Question about Gaussian Intergal Underlying the"Central Identity of Quantum Field Theory"

[Jay R. Yablon]

I am trying to pinpoint the precise origins of the the term d/dJ which
appears as the argument in the potential V(d/dJ) in the so-called
"Central Identity of Quantum Field Theory," given on page 460 of Zee's
QFT in an Nutshell, and especially how one gets from V(x) --> V(d/dJ).

I have outlined my queries about this in a one page file linked below,
which also ties this query together with some of my other recent
queries:

http://jayryablon.files.wordpress.com/2008/04/central-identity.pdf .

Any help is appreciated. If clicking the link above does not work, then
right click and download the file, then open.

Thanks,

Jay.
____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.nycap.rr.com/jry/FermionMass.htm

[neuropulp@yahoo.com.au]

Jay R. Yablon wrote:

> I am trying to pinpoint the precise origins of the the term d/dJ which
> appears as the argument in the potential V(d/dJ) in the so-called
> "Central Identity of Quantum Field Theory," given on page 460 of Zee's
> QFT in an Nutshell, and especially how one gets from V(x) --> V(d/dJ).

Hey J-boy!

Isn't this done in Zee's "A baby problem" on pp42-43 ?
The steps leading up to formulas (3) and (4) ?

LOL with Neuropulp!

[Jay R. Yablon]

<neuropulp@yahoo.com.au> wrote in message
news:417d1152-4cf1-4648-ac8b-ddf20d4e621c@25g2000hsx.googlegroups.com...
> Jay R. Yablon wrote:
>
>> I am trying to pinpoint the precise origins of the the term d/dJ
>> which
>> appears as the argument in the potential V(d/dJ) in the so-called
>> "Central Identity of Quantum Field Theory," given on page 460 of
>> Zee's
>> QFT in an Nutshell, and especially how one gets from V(x) -->
>> V(d/dJ).
>
> Hey J-boy!
>
> Isn't this done in Zee's "A baby problem" on pp42-43 ?
> The steps leading up to formulas (3) and (4) ?
>
> LOL with Neuropulp!
>

Hey pulp-boy! ;-)

Yes it is. I was mulling though exactly that when I first made the
post, because I was looking for a good way to frame that derivation in
the most general way possible, and not be tied to that specific "baby
problem" in Zee. I think I have succeeded in that complete
generalization, which I have laid out in the ~1 page file linked below.
(If left click does not work, then right click to download, then open.)

http://jayryablon.files.wordpress.com/2008/05/zee-baby-problem.pdf

Does this pretty much answer the original question?

Thanks,

Jay.

[Jay R. Yablon]

One other question:

The identity (6) at
http://jayryablon.files.wordpress.com/2008/05/zee-baby-problem.pdf is
based on B<>0 in (4). Does this dependence on non-zero B still apply to
(6)?

In other words: if B=0, then (6) transparently reduces to
($=integral -oo to +oo):

$exp[Ax^2-V(x)] = exp[-V(d/dB)] sqrt(2pi/A) (7)

But, what happens to the V(d/dB), since this only arises from (4) based
on assuming non-zero B. In (7), exp[-V(d/dB)] is only operating on
sqrt(2pi/A), with B=0. So, is (7) above a valid expression, and if so,
am I to conclude from taking a series expansion of exp[-V(d/dB)], then
having it operate on sqrt(2pi/A), that the whole expression

(7) = 0?

Thanks.

Jay.

[Avodyne]

First of all, in (6), you are not allowed to combine the exponentials in the last expression.

Then you are not allowed to set B=0 in (6), because you have a derivative acting on it. Just like, if you write (d/dx)x=1, you are not allowed to set x=0 on the left; if you do, you will get 0=1, which is pretty obviously wrong.