Physics:Quantum Creation and annihilation operators - Revision history

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	In quantum field theories and many-body problems one works with creation and annihilation operators of quantum states, <math>a^\dagger_i</math> and <math>a^{\,}_i</math>. These operators change the eigenvalues of the number operator,		In quantum field theories and many-body problems one works with creation and annihilation operators of quantum states, <math>a^\dagger_i</math> and <math>a^{\,}_i</math>. These operators change the eigenvalues of the number operator,
	<math display="block">N = \sum_i n_i = \sum_i a^\dagger_i a^{\,}_i,</math>		<math display="block">N = \sum_i n_i = \sum_i a^\dagger_i a^{\,}_i,</math>
	by one, in analogy to the harmonic oscillator. The indices (such as <math>i</math>) represent quantum numbers that label the single-particle states of the system; hence, they are not necessarily single numbers. For example, a tuple of quantum numbers <math>(n, \ell, m, s)</math> is used to label states in the [[Physics:~~Hydrogen atom~~\|hydrogen atom]].		by one, in analogy to the harmonic oscillator. The indices (such as <math>i</math>) represent quantum numbers that label the single-particle states of the system; hence, they are not necessarily single numbers. For example, a tuple of quantum numbers <math>(n, \ell, m, s)</math> is used to label states in the [[Physics:Quantum atoms/hydrogen\|hydrogen atom]].

	The commutation relations of creation and annihilation operators in a multiple-boson system are,		The commutation relations of creation and annihilation operators in a multiple-boson system are,

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	{{Short description\|Operators useful in quantum mechanics}}~~Book I~~		{{Short description\|Operators useful in quantum mechanics}}

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	==Ladder operators for the quantum harmonic oscillator==		==Ladder operators for the quantum harmonic oscillator==
	~~{{see also\|Physics~~:Ladder operator}}		''Related topic:'' Ladder operator

	In the context of the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]], one reinterprets the ladder operators as creation and annihilation operators, adding or subtracting fixed [[Physics:Quantum\|quanta]] of energy to the oscillator system.		In the context of the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]], one reinterprets the ladder operators as creation and annihilation operators, adding or subtracting fixed [[Physics:Quantum\|quanta]] of energy to the oscillator system.
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	== Generalized creation and annihilation operators ==		== Generalized creation and annihilation operators ==

	~~{{Main\|Physics~~:CCR and CAR algebras}}		''Related topic:'' CCR and CAR algebras
	~~{{see also\|Physics~~:Ladder operator}}		''Related topic:'' Ladder operator

	Thanks to representation theory and C*-algebras the operators derived above are actually a specific instance of a more generalized notion of creation and annihilation operators in the context of CCR and CAR algebras. Mathematically and even more generally ladder operators can be understood in the context of a root system of a semisimple Lie group and the associated semisimple Lie algebra without the need of realizing the representation as operators on a functional Hilbert space.<ref>{{citation\| first=Fulton \|last= Harris\|title=Representation Theory}} pp. 164</ref>		Thanks to representation theory and C*-algebras the operators derived above are actually a specific instance of a more generalized notion of creation and annihilation operators in the context of CCR and CAR algebras. Mathematically and even more generally ladder operators can be understood in the context of a root system of a semisimple Lie group and the associated semisimple Lie algebra without the need of realizing the representation as operators on a functional Hilbert space.<ref>{{citation\| first=Fulton \|last= Harris\|title=Representation Theory}} pp. 164</ref>
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	==In quantum field theories==		==In quantum field theories==

	~~{{Main\|Physics~~:Second quantization~~\|Quantum field theory#Physics:~~Second quantization}}		''Related topic:'' Second quantization, Second quantization

	In quantum field theories and many-body problems one works with creation and annihilation operators of quantum states, <math>a^\dagger_i</math> and <math>a^{\,}_i</math>. These operators change the eigenvalues of the number operator,		In quantum field theories and many-body problems one works with creation and annihilation operators of quantum states, <math>a^\dagger_i</math> and <math>a^{\,}_i</math>. These operators change the eigenvalues of the number operator,

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	Creation and annihilation operators can act on states of various types of particles. For example, in quantum chemistry and many-body theory the creation and annihilation operators often act on electron states. They can also refer specifically to the ladder operators for the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]]. In the latter case, the creation operator is interpreted as a raising operator, adding a quantum of energy to the oscillator system (similarly for the lowering operator). They can be used to represent phonons. Constructing Hamiltonians using these operators has the advantage that the theory automatically satisfies the cluster decomposition theorem.<ref>{{cite book\|first=Steven\|last=Weinberg\|title=The Quantum Theory of Fields Volume 1\|publisher=Cambridge University Press\|date=1995\|chapter=4\|page=169\|isbn=9780521670531}}</ref>		Creation and annihilation operators can act on states of various types of particles. For example, in quantum chemistry and many-body theory the creation and annihilation operators often act on electron states. They can also refer specifically to the ladder operators for the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]]. In the latter case, the creation operator is interpreted as a raising operator, adding a quantum of energy to the oscillator system (similarly for the lowering operator). They can be used to represent phonons. Constructing Hamiltonians using these operators has the advantage that the theory automatically satisfies the cluster decomposition theorem.<ref>{{cite book\|first=Steven\|last=Weinberg\|title=The Quantum Theory of Fields Volume 1\|publisher=Cambridge University Press\|date=1995\|chapter=4\|page=169\|isbn=9780521670531}}</ref>

	The mathematics for the creation and annihilation operators for bosons is the same as for the ladder operators of the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]].<ref name='Feynman1998p167'>{{harvnb\|Feynman\|1998\|p=167}}</ref> For example, the ~~[[Commutator#Ring theory\|~~commutator]] of the creation and annihilation operators that are associated with the same boson state equals one, while all other commutators vanish. However, for fermions the mathematics is different, involving anticommutators instead of commutators.<ref name='Feynman1998p174-5'>{{harvnb\|Feynman\|1998\|pp=174–5}}</ref>		The mathematics for the creation and annihilation operators for bosons is the same as for the ladder operators of the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]].<ref name='Feynman1998p167'>{{harvnb\|Feynman\|1998\|p=167}}</ref> For example, the commutator of the creation and annihilation operators that are associated with the same boson state equals one, while all other commutators vanish. However, for fermions the mathematics is different, involving anticommutators instead of commutators.<ref name='Feynman1998p174-5'>{{harvnb\|Feynman\|1998\|pp=174–5}}</ref>
	</div>		</div>

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	===Normalization conventions===		===Normalization conventions===
	While Zee<ref name=Zee>{{cite book \|last1=Zee \|first1=A. \|title=Quantum field theory in a nutshell \|date=2003 \|publisher=Princeton University Press \|isbn=978-0691010199 \|page=63}}</ref> obtains the momentum space normalization <math>[\hat a_{\mathbf p},\hat a_{\mathbf q}^\dagger] = \delta(\mathbf{p} - \mathbf{q})</math> via the ~~[[Fourier transform#Other conventions\|~~symmetric convention]] for Fourier transforms, Tong<ref name=Tong>{{cite book \|last1=Tong \|first1=David \|title=Quantum Field Theory \|date=2007 \|page=24,31 \|url=https://www.damtp.cam.ac.uk/user/tong/qft.html \|access-date=3 December 2019}}</ref> and Peskin & Schroeder<ref name="peskin">{{cite book \|last1=Peskin \|first1=M. \|author-link=Michael Peskin \|last2=Schroeder \|first2=D.		While Zee<ref name=Zee>{{cite book \|last1=Zee \|first1=A. \|title=Quantum field theory in a nutshell \|date=2003 \|publisher=Princeton University Press \|isbn=978-0691010199 \|page=63}}</ref> obtains the momentum space normalization <math>[\hat a_{\mathbf p},\hat a_{\mathbf q}^\dagger] = \delta(\mathbf{p} - \mathbf{q})</math> via the symmetric convention for Fourier transforms, Tong<ref name=Tong>{{cite book \|last1=Tong \|first1=David \|title=Quantum Field Theory \|date=2007 \|page=24,31 \|url=https://www.damtp.cam.ac.uk/user/tong/qft.html \|access-date=3 December 2019}}</ref> and Peskin & Schroeder<ref name="peskin">{{cite book \|last1=Peskin \|first1=M. \|author-link=Michael Peskin \|last2=Schroeder \|first2=D.
	\|year=1995		\|year=1995
	\|title=An Introduction to Quantum Field Theory		\|title=An Introduction to Quantum Field Theory

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	[[Category:Quantum field theory]]		[[Category:Quantum field theory]]

	{{Sourceattribution\|Quantum Creation and annihilation operators\|1}}		{{Sourceattribution\|Physics:Quantum Creation and annihilation operators\|1}}

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