Physics:Quantum S-matrix - Revision history

Maintenance script: Normalize quantum page header order

2026-05-22T11:32:19Z

Normalize quantum page header order

← Older revision		Revision as of 11:32, 22 May 2026
Line 2:		Line 2:
	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}
	{{Quantum article nav\|previous=Physics:Quantum Scattering matrix\|previous label=Scattering matrix\|next=Physics:Quantum tunnelling\|next label=Tunnelling}}		{{Quantum article nav\|previous=Physics:Quantum Scattering matrix\|previous label=Scattering matrix\|next=Physics:Quantum tunnelling\|next label=Tunnelling}}

	<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

2026-05-22T11:10:39Z

Clean Book label remnants and backlink spacing

← Older revision		Revision as of 11:10, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Quantum Collection topic on Quantum S-matrix}}		{{Short description\|Quantum Collection topic on Quantum S-matrix}}

	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}
	{{Quantum article nav\|previous=Physics:Quantum Scattering matrix\|previous label=Scattering matrix\|next=Physics:Quantum tunnelling\|next label=Tunnelling}}		{{Quantum article nav\|previous=Physics:Quantum Scattering matrix\|previous label=Scattering matrix\|next=Physics:Quantum tunnelling\|next label=Tunnelling}}

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

2026-05-22T11:02:51Z

Remove hidden BOM characters and direct Book label after Short description

← Older revision		Revision as of 11:02, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Quantum Collection topic on Quantum S-matrix}}~~Book I~~		{{Short description\|Quantum Collection topic on Quantum S-matrix}}

	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}

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

2026-05-22T10:47:54Z

Clean Short description prefix and add Book I label

← Older revision		Revision as of 10:47, 22 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Quantum Collection topic on Quantum S-matrix}}		{{Short description\|Quantum Collection topic on Quantum S-matrix}}Book I

	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}

Maintenance script: Apply continuous Quantum previous-next navigation

2026-05-20T21:59:16Z

Apply continuous Quantum previous-next navigation

← Older revision		Revision as of 21:59, 20 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Quantum Collection topic on Quantum S-matrix}}		{{Short description\|Quantum Collection topic on Quantum S-matrix}}

	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}
	{{Quantum article nav\|previous=Physics:Quantum Scattering ~~theory~~\|previous label=Scattering ~~theory~~\|next=Physics:Quantum tunnelling\|next label=Tunnelling}}		{{Quantum article nav\|previous=Physics:Quantum Scattering matrix\|previous label=Scattering matrix\|next=Physics:Quantum tunnelling\|next label=Tunnelling}}

	<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: Apply Quantum previous-next navigation

2026-05-20T12:24:32Z

Apply Quantum previous-next navigation

← Older revision		Revision as of 12:24, 20 May 2026
Line 1:		Line 1:
	{{Short description\|Quantum Collection topic on Quantum S-matrix}}		{{Short description\|Quantum Collection topic on Quantum S-matrix}}

	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}
			{{Quantum article nav\|previous=Physics:Quantum Scattering theory\|previous label=Scattering theory\|next=Physics:Quantum tunnelling\|next label=Tunnelling}}

	<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 missing Book I links and template output

2026-05-20T08:47:12Z

Clean missing Book I links and template output

← Older revision		Revision as of 08:47, 20 May 2026
Line 1:		Line 1:
	{{Short description\|Quantum Collection topic on Quantum S-matrix}}		{{Short description\|Quantum Collection topic on Quantum S-matrix}}

	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}
Line 10:		Line 10:

	<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;">
	'''''S''-matrix theory''' was a proposal for replacing local [[quantum field theory]] as the basic principle of elementary [[particle physics]]. It avoids the notion of space and time by replacing it with abstract mathematical properties of the ''S''-matrix, which relates the infinite past to the infinite future in a single step without intermediate time evolution.		'''''S''-matrix theory''' was a proposal for replacing local quantum field theory as the basic principle of elementary particle physics. It avoids the notion of space and time by replacing it with abstract mathematical properties of the ''S''-matrix, which relates the infinite past to the infinite future in a single step without intermediate time evolution.

	The approach was influential in the 1960s as an alternative to quantum field theory, which at the time faced problems such as the Landau pole at strong coupling. Applied to strong interactions, it led to the development of [[string theory]]. Although later superseded by [[quantum chromodynamics]], the ideas of ''S''-matrix theory remain important, especially in modern approaches to [[quantum gravity]] and in connections with the [[holographic principle]] and the AdS/CFT correspondence.<ref>{{cite journal \|author1-link=Steven Giddings\| last=Giddings \| first=Steven B. \| title=Boundary S-Matrix and the Anti–de Sitter Space to Conformal Field Theory Dictionary \| journal=Physical Review Letters \| volume=83 \| issue=14 \| date=1999-10-04 \| doi=10.1103/physrevlett.83.2707 \| pages=2707–2710\|arxiv=hep-th/9903048}}</ref>		The approach was influential in the 1960s as an alternative to quantum field theory, which at the time faced problems such as the Landau pole at strong coupling. Applied to strong interactions, it led to the development of string theory. Although later superseded by quantum chromodynamics, the ideas of ''S''-matrix theory remain important, especially in modern approaches to quantum gravity and in connections with the holographic principle and the AdS/CFT correspondence.<ref>{{cite journal \|author1-link=Steven Giddings\| last=Giddings \| first=Steven B. \| title=Boundary S-Matrix and the Anti–de Sitter Space to Conformal Field Theory Dictionary \| journal=Physical Review Letters \| volume=83 \| issue=14 \| date=1999-10-04 \| doi=10.1103/physrevlett.83.2707 \| pages=2707–2710\|arxiv=hep-th/9903048}}</ref>
	</div>		</div>

Line 24:		Line 24:
	''S''-matrix theory was proposed by Werner Heisenberg in 1943,<ref>{{cite journal \| last=Heisenberg \| first=W. \| title=Die beobachtbaren Größen in der Theorie der Elementarteilchen \| journal=Zeitschrift für Physik \| volume=120 \| issue=7–10 \| year=1943 \| pages=513–538}}</ref> building on earlier work by John Archibald Wheeler, who introduced the concept of the ''S''-matrix in 1937.<ref>{{cite journal \| last=Wheeler \| first=John A. \| title=On the Mathematical Description of Light Nuclei \| journal=Physical Review \| volume=52 \| issue=11 \| date=1937-12-01 \| pages=1107–1122}}</ref>		''S''-matrix theory was proposed by Werner Heisenberg in 1943,<ref>{{cite journal \| last=Heisenberg \| first=W. \| title=Die beobachtbaren Größen in der Theorie der Elementarteilchen \| journal=Zeitschrift für Physik \| volume=120 \| issue=7–10 \| year=1943 \| pages=513–538}}</ref> building on earlier work by John Archibald Wheeler, who introduced the concept of the ''S''-matrix in 1937.<ref>{{cite journal \| last=Wheeler \| first=John A. \| title=On the Mathematical Description of Light Nuclei \| journal=Physical Review \| volume=52 \| issue=11 \| date=1937-12-01 \| pages=1107–1122}}</ref>

	The theory was further developed by physicists such as Geoffrey Chew, Steven Frautschi, Stanley Mandelstam, Vladimir Gribov, and Tullio Regge. Related ideas were also promoted by Lev Landau and Murray Gell-Mann. In 1979, Steven Weinberg connected ''S''-matrix ideas with [[effective field theory]] through his "folk theorem".<ref>{{Cite book \|last=Cushing \|first=James T. \|title=Theory Construction and Selection in Modern Physics: The S Matrix \|date=1990 \|publisher=Cambridge University Press}}</ref>		The theory was further developed by physicists such as Geoffrey Chew, Steven Frautschi, Stanley Mandelstam, Vladimir Gribov, and Tullio Regge. Related ideas were also promoted by Lev Landau and Murray Gell-Mann. In 1979, Steven Weinberg connected ''S''-matrix ideas with effective field theory through his "folk theorem".<ref>{{Cite book \|last=Cushing \|first=James T. \|title=Theory Construction and Selection in Modern Physics: The S Matrix \|date=1990 \|publisher=Cambridge University Press}}</ref>

	== Basic principles ==		== Basic principles ==
Line 43:		Line 43:

	== Bootstrap models ==		== Bootstrap models ==
	Because the general principles were too broad, additional assumptions were introduced in so-called [[bootstrap ~~model]]s~~. These models attempted to determine particle properties self-consistently using experimental data and dispersion relations.		Because the general principles were too broad, additional assumptions were introduced in so-called bootstrap models. These models attempted to determine particle properties self-consistently using experimental data and dispersion relations.

	Although successful in some cases, the resulting equations were mathematically complex and lacked a clear spacetime interpretation, limiting their usefulness.		Although successful in some cases, the resulting equations were mathematically complex and lacked a clear spacetime interpretation, limiting their usefulness.
Line 50:		Line 50:
	In Regge theory, strongly interacting particles are organized along Regge trajectories. This suggested that hadrons are related through a unified structure, rather than being fundamental.		In Regge theory, strongly interacting particles are organized along Regge trajectories. This suggested that hadrons are related through a unified structure, rather than being fundamental.

	These ideas led to the development of early string models, where particle interactions were interpreted as arising from extended one-dimensional objects. This provided a bridge between ''S''-matrix theory and modern [[string theory]].		These ideas led to the development of early string models, where particle interactions were interpreted as arising from extended one-dimensional objects. This provided a bridge between ''S''-matrix theory and modern string theory.

	== Scattering interpretation ==		== Scattering interpretation ==

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

2026-05-20T08:14:07Z

Clean Book I red links, intro, and image slots

Harold: Clean Quantum header BOM and legacy backlink

2026-05-17T22:05:00Z

Clean Quantum header BOM and legacy backlink

← Older revision		Revision as of 22:05, 17 May 2026
Line 1:		Line 1:
	{{Short description\|Quantum Collection topic on Quantum S-matrix}}		{{Short description\|Quantum Collection topic on Quantum S-matrix}}

	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}

Harold: Restore Quantum article header template

2026-05-17T21:51:31Z

Restore Quantum article header template

← Older revision		Revision as of 21:51, 17 May 2026
Line 1:		Line 1:
	{{Short description\|Quantum Collection topic on Quantum S-matrix}}		{{Short description\|Quantum Collection topic on Quantum S-matrix}}

	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}
Line 21:		Line 21:
	</div>		</div>

	= History =		== History ==
	''S''-matrix theory was proposed by [[Werner Heisenberg]] in 1943,<ref>{{cite journal \| last=Heisenberg \| first=W. \| title=Die beobachtbaren Größen in der Theorie der Elementarteilchen \| journal=Zeitschrift für Physik \| volume=120 \| issue=7–10 \| year=1943 \| pages=513–538}}</ref> building on earlier work by [[John Archibald Wheeler]], who introduced the concept of the ''S''-matrix in 1937.<ref>{{cite journal \| last=Wheeler \| first=John A. \| title=On the Mathematical Description of Light Nuclei \| journal=Physical Review \| volume=52 \| issue=11 \| date=1937-12-01 \| pages=1107–1122}}</ref>		''S''-matrix theory was proposed by [[Werner Heisenberg]] in 1943,<ref>{{cite journal \| last=Heisenberg \| first=W. \| title=Die beobachtbaren Größen in der Theorie der Elementarteilchen \| journal=Zeitschrift für Physik \| volume=120 \| issue=7–10 \| year=1943 \| pages=513–538}}</ref> building on earlier work by [[John Archibald Wheeler]], who introduced the concept of the ''S''-matrix in 1937.<ref>{{cite journal \| last=Wheeler \| first=John A. \| title=On the Mathematical Description of Light Nuclei \| journal=Physical Review \| volume=52 \| issue=11 \| date=1937-12-01 \| pages=1107–1122}}</ref>

← Older revision		Revision as of 11:10, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Quantum Collection topic on Quantum S-matrix}}		{{Short description\|Quantum Collection topic on Quantum S-matrix}}

	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}
	{{Quantum article nav\|previous=Physics:Quantum Scattering matrix\|previous label=Scattering matrix\|next=Physics:Quantum tunnelling\|next label=Tunnelling}}		{{Quantum article nav\|previous=Physics:Quantum Scattering matrix\|previous label=Scattering matrix\|next=Physics:Quantum tunnelling\|next label=Tunnelling}}

← Older revision		Revision as of 08:14, 20 May 2026
Line 1:		Line 1:
	{{Short description\|Quantum Collection topic on Quantum S-matrix}}		{{Short description\|Quantum Collection topic on Quantum S-matrix}}

	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}
Line 10:		Line 10:

	<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;">
	'''''S''-matrix theory''' was a proposal for replacing local [[quantum field theory]] as the basic principle of elementary [[particle physics]]. It avoids the notion of space and time by replacing it with abstract mathematical properties of the ~~[[S-matrix\|~~''S''-matrix]], which relates the infinite past to the infinite future in a single step without intermediate time evolution.		'''''S''-matrix theory''' was a proposal for replacing local [[quantum field theory]] as the basic principle of elementary [[particle physics]]. It avoids the notion of space and time by replacing it with abstract mathematical properties of the ''S''-matrix, which relates the infinite past to the infinite future in a single step without intermediate time evolution.

	The approach was influential in the 1960s as an alternative to quantum field theory, which at the time faced problems such as the [[Landau pole]] at strong coupling. Applied to strong interactions, it led to the development of [[string theory]]. Although later superseded by [[quantum chromodynamics]], the ideas of ''S''-matrix theory remain important, especially in modern approaches to [[quantum gravity]] and in connections with the [[holographic principle]] and the [[AdS/CFT correspondence]].<ref>{{cite journal \|author1-link=Steven Giddings\| last=Giddings \| first=Steven B. \| title=Boundary S-Matrix and the Anti–de Sitter Space to Conformal Field Theory Dictionary \| journal=Physical Review Letters \| volume=83 \| issue=14 \| date=1999-10-04 \| doi=10.1103/physrevlett.83.2707 \| pages=2707–2710\|arxiv=hep-th/9903048}}</ref>		The approach was influential in the 1960s as an alternative to quantum field theory, which at the time faced problems such as the Landau pole at strong coupling. Applied to strong interactions, it led to the development of [[string theory]]. Although later superseded by [[quantum chromodynamics]], the ideas of ''S''-matrix theory remain important, especially in modern approaches to [[quantum gravity]] and in connections with the [[holographic principle]] and the AdS/CFT correspondence.<ref>{{cite journal \|author1-link=Steven Giddings\| last=Giddings \| first=Steven B. \| title=Boundary S-Matrix and the Anti–de Sitter Space to Conformal Field Theory Dictionary \| journal=Physical Review Letters \| volume=83 \| issue=14 \| date=1999-10-04 \| doi=10.1103/physrevlett.83.2707 \| pages=2707–2710\|arxiv=hep-th/9903048}}</ref>
	</div>		</div>

Line 22:		Line 22:

	== History ==		== History ==
	''S''-matrix theory was proposed by [[Werner Heisenberg]] in 1943,<ref>{{cite journal \| last=Heisenberg \| first=W. \| title=Die beobachtbaren Größen in der Theorie der Elementarteilchen \| journal=Zeitschrift für Physik \| volume=120 \| issue=7–10 \| year=1943 \| pages=513–538}}</ref> building on earlier work by [[John Archibald Wheeler]], who introduced the concept of the ''S''-matrix in 1937.<ref>{{cite journal \| last=Wheeler \| first=John A. \| title=On the Mathematical Description of Light Nuclei \| journal=Physical Review \| volume=52 \| issue=11 \| date=1937-12-01 \| pages=1107–1122}}</ref>		''S''-matrix theory was proposed by Werner Heisenberg in 1943,<ref>{{cite journal \| last=Heisenberg \| first=W. \| title=Die beobachtbaren Größen in der Theorie der Elementarteilchen \| journal=Zeitschrift für Physik \| volume=120 \| issue=7–10 \| year=1943 \| pages=513–538}}</ref> building on earlier work by John Archibald Wheeler, who introduced the concept of the ''S''-matrix in 1937.<ref>{{cite journal \| last=Wheeler \| first=John A. \| title=On the Mathematical Description of Light Nuclei \| journal=Physical Review \| volume=52 \| issue=11 \| date=1937-12-01 \| pages=1107–1122}}</ref>

	The theory was further developed by physicists such as [[Geoffrey Chew]], [[Steven Frautschi]], [[Stanley Mandelstam]], [[Vladimir Gribov]], and [[Tullio Regge]]. Related ideas were also promoted by [[Lev Landau]] and [[Murray Gell-Mann]]. In 1979, [[Steven Weinberg]] connected ''S''-matrix ideas with [[effective field theory]] through his "folk theorem".<ref>{{Cite book \|last=Cushing \|first=James T. \|title=Theory Construction and Selection in Modern Physics: The S Matrix \|date=1990 \|publisher=Cambridge University Press}}</ref>		The theory was further developed by physicists such as Geoffrey Chew, Steven Frautschi, Stanley Mandelstam, Vladimir Gribov, and Tullio Regge. Related ideas were also promoted by Lev Landau and Murray Gell-Mann. In 1979, Steven Weinberg connected ''S''-matrix ideas with [[effective field theory]] through his "folk theorem".<ref>{{Cite book \|last=Cushing \|first=James T. \|title=Theory Construction and Selection in Modern Physics: The S Matrix \|date=1990 \|publisher=Cambridge University Press}}</ref>

	== Basic principles ==		== Basic principles ==
	The theory is based on a set of general principles:		The theory is based on a set of general principles:

	* '''Relativity''': the ''S''-matrix forms a representation of the [[Poincaré group]].		* '''Relativity''': the ''S''-matrix forms a representation of the Poincaré group.
	* '''[[Unitarity]]''': <math>S S^{\dagger} = 1</math>, ensuring probability conservation.		* '''Unitarity''': <math>S S^{\dagger} = 1</math>, ensuring probability conservation.
	* '''Analyticity''': scattering amplitudes are analytic functions with well-defined singularities.		* '''Analyticity''': scattering amplitudes are analytic functions with well-defined singularities.

	Additional analyticity conditions include:		Additional analyticity conditions include:

	* ~~[[Crossing (physics)\|~~Crossing symmetry]], relating particle and antiparticle processes.		* Crossing symmetry, relating particle and antiparticle processes.
	* ~~[[Kramers–Kronig relations\|~~Dispersion relations]], connecting real and imaginary parts of amplitudes.		* Dispersion relations, connecting real and imaginary parts of amplitudes.
	* Causality conditions restricting the allowed singularities.		* Causality conditions restricting the allowed singularities.
	* The '''Landau principle''', stating that singularities correspond to physical particle production thresholds.<ref>{{cite journal \| last=Landau \| first=L.D. \| title=On analytic properties of vertex parts in quantum field theory \| journal=Nuclear Physics \| volume=13 \| year=1959 \| pages=181–192}}</ref>		* The '''Landau principle''', stating that singularities correspond to physical particle production thresholds.<ref>{{cite journal \| last=Landau \| first=L.D. \| title=On analytic properties of vertex parts in quantum field theory \| journal=Nuclear Physics \| volume=13 \| year=1959 \| pages=181–192}}</ref>
Line 48:		Line 48:

	== Regge theory ==		== Regge theory ==
	In [[Regge theory]], strongly interacting particles are organized along ~~[[Regge trajectory\|~~Regge trajectories]]. This suggested that hadrons are related through a unified structure, rather than being fundamental.		In Regge theory, strongly interacting particles are organized along Regge trajectories. This suggested that hadrons are related through a unified structure, rather than being fundamental.

	These ideas led to the development of early string models, where particle interactions were interpreted as arising from extended one-dimensional objects. This provided a bridge between ''S''-matrix theory and modern [[string theory]].		These ideas led to the development of early string models, where particle interactions were interpreted as arising from extended one-dimensional objects. This provided a bridge between ''S''-matrix theory and modern [[string theory]].