Propagator
Encyclopedia
In quantum mechanics
and quantum field theory
, the propagator gives the probability amplitude
for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particle
s on the internal lines of Feynman diagram
s. They also can be viewed as the inverse of the wave operator appropriate to the particle, and are therefore often called Green's function
s.
to travel from one spatial point at one time to another spatial point at a later time. It is a Green's function
for the Schrödinger equation
. This means that if a system has Hamiltonian
then the appropriate propagator is a function 
satisfying
where
denotes the Hamiltonian written in terms of the 
coordinates and 
denotes the Dirac delta-function.
This can also be written as
where
is the unitary
time-evolution operator for the system taking states at time
to states at time 
.
.
where the boundary conditions of the path integral include q(t)=x, q(t')=x'. Here
denotes the Lagrangian
of the system. The paths that are summed over move only forwards in time.
, the propagator lets you find the state of a system given an initial state and a time interval. The new state is given by the equation:
If
only depends on the difference 
this is a convolution
of the initial state and the propagator.
.
The propagator of one-dimensional free particle, with the far-right expression obtained via saddle-point approximation, is then
.
The propagator of one-dimensional harmonic oscillator is
.
For N-dimensional case, the propagator can be simply obtained by the product
.
the propagators are Lorentz invariant. They give the amplitude for a particle
to travel between two spacetime
points.
is a useful and simple example which serves to illustrate the concepts needed for more complicated theories. It describes spin
zero particles. There are a number of possible propagators for free scalar field theory. We now describe the most common ones.
s for the Klein–Gordon equation. This means they are functions
which satisfy
where:
(As typical in relativistic
quantum field theory calculations, we use units where the speed of light
,
, is 1.)
We shall restrict attention to 4-dimensional Minkowski spacetime. We can perform a Fourier transform
of the equation for the propagator being this linear obtaining
This equation can be inverted in the sense of distributions
noting that the equation
has the solution 
, with 
implying the limit to zero. Below we discuss the right choice of the sign arising from causality requirements and finally we can write down for the solution
where
is the 4-vector inner product.
The different choices for how to deform the integration contour
in the above expression lead to different forms for the propagator. The choice of contour is usually phrased in terms of the
integral.
The integrand then has two poles at
so different choices of how to avoid these lead to different propagators.
A contour going clockwise over both poles gives the causal retarded propagator. This is zero if
and 
are spacelike or if 
(i.e. if 
is to the future of 
).
This choice of contour is equivalent to calculating the limit
:
Here
is the proper time
from
to 
and 
is a Bessel function of the first kind. The expression 
means 
causally precedes
which, for Minkowski spacetime, means
and 
.
This expression can also be expressed in terms of the vacuum expectation value
of the commutator
of the free scalar field.
Here
where
is the Heaviside step function
and
is the commutator
.
Advanced propagator:
A contour going anti-clockwise under both poles gives the causal advanced propagator. This is zero if
and 
are spacelike or if 
(i.e. if 
is to the past of 
).
This choice of contour is equivalent to calculating the limit:
This expression can also be expressed in terms of the vacuum expectation value
of the commutator
of the free scalar field.
Here
A contour going under the left pole and over the right pole gives the Feynman propagator.
This choice of contour is equivalent to calculating the limit (see Huang p30):
Here
Here
and 
are two points in Minkowski spacetime, and the dot in the exponent is a four-vector
inner product.
is a Hankel function and 
is a modified Bessel function.
This expression can be derived directly from the field theory as the vacuum expectation value
of the time-ordered product of the free scalar field, that is, the product always taken such that the time ordering of the spacetime points is the same:
This expression is Lorentz invariant as long as the field operators commute with one another when the points
and 
are separated by a spacelike interval.
The usual derivation is to insert a complete set of single-particle momentum states between the fields with Lorentz covariant normalization, then show that the
functions providing the causal time ordering may be obtained by a contour integral
along the energy axis if the integrand is as above (hence the infinitesimal imaginary part, to move the pole off the real line).
The propagator may also be derived using the path integral formulation
of quantum theory.
of the position space propagators can be thought of as propagators in momentum space. These take a much simpler form than the position space propagators.
They are often written with an explicit
term although this is understood to be a reminder about which integration contour is appropriate (see above). This 
term is included to incorporate boundary conditions and causality
(see below).
For a 4-momentum
the causal and Feynman propagators in momentum space are:
For purposes of Feynman diagram calculations it is usually convenient to write these with an additional overall factor of
(conventions vary).
, though it falls off rapidly for spacelike intervals. Interpreted as an amplitude for particle motion, this translates to the virtual particle traveling faster than light. It is not immediately obvious how this can be reconciled with causality: can we use faster-than-light virtual particles to send faster-than-light messages?
The answer is no: while in classical mechanics
the intervals along which particles and causal effects can travel are the same, this is no longer true in quantum field theory, where it is commutator
s that determine which operators can affect one another.
So what does the spacelike part of the propagator represent? In QFT the vacuum
is an active participant, and particle number
s and field values are related by an uncertainty principle
; field values are uncertain even for particle number zero. There is a nonzero probability amplitude
to find a significant fluctuation in the vacuum value of the field
if one measures it locally (or, to be more precise, if one measures an operator obtained by averaging the field over a small region). Furthermore, the dynamics of the fields tend to favor spatially correlated fluctuations to some extent. The nonzero time-ordered product for spacelike-separated fields then just measures the amplitude for a nonlocal correlation in these vacuum fluctuations, analogous to an EPR correlation
. Indeed, the propagator is often called a two-point correlation function for the free field.
Since, by the postulates of quantum field theory, all observable
operators commute with each other at spacelike separation, messages can no more be sent through these correlations than they can through any other EPR correlations; the correlations are in random variables.
In terms of virtual particles, the propagator at spacelike separation can be thought of as a means of calculating the amplitude for creating a virtual particle-antiparticle
pair that eventually disappear into the vacuum, or for detecting a virtual pair emerging from the vacuum. In Feynman
's language, such creation and annihilation processes are equivalent to a virtual particle wandering backward and forward through time, which can take it outside of the light cone. However, no causality violation is involved.
s for particle interactions using Feynman diagram
s. These calculations are usually carried out in momentum space. In general, the amplitude gets a factor of the propagator for every internal line, that is, every line that does not represent an incoming or outgoing particle in the initial or final state. It will also get a factor proportional to, and similar in form to, an interaction term in the theory's Lagrangian
for every internal vertex where lines meet. These prescriptions are known as Feynman rules.
Internal lines correspond to virtual particles. Since the propagator does not vanish for combinations of energy and momentum disallowed by the classical equations of motion, we say that the virtual particles are allowed to be off shell. In fact, since the propagator is obtained by inverting the wave equation, in general it will have singularities on shell.
The energy carried by the particle in the propagator can even be negative. This can be interpreted simply as the case in which, instead of a particle going one way, its antiparticle
is going the other way, and therefore carrying an opposing flow of positive energy. The propagator encompasses both possibilities. It does mean that one has to be careful about minus signs for the case of fermions, whose propagators are not even functions in the energy and momentum (see below).
Virtual particles conserve energy and momentum. However, since they can be off shell, wherever the diagram contains a closed loop, the energies and momenta of the virtual particles participating in the loop will be partly unconstrained, since a change in a quantity for one particle in the loop can be balanced by an equal and opposite change in another. Therefore, every loop in a Feynman diagram requires an integral over a continuum of possible energies and momenta. In general, these integrals of products of propagators can diverge, a situation that must be handled by the process of renormalization
.
field representing the electron
in quantum electrodynamics
has the form
where the
are the gamma matrices appearing in the covariant formulation of the Dirac equation. It is sometimes written, using Feynman slash notation
,
for short. In position space we have:
This is related to the Feynman propagator by
where
.
The propagator for a gauge boson
in a gauge theory
depends on the choice of convention to fix the gauge. For the gauge used by Feynman and Stueckelberg
, the propagator for a photon
is
. We follow the notation in Bjorken and Drell. See also Bogolyubov and Shirkov (Appendix A). These function are most simply defined in terms of the vacuum expectation value
of products of field operators.
by
with
This satisfies
and is zero if 
.
, sometimes called cut propagators, in a relativistically invariant way.
This allows us to define the positive frequency part:
,
and the negative frequency part:
.
These satisfy
and
function by
with
This satisfies
where
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and 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...
, the propagator gives the probability amplitude
Probability amplitude
In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...
for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particle
Virtual particle
In physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...
s on the internal lines of Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s. They also can be viewed as the inverse of the wave operator appropriate to the particle, and are therefore often called Green's function
Green's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
s.
Non-relativistic propagators
In non-relativistic quantum mechanics the propagator gives the amplitude for a particleElementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
to travel from one spatial point at one time to another spatial point at a later time. It is a Green's function
Green's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
for the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
. This means that if a system has Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
then the appropriate propagator is a function satisfyingwhere
denotes the Hamiltonian written in terms of the coordinates and denotes the Dirac delta-function.This can also be written as
where
is the unitaryUnitary
Unitary may refer to:* Unitary construction, in automotive design, another common term for a unibody or monocoque construction**Unitary as chemical weapons opposite of Binary...
time-evolution operator for the system taking states at time
to states at time .Path integral in quantum mechanics
The quantum mechanical propagator may also be found by using a path integralPath integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
.
where the boundary conditions of the path integral include q(t)=x, q(t')=x'. Here
denotes the LagrangianLagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
of the system. The paths that are summed over move only forwards in time.
Using the quantum mechanical propagator
In non-relativistic quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, the propagator lets you find the state of a system given an initial state and a time interval. The new state is given by the equation:
If
only depends on the difference this is a convolutionConvolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
of the initial state and the propagator.
Propagator of Free Particle and Harmonic Oscillator
For time translational invariant system, the propagator only depends on the time difference (t-t'), thus it may be rewritten as.The propagator of one-dimensional free particle, with the far-right expression obtained via saddle-point approximation, is then
.The propagator of one-dimensional harmonic oscillator is
.For N-dimensional case, the propagator can be simply obtained by the product
.Relativistic propagators
In relativistic quantum mechanics and 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...
the propagators are Lorentz invariant. They give the amplitude for a particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
to travel between two spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
points.
Scalar propagator
In quantum field theory the theory of a free (non-interacting) scalar fieldScalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
is a useful and simple example which serves to illustrate the concepts needed for more complicated theories. It describes spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
zero particles. There are a number of possible propagators for free scalar field theory. We now describe the most common ones.
Position space
The position space propagators are Green's functionGreen's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
s for the Klein–Gordon equation. This means they are functions
which satisfywhere:
-
are two points in Minkowski spacetime. -
is the d'Alembertian operator acting on the 
coordinates. -
is the Dirac delta-function.
(As typical in relativistic
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
quantum field theory calculations, we use units where the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
,
, is 1.)We shall restrict attention to 4-dimensional Minkowski spacetime. We can perform a Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of the equation for the propagator being this linear obtaining
This equation can be inverted in the sense of distributions
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
noting that the equation
has the solution , with implying the limit to zero. Below we discuss the right choice of the sign arising from causality requirements and finally we can write down for the solutionwhere
is the 4-vector inner product.The different choices for how to deform the integration contour
Methods of contour integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.Contour integration is closely related to the calculus of residues, a methodology of complex analysis....
in the above expression lead to different forms for the propagator. The choice of contour is usually phrased in terms of the
integral.The integrand then has two poles at
so different choices of how to avoid these lead to different propagators.Causal propagator
Retarded propagator:A contour going clockwise over both poles gives the causal retarded propagator. This is zero if
and are spacelike or if (i.e. if is to the future of ).This choice of contour is equivalent to calculating the limit
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
:
Here
is the proper time
Proper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...
from
to and is a Bessel function of the first kind. The expression means causally precedesCausal structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.- Introduction :In modern physics spacetime is represented by a Lorentzian manifold...
which, for Minkowski spacetime, means and .This expression can also be expressed in terms of the vacuum expectation value
Vacuum expectation value
In quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
of the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
of the free scalar field.
Here
where
is the Heaviside step function
Heaviside 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
is the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
.
Advanced propagator:
A contour going anti-clockwise under both poles gives the causal advanced propagator. This is zero if
and are spacelike or if (i.e. if is to the past of ).This choice of contour is equivalent to calculating the limit:
This expression can also be expressed in terms of the vacuum expectation value
Vacuum expectation value
In quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
of the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
of the free scalar field.
Here
Feynman propagator
This choice of contour is equivalent to calculating the limit (see Huang p30):
 |
 |
 |
Here
Here
and are two points in Minkowski spacetime, and the dot in the exponent is a four-vectorFour-vector
In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the four-vector name tacitly assumes that its components refer to a standard basis...
inner product.
is a Hankel function and is a modified Bessel function.This expression can be derived directly from the field theory as the vacuum expectation value
Vacuum expectation value
In quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
of the time-ordered product of the free scalar field, that is, the product always taken such that the time ordering of the spacetime points is the same:
 |
|
This expression is Lorentz invariant as long as the field operators commute with one another when the points
and are separated by a spacelike interval.The usual derivation is to insert a complete set of single-particle momentum states between the fields with Lorentz covariant normalization, then show that the
functions providing the causal time ordering may be obtained by a contour integralLine integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
along the energy axis if the integrand is as above (hence the infinitesimal imaginary part, to move the pole off the real line).
The propagator may also be derived using the path integral formulation
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
of quantum theory.
Momentum space propagator
The Fourier transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of the position space propagators can be thought of as propagators in momentum space. These take a much simpler form than the position space propagators.
They are often written with an explicit
term although this is understood to be a reminder about which integration contour is appropriate (see above). This term is included to incorporate boundary conditions and causalityCausality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
(see below).
For a 4-momentum
the causal and Feynman propagators in momentum space are:For purposes of Feynman diagram calculations it is usually convenient to write these with an additional overall factor of
(conventions vary).Faster than light?
The Feynman propagator has some properties that seem baffling at first. In particular, unlike the commutator, the propagator is nonzero outside of the light coneLight cone
A light cone is the path that a flash of light, emanating from a single event and traveling in all directions, would take through spacetime...
, though it falls off rapidly for spacelike intervals. Interpreted as an amplitude for particle motion, this translates to the virtual particle traveling faster than light. It is not immediately obvious how this can be reconciled with causality: can we use faster-than-light virtual particles to send faster-than-light messages?
The answer is no: while in classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
the intervals along which particles and causal effects can travel are the same, this is no longer true in quantum field theory, where it is commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
s that determine which operators can affect one another.
So what does the spacelike part of the propagator represent? In QFT the vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
is an active participant, and particle number
Particle number
The particle number of a thermodynamic system, conventionally indicated with the letter N, is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is conjugate to the chemical potential. Unlike most physical quantities, particle...
s and field values are related by an uncertainty principle
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
; field values are uncertain even for particle number zero. There is a nonzero probability amplitude
Probability amplitude
In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...
to find a significant fluctuation in the vacuum value of the field
if one measures it locally (or, to be more precise, if one measures an operator obtained by averaging the field over a small region). Furthermore, the dynamics of the fields tend to favor spatially correlated fluctuations to some extent. The nonzero time-ordered product for spacelike-separated fields then just measures the amplitude for a nonlocal correlation in these vacuum fluctuations, analogous to an EPR correlationEPR paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...
. Indeed, the propagator is often called a two-point correlation function for the free field.
Since, by the postulates of quantum field theory, all observable
Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
operators commute with each other at spacelike separation, messages can no more be sent through these correlations than they can through any other EPR correlations; the correlations are in random variables.
In terms of virtual particles, the propagator at spacelike separation can be thought of as a means of calculating the amplitude for creating a virtual particle-antiparticle
Antiparticle
Corresponding to most kinds of particles, there is an associated antiparticle with the same mass and opposite electric charge. For example, the antiparticle of the electron is the positively charged antielectron, or positron, which is produced naturally in certain types of radioactive decay.The...
pair that eventually disappear into the vacuum, or for detecting a virtual pair emerging from the vacuum. In Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
's language, such creation and annihilation processes are equivalent to a virtual particle wandering backward and forward through time, which can take it outside of the light cone. However, no causality violation is involved.
Propagators in Feynman diagrams
The most common use of the propagator is in calculating probability amplitudeProbability amplitude
In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...
s for particle interactions using Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s. These calculations are usually carried out in momentum space. In general, the amplitude gets a factor of the propagator for every internal line, that is, every line that does not represent an incoming or outgoing particle in the initial or final state. It will also get a factor proportional to, and similar in form to, an interaction term in the theory's Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
for every internal vertex where lines meet. These prescriptions are known as Feynman rules.
Internal lines correspond to virtual particles. Since the propagator does not vanish for combinations of energy and momentum disallowed by the classical equations of motion, we say that the virtual particles are allowed to be off shell. In fact, since the propagator is obtained by inverting the wave equation, in general it will have singularities on shell.
The energy carried by the particle in the propagator can even be negative. This can be interpreted simply as the case in which, instead of a particle going one way, its antiparticle
Antiparticle
Corresponding to most kinds of particles, there is an associated antiparticle with the same mass and opposite electric charge. For example, the antiparticle of the electron is the positively charged antielectron, or positron, which is produced naturally in certain types of radioactive decay.The...
is going the other way, and therefore carrying an opposing flow of positive energy. The propagator encompasses both possibilities. It does mean that one has to be careful about minus signs for the case of fermions, whose propagators are not even functions in the energy and momentum (see below).
Virtual particles conserve energy and momentum. However, since they can be off shell, wherever the diagram contains a closed loop, the energies and momenta of the virtual particles participating in the loop will be partly unconstrained, since a change in a quantity for one particle in the loop can be balanced by an equal and opposite change in another. Therefore, every loop in a Feynman diagram requires an integral over a continuum of possible energies and momenta. In general, these integrals of products of propagators can diverge, a situation that must be handled by the process of renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
.
Other theories
If the particle possesses spin then its propagator is in general somewhat more complicated, as it will involve the particle's spin or polarization indices. The momentum-space propagator used in Feynman diagrams for a DiracDirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
field representing the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
in quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
has the form
where the
are the gamma matrices appearing in the covariant formulation of the Dirac equation. It is sometimes written, using Feynman slash notationFeynman slash notation
In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation...
,
for short. In position space we have:
This is related to the Feynman propagator by
where
.The propagator for a gauge boson
Gauge boson
In particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental forces of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles.-...
in a gauge theory
Gauge theory
In 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...
depends on the choice of convention to fix the gauge. For the gauge used by Feynman and Stueckelberg
Ernst Stueckelberg
Ernst Carl Gerlach Stueckelberg was a Swiss mathematician and physicist.- Career :In 1927 Stueckelberg got his Ph. D. at the University of Basel under August Hagenbach...
, the propagator for a photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
is
Related singular functions
The scalar propagators are Green's functions for the Klein–Gordon equation. There are related singular functions which are important 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...
. We follow the notation in Bjorken and Drell. See also Bogolyubov and Shirkov (Appendix A). These function are most simply defined in terms of the vacuum expectation value
Vacuum expectation value
In quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
of products of field operators.
Pauli–Jordan function
The commutator of two scalar field operators defines the Pauli–Jordan function bywith
This satisfies
and is zero if .Positive and negative frequency parts (cut propagators)
We can define the positive and negative frequency parts of, sometimes called cut propagators, in a relativistically invariant way.This allows us to define the positive frequency part:
,and the negative frequency part:
.These satisfy
and
Auxiliary function
The anti-commutator of two scalar field operators defines function bywith
This satisfies
Green's functions for the Klein-Gordon equation
The retarded, advanced and Feynman propagators defined above are all Green's functions for the Klein-Gordon equation. They are related to the singular functions by
where