Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze
quantum
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
many-body
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
systems. In
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, it is known as
canonical quantization, in which the fields (typically as the wave functions of matter) are thought of as
field operator
In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible.
Historically, this was not quite ...
s, in a manner similar to how the physical quantities (position, momentum, etc.) are thought of as operators in
first quantization
A first quantization of a physical system is a possibly semiclassical treatment of quantum mechanics, in which particles or physical objects are treated using quantum wave functions but the surrounding environment (for example a potential well o ...
. The key ideas of this method were introduced in 1927 by
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
,
and were later developed, most notably, by
Pascual Jordan
Ernst Pascual Jordan (; 18 October 1902 – 31 July 1980) was a German theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matri ...
and
Vladimir Fock
Vladimir Aleksandrovich Fock (or Fok; russian: Влади́мир Алекса́ндрович Фок) (December 22, 1898 – December 27, 1974) was a Soviet Union, Soviet physicist, who did foundational work on quantum mechanics and quantum ...
.
In this approach, the quantum many-body states are represented in the
Fock state
In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an impo ...
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
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually ...
to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory.
Quantum many-body states
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
and different configurations of the set of
s correspond to different many-body states, ''in quantum mechanics, the particles are identical, such that exchanging two particles, i.e.
, does not lead to a different many-body 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
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
of the particles, the many-body wave function can either be symmetric or antisymmetric under the particle exchange:
:
if the particles are
bosons
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
,
:
if the particles are
fermions
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
.
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
permanent
Permanent may refer to:
Art and entertainment
* ''Permanent'' (film), a 2017 American film
* ''Permanent'' (Joy Division album)
* "Permanent" (song), by David Cook
Other uses
* Permanent (mathematics), a concept in linear algebra
* Permanent (cy ...
s (for bosons) or
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
s (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
Consider a complete set of single-particle wave functions
labeled by
(which may be a combined index of a number of quantum numbers). The following wave function
:
represents an ''N''-particle state with the ''i''th particle occupying the single-particle state
. In the shorthanded notation, the position argument of the wave function may be omitted, and it is assumed that the ''i''th single-particle wave function describes the state of the ''i''th particle. The wave function
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 operators
for symmetrizer, and
for
antisymmetrizer
In quantum mechanics, an antisymmetrizer \mathcal (also known as antisymmetrizing operatorP.A.M. Dirac, ''The Principles of Quantum Mechanics'', 4th edition, Clarendon, Oxford UK, (1958) p. 248) is a linear operator that makes a wave function of ...
.
For bosons, the many-body wave function must be symmetrized,
:
while for fermions, the many-body wave function must be anti-symmetrized,
:
Here
is an element in the ''N''-body permutation group (or
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
)
, which performs a
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
among the state labels
, and
denotes the corresponding
permutation sign.
is the normalization operator that normalizes the wave function. (It is the operator that applies a suitable numerical normalization factor to the symmetrized tensors of degree ''n''; see the next section for its value.)
If one arranges the single-particle wave functions in a matrix
, such that the row-''i'' column-''j'' matrix element is
, then the boson many-body wave function can be simply written as a
permanent
Permanent may refer to:
Art and entertainment
* ''Permanent'' (film), a 2017 American film
* ''Permanent'' (Joy Division album)
* "Permanent" (song), by David Cook
Other uses
* Permanent (mathematics), a concept in linear algebra
* Permanent (cy ...
, and the fermion many-body wave function as a
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
(also known as the
Slater determinant
In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two elect ...
).
Second-quantized Fock states
First quantized wave functions involve complicated symmetrization procedures to describe physically realizable many-body states because the language of first quantization is redundant for indistinguishable particles. In the first quantization language, the many-body state is described by answering a series of questions like ''"Which particle is in 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 seemingly different states
and
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 in 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
:
meaning that there are
particles in the single-particle state
(or as
). The occupation numbers sum to the total number of particles, i.e.
. For
fermions
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
, the occupation number
can only be 0 or 1, due to the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
; while for
bosons
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
it can be any non-negative integer
:
The occupation number states
are also known as Fock states. All the Fock states form a complete basis of the many-body Hilbert space, or
Fock space
The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intr ...
. Any generic quantum many-body state can be expressed as a linear combination of Fock states.
Note that besides providing a more efficient language, Fock space allows for a variable number of particles. As a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, it is isomorphic to the sum of the ''n''-particle bosonic or fermionic tensor spaces described in the previous section, including a one-dimensional zero-particle space C.
The Fock state with all occupation numbers equal to zero is called the
vacuum state
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
, denoted
. The Fock state with only one non-zero occupation number is a single-mode Fock state, denoted
. In terms of the first quantized wave function, the vacuum state is the unit tensor product and can be denoted
. The single-particle state is reduced to its wave function
. Other single-mode many-body (boson) states are just the tensor product of the wave function of that mode, such as
and
. For multi-mode Fock states (meaning more than one single-particle state
is involved), the corresponding first-quantized wave function will require proper symmetrization according to the particle statistics, e.g.
for a boson state, and
for a fermion state (the symbol
between
and
is omitted for simplicity). In general, the normalization is found to be
, where ''N'' is the total number of particles. For fermion, this expression reduces to
as
can only be either zero or one. So the first-quantized wave function corresponding to the Fock state reads
:
for bosons and
:
for fermions. Note that for fermions,
only, so the tensor product above is effectively just a product over all occupied single-particle states.
Creation and annihilation operators
The
creation and annihilation operators
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually ...
are introduced to add or remove a particle from the many-body system. These operators lie 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
as the raising and lowering operators, which are then generalized to the field operators in the quantum field theory.
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
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
be a single-particle state, let 1 be the tensor identity (it is the generator of the zero-particle space C and satisfies
in the
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
over the fundamental Hilbert space), and let
be a generic tensor product state. The insertion
and the deletion
operators are linear operators defined by the following recursive equations
:
:
Here
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
symbol, which gives 1 if
, and 0 otherwise. The subscript
of the insertion or deletion operators indicates whether symmetrization (for bosons) or anti-symmetrization (for fermions) is implemented.
Boson creation and annihilation operators
The boson creation (resp. annihilation) operator is usually denoted as
(resp.
). The creation operator
adds a boson to the single-particle state
, and the annihilation operator
removes a boson from the single-particle state
. The creation and annihilation operators are Hermitian conjugate to each other, but neither of them are Hermitian operators (
).
Definition
The boson creation (annihilation) operator is a linear operator, whose action on a ''N''-particle first-quantized wave function
is defined as
:
:
where
inserts the single-particle state
in
possible insertion positions symmetrically, and
deletes the single-particle state
from
possible deletion positions symmetrically.
Hereinafter the tensor symbol
between single-particle states is omitted for simplicity. Take the state
, create one more boson on the state
,
:
Then annihilate one boson from the state
,
:
Action on Fock states
Starting from the single-mode vacuum state
, applying the creation operator
repeatedly, one finds
:
:
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
:
On the other hand, the annihilation operator
lowers the boson occupation number by 1
:
It will also quench the vacuum state
as there has been no boson left in the vacuum state to be annihilated. Using the above formulae, it can be shown that
:
meaning that
defines the boson number operator.
The above result can be generalized to any Fock state of bosons.
:
:
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 taken care of by the creation and annihilation operators (when acting on the first-quantized wave function), so that the complexity is not revealed on the second-quantized level, and the second-quantization formulae are simple and neat.
Operator identities
The following operator identities follow from the action of the boson creation and annihilation operators on the Fock state,
:
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
also satisfy the same set of commutation relations, implying that the bosons can be interpreted as the energy quanta (phonons) of an oscillator. The position and momentum operators of a Harmonic oscillator (or a collection of Harmonic oscillating modes) are given by Hermitian combinations of phonon creation and annihilation operators,
:
which reproduce the canonical commutation relation between position and momentum operators (with
)
:
This idea is generalized in the
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, 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
The fermion creation (annihilation) operator is usually denoted as
(
). The creation operator
adds a fermion to the single-particle state
, and the annihilation operator
removes a fermion from the single-particle state
.
Definition
The fermion creation (annihilation) operator is a linear operator, whose action on a ''N''-particle first-quantized wave function
is defined as
:
:
where
inserts the single-particle state
in
possible insertion positions anti-symmetrically, and
deletes the single-particle state
from
possible deletion positions anti-symmetrically.
It is particularly instructive to view the results of creation and annihilation operators on states of two (or more) fermions, because they demonstrate the effects of exchange. A few illustrative operations are given in the example below. The complete algebra for creation and annihilation operators on a two-fermion state can be found in ''Quantum Photonics''.
Hereinafter the tensor symbol
between single-particle states is omitted for simplicity. Take the state
, attempt to create one more fermion on the occupied
state will quench the whole many-body wave function,
:
Annihilate a fermion on the
state,
take the state
,
:
The minus sign (known as the fermion sign) appears due to the anti-symmetric property of the fermion wave function.
Action on Fock states
Starting from the single-mode vacuum state
, applying the fermion creation operator
,
:
:
If the single-particle state
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
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
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
,
:
:
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
:
It is easy to check (by enumeration) that
:
meaning that
defines the fermion number operator.
The above result can be generalized to any Fock state of fermions.
:
:
Recall that the occupation number
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 that the fermion sign structure
, also known as the
Jordan-Wigner string, requires there to exist a predefined ordering of the single-particle states (the
spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathematical ...
) 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
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
system.
Operator identities
The following operator identities follow from the action of the fermion creation and annihilation operators on the Fock state,
:
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.
The creation and annihilation operators are Hermitian conjugate to each other, but neither of them are Hermitian operators (
). The Hermitian combination of the fermion creation and annihilation operators
:
are called
Majorana fermion operators. They can be viewed as the fermionic analog of position and momentum operators of a "fermionic" Harmonic oscillator. They satisfy the anticommutation relation
:
where
labels any Majorana fermion operators on equal footing (regardless their origin from Re or Im combination of complex fermion operators
). The anticommutation relation indicates that Majorana fermion operators generates a
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
, which can be systematically represented as Pauli operators in the many-body Hilbert space.
Quantum field operators
Defining
as a general annihilation (creation) operator for a single-particle state
that could be either fermionic
or bosonic
, the
real space representation of the operators defines the
quantum
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
field
operators
and
by
:
:
These are second quantization operators, with coefficients
and
that are ordinary
first-quantization wavefunctions. Thus, for example, any expectation values will be ordinary first-quantization wavefunctions. Loosely speaking,
is the sum of all possible ways to add a particle to the system at position r through any of the basis states
, not necessarily plane waves, as below.
Since
and
are second quantization operators defined in every point in space they are called
quantum field
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
operators. They obey the following fundamental commutator and anti-commutator relations,
:
boson fields,
:
fermion fields.
For homogeneous systems it is often desirable to transform between real space and the momentum representations, hence, the quantum fields operators in
Fourier basis
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be repres ...
yields:
:
:
Comment on nomenclature
The term "second quantization", introduced by Jordan,
is a misnomer that has persisted for historical reasons. At the origin of quantum field theory, it was inappositely thought that the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
described a relativistic wavefunction (hence the obsolete "Dirac sea" interpretation), rather than a classical spinor field which, when quantized (like the scalar field), yielded a fermionic quantum field (vs. a bosonic quantum field).
One is not quantizing "again", as the term "second" might suggest; the field that is being quantized is not a
Schrödinger wave function that was produced as the result of quantizing a particle, but is a classical field (such as the electromagnetic field or
Dirac spinor
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain com ...
field), essentially an assembly of coupled oscillators, that was not previously quantized. One is merely quantizing each oscillator in this assembly, shifting from a
semiclassical treatment of the system to a fully quantum-mechanical one.
See also
*
Canonical quantization
*
First quantization
A first quantization of a physical system is a possibly semiclassical treatment of quantum mechanics, in which particles or physical objects are treated using quantum wave functions but the surrounding environment (for example a potential well o ...
*
Geometric quantization
In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a wa ...
*
Quantization (physics)
In physics, quantization (in British English quantisation) is the systematic transition procedure from a classical understanding of physical phenomena to a newer understanding known as quantum mechanics. It is a procedure for constructing qua ...
*
Schrödinger functional
*
Scalar field theory
In theoretical physics, scalar field theory can refer to a relativistically invariant classical or quantum theory of scalar fields. A scalar field is invariant under any Lorentz transformation.
The only fundamental scalar quantum field that has b ...
References
{{Reflist , 3
Quantum field theory
Mathematical quantization