- #1
Imanbk
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Hello Everyone!
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.
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).
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
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