- #1
Sangoku
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given the functional integral with 'g' small coupling constant
[tex] \int \mathcal D [\phi]exp(iS_{0}[\phi]+\int d^{4}x \phi ^{k}) [/tex]
so k >2 then could we use a similar 'Functional determinant approach' to this Feynman integral ?? in the sense that the integral above will be equal to
[tex] Cx(Det[\partial _{\mu}+k\phi^{k-1})^{-1/b} [/tex]
where C and 'b' are constant and the determinant is defined as an infinite product of eigenvalues
[tex] \zeta (s) \Gamma(s)= \int_{0}^{\infty}dt t^{s-1}Tr[e^{-sH_{0}-sgV_{int}] [/tex]
where the index '0' means the quadratic part of our Hamiltonian /action and so on..
since 'g' is small then we can express for every eigenvalue:
[tex] \lambda _{n} =\lambda _{0}+g\lambda _{n}^{1} +g^{2}\lambda_{n}^{2}+... [/tex]
so [tex] det= \prod_{n} \lambda_{n} [/tex]
[tex] \int \mathcal D [\phi]exp(iS_{0}[\phi]+\int d^{4}x \phi ^{k}) [/tex]
so k >2 then could we use a similar 'Functional determinant approach' to this Feynman integral ?? in the sense that the integral above will be equal to
[tex] Cx(Det[\partial _{\mu}+k\phi^{k-1})^{-1/b} [/tex]
where C and 'b' are constant and the determinant is defined as an infinite product of eigenvalues
[tex] \zeta (s) \Gamma(s)= \int_{0}^{\infty}dt t^{s-1}Tr[e^{-sH_{0}-sgV_{int}] [/tex]
where the index '0' means the quadratic part of our Hamiltonian /action and so on..
since 'g' is small then we can express for every eigenvalue:
[tex] \lambda _{n} =\lambda _{0}+g\lambda _{n}^{1} +g^{2}\lambda_{n}^{2}+... [/tex]
so [tex] det= \prod_{n} \lambda_{n} [/tex]