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 $U : Cob \to Vect$ 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 data $x$ and $y$. Formally one writes

$$K(x,y) = \int \exp\big( \mathrm{i} S(\phi)\big) \; D\phi$$

and calls this the path integral. Here the expression $D \phi$ is supposed to allude to a 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 Feynman 1942 (and, less famously so, in fact to Dirac 1933), who motivated the idea in non-relativistic quantum mechanics. In that context the notion can typically be made precise and can be 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

• Elementary description in quantum mechanics.

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

• As an integral against the Wiener measure.

Then we indicate a formulation in perturbation theory and BV-formalism in

• Perturbatively in the BV-formalism

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:

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

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

$$\langle \psi_F|e^{-iHt}|\psi_I\rangle.$$

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

$$\int_{-\infty}^{\infty}|q\rangle\langle q| = 1$$

to get

$$\langle \psi_F| e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} |q_{N-1} \rangle \langle q_{N-1}| dq_{N-1}\right) e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} |q_{N-2} \rangle \langle q_{N-2}| dq_{N-2}\right) e^{-iH\Delta t} \cdots e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} |q_1 \rangle \langle q_1| dq_1\right) e^{-iH\Delta t} |\psi_I\rangle$$

$$= \int_{q_1} \cdots \int_{q_{N-2}} \int_{q_{N-1}} \langle \psi_F| e^{-iH\Delta t} |q_{N-1} \rangle \langle q_{N-1}| e^{-iH\Delta t} |q_{N-2} \rangle \langle q_{N-2}| e^{-iH\Delta t} \cdots e^{-iH\Delta t} |q_1 \rangle \langle q_1| e^{-iH\Delta t} |\psi_I\rangle dq_{N-1} dq_{N-2} \cdots dq_1$$

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

$$\array{\langle q_{n+1}| e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} \frac{dp}{2\pi}|p\rangle \langle p|\right)|q_{n} \rangle &=& \int_{-\infty}^{\infty} \frac{dp}{2\pi} e^{-ip^2\Delta t/2m} \langle q_{n+1}|p\rangle \langle p|q_{n} \rangle \\ &=& \int_{-\infty}^{\infty} \frac{dp}{2\pi} e^{-ip^2\Delta t/2m} e^{ip(q_{n+1}-q_n)} \\ &=& \left(\frac{-i 2\pi m}{\Delta t}\right)^{\frac{1}{2}} e^{i \Delta t (m/2)[(q_{n+1}-q_n)/\Delta t]^2}.}$$

Defining

$$\int Dq = \lim_{N \to \infty} \left(\frac{-i 2\pi m}{\Delta t}\right)^{\frac{N}{2}} \prod_{n=0}^{N-1} \int dq_n,$$

and letting $\Delta t \to 0, N \to \infty,$ we get

$$\langle \psi_F|e^{-iHt}|\psi_I\rangle = \int Dq e^{i \int_0^t dt \frac{1}{2}m \dot{q}^2}.$$

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

$$\array{\langle \psi_F|e^{-iHt}|\psi_I\rangle &=& \int Dq e^{i \int_0^t dt \frac{1}{2}m \dot{q}^2 - V(x)} \\ &=& \int Dq e^{i\int_0^t\mathcal{L}(\dot{q},q) dt} \\ &=& \int Dq e^{iS(q)}, }$$

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

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 of the gauge-coupling action against 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

$$\exp(i S) \coloneqq \exp(2\pi i \int_{(-)} [(-),\nabla] ) \colon [I, X]_{x_0,x_1} \to Hom(E_{x_0}, E_{x_1}) \,,$$

where $E \to X$ 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

$$d\mu_W = [\exp(-S_{kin})D\gamma]$$

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

$$U(x_0,x_1) = \int_{\gamma} tra(\nabla)(\gamma) \, [\exp(-S_{kin}(\gamma)) D\gamma] \,,$$

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.)

Remark

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 $\exp(i \int_\gamma A)$).

Remark

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 $\mathbf{B}U(1)$

$$\array{ && [I,X] \\ & {}^{(-)|_0}\swarrow && \searrow^{(-)|_1} \\ X && \swArrow_{\exp(i S)} && X \\ & {}_{\mathllap{\chi(\nabla)}}\searrow && \swarrow_{\mathrlap{\chi(\nabla)}} \\ && \mathbf{B}U(1) }$$

(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:

$$\array{ && X \times X \\ & {}^{p_1}\swarrow && \searrow^{p_2} \\ X && \swArrow_{\int_{\gamma}\exp(i S(\gamma)) [\exp(-S_{kin}(\gamma))D\gamma]} && X \\ & {}_{\mathllap{\rho(\chi(\nabla))}}\searrow && \swarrow_{\mathrlap{\rho(\chi(\nabla))}} \\ && \mathbb{C}\mathbf{Mod} \,. }$$

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.

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
$d_q =$	$Q$ +	$\{I,-\}$ +	$\hbar \Delta$

Properties

Quantum commutators are time-ordered ordinary products of observables

The path integral formulation (and generally the notion of time-ordered products satisfying the Schwinger-Dyson equation) reveals the following foundational fact of quantum physics, which is “well known” but not widely appreciated (most textbooks don’t mention it).

As slogans, in slightly increasing order of accuracy:

Slogan: The quantum (operator) product of observables is their ordinary product after slightly shifting their time domains into operator order.

Or more technically:

Slogan: The operator product $O_2(t) \star O_1(t)$ of observables at equal time $t$ is their ordinary product after slightly shifting the observation $O_2$ to after $O_1$, hence is $\underset{\underset{\epsilon \to_+ 0}{\longrightarrow}}{\lim} O_2(t + \epsilon) O_1(t)$.

Or rather:

Slogan: The non-commutativity of quantum observables (such as witnessed by the canonical commutator between field observables and their canonical momenta) reflects that the temporal order of observation matters, hence reflects the difference $\underset{\underset{\epsilon \to_+ 0}{\longrightarrow}}{\lim} \big( O_2(t + \epsilon) O_1(t) - O_1(t + \epsilon) O_2(t)\big)$.

Here these (limits of) ordinary products of ordinary observables (on $\mathbb{C}$-valued functions of physical configurations) are to be understood as expectation values as produced by a path integral with respect to some (arbitrary) state. We proceed to say this in more technical detail.

This insight goes back to Feynman 1948 p. 381, who considered it in the context of non-relativistic quantum mechanics, reviewed below in:

• In Quantum Mechanics

But this generalizes to relativistic quantum field theory, discussed below in:

• In Quantum Field Theory.

In fact, the analogous statement remains true also in light-front quantization (cf. Rem. below), where it says that the canonical commutators are given by ordinary products of observables after shifting their light-front-parameter domain into operator order.

In Quantum Mechanics

The following is the original observation of Feynman 1948, p. 381.

(This has been recalled by Feynman, Hibbs & Styer 2010 (7.45); Schulman 1981, Ch. 8; Nagaosa 1999, pp. 33; Ong 2012; Rischke 2021 Section 5.6, but all these authors follow Feynman 1948 essentially verbatim. In particular, none actively recognizes the Schwinger-Dyson equation in the argument nor comments on generalization beyond the 1d discretized nonrelativistic path integral that Feynman considered and which we recall now.)

Consider the path integral for a particle propagating on a circle $S^1$, and approximated by an ordinary integral over positions $x_t$ at $N$ discrete time steps $t \in \mathbf{N} \coloneqq \{0, 1, \cdots, N-1\}$, hence over discretized trajectories

$$x \colon \mathbf{N} \longrightarrow S^1 \mathrlap\,.$$

To recall that the quantum expectation value of an observable $O \colon (S^1) ^{\mathbf{N}} \longrightarrow \mathbb{C}$ with respect to a pure quantum state $\psi \colon S^1 \longrightarrow \mathbb{C}$ is expressed as the following (discretized) path integral:

(1)$$\big\langle O \big\rangle \;\coloneqq\; \tfrac{1}{\mathcal{N}} \int O(x) \, \exp\big(\tfrac{\mathrm{i}}{\hbar} S(x)\big) \, \psi^\ast(x_N) \psi(x_0) \, D x \mathrlap{\,,}$$

where

$$\mathcal{N} \coloneqq \int \exp\big(\tfrac{\mathrm{i}}{\hbar} S(x)\big) \, \psi^\ast(x_N) \psi(x_0) \, D x$$

is the normalization factor (the “partition function”), and where

$$\int D x \,\coloneqq\, \int_{S^1} \cdots \int_{S^1} \mathrm{d}x_0 \cdots \mathrm{d}x_{\mathbf{N}-1} \mathrlap{\,.}$$

With that simple setup, ordinary integration by parts gives for an observable which is a partial derivative,

$$O(x) \,=\, \tfrac{\partial F}{\partial x_t} (x) \,, \phantom{--} 1 \lt t \lt N \mathrlap{\,,}$$

that its expectation value is equivalently expressed as:

(2)$$\begin{aligned} \big\langle O \big\rangle & \equiv \big\langle \tfrac{\partial F}{\partial x_t} \big\rangle \\ & -\tfrac{\mathrm{i}}{\hbar} \big\langle F \tfrac{\partial S}{\partial x_t} \big\rangle \end{aligned}$$

(which we may recognize as the 1d discretized form of what is now called the Schwinger-Dyson equation in quantum field theory more generally).

Specializing this to the free non-relativistic particle of mass $m \gt 0$, for which the discretized action functional is

$$S(x) \,=\, \sum_{1 \leq t \lt N} \tfrac{m}{2} ( x_{t+1} - x_{t} )^2 \tfrac{1}{N} \mathrlap{\,,}$$

the key point to observe is that

$$\tfrac{\partial S}{\partial x_t} \,=\, m \tfrac{ (x_{t} - x_{t-1}) }{1/N} - m \tfrac{ (x_{t+1} - t_n) }{1/N} \mathrlap{\,.}$$

Using this when entering equation (2) with the choice

$$F \coloneqq x_t$$

gives:

$$\mathrm{i}\hbar = \big\langle x_t \, m \tfrac{ (x_{t} - x_{t-1}) }{1/N} \big\rangle - \big\langle m \tfrac{ (x_{t+1} - x_{t}) }{1/N} \, x_t \big\rangle \mathrlap{\,.}$$

Here we recognize

$$p_{t+1/2} \coloneqq m \tfrac{ (x_{t+1} - x_{t}) } {1/N}$$

as the discrete approximation to the momentum observable at time $t + 1/2$, in terms of which we have found that:

(3)$$\begin{aligned} \mathrm{i}\hbar & = \big\langle x_t \cdot p_{t - 1/2} \,-\, p_{t + 1/2} \cdot x_t \big\rangle \,. \end{aligned}$$

In the time continuum limit, this becomes

$$\mathrm{i}\hbar \,=\, \big\langle x_t \cdot p_{t - \epsilon} \,-\, p_{t + \epsilon} \cdot x_t \big\rangle$$

for $\epsilon \to 0$.

But this is clearly the path integral expression for what in operator formalism is the canonical commutation relation

$$\mathrm{i}\hbar = \hat x \cdot \hat p - \hat p \cdot \hat x \,.$$

In conclusion, the observable corresponding to a quantum operator product $B \cdot A$ of observables at times $t$ may be thought of as the result of first shifting the temporal supports of the observables so that $B$ is observation at a time just a little after that of $A$, and then forming the ordinary product of observed values.

As Feynman 1948 also noticed, the same conclusion holds with an ordinary potential energy term included in the action functional, since its contribution is non-singular and hence vanishes in the final $\epsilon\to 0$ limit.

In Quantum Field Theory

In fact, by using the Schwinger-Dyson equation, this argument generalizes (cf. physics.SE:685812) from the quantum mechanics of a nonrelativistic particle to general quantum field theories with ordinary potential energy terms, as follows.

(Conversely, the product of observable-values in the path integral corresponds to the time-ordered product of the corresponding linear operators (eg. Polchinski 1998 (A.1.17); Rischke 2021 (5.63).)

Imagine a path integral-formulation exists of some $1+d$-dimensional quantum field theory determined by a Lagrangian density $L$ with an ordinary potential energy term and denote the corresponding expectation values in some state by $\langle-\rangle$ — or else regard $\langle-\rangle$ as denoting the time-ordered product of its arguments, that’s all we need.

Let $\phi$ be one of the field species. (It could be a scalar field but it may just as well be a component of any more complex field.)

Assuming we are on cylindrical Minkowski spacetime $\mathbb{R}^{1,d} / \mathbb{Z}^{d}$ — just for notational simplicity — then the Schwinger-Dyson equation for field insertion $\phi(y)$ says that

(4)$$\bigg\langle \underset { \delta S / \delta \phi(x) }{ \underbrace{ \Big( \frac{ \partial L }{ \partial \phi } (x) - \partial_\mu \frac{ \partial L }{ \partial (\partial_\mu \phi) } \Big) } } \, \phi(y) \bigg\rangle \;=\; \mathrm{i}\hbar \left\langle \frac{ \partial \phi(y) }{ \partial \phi}(x) \right\rangle \;=\; \mathrm{i}\hbar \, \delta^{1+d}(x-y) \mathrlap{\,.}$$

This is the field theoretic version of Feynman’s equation (2) above.

Now consider the integration of this expression in the variable $x$ over the spacetime region in a small time interval $(y^0- \epsilon, y^0 + \epsilon) \times \mathbb{R}^{d}/\mathbb{Z}^{d}$ and let $\epsilon \to 0$. Then:

1. the first summand on the left of (4) vanishes (being asymptotically proportional to $\epsilon$ since we are assuming that the potential term and hence the $\phi$-dependence of $L$ is that of an ordinary smooth function),

2. by Stokes's theorem the spatial integral over the spatial components of the second summand vanishes and

3. the remaining temporal integral of its temporal component gives two boundary terms (where we now decompose $x = (x^0, \vec x)$):

(5)$$\underset{\underset{\epsilon\to 0}{\longrightarrow}}{\lim} \int_{\mathbb{R}^{d}/\mathbb{Z}^{d}} \mathrm{d}^d \vec x \, \left\langle \frac{ \partial L }{ \partial (\partial_0 \phi) } (y^0 + \epsilon, \vec x) \, \phi(y^0, \vec y) \;-\; \frac{ \partial L }{ \partial (\partial_0 \phi) } (y^0 - \epsilon, \vec x) \, \phi(y^0, \vec y) \right\rangle \;=\; -\mathrm{i}\hbar \mathrlap{\,.}$$

Here we recognize the canonical momentum $\pi$ to the field $\phi$:

$$\pi(x) \;\coloneqq\; \frac{ \partial L }{ \partial (\partial_0 \phi) }(x) \mathrlap{\,,}$$

so that

(6)$$\underset{\underset{\epsilon\to 0}{\longrightarrow}}{\lim} \int_{\mathbb{R}^{d}/\mathbb{Z}^{d}} \mathrm{d}^d x \, \Big\langle \pi(y^0 + \epsilon, \vec x) \, \phi(y^0, \vec y) \;-\; \pi(y^0 - \epsilon, \vec x) \, \phi(y^0, \vec y) \Big\rangle \;=\; -\mathrm{i}\hbar \mathrlap{\,.}$$

This is the field-theoretic version of Feynman’s equation (3) above.

We may redo this derivation after multiplication of the original Schwinger-Dyson equation (4) with any “smearing function” $f(\vec x)$ (a spatial bump function). Then where we used Stokes' theorem above we are now faced with an integration by parts that picks up terms proportional to the gradient of $f$ — but if the dependence of $L$ on spatial derivatives of $\phi$ does not have unusual singularities (i.e. if the kinetic energy term in $L$ is a standard one) then these terms vanish with $\epsilon$ just as the potential energy term does, and hence we end up with

(7)$$\underset{\underset{\epsilon\to 0}{\longrightarrow}}{\lim} \int_{\mathbb{R}^{d}/\mathbb{Z}^{d}} \int \mathrm{d}^d \vec x \, f(\vec x) \, \Big\langle \pi(y^0 + \epsilon, \vec x) \, \phi(y^0, \vec y) \;-\; \pi(y^0 - \epsilon, \vec x) \, \phi(y^0, \vec y) \Big\rangle \;=\; -\mathrm{i}\hbar f(\vec y) \mathrlap{\,.}$$

But since this holds for all smearing functions $f$, this is equivalent to the distributional equation

(8)$$\underset{\underset{\epsilon\to 0}{\longrightarrow}}{\lim} \Big\langle \pi(y^0 + \epsilon, \vec x) \, \phi(y^0, \vec y) \;-\; \pi(y^0 - \epsilon, \vec x) \, \phi(y^0, \vec y) \Big\rangle \;=\; -\mathrm{i}\hbar \, \delta^d(\vec x - \vec y) \mathrlap{\,,}$$

which is the claimed incarnation of the canonical commutation relation of field operators at equal times,

$$\big[ \widehat{\pi}(\vec x), \widehat{\phi}(\vec x) \big] \,=\, -\mathrm{i}\hbar \, \delta^d(\vec x - \vec y) \mathrlap{\,,}$$

now re-expressed as an expectation value of ordinary products of observables after shifting their temporal domains into operator order.

Remark

The analogous conclusion holds also for light front quantization, with the role of the time coordinate $x^0$ now played by the light front parameter $x^+$, for

$$x^\pm \coloneqq (x^0 \pm x^d)/\sqrt{2} \,.$$

Here the light-front canonical momentum to a field $\phi$ is (cf. Burkardt 1996 table 2.1 for the following equations):

$$\pi \,=\, \frac { \partial L } { \partial (\partial_+ \phi) } \mathrlap{\,,}$$

which for Lagrangian densities with standard kinetic energy term

$$L \,=\, \partial_+ \phi \, \partial_- \phi \,-\, \tfrac{1}{2} (\vec \nabla_\perp \phi)^2 - V(\phi)$$

comes out as

$$\pi \,=\, \partial_- \phi \mathrlap{\,.}$$

While the nature of this light front momentum in canonical quantization (where it is a second class constraint) is quite different from the nature of the canonical momentum in instant form, at the end the equal-LF-parameter commutation relation has the same form as the usual equal-time commutator:

$$\Big[ \widehat{\pi}\big(x^+, x^-, \vec x_\perp\big) ,\, \widehat{\phi}\big(x^+, y^-, \vec y_\perp\big) \Big] \,=\, -\mathrm{i}\hbar \tfrac{1}{2} \delta\big(x^- - y^-\big) \delta^{d-1}\big(\vec x_\perp - \vec y_\perp\big) \mathrlap{\,.}$$

And so the above Schwinger-Dyson argument, just with the time coordinate $x^0$ replaced by the light front parameter $x^+$, reproduces this in the form:

(9)$$\begin{array}{l} \underset{\underset{\epsilon\to 0}{\longrightarrow}}{\lim} \Big\langle \pi\big(y^+ + \epsilon, x^-, \vec x_\perp\big) \, \phi\big(y^+, y^-, \vec y_\perp\big) \;-\; \pi\big(y^+ - \epsilon, x^-, \vec x_\perp\big) \, \phi\big(y^+, y^-, \vec y_\perp\big) \Big\rangle \\ \;=\; -\mathrm{i}\hbar \, \tfrac{1}{2} \delta(x^- - y^-) \delta^{d-1}(\vec x_\perp - \vec y_{\perp}) \mathrlap{\,.} \end{array}$$

(Just beware the somewhat subtle factor of $1/2$ on the right of (9). In the constrained canonical quantization this factor may be found discussed carefully in Burkardt 1996 §A p. 76. In the path integral picture the factor arises more transparently as a factor of $2$ on the left, originating in: $\delta(\partial_+ \phi \partial_- \phi)/\delta\phi = -\partial_+ \partial_- \phi - \partial_- \partial_+ \phi = -2 \partial_+ \partial_- \phi$.)

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: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;

  • 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

• Feynman-Kac formula

• S-matrix

• Dyson formula

• fermionic path integral

• principle of extremal action

• integration over infinite-dimensional manifolds

• cohomological integration, BV-BRST quantization

• Schwinger-Dyson equation

References

General

The original articles:

• Paul A. M. Dirac: The Lagrangian in Quantum Mechanics, Physikalische Zeitschrift der Sowjetunion, Band 3, Heft 1 (1933) 64–72 [pdf], reprinted in Brown 2005 [doi:10.1142/9789812567635_0003]

• Richard P. Feynman: The Principles of Least Action in Quantum Mechanics, PhD thesis (1942), reprinted in Brown 2005 [doi:10.1142/9789812567635_0001]

• Richard P. Feynman: Space-Time Approach to Non-Relativistic Quantum Mechanics, Rev. Mod. Phys. 20 (1948) 367 [doi:10.1103/RevModPhys.20.367, pdf, pdf]

reprinted in:

• Laurie M. Brown, Feynman’s Thesis — A New Approach to Quantum Theory, World Scientific (2005) [doi:10.1142/5852]

and then the original monograph:

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

• Daniel F. Styer, The Errors of Feynman and Hibbs [pdf]

• Richard P. Feynman, Albert R. Hibbs, Daniel F. Styer: Quantum Mechanics and Path Integrals: Emended Edition, Dover (2010) [ISBN:0486477223, pdf]

See also:

• Ted Jacobson: Feynman 1947 letter on path integral for the Dirac equation [arXiv:2408.15070]

  (on early attempts at a path integral for spinning particles)

• Daniel Parrochia: Some remarks on history and pre-history of Feynman path integral [arXiv:1907.11168]

Further textbook accounts:

• L. S. Schulman: Techniques and Applications of Path Integrals, Wiley & Sons (1981, 1996), Dover (2005) [ISBN:9780486445281, ch. 8: pdf]

• John W. Negele, Henri Orland, Functional Integral Formulation, §2.2 in: Quantum Many-Particle Systems, Westview Press (1988, 1998) [doi:10.1201/9780429497926]

• Hagen Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific (1990, 1994, 2003, 2006, 2009) [doi:10.1142/7305, pdf]

• Joseph Polchinski, A short course on path integrals, Appendix A in: String theory part I: An Introduction to the Bosonic String, Cambridge University Press (1998) [doi:10.1017/CBO9780511816079, pdf]

• Naoto Nagaosa, Quantization with Path Integral Methods, Section 2 of: Quantum Field Theory in Condensed Matter Physics, Texts and Monographs in Physics, Springer (1999) [doi:10.1007/978-3-662-03774-4_2, pdf]

• G. Johnson, M. Lapidus, The Feynman integral and Feynman’s operational calculus, Oxford University Press (2000)

• Barry Simon: Functional integration and quantum physics, AMS Chelsea Publishing 351, AMS (2005) [ISBN:978-0-8218-3582-1, pdf]

• Pierre Cartier, Cécile DeWitt-Morette, Functional integration: action and symmetries, Cambridge Monographs on Mathematical Physics (2006) [ISBN:9780521143578]

• Daisuke Fujiwara, Rigorous Time Slicing Approach to Feynman Path Integrals (2017) Springer [doi:/10.1007/978-4-431-56553-6]

See also:

• Yen Chin Ong: Where is the Commutation Relation Hiding in the Path Integral Formulation? (2012) [pdf]

• Giampiero Esposito, Tepper L. Gill: On the physical and mathematical foundations of quantum physics via functional integrals [arXiv:2408.16404]

Further lecture notes:

• R. Rosenfelder, Path Integrals in Quantum Physics [arXiv:1209.1315]

• Dirk Rischke, Path Integrals in Quantum Mechanics, lecture notes (2021) [pdf]

Discussion in constructive quantum field theory:

• James Glimm, Arthur Jaffe, Quantum physics: a functional integral point of view, Springer (1981, 1987) [doi:10.1007/978-1-4612-4728-9]

• 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

• Matthew Strassler, Field Theory Without Feynman Diagrams: One-Loop Effective Actions, Nucl. Phys. B385:145-184,1992 (arXiv:hep-ph/9205205)

On discretization of path integrals for AKSZ sigma-models:

• Joshua Lackman, A Groupoid Construction of Functional Integrals: Brownian Motion and Some TQFTs [arXiv:2402.05866]

• Joshua Lackman, A Groupoid Construction of Functional Integrals, talk at CQTS (Feb 2024) [video:kt]

• Joshua Lackman, A Mathematical Definition of Path Integrals on Symplectic Manifolds [arXiv:2406.14547]

• Joshua Lackman: On an Axiomatization of Path Integral Quantization and its Equivalence to Berezin’s Quantization [arXiv:2410.02739]

Conversely, on using an AKSZ sigma-model for understanding path integral quantization of symplectic manifolds:

• Thomas Basile, Nicolas Boulanger, Arghya Chattopadhyay: Reframing classical mechanics: An AKSZ sigma model perspective [arXiv:2504.18826]

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 F. Sawin: A Rigorous Path Integral for Supersymmetric Quantum Mechanics and the Heat Kernel, Commun. Math. Phys. 284 (2008) 79–91 [arXiv:0705.0638, doi:10.1007/s00220-008-0606-2]

  (for supersymmetric quantum mechanics)

• 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

• Pierre Cartier, Cécile DeWitt-Morette, Functional integration: action and symmetries, Cambridge Monographs on Mathematical Physics, 2006. ISBN: 9780521866965. doi.

• Marc Henneaux, Claudio Teitelboim, Quantization of Gauge Systems, Princeton University Press 1992

• Nikolai Reshetikhin, Lectures on quantization of gauge systems, arxiv/1008.1411

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

Detailed rigorous discussion for quadratic Hamiltonians and for phase space paths in in

• Joel Robbin, Dietmar Salamon, Feynman path integrals on phase space and the metaplectic representation in Dietmar Salamon (ed.), Symplectic Geometry, LMS Lecture Note series 192 (1993) (pdf)

Discussion of quantization of Chern-Simons theory via a Wiener measure is in

• Adrian P. C. Lim. Chern-Simons Path Integral on $\mathbb{R}^3$ Using Abstract Wiener Measure. Commun. Math. Anal., vol. 11, num. 2; 1-22 (2011). (Euclid, MR2780879).

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

• Theo Johnson-Freyd, The formal path integral and quantum mechanics, J. Math. Phys. 51, 122103 (2010) arxiv/1004.4305, doi; On the coordinate (in)dependence of the formal path integral, arxiv/1003.5730