In
quantum field theory a product of quantum fields, or equivalently their
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 d ...
, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operators in the product. The process of putting a product into normal order is called normal ordering (also called Wick ordering). The terms antinormal order and antinormal ordering are analogously defined, where the annihilation operators are placed to the left of the creation operators.
Normal ordering of a product quantum fields or
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 d ...
can also be defined in many
other ways. Which definition is most appropriate depends on the expectation values needed for a given calculation. Most of this article uses the most common definition of normal ordering as given above, which is appropriate when taking
expectation value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
s using the vacuum state of 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 d ...
.
The process of normal ordering is particularly important for a
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
. When quantizing a
classical Hamiltonian there is some freedom when choosing the operator order, and these choices lead to differences in the
ground state energy.
Notation
If
denotes an arbitrary product of creation and/or annihilation operators (or equivalently, quantum fields), then the normal ordered form of
is denoted by
.
An alternative notation is
.
Note that normal ordering is a concept that only makes sense for products of operators. Attempting to apply normal ordering to a sum of operators is not useful as normal ordering is not a linear operation.
Bosons
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 ...
are particles which satisfy
Bose–Einstein statistics
In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic ...
. We will now examine the normal ordering of bosonic creation and annihilation operator products.
Single bosons
If we start with only one type of boson there are two operators of interest:
*
: the boson's creation operator.
*
: the boson's annihilation operator.
These satisfy the
commutator relationship
:
:
:
where
denotes the
commutator. We may rewrite the last one as:
Examples
1. We'll consider the simplest case first. This is the normal ordering of
:
:
The expression
has not been changed because it is ''already'' in normal order - the creation operator
is already to the left of the annihilation operator
.
2. A more interesting example is the normal ordering of
:
:
Here the normal ordering operation has ''reordered'' the terms by placing
to the left of
.
These two results can be combined with the commutation relation obeyed by
and
to get
:
or
:
This equation is used in defining the contractions used in
Wick's theorem
Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem. It is named after Italian physicist Gian-Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihila ...
.
3. An example with multiple operators is:
:
4. A simple example shows that normal ordering cannot be extended by linearity from the monomials to all operators in a self-consistent way:
:
The implication is that normal ordering is not a linear function on operators.
Multiple bosons
If we now consider
different bosons there are
operators:
*
: the
boson's creation operator.
*
: the
boson's annihilation operator.
Here
.
These satisfy the commutation relations:
:
:
:
where
and
denotes 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 & ...
.
These may be rewritten as:
:
:
:
Examples
1. For two different bosons (
) we have
:
:
2. For three different bosons (
) we have
:
Notice that since (by the commutation relations)
the order in which we write the annihilation operators does not matter.
:
:
Bosonic operator functions
Normal ordering of bosonic operator functions
, with occupation number operator
, can be accomplished using
(falling) factorial powers and
Newton series
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
instead of
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
:
It is easy to show
that factorial powers
are equal to normal-ordered (raw)
powers and are therefore normal ordered by construction,
:
such that the Newton series expansion
:
of an operator function
, with
-th
forward difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
at
, is always normal ordered. Here, the
eigenvalue equation
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then
T( \mathbf x ) = A \mathbf x
for some m \times n matrix ...
relates
and
.
As a consequence, the normal-ordered Taylor series of an arbitrary function
is equal to the Newton series of an associated function
, fulfilling
:
if the series coefficients of the Taylor series of
, with continuous
, match the coefficients of the Newton series of
, with integer
,
:
with
-th
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
at
.
The functions
and
are related through the so-called
normal-order transform
In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operato ...