Editing Path integral formulation (section)

== Path integral in quantum mechanics ==

=== Time-slicing derivation ===
{{main|Relation between Schrödinger%27s equation and the path integral formulation of quantum mechanics}}
One common approach to deriving the path integral formula is to divide the time interval into small pieces. Once this is done, the [[Lie product formula|Trotter product formula]] tells us that the noncommutativity of the kinetic and potential energy operators can be ignored.

For a particle in a smooth potential, the path integral is approximated by [[zigzag]] paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position {{mvar|x<sub>a</sub>}} at time {{mvar|t<sub>a</sub>}} to {{mvar|x<sub>b</sub>}} at time {{mvar|t<sub>b</sub>}}, the time sequence
: <math>t_a = t_0 < t_1 < \cdots < t_{n-1} < t_n < t_{n+1} = t_b</math>
can be divided up into {{math|''n'' + 1}} smaller segments {{math|''t<sub>j</sub>'' − ''t''<sub>''j'' − 1</sub>}}, where {{math|''j'' {{=}} 1, ..., ''n'' + 1}}, of fixed duration
: <math>\varepsilon = \Delta t = \frac{t_b - t_a}{n + 1}.</math>

This process is called ''time-slicing''.

An approximation for the path integral can be computed as proportional to
: <math>\int\limits_{-\infty}^{+\infty} \cdots \int\limits_{-\infty}^{+\infty}
 \exp \left(\frac{i}{\hbar}\int_{t_a}^{t_b} L\big(x(t), v(t)\big) \,dt\right) \,dx_0 \, \cdots \, dx_n, </math>
where {{math|''L''(''x'', ''v'')}} is the Lagrangian of the one-dimensional system with position variable {{math|''x''(''t'')}} and velocity {{math|''v'' {{=}} ''ẋ''(''t'')}} considered (see below), and {{mvar|dx<sub>j</sub>}} corresponds to the position at the {{mvar|j}}th time step, if the time integral is approximated by a sum of {{mvar|n}} terms.<ref group=nb>For a simplified, step-by-step derivation of the above relation, see [http://www.quantumfieldtheory.info/website_Chap18.pdf Path Integrals in Quantum Theories: A Pedagogic 1st Step].</ref>

In the limit {{mvar|''n'' → ∞}}, this becomes a [[functional integral]], which, apart from a nonessential factor, is directly the product of the probability amplitudes {{math|{{bra-ket|''x<sub>b</sub>'', ''t<sub>b</sub>''|''x<sub>a</sub>'', ''t<sub>a</sub>''}}}} (more precisely, since one must work with a continuous spectrum, the respective densities) to find the quantum mechanical particle at {{mvar|t<sub>a</sub>}} in the initial state {{mvar|x<sub>a</sub>}} and at {{mvar|t<sub>b</sub>}} in the final state {{mvar|x<sub>b</sub>}}.

Actually {{mvar|L}} is the classical [[Lagrangian mechanics|Lagrangian]] of the one-dimensional system considered,
: <math> L(x, \dot x) = T-V=\frac{1}{2}m|\dot{x}|^2-V(x)</math>
and the abovementioned "zigzagging" corresponds to the appearance of the terms
: <math>\exp\left(\frac{i}{\hbar}\varepsilon \sum_{j=1}^{n+1} L \left(\tilde x_j, \frac{x_j - x_{j-1}}{\varepsilon}, j \right)\right)</math>
in the [[Riemann sum]] approximating the time integral, which are finally integrated over {{math|''x''<sub>1</sub>}} to {{mvar|x<sub>n</sub>}} with the integration measure {{math|''dx''<sub>1</sub>...''dx<sub>n</sub>''}}, {{mvar|x̃<sub>j</sub>}} is an arbitrary value of the interval corresponding to {{mvar|j}}, e.g. its center, {{math|{{sfrac|''x<sub>j</sub>'' + ''x''<sub>''j''−1</sub>|2}}}}.

Thus, in contrast to classical mechanics, not only does the stationary path contribute, but actually all virtual paths between the initial and the final point also contribute.

=== Path integral ===

In terms of the wave function in the position representation, the path integral formula reads as follows:
: <math>\psi(x,t)=\frac{1}{Z}\int_{\mathbf{x}(0)=x}\mathcal{D}\mathbf{x}\, e^{iS[\mathbf{x},\dot{\mathbf{x}}]}\psi_0(\mathbf{x}(t))\,</math>
where <math>\mathcal{D}\mathbf{x}</math> denotes integration over all paths <math>\mathbf{x}</math> with <math>\mathbf{x}(0)=x</math> and where <math>Z</math> is a normalization factor. Here <math>S</math> is the action, given by
: <math>S[\mathbf{x},\dot\mathbf{x}]=\int dt\, L(\mathbf{x}(t),\dot\mathbf{x}(t))</math>

[[File:Path integral example.webm|thumb|The diagram shows the contribution to the path integral of a free particle for a set of paths, eventually drawing a [[Cornu Spiral]].]]

=== Free particle ===

The path integral representation gives the quantum amplitude to go from point {{mvar|x}} to point {{mvar|y}} as an integral over all paths. For a free-particle action (for simplicity let {{math|''m'' {{=}} 1}}, {{math|''ħ'' {{=}} 1}})
: <math>S = \int \frac{\dot{x}^2}{2}\, \mathrm{d}t,</math>
the integral can be evaluated explicitly.

To do this, it is convenient to start without the factor {{mvar|i}} in the exponential, so that large deviations are suppressed by small numbers, not by cancelling oscillatory contributions. The amplitude (or Kernel) reads:
: <math>K(x - y; T) = \int_{x(0) = x}^{x(T) = y} \exp\left(-\int_0^T \frac{\dot{x}^2}{2} \,\mathrm{d}t\right) \,\mathcal{D}x.</math>

Splitting the integral into time slices:
: <math>K(x - y; T) = \int_{x(0) = x}^{x(T) = y} \prod_t \exp\left(-\tfrac12 \left(\frac{x(t + \varepsilon) - x(t)}{\varepsilon}\right)^2 \varepsilon \right) \,\mathcal{D}x,</math>
where the {{mathcal|D}} is interpreted as a finite collection of integrations at each integer multiple of {{mvar|ε}}. Each factor in the product is a Gaussian as a function of {{math|''x''(''t'' + ''ε'')}} centered at {{math|''x''(''t'')}} with variance {{mvar|ε}}. The multiple integrals are a repeated [[convolution]] of this Gaussian {{mvar|G<sub>ε</sub>}} with copies of itself at adjacent times:
: <math>K(x - y; T) = G_\varepsilon * G_\varepsilon * \cdots * G_\varepsilon,</math>
where the number of convolutions is {{math|{{sfrac|''T''|''ε''}}}}. The result is easy to evaluate by taking the Fourier transform of both sides, so that the convolutions become multiplications:
: <math>\tilde{K}(p; T) = \tilde{G}_\varepsilon(p)^{T/\varepsilon}.</math>

The Fourier transform of the Gaussian {{mvar|G}} is another Gaussian of reciprocal variance:
: <math>\tilde{G}_\varepsilon(p) = e^{-\frac{\varepsilon p^2}{2}},</math>
and the result is
: <math>\tilde{K}(p; T) = e^{-\frac{T p^2}{2}}.</math>

The Fourier transform gives {{mvar|K}}, and it is a Gaussian again with reciprocal variance:
: <math>K(x - y; T) \propto e^{ -\frac{(x - y)^2}{2T}}.</math>

The proportionality constant is not really determined by the time-slicing approach, only the ratio of values for different endpoint choices is determined. The proportionality constant should be chosen to ensure that between each two time slices the time evolution is quantum-mechanically unitary, but a more illuminating way to fix the normalization is to consider the path integral as a description of a [[stochastic process]].

The result has a probability interpretation. The sum over all paths of the exponential factor can be seen as the sum over each path of the probability of selecting that path. The probability is the product over each segment of the probability of selecting that segment, so that each segment is probabilistically independently chosen. The fact that the answer is a Gaussian spreading linearly in time is the [[central limit theorem]], which can be interpreted as the first historical evaluation of a statistical path integral.

The probability interpretation gives a natural normalization choice. The path integral should be defined so that
: <math>\int K(x - y; T) \,dy = 1.</math>

This condition normalizes the Gaussian and produces a kernel that obeys the diffusion equation:
: <math>\frac{d}{dt} K(x; T) = \frac{\nabla^2}{2} K.</math>

For oscillatory path integrals, ones with an {{mvar|i}} in the numerator, the time slicing produces convolved Gaussians, just as before. Now, however, the convolution product is marginally singular, since it requires careful limits to evaluate the oscillating integrals. To make the factors well defined, the easiest way is to add a small imaginary part to the time increment {{mvar|ε}}. This is closely related to [[Wick rotation]]. Then the same convolution argument as before gives the propagation kernel:
: <math>K(x - y; T) \propto e^\frac{i(x - y)^2}{2T},</math>
which, with the same normalization as before (not the sum-squares normalization – this function has a divergent norm), obeys a free Schrödinger equation:
: <math>\frac{d}{dt} K(x; T) = i \frac{\nabla^2}{2} K.</math>

This means that any superposition of {{mvar|K}}s will also obey the same equation, by linearity. Defining
: <math>\psi_t(y) = \int \psi_0(x) K(x - y; t) \,dx = \int \psi_0(x) \int_{x(0) = x}^{x(t) = y} e^{iS} \,\mathcal{D}x,</math>

then {{mvar|ψ<sub>t</sub>}} obeys the free Schrödinger equation just as {{mvar|K}} does:
: <math>i\frac{\partial}{\partial t} \psi_t = -\frac{\nabla^2}{2} \psi_t.</math>

=== Simple harmonic oscillator ===
{{see also|Propagator#Basic examples: propagator of free particle and harmonic oscillator| Mehler kernel}}
The Lagrangian for the simple harmonic oscillator is<ref>{{cite web |last1=Hilke |first1=M. |title=Path Integral |work=221A Lecture Notes |url=http://hitoshi.berkeley.edu/221A/pathintegral.pdf}}</ref>
: <math>\mathcal{L} = \tfrac12 m \dot{x}^2 - \tfrac12 m \omega^2 x^2.</math>

Write its trajectory {{math|''x''(''t'')}} as the classical trajectory plus some perturbation, {{math|''x''(''t'') {{=}} ''x''<sub>c</sub>(''t'') + ''δx''(''t'')}} and the action as {{math|''S'' {{=}} ''S''<sub>c</sub> + ''δS''}}. The classical trajectory can be written as

: <math>x_\text{c}(t) = x_i \frac{\sin\omega(t_f - t)}{\sin\omega(t_f - t_i)} + x_f \frac{\sin\omega(t - t_i)}{\sin\omega(t_f - t_i)}.</math>

This trajectory yields the classical action
: <math>
\begin{align}
S_\text{c} & = \int_{t_i}^{t_f} \mathcal{L} \,dt = \int_{t_i}^{t_f} \left(\tfrac12 m\dot{x}^2 - \tfrac12 m\omega^2 x^2 \right) \,dt \\[6pt]
& = \frac 1 2 m\omega \left( \frac{(x_i^2 + x_f^2) \cos\omega(t_f - t_i) - 2 x_i x_f}{\sin\omega(t_f - t_i)} \right)~.
\end{align}
</math>

Next, expand the deviation from the classical path as a Fourier series, and calculate the contribution to the action {{mvar|δS}}, which gives
: <math>S = S_\text{c} + \sum_{n = 1}^\infty \tfrac12 a_n^2 \frac{m}{2} \left( \frac{(n \pi)^2}{t_f - t_i} - \omega^2(t_f - t_i) \right).</math>

This means that the propagator is
: <math>
\begin{align}
K(x_f, t_f; x_i, t_i) & = Q e^\frac{i S_\text{c}}{\hbar} \prod_{j=1}^\infty \frac{j \pi}{\sqrt{2}} \int da_j \exp{\left( \frac{i}{2\hbar}a_j^2 \frac{m}{2} \left( \frac{(j \pi)^2}{t_f - t_i} - \omega^2(t_f - t_i) \right) \right)} \\[6pt]
& = e^\frac{i S_\text{c}}{\hbar} Q \prod_{j=1}^\infty \left( 1 - \left( \frac{\omega(t_f - t_i)}{j \pi} \right)^2 \right)^{-\frac12}
\end{align}
</math>
for some normalization
: <math> Q = \sqrt{\frac{m}{2\pi i \hbar (t_f - t_i)}}~. </math>

Using the infinite-product representation of the [[sinc function]],
: <math>\prod_{j=1}^\infty \left( 1 - \frac{x^2}{j^2} \right) = \frac{\sin\pi x}{\pi x}, </math>
the propagator can be written as
: <math> K(x_f, t_f; x_i, t_i) = Q e^\frac{i S_\text{c}}{\hbar} \sqrt{ \frac{\omega(t_f - t_i)}{\sin\omega(t_f - t_i)} } = e^\frac{i S_c}{\hbar} \sqrt{ \frac{m\omega}{2\pi i \hbar \sin\omega(t_f - t_i)}}.</math>

Let {{math|''T'' {{=}} ''t<sub>f</sub>'' − ''t<sub>i</sub>''}}.  One may write this propagator in terms of energy eigenstates as
: <math>
\begin{align}
K(x_f, t_f; x_i, t_i) & = \left( \frac{m \omega}{2 \pi i \hbar \sin\omega T } \right)^\frac12 \exp{ \left( \frac{i}{\hbar} \tfrac12 m \omega  \frac{ (x_i^2 + x_f^2) \cos \omega T - 2 x_i x_f }{ \sin \omega T } \right) } \\[6pt]
& = \sum_{n = 0}^\infty \exp{ \left( - \frac{i E_n T}{\hbar} \right) } \psi_n(x_f) \psi_n(x_i)^{*}~.
\end{align}
</math>

Using the identities {{math|''i'' sin ''ωT'' {{=}} {{sfrac|1|2}}''e''<sup>''iωT''</sup> (1 − ''e''<sup>−2''iωT''</sup>)}} and {{math|cos ''ωT'' {{=}} {{sfrac|1|2}}''e''<sup>''iωT''</sup> (1 + ''e''<sup>−2''iωT''</sup>)}}, this amounts to
: <math>K(x_f, t_f; x_i, t_i) = \left( \frac{m \omega}{\pi \hbar} \right)^\frac12 e^\frac{-i \omega T} 2 \left( 1 - e^{-2 i \omega T} \right)^{-\frac12} \exp{ \left( - \frac{m \omega}{2 \hbar} \left( \left(x_i^2 + x_f^2\right) \frac{ 1 + e^{-2 i \omega T} }{ 1 - e^{- 2 i \omega T}} - \frac{4 x_i x_f e^{-i \omega T}}{1 - e^{ - 2 i \omega T} }\right) \right) }.</math>

One may absorb all terms after the first {{math|''e''<sup>−''iωT''/2</sup>}} into {{math|''R''(''T'')}}, thereby  obtaining
: <math> K(x_f, t_f; x_i, t_i) = \left( \frac{m \omega}{\pi \hbar} \right)^\frac12 e^\frac{-i \omega T } 2 \cdot R(T).</math>

One may  finally expand {{math|''R''(''T'')}} in  powers of {{math|''e''<sup>−''iωT''</sup>}}: All  terms in this expansion get multiplied by the {{math|''e''<sup>−''iωT''/2</sup>}} factor in the front,  yielding terms  of the form
: <math>e^\frac{-i\omega T}{2} e^{-i n\omega T} = e^{-i \omega T \left( \frac12 + n\right) } \quad\text{for } n = 0, 1, 2, \ldots.</math>
Comparison  to the above eigenstate expansion  yields the  standard energy spectrum for the simple harmonic oscillator,
: <math>E_n = \left( n + \tfrac12 \right) \hbar \omega~.</math>

=== Coulomb potential ===
Feynman's time-sliced approximation does not, however, exist for the most important quantum-mechanical path integrals of atoms, due to the singularity of the [[Coulomb potential]] {{math|{{sfrac|''e''<sup>2</sup>|''r''}}}} at the origin. Only after replacing the time {{mvar|t}} by another path-dependent pseudo-time parameter
: <math>s = \int \frac{dt}{r(t)}</math>
the singularity is removed and a time-sliced approximation exists, which is exactly integrable, since it can be made harmonic by a simple coordinate transformation, as discovered in 1979 by [[İsmail Hakkı Duru]] and [[Hagen Kleinert]].<ref>{{harvnb|Duru|Kleinert|1979|loc=Chapter 13.}}</ref> The combination of a path-dependent time transformation and a coordinate transformation is an important tool to solve many path integrals and is called generically the [[Duru–Kleinert transformation]].

=== The Schrödinger equation ===

{{Main|Relation between Schrödinger's equation and the path integral formulation of quantum mechanics}}

The path integral reproduces the Schrödinger equation for the initial and final state even when a potential is present. This is easiest to see by taking a path-integral over infinitesimally separated times.
: <math>\psi(y;t+\varepsilon) = \int_{-\infty}^\infty \psi(x;t)\int_{x(t)=x}^{x(t+\varepsilon)=y} e^{i\int_t^{t+\varepsilon} \bigl(\frac{1}{2}\dot{x}^2 - V(x)\bigr)dt} Dx(t)\,dx\qquad (1)</math>

Since the time separation is infinitesimal and the cancelling oscillations become severe for large values of {{mvar|ẋ}}, the path integral has most weight for {{mvar|y}} close to {{mvar|x}}. In this case, to lowest order the potential energy is constant, and only the kinetic energy contribution is nontrivial. (This separation of the kinetic and potential energy terms in the exponent is essentially the [[Lie product formula|Trotter product formula]].) The exponential of the action is
: <math>e^{-i\varepsilon V(x)} e^{i\frac{\dot{x}^2}{2}\varepsilon}</math>

The first term rotates the phase of {{math|''ψ''(''x'')}} locally by an amount proportional to the potential energy. The second term is the free particle propagator, corresponding to {{mvar|i}} times a diffusion process. To lowest order in {{mvar|ε}} they are additive; in any case one has with (1):
: <math>\psi(y;t+\varepsilon) \approx \int \psi(x;t) e^{-i\varepsilon V(x)} e^\frac{i(x-y)^2 }{ 2\varepsilon} \,dx\,.</math>

As mentioned, the spread in {{mvar|ψ}} is diffusive from the free particle propagation, with an extra infinitesimal rotation in phase that slowly varies from point to point from the potential:
: <math>\frac{\partial\psi}{\partial t} = i\cdot \left(\tfrac12\nabla^2 - V(x)\right)\psi\,</math>
and this is the Schrödinger equation. The normalization of the path integral needs to be fixed in exactly the same way as in the free particle case. An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment.

=== Equations of motion ===

Since the states obey the Schrödinger equation, the path integral must reproduce the Heisenberg equations of motion for the averages of {{mvar|x}} and  {{mvar|ẋ}} variables, but it is instructive to see this directly. The direct approach shows that the expectation values calculated from the path integral reproduce the usual ones of quantum mechanics.

Start by considering the path integral with some fixed initial state
: <math>\int \psi_0(x) \int_{x(0)=x} e^{iS(x,\dot{x})}\, Dx\,</math>

Now {{mvar|''x''(''t'')}} at each separate time is a separate integration variable. So it is legitimate to change variables in the integral by shifting: {{math|''x''(''t'') {{=}} ''u''(''t'') + ''ε''(''t'')}} where {{math|''ε''(''t'')}} is a different shift at each time but {{math|''ε''(0) {{=}} ''ε''(''T'') {{=}} 0}}, since the endpoints are not integrated:
: <math>\int \psi_0(x) \int_{u(0)=x} e^{iS(u+\varepsilon,\dot{u}+\dot{\varepsilon})}\, Du\,</math>

The change in the integral from the shift is, to first infinitesimal order in {{mvar|ε}}:
: <math>\int \psi_0(x) \int_{u(0)=x} \left( \int \frac{\partial S }{ \partial u } \varepsilon + \frac{ \partial S }{ \partial \dot{u} } \dot{\varepsilon}\, dt \right) e^{iS} \,Du\,</math>
which, integrating by parts in {{mvar|t}}, gives:
: <math>\int \psi_0(x) \int_{u(0)=x} -\left( \int \left(\frac{d}{dt} \frac{\partial S}{\partial \dot{u}} - \frac{\partial S}{\partial u}\right)\varepsilon(t)\, dt \right) e^{iS}\, Du\,</math>

But this was just a shift of integration variables, which doesn't change the value of the integral for any choice of {{mvar|''ε''(''t'')}}. The conclusion is that this first order variation is zero for an arbitrary initial state and at any arbitrary point in time:
: <math>\left\langle \psi_0\left| \frac{\delta S}{\delta x}(t) \right|\psi_0 \right\rangle = 0</math>
this is the Heisenberg equation of motion.

If the action contains terms that multiply {{mvar|ẋ}} and {{mvar|x}}, at the same moment in time, the manipulations above are only heuristic, because the multiplication rules for these quantities is just as noncommuting in the path integral as it is in the operator formalism.

=== Stationary-phase approximation ===

If the variation in the action exceeds {{mvar|ħ}} by many orders of magnitude, we typically have destructive interference other than in the vicinity of those trajectories satisfying the [[Euler–Lagrange equation]], which is now reinterpreted as the condition for constructive interference. This can be shown using the method of stationary phase applied to the propagator. As {{mvar|ħ}} decreases, the exponential in the integral oscillates rapidly in the complex domain for any change in the action. Thus, in the limit that {{mvar|ħ}} goes to zero, only points where the classical action does not vary contribute to the propagator.

=== Canonical commutation relations ===

The formulation of the path integral does not make it clear at first sight that the quantities {{mvar|x}} and {{mvar|p}} do not commute. In the path integral, these are just integration variables and they have no obvious ordering. Feynman discovered that the non-commutativity is still present.<ref>{{harvnb|Feynman|1948}}</ref>

To see this, consider the simplest path integral, the brownian walk. This is not yet quantum mechanics, so in the path-integral the action is not multiplied by {{mvar|i}}:
: <math>S=  \int \left( \frac{dx}{dt} \right)^2\, dt</math>

The quantity {{mvar|''x''(''t'')}} is fluctuating, and the derivative is defined as the limit of a discrete difference.
: <math>\frac{dx}{dt} = \frac{x(t+\varepsilon) - x(t)} \varepsilon </math>

The distance that a random walk moves is proportional to {{math|{{sqrt|''t''}}}}, so that:
: <math>x(t+\varepsilon) - x(t) \approx \sqrt{\varepsilon}</math>
This shows that the random walk is not differentiable, since the ratio that defines the derivative diverges with probability one.

The quantity {{mvar|xẋ}} is ambiguous, with two possible meanings:
: <math>[1] = x \frac{dx}{dt}  = x(t) \frac{x(t+\varepsilon) - x(t) }{\varepsilon } </math>
: <math>[2] = x \frac{dx}{dt} = x(t+\varepsilon) \frac{x(t+\varepsilon) - x(t) }{\varepsilon} </math>

In elementary calculus, the two are only different by an amount that goes to 0 as {{mvar|ε}} goes to 0. But in this case, the difference between the two is not 0:
: <math>[2] - [1] =  \frac{\big( x(t + \varepsilon) - x(t)\big )^2}{\varepsilon} \approx \frac \varepsilon \varepsilon</math>

Let
: <math>f(t) = \frac{\big(x(t+\varepsilon)- x(t)\big)^2 }{\varepsilon}</math>

Then {{math|''f''(''t'')}} is a rapidly fluctuating statistical quantity, whose average value is 1, i.e. a normalized "Gaussian process". The fluctuations of such a quantity can be described by a statistical Lagrangian
: <math>\mathcal L =  (f(t)-1)^2 \,,</math>
and the equations of motion for {{mvar|f}} derived from extremizing the action {{mvar|S}} corresponding to {{mathcal|L}} just set it equal to 1. In physics, such a quantity is "equal to 1 as an operator identity". In mathematics, it "weakly converges to 1". In either case, it is 1 in any expectation value, or when averaged over any interval, or for all practical purpose.

Defining the time order to ''be'' the operator order:
: <math>[x, \dot x] = x \frac{dx}{dt} - \frac{dx}{dt} x = 1</math>

This is called the [[Itō lemma]] in [[stochastic calculus]], and the (euclideanized) canonical commutation relations in physics.

For a general statistical action, a similar argument shows that
: <math>\left[x , \frac{\partial S }{ \partial \dot x} \right] = 1</math>
and in quantum mechanics, the extra imaginary unit in the action converts this to the canonical commutation relation,
: <math>[x,p ] = i</math>

=== Particle in curved space ===

For a particle in curved space the [[kinetic term]] depends on the position, and the above time slicing cannot be applied, this being a manifestation of the notorious [[operator ordering problem]] in Schrödinger quantum mechanics. One may, however, solve this problem by transforming the time-sliced flat-space path integral to curved space using a multivalued coordinate transformation ([[nonholonomic mapping]] explained [http://www.physik.fu-berlin.de/~kleinert/b5/psfiles/pthic10.pdf here]).

=== Measure-theoretic factors ===
Sometimes (e.g. a particle moving in curved space) we also have measure-theoretic factors in the functional integral:
: <math>\int \mu[x] e^{iS[x]} \,\mathcal{D}x.</math>
This factor is needed to restore unitarity.

For instance, if
: <math>S = \int \left( \frac{m}{2} g_{ij} \dot{x}^i \dot{x}^j - V(x) \right) \,dt,</math>
then it means that each spatial slice is multiplied by the measure {{math|{{sqrt|''g''}}}}. This measure cannot be expressed as a functional multiplying the {{math|{{mathcal|D}}''x''}} measure because they belong to entirely different classes.

=== Expectation values and matrix elements ===

Matrix elements of the kind <math>\langle x_f|e^{-\frac{i}{\hbar}\hat{H}(t-t')} F(\hat{x}) e^{-\frac{i}{\hbar}\hat{H}(t')}|x_i\rangle</math> take the form
: <math>\int_{x(0)=x_i}^{x(t)=x_f} \mathcal{D}[x] F(x(t')) e^{\frac{i}{\hbar}\int dt L(x(t),\dot{x}(t))}</math>.

This generalizes to multiple operators, for example
: <math>\langle x_f|e^{-\frac{i}{\hbar}\hat{H}(t-t_1)} F_1(\hat{x}) e^{-\frac{i}{\hbar}\hat{H}(t_1-t_2)} F_2(\hat{x}) e^{-\frac{i}{\hbar}\hat{H}(t_2)}|x_i\rangle =
\int_{x(0)=x_i}^{x(t)=x_f} \mathcal{D}[x] F_1(x(t_1)) F_2(x(t_2)) e^{\frac{i}{\hbar}\int dt L(x(t),\dot{x}(t))}</math>,
and to the general vacuum expectation value (in the large time limit)
: <math>\langle F\rangle=\frac{\int \mathcal{D}[\phi] F(\phi) e^{\frac{i}{\hbar}S[\phi]}}{\int \mathcal{D}[\phi] e^{\frac{i}{\hbar}S[\phi]}}</math>.