Second quantization

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Second quantization is a formalism used to describe and analyze quantum many-body systems. It is also known as canonical quantization in quantum field theory, in which the fields (typically as the wave functions of matters) are upgraded into field operators, following the similar idea that the physical quantities (position, momentum etc.) are upgraded into operators in first quantization. The key ideas of this method were introduced in 1927 by Dirac,[1] and were developed, most notably, by Fock and Jordan later.[2][3]

In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed by filling up each single-particle state with a certain number of identical particles. The second quantization formalism introduces the creation and annihilation operators to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory.

Quantum Many-Body States[edit]

The starting point of the second quantization formalism is the notion of indistinguishability of particles in quantum mechanics. Unlike in classical mechanics, where each particle is labeled by a distinct position vector \bold{r}_i and different configurations of the set of \bold{r}_i's correspond to different many-body states. In quantum mechanics, the particles are identical, such that exchanging two particles, i.e. \bold{r}_i\leftrightarrow\bold{r}_j, does not lead to a different quantum state. This implies that the quantum many-body wave function must be invariant (up to a phase factor) under the exchange of two particles. According to the statistics of the particles, the many-body wave function can either be symmetric or antisymmetric under the particle exchange:

\Psi_B(\cdots,\bold{r}_i,\cdots,\bold{r}_j,\cdots)=+\Psi_B(\cdots,\bold{r}_j,\cdots,\bold{r}_i,\cdots) if the particles are bosons,
\Psi_F(\cdots,\bold{r}_i,\cdots,\bold{r}_j,\cdots)=-\Psi_F(\cdots,\bold{r}_j,\cdots,\bold{r}_i,\cdots) if the particles are fermions.

This exchange symmetry property imposes a constraint on the many-body wave function. Each time a particle is added or removed from the many-body system, the wave function must be properly symmetrized or anti-symmetrized to satisfy the symmetry constraint. In the first quantization formalism, this constraint is guaranteed by representing the wave function as linear combination of permanents (for bosons) or determinants (for fermions) of single-particle states. In the second quantization formalism, the issue of symmetrization is automatically taken care of by the creation and annihilation operators, such that its notation can be much simpler.

First Quantized Many-Body Wave Function[edit]

Consider a complete set of single-particle wave functions \psi_{\alpha}(\bold{r}) labeled by \alpha (which may be a combined index of a bunch of quantum numbers). The following wave function

\Psi[\bold{r}_i]=\prod_{i=1}^{N}\psi_{\alpha_i}(\bold{r}_i)\equiv \psi_{\alpha_1}\otimes\psi_{\alpha_2}\otimes\cdots\otimes\psi_{\alpha_N}

represents an N-particle state with the ith particle occupying the state |{\alpha_i}\rangle. In the shorthanded notation, the position argument of the wave function may be omitted, and it is assumed that the ith single-particle wave function describes the state of the ith particle. The wave function \Psi has not been symmetrized or anti-symmetrized, thus in general not qualified as a many-body wave function for identical particles. However, it can be brought to the symmetrized (anti-symmetrized) form by the symmetrization (anti-symmetrization) operators, denoted \mathcal{S} (\mathcal{A}).

For bosons, the many-body wave function must be symmetrized,

\Psi_B[\bold{r}_i]=\mathcal{N}\mathcal{S}\Psi[\bold{r}_i]=\mathcal{N}\sum_{\pi\in S_N}\prod_{i=1}^{N}\psi_{\alpha_{\pi(i)}}(\bold{r}_i)=\mathcal{N}\sum_{\pi\in S_N}\psi_{\alpha_{\pi(1)}}\otimes\psi_{\alpha_{\pi(2)}}\otimes\cdots\otimes\psi_{\alpha_{\pi(N)}};

while for fermions, the many-body wave function must be anti-symmetrized,

\Psi_F[\bold{r}_i]=\mathcal{N}\mathcal{A}\Psi[\bold{r}_i]=\mathcal{N}\sum_{\pi\in S_N}(-1)^\pi\prod_{i=1}^{N}\psi_{\alpha_{\pi(i)}}(\bold{r}_i)=\mathcal{N}\sum_{\pi\in S_N}(-1)^\pi\psi_{\alpha_{\pi(1)}}\otimes\psi_{\alpha_{\pi(2)}}\otimes\cdots\otimes\psi_{\alpha_{\pi(N)}}.

Here \pi is an element in the N-body permutation group (or symmetric group) S_{N}, which performs a permutation among the state labels \alpha_i, and (-1)^\pi denotes the corresponding permutation sign. \mathcal{N} is the normalization operator that normalize the wave function. If one arranges the wave function in a matrix U, such that the row-i column-j matrix element is U_{ij}=\psi_{\alpha_{j}}(\bold{r}_i)\equiv \langle\bold{r}_i|\alpha_j\rangle, then the boson many-body wave function can be simply written as a permanent \Psi_B=\mathcal{N}\operatorname{perm} U, while the fermion many-body wave function as a determinant \Psi_F=\mathcal{N}\operatorname{det} U (also known as the Slater determinant).

Second Quantized Fock States[edit]

The reason that the first quantized wave function involves complicated symmetrization procedures is because that it is using a redundant language to formulate the many-body state. In the first quantization language, the many-body state is described by answering a series of questions like "which particle is on which state". However these are not physical questions, because the particles are identical, and it is impossible to tell which particle is which in the first place. The seemly different states \psi_1\psi_2 and \psi_2\psi_1 are actually redundant names of the same quantum many-body state. So the symmetrization (or anti-symmetrization) must be introduced to eliminate this redundancy in the first quantization description.

In the second quantization language, instead of asking "each particle on which state", one asks "how many particles are there on each state". Because this description does not refer to the labeling of particles, it contains no redundant information, and hence leads to a precise and simpler description of the quantum many-body state. In this approach, the many-body state is represented in the occupation number basis, and the basis state is labeled by the set of occupation numbers, denoted

|[n_{\alpha}]\rang\equiv|n_1,n_2,\cdots, n_{\alpha}, \cdots \rang,

meaning that there are  n_{\alpha} particles in the state |\alpha\rangle. The occupation numbers sum up to the total number of particles, i.e.  \sum_\alpha n_{\alpha} = N. For fermions, the occupation number  n_{\alpha} can only be 0 or 1, due to the Pauli exclusion principle; while for bosons it can be any non negative integer

n_{\alpha}= \begin{cases}
  0, 1 &\text{fermions,}\\
  0,1,2,3,...           &\text{bosons.}
\end{cases}

The occupation number states |[n_{\alpha}]\rang are also known as the Fock states. All the Fock states form a complete set of basis of the many-body Hilbert space, or the Fock space. Any generic quantum many-body state can be expressed as a linear combination of Fock states.

The Fock state with all occupation numbers being zero is called the vacuum state, denoted |0\rangle\equiv|\cdots,0_\alpha,\cdots\rangle. The Fock state with only one non-zero occupation number is a single-mode Fock state, denoted |n_\alpha\rangle\equiv|\cdots,0,n_\alpha,0,\cdots\rangle. In terms of the first quantized wave function, the vacuum state is the unit of tensor product, and can be denoted as |0\rangle=1. The single-particle state is reduced to its wave function |1_\alpha\rangle=\psi_\alpha. Other single-mode many-body (boson) state are just the tensor product of the wave function of that mode, such as |2_\alpha\rangle=\psi_\alpha\otimes\psi_\alpha and |n_\alpha\rangle=\psi_\alpha^{\otimes n}. For multi-mode Fock states, the first quantized wave function will require proper symmetrization according to the particle statistics, e.g. |1_1,1_2\rangle=(\psi_1\psi_2+\psi_2\psi_1)/\sqrt{2} for boson state, and |1_1,1_2\rangle=(\psi_1\psi_2-\psi_2\psi_1)/\sqrt{2} for fermion state (\otimes between \psi_1 and \psi_2 is omitted for simplicity). In general, the first quantized wave function of the Fock state reads

|[n_\alpha]\rangle_B=\left(\frac{\prod_{\alpha}n_\alpha!}{N!}\right)^{1/2}\mathcal{S}\bigotimes\limits_\alpha\psi_\alpha^{\otimes n_\alpha} for bosons,
|[n_\alpha]\rangle_F=\frac{1}{\sqrt{N!}}\mathcal{A}\bigotimes\limits_\alpha\psi_\alpha^{\otimes n_\alpha} for fermions,

where N is the total number of particles.

Creation and Annihilation Operators[edit]

The creation and annihilation operators are introduced to add or remove a particle from the many-body system. These operators are lying at the core of the second quantization formalism, bridging the gap between the first and the second quantized states. Applying the creation (annihilation) operator to a first quantized many-body wave function will insert (delete) a single-particle state from the wave function in a symmetrized way depending on the particle statistics. On the other hand, all the second quantized Fock states can be constructed by applying the creation operators to the vacuum state repeatedly.

The creation and annihilation operators (for bosons) are originally constructed in the context of the quantum harmonic oscillator as the raising and lowering operators, which are then generalized to the field operators in the quantum field theory.[4] They are fundamental to the quantum many-body theory, in the sense that every many-body operator (including the Hamiltonian of the many-body system and all the physical observables) can be expressed in terms of them.

Insertion and Deletion Operation[edit]

The creation and annihilation of a particle is implemented by the insertion and deletion of the single-particle state from the first quantized wave function in an either symmetric or anti-symmetric manner. Let \psi_\alpha be a single-particle state, 1 be the (right) tensor identity (such that \psi_\alpha\equiv\psi_\alpha\otimes1), and \Psi =\psi_{\alpha_1}\otimes\psi_{\alpha_2}\otimes\cdots be a generic tensor product state. The insertion \otimes_\pm is and the deletion \oslash_\pm operators are defined by the following recursive equations

\psi_\alpha\otimes_\pm 1=\psi_\alpha,\quad\psi_\alpha\otimes_\pm(\psi_\beta\otimes\Psi)= \psi_\alpha\otimes\psi_\beta\otimes\Psi\pm\psi_\beta\otimes(\psi_\alpha\otimes_\pm\Psi);
\psi_\alpha\oslash_\pm 1=0,\quad\psi_\alpha\oslash_\pm(\psi_\beta\otimes\Psi)= \delta_{\alpha\beta}\Psi\pm\psi_\beta\otimes(\psi_\alpha\oslash_\pm\Psi).

Here \delta_{\alpha\beta} is the Kronecker delta symbol, which gives 1 if \alpha=\beta, and 0 otherwise.

Boson Creation and Annihilation Operators[edit]

The boson creation (annihilation) operator is usually denoted as b_{\alpha}^\dagger (b_{\alpha}). The creation operator b_{\alpha}^\dagger adds a boson to the single-particle state |\alpha\rangle, and the annihilation operator b_{\alpha} removes a boson from the single-particle state |\alpha\rangle. The creation and annihilation operators are Hermitian conjugate to each other, but neither of them are Hermitian operators (b_\alpha\neq b_\alpha^\dagger).

Definition[edit]

The boson creation (annihilation) operator is a linear operator, whose action on a N-particle first quantized wave function \Psi is defined as

b_\alpha^\dagger \Psi = \frac{1}{\sqrt{N+1}}\psi_\alpha\otimes_+\Psi,
b_\alpha\Psi = \frac{1}{\sqrt{N}}\psi_\alpha\oslash_+\Psi,

where \psi_\alpha\otimes_+ inserts the single-particle state \psi_\alpha in N+1 possible insertion positions symmetrically, and \psi_\alpha\oslash_+ deletes the single-particle state \psi_\alpha from N possible deletion positions symmetrically.

Action on Fock States[edit]

Starting from the single-mode vacuum state |0_\alpha\rangle=1, applying the creation operator b_\alpha^\dagger repeatedly, one finds

b_\alpha^\dagger|0_\alpha\rangle=\psi_\alpha\otimes_+ 1=\psi_\alpha=|1_\alpha\rangle,
b_\alpha^\dagger|n_\alpha\rangle=\frac{1}{\sqrt{n_\alpha+1}}\psi_\alpha\otimes_+ \psi_\alpha^{\otimes n_\alpha}=\sqrt{n_\alpha+1}\psi_\alpha^{\otimes (n_\alpha+1)}=\sqrt{n_\alpha+1}|n_\alpha+1\rangle.

The creation operator raises the boson occupation number by 1. Therefore all the occupation number states can be constructed by the boson creation operator from the vacuum state

|n_\alpha\rangle=\frac{1}{\sqrt{n_\alpha!}}(b_{\alpha}^\dagger)^{n_\alpha}|0_\alpha\rangle.

On the other hand, the annihilation operator b_\alpha lowers the boson occupation number by 1

b_\alpha|n_\alpha\rangle=\frac{1}{\sqrt{n_\alpha}}\psi_\alpha\oslash_+ \psi_\alpha^{\otimes n_\alpha}=\sqrt{n_\alpha}\psi_\alpha^{\otimes (n_\alpha-1)}=\sqrt{n_\alpha}|n_\alpha-1\rangle.

It will also quench the vacuum state b_\alpha|0_\alpha\rangle=0 as there has been no boson left in the vacuum state to be annihilated. Using the above formulae, it can be shown that

b_\alpha^\dagger b_\alpha|n_\alpha\rangle=n_\alpha|n_\alpha\rangle,

meaning that \hat{n}_\alpha = b_\alpha^\dagger b_\alpha defines the boson number operator.

The above result can be generalized to any Fock state of bosons.

b_\alpha^\dagger|\cdots,n_\beta,n_\alpha,n_\gamma,\cdots\rangle= \sqrt{n_\alpha+1}|\cdots,n_\beta,n_\alpha+1,n_\gamma,\cdots\rangle.
b_\alpha|\cdots,n_\beta,n_\alpha,n_\gamma,\cdots\rangle= \sqrt{n_\alpha}|\cdots,n_\beta,n_\alpha-1,n_\gamma,\cdots\rangle.

These two equations can be considered as the defining properties of boson creation and annihilation operators in the second quantization formalism. The complicated symmetrization of the underlying first quantized wave function is automatically performed by the creation and annihilation operators, so that the complexity is not revealed on the second quantized level, and the second quantization formulae are simple and neat.

Operator Identities[edit]

The following operator identities follows from the action of the boson creation and annihilation operators on the Fock state,

[b_\alpha^\dagger,b_\beta^\dagger]=[b_\alpha,b_\beta]=0,\quad [b_\alpha,b_\beta^\dagger]=\delta_{\alpha\beta}.

These commutation relations can be considered as the algebraic definition of the boson creation and annihilation operators. The fact that the boson many-body wave function is symmetric under particle exchange is also manifested by the commutation of the boson operators.

The raising and lowering operators of the quantum harmonic oscillator also satisfies the same set of commutation relations, implying that the bosons can be interpreted as the energy quanta (phonons) of an oscillator. This is indeed the idea of quantum field theory, which considers each mode of the matter field as an oscillator subject to quantum fluctuations, and the bosons are treated as the excitations (or energy quanta) of the field.

Fermion Creation and Annihilation Operators[edit]

The fermion creation (annihilation) operator is usually denoted as c_{\alpha}^\dagger (c_{\alpha}). The creation operator c_{\alpha}^\dagger adds a fermion to the single-particle state |\alpha\rangle, and the annihilation operator c_{\alpha} removes a fermion from the single-particle state |\alpha\rangle. The creation and annihilation operators are Hermitian conjugate to each other, but neither of them are Hermitian operators (c_\alpha\neq c_\alpha^\dagger). The Hermitian combination of the fermion creation and annihilation operators

\chi_{\alpha,\text{Re}}=(c_\alpha+c_\alpha^\dagger)/2, \quad \chi_{\alpha,\text{Im}}=(c_\alpha-c_\alpha^\dagger)/(2\mathrm{i}),

are called Majorana fermion operators.

Definition[edit]

The fermion creation (annihilation) operator is a linear operator, whose action on a N-particle first quantized wave function \Psi is defined as

c_\alpha^\dagger \Psi = \frac{1}{\sqrt{N+1}}\psi_\alpha\otimes_-\Psi,
c_\alpha\Psi = \frac{1}{\sqrt{N}}\psi_\alpha\oslash_-\Psi,

where \psi_\alpha\otimes_- inserts the single-particle state \psi_\alpha in N+1 possible insertion positions anti-symmetrically, and \psi_\alpha\oslash_- deletes the single-particle state \psi_\alpha from N possible deletion positions anti-symmetrically.

Action on Fock States[edit]

Starting from the single-mode vacuum state |0_\alpha\rangle=1, applying the fermion creation operator c_\alpha^\dagger,

c_\alpha^\dagger|0_\alpha\rangle=\psi_\alpha\otimes_- 1=\psi_\alpha=|1_\alpha\rangle,
c_\alpha^\dagger|1_\alpha\rangle=\frac{1}{\sqrt{2}}\psi_\alpha\otimes_- \psi_\alpha=0.

If the single-particle state |\alpha\rangle is empty, the creation operator will fill the state with a fermion. However if the state is already occupied by a fermion, further application of the creation operator will quench the state, demonstrating the Pauli exclusion principle that two identical fermions can not occupy the same state simultaneously. Nevertheless the fermion can be removed from the occupied state by the fermion annihilation operator c_\alpha,

c_\alpha|1_\alpha\rangle=\psi_\alpha\oslash_-\psi_\alpha=1=|0_\alpha\rangle,
c_\alpha|0_\alpha\rangle =0.

The vacuum state is quenched by the action of the annihilation operator.

Similar to the boson case, the fermion Fock state can be constructed from the vacuum state using the fermion creation operator

|n_\alpha\rangle=(c_{\alpha}^\dagger)^{n_\alpha}|0_\alpha\rangle.

It is easy to check (by enumeration) that

c_\alpha^\dagger c_\alpha|n_\alpha\rangle=n_\alpha|n_\alpha\rangle,

meaning that \hat{n}_\alpha = c_\alpha^\dagger c_\alpha defines the fermion number operator.

The above result can be generalized to any Fock state of fermions.

c_\alpha^\dagger|\cdots,n_\beta,n_\alpha,n_\gamma,\cdots\rangle=(-1)^{\sum_{\beta<\alpha}n_\beta} (1-n_\alpha)|\cdots,n_\beta,1-n_\alpha,n_\gamma,\cdots\rangle.
c_\alpha|\cdots,n_\beta,n_\alpha,n_\gamma,\cdots\rangle= (-1)^{\sum_{\beta<\alpha}n_\beta} n_\alpha|\cdots,n_\beta,1-n_\alpha,n_\gamma,\cdots\rangle.

Remind that the occupation number n_\alpha can only take 0 or 1 for fermions. These two equations can be considered as the defining properties of fermion creation and annihilation operators in the second quantization formalism. Note the fermion sign structure (also known as the Jordan-Wigner string) (-1)^{\sum_{\beta<\alpha}n_\beta} in the formula, which requires an ordering of the single-particle states (the spin structure) and involves a counting of the fermion occupation numbers of all the preceding states, therefore the fermion creation and annihilation operators are considered non-local in some sense. This observation leads to the idea that fermions are emergent particles in the long-range entangled local qubit system.[5]

Operator Identities[edit]

The following operator identities follows from the action of the fermion creation and annihilation operators on the Fock state,

\{c_\alpha^\dagger,c_\beta^\dagger\}=\{c_\alpha,c_\beta\}=0,\quad \{c_\alpha,c_\beta^\dagger\}=\delta_{\alpha\beta}.

These anti-commutation relations can be considered as the algebraic definition of the fermion creation and annihilation operators. The fact that the fermion many-body wave function is anti-symmetric under particle exchange is also manifested by the anti-commutation of the fermion operators.

Quantum field operators[edit]

Defining {a^{(\dagger)}}_{\nu} as a general annihilation(creation) operator that could be either fermionic ({c^{(\dagger)}}_{\nu}) or bosonic ({b^{(\dagger)}}_{\nu}), the real space representation of the operators defines the quantum field operators  \Psi(\bold{r}) and \Psi^{\dagger}(\bold{r}) by

 \Psi(\bold{r})=\sum_{\nu} \psi_{\nu} \left( \bold{r} \right) a_{\nu}
 \Psi^{\dagger}(\bold{r})=\sum_{\nu} {\psi^*}_{\nu} \left( \bold{r} \right) {a^{\dagger}}_{\nu}

Second quantization operators, while the coefficients \psi_{\nu} \left( \bold{r} \right) and  {\psi^*}_{\nu} \left( \bold{r} \right) are the ordinary first quantization wavefunctions. Loosely speaking, \Psi^{\dagger}(\bold{r}) is the sum of all possible ways to add a particle to the system at position r through any of the basis states \psi_{\nu}\left(\bold{r}\right). Since  \Psi(\bold{r}) and \Psi^{\dagger}(\bold{r}) are second quantization operators defined in every point in space they are called quantum field operators. They obey the following fundamental commutator and anti-commutator,

 \left[\Psi(\bold{r}_1),\Psi^\dagger(\bold{r}_2)\right]=\delta (\bold{r}_1-\bold{r}_2) boson fields,
 \{\Psi(\bold{r}_1),\Psi^\dagger(\bold{r}_2)\}=\delta (\bold{r}_1-\bold{r}_2) fermion fields.

In homogeneous systems it is often desirable to transform between real space and the momentum representations, hence, the quantum fields operators in Fourier basis yields:

 \Psi(\bold{r})={1\over \sqrt {V}} \sum_{\bold{k}} e^{i\bold{k\cdot r}}a_{\bold{k}}
 \Psi^{\dagger}(\bold{r})={ 1\over \sqrt{V}} \sum_{\bold{k}} e^{-i\bold{k\cdot r}}{a^{\dagger}}_{\bold{k}}

Comments[edit]

The term is objected to by many working physicists, and is considered deprecated. One is not quantizing "again", as the term "second" would suggest; one is merely shifting from a semiclassical treatment of a problem to a fully quantum-mechanical one.

See also[edit]

References[edit]

  1. ^ Dirac, P. A. M. (1927). "The Quantum Theory of the Emission and Absorption of Radiation". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 114 (767): 243. doi:10.1098/rspa.1927.0039.  edit
  2. ^ V. Fock, Z. Phys. 75 (1932), 622-647
  3. ^ M.C. Reed, B. Simon, "Methods of Modern Mathematical Physics, Volume II", Academic Press 1975. Page 328.
  4. ^ Mahan, GD (1981). Many Particle Physics. New York: Springer. ISBN 0306463385. 
  5. ^ Levin, M.; Wen, X. G. (2003). "Fermions, strings, and gauge fields in lattice spin models". Physical Review B 67 (24). doi:10.1103/PhysRevB.67.245316.  edit

External links[edit]