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algori
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There are two naive approaches to Feynman integrals -- finite-dimensional approximations and analytic continuation. Both are discussed in S. Albeverio, R. Hoegh-Krohn, S. Mazzucchi, Mathematical theory of Feynman path integrals, Springer LNM 523, 2 ed. As far as I understand, both work fine as long as $L$ is a non-degenerate quadratic function plus a bounded one. Moreover, in the 1st edition of the above-mentioned book Albeverio and Hoegh-Krohn give yet another definition, which works under the same hypotheses.

Finally, let me mention yet another approach to Feynman integrals: the "white noise analysis" (see e.g. Lectures on White noise functionals by T. Hida and S. Si). It was used by S. Albeverio and A. Sengupta (Comm Math Phys, 186, 1997) to define rigorously the Chern-Simons integrals when the ambient manifold is $\mathbf{R}^3$ (they have to use gauge-fixing though). Since I don't know much about this I'd like to ask a couple of naive questions: Is there a way this could help one handle the cases other approaches can't? Can this help eliminate the necessity to start with a non-degenerate quadratic function?

There are two naive approaches to Feynman integrals -- finite-dimensional approximations and analytic continuation. Both are discussed in Albeverio, Hoegh-Krohn, Mazzucchi, Mathematical theory of Feynman path integrals, Springer LNM 523, 2 ed. As far as I understand, both work fine as long as $L$ is a non-degenerate quadratic function plus a bounded one. Moreover, in the 1st edition of the above-mentioned book Albeverio and Hoegh-Krohn give yet another definition, which works under the same hypotheses.

Finally, let me mention yet another approach to Feynman integrals: the "white noise analysis" (see e.g. Lectures on White noise functionals by Hida and Si). It was used by Albeverio and Sengupta to define rigorously the Chern-Simons integrals when the ambient manifold is $\mathbf{R}^3$ (they have to use gauge-fixing though). Since I don't know much about this I'd like to ask a couple of naive questions: Is there a way this could help one handle the cases other approaches can't? Can this help eliminate the necessity to start with a non-degenerate quadratic function?

There are two naive approaches to Feynman integrals -- finite-dimensional approximations and analytic continuation. Both are discussed in S. Albeverio, R. Hoegh-Krohn, S. Mazzucchi, Mathematical theory of Feynman path integrals, Springer LNM 523, 2 ed. As far as I understand, both work fine as long as $L$ is a non-degenerate quadratic function plus a bounded one. Moreover, in the 1st edition of the above-mentioned book Albeverio and Hoegh-Krohn give yet another definition, which works under the same hypotheses.

Finally, let me mention yet another approach to Feynman integrals: the "white noise analysis" (see e.g. Lectures on White noise functionals by T. Hida and S. Si). It was used by S. Albeverio and A. Sengupta (Comm Math Phys, 186, 1997) to define rigorously the Chern-Simons integrals when the ambient manifold is $\mathbf{R}^3$ (they have to use gauge-fixing though). Since I don't know much about this I'd like to ask a couple of naive questions: Is there a way this could help one handle the cases other approaches can't? Can this help eliminate the necessity to start with a non-degenerate quadratic function?

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algori
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Since it is so difficult to define Feynman integrals directly, people have resorted to various tricks. For example, suppose $0$ is a non-denenerate critical point of $L$, so $L$ can be written $L(x)=Q(x)+U(x)$ where $Q$ is a non-degenerate quadratic function and $U$ is formed by the higher order terms. Introducing a parameter $h$ one can write (in finite dimensions and for a sufficiently nice $U$) the Taylor series for $\int_E e^{i (Q(x)+hU(x)}f(x) dx$$ at $h=0$ in terms of Feynman diagrams. This is explained in e.g. in section 2 of the paper "Feynman integrals for pedestrians" by M. Polyak, arXiv:math/0406251. When one tries to mimic this in infinite dimensions, one runs into difficulties: the coefficients of Feynman diagrams are given by finite-dimensional but divergent integrals (e.g. if $E$$\int_E e^{i (Q(x)+hU(x))}f(x) dx$ at $h=0$ in terms of Feynman diagrams. This is explained in e.g. in section 2 of the paper "Feynman integrals for pedestrians" by M. Polyak, arXiv:math/0406251. When one tries to mimic this in infinite dimensions, one runs into difficulties: the coefficients of Feynman diagrams are given by finite-dimensional but divergent integrals (e.g. if $E$ is the space of functions of some kind on some manifold, then the integrals are taken along the configuration spaces of that manifold).

Since it is so difficult to define Feynman integrals directly, people have resorted to various tricks. For example, suppose $0$ is a non-denenerate critical point of $L$, so $L$ can be written $L(x)=Q(x)+U(x)$ where $Q$ is a non-degenerate quadratic function and $U$ is formed by the higher order terms. Introducing a parameter $h$ one can write (in finite dimensions and for a sufficiently nice $U$) the Taylor series for $\int_E e^{i (Q(x)+hU(x)}f(x) dx$$ at $h=0$ in terms of Feynman diagrams. This is explained in e.g. in section 2 of the paper "Feynman integrals for pedestrians" by M. Polyak, arXiv:math/0406251. When one tries to mimic this in infinite dimensions, one runs into difficulties: the coefficients of Feynman diagrams are given by finite-dimensional but divergent integrals (e.g. if $E$ is the space of functions of some kind on some manifold, then the integrals are taken along the configuration spaces of that manifold).

Since it is so difficult to define Feynman integrals directly, people have resorted to various tricks. For example, suppose $0$ is a non-denenerate critical point of $L$, so $L$ can be written $L(x)=Q(x)+U(x)$ where $Q$ is a non-degenerate quadratic function and $U$ is formed by the higher order terms. Introducing a parameter $h$ one can write (in finite dimensions and for a sufficiently nice $U$) the Taylor series for $\int_E e^{i (Q(x)+hU(x))}f(x) dx$ at $h=0$ in terms of Feynman diagrams. This is explained in e.g. in section 2 of the paper "Feynman integrals for pedestrians" by M. Polyak, arXiv:math/0406251. When one tries to mimic this in infinite dimensions, one runs into difficulties: the coefficients of Feynman diagrams are given by finite-dimensional but divergent integrals (e.g. if $E$ is the space of functions of some kind on some manifold, then the integrals are taken along the configuration spaces of that manifold).

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algori
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I should admit that my interest in all of this comes from the attempts to understand Witten's definition of invariants of links in 3-manifolds (Comm Math Phys 121, 1989).

I should admit that my interest in all of this comes from the attempts to understand Witten's definition of invariants of links in 3-manifolds (Comm Math Phys 121, 1989).

I should admit that my interest in all of this comes from attempts to understand Witten's definition of invariants of links in 3-manifolds (Comm Math Phys 121, 1989).

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algori
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