Path-ordering
Encyclopedia
In theoretical physics
, path-ordering is the procedure (or a meta-operator
) of ordering a product of many operators according to the value of one chosen parameter
:
Here
is a permutation
that orders the parameters:
For example:
is not simply expressed as a product, but as a function of another operator, we must first perform Taylor expansion of this function. This is the case of the Wilson loop
that is defined as a path-ordered exponential
; this guarantees that the Wilson loop encodes the holonomy
of the gauge connection. The parameter
that determines the ordering is a parameter describing the contour
, and because the contour is closed, the Wilson loop must be defined as a trace
in order to become gauge-invariant.
it is useful to take the time-ordered product of operators. This operation is denoted by
. For two operators 
and 
that depend on spacetime locations x and y we define:
Here
and 
denote the time-coordinates of the points x and y.
Explicitly we have
where
denotes the Heaviside step function
and the
depends on if the operators are bosonic or fermionic in nature. If bosonic, then the + sign is always chosen, if fermionic then the sign will depend on the number of operator interchanges necessary to achieve the proper time ordering. Note that the statistical factors do not enter here.
Since the operators depend on their location in spacetime (i.e. not just time) this time-ordering operation is only coordinate independent if operators at spacelike separated points commute
. Note that the time-ordering is usually written with the time argument increasing from right to left.
The S-matrix in quantum field theory
is an example of a time-ordered product. The S-matrix, transforming the state at
to a state at 
, can also be thought of as a kind of "holonomy
", analogous to the Wilson loop. We obtain a time-ordered expression because of the following reason:
We start with this simple formula for the exponential
Now consider the discretized evolution operator
where
is the evolution operator over an infinitesimal time interval 
. The higher order terms can be neglected in the limit 
. The operator 
is defined by
Note that the evolution operators over the "past" time intervals appears on the right side of the product. We see that the formula is analogous to the identity above satisfied by the exponential, and we may write
The only subtlety we had to include was the time-ordering operator
because the factors in the product defining 
above were time-ordered, too (and operators do not commute in general) and the operator 
guarantees that this ordering will be preserved.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, path-ordering is the procedure (or a meta-operator
Operator (physics)
In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
) of ordering a product of many operators according to the value of one chosen parameterParameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
:
Here
is a permutationPermutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
that orders the parameters:
For example:
Examples
If an operatorOperator (physics)
In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
is not simply expressed as a product, but as a function of another operator, we must first perform Taylor expansion of this function. This is the case of the Wilson loop
Wilson loop
In gauge theory, a Wilson loop is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop...
that is defined as a path-ordered exponential
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
; this guarantees that the Wilson loop encodes the holonomy
Holonomy
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections,...
of the gauge connection. The parameter
that determines the ordering is a parameter describing the contourContour
Contour may refer to:* an outline or silhouette* a contour line on a contour map, or the corresponding line on the ground or sea bed* Contour , a phonetic sound* Pitch contour , a melody shape...
, and because the contour is closed, the Wilson loop must be defined as a trace
Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
in order to become gauge-invariant.
Time ordering
In quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
it is useful to take the time-ordered product of operators. This operation is denoted by
. For two operators and that depend on spacetime locations x and y we define:Here
and denote the time-coordinates of the points x and y.Explicitly we have
where
denotes the Heaviside step functionHeaviside step function
The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....
and the
depends on if the operators are bosonic or fermionic in nature. If bosonic, then the + sign is always chosen, if fermionic then the sign will depend on the number of operator interchanges necessary to achieve the proper time ordering. Note that the statistical factors do not enter here.Since the operators depend on their location in spacetime (i.e. not just time) this time-ordering operation is only coordinate independent if operators at spacelike separated points commute
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
. Note that the time-ordering is usually written with the time argument increasing from right to left.
The S-matrix in quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
is an example of a time-ordered product. The S-matrix, transforming the state at
to a state at , can also be thought of as a kind of "holonomyHolonomy
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections,...
", analogous to the Wilson loop. We obtain a time-ordered expression because of the following reason:
We start with this simple formula for the exponential
Now consider the discretized evolution operator
where
is the evolution operator over an infinitesimal time interval . The higher order terms can be neglected in the limit . The operator is defined byNote that the evolution operators over the "past" time intervals appears on the right side of the product. We see that the formula is analogous to the identity above satisfied by the exponential, and we may write
The only subtlety we had to include was the time-ordering operator
because the factors in the product defining above were time-ordered, too (and operators do not commute in general) and the operator guarantees that this ordering will be preserved.See also
- Ordered exponentialOrdered exponentialThe ordered exponential is a mathematical object, defined in non-commutative algebras, which is equivalent to the exponential function of the integral in the commutative algebras. Therefore it is a function, defined by means of a function from real numbers to a real or complex associative algebra...
describes essentially the same concept. - Gauge theoryGauge theoryIn physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
- S-matrix