nLab
path integral

under construction

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 $U:\mathrm{Cob}\to \mathrm{Vect}$ 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 $\varphi$ with prescribed boundary datat $x$ and $y$. Formally one writes

$$K(x,y)=\int \exp (iS(\varphi ))D\varphi$$K(x,y) = \int \exp(i S(\phi))\; D\phi

and calls this the path integral. Here the expression $D\varphi$ is supposed to allude to an measure integral on the space of all $\varphi$. 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 $\varphi$ is a path $\varphi :[0,1]\to X$ 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.

Examples

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.

The Schrödinger equation says that the rate at which the phase of an energy eigenvector rotates is proportional to its energy:

(1)$$i\hslash \frac{d}{\mathrm{dt}}\psi =H\psi .$$i \hbar \frac{d}{dt} \psi = H \psi.

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

(2)$$⟨{\psi }_F\mid e^{-\mathrm{iHt}}\mid {\psi }_I⟩.$$\langle \psi_F|e^{-iHt}|\psi_I\rangle.

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

(3)$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mid q⟩⟨q\mid =1$$\int_{-\infty}^{\infty}|q\rangle\langle q| = 1

to get

(4)$$⟨{\psi }_F\mid e^{-\mathrm{iH}\Delta t}({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mid q_{N-1}⟩⟨q_{N-1}\mid {\mathrm{dq}}_{N-1})e^{-\mathrm{iH}\Delta t}({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mid q_{N-2}⟩⟨q_{N-2}\mid {\mathrm{dq}}_{N-2})e^{-\mathrm{iH}\Delta t}\cdots e^{-\mathrm{iH}\Delta t}({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mid q_1⟩⟨q_1\mid {\mathrm{dq}}_1)e^{-\mathrm{iH}\Delta t}\mid {\psi }_I⟩$$\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

(5)$$={\int }_{q_1}\cdots {\int }_{q_{N-2}}{\int }_{q_{N-1}}⟨{\psi }_F\mid e^{-\mathrm{iH}\Delta t}\mid q_{N-1}⟩⟨q_{N-1}\mid e^{-\mathrm{iH}\Delta t}\mid q_{N-2}⟩⟨q_{N-2}\mid e^{-\mathrm{iH}\Delta t}\cdots e^{-\mathrm{iH}\Delta t}\mid q_1⟩⟨q_1\mid e^{-\mathrm{iH}\Delta t}\mid {\psi }_I⟩{\mathrm{dq}}_{N-1}{\mathrm{dq}}_{N-2}\cdots {\mathrm{dq}}_1$$= \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 $H=p^2/2m.$ Looking at an individual term $⟨q_{n+1}\mid e^{-\mathrm{iH}\Delta t}\mid q_n⟩,$ we can insert a factor of 1 and solve to get

(6)$$\begin{array}{ccc}⟨q_{n+1}\mid e^{-\mathrm{iH}\Delta t}({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{dp}}{2\pi }\mid p⟩⟨p\mid )\mid q_n⟩& =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{dp}}{2\pi }{e}^{-{\mathrm{ip}}^2\Delta t/2m}⟨q_{n+1}\mid p⟩⟨p\mid q_n⟩\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{dp}}{2\pi }{e}^{-{\mathrm{ip}}^2\Delta t/2m}{e}^{\mathrm{ip}(q_{n+1}-q_n)}\\ & =& {\left(\frac{-i2\pi m}{\Delta t}\right)}^{\frac{1}{2}}{e}^{i\Delta t(m/2)[(q_{n+1}-q_n)/\Delta t{]}^2}.\end{array}$$\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

(7)$$\int \mathrm{Dq}=\underset{N\to \mathrm{\infty }}{\lim }{\left(\frac{-i2\pi m}{\Delta t}\right)}^{\frac{N}{2}}\prod _{n=0}^{N-1}\int {\mathrm{dq}}_n,$$\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 $\Delta t\to 0,N\to \mathrm{\infty },$ we get

(8)$$⟨{\psi }_F\mid e^{-\mathrm{iHt}}\mid {\psi }_I⟩=\int \mathrm{Dq}{e}^{i{\int }_0^t\mathrm{dt}\frac{1}{2}m{\dot{q}}^2}.$$\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 $H=\frac{p^2}{2m}+V(x),$ we get

(9)$$\begin{array}{ccc}⟨{\psi }_F\mid e^{-\mathrm{iHt}}\mid {\psi }_I⟩& =& \int \mathrm{Dq}{e}^{i{\int }_0^t\mathrm{dt}\frac{1}{2}m{\dot{q}}^2-V(x)}\\ & =& \int \mathrm{Dq}{e}^{i{\int }_0^tℒ(\dot{q},q)\mathrm{dt}}\\ & =& \int \mathrm{Dq}{e}^{\mathrm{iS}(q)},\end{array}$$\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 $S(q)$ is the action functional.

Is there an easy way to see how the Hamiltonian transforms into the Lagrangian in the exponent?

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

• Dan Freed

  • Quantum groups from path integrals (arXiv)

  • Higher algebraic structures and quantization (arXiv)

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;

  • 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:

• 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;

• see path integral as a pull-push transform

• and something similar or is indicated in section 3 and section 6 of

  • Dan Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups (arXiv:0905.0731)

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

Related concepts

• fermionic path integral

References

The original textbook reference is

• Richard Feynman, A. R. Hibbs, , Quantum Mechanics and Path Integrals , New York: McGraw-Hill, (1965)

Modern precise formulations of path integral technology for quantum mechanics can be found for instance in

• Christian Bär, Frank Pfäffle, Path integrals on manifolds by finite dimensional approximation, J. reine angew. Math., (2008), 625: 29-57. (arXiv:math.AP/0703272)

This discusses the path integral for the sigma-model given by a particle propagating on a Riemannian manifold and charged under a gauge field given by a connection on a bundle.

• Dana Fine, Stephen Sawin, A Rigorous Path Integral for Supersymmetric Quantum Mechanics and the Heat Kernel (arXiv:0705.0638)

(Usually the quadratic Hamiltonians are the ones for which the integral is well understood in several approaches; and of course many cases corresponding to the exactly solvable models. )

Other references on mathematical aspects of path integrals include

• Sergio Albeverio?, Raphael Hoegh-Krohn?, Sonia Mazzucchi, Mathematical theory of Feynman path integrals. An introduction (ZMATH

• Pierre Cartier, Cecile DeWitt-Morette?, Functional integration: action and symmetries (ZMATH)

• Edward Witten, A new look at the path integral of quantum mechanics, (arxiv/1009.6032)

Lecture notes on quantum field theory, emphasizing mathematics of the Euclidean path integrals and the relation to statistical physics are at

• AJ Tolland, Wilsonian QFT for Mathematicians

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