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The path integral also relates quantum and [[stochastic]] processes, and this provided the basis for the grand synthesis of the 1970s, which unified [[quantum field theory]] with the [[statistical field theory]] of a fluctuating field near a [[second-order phase transition]]. The [[Schrödinger equation]] is a [[diffusion equation]] with an imaginary diffusion constant, and the path integral is an [[analytic continuation]] of a method for summing up all possible [[random walk]]s.<ref>{{cite web |last=Vinokur |first=V. M. |date=2015-02-27 |url=https://www.gc.cuny.edu/CUNY_GC/media/CUNY-Graduate-Center/PDF/ITS/Vinokur_Spring2015.pdf |title=Dynamic Vortex Mott Transition |access-date=2018-12-15 |archive-date=2017-08-12 |archive-url=https://web.archive.org/web/20170812032227/http://www.gc.cuny.edu/CUNY_GC/media/CUNY-Graduate-Center/PDF/ITS/Vinokur_Spring2015.pdf |url-status=dead }}</ref>
The path integral has impacted a wide array of sciences, including [[polymer physics]], quantum field theory, [[string theory]] and [[cosmology]].
The basic idea of the path integral formulation can be traced back to [[Norbert Wiener]], who introduced the [[Wiener integral]] for solving problems in diffusion and [[Brownian motion]].<ref>{{harvnb|Chaichian|Demichev|2001}}</ref> This idea was extended to the use of the [[Lagrangian (field theory)|Lagrangian]] in quantum mechanics by [[Paul Dirac]], whose 1933 paper gave birth to path integral formulation.<ref>{{harvnb|Dirac|1933}}</ref><ref>{{harvnb|Van Vleck|1928}}</ref><ref name=":0">{{cite arXiv |eprint=1004.3578 |class=physics.hist-ph |first=Jeremy |last=Bernstein |title=Another Dirac |date=2010-04-20}}</ref><ref name=":02">{{cite arXiv |eprint=2003.12683 |class=physics.hist-ph |first=N. D. |last=Hari Dass |title=Dirac and the Path Integral |date=2020-03-28}}</ref> The complete method was developed in 1948 by [[Richard Feynman]].{{sfn|Feynman|1948}} Some preliminaries were worked out earlier in his doctoral work under the supervision of [[John Archibald Wheeler]]. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the [[Wheeler–Feynman absorber theory]] using a [[Lagrangian (field theory)|Lagrangian]] (rather than a [[Hamiltonian (quantum mechanics)|Hamiltonian]]) as a starting point.
[[File:Feynman paths.png|alt=|thumb|These are five of the infinitely many paths available for a particle to move from point A at time t to point B at time t’(>t)
== Quantum action principle ==
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In quantum mechanics, as in classical mechanics, the [[Hamiltonian (quantum mechanics)|Hamiltonian]] is the generator of time translations. This means that the state at a slightly later time differs from the state at the current time by the result of acting with the Hamiltonian operator (multiplied by the negative [[imaginary unit]], {{math|−''i''}}). For states with a definite energy, this is a statement of the [[de Broglie relation]] between frequency and energy, and the general relation is consistent with that plus the [[superposition principle]].
The Hamiltonian in classical mechanics is derived from a [[Lagrangian (field theory)|Lagrangian]], which is a more fundamental quantity
The Hamiltonian is a function of the position and momentum at one time, and it determines the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later (or, equivalently for infinitesimal time separations, it is a function of the position and velocity). The relation between the two is by a [[Legendre transformation]], and the condition that determines the classical equations of motion (the [[Euler–Lagrange equation]]s) is that the [[action (physics)|action]] has an extremum.
In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. In classical mechanics, with [[discretization]] in time, the Legendre transform becomes
: <math> \varepsilon H = p(t)\big(q(t + \varepsilon) - q(t)\big) - \varepsilon L</math>
and
: <math> p = \frac{\partial L}{\partial \dot{q}},</math>
where the [[partial derivative]] with respect to <math>\dot q</math> holds {{math|''q''(''t'' + ''ε'')}} fixed. The inverse Legendre transform is
: <math> \varepsilon L = \varepsilon p \dot{q} - \varepsilon H,</math>
where
: <math> \dot q = \frac{\partial H}{\partial p},</math>
and the partial derivative now is with respect to {{mvar|p}} at fixed {{mvar|q}}.
In quantum mechanics, the state is a [[quantum superposition|superposition of different states]] with different values of {{mvar|q}}, or different values of {{mvar|p}}, and the quantities {{mvar|p}} and {{mvar|q}} can be interpreted as noncommuting operators. The operator {{mvar|p}} is only definite on states that are indefinite with respect to {{mvar|q}}. So consider two states separated in time and act with the operator corresponding to the Lagrangian:
: <math> e^{i\big[p \big(q(t + \varepsilon) - q(t)\big) - \varepsilon H(p, q) \big]}.</math>
If the multiplications implicit in this formula are reinterpreted as ''matrix'' multiplications, the first factor is
: <math> e^{-ip q(t)},</math>
and if this is also interpreted as a matrix multiplication, the sum over all states integrates over all {{math|''q''(''t'')}}, and so it takes the [[Fourier transform]] in {{math|''q''(''t'')}} to change basis to {{math|''p''(''t'')}}. That is the action on the Hilbert space – <em>change basis to {{mvar|p}} at time {{mvar|t}}</em>.
Next comes
: <math>e^{-i\varepsilon H(p,q)},</math>
or <em>evolve an infinitesimal time into the future</em>.
Finally, the last factor in this interpretation is
: <math>e^{i p q(t + \varepsilon)},</math>
which means <em>change basis back to {{mvar|q}} at a later time</em>.
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Dirac further noted that one could square the time-evolution operator in the {{mvar|S}} representation:
: <math> e^{i\varepsilon S},</math>
and this gives the time-evolution operator between time {{mvar|t}} and time {{math|''t'' + 2''ε''}}. While in the {{mvar|H}} representation the quantity that is being summed over the intermediate states is an obscure matrix element, in the {{mvar|S}} representation it is reinterpreted as a quantity associated to the path. In the limit that one takes a large power of this operator, one reconstructs the full quantum evolution between two states, the early one with a fixed value of {{math|''q''(0)}} and the later one with a fixed value of {{math|''q''(''t'')}}. The result is a sum over paths with a phase, which is the quantum action.
== Classical limit ==
Crucially, Dirac identified the effect of the [[classical limit]] on the quantum form of the action principle:
{{blockquote|...we see that the integrand in (11) must be of the form {{math|''e''<sup>''iF''/''h''</sup>}}, where {{mvar|F}} is a function of {{math|''q''<sub>''T''</sub>, ''q''<sub>1</sub>, ''q''<sub>2</sub>, … ''q''<sub>''m''</sub>, ''q''<sub>''t''</sub>}}, which remains finite as {{mvar|h}} tends to zero. Let us now picture one of the intermediate {{mvar|q}}s, say {{mvar|q<sub>k</sub>}}, as varying continuously while the other ones are fixed. Owing to the smallness of {{mvar|h}}, we shall then in general have ''F''/''h'' varying extremely rapidly. This means that {{math|''e''<sup>''iF''/''h''</sup>}} will vary periodically with a very high frequency about the value zero, as a result of which its integral will be practically zero. The only important part in the ___domain of integration of {{mvar|q<sub>k</sub>}} is thus that for which a comparatively large variation in {{mvar|q<sub>k</sub>}} produces only a very small variation in {{mvar|F}}. This part is the neighbourhood of a point for which {{mvar|F}} is stationary with respect to small variations in {{mvar|q<sub>k</sub>}}. We can apply this argument to each of the variables of integration ... and obtain the result that the only important part in the ___domain of integration is that for which {{mvar|F}} is stationary for small variations in all intermediate {{mvar|q}}s. ... We see that {{mvar|F}} has for its classical analogue {{math|{{intmath|int|''T''|''t''}} ''L dt''}}, which is just the action function, which classical mechanics requires to be stationary for small variations in all the intermediate {{mvar|q}}s. This shows the way in which equation (11) goes over into classical results when {{mvar|h}} becomes extremely small. |source=Dirac (1933), p. 69}}
That is, in the limit of action that is large compared to the [[Planck
== Feynman's interpretation ==
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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>
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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]].]]
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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}\,
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} \,
Splitting the integral into time slices:
: <math>K(x
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
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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
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</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
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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]].
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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.
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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
=== Stationary-phase approximation ===
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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.
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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>.
== Euclidean path integrals ==
It is very common in path integrals to perform a [[Wick rotation]] from real to imaginary times. In the setting of quantum field theory, the Wick rotation changes the geometry of space-time from Lorentzian to Euclidean; as a result, Wick-rotated path integrals are often called Euclidean path integrals.
=== Wick rotation and the Feynman–Kac formula ===
If we replace <math>t</math> by <math>-it</math>, the time-evolution operator <math>e^{-it\hat{H}/\hbar}</math> is replaced by <math>e^{-t\hat{H}/\hbar}</math>. (This change is known as a [[Wick rotation]].) If we repeat the derivation of the path-integral formula in this setting, we obtain<ref>{{harvnb|Hall|2013|loc=Section 20.3.}}</ref>
: <math>\psi(x,t)=\frac{1}{Z}\int_{\mathbf{x}(0)=x} e^{-S_{\mathrm{Euclidean}}(\mathbf{x},\dot{\mathbf{x}})/\hbar}\psi_0(\mathbf{x}(t))\, \mathcal{D}\mathbf{x}\,</math>,
where <math>S_{\mathrm{Euclidean}}</math> is the Euclidean action, given by
: <math>S_{\mathrm{Euclidean}}(\mathbf{x},\dot{\mathbf{x}})=\int\left[ \frac{m}{2}|\dot\mathbf{x}(t)|^2+V(\mathbf{x}(t))\right] \,dt</math>.
Note the sign change between this and the normal action, where the potential energy term is negative. (The term ''Euclidean'' is from the context of quantum field theory, where the change from real to imaginary time changes the space-time geometry from Lorentzian to Euclidean.)
Now, the contribution of the kinetic energy to the path integral is as follows:
: <math>\frac{1}{Z}\int_{\mathbf{x}(0)=x} f(\mathbf{x})e^{-\frac{m}{2}\int |\dot\mathbf{x}|^2dt}\, \mathcal{D}\mathbf{x}\,</math>
where <math>f(\mathbf{x})</math> includes all the remaining dependence of the integrand on the path. This integral has a rigorous mathematical interpretation as integration against the [[Wiener process|Wiener measure]], denoted <math>\mu_{x}</math>. The Wiener measure, constructed by [[Norbert Wiener]] gives a rigorous foundation to [[Brownian motion#Einstein.27s theory|Einstein's mathematical model of Brownian motion]]. The subscript <math>x</math> indicates that the measure <math>\mu_x</math> is supported on paths <math>\mathbf{x}</math> with <math>\mathbf{x}(0)=x</math>.
We then have a rigorous version of the Feynman path integral, known as the [[Feynman–Kac formula]]:<ref>{{harvnb|Hall|2013|loc=Theorem 20.3.}}</ref>
: <math>\psi(x,t)=\int e^{-\int V(\mathbf{x}(t))\,dt/\hbar}\,\psi_0(\mathbf{x}(t)) \,d\mu_x(\mathbf{x})</math>,
where now <math>\psi(x,t)</math> satisfies the Wick-rotated version of the Schrödinger equation,
: <math>\hbar \frac{\partial}{\partial t}\psi(x,t) = -\hat H \psi(x,t)</math>.
Although the Wick-rotated Schrödinger equation does not have a direct physical meaning, interesting properties of the Schrödinger operator <math>\hat{H}</math> can be extracted by studying it.<ref>{{harvnb|Simon|1979}}</ref>
Much of the study of quantum field theories from the path-integral perspective, in both the mathematics and physics literatures, is done in the Euclidean setting, that is, after a Wick rotation. In particular, there are various results showing that if a Euclidean field theory with suitable properties can be constructed, one can then undo the Wick rotation to recover the physical, Lorentzian theory.<ref>{{harvnb|Glimm|Jaffe|1981|loc=Chapter 19.}}</ref> On the other hand, it is much more difficult to give a meaning to path integrals (even Euclidean path integrals) in quantum field theory than in quantum mechanics.<ref group=nb>For a brief account of the origins of these difficulties, see {{harvnb|Hall|2013|loc=Section 20.6.}}</ref>
===
The path integral is just the generalization of the integral above to all quantum mechanical problems—
: <math>Z = \int e^\frac{i\mathcal{S}[\mathbf{x}]}{\hbar}\, \mathcal{D}\mathbf{x} \quad\text{where }\mathcal{S}[\mathbf{x}]=\int_0^{t_f} L[\mathbf{x}(t),\dot\mathbf{x}(t)]\, dt</math>
is the [[action (physics)|action]] of the classical problem in which one investigates the path starting at time {{math|''t'' {{=}} 0}} and ending at time {{math|''t'' {{=}} t<sub>f</sub>}}, and <math>\mathcal{D}\mathbf{x}</math> denotes the integration measure over all paths. In the classical limit, <math>\mathcal{S}[\mathbf{x}]\gg\hbar</math>, the path of minimum action dominates the integral, because the phase of any path away from this fluctuates rapidly and different contributions cancel.<ref name="Feynman-Hibbs">{{harvnb|Feynman|Hibbs|Styer|2010|pp=29–31}}</ref>
The connection with [[statistical mechanics]] follows. Considering only paths
Clearly, such a deep analogy between quantum mechanics and statistical mechanics cannot be dependent on the formulation. In the canonical formulation, one sees that the unitary evolution operator of a state is given by
: <math>|\alpha;t\rangle=e^{-\frac{iHt}{\hbar}}|\alpha;0\rangle</math>
where the state {{mvar|α}} is evolved from time {{math|''t'' {{=}} 0}}. If one makes a Wick rotation here, and finds the amplitude to go from any state, back to the same state in (imaginary) time {{mvar|iβ}} is given by
: <math>Z = \operatorname{Tr} \left[e^{-H\beta}\right]</math>
which is precisely the partition function of statistical mechanics for the same system at the temperature quoted earlier. One aspect of this equivalence was also known to [[Erwin Schrödinger]] who remarked that the equation named after him looked like the [[diffusion equation]] after Wick rotation. Note, however, that the Euclidean path integral is actually in the form of a ''classical'' statistical mechanics model.
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{{Quantum field theory}}
Both the Schrödinger and Heisenberg approaches to quantum mechanics single out time and are not in the spirit of relativity. For example, the Heisenberg approach requires that scalar field operators obey the commutation relation
: <math>[\varphi(x), \partial_t \varphi(y)] = i \delta^3(x - y)</math>
for two simultaneous spatial positions {{mvar|x}} and {{mvar|y}}, and this is not a relativistically invariant concept. The results of a calculation ''are'' covariant, but the symmetry is not apparent in intermediate stages. If naive field-theory calculations did not produce infinite answers in the [[continuum limit]], this would not have been such a big problem – it would just have been a bad choice of coordinates. But the lack of symmetry means that the infinite quantities must be cut off, and the bad coordinates make it nearly impossible to cut off the theory without spoiling the symmetry. This makes it difficult to extract the physical predictions, which require a [[renormalization|careful limiting procedure]].
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The integration variables in the path integral are subtly non-commuting. The value of the product of two field operators at what looks like the same point depends on how the two points are ordered in space and time. This makes some naive identities [[Anomaly (physics)|fail]].
===
In relativistic theories, there is both a particle and field representation for every theory. The field representation is a sum over all field configurations, and the particle representation is a sum over different particle paths.
The nonrelativistic formulation is traditionally given in terms of particle paths, not fields. There, the path integral in the usual variables, with fixed boundary conditions, gives the probability amplitude for a particle to go from point {{mvar|x}} to point {{mvar|y}} in time {{mvar|T}}:
: <math>K(x, y; T) = \langle y; T \mid x; 0 \rangle = \int_{x(0)=x}^{x(T)=y} e^{i S[x]} \,Dx.</math>
This is called the [[propagator]]. To obtain the final state at {{math|''y''}} we simply apply {{math|''K''(''x'',''y''; ''T'')}} to the initial state and integrate over {{math|''x''}} resulting in:
: <math>\psi_T(y) = \int_x \psi_0(x) K(x, y; T) \,dx = \int^{x(T)=y} \psi_0(x(0)) e^{i S[x]} \,Dx.</math>
For a spatially homogeneous system, where {{math|''K''(''x'', ''y'')}} is only a function of {{math|(''x'' − ''y'')}}, the integral is a [[convolution]], the final state is the initial state convolved with the propagator:
: <math>\psi_T = \psi_0 * K(;T).</math>
For a free particle of mass {{mvar|m}}, the propagator can be evaluated either explicitly from the path integral or by noting that the Schrödinger equation is a diffusion equation in imaginary time, and the solution must be a normalized Gaussian:
: <math>K(x, y; T) \propto e^\frac{i m(x - y)^2}{2T}.</math>
Taking the Fourier transform in {{math|(''x'' − ''y'')}} produces another Gaussian:
: <math>K(p; T) = e^\frac{i T p^2}{2m},</math>
and in {{mvar|p}}-space the proportionality factor here is constant in time, as will be verified in a moment. The Fourier transform in time, extending {{math|''K''(''p''; ''T'')}} to be zero for negative times, gives Green's function, or the frequency-space propagator:
: <math>G_\text{F}(p, E) = \frac{-i}{E - \frac{\vec{p}^2}{2m} + i\varepsilon},</math>
which is the reciprocal of the operator that annihilates the wavefunction in the Schrödinger equation, which wouldn't have come out right if the proportionality factor weren't constant in the {{mvar|p}}-space representation.
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It is also possible to reexpress the nonrelativistic time evolution in terms of propagators going toward the past, since the Schrödinger equation is time-reversible. The past propagator is the same as the future propagator except for the obvious difference that it vanishes in the future, and in the Gaussian {{mvar|t}} is replaced by {{math|−''t''}}. In this case, the interpretation is that these are the quantities to convolve the final wavefunction so as to get the initial wavefunction:
: <math>G_\text{B}(p, E) = \frac{-i}{-E - \frac{i\vec{p}^2}{2m} + i\varepsilon}.</math>
Given the nearly identical only change is the sign of {{mvar|E}} and {{mvar|ε}}, the parameter {{mvar|E}} in Green's function can either be the energy if the paths are going toward the future, or the negative of the energy if the paths are going toward the past.
For a nonrelativistic theory, the time as measured along the path of a moving particle and the time as measured by an outside observer are the same. In relativity, this is no longer true. For a relativistic theory the propagator should be defined as the sum over all paths that travel between two points in a fixed proper time, as measured along the path (these paths describe the trajectory of a particle in space and in time):
: <math>K(x - y, \Tau) = \int_{x(0)=x}^{x(\Tau)=y} e^{i \int_0^\Tau \sqrt{\dot{x}^2 - \alpha} \,d\tau}.</math>
The integral above is not trivial to interpret because of the square root. Fortunately, there is a heuristic trick. The sum is over the relativistic arc length of the path of an oscillating quantity, and like the nonrelativistic path integral should be interpreted as slightly rotated into imaginary time. The function {{math|''K''(''x'' − ''y'', ''τ'')}} can be evaluated when the sum is over paths in Euclidean space:
: <math>K(x - y, \Tau) = e^{-\alpha \Tau} \int_{x(0)=x}^{x(\Tau)=y} e^{-L}.</math>
This describes a sum over all paths of length {{math|Τ}} of the exponential of minus the length. This can be given a probability interpretation. The sum over all paths is a probability average over a path constructed step by step. The total number of steps is proportional to {{math|Τ}}, and each step is less likely the longer it is. By the [[central limit theorem]], the result of many independent steps is a Gaussian of variance proportional to {{math|Τ}}:
: <math>K(x - y,\Tau) = e^{-\alpha \Tau} e^{-\frac{(x - y)^2}{\Tau}}.</math>
The usual definition of the relativistic propagator only asks for the amplitude
: <math>K(x - y) = \int_0^\infty K(x - y, \Tau) W(\Tau) \,d\Tau,</math>
where {{math|''W''(Τ)}} is a weight factor, the relative importance of paths of different proper time. By the translation symmetry in proper time, this weight can only be an exponential factor and can be absorbed into the constant {{mvar|α}}:
: <math>K(x - y) = \int_0^\infty e^{-\frac{(x - y)^2}{\Tau} -\alpha \Tau} \,d\Tau.</math>
This is the [[Feynman diagram#Schwinger representation|Schwinger representation]]. Taking a Fourier transform over the variable {{math|(''x'' − ''y'')}} can be done for each value of {{math|Τ}} separately, and because each separate {{math|Τ}} contribution is a Gaussian, gives whose Fourier transform is another Gaussian with reciprocal width. So in {{mvar|p}}-space, the propagator can be reexpressed simply:
: <math>K(p) = \int_0^\infty e^{-\Tau p^2 - \Tau \alpha} \,d\Tau = \frac{1}{p^2 + \alpha},</math>
which is the Euclidean propagator for a scalar particle. Rotating {{math|''p''<sub>0</sub>}} to be imaginary gives the usual relativistic propagator, up to a factor of {{math|−''i''}} and an ambiguity, which will be clarified below:
: <math>K(p) = \frac{i}{p_0^2 - \vec{p}^2 - m^2}.</math>
This expression can be interpreted in the nonrelativistic limit, where it is convenient to split it by [[partial fractions]]:
: <math>2 p_0 K(p) = \frac{i}{p_0 - \sqrt{\vec{p}^2 + m^2}} + \frac{i}{p_0 + \sqrt{\vec{p}^2 + m^2}}.</math>
For states where one nonrelativistic particle is present, the initial wavefunction has a frequency distribution concentrated near {{math|''p''<sub>0</sub> {{=}} ''m''}}. When convolving with the propagator, which in {{mvar|p}} space just means multiplying by the propagator, the second term is suppressed and the first term is enhanced. For frequencies near {{math|''p''<sub>0</sub> {{=}} ''m''}}, the dominant first term has the form
: <math>2m K_\text{NR}(p) = \frac{i}{(p_0 - m) - \frac{\vec{p}^2}{2m}}.</math>
This is the expression for the nonrelativistic [[Green's function]] of a free Schrödinger particle.
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The proper way to express this mathematically is that, adding a small suppression factor in proper time, the limit where {{math|''t'' → −∞}} of the first term must vanish, while the {{math|''t'' → +∞}} limit of the second term must vanish. In the Fourier transform, this means shifting the pole in {{math|''p''<sub>0</sub>}} slightly, so that the inverse Fourier transform will pick up a small decay factor in one of the time directions:
: <math>K(p) = \frac{i}{p_0 - \sqrt{\vec{p}^2 + m^2} + i\varepsilon} + \frac{i}{p_0 - \sqrt{\vec{p}^2+m^2} - i\varepsilon}.</math>
Without these terms, the pole contribution could not be unambiguously evaluated when taking the inverse Fourier transform of {{math|''p''<sub>0</sub>}}. The terms can be recombined:
: <math>K(p) = \frac{i}{p^2 - m^2 + i\varepsilon},</math>
which when factored, produces opposite-sign infinitesimal terms in each factor. This is the mathematically precise form of the relativistic particle propagator, free of any ambiguities. The {{mvar|ε}} term introduces a small imaginary part to the {{math|''α'' {{=}} ''m''<sup>2</sup>}}, which in the Minkowski version is a small exponential suppression of long paths.
So in the relativistic case, the Feynman path-integral representation of the propagator includes paths going backwards in time, which describe antiparticles. The paths that contribute to the relativistic propagator go forward and backwards in time, and the [[Feynman–Stueckelberg interpretation|interpretation]] of this is that the amplitude for a free particle to travel between two points includes amplitudes for the particle to fluctuate into an antiparticle, travel back in time, then forward again.
Unlike the nonrelativistic case, it is impossible to produce a relativistic theory of local particle propagation without including antiparticles. All local differential operators have inverses that are nonzero outside the light cone, meaning that it is impossible to keep a particle from travelling faster than light. Such a particle cannot have a Green's function
=== Functionals of fields ===
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In [[quantum field theory]], if the [[action (physics)|action]] is given by the [[functional (mathematics)|functional]] {{mathcal|S}} of field configurations (which only depends locally on the fields), then the [[time-ordered]] [[vacuum expectation value]] of [[polynomially bounded]] functional {{mvar|F}}, {{math|{{angbr|''F''}}}}, is given by
: <math>\langle F \rangle = \frac{\int\mathcal{D}\varphi F[\varphi]e^{i\mathcal{S}[\varphi]}}{\int\mathcal{D}\varphi e^{i\mathcal{S}[\varphi]}}.</math>
The symbol {{math|∫{{mathcal|D}}''ϕ''}} here is a concise way to represent the infinite-dimensional integral over all possible field configurations on all of space-time. As stated above, the unadorned path integral in the denominator ensures proper normalization.
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=== As a probability ===
Strictly speaking, the only question that can be asked in physics is: ''What fraction of states satisfying condition {{mvar|A}} also satisfy condition {{mvar|B}}?'' The answer to this is a number between 0 and 1, which can be interpreted as a [[conditional probability]], written as {{math|P(''B''{{!}}''A'')}}. In terms of path integration, since {{math|P(''B''{{!}}''A'') {{=}} {{sfrac|P(''A''∩''B'') | P(''A'')}}}}, this means
: <math>\operatorname{P}(B\mid A) = \frac
{\sum_{F \subset A \cap B}\left| \int\mathcal{D}\varphi O_\text{in}[\varphi]e^{i\mathcal{S}[\varphi]} F[\varphi]\right|^2}
{\sum_{F \subset A} \left|\int\mathcal{D}\varphi O_\text{in}[\varphi] e^{i\mathcal{S}[\varphi]} F[\varphi]\right|^2},</math>
where the functional {{math|''O''<sub>in</sub>[''ϕ'']}} is the superposition of all incoming states that could lead to the states we are interested in. In particular, this could be a state corresponding to the state of the Universe just after the [[Big Bang]], although for actual calculation this can be simplified using heuristic methods. Since this expression is a quotient of path integrals, it is naturally normalised.
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In the language of functional analysis, we can write the [[Euler–Lagrange equation]]s as
: <math>\frac{\delta \mathcal{S}[\varphi]}{\delta \varphi} = 0</math>
(the left-hand side is a [[functional derivative]]; the equation means that the action is stationary under small changes in the field configuration). The quantum analogues of these equations are called the [[Schwinger–Dyson equation]]s.
If the [[functional measure]] {{math|{{mathcal|D}}''ϕ''}} turns out to be [[Translational symmetry|translationally invariant]] (we'll assume this for the rest of this article, although this does not hold for, let's say [[nonlinear sigma model]]s), and if we assume that after a [[Wick rotation]]
: <math>e^{i\mathcal{S}[\varphi]},</math>
which now becomes
: <math>e^{-H[\varphi]}</math>
for some {{mvar|H}}, it goes to zero faster than a [[Multiplicative inverse|reciprocal]] of any [[polynomial]] for large values of {{mvar|φ}}, then we can [[integration by parts|integrate by parts]] (after a Wick rotation, followed by a Wick rotation back) to get the following Schwinger–Dyson equations for the expectation:
: <math>\left\langle \frac{\delta F[\varphi]}{\delta \varphi} \right\rangle = -i \left\langle F[\varphi]\frac{\delta \mathcal{S}[\varphi]}{\delta\varphi} \right\rangle</math>
for any polynomially-bounded functional {{mvar|F}}. In the [[deWitt notation]] this looks like<ref>{{cite journal |first=Jean |last=Zinn-Justin |date=2009 |title=Path integral |journal=Scholarpedia |volume=4 |issue=2 |doi=10.4249/scholarpedia.8674 |bibcode=2009SchpJ...4.8674Z |at=8674|doi-access=free }}</ref>
: <math>\left\langle F_{,i} \right\rangle = -i \left\langle F \mathcal{S}_{,i} \right\rangle.</math>
These equations are the analog of the [[on-shell]] EL equations. The time ordering is taken before the time derivatives inside the {{math|{{mathcal|S}}<sub>,''i''</sub>}}.
If {{mvar|J}} (called the [[source field]]) is an element of the [[dual space]] of the field configurations (which has at least an [[affine structure]] because of the assumption of the [[translational invariance]] for the functional measure), then the [[generating functional]] {{mvar|Z}} of the source fields is '''defined''' to be
: <math>Z[J] = \int \mathcal{D}\varphi e^{i\left(\mathcal{S}[\varphi] + \langle J,\varphi \rangle\right)}.</math>
Note that
: <math>\frac{\delta^n Z}{\delta J(x_1) \cdots \delta J(x_n)}[J] = i^n \, Z[J] \, \left\langle \varphi(x_1)\cdots \varphi(x_n)\right\rangle_J,</math>
or
: <math>Z^{,i_1\cdots i_n}[J] = i^n Z[J] \left \langle \varphi^{i_1}\cdots \varphi^{i_n}\right\rangle_J,</math>
where
: <math>\langle F \rangle_J = \frac{\int \mathcal{D}\varphi F[\varphi]e^{i\left(\mathcal{S}[\varphi] + \langle J,\varphi \rangle\right)}}{\int\mathcal{D}\varphi e^{i\left(\mathcal{S}[\varphi] + \langle J,\varphi \rangle\right)}}.</math>
Basically, if {{math|{{mathcal|D}}''φ'' ''e''<sup>''i''{{mathcal|S}}[''φ'']</sup>}} is viewed as a functional distribution (this shouldn't be taken too literally as an interpretation of [[Quantum field theory|QFT]], unlike its Wick-rotated [[statistical mechanics]] analogue, because we have [[time ordering]] complications here!), then {{math|{{angbr|''φ''(''x''<sub>1</sub>) ... ''φ''(''x<sub>n</sub>'')}}}} are its [[moment (mathematics)|moments]], and {{mvar|Z}} is its [[Fourier transform]].
If {{mvar|F}} is a functional of {{mvar|φ}}, then for an [[Operator (mathematics)|operator]] {{mvar|K}}, {{math|''F''[''K'']}} is defined to be the operator that substitutes {{mvar|K}} for {{mvar|φ}}. For example, if
: <math>F[\varphi] = \frac{\partial^{k_1}}{\partial x_1^{k_1}}\varphi(x_1)\cdots \frac{\partial^{k_n}}{\partial x_n^{k_n}}\varphi(x_n),</math>
and {{mvar|G}} is a functional of {{mvar|J}}, then
: <math>F\left[-i\frac{\delta}{\delta J}\right] G[J] = (-i)^n \frac{\partial^{k_1}}{\partial x_1^{k_1}}\frac{\delta}{\delta J(x_1)} \cdots \frac{\partial^{k_n}}{\partial x_n^{k_n}}\frac{\delta}{\delta J(x_n)} G[J].</math>
Then, from the properties of the [[functional integral]]s
: <math>\left \langle \frac{\delta \mathcal{S}}{\delta \varphi(x)} [\varphi] + J(x)\right\rangle_J = 0</math>
we get the "master" Schwinger–Dyson equation:
: <math>\frac{\delta \mathcal{S}}{\delta \varphi(x)}\left[-i \frac{\delta}{\delta J}\right]Z[J] + J(x)Z[J] = 0,</math>
or
: <math>\mathcal{S}_{,i}[-i\partial]Z + J_i Z = 0.</math>
If the functional measure is not translationally invariant, it might be possible to express it as the product {{math|''M''[''φ''] {{mathcal|D}}''φ''}}, where {{mvar|M}} is a functional and {{math|{{mathcal|D}}''φ''}} is a translationally invariant measure. This is true, for example, for nonlinear sigma models where the [[target space]] is diffeomorphic to {{math|'''R'''<sup>''n''</sup>}}. However, if the [[target manifold]] is some topologically nontrivial space, the concept of a translation does not even make any sense.
In that case, we would have to replace the {{mathcal|S}} in this equation by another functional
: <math>\hat{\mathcal{S}} = \mathcal{S} - i\ln M.</math>
If we expand this equation as a [[Taylor series]] about ''J'' {{=}} 0, we get the entire set of Schwinger–Dyson equations.
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Let's just assume for simplicity here that the symmetry in question is local (not local in the sense of a [[gauge symmetry]], but in the sense that the transformed value of the field at any given point under an infinitesimal transformation would only depend on the field configuration over an arbitrarily small neighborhood of the point in question). Let's also assume that the action is local in the sense that it is the integral over spacetime of a [[Lagrangian (field theory)|Lagrangian]], and that
: <math>Q[\mathcal{L}(x)]=\partial_\mu f^\mu (x)</math>
for some function {{mvar|f}} where {{mvar|f}} only depends locally on {{mvar|φ}} (and possibly the spacetime position).
If we don't assume any special boundary conditions, this would not be a "true" symmetry in the true sense of the term in general unless {{math|''f'' {{=}} 0}} or something. Here, {{mvar|Q}} is a [[Derivation (abstract algebra)|derivation]]
Let's also assume
: <math>\int \mathcal{D}\varphi\, Q[F][\varphi]=0</math>
for any polynomially-bounded functional {{mvar|F}}. This property is called the invariance of the measure, and this does not hold in general. (See ''[[anomaly (physics)]]'' for more details.)
Then,
: <math>\int \mathcal{D}\varphi\, Q\left[F e^{iS}\right][\varphi]=0,</math>
which implies
: <math>\langle Q[F]\rangle +i\left\langle F\int_{\partial V} f^\mu\, ds_\mu\right\rangle=0</math>
where the integral is over the boundary. This is the quantum analog of Noether's theorem.
Now, let's assume even further that {{mvar|Q}} is a local integral
: <math>Q=\int d^dx\, q(x)</math>
where
: <math>q(x)[\varphi(y)] = \delta^{(d)}(X-y)Q[\varphi(y)] \,</math>
so that\
: <math>q(x)[S]=\partial_\mu j^\mu (x) \,</math>
where
: <math>j^{\mu}(x)=f^\mu(x)-\frac{\partial}{\partial (\partial_\mu \varphi)}\mathcal{L}(x) Q[\varphi] \,</math>
(this is assuming the Lagrangian only depends on {{mvar|φ}} and its first partial derivatives! More general Lagrangians would require a modification to this definition!). We're not insisting that {{math|''q''(''x'')}} is the generator of a symmetry (i.e. we are ''not'' insisting upon the [[gauge principle]]), but just that {{mvar|Q}} is. And we also assume the even stronger assumption that the functional measure is locally invariant:
: <math>\int \mathcal{D}\varphi\, q(x)[F][\varphi]=0.</math>
Then, we would have
: <math>\langle q(x)[F] \rangle +i\langle F q(x)[S]\rangle=\langle q(x)[F]\rangle +i\left\langle F\partial_\mu j^\mu(x)\right\rangle=0.</math>
Alternatively,
: <math>q(x)[S]\left[-i \frac{\delta}{\delta J}\right]Z[J]+J(x)Q[\varphi(x)]\left[-i \frac{\delta}{\delta J}\right]Z[J]=\partial_\mu j^\mu(x)\left[-i \frac{\delta}{\delta J}\right]Z[J]+J(x)Q[\varphi(x)]\left[-i \frac{\delta}{\delta J}\right]Z[J]=0.</math>
The above two equations are the Ward–Takahashi identities.
Now for the case where {{math|''f'' {{=}} 0}}, we can forget about all the boundary conditions and locality assumptions. We'd simply have
: <math>\left\langle Q[F]\right\rangle =0.</math>
Alternatively,
: <math>\int d^dx\, J(x)Q[\varphi(x)]\left[-i \frac{\delta}{\delta J}\right]Z[J]=0.</math>
== Caveats ==
===
Path integrals as they are defined here require the introduction of [[Regularization (physics)|regulators]]. Changing the scale of the regulator leads to the [[renormalization group]]. In fact, renormalization is the major obstruction to making path integrals well-defined.
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Regardless of whether one works in configuration space or phase space, when equating the [[Mathematical formulation of quantum mechanics|operator formalism]] and the path integral formulation, an ordering prescription is required to resolve the ambiguity in the correspondence between non-commutative operators and the commutative functions that appear in path integrands. For example, the operator <math>\frac{1}{2}(\hat{q}\hat{p}+\hat{p}\hat{q})</math> can be translated back as either <math>qp-\frac{i\hbar}{2}</math>, <math>qp+\frac{i\hbar}{2}</math>, or <math>qp</math> depending on whether one chooses the <math>\hat{q}\hat{p}</math>, <math>\hat{p}\hat{q}</math>, or Weyl ordering prescription; conversely, <math>qp</math> can be translated to either <math>\hat{q}\hat{p}</math>, <math>\hat{p}\hat{q}</math>, or <math>\frac{1}{2}(\hat{q}\hat{p}+\hat{p}\hat{q})</math> for the same respective choice of ordering prescription.
==
In one [[interpretation of quantum mechanics]], the "sum over histories" interpretation, the path integral is taken to be fundamental, and reality is viewed as a single indistinguishable "class" of paths that all share the same events.<ref>{{Cite web |last=Pössel |first=Markus |title=The sum over all possibilities: The path integral formulation of quantum theory |website=Einstein Online |url=https://www.einstein-online.info/en/spotlight/path_integrals/ |access-date=2021-07-16 |date=2006 |id=02-1020}}</ref> For this interpretation, it is crucial to understand what exactly an event is. The sum-over-histories method gives identical results to canonical quantum mechanics, and Sinha and Sorkin<ref>{{harvnb|Sinha|Sorkin|1991}}</ref> claim the interpretation explains the [[Einstein–Podolsky–Rosen paradox]] without resorting to [[action at a distance|nonlocality]].
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== Quantum tunneling ==
[[Quantum tunnelling]] can be modeled by using the path integral formation to determine the action of the trajectory through a potential barrier. Using the [[WKB approximation]], the tunneling rate ({{math|Γ}}) can be determined to be of the form
: <math> \Gamma = A_\mathrm{o} \exp \left(-\frac{S_\mathrm{eff}}{\hbar}\right) </math>
with the effective action {{math|''S''<sub>eff</sub>}} and pre-exponential factor {{math|''A''<sub>o</sub>}}. This form is specifically useful in a [[dissipative system]], in which the systems and surroundings must be modeled together. Using the [[Langevin equation]] to model [[Brownian motion]], the path integral formation can be used to determine an effective action and pre-exponential model to see the effect of dissipation on tunnelling.<ref>{{harvnb|Caldeira|Leggett|1983}}</ref> From this model, tunneling rates of macroscopic systems (at finite temperatures) can be predicted.
== See also ==
{{cols}}
* [[Static forces and virtual-particle exchange]]
* [[Feynman checkerboard]]
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== Remarks ==
{{
== References ==
{{
== Bibliography ==
{{divcol}}
* {{cite book |last=Ahmad |first=Ishfaq |author-link=Ishfaq Ahmad |title=Mathematical Integrals in Quantum Nature |series=The Nucleus |year=1971 |pages=189–209}}
* {{cite book |last1=Albeverio |first1=S. |last2=Hoegh-Krohn |first2=R. |last3=Mazzucchi |first3=S |name-list-style=amp |title=Mathematical Theory of Feynman Path Integrals |series=Lecture Notes in Mathematics 523 |publisher=Springer-Verlag |year=2008 |isbn=9783540769569}}
* {{cite journal |first1=A. O. |last1=Caldeira |author-link1=Amir Caldeira |first2=A. J. |last2=Leggett |author-link2=Anthony James Leggett |title=Quantum tunnelling in a dissipative system |journal=Annals of Physics |volume=149 |year=1983 |issue=2 |pages=374–456 |doi=10.1016/0003-4916(83)90202-6 |bibcode=1983AnPhy.149..374C}}
* {{cite journal |last1=Cartier |first1=P. C. |author-link=Pierre Cartier (mathematician) |last2=DeWitt-Morette |first2=Cécile |title=A new perspective on Functional Integration |journal=Journal of Mathematical Physics |volume=36 |year=1995 |issue=5 |pages=2137–2340 |doi=10.1063/1.531039 |arxiv=funct-an/9602005 |bibcode=1995JMP....36.2237C |s2cid=119581543}}
* {{cite book |title=Path Integrals in Physics Volume 1: Stochastic Process & Quantum Mechanics |chapter-url=https://books.google.com/books?id=-XDP-8mrmQYC&pg=PA1 |chapter=Introduction |page=1ff |isbn=978-0-7503-0801-4 |year=2001 |publisher=Taylor & Francis |first1=M. |last1=Chaichian |first2=A. P. |last2=Demichev}}
* {{cite journal |author-link=Cécile DeWitt-Morette |last=DeWitt-Morette |first=C. |title=Feynman's path integral: Definition without limiting procedure |journal=Communications in Mathematical Physics |volume=28 |issue=1 |year=1972 |pages=47–67 |mr=0309456 |doi=10.1007/BF02099371 |bibcode=1972CMaPh..28...47D |s2cid=119669964}}
* {{cite journal |last=Dirac |first=Paul A. M. |author-link=Paul Dirac |year=1933 |title=The Lagrangian in Quantum Mechanics |journal=Physikalische Zeitschrift der Sowjetunion |volume=3 |pages=64–72 |url=http://www.hep.anl.gov/czachos/soysoy/Dirac33.pdf}}
* {{cite journal |first1=İ. H. |last1=Duru |author-link1=İsmail Hakkı Duru |first2=Hagen |last2=Kleinert |author-link2=Hagen Kleinert |title=Solution of the path integral for the H-atom |year=1979 |journal=Physics Letters |volume=84B |issue=2 |pages=185–188 |url=http://www.physik.fu-berlin.de/~kleinert/kleiner_re65/65.pdf |access-date=2007-11-25 |bibcode=1979PhLB...84..185D |doi=10.1016/0370-2693(79)90280-6 |archive-date=2008-03-09 |archive-url=https://web.archive.org/web/20080309160840/http://www.physik.fu-berlin.de/~kleinert/kleiner_re65/65.pdf |url-status=dead }}
* {{cite web |last=Etingof |first=P. |author-link=Pavel Etingof |title=Geometry and Quantum Field Theory |publisher=MIT OpenCourseWare |year=2002 |url=http://ocw.mit.edu/courses/mathematics/18-238-geometry-and-quantum-field-theory-fall-2002/index.htm}} <small>This course, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals.</small>
* {{cite book |last=Feynman |first=R. P. |author-link=Richard Feynman |editor-last=Brown |editor-first=L. M. |title=Feynman's Thesis — A New Approach to Quantum Theory |year=2005 |orig-year=1942/1948 |publisher=World Scientific |doi=10.1142/5852 |bibcode=2005ftna.book.....B |url=https://cds.cern.ch/record/910611 |isbn=978-981-256-366-8}} <small>The 1942 thesis. Also includes Dirac's 1933 paper and Feynman's 1948 publication.</small>
* {{cite journal |last=Feynman |first=R. P. |title=Space-Time Approach to Non-Relativistic Quantum Mechanics |journal=Reviews of Modern Physics |volume=20 |issue=2 |pages=367–387 |year=1948 |doi=10.1103/RevModPhys.20.367 |bibcode=1948RvMP...20..367F |url=https://authors.library.caltech.edu/47756/1/FEYrmp48.pdf}}
* {{cite book |last1=Feynman |first1=R. P. |last2=Hibbs |first2=A. R. |year=1965 |title=Quantum Mechanics and Path Integrals |place=New York |publisher=McGraw-Hill |isbn=978-0-07-020650-2 |url-access=registration |url=https://archive.org/details/quantummechanics0000feyn }} <small>The historical reference written by the inventor of the path integral formulation himself and one of his students.</small>
* {{cite book |last1=Feynman |first1=R. P. |last2=Hibbs |first2=A. R. |author-link2=Albert Hibbs |last3=Styer |first3=D. F. |author-link3=Daniel F. Styer |title=Quantum Mechanics and Path Integrals |year=2010 |publisher=Dover Publications |___location=Mineola, NY |isbn=978-0-486-47722-0 |pages=29–31}}
* {{cite book |contribution=Most of the Good Stuff |title=Memories Of Richard Feynman |editor1-first=Laurie M. |editor1-last=Brown |editor2-first=John S. |editor2-last=Rigden |publisher=American Institute of Physics |first=Murray |last=Gell-Mann |author-link=Murray Gell-Mann |isbn=978-0883188705 |year=1993}}
* {{cite book |last1=Glimm |first1=J. |last2=Jaffe |first2=A. |name-list-style=amp |title=Quantum Physics: A Functional Integral Point of View |url=https://archive.org/details/quantumphysicsfu0000glim |url-access=registration |place=New York |publisher=Springer-Verlag |year=1981 |isbn=978-0-387-90562-4}}
* {{cite book |last1=Grosche |first1= Christian |last2=Steiner |first2= Frank |name-list-style=amp |year=1998 |title=Handbook of Feynman Path Integrals |series=Springer Tracts in Modern Physics 145 |publisher=Springer-Verlag |isbn=978-3-540-57135-3}}
* {{cite arXiv |last=Grosche |first=Christian |title=An Introduction into the Feynman Path Integral |year=1992 |eprint=hep-th/9302097}}
* {{cite book |last=Hall |first=Brian C. |year=2013 |title=Quantum Theory for Mathematicians |series=Graduate Texts in Mathematics |volume=267 |publisher=Springer |isbn=978-1-4614-7115-8 |doi=10.1007/978-1-4614-7116-5 |bibcode=2013qtm..book.....H |s2cid=117837329 }}
* {{cite book |last1=Inomata |first1= Akira |last2= Kuratsuji |first2= Hiroshi |last3= Gerry |first3= Christopher |title=Path Integrals and Coherent States of SU(2) and SU(1,1) |place=Singapore |publisher=World Scientific |year=1992 |isbn=978-981-02-0656-7}}
* {{cite book |editor-last1=Janke |editor-first1=W. |editor-last2=Pelster |editor-first2=Axel |title=Path Integrals--New Trends And Perspectives |year=2008 |series=Proceedings Of The 9Th International Conference |publisher=World Scientific Publishing |isbn=978-981-283-726-4}}
* {{cite book |first1=Gerald W. |last1=Johnson |first2=Michel L. |last2= Lapidus |title=The Feynman Integral and Feynman's Operational Calculus |series=Oxford Mathematical Monographs |publisher=Oxford University Press |year=2002 |isbn=978-0-19-851572-2}}
* {{cite book |author-link=John R. Klauder |last=Klauder |first=John R. |title=A Modern Approach to Functional Integration |place=New York |publisher=Birkhäuser |year=2010 |isbn=978-0-8176-4790-2}}
* {{cite book |author-link=Hagen Kleinert |last=Kleinert |first=Hagen |year=2004 |title=Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets |edition=4th |place=Singapore |publisher=World Scientific |isbn=978-981-238-107-1 |url=http://www.physik.fu-berlin.de/~kleinert/b5 |archive-date=2008-06-15 |access-date=2005-02-16 |archive-url=https://web.archive.org/web/20080615134934/http://www.physik.fu-berlin.de/~kleinert/b5 |url-status=dead }}
* {{cite arXiv |last=MacKenzie |first=Richard |year=2000 |title=Path Integral Methods and Applications |eprint=quant-ph/0004090}}
* {{cite book |last= Mazzucchi |first= S. |title=Mathematical Feynman path integrals and their applications |publisher=World Scientific |year=2009 |isbn=978-981-283-690-8}}
* {{cite book |first=Harald J. W. |last=Müller-Kirsten |year=2012 |title=Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral |edition=2nd |place=Singapore |publisher=World Scientific}}
* {{cite book |last=Rivers |first= R. J. |title=Path Integrals Methods in Quantum Field Theory |publisher=Cambridge University Press |year=1987 |isbn=978-0-521-25979-8}}
* {{cite book |last=Ryder |first= Lewis H. |title=Quantum Field Theory |url=https://archive.org/details/quantumfieldtheo0000ryde |url-access=registration |publisher=Cambridge University Press |year=1985 |isbn=978-0-521-33859-2}} Highly readable textbook; introduction to relativistic QFT for particle physics.
* {{cite book |last=Schulman |first= L S. |year=1981 |title=Techniques & Applications of Path Integration |place=New York |publisher=John Wiley & Sons |isbn=978-0-486-44528-1}}
* {{cite book |author-link=Barry Simon |last=Simon |first=B. |title=Functional Integration and Quantum Physics |place=New York |publisher=Academic Press |year=1979 |isbn=978-0-8218-6941-3}}
* {{cite journal |first1=Sukanya |last1=Sinha |first2=Rafael D. |last2=Sorkin |title=A Sum-over-histories Account of an EPR(B) Experiment |journal=Foundations of Physics Letters |volume=4 |issue=4 |pages=303–335 |year=1991 |doi=10.1007/BF00665892 |url=https://www.perimeterinstitute.ca/personal/rsorkin/some.papers/63.eprb.pdf |bibcode=1991FoPhL...4..303S |s2cid=121370426 }}
* {{cite book |last=Tomé |first=W. A. |author-link= Wolfgang A. Tomé |year=1998 |title=Path Integrals on Group Manifolds |place=Singapore |publisher=World Scientific |isbn=978-981-02-3355-6}} Discusses the definition of Path Integrals for systems whose kinematical variables are the generators of a real separable, connected Lie group with irreducible, square integrable representations.
* {{cite journal |last=Van Vleck |first=J. H. |author-link=John Hasbrouck Van Vleck |title=The correspondence principle in the statistical interpretation of quantum mechanics |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=14 |issue=2 |pages=178–188 |year=1928 |doi=10.1073/pnas.14.2.178 |pmid=16577107 |pmc=1085402 |bibcode=1928PNAS...14..178V |doi-access=free}}
* {{citation |last=Weinberg |first=S. |year=2002 |orig-year=1995 |title=Foundations |series=The Quantum Theory of Fields |volume=1 |isbn=978-0-521-55001-7 |author-link=Steven Weinberg |___location=Cambridge |publisher=[[Cambridge University Press]] |url-access=registration |url=https://archive.org/details/quantumtheoryoff00stev}}
* {{cite book |last=Zee |first=A. |author-link=Anthony Zee |title=Quantum Field Theory in a Nutshell |edition=Second |publisher=Princeton University Press |isbn=978-0-691-14034-6 |date=2010-02-21 |url-access=registration |url=https://archive.org/details/isbn_9780691140346 }} A great introduction to Path Integrals (Chapter 1) and QFT in general.
* {{cite book |last=Zinn Justin |first= J. |author-link=Jean Zinn-Justin |year=2004 |title=Path Integrals in Quantum Mechanics |publisher=Oxford University Press |isbn=978-0-19-856674-8}}
* {{cite book |last=Deshmukh |first= P. C. |author-link=Pranawachandra Deshmukh |year=2023 |title=Quantum Mechanics Formalism, Methodologies, and Applications |publisher=Cambridge University Press |isbn=978-1316512258}}
{{divcol end}}
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* [http://www.quantumfieldtheory.info/website_Chap18.pdf Path Integrals in Quantum Theories: A Pedagogic 1st Step]
* [https://www.youtube.com/watch?v=QTjmLBzAdAA A mathematically rigorous approach to perturbative path integrals] via animation on YouTube
* [https://www.youtube.com/watch?v=vSFRN-ymfgE Feynman's Infinite Quantum Paths] | PBS Space Time. July 7, 2017. (Video, 15:48)
{{Quantum mechanics topics|state=expanded}}
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