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path integral in nLab

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Context

Quantum systems

quantum logic

quantum physics

quantum probability theory – observables and states

quantum information

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quantum computing

Quantum field theory

Measure and probability theory

Integration theory

under construction

Contents

Idea

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→VectU : Cob \to Vect on a certain cobordism – is given by an integral kernel U:ψ↦∫K(−,y)ψ(y)dμU : \psi \mapsto \int K(-,y) \psi(y) d\mu where K(x,y)K(x,y) is something like the integral of the exponentiated action functional SS over all field configurations ϕ\phi with prescribed boundary data xx and yy. Formally one writes

K(x,y)=∫exp(iS(ϕ))Dϕ K(x,y) = \int \exp(i S(\phi))\; D\phi

and calls this the path integral. Here the expression DϕD \phi is supposed to allude to a measure integral on the space of all ϕ\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 XX. In this case a field configuration ϕ\phi is a path ϕ:[0,1]→X\phi : [0,1] \to X in XX, 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 literature 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.

Realizations

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

Elementary description in quantum mechanics

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:

iℏddtψ=Hψ. i \hbar \frac{d}{dt} \psi = H \psi.

Therefore, the probability that the system evolves to the final state ψ F\psi_F after evolving for time tt from the initial state ψ I\psi_I is

⟨ψ F|e −iHt|ψ I⟩. \langle \psi_F|e^{-iHt}|\psi_I\rangle.

Chop this up into time steps Δt=t/N\Delta t = t/N and use the fact that

∫ −∞ ∞|q⟩⟨q|=1\int_{-\infty}^{\infty}|q\rangle\langle q| = 1

to get

⟨ψ F|e −iHΔt(∫ −∞ ∞|q N−1⟩⟨q N−1|dq N−1)e −iHΔt(∫ −∞ ∞|q N−2⟩⟨q N−2|dq N−2)e −iHΔt⋯e −iHΔt(∫ −∞ ∞|q 1⟩⟨q 1|dq 1)e −iHΔt|ψ I⟩ \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

=∫ q 1⋯∫ q N−2∫ q N−1⟨ψ F|e −iHΔt|q N−1⟩⟨q N−1|e −iHΔt|q N−2⟩⟨q N−2|e −iHΔt⋯e −iHΔt|q 1⟩⟨q 1|e −iHΔt|ψ I⟩dq N−1dq N−2⋯dq 1 = \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

Assume we have the free Hamiltonian H=p 2/2m.H=p^2/2m. Looking at an individual term ⟨q n+1|e −iHΔt|q n⟩,\langle q_{n+1}| e^{-iH\Delta t} |q_{n} \rangle, we can insert a factor of 1 and solve to get

⟨q n+1|e −iHΔt(∫ −∞ ∞dp2π|p⟩⟨p|)|q n⟩ = ∫ −∞ ∞dp2πe −ip 2Δt/2m⟨q n+1|p⟩⟨p|q n⟩ = ∫ −∞ ∞dp2πe −ip 2Δt/2me ip(q n+1−q n) = (−i2πmΔt) 12e iΔt(m/2)[(q n+1−q n)/Δt] 2. \array{\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}.}

Defining

∫Dq=lim N→∞(−i2πmΔt) N2∏ n=0 N−1∫dq n,\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,

and letting Δt→0,N→∞,\Delta t \to 0, N \to \infty, we get

⟨ψ F|e −iHt|ψ I⟩=∫Dqe i∫ 0 tdt12mq˙ 2. \langle \psi_F|e^{-iHt}|\psi_I\rangle = \int Dq e^{i \int_0^t dt \frac{1}{2}m \dot{q}^2}.

For arbitrary Hamiltonians H=p 22m+V(x),H = \frac{p^2}{2m} + V(x), we get

⟨ψ F|e −iHt|ψ I⟩ = ∫Dqe i∫ 0 tdt12mq˙ 2−V(x) = ∫Dqe i∫ 0 tℒ(q˙,q)dt = ∫Dqe iS(q), \array{\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)}, }

where S(q)S(q) is the action functional.

As an integral against the Wiener measure

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)(X,g) – hence a background field of gravity – and a connection ∇:X→BU(1) conn\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

exp(iS)≔exp(2πi∫ (−)[(−),∇]):[I,X] x 0,x 1→Hom(E x 0,E x 1), \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}) \,,

where E→XE \to X is the complex line bundle which is associated to ∇\nabla.

The Wiener measure dμ Wd\mu_W on the space of stochastic paths in XX,we may write suggestively write as

dμ W=[exp(−S kin)Dγ] d\mu_W = [\exp(-S_{kin})D\gamma]

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

U(x 0,x 1)=∫ γtra(∇)(γ)[exp(−S kin(γ))Dγ], U(x_0,x_1) = \int_{\gamma} tra(\nabla)(\gamma) \, [\exp(-S_{kin}(\gamma)) D\gamma] \,,

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.)

For more discussion along these lines see at motivic quantization.

Perturbatively for free field theory in BV-formalism

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.

The path integral in the bigger picture

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

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 dd-dimensional quantum field theory in dimension dd: also the assignment to higher codimensions is to be regarded as part of the path integral;

    • 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 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

More recently, one sees attempts to formalize this observation of Freed’s, notably in the context of the cobordism hypothesis:

based on material (on categories of “families”) in On the Classification of Topological Field Theories .

References

General

The original articles:

reprinted in:

and then the original monograph:

Textbook accounts:

See also:

  • Yen Chin Ong, Where is the Commutation Relation Hiding in the Path Integral Formulation? [pdf, pdf]

Further lecture notes:

Discussion in constructive quantum field theory:

The worldline path integral as a way to compute scattering amplitudes in QFT was understood in

On dicretization of path integrals for AKSZ sigma-models:

Stochastic integration theory

The following articles use the integration over Wiener measures on stochastic processes for formalizing the path ingegral.

  • James Norris, A complete differential formalism for stochastic calculus in manifolds, Séminaire de probabilités de Strasbourg, 26 (1992), p. 189-209 (NUMDAM)

  • 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)

For charged particle/path integral of holonomy functional

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)

  • Florian Hanisch, Matthias Ludewig, A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold, (arXiv:1709.10027)

A discussion for phase spaces equipped with a Kähler polarization and a prequantum line bundle is in

  • Laurent Charles, Feynman path integral and Toeplitz Quantization, Helv. Phys. Acta 72 (1999) 341., (pdf)

following Norris 92, theorem (34).

More

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

  • Adrian P. C. Lim, Chern-Simons Path Integral on ℝ 3\mathbb{R}^3 using Abstract Wiener Measure (pdf)

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

Last revised on February 22, 2025 at 21:37:44. See the history of this page for a list of all contributions to it.