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integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
under construction
The notion of 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
and calls this the path integral. Here the expression $D \phi$ is supposed to allude to an 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 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 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 definition of the perturbative path integral.
We start with stating the elementary description of the Feynman-Kac formula? as traditional in physics textbooks in
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
Then we indicate a formulation in perturbation theory and BV-formalism in
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 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:
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
Chop this up into time steps $\Delta t = t/N$ and use the fact that
to get
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
Defining
and letting $\Delta t \to 0, N \to \infty,$ we get
For arbitrary Hamiltonians $H = \frac{p^2}{2m} + V(x),$ we get
where $S(q)$ is the action functional.
Is there an easy way to see how the Hamiltonian transforms into the Lagrangian in the exponent?
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 $\nabla : X \to \mathbf{B}U(1)_{conn}$ – hence an electromagnetic background gauge field.
The gauge-coupling interaction term is given by the parallel transport of this connection
where $E \to X$ is the complex line bundle which is associated to $\nabla$.
The Wiener measure $d\mu_W$ on the space of stochastic paths in $X$,we may write suggestively write as
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. Norris92, theorem (34), Charles 99, theorem 6.1):
the integral kernel for the time evolution propagator is
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 below.)
This “holonomy integrated against the Wiener measure” is the path integral in the form in which it notably appears in the worldline formalism for computing scattering amplitudes in quantum field theory. See (Strassler 92, (2.9), (2.10)). Notice in particular that by the discussion there this is the correct Wick rotated form: the kinetic action is not a complex phase but a real exponential $\exp(- S_{kin})$ while the gauge interaction term (the holonomy) is a complex phase (locally $\exp(i \int_\gamma A)$).
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 $\mathbf{B}U(1)$
(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:
For more discussion along these lines see at motivic quantization.
BV-BRST formalism is a means to formalize the path integral in perturbation theory as the passage to cochain cohomology in a quantum BV-complex. See at The BV-complex and homological integration for more details.
action functional | kinetic action | interaction | path integral measure |
---|---|---|---|
$\exp(-S(\phi)) \cdot \mu =$ | $\exp(-(\phi, Q \phi)) \cdot$ | $\exp(I(\phi)) \cdot$ | $\mu$ |
BV differential | elliptic complex + | antibracket with interaction + | BV-Laplacian |
$d_q =$ | $Q$ + | $\{I,-\}$ + | $\hbar \Delta$ |
Ours is the age whose central fundamental theoretical physics question is:
What is quantum field theory?
A closely related question is:
What is the path integral ?
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 functorial quantum field theory is the formalization of the properties that the locality and the 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 categorical formalization of the path integral is probably the idea indicated in
Quantum groups from path integrals (arXiv:q-alg/9501025)
Higher Algebraic Structures and Quantization (arXiv:hep-th/9212115)
which says that
it is wrong to think of the action functional that the path integral integrates over as just a function: it is a higher categorical object;
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;
More recently, one sees attempts to formalize this observation of Freed’s, notably in the context of the cobordism hypothesis:
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;
and something similar or is indicated in section 3 and section 6 of
based on material (on categories of “families”) in On the Classification of Topological Field Theories .
The original textbook reference is
Lecture notes include
Textbook accounts include
G. Johnson, M. Lapidus, The Feynman integral and Feynman’s operational calculus, Oxford University Press, Oxford, 2000.
Barry Simon, Functional integration and quantum physics AMS Chelsea Publ., Providence, 2005
Joseph Polchinski, String theory, part I, appendix A
Discussion in constructive quantum field theory includes
James Glimm, Arthur Jaffe, Quantum physics -- A functional integral point of view, 535 pages, Springer
Simon, Functional Integration in Quantum Physics (AMS, 2005)
Sergio Albeverio, Raphael Høegh-Krohn, Sonia Mazzucchi. Mathematical theory of Feynman path integrals - An Introduction, 2 nd corrected and enlarged edition, Lecture Notes in Mathematics, Vol. 523. Springer, Berlin, 2008 (ZMATH)
Sonia Mazzucchi, Mathematical Feynman Path Integrals and Their Applications, World Scientific, Singapore, 2009.
The worldline path integral as a way to compute scattering amplitudes in QFT was understood in
The following articles use the integration over Wiener measures on stochastic processes? for formalizing the path ingegral.
Vassili Kolokoltsov, Path integration: connecting pure jump and Wiener processes (pdf)
Bruce Driver, Anton Thalmaier, Heat equation derivative formulas for vector bundles, Journal of Functional Analysis 183, 42-108 (2001) (pdf)
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.
Marc Arnaudon and Anton Thalmaier, Yang–Mills fields and random holonomy along Brownian bridges, Ann. Probab. Volume 31, Number 2 (2003), 769-790. (Euclid)
Mikhail Kapranov, Noncommutative geometry and path integrals, in Algebra, Arithmetic and Geometry, Birkhäuser Progress in Mathematics 27 (2009) (arXiv:math/0612411)
Christian Bär, Frank Pfäffle, Path integrals on manifolds by finite dimensional approximation, J. reine angew. Math., (2008), 625: 29-57. (arXiv:math.AP/0703272)
Dana Fine, Stephen Sawin, A Rigorous Path Integral for Supersymmetric Quantum Mechanics and the Heat Kernel (arXiv:0705.0638)
A discussion for phase spaces equipped with a Kähler polarization and a prequantum line bundle is in
following Norris 92, theorem (34).
Other references on mathematical aspects of path integrals include
Detailed rigorous discussion for quadratic Hamiltonians and for phase space paths in in
Discussion of quantization of Chern-Simons theory via a Wiener measure is in
Lecture notes on quantum field theory, emphasizing mathematics of the Euclidean path integrals and the relation to statistical physics are at
MathOverflow questions: mathematics-of-path-integral-state-of-the-art,path-integrals-outside-qft, doing-geometry-using-feynman-path-integral, path-integrals-localisation, finite-dimensional-feynman-integrals, the-mathematical-theory-of-feynman-integrals