- #1

- 24

- 0

I have a problem I am solving through a self study project from Lowell Brown's book entitled: Quantum Field Theory". It is a math question (basically) on recursion relations.

## Homework Statement

The variational definition gives us the relation:

det[1-λK] = exp{tr ln[1-λK]}.

Where λ is a "small" number and K is the kernel.

The variational definition shows us that λ needs to be small in order for the power series of ln[1-λK] to converge. But on the other hand, one can show that the power series:

d(λ) = det[1-λK] = Ʃ(n=0 to inf) d_(n)λ^(n)

Always converges provided that K (the kernel) is sufficiently "well behaved".

**Relavant question:**

What we are are asked to do is to plug the above power series into the following differential eqaution:

d/dλ d(λ) = -d(λ)*Ʃ(m=0 to inf) [λ^(m)*trK^(m+1)]

to find a recursion relation relating d_(n+1), d_(n), and trK^(n+1).

## The Attempt at a Solution

I showed why the differential equation has the form it has by integrating over λ and using boundary condition det1=1 (so that ln(det1)=0) to give us back the variational definition.

To solve for the recursion relation I tried several approaches. My closest approach was writing out the sum over n as:

Ʃ(n=0 to inf) [(n+1)*d_(n+1)*λ^(n)+d_(n)*λ^(n+m)*trK^(n+m)] = 0.

I tried to get the sum with all variables λ factored out so we get the sum in the following form:

Ʃ(...)*λ^(some power) = 0

so that I can use the theorem which states that: a polynomial is identically zero if and only if all of its coefficents are zero but I can't factor out λ^(m) from the second term.

I also supposed that for n ≠ m we have zero contributions to get:

Ʃ(n=0 to inf) [(n+1)*d_(n+1)+d_(n)*λ^(2)*trK^(n+m)]*λ^(n) = 0.

I have no reason to do this (yet) but I was just playing around to see if I can get the trK^(n+1) term which was asked for in the question.

I hope I gave enough info about this question. If not please let me know! I tried hard at this question with no solution. I was taught to solve recursion relations for terms in powers of λ which differ by constants such as λ^(n), λ^(n-1), λ^(n-2), etc. but not terms which differ by powers of λ of another summation variable (m).

Thanks a lot for help on this!

Imankb