Creation and annihilation operators#Matrix representation

{{main|Quantum harmonic oscillator#Ladder operator method}}

{{see also|Ladder operator}}

In the context of the quantum harmonic oscillator, one reinterprets the ladder operators as creation and annihilation operators, adding or subtracting fixed quanta of energy to the oscillator system.

Creation/annihilation operators are different for bosons (integer spin) and fermions (half-integer spin). This is because their wavefunctions have different symmetry properties.

First consider the simpler bosonic case of the photons of the quantum harmonic oscillator.

Start with the Schrödinger equation for the one-dimensional time independent quantum harmonic oscillator,

$$\backslash left(-\backslash frac\{\backslash hbar^2\}\{2m\}\; \backslash frac\{d^2\}\{d\; x^2\}\; +\; \backslash frac\{1\}\{2\}m\; \backslash omega^2\; x^2\backslash right)\; \backslash psi(x)\; =\; E\; \backslash psi(x).$$

Make a coordinate substitution to nondimensionalize the differential equation

$$x\; \backslash \; =\; \backslash \; \backslash sqrt\{\; \backslash frac\{\backslash hbar\}\{m\; \backslash omega\}\}\; q.$$

The Schrödinger equation for the oscillator becomes

$$\backslash frac\{\backslash hbar\; \backslash omega\}\{2\}\; \backslash left(-\backslash frac\{d^2\}\{d\; q^2\}\; +\; q^2\; \backslash right)\; \backslash psi(q)\; =\; E\; \backslash psi(q).$$

Note that the quantity $\backslash hbar\; \backslash omega\; =\; h\; \backslash nu$ is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian can be written as

$$-\backslash frac\{d^2\}\{dq^2\}\; +\; q^2\; =\; \backslash left(-\backslash frac\{d\}\{dq\}+q\; \backslash right)\; \backslash left(\backslash frac\{d\}\{dq\}+\; q\; \backslash right)\; +\; \backslash frac\; \{d\}\{dq\}q\; -\; q\; \backslash frac\; \{d\}\{dq\}\; .$$

The last two terms can be simplified by considering their effect on an arbitrary differentiable function $f(q),$

$$\backslash left(\backslash frac\{d\}\{dq\}\; q-\; q\; \backslash frac\{d\}\{dq\}\; \backslash right)f(q)\; =\; \backslash frac\{d\}\{dq\}(q\; f(q))\; -\; q\; \backslash frac\{df(q)\}\{dq\}\; =\; f(q)$$

which implies,

$$\backslash frac\{d\}\{dq\}\; q-\; q\; \backslash frac\{d\}\{dq\}\; =\; 1\; ,$$

coinciding with the usual canonical commutation relation $-i[q,p]=1$, in position space representation: $p:=-i\backslash frac\{d\}\{dq\}$.

Therefore,

$$-\backslash frac\{d^2\}\{dq^2\}\; +\; q^2\; =\; \backslash left(-\backslash frac\{d\}\{dq\}+q\; \backslash right)\; \backslash left(\backslash frac\{d\}\{dq\}+\; q\; \backslash right)\; +\; 1$$

and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2,

$$\backslash hbar\; \backslash omega\; \backslash left[\backslash frac\{1\}\{\backslash sqrt\{2\}\}\; \backslash left(-\backslash frac\{d\}\{dq\}+q\; \backslash right)\backslash frac\{1\}\{\backslash sqrt\{2\}\}\; \backslash left(\backslash frac\{d\}\{dq\}+\; q\; \backslash right)\; +\; \backslash frac\{1\}\{2\}\; \backslash right]\; \backslash psi(q)\; =\; E\; \backslash psi(q).$$

If one defines

$$a^\backslash dagger\; \backslash \; =\; \backslash \; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; \backslash left(-\backslash frac\{d\}\{dq\}\; +\; q\backslash right)$$

as the "creation operator" or the "raising operator" and

$$a\; \backslash \; \backslash \; =\; \backslash \; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; \backslash left(\backslash frac\{d\}\{dq\}\; +\; q\backslash right)$$

as the "annihilation operator" or the "lowering operator", the Schrödinger equation for the oscillator reduces to

$$\backslash hbar\; \backslash omega\; \backslash left(\; a^\backslash dagger\; a\; +\; \backslash frac\{1\}\{2\}\; \backslash right)\; \backslash psi(q)\; =\; E\; \backslash psi(q).$$

This is significantly simpler than the original form. Further simplifications of this equation enable one to derive all the properties listed above thus far.

Letting $p\; =\; -\; i\; \backslash frac\{d\}\{dq\}$, where $p$ is the nondimensionalized momentum operator

one has

$$[q,\; p]\; =\; i\; \backslash ,$$

and

$$\backslash begin\{align\}$$

a &= \frac{1}{\sqrt{2}}(q + i p) = \frac{1}{\sqrt{2}}\left( q + \frac{d}{dq}\right) \\[1ex]

a^\dagger &= \frac{1}{\sqrt{2}}(q - i p) = \frac{1}{\sqrt{2}}\left( q - \frac{d}{dq}\right).

\end{align}

Note that these imply

$$[a,\; a^\backslash dagger\; ]\; =\; \backslash frac\{1\}\{2\}\; [\; q\; +\; ip\; ,\; q-i\; p]\; =\; \backslash frac\{1\}\{2\}\; ([q,-ip]\; +\; [ip,\; q])\; =\; -\backslash frac\{i\}\{2\}\; ([q,\; p]\; +\; [q,\; p])\; =\; 1.$$

The operators $a\backslash ,$ and $a^\backslash dagger\backslash ,$ may be contrasted to normal operators, which commute with their adjoints.A normal operator has a representation {{math|A{{=}} B + i C}}, where {{math|B,C}} are self-adjoint and commute, i.e. $BC=CB$. By contrast, {{mvar|a}} has the representation $a=q+ip$ where $p,q$ are self-adjoint but $[p,q]=1$. Then {{mvar|B}} and {{mvar|C}} have a common set of eigenfunctions (and are simultaneously diagonalizable), whereas {{mvar|p}} and {{mvar|q}} famously don't and aren't.

Using the commutation relations given above, the Hamiltonian operator can be expressed as

$$\backslash hat\; H\; =\; \backslash hbar\; \backslash omega\; \backslash left(\; a\; \backslash ,\; a^\backslash dagger\; -\; \backslash frac\{1\}\{2\}\backslash right)\; =\; \backslash hbar\; \backslash omega\; \backslash left(\; a^\backslash dagger\; \backslash ,\; a\; +\; \backslash frac\{1\}\{2\}\backslash right).\backslash qquad\backslash qquad(*)$$

One may compute the commutation relations between the $a\backslash ,$ and $a^\backslash dagger\backslash ,$ operators and the Hamiltonian:{{cite web|last=Branson|first=Jim|title=Quantum Physics at UCSD|url=http://quantummechanics.ucsd.edu/ph130a/130_notes/node170.html#section:HOraise|access-date=16 May 2012}}

$$\backslash begin\{align\}$$

\left[\hat H, a \right] &= \left[\hbar \omega \left ( a a^\dagger - \tfrac{1}{2}\right ) , a\right] = \hbar \omega \left[ a a^\dagger, a\right] = \hbar \omega \left( a [a^\dagger,a] + [a,a] a^\dagger\right) = -\hbar \omega a. \\[1ex]

\left[\hat H, a^\dagger \right] &= \hbar \omega \, a^\dagger .

\end{align}

These relations can be used to easily find all the energy eigenstates of the quantum harmonic oscillator as follows.

Assuming that $\backslash psi\_n$ is an eigenstate of the Hamiltonian $\backslash hat\; H\; \backslash psi\_n\; =\; E\_n\backslash ,\; \backslash psi\_n$. Using these commutation relations, it follows that

$$\backslash begin\{align\}$$

\hat H\, a\psi_n &= (E_n - \hbar \omega)\, a\psi_n . \\[1ex]

\hat H\, a^\dagger\psi_n &= (E_n + \hbar \omega)\, a^\dagger\psi_n .

\end{align}

This shows that $a\backslash psi\_n$ and $a^\backslash dagger\backslash psi\_n$ are also eigenstates of the Hamiltonian, with eigenvalues $E\_n\; -\; \backslash hbar\; \backslash omega$ and $E\_n\; +\; \backslash hbar\; \backslash omega$ respectively. This identifies the operators $a$ and $a^\backslash dagger$ as "lowering" and "raising" operators between adjacent eigenstates. The energy difference between adjacent eigenstates is $\backslash Delta\; E\; =\; \backslash hbar\; \backslash omega$.

The ground state can be found by assuming that the lowering operator possesses a nontrivial kernel: $a\backslash ,\; \backslash psi\_0\; =\; 0$ with $\backslash psi\_0\backslash ne0$. Applying the Hamiltonian to the ground state,

$$\backslash hat\; H\backslash psi\_0\; =\; \backslash hbar\backslash omega\backslash left(a^\backslash dagger\; a+\backslash frac\{1\}\{2\}\backslash right)\backslash psi\_0\; =\; \backslash hbar\backslash omega\; a^\backslash dagger\; a\; \backslash psi\_0\; +\; \backslash frac\{\backslash hbar\backslash omega\}\{2\}\backslash psi\_0=0+\backslash frac\{\backslash hbar\backslash omega\}\{2\}\backslash psi\_0=E\_0\backslash psi\_0.$$

So $\backslash psi\_0$ is an eigenfunction of the Hamiltonian.

This gives the ground state energy $E\_0\; =\; \backslash hbar\; \backslash omega\; /2$, which allows one to identify the energy eigenvalue of any eigenstate $\backslash psi\_n$ as

$$E\_n\; =\; \backslash left(n\; +\; \backslash tfrac\{1\}\{2\}\backslash right)\backslash hbar\; \backslash omega.$$

Furthermore, it turns out that the first-mentioned operator in (*), the number operator $N=a^\backslash dagger\; a\backslash ,,$ plays the most important role in applications, while the second one, $a\; a^\backslash dagger\; \backslash ,$ can simply be replaced by $N+1$.

Consequently,

$$\backslash hbar\backslash omega\; \backslash ,\backslash left(N+\backslash tfrac\{1\}\{2\}\backslash right)\backslash ,\backslash psi\; (q)\; =E\backslash ,\backslash psi\; (q)~.$$

The time-evolution operator is then

$$\backslash begin\{align\}$$

U(t)

&= \exp ( -it \hat{H}/\hbar)

= \exp (-it\omega (a^\dagger a+1/2)) ~, \\[1ex]

&= e^{-it \omega /2} ~ \sum_{k=0}^{\infty} {(e^{-i\omega t}-1)^k \over k!} a^{{\dagger} {k}} a^k ~.

\end{align}

The ground state $\backslash \; \backslash psi\_0(q)$ of the quantum harmonic oscillator can be found by imposing the condition that

$$a\; \backslash \; \backslash psi\_0(q)\; =\; 0.$$

Written out as a differential equation, the wavefunction satisfies

$$q\; \backslash psi\_0\; +\; \backslash frac\{d\backslash psi\_0\}\{dq\}\; =\; 0$$

with the solution

$$\backslash psi\_0(q)\; =\; C\; \backslash exp\backslash left(-\backslash tfrac\; 1\; 2\; q^2\backslash right).$$

The normalization constant {{mvar|C}} is found to be $1/\; \backslash sqrt[4]\{\backslash pi\}$ from $\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash psi\_0^*\; \backslash psi\_0\; \backslash ,dq\; =\; 1$,  using the Gaussian integral. Explicit formulas for all the eigenfunctions can now be found by repeated application of $a^\backslash dagger$ to $\backslash psi\_0$.This, and further operator formalism, can be found in Glimm and Jaffe, Quantum Physics, pp. 12–20.

The matrix expression of the creation and annihilation operators of the quantum harmonic oscillator with respect to the above orthonormal basis is

$$\backslash begin\{align\}$$

a^\dagger &= \begin{pmatrix}

0 & 0 & 0 & 0 & \dots & 0 & \dots \\

\sqrt{1} & 0 & 0 & 0 & \dots & 0 & \dots \\

0 & \sqrt{2} & 0 & 0 & \dots & 0 & \dots \\

0 & 0 & \sqrt{3} & 0 & \dots & 0 & \dots \\

\vdots & \vdots & \vdots & \ddots & \ddots & \dots & \dots \\

0 & 0 & 0 & \dots & \sqrt{n} & 0 & \dots & \\

\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots \end{pmatrix}

\\[1ex]

a &= \begin{pmatrix}

0 & \sqrt{1} & 0 & 0 & \dots & 0 & \dots \\

0 & 0 & \sqrt{2} & 0 & \dots & 0 & \dots \\

0 & 0 & 0 & \sqrt{3} & \dots & 0 & \dots \\

0 & 0 & 0 & 0 & \ddots & \vdots & \dots \\

\vdots & \vdots & \vdots & \vdots & \ddots & \sqrt{n} & \dots \\

0 & 0 & 0 & 0 & \dots & 0 & \ddots \\

\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}

\end{align}

These can be obtained via the relationships $a^\backslash dagger\_\{ij\}\; =\; \backslash left\backslash langle\backslash psi\_i\; \backslash right|\; a^\backslash dagger\; \backslash left|\; \backslash psi\_j\backslash right\backslash rangle$ and $a\_\{ij\}\; =\; \backslash left\backslash langle\backslash psi\_i\; \backslash right|\; a\; \backslash left|\; \backslash psi\_j\backslash right\backslash rangle$. The eigenvectors $\backslash psi\_i$ are those of the quantum harmonic oscillator, and are sometimes called the "number basis".

{{Main|CCR and CAR algebras}}

{{see also|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.{{citation| first=Fulton |last= Harris|title=Representation Theory}} pp. 164

In the Hilbert space representation case the operators are constructed as follows: Let $H$ be a one-particle Hilbert space (that is, any Hilbert space, viewed as representing the state of a single particle).

The (bosonic) CCR algebra over $H$ is the algebra-with-conjugation-operator (called *) abstractly generated by elements $a(f)$, where $f\backslash ,$ranges freely over $H$, subject to the relations

$$\backslash begin\{align\}$$

\left[a(f), a(g)\right] &= \left[a^\dagger(f), a^\dagger(g)\right] = 0 \\[1ex]

\left[a(f), a^\dagger(g)\right] &= \langle f\mid g \rangle,

\end{align}

in bra–ket notation.

The map $a:\; f\; \backslash to\; a(f)$ from $H$ to the bosonic CCR algebra is required to be complex antilinear (this adds more relations). Its adjoint is $a^\backslash dagger(f)$, and the map $f\backslash to\; a^\backslash dagger(f)$ is complex linear in {{mvar|H}}. Thus $H$ embeds as a complex vector subspace of its own CCR algebra. In a representation of this algebra, the element $a(f)$ will be realized as an annihilation operator, and $a^\backslash dagger(f)$ as a creation operator.

In general, the CCR algebra is infinite dimensional. If we take a Banach space completion, it becomes a C*-algebra. The CCR algebra over $H$ is closely related to, but not identical to, a Weyl algebra.{{Clarify|date=December 2023|reason=Obscure reference: How ?|text= |pre-text= |post-text= }}

For fermions, the (fermionic) CAR algebra over $H$ is constructed similarly, but using anticommutator relations instead, namely

$$\backslash begin\{align\}$$

\{a(f),a(g)\} &= \{a^\dagger(f),a^\dagger(g)\} = 0 \\[1ex]

\{a(f),a^\dagger(g)\} &= \langle f\mid g \rangle.

\end{align}

The CAR algebra is finite dimensional only if $H$ is finite dimensional. If we take a Banach space completion (only necessary in the infinite dimensional case), it becomes a $C^*$ algebra. The CAR algebra is closely related, but not identical to, a Clifford algebra.

{{Clarify|date=December 2023|reason=Obscure reference: How ?|text= |pre-text= |post-text= }}

Physically speaking, $a(f)$ removes (i.e. annihilates) a particle in the state $|f\backslash rangle$ whereas $a^\backslash dagger(f)$ creates a particle in the state $|f\backslash rangle$.

The free field vacuum state is the state $\backslash left\backslash vert0\backslash right\backslash rangle$ with no particles, characterized by

$$a(f)\; \backslash left|\; 0\backslash right\backslash rangle=0.$$

If $|f\backslash rangle$ is normalized so that $\backslash langle\; f|f\backslash rangle\; =\; 1$, then $N=a^\backslash dagger(f)a(f)$ gives the number of particles in the state $|f\backslash rangle$.

The annihilation and creation operator description has also been useful to analyze classical reaction diffusion equations, such as the situation when a gas of molecules $A$ diffuse and interact on contact, forming an inert product: $A+A\backslash to\; \backslash empty$. To see how this kind of reaction can be described by the annihilation and creation operator formalism, consider $n\_\{i\}$ particles at a site {{mvar|i}} on a one dimensional lattice. Each particle moves to the right or left with a certain probability, and each pair of particles at the same site annihilates each other with a certain other probability.

The probability that one particle leaves the site during the short time period {{math|dt}} is proportional to $n\_i\; \backslash ,\; dt$, let us say a probability $\backslash alpha\; n\_\{i\}dt$ to hop left and $\backslash alpha\; n\_i\; \backslash ,\; dt$ to hop right. All $n\_i$ particles will stay put with a probability $1-2\backslash alpha\; n\_i\; \backslash ,\; dt$. (Since {{math|dt}} is so short, the probability that two or more will leave during {{math|dt}} is very small and will be ignored.)

We can now describe the occupation of particles on the lattice as a 'ket' of the form $|\backslash dots,\; n\_\{-1\},\; n\_0,\; n\_1,\; \backslash dots\backslash rangle$. It represents the juxtaposition (or conjunction, or tensor product) of the number states $\backslash dots,\; |n\_\{-1\}\backslash rangle$, $|n\_\{0\}\backslash rangle$, $|n\_\{1\}\backslash rangle,\; \backslash dots$ located at the individual sites of the lattice. Recall that

$$a\backslash left|\; n\; \backslash right\backslash rangle\; =\; \backslash sqrt\{n\}\; \backslash left|n-1\backslash right\backslash rangle$$

and

$$a^\backslash dagger\; \backslash left|\; n\backslash right\backslash rangle=\; \backslash sqrt\{n+1\}\backslash left|\; n+1\backslash right\backslash rangle,$$

for all {{math|n ≥ 0}}, while

$$[a,a^\{\backslash dagger\}]\; =\; \backslash mathbf\; 1$$

This definition of the operators will now be changed to accommodate the "non-quantum" nature of this problem and we shall use the following definition:{{cite web |last1=Pruessner |first1=Gunnar |title=Analysis of Reaction-Diffusion Processes by Field Theoretic Methods |url=http://wwwf.imperial.ac.uk/~pruess/publications/talk_usp_reduced.pdf |access-date=31 May 2021}}

$$\backslash begin\{align\}$$

a \left|n\right\rangle &= (n) \left|n{-}1\right\rangle \\[1ex]

a^\dagger \left|n\right\rangle &= \left| n{+}1\right\rangle

\end{align}

note that even though the behavior of the operators on the kets has been modified, these operators still obey the commutation relation

$$[a,a^\{\backslash dagger\}]=\backslash mathbf\; 1$$

Now define $a\_i$ so that it applies $a$ to $|n\_i\backslash rangle$. Correspondingly, define $a^\backslash dagger\_i$ as applying $a^\backslash dagger$ to $|n\_i\backslash rangle$. Thus, for example, the net effect of $a\_\{i-1\}\; a^\backslash dagger\_i$ is to move a particle from the {{nowrap|$(i-1)$-th}} to the {{mvar|i}}-th site while multiplying with the appropriate factor.

This allows writing the pure diffusive behavior of the particles as

$$\backslash partial\_\{t\}\backslash left|\; \backslash psi\backslash right\backslash rangle$$

= -\alpha \sum_i \left(2a_i^\dagger a_i-a_{i-1}^\dagger a_i-a_{i+1}^\dagger a_i\right) \left|\psi\right\rangle

= -\alpha\sum_i \left(a_i^\dagger-a_{i-1}^\dagger\right)(a_i-a_{i-1}) \left|\psi\right\rangle.

The reaction term can be deduced by noting that $n$ particles can interact in $n(n-1)$ different ways, so that the probability that a pair annihilates is $\backslash lambda\; n(n-1)dt$, yielding a term

$$\backslash lambda\; \backslash sum\_i\; (a\_i\; a\_i-a\_i^\backslash dagger\; a\_i^\backslash dagger\; a\_i\; a\_i)$$

where number state {{mvar|n}} is replaced by number state {{math|n − 2}} at site $i$ at a certain rate.

Thus the state evolves by

$$\backslash partial\_t\backslash left|\backslash psi\backslash right\backslash rangle\; =\; -\backslash alpha\backslash sum\_i\; \backslash left(a\_i^\backslash dagger-a\_\{i-1\}^\backslash dagger\backslash right)\; \backslash left(a\_i-a\_\{i-1\}\backslash right)\; \backslash left|\backslash psi\backslash right\backslash rangle\; +\; \backslash lambda\backslash sum\_i\; \backslash left(a\_i^2-a\_i^\{\backslash dagger\; 2\}a\_i^2\backslash right)\; \backslash left|\backslash psi\backslash right\backslash rangle$$

Other kinds of interactions can be included in a similar manner.

This kind of notation allows the use of quantum field theoretic techniques to be used in the analysis of reaction diffusion systems.Baez, John Carlos (2011). Network theory (blog post series; [https://math.ucr.edu/home/baez/networks/networks_1.html first post]). Later adapted into {{cite book|last1=Baez|first1=John Carlos|last2=Biamonte|first2=Jacob D.|title=Quantum Techniques in Stochastic Mechanics|date=April 2018|doi=10.1142/10623}}

{{Main|Second quantization|Quantum field theory#Second quantization}}

In quantum field theories and many-body problems one works with creation and annihilation operators of quantum states, $a^\backslash dagger\_i$ and $a^\{\backslash ,\}\_i$. These operators change the eigenvalues of the number operator,

$$N\; =\; \backslash sum\_i\; n\_i\; =\; \backslash sum\_i\; a^\backslash dagger\_i\; a^\{\backslash ,\}\_i,$$

by one, in analogy to the harmonic oscillator. The indices (such as $i$) 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 $(n,\; \backslash ell,\; m,\; s)$ is used to label states in the hydrogen atom.

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

$$\backslash begin\{align\}$$

\left[a^{\,}_i, a^\dagger_j\right] &\equiv a^{\,}_i a^\dagger_j - a^\dagger_ja^{\,}_i = \delta_{i j}, \\[1ex]

\left[a^\dagger_i, a^\dagger_j\right] &= [a^{\,}_i, a^{\,}_j] = 0,

\end{align}

where $[\backslash cdot\; ,\; \backslash cdot\; ]$ is the commutator and $\backslash delta\_\{i\; j\}$ is the Kronecker delta.

For fermions, the commutator is replaced by the anticommutator {{nowrap|$\backslash \{\backslash cdot\; ,\; \backslash cdot\; \backslash \}$,}}

$$\backslash begin\{align\}$$

\{a^{\,}_i, a^\dagger_j\} &\equiv a^{\,}_i a^\dagger_j +a^\dagger_j a^{\,}_i = \delta_{i j}, \\[1ex]

\{a^\dagger_i, a^\dagger_j\} &= \{a^{\,}_i, a^{\,}_j\} = 0.

\end{align}

Therefore, exchanging disjoint (i.e. $i\; \backslash ne\; j$) operators in a product of creation or annihilation operators will reverse the sign in fermion systems, but not in boson systems.

If the states labelled by i are an orthonormal basis of a Hilbert space H, then the result of this construction coincides with the CCR algebra and CAR algebra construction in the previous section but one. If they represent "eigenvectors" corresponding to the continuous spectrum of some operator, as for unbound particles in QFT, then the interpretation is more subtle.

While 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}} obtains the momentum space normalization $[\backslash hat\; a\_\{\backslash mathbf\; p\},\backslash hat\; a\_\{\backslash mathbf\; q\}^\backslash dagger]\; =\; \backslash delta(\backslash mathbf\{p\}\; -\; \backslash mathbf\{q\})$ via the symmetric convention for Fourier transforms, Tong{{cite book |last1=Tong |first1=David |title=Quantum Field Theory |date=2007 |page=24,31 |url=http://www.damtp.cam.ac.uk/user/tong/qft.html |access-date=3 December 2019}} and Peskin & Schroeder{{cite book |last1=Peskin |first1=M. |author-link=Michael Peskin |last2=Schroeder |first2=D.

|year=1995

|title=An Introduction to Quantum Field Theory

|url=https://books.google.com/books?id=i35LALN0GosC

|publisher=Westview Press

|isbn=978-0-201-50397-5 }} use the common asymmetric convention to obtain $[\backslash hat\; a\_\{\backslash mathbf\; p\},\backslash hat\; a\_\{\backslash mathbf\; q\}^\backslash dagger]\; =\; (2\backslash pi)^3\backslash delta(\backslash mathbf\{p\}\; -\; \backslash mathbf\{q\})$. Each derives $[\backslash hat\; \backslash phi(\backslash mathbf\; x),\; \backslash hat\; \backslash pi(\backslash mathbf\; x\text{'})]\; =\; i\backslash delta(\backslash mathbf\; x\; -\; \backslash mathbf\; x\text{'})$.

Srednicki additionally merges the Lorentz-invariant measure into his asymmetric Fourier measure, $\backslash tilde\{dk\}=\backslash frac\{d^3k\}\{(2\backslash pi)^3\; 2\backslash omega\}$, yielding $[\backslash hat\; a\_\{\backslash mathbf\; k\},\backslash hat\; a\_\{\backslash mathbf\; k\text{'}\}^\backslash dagger]\; =\; (2\backslash pi)^3\; 2\backslash omega\backslash ,\backslash delta(\backslash mathbf\{k\}\; -\; \backslash mathbf\{k\}\text{'})$.{{cite book |last1=Srednicki |first1=Mark |title=Quantum field theory |date=2007 |publisher=Cambridge University Press |isbn=978-0521-8644-97 |pages=39,41 |url=https://web.physics.ucsb.edu/~mark/qft.html |access-date=3 December 2019}}

{{div col}}

• Fock space
• Segal–Bargmann space
• Optical phase space
• Coherent state
• Bogoliubov–Valatin transformation
• Holstein–Primakoff transformation
• Jordan–Wigner transformation
• Jordan–Schwinger transformation
• Klein transformation
• Canonical commutation relations

{{div col end}}

{{refbegin}}

• {{cite book | last = Feynman | first = Richard P. | author-link = Richard Feynman | title = Statistical Mechanics: A Set of Lectures | publisher = Addison-Wesley | year = 1998 | orig-year = 1972 | edition=2nd | location = Reading, Massachusetts | url = https://books.google.com/books?id=Ou4ltPYiXPgC&pg=Front | isbn = 978-0-201-36076-9}}
• Albert Messiah, 1966. Quantum Mechanics (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Ch. XII. [https://archive.org/details/QuantumMechanicsVolumeI online]

{{refend}}

{{Physics operator}}

Category:Quantum operators

Category:Quantum field theory