Help Solve QFT Questions from Jean Zinn Justin P.22

[alfredblase]

Three related problems in this one.

1. Show that: $$\frac{\partial}{\partial t} \int_{t'}^t d\tau V \left (\mathbf{q'}+ \frac{\tau-t'}{t-t'} \mathbf{q-q'},\tau \right )=V \left (\mathbf{q},t \right ) + \int_{t'}^t d\tau \frac{\partial}{\partial t}V \left (\mathbf{q'}+ \frac{\tau-t'}{t-t'}\mathbf{q-q'},\tau \right )$$

where $$\frac{\partial \tau}{\partial t}\neq 0$$.

2. Whats is $$A_i$$?

3. Show that $$\frac{\partial V}{\partial A_i}=\nabla_{q_i} V\frac{t-t'}{\tau-t'}$$

They are probelms that I need to solve in reading a QFT text (Jean Zinn Justin p 22)

 

[D H]

Dig way back to the fundamental theorem of calculus.

 

[dextercioby]

What book by Zinn Justin ? The one on QFT and critical phenomena?

Daniel.

 

[alfredblase]

Dig way back to the fundamental theorem of calculus.

I have done but I could only get the V(q,t) part out of that.. been working on these problems for 3 days :/

@ dextercioby: yes that one

 

[alfredblase]

pretty please?

 

[D H]

Use this formulation of the fundamental theorem of calculus,

$$\frac d{dt} \int_{a(t)}^{b(t)} f(t,\tau)d\tau = \int_{a(t)}^{b(t)} \frac d{dt} f(t,\tau) d\tau + (\frac d{dt}a(t))f(t,a(t)) - (\frac d{dt}b(t))f(t,b(t))$$

which is also known as LaGrange's Formula.

 

[alfredblase]

Use this formulation of the fundamental theorem of calculus,

$$\frac d{dt} \int_{a(t)}^{b(t)} f(t,\tau)d\tau = \int_{a(t)}^{b(t)} \frac d{dt} f(t,\tau) d\tau + (\frac d{dt}a(t))f(t,a(t)) - (\frac d{dt}b(t))f(t,b(t))$$

which is also known as LaGrange's Formula.

Thank you DH! ^^

 

[dextercioby]

Use this formulation of the fundamental theorem of calculus,

$$\frac d{dt} \int_{a(t)}^{b(t)} f(t,\tau)d\tau = \int_{a(t)}^{b(t)} \frac d{dt} f(t,\tau) d\tau + (\frac d{dt}a(t))f(t,a(t)) - (\frac d{dt}b(t))f(t,b(t))$$

which is also known as LaGrange's Formula.

I think you mean the French mathematician (Joseph Louis) Lagrange.

Daniel.

 

[alfredblase]

hi again. OK so it seems most of my problems with the book I'm reading come from not knowing certain bits of math, whether it be advanced linear algebra, unusual calculus methods (I class the above as unusual :P )

The book I was recommended was Webber's Mathematical Method's for Physicists. Great book no doubt, but so far it hasn't contained any of the relevant bits that have been/are missing in my knowledge.

Can anyone recommend a book/books that contains the Lagrange formula posted above, linear algebra, and in general most if not all of the maths required to read QFT from a statistical physics approach, such as is presented in Jean Zinn Justin's QFT and critical phenomena? Comprehensive, pedagogical and relevant coverage of the necessary maths is what I'm looking for here :D (a lot to ask i guess :P )

regards, alf

 

[D H]

This is basic stuff. Does your book cover integral equations (Fredholm equations, Green's theorem, ...)? This relation is essential in the treatment of integral equations. However, in the section on integral equations in Hildebrand, "Methods of Applied Mathematics", the author just identifies this is a "known formula" and proceeds from there.

BTW, the equation is more correctly written as
$$\frac d{dt} \int_{a(t)}^{b(t)} f(t,\tau)d\tau = \int_{a(t)}^{b(t)} \frac {\partial f(t,\tau)}{\partial t} d\tau \,+\, \frac {da(t)}{dt}f(t,a(t)) \,-\, \frac {db(t)}{dt}f(t,b(t))$$

 

[alfredblase]

This is basic stuff.

Depends who you ask I guess I've never seen the equation for the fundamental theorem of calculus (FTC) for integrands with 2 variables before in my life.

Does your book cover integral equations (Fredholm equations, Green's theorem, ...)?

yes.. so? It doesn't have the FTC for integrands with two variables (or one variable for that matter ) :/

This relation is essential in the treatment of integral equations.

yes.

However, in the section on integral equations in Hildebrand, "Methods of Applied Mathematics", the author just identifies this is a "known formula" and proceeds from there.

Do you know any book where this formula is shown to be true for integrands with 2 variables? (of course I trust it and yourself but you understand that i want to understand :D ) You see I can show the FTC to be true when the integrand is a function of only one variable but obviously can't for the form you supplied where it was a funciton of two variables.. Then again Hildebrand's seems like a better book for me since it actually had that formula. Thanks I'll look it up.

BTW, the equation is more correctly written as
$$\frac d{dt} \int_{a(t)}^{b(t)} f(t,\tau)d\tau = \int_{a(t)}^{b(t)} \frac {\partial f(t,\tau)}{\partial t} d\tau \,+\, \frac {da(t)}{dt}f(t,a(t)) \,-\, \frac {db(t)}{dt}f(t,b(t))$$

ok. thank you :)