In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, path-ordering is the procedure (or a
meta-operator ) that orders a product of operators according to the value of a chosen
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
:
:
Here ''p'' is 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 pro ...
that orders the parameters by value:
:
:
For example:
:
Examples
If an
operator is not simply expressed as a product, but as a function of another operator, we must first perform a
Taylor expansion of this function. This is the case of the
Wilson loop, which is defined as a path-
ordered exponential
The ordered exponential, also called the path-ordered exponential, is a mathematical operation defined in non-commutative algebras, equivalent to the exponential of the integral in the commutative algebras. In practice the ordered exponential is ...
to guarantee that the Wilson loop encodes the
holonomy of the
gauge connection
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
. The parameter ''σ'' 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 be
gauge-invariant.
Time ordering
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 useful to take the time-ordered product of operators. This operation is denoted by
. (Although
is often called the "time-ordering operator", strictly speaking it is neither an
operator on states nor a
superoperator
In physics, a superoperator is a linear operator acting on a vector space of linear operators.John Preskill, Lecture notes for Quantum Computation course at CaltechCh. 3
Sometimes the term refers more specially to a completely positive map whic ...
on operators.)
For two operators ''A''(''x'') and ''B''(''y'') that depend on spacetime locations x and y we define:
:
Here
and
denote the ''invariant'' scalar time-coordinates of the points x and y.
[ Steven Weinberg, ''The Quantum Theory of Fields'', Vol. 3, Cambridge University Press, 1995, , p. 143.]
Explicitly we have
:
where
denotes the
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...
and the
depends on if the operators are
bosonic or
fermion
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 ...
ic 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
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
. This is why it is necessary to use
rather than
, since
usually indicates the coordinate dependent time-like index of the spacetime point. Note that the time-ordering is usually written with the time argument increasing from right to left.
In general, for the product of ''n'' field operators the time-ordered product of operators are defined as follows:
:
where the sum runs all over ''ps and over the
symmetric group of ''n'' degree permutations and
:
The
S-matrix 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 ...
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