Physics:Quantum Noether theorem - Revision history

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	{{Short description\|Connection between continuous symmetries and conservation laws}}		{{Short description\|Connection between continuous symmetries and conservation laws}}
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	{{Quantum article nav\|previous=Physics:Quantum Symmetry in quantum mechanics\|previous label=Symmetry in quantum mechanics\|next=Physics:Quantum Angular momentum operator\|next label=Angular momentum operator}}		{{Quantum article nav\|previous=Physics:Quantum Symmetry in quantum mechanics\|previous label=Symmetry in quantum mechanics\|next=Physics:Quantum Angular momentum operator\|next label=Angular momentum operator}}
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	[[File:Quantum_noether_theorem_yellow.png\|~~thumb\|right\|320px~~\|Noether theorem linking symmetry transformations with conserved quantities.]]		\|text='''Quantum Noether theorem''' refers to the use of Noether's theorem in quantum mechanics and quantum field theory. The theorem states that every continuous symmetry of an action corresponds to a conservation law. It is one of the central bridges between mathematical symmetry and physical observables.<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Noether%27s_theorem \|title=Noether's theorem \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>

	'''Quantum Noether theorem''' refers to the use of Noether's theorem in quantum mechanics and quantum field theory. The theorem states that every continuous symmetry of an action corresponds to a conservation law. It is one of the central bridges between mathematical symmetry and physical observables.<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Noether%27s_theorem \|title=Noether's theorem \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>

	Although first formulated in classical field theory, Noether's theorem remains essential in quantum physics. Time-translation symmetry is associated with conservation of energy, spatial translation with momentum, rotation with angular momentum, and gauge symmetry with conserved charge.		Although first formulated in classical field theory, Noether's theorem remains essential in quantum physics. Time-translation symmetry is associated with conservation of energy, spatial translation with momentum, rotation with angular momentum, and gauge symmetry with conserved charge.
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	== Symmetry and conservation ==		== Symmetry and conservation ==

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	~~<div style="display~~:~~flex; gap:24px; align-items:flex-start; max-width:1280px; clear:both;">~~		[[File:Quantum_noether_theorem_yellow.png\|thumb\|right\|320px\|Noether theorem linking symmetry transformations with conserved quantities.]]
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	'''Quantum Noether theorem''' refers to the use of Noether's theorem in quantum mechanics and quantum field theory. The theorem states that every continuous symmetry of an action corresponds to a conservation law. It is one of the central bridges between mathematical symmetry and physical observables.<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Noether%27s_theorem \|title=Noether's theorem \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>		'''Quantum Noether theorem''' refers to the use of Noether's theorem in quantum mechanics and quantum field theory. The theorem states that every continuous symmetry of an action corresponds to a conservation law. It is one of the central bridges between mathematical symmetry and physical observables.<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Noether%27s_theorem \|title=Noether's theorem \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>

	Although first formulated in classical field theory, Noether's theorem remains essential in quantum physics. Time-translation symmetry is associated with conservation of energy, spatial translation with momentum, rotation with angular momentum, and gauge symmetry with conserved charge.		Although first formulated in classical field theory, Noether's theorem remains essential in quantum physics. Time-translation symmetry is associated with conservation of energy, spatial translation with momentum, rotation with angular momentum, and gauge symmetry with conserved charge.
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	== Symmetry and conservation ==		== Symmetry and conservation ==

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	{{Quantum article nav\|previous=Physics:Quantum Symmetry in quantum mechanics\|previous label=Symmetry in quantum mechanics\|next=Physics:Quantum Angular momentum operator\|next label=Angular momentum operator}}		{{Quantum article nav\|previous=Physics:Quantum Symmetry in quantum mechanics\|previous label=Symmetry in quantum mechanics\|next=Physics:Quantum Angular momentum operator\|next label=Angular momentum operator}}

	~~[[File~~:~~Quantum_noether_theorem_yellow~~.~~png\|thumb\|280px\|Noether theorem linking symmetry transformations with conserved quantities.]]~~		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1280px; clear:both;">
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	'''Quantum Noether theorem''' refers to the use of Noether's theorem in quantum mechanics and quantum field theory. The theorem states that every continuous symmetry of an action corresponds to a conservation law. It is one of the central bridges between mathematical symmetry and physical observables.<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Noether%27s_theorem \|title=Noether's theorem \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>		'''Quantum Noether theorem''' refers to the use of Noether's theorem in quantum mechanics and quantum field theory. The theorem states that every continuous symmetry of an action corresponds to a conservation law. It is one of the central bridges between mathematical symmetry and physical observables.<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Noether%27s_theorem \|title=Noether's theorem \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>

	Although first formulated in classical field theory, Noether's theorem remains essential in quantum physics. Time-translation symmetry is associated with conservation of energy, spatial translation with momentum, rotation with angular momentum, and gauge symmetry with conserved charge.		Although first formulated in classical field theory, Noether's theorem remains essential in quantum physics. Time-translation symmetry is associated with conservation of energy, spatial translation with momentum, rotation with angular momentum, and gauge symmetry with conserved charge.
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	== Symmetry and conservation ==		== Symmetry and conservation ==

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			{{Quantum article nav\|previous=Physics:Quantum Symmetry in quantum mechanics\|previous label=Symmetry in quantum mechanics\|next=Physics:Quantum Angular momentum operator\|next label=Angular momentum operator}}

	[[File:Quantum_noether_theorem_yellow.png\|thumb\|280px\|Noether theorem linking symmetry transformations with conserved quantities.]]		[[File:Quantum_noether_theorem_yellow.png\|thumb\|280px\|Noether theorem linking symmetry transformations with conserved quantities.]]

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{{Short description|Connection between continuous symmetries and conservation laws}}
{{Quantum book backlink|Mathematical structure and systems}}

[[File:Quantum_noether_theorem_yellow.png|thumb|280px|Noether theorem linking symmetry transformations with conserved quantities.]]

'''Quantum Noether theorem''' refers to the use of Noether's theorem in quantum mechanics and quantum field theory. The theorem states that every continuous symmetry of an action corresponds to a conservation law. It is one of the central bridges between mathematical symmetry and physical observables.<ref>{{cite web |url=https://en.wikipedia.org/wiki/Noether%27s_theorem |title=Noether's theorem |publisher=Wikipedia |access-date=20 May 2026}}</ref>

Although first formulated in classical field theory, Noether's theorem remains essential in quantum physics. Time-translation symmetry is associated with conservation of energy, spatial translation with momentum, rotation with angular momentum, and gauge symmetry with conserved charge.

== Symmetry and conservation ==
In quantum mechanics, observables are represented by operators. Continuous symmetries are generated by operators such as the Hamiltonian, momentum, or angular momentum. Conservation follows when a generator commutes with the dynamics of the system.

In quantum field theory, Noether currents encode conserved quantities locally. These currents help organize particle interactions, selection rules, and gauge theories.

== Importance ==
Noether's theorem explains why conservation laws are not isolated facts, but consequences of symmetry. This insight underlies much of modern theoretical physics, from atomic spectra to the [[Physics:Quantum Standard Model|Standard Model]].

It is especially useful when deciding which interactions are allowed by the symmetries of a quantum theory.

== See also ==
* [[Physics:Quantum Symmetry in quantum mechanics]]
* [[Physics:Quantum Angular momentum operator]]
* [[Physics:Quantum Field Theory Gauge symmetry]]
* [[Biography:Eugene Wigner]]

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Physics:Quantum Noether theorem|1}}