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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{path integral} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory} [[!include functorial quantum field theory - contents]] \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory} [[!include measure theory - contents]] \hypertarget{integration_theory}{}\paragraph*{{Integration theory}}\label{integration_theory} [[!include integration theory - contents]] \begin{quote}% under construction \end{quote} \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{realizations}{Realizations}\dotfill \pageref*{realizations} \linebreak \noindent\hyperlink{ElementaryDescription}{Elementary description in quantum mechanics}\dotfill \pageref*{ElementaryDescription} \linebreak \noindent\hyperlink{AsAnIntegralAgainstTheWienerMeasure}{As an integral against the Wiener measure}\dotfill \pageref*{AsAnIntegralAgainstTheWienerMeasure} \linebreak \noindent\hyperlink{PerturbativelyInBVFormalism}{Perturbatively for free field theory in BV-formalism}\dotfill \pageref*{PerturbativelyInBVFormalism} \linebreak \noindent\hyperlink{the_path_integral_in_the_bigger_picture}{The path integral in the bigger picture}\dotfill \pageref*{the_path_integral_in_the_bigger_picture} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{stochastic_integration_theory}{Stochastic integration theory}\dotfill \pageref*{stochastic_integration_theory} \linebreak \noindent\hyperlink{ReferencesForChargedParticle}{For charged particle/path integral of holonomy functional}\dotfill \pageref*{ReferencesForChargedParticle} \linebreak \noindent\hyperlink{more}{More}\dotfill \pageref*{more} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The notion of \textbf{path integral} originates in and is mainly used in the context of [[quantum mechanics]] and [[quantum field theory]], where it is a certain operation supposed to model the notion of [[quantization]]. The idea is that the quantum propagator -- in [[FQFT]] the value of the functor $U : Cob \to Vect$ on a certain [[cobordism]] -- is given by an [[integral kernel]] $U : \psi \mapsto \int K(-,y) \psi(y) d\mu$ where $K(x,y)$ is something like the integral of the exponentiated [[action functional]] $S$ over all field configurations $\phi$ with prescribed boundary datat $x$ and $y$. Formally one writes \begin{displaymath} K(x,y) = \int \exp(i S(\phi))\; D\phi \end{displaymath} and calls this the \textbf{path integral}. Here the expression $D \phi$ is supposed to allude to a [[measure space|measure integral]] on the space of all $\phi$. The main problem with the path integral idea is that it is typically unclear what this measure should be, or, worse, it is typically clear that no suitable such measure does exist. The name \emph{path integral} originates from the special case where the system is the [[sigma model]] describing a particle on a target space manifold $X$. In this case a field configuration $\phi$ is a path $\phi : [0,1] \to X$ in $X$, hence the integral over all field configurations is an integral over all paths. The idea of the path integral famously goes back to [[Richard Feynman]], who motivated the idea in [[quantum mechanics]]. In that context the notion can typically be made precise and shown to be equivalent to various other [[quantization]] prescriptions. The central impact of the idea of the path integral however is in its application to [[quantum field theory]], where it is often taken in the physics literatire as the \emph{definition} of what the quantum field theory encoded by an [[action functional]] should be, disregarding the fact that in these contexts it is typically quite unclear what the path integral actually means, precisely. Notably the [[Feynman perturbation series]] summing over [[Feynman graphs]] is motivated as one way to make sense of the path integral in quantum field theory and in practice usually serves as a \emph{definition} of the perturbative path integral. \hypertarget{realizations}{}\subsection*{{Realizations}}\label{realizations} We start with stating the elementary description of the [[Feynman-Kac formula]] as traditional in physics textbooks in \begin{itemize}% \item \hyperlink{ElementaryDescription}{Elementary description in quantum mechanics}. \end{itemize} Then we indicate the more abstract formulation of this in terms of [[integration]] against the [[Wiener measure]] on the space of paths (for the Euclidean path integral) in \begin{itemize}% \item \emph{\hyperlink{AsAnIntegralAgainstTheWienerMeasure}{As an integral against the Wiener measure}}. \end{itemize} Then we indicate a formulation in [[perturbation theory]] and [[BV-formalism]] in \begin{itemize}% \item \emph{\hyperlink{PerturbativelyInBVFormalism}{Perturbatively in the BV-formalism}} \end{itemize} \hypertarget{ElementaryDescription}{}\subsubsection*{{Elementary description in quantum mechanics}}\label{ElementaryDescription} A simple form of the path integral is realized in [[quantum mechanics]], where it was originally dreamed up by [[Richard Feynman]] and then made precise using the [[Feynman-Kac formula]]. (Most calculations in practice are still done using [[perturbation theory]], see the section \emph{\hyperlink{PerturbativelyInBVFormalism}{Perturbatively in BV-formalism}} below). The [[Schrödinger equation]] says that the rate at which the phase of an energy eigenvector rotates is proportional to its energy: \begin{displaymath} i \hbar \frac{d}{dt} \psi = H \psi. \end{displaymath} Therefore, the probability that the system evolves to the final state $\psi_F$ after evolving for time $t$ from the initial state $\psi_I$ is \begin{displaymath} \langle \psi_F|e^{-iHt}|\psi_I\rangle. \end{displaymath} Chop this up into time steps $\Delta t = t/N$ and use the fact that \begin{displaymath} \int_{-\infty}^{\infty}|q\rangle\langle q| = 1 \end{displaymath} to get \begin{displaymath} \langle \psi_F| e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} |q_{N-1} \rangle \langle q_{N-1}| dq_{N-1}\right) e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} |q_{N-2} \rangle \langle q_{N-2}| dq_{N-2}\right) e^{-iH\Delta t} \cdots e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} |q_1 \rangle \langle q_1| dq_1\right) e^{-iH\Delta t} |\psi_I\rangle \end{displaymath} \begin{displaymath} = \int_{q_1} \cdots \int_{q_{N-2}} \int_{q_{N-1}} \langle \psi_F| e^{-iH\Delta t} |q_{N-1} \rangle \langle q_{N-1}| e^{-iH\Delta t} |q_{N-2} \rangle \langle q_{N-2}| e^{-iH\Delta t} \cdots e^{-iH\Delta t} |q_1 \rangle \langle q_1| e^{-iH\Delta t} |\psi_I\rangle dq_{N-1} dq_{N-2} \cdots dq_1 \end{displaymath} Assume we have the free Hamiltonian $H=p^2/2m.$ Looking at an individual term $\langle q_{n+1}| e^{-iH\Delta t} |q_{n} \rangle,$ we can insert a factor of 1 and solve to get \begin{displaymath} \itexarray{\langle q_{n+1}| e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} \frac{dp}{2\pi}|p\rangle \langle p|\right)|q_{n} \rangle &=& \int_{-\infty}^{\infty} \frac{dp}{2\pi} e^{-ip^2\Delta t/2m} \langle q_{n+1}|p\rangle \langle p|q_{n} \rangle \\ &=& \int_{-\infty}^{\infty} \frac{dp}{2\pi} e^{-ip^2\Delta t/2m} e^{ip(q_{n+1}-q_n)} \\ &=& \left(\frac{-i 2\pi m}{\Delta t}\right)^{\frac{1}{2}} e^{i \Delta t (m/2)[(q_{n+1}-q_n)/\Delta t]^2}.} \end{displaymath} Defining \begin{displaymath} \int Dq = \lim_{N \to \infty} \left(\frac{-i 2\pi m}{\Delta t}\right)^{\frac{N}{2}} \prod_{n=0}^{N-1} \int dq_n, \end{displaymath} and letting $\Delta t \to 0, N \to \infty,$ we get \begin{displaymath} \langle \psi_F|e^{-iHt}|\psi_I\rangle = \int Dq e^{i \int_0^t dt \frac{1}{2}m \dot{q}^2}. \end{displaymath} For arbitrary Hamiltonians $H = \frac{p^2}{2m} + V(x),$ we get \begin{displaymath} \itexarray{\langle \psi_F|e^{-iHt}|\psi_I\rangle &=& \int Dq e^{i \int_0^t dt \frac{1}{2}m \dot{q}^2 - V(x)} \\ &=& \int Dq e^{i\int_0^t\mathcal{L}(\dot{q},q) dt} \\ &=& \int Dq e^{iS(q)}, } \end{displaymath} where $S(q)$ is the [[action]] functional. Is there an easy way to see how the Hamiltonian transforms into the Lagrangian in the exponent? \hypertarget{AsAnIntegralAgainstTheWienerMeasure}{}\subsubsection*{{As an integral against the Wiener measure}}\label{AsAnIntegralAgainstTheWienerMeasure} More abstractly, the Euclidean path integral for the [[quantum mechanics]] of a [[charged particle]] may be defined by [[integration]] the gauge-coupling action again the [[Wiener measure]] on the space of paths. Consider a [[Riemannian manifold]] $(X,g)$ -- hence a [[background field]] of [[gravity]] -- and a [[connection on a bundle|connection]] $\nabla : X \to \mathbf{B}U(1)_{conn}$ -- hence an [[electromagnetic field|electromagnetic]] [[background gauge field]]. The gauge-coupling [[interaction]] term is given by the [[parallel transport]] of this connection \begin{displaymath} \exp(i S) \coloneqq \exp(2\pi i \int_{(-)} [(-),\nabla] ) \colon [I, X]_{x_0,x_1} \to Hom(E_{x_0}, E_{x_1}) \,, \end{displaymath} where $E \to X$ is the [[complex line bundle]] which is [[associated bundle|associated]] to $\nabla$. The [[Wiener measure]] $d\mu_W$ on the space of stochastic paths in $X$,we may write suggestively write as \begin{displaymath} d\mu_W = [\exp(-S_{kin})D\gamma] \end{displaymath} for it combines what in the physics literature is the [[kinetic action]] and a canonical measure on paths. (This is a general phenomenon in formalizations of the process of [[quantization]]: the [[kinetic action]] (the [[free field theory]]-part of the [[action functional]]) is absorbed as part of the integration [[measure]] against with the remaining [[interaction]] terms are integrated. ) Then one has (e.g. \hyperlink{Norris}{Norris92, theorem (34)}, \hyperlink{Charles99}{Charles 99, theorem 6.1}): the [[integral kernel]] for the time evolution propagator is \begin{displaymath} U(x_0,x_1) = \int_{\gamma} tra(\nabla)(\gamma) \, [\exp(-S_{kin}(\gamma)) D\gamma] \,, \end{displaymath} hence the [[integration]] of the [[parallel transport]]/[[holonomy]] against the [[Wiener measure]]. (To make sense of this one first needs to extend the [[parallel transport]] from smooth paths to stochastic paths, see the references \hyperlink{ReferencesForChargedParticle}{below}.) \begin{remark} \label{WorldlineFormalism}\hypertarget{WorldlineFormalism}{} This ``holonomy integrated against the Wiener measure'' is the path integral in the form in which it notably appears in the \emph{[[worldline formalism]]} for computing [[scattering amplitudes]] in [[quantum field theory]]. See (\hyperlink{Strassler92}{Strassler 92, (2.9), (2.10)}). Notice in particular that by the discussion there this is the correct [[Wick rotation|Wick rotated]] form: the [[kinetic action]] is not a [[complex phase]] but a real exponential $\exp(- S_{kin})$ while the [[gauge field|gauge]] [[interaction]] term (the [[holonomy]]) is a complex phase (locally $\exp(i \int_\gamma A)$). \end{remark} \begin{remark} \label{}\hypertarget{}{} From the point of view of [[higher prequantum field theory]] this means that the path integral sends a [[correspondence]] in the [[slice (infinity,1)-topos]] of [[smooth infinity-groupoids]] over the [[delooping]] groupoid $\mathbf{B}U(1)$ \begin{displaymath} \itexarray{ && [I,X] \\ & {}^{(-)|_0}\swarrow && \searrow^{(-)|_1} \\ X && \swArrow_{\exp(i S)} && X \\ & {}_{\mathllap{\chi(\nabla)}}\searrow && \swarrow_{\mathrlap{\chi(\nabla)}} \\ && \mathbf{B}U(1) } \end{displaymath} (essentially a [[prequantized Lagrangian correspondence]]) to another [[correspondence]], now in the slice over the [[stack]] (now an actual [[2-sheaf]]) $\mathbb{C}\mathbf{Mod}$ of [[modules]] over the [[complex numbers]], hence of [[complex vector bundles]]: \begin{displaymath} \itexarray{ && X \times X \\ & {}^{p_1}\swarrow && \searrow^{p_2} \\ X && \swArrow_{\int_{\gamma}\exp(i S(\gamma)) [\exp(-S_{kin}(\gamma))D\gamma]} && X \\ & {}_{\mathllap{\rho(\chi(\nabla))}}\searrow && \swarrow_{\mathrlap{\rho(\chi(\nabla))}} \\ && \mathbb{C}\mathbf{Mod} \,. } \end{displaymath} \end{remark} For more discussion along these lines see at \emph{[[motivic quantization]]}. \hypertarget{PerturbativelyInBVFormalism}{}\subsubsection*{{Perturbatively for free field theory in BV-formalism}}\label{PerturbativelyInBVFormalism} [[BV-BRST formalism]] is a means to formalize the path integral in [[perturbation theory]] as the passage to [[cochain cohomology]] in a \emph{[[quantum BV-complex]]}. See at \emph{\href{BV-BRST+formalism#HomologicalIntegration}{The BV-complex and homological integration}} for more details. [[!include action (physics) - table]] \hypertarget{the_path_integral_in_the_bigger_picture}{}\subsection*{{The path integral in the bigger picture}}\label{the_path_integral_in_the_bigger_picture} Ours is the age whose central fundamental theoretical physics question is: \begin{quote}% \emph{What is [[quantum field theory]]}? \end{quote} A closely related question is: \begin{quote}% \emph{What is the path integral} ? \end{quote} After its conception by [[Richard Feynman]] in the middle of the 20th century It was notably [[Edward Witten]]`s achievement in the late 20th century to make clear the vast potential for fundamental physics and pure math underlying the concept of the quantum field theoretic path integral. And yet, among all the aspects of QFT, the notion of the path integral is the one that has resisted attempts at formalization the most. While [[FQFT|functorial quantum field theory]] is the formalization of the properties that the \emph{locality} and the \emph{sewing law} of the path integral is demanded to have -- whatever the path integral is, it is a process that in the end yields a [[functor]] on a [[(infinity,n)-category of cobordisms]] -- by itself, this sheds no light on what that procedure called ``path integration'' or ``path integral quantization'' is. The single major insight into the right [[higher category theory|higher categorical]] formalization of the path integral is probably the idea indicated in \begin{itemize}% \item [[Dan Freed]] \begin{itemize}% \item \emph{Quantum groups from path integrals} (\href{http://xxx.lanl.gov/abs/q-alg/9501025}{arXiv:q-alg/9501025}) \item \emph{[[Higher Algebraic Structures and Quantization]]} (\href{http://arxiv.org/abs/hep-th/9212115}{arXiv:hep-th/9212115}) \end{itemize} \end{itemize} which says that \begin{itemize}% \item it is wrong to think of the \emph{action functional} that the path integral integrates over as just a \emph{function}: it is a higher categorical object; \item accordingly, the path integral is not something that just controls the numbers or linear maps assigned by a $d$-dimensional [[quantum field theory]] in dimension $d$: also the assignment to higher codimensions is to be regarded as part of the path integral; \begin{itemize}% \item notably: the fact that quantum mechanics assigns a (Hilbert) space of sections of a vector bundle to codimension 1 is to be regarded as due to a \emph{summing operation} in the sense of the path integral, too: the space of sections of a vector bundle is the continuum equivalent of the direct sum of its fibers \end{itemize} \end{itemize} More recently, one sees attempts to formalize this observation of Freed's, notably in the context of the [[cobordism hypothesis]]: \begin{itemize}% \item [[geometric infinity-function theory]] is used to compute at least something like a path integral in codimension 1 and 2 in the context of [[sigma-model]] QFT; \item see [[path integral as a pull-push transform]] \item and something similar or is indicated in section 3 and section 6 of \begin{itemize}% \item [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], \emph{[[Topological Quantum Field Theories from Compact Lie Groups]]} (\href{http://arxiv.org/abs/0905.0731}{arXiv:0905.0731}) \end{itemize} \end{itemize} based on material (on categories of ``families'') in \emph{[[On the Classification of Topological Field Theories]]} . \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[S-matrix]] \item [[Dyson formula]] \item [[fermionic path integral]] \item [[principle of extremal action]] \item [[integration over infinite-dimensional manifolds]] \item [[cohomological integration]], [[BV-BRST quantization]] \item [[Schwinger-Dyson equation]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} \hypertarget{general}{}\subsubsection*{{General}}\label{general} The original textbook reference is \begin{itemize}% \item [[Richard Feynman]], A. R. Hibbs, , \emph{Quantum Mechanics and Path Integrals} , New York: McGraw-Hill, (1965) \end{itemize} Lecture notes include \begin{itemize}% \item R. Rosenfelder, \emph{Path Integrals in Quantum Physics} (\href{http://arxiv.org/abs/1209.1315}{arXiv:1209.1315}) \end{itemize} Textbook accounts include \begin{itemize}% \item G. Johnson, M. Lapidus, \emph{The Feynman integral and Feynman's operational calculus}, Oxford University Press, Oxford, 2000. \item [[Barry Simon]], \emph{Functional integration and quantum physics} AMS Chelsea Publ., Providence, 2005 \item [[Joseph Polchinski]], \emph{[[String theory]]}, part I, appendix A \item Daisuke Fujiwara, \emph{Rigorous Time Slicing Approach to Feynman Path Integrals} 2017, Springer (\href{https://dx.doi.org/10.1007/978-4-431-56553-6}{doi:/10.1007/978-4-431-56553-6}) \end{itemize} Discussion in [[constructive quantum field theory]] includes \begin{itemize}% \item [[James Glimm]], [[Arthur Jaffe]], \emph{[[Quantum physics -- A functional integral point of view]]}, 535 pages, Springer \item Simon, \emph{Functional Integration in Quantum Physics} (AMS, 2005) \item [[Sergio Albeverio]], Raphael H\o{}egh-Krohn, Sonia Mazzucchi. \emph{Mathematical theory of Feynman path integrals - An Introduction}, 2 nd corrected and enlarged edition, Lecture Notes in Mathematics, Vol. 523. Springer, Berlin, 2008 (\href{href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:pre05233292&format=complete}{ZMATH}) \item Sonia Mazzucchi, \emph{Mathematical Feynman Path Integrals and Their Applications}, World Scientific, Singapore, 2009. \end{itemize} The [[worldline]] path integral as a way to compute [[scattering amplitudes]] in [[QFT]] was understood in \begin{itemize}% \item [[Matthew Strassler]], \emph{Field Theory Without Feynman Diagrams: One-Loop Effective Actions}, Nucl. Phys. B385:145-184,1992 (\href{http://arxiv.org/abs/hep-ph/9205205}{arXiv:hep-ph/9205205}) \end{itemize} \hypertarget{stochastic_integration_theory}{}\subsubsection*{{Stochastic integration theory}}\label{stochastic_integration_theory} The following articles use the integration over [[Wiener measures]] on [[stochastic processes]] for formalizing the path ingegral. \begin{itemize}% \item [[James Norris]], \emph{A complete differential formalism for stochastic calculus in manifolds}, S\'e{}minaire de probabilit\'e{}s de Strasbourg, 26 (1992), p. 189-209 (\href{http://www.numdam.org/item?id=SPS_1992__26__189_0}{NUMDAM}) \end{itemize} \begin{itemize}% \item Vassili Kolokoltsov, \emph{Path integration: connecting pure jump and Wiener processes} (\href{http://www2.warwick.ac.uk/fac/sci/statistics/staff/academic-research/kolokoltsov/minnesota.pdf}{pdf}) \item Bruce Driver, Anton Thalmaier, \emph{Heat equation derivative formulas for vector bundles}, Journal of Functional Analysis 183, 42-108 (2001) (\href{http://www.math.ucsd.edu/~bdriver/DRIVER/Papers/Drivers_Papers/A26-Heat-Derivative-Formula.pdf}{pdf}) \end{itemize} \hypertarget{ReferencesForChargedParticle}{}\subsubsection*{{For charged particle/path integral of holonomy functional}}\label{ReferencesForChargedParticle} The following articles discuss (aspects of) the path integral for the [[charged particle]] coupled to a [[background gauge field]], in which case the path integral is essentially the integration of the [[holonomy]]/[[parallel transport]] functional against the [[Wiener measure]]. \begin{itemize}% \item Marc Arnaudon and Anton Thalmaier, \emph{Yang--Mills fields and random holonomy along Brownian bridges}, Ann. Probab. Volume 31, Number 2 (2003), 769-790. (\href{http://projecteuclid.org/euclid.aop/1048516535}{Euclid}) \item [[Mikhail Kapranov]], \emph{Noncommutative geometry and path integrals}, in \emph{Algebra, Arithmetic and Geometry}, Birkh\"a{}user Progress in Mathematics 27 (2009) (\href{http://arxiv.org/abs/math/0612411}{arXiv:math/0612411}) \item [[Christian Bär]], [[Frank Pfäffle]], \emph{Path integrals on manifolds by finite dimensional approximation}, J. reine angew. Math., (2008), 625: 29-57. (\href{http://arxiv.org/abs/math.AP/0703272}{arXiv:math.AP/0703272}) \item Dana Fine, Stephen Sawin, \emph{A Rigorous Path Integral for Supersymmetric Quantum Mechanics and the Heat Kernel} (\href{http://arxiv.org/abs/0705.0638}{arXiv:0705.0638}) \end{itemize} A discussion for [[phase spaces]] equipped with a [[Kähler polarization]] and a [[prequantum line bundle]] is in \begin{itemize}% \item Laurent Charles, \emph{Feynman path integral and Toeplitz Quantization}, Helv. Phys. Acta \textbf{72} (1999) 341., (\href{http://ipht.cea.fr/DocsphtV2/articles/t98/093/public/publi.pdf}{pdf}) \end{itemize} following \hyperlink{Norris92}{Norris 92, theorem (34)}. \hypertarget{more}{}\subsubsection*{{More}}\label{more} Other references on mathematical aspects of path integrals include \begin{itemize}% \item [[Pierre Cartier]], [[Cecile DeWitt-Morette]], \emph{Functional integration: action and symmetries} (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1122.81004&f ormat=complete}{ZMATH}) \item [[Marc Henneaux]], [[Claudio Teitelboim]], \emph{[[Quantization of Gauge Systems]]}, Princeton University Press 1992 \item [[Nikolai Reshetikhin]], \emph{Lectures on quantization of gauge systems}, \href{http://arxiv.org/abs/1008.1411}{arxiv/1008.1411} \item [[Edward Witten]], \emph{A new look at the path integral of quantum mechanics}, (\href{http://arxiv.org/abs/1009.6032}{arxiv/1009.6032}) \end{itemize} Detailed rigorous discussion for [[quadratic Hamiltonians]] and for [[phase space]] paths in in \begin{itemize}% \item [[Joel Robbin]], [[Dietmar Salamon]], \emph{Feynman path integrals on phase space and the metaplectic representation} in [[Dietmar Salamon]] (ed.), \emph{Symplectic Geometry}, LMS Lecture Note series 192 (1993) ([[RobbinSalamonMetaplectic.pdf:file]]) \end{itemize} Discussion of quantization of [[Chern-Simons theory]] via a [[Wiener measure]] is in \begin{itemize}% \item Adrian P. C. Lim, \emph{Chern-Simons Path Integral on $\mathbb{R}^3$ using Abstract Wiener Measure} (\href{http://www.math.cornell.edu/~pclim/Docs/papers/CSabe01.pdf}{pdf}) \end{itemize} Lecture notes on [[quantum field theory]], emphasizing mathematics of the Euclidean path integrals and the relation to statistical physics are at \begin{itemize}% \item [[AJ Tolland]], \emph{\href{http://www.math.sunysb.edu/~ajt/Teaching/560spring2011/}{Wilsonian QFT for Mathematicians}} \end{itemize} MathOverflow questions: \href{http://mathoverflow.net/questions/19495/mathematics-of-path-integral-state-of-the-art}{mathematics-of-path-integral-state-of-the-art},\href{http://mathoverflow.net/questions/20393/path-integrals-outside-qft}{path-integrals-outside-qft}, \href{http://mathoverflow.net/questions/19490/doing-geometry-using-feynman-path-integral}{doing-geometry-using-feynman-path-integral}, \href{http://mathoverflow.net/questions/17577/path-integrals-localisation}{path-integrals-localisation}, \href{http://mathoverflow.net/questions/31966/finite-dimensional-feynman-integrals}{finite-dimensional-feynman-integrals}, \href{http://mathoverflow.net/questions/24823/the-mathematical-theory-of-feynman-integrals}{the-mathematical-theory-of-feynman-integrals} \begin{itemize}% \item [[Theo Johnson-Freyd]], \emph{The formal path integral and quantum mechanics}, J. Math. Phys. \textbf{51}, 122103 (2010) \href{http://arxiv.org/abs/1004.4305}{arxiv/1004.4305}, \href{http://dx.doi.org/10.1063/1.3503472}{doi}; \emph{On the coordinate (in)dependence of the formal path integral}, \href{http://arxiv.org/abs/1003.5730}{arxiv/1003.5730} \end{itemize} [[!redirects path integrals]] [[!redirects Feynman path integral]] [[!redirects Feynman path integrals]] [[!redirects path-integral]] [[!redirects path-integrals]] [[!redirects Feynman path-integral]] [[!redirects Feynman path-integrals]] \end{document}