Physics:Quantum Path integral formulation - Revision history

WikiHarold: Clean Quantum page image and red links

2026-05-23T23:33:10Z

Clean Quantum page image and red links

← Older revision		Revision as of 23:33, 23 May 2026
Line 660:		Line 660:
	* [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=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)		* [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}}~~
	{{Richard Feynman\|state=collapsed}}		{{Richard Feynman\|state=collapsed}}
	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

Maintenance script: Normalize quantum page header order

2026-05-22T11:33:31Z

Normalize quantum page header order

← Older revision		Revision as of 11:33, 22 May 2026
Line 3:		Line 3:
	{{Quantum article nav\|previous=Physics:Quantum Feynman diagrams\|previous label=Feynman diagrams\|next=Physics:Quantum Renormalization in field theory\|next label=Renormalization in field theory}}		{{Quantum article nav\|previous=Physics:Quantum Feynman diagrams\|previous label=Feynman diagrams\|next=Physics:Quantum Renormalization in field theory\|next label=Renormalization in field theory}}
	{{about\|a formulation of quantum mechanics\|integrals along a path, also known as line or contour integrals\|line integral}}		{{about\|a formulation of quantum mechanics\|integrals along a path, also known as line or contour integrals\|line integral}}

	<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

Maintenance script: Clean Book label remnants and backlink spacing

2026-05-22T11:11:50Z

Clean Book label remnants and backlink spacing

← Older revision		Revision as of 11:11, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Formulation of quantum mechanics}}		{{Short description\|Formulation of quantum mechanics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
	{{Quantum article nav\|previous=Physics:Quantum Feynman diagrams\|previous label=Feynman diagrams\|next=Physics:Quantum Renormalization in field theory\|next label=Renormalization in field theory}}		{{Quantum article nav\|previous=Physics:Quantum Feynman diagrams\|previous label=Feynman diagrams\|next=Physics:Quantum Renormalization in field theory\|next label=Renormalization in field theory}}

Maintenance script: Remove hidden BOM characters and direct Book label after Short description

2026-05-22T11:04:03Z

Remove hidden BOM characters and direct Book label after Short description

← Older revision		Revision as of 11:04, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Formulation of quantum mechanics}}~~Book I~~		{{Short description\|Formulation of quantum mechanics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}

Maintenance script: Clean Short description prefix and add Book I label

2026-05-22T10:49:06Z

Clean Short description prefix and add Book I label

← Older revision		Revision as of 10:49, 22 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Formulation of quantum mechanics}}		{{Short description\|Formulation of quantum mechanics}}Book I

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}

Maintenance script: Apply Quantum previous-next navigation

2026-05-20T12:26:08Z

Apply Quantum previous-next navigation

← Older revision		Revision as of 12:26, 20 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Formulation of quantum mechanics}}		{{Short description\|Formulation of quantum mechanics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
			{{Quantum article nav\|previous=Physics:Quantum Feynman diagrams\|previous label=Feynman diagrams\|next=Physics:Quantum Renormalization in field theory\|next label=Renormalization in field theory}}
	{{about\|a formulation of quantum mechanics\|integrals along a path, also known as line or contour integrals\|line integral}}		{{about\|a formulation of quantum mechanics\|integrals along a path, also known as line or contour integrals\|line integral}}

Maintenance script: Improve Book I intro quality

2026-05-20T09:05:01Z

Improve Book I intro quality

← Older revision		Revision as of 09:05, 20 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Formulation of quantum mechanics}}		{{Short description\|Formulation of quantum mechanics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
Line 11:		Line 11:

	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	'''Path integral formulation''' ~~is a Book I topic in~~ the ~~Quantum Collection. The~~ path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. [[File:Path_integral_formulation.jpg\|thumb\|400px\|Path integral formulation showing the propagation amplitude as a sum over all possible paths between two space-time points, \langle \psi_f \| \psi_i \rangle = \int \mathcal{D}[x(t)] e^{iS[x(t)]/\hbar}.]] The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.		'''Path integral formulation''' the path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical quantization.
	</div>		</div>

Maintenance script: Clean missing Book I links and template output

2026-05-20T08:47:43Z

Clean missing Book I links and template output

← Older revision		Revision as of 08:47, 20 May 2026
Line 1:		Line 1:
	{{Short description\|Formulation of quantum mechanics}}		{{Short description\|Formulation of quantum mechanics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
Line 233:		Line 233:
	=== The Schrödinger equation ===		=== The Schrödinger equation ===

	~~{{Main\|Physics~~:Relation between Schrödinger's equation and the path integral formulation of quantum mechanics}}		''Related topic:'' 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.		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.
Line 476:		Line 476:

	=== Schwinger–Dyson equations ===		=== Schwinger–Dyson equations ===
	~~{{Main\|Physics~~:Schwinger–Dyson equation}}		''Related topic:'' Schwinger–Dyson equation

	Since this formulation of quantum mechanics is analogous to classical action principle, one might expect that identities concerning the action in classical mechanics would have quantum counterparts derivable from a functional integral. This is often the case.		Since this formulation of quantum mechanics is analogous to classical action principle, one might expect that identities concerning the action in classical mechanics would have quantum counterparts derivable from a functional integral. This is often the case.
Line 530:		Line 530:

	=== Ward–Takahashi identities ===		=== Ward–Takahashi identities ===
	~~{{main\|Physics~~:Ward–Takahashi identity}}		''Related topic:'' Ward–Takahashi identity

	Now how about the on shell Noether's theorem for the classical case? Does it have a quantum analog as well? Yes, but with a caveat. The functional measure would have to be invariant under the one parameter group of symmetry transformation as well.		Now how about the on shell Noether's theorem for the classical case? Does it have a quantum analog as well? Yes, but with a caveat. The functional measure would have to be invariant under the one parameter group of symmetry transformation as well.

Maintenance script: Clean Book I red links, intro, and image slots

2026-05-20T08:36:47Z

Clean Book I red links, intro, and image slots

Maintenance script: Clean Book I red links, intro, and image slots

2026-05-20T08:16:35Z

Clean Book I red links, intro, and image slots

Show changes

← Older revision		Revision as of 23:33, 23 May 2026
Line 660:		Line 660:
	* [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=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)		* [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}}~~
	{{Richard Feynman\|state=collapsed}}		{{Richard Feynman\|state=collapsed}}
	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

← Older revision		Revision as of 11:11, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Formulation of quantum mechanics}}		{{Short description\|Formulation of quantum mechanics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
	{{Quantum article nav\|previous=Physics:Quantum Feynman diagrams\|previous label=Feynman diagrams\|next=Physics:Quantum Renormalization in field theory\|next label=Renormalization in field theory}}		{{Quantum article nav\|previous=Physics:Quantum Feynman diagrams\|previous label=Feynman diagrams\|next=Physics:Quantum Renormalization in field theory\|next label=Renormalization in field theory}}

← Older revision		Revision as of 12:26, 20 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Formulation of quantum mechanics}}		{{Short description\|Formulation of quantum mechanics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
			{{Quantum article nav\|previous=Physics:Quantum Feynman diagrams\|previous label=Feynman diagrams\|next=Physics:Quantum Renormalization in field theory\|next label=Renormalization in field theory}}
	{{about\|a formulation of quantum mechanics\|integrals along a path, also known as line or contour integrals\|line integral}}		{{about\|a formulation of quantum mechanics\|integrals along a path, also known as line or contour integrals\|line integral}}

← Older revision		Revision as of 08:36, 20 May 2026
Line 1:		Line 1:
	{{Short description\|Formulation of quantum mechanics}}		{{Short description\|Formulation of quantum mechanics}}

	{{Quantum book backlink\|Quantum field theory}}		{{Quantum book backlink\|Quantum field theory}}
Line 126:		Line 126:
	: <math>S[\mathbf{x},\dot\mathbf{x}]=\int dt\, L(\mathbf{x}(t),\dot\mathbf{x}(t))</math>		: <math>S[\mathbf{x},\dot\mathbf{x}]=\int dt\, L(\mathbf{x}(t),\dot\mathbf{x}(t))</math>

	[[File:Quantum_book1_path_integral_formulation_yellow.png\|thumb\|The diagram shows the contribution to the path integral of a free particle for a set of paths, eventually drawing a [[Cornu Spiral]].]]		[[File:Quantum_book1_path_integral_formulation_yellow.png\|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 ===		=== Free particle ===
Line 347:		Line 347:
	Now, the contribution of the kinetic energy to the path integral is as follows:		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>		: <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 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>.		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 measure, denoted <math>\mu_{x}</math>. The Wiener measure, constructed by Norbert Wiener gives a rigorous foundation to 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>		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>
Line 427:		Line 427:
	: <math>K(x - y) = \int_0^\infty e^{-\frac{(x - y)^2}{\Tau} -\alpha \Tau} \,d\Tau.</math>		: <math>K(x - y) = \int_0^\infty e^{-\frac{(x - y)^2}{\Tau} -\alpha \Tau} \,d\Tau.</math>

	This is the ~~[[Physics: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:		This is the 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>		: <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:		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: