\phi(x)e^{i\int \phi(t)f(t)dt}\right)\mathscr{D}\phi$$
Even if we took this expression at face value, something seems to be very amiss since usually ##\delta F/\delta f## is a function, whereas here we just have a number (due to the integration over all ##\phi##)...
Some help would be appreciated @_@
Last edited: