Quantum theory, show variation of S zero, integrate by parts

[binbagsss]

Homework Statement

Hi,

Please see attached.

I am trying to show the second equality , expressing all as a total derivative (I can then show that $\delta S =$)

Homework Equations

See above

The Attempt at a Solution

So the $m$ term is pretty obvious, simply using the chain rule.

It is the first term I am stuck on. So looking by the sign, it looks like we have done integration by parts twice.

My working so far is to go by parts initially as:

$w=\partial^{u}\phi$
$\partial w = \partial_{v}\partial^{u} \phi$
$\partial z = \partial_{u}\partial_{v} \phi$
$z= \partial_{u} \phi$

to get, since we are allowed to assume vanishing of the field $\phi$ at inifnity:

$- \int \partial_{u} \phi ( \partial_{v} \partial^{u} \phi )$

I am now stuck of what to do, I can't see a move that will get the desired expression for a choice of integration by parts, which makes me question whether this was the correct first move to make.

Many thanks in advance.

 

[Orodruin]

Is $a$ constant? It is just the same type of relation as the mass term. Just switch the order of the derivatives.

 

[binbagsss]

Is $a$ constant? It is just the same type of relation as the mass term. Just switch the order of the derivatives.

The $\partial_{u}$ is not hitting the $\partial^{u} \phi$ though?

I have the first line as $a^{v}\partial_{u}\partial_{v}\phi \partial^{u} \phi = a^{v}(\partial^2 _{uv} \phi)(\partial^{u} \phi )$,
whereas I have the bottom line as $(a^{v}\partial^{2}_{uv}\phi)(\partial_{v}\partial^{u} \phi)$..

thanks

 

[Orodruin]

The $\partial_{u}$ is not hitting the $\partial^{u} \phi$ though?

I have the first line as $a^{v}\partial_{u}\partial_{v}\phi \partial^{u} \phi = a^{v}(\partial^2 _{uv} \phi)(\partial^{u} \phi )$,
whereas I have the bottom line as $(a^{v}\partial^{2}_{uv}\phi)(\partial_{v}\partial^{u} \phi)$..

thanks

No this is not the bottom line - it does not contain four derivatives ... Just use the same argument as for the mass term, or more generally $f\, df/dx = 0.5 df^2/dx$.

 

[binbagsss]

No this is not the bottom line - it does not contain four derivatives ... Just use the same argument as for the mass term, or more generally $f\, df/dx = 0.5 df^2/dx$.

erm, so I should have used the product rule on the bottom line?
Which would give arise to two terms, whereas the top line has one term..?

 

[Orodruin]

erm, so I should have used the product rule on the bottom line?
Which would give arise to two terms, whereas the top line has one term..?

I don't get your meaning. Both the bottom and top lines have two terms. One from the mass and one from the kinetic term.

 

[binbagsss]

I don't get your meaning. Both the bottom and top lines have two terms. One from the mass and one from the kinetic term.

apologies, corresponding to the kinetic term, I am only referring to here.

 

[Orodruin]

Is your problem that you do not see that
$$
(\partial_a \partial_b \phi) (\partial^a \phi) = \frac{1}{2} \partial_b [(\partial_a \phi)(\partial^a\phi)] ?
$$

 

[Orodruin]

apologies, corresponding to the kinetic term, I am only referring to here.

The kinetic term has one term only in both lines in your OP.

 

[binbagsss]

The kinetic term has one term only in both lines in your OP.

oh right got it !
by two terms I was referring to the product rule, but then you sea-saw giving the factor of $1/2$, thank you !

 

[binbagsss]

[QUOTE="binbagsss, post: 5663557, member: 252335"

My working so far is to go by parts initially as:

$w=\partial^{u}\phi$
$\partial w = \partial_{v}\partial^{u} \phi$
$\partial z = \partial_{u}\partial_{v} \phi$
$z= \partial_{u} \phi$

[/QUOTE]

May I ask, in general how should you approach integration by parts when four -derivaitves are involved.
Should I choose a different index like $\partial_a$ than u and v which are already used in the problem
Am I okay to use $\partial_v$, but then I must be consistent throughout? i.e. not change to $\partial_u$
I'm a bit confused..
Many thanks !

 

[Orodruin]

There is nothing different from the usual integration by parts, you just do it in the direction of the given index. In other words, $\partial_\mu$ is really a lot of different terms. You can do partial integration with respect to each direction separately. In reality, it really is just the divergence theorem.