In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. Propagators may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called (causal) Green's functions (called "causal" to distinguish it from the elliptic Laplacian Green's function).
Non-relativistic propagators In non-relativistic quantum mechanics, the propagator gives the probability amplitude for a particle to travel from one spatial point (x') at one time (t') to another spatial point (x) at a later time (t). The Green's function G for the Schrödinger equation is a function
G ( x , t ; x ′ , t ′ ) = 1 i ℏ Θ ( t − t ′ ) K ( x , t ; x ′ , t ′ ) {\displaystyle G(x,t;x',t')={\frac {1}{i\hbar }}\Theta (t-t')K(x,t;x',t')}
satisfying
( i ℏ ∂ ∂ t − H x ) G ( x , t ; x ′ , t ′ ) = δ ( x − x ′ ) δ ( t − t ′ ) , {\displaystyle \left(i\hbar {\frac {\partial }{\partial t}}-H_{x}\right)G(x,t;x',t')=\delta (x-x')\delta (t-t'),}
where H denotes the Hamiltonian, δ(x) denotes the Dirac delta-function and Θ(t) is the Heaviside step function. The kernel of the above Schrödinger differential operator in the big parentheses is denoted by K(x, t ;x′, t′) and called the propagator. This propagator may also be written as the transition amplitude
K ( x , t ; x ′ , t ′ ) = ⟨ x | U ( t , t ′ ) | x ′ ⟩ , {\displaystyle K(x,t;x',t')={\big \langle }x{\big |}U(t,t'){\big |}x'{\big \rangle },}
where U(t, t′) is the unitary time-evolution operator for the system taking states at time t′ to states at time t. Note the initial condition enforced by
lim t → t ′ K ( x , t ; x ′ , t ′ ) = δ ( x − x ′ ) . {\displaystyle \lim _{t\to t'}K(x,t;x',t')=\delta (x-x').}
The propagator may also be found by using a path integral:
K ( x , t ; x ′ , t ′ ) = ∫ exp [ i ℏ ∫ t ′ t L ( q ˙ , q , t ) d t ] D [ q ( t ) ] , {\displaystyle K(x,t;x',t')=\int \exp \left[{\frac {i}{\hbar }}\int _{t'}^{t}L({\dot {q}},q,t)\,dt\right]D[q(t)],}
where L denotes the Lagrangian and the boundary conditions are given by q(t) = x, q(t′) = x′. The paths that are summed over move only forwards in time and are integrated with the differential D [ q ( t ) ] {\displaystyle D[q(t)]} following the path in time. The propagator lets one find the wave function of a system, given an initial wave function and a time interval. The new wave function is given by
ψ ( x , t ) = ∫ − ∞ ∞ ψ ( x ′ , t ′ ) K ( x , t ; x ′ , t ′ ) d x ′ . {\displaystyle \psi (x,t)=\int _{-\infty }^{\infty }\psi (x',t')K(x,t;x',t')\,dx'.}
If K(x, t; x′, t′) only depends on the difference x − x′, this is a convolution of the initial wave function and the propagator.
Examples
For a time-translationally invariant system, the propagator only depends on the time difference t − t′, so it may be rewritten as
K ( x , t ; x ′ , t ′ ) = K ( x , x ′ ; t − t ′ ) . {\displaystyle K(x,t;x',t')=K(x,x';t-t').}
The propagator of a one-dimensional free particle, obtainable from, e.g., the path integral, is then
Similarly, the propagator of a one-dimensional quantum harmonic oscillator is the Mehler kernel,
The latter may be obtained from the previous free-particle result upon making use of van Kortryk's SU(1,1) Lie-group identity,
exp ( − i t ℏ ( 1 2 m p 2 + 1 2 m ω 2 x 2 ) ) = exp ( − i m ω 2 ℏ x 2 tan ω t 2 ) exp ( − i 2 m ω ℏ p 2 sin ( ω t ) ) exp ( − i m ω 2 ℏ x 2 tan ω t 2 ) , {\displaystyle {\begin{aligned}&\exp \left(-{\frac {it}{\hbar }}\left({\frac {1}{2m}}{\mathsf {p}}^{2}+{\frac {1}{2}}m\omega ^{2}{\mathsf {x}}^{2}\right)\right)\\&=\exp \left(-{\frac {im\omega }{2\hbar }}{\mathsf {x}}^{2}\tan {\frac {\omega t}{2}}\right)\exp \left(-{\frac {i}{2m\omega \hbar }}{\mathsf {p}}^{2}\sin(\omega t)\right)\exp \left(-{\frac {im\omega }{2\hbar }}{\mathsf {x}}^{2}\tan {\frac {\omega t}{2}}\right),\end{aligned}}}
valid for operators x {\displaystyle {\mathsf {x}}} and p {\displaystyle {\mathsf {p}}} satisfying the Heisenberg relation [ x , p ] = i ℏ {\displaystyle [{\mathsf {x}},{\mathsf {p}}]=i\hbar } . For the N-dimensional case, the propagator can be simply obtained by the product
K ( x → , x → ′ ; t ) = ∏ q = 1 N K ( x q , x q ′ ; t ) . {\displaystyle K({\vec {x}},{\vec {x}}';t)=\prod _{q=1}^{N}K(x_{q},x_{q}';t).}
Relativistic propagators In relativistic quantum mechanics and quantum field theory the propagators are Lorentz-invariant. They give the amplitude for a particle to travel between two spacetime events.
Scalar propagator In quantum field theory, the theory of a free (or non-interacting) scalar field 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.
Position space The position space propagators are Green's functions for the Klein–Gordon equation. This means that they are functions G(x, y) satisfying
( ◻ x + m 2 ) G ( x , y ) = − δ ( x − y ) , {\displaystyle \left(\square _{x}+m^{2}\right)G(x,y)=-\delta (x-y),}
where
x, y are two points in Minkowski spacetime with metric signature (+, −, −, −),
◻ x = ∂ 2 ∂ t 2 − ∇ 2 {\displaystyle \square _{x}={\tfrac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}} is the d'Alembertian operator acting on the x coordinates, δ(x − y) is the Dirac delta function. (As typical in relativistic quantum field theory calculations, we use units where the speed of light c and the reduced Planck constant ħ are set to unity.) We shall restrict attention to 4-dimensional Minkowski spacetime. We can perform a Fourier transform of the equation for the propagator, obtaining
( − p 2 + m 2 ) G ( p ) = − 1. {\displaystyle \left(-p^{2}+m^{2}\right)G(p)=-1.}
This equation can be inverted in the sense of distributions, noting that the equation xf(x) = 1 has the solution (see Sokhotski–Plemelj theorem)
f ( x ) = 1 x ± i ε = 1 x ∓ i π δ ( x ) , {\displaystyle f(x)={\frac {1}{x\pm i\varepsilon }}={\frac {1}{x}}\mp i\pi \delta (x),}
with ε implying the limit to zero. Below, we discuss the right choice of the sign arising from causality requirements. The solution is
where
p ( x − y ) := p 0 ( x 0 − y 0 ) − p → ⋅ ( x → − y → ) {\displaystyle p(x-y):=p_{0}(x^{0}-y^{0})-{\vec {p}}\cdot ({\vec {x}}-{\vec {y}})} is the 4-vector Minkowski inner product. The different choices for how to deform the integration contour in the above expression lead to various forms for the propagator. The choice of contour is usually phrased in terms of the p 0 {\displaystyle p_{0}} integral. The integrand then has two poles at
p 0 = ± p → 2 + m 2 , {\displaystyle p_{0}=\pm {\sqrt {{\vec {p}}^{2}+m^{2}}},} so different choices of how to avoid these lead to different propagators.
Causal propagators
Retarded propagator
A contour going clockwise over both poles gives the causal retarded propagator. This is zero if x-y is spacelike or y is to the future of x, so it is zero if x ⁰< y ⁰. This choice of contour is equivalent to calculating the limit,
G ret ( x , y ) = lim ε → 0 1 ( 2 π ) 4 ∫ d 4 p e − i p ( x − y ) ( p 0 + i ε ) 2 − p → 2 − m 2 = − Θ ( x 0 − y 0 ) 2 π δ ( τ x y 2 ) + Θ ( x 0 − y 0 ) Θ ( τ x y 2 ) m J 1 ( m τ x y ) 4 π τ x y . {\displaystyle G_{\text{ret}}(x,y)=\lim _{\varepsilon \to 0}{\frac {1}{(2\pi )^{4}}}\int d^{4}p\,{\frac {e^{-ip(x-y)}}{(p_{0}+i\varepsilon )^{2}-{\vec {p}}^{2}-m^{2}}}=-{\frac {\Theta (x^{0}-y^{0})}{2\pi }}\delta (\tau _{xy}^{2})+\Theta (x^{0}-y^{0})\Theta (\tau _{xy}^{2}){\frac {mJ_{1}(m\tau _{xy})}{4\pi \tau _{xy}}}.}
Here
Θ ( x ) := { 1 x ≥ 0 0 x < 0 {\displaystyle \Theta (x):={\begin{cases}1&x\geq 0\\0&x<0\end{cases}}}
is the Heaviside step function,
τ x y := ( x 0 − y 0 ) 2 − ( x → − y → ) 2 {\displaystyle \tau _{xy}:={\sqrt {(x^{0}-y^{0})^{2}-({\vec {x}}-{\vec {y}})^{2}}}}
is the proper time from x to y, and J 1 {\displaystyle J_{1}} is a Bessel function of the first kind. The propagator is non-zero only if y ≺ x {\displaystyle y\prec x} , i.e., y causally precedes x, which, for Minkowski spacetime, means
y 0 ≤ x 0 {\displaystyle y^{0}\leq x^{0}} and τ x y 2 ≥ 0 . {\displaystyle \tau _{xy}^{2}\geq 0~.}
This expression can be related to the vacuum expectation value of the commutator of the free scalar field operator,
G ret ( x , y ) = − i ⟨ 0 | [ Φ ( x ) , Φ ( y ) ] | 0 ⟩ Θ ( x 0 − y 0 ) , {\displaystyle G_{\text{ret}}(x,y)=-i\langle 0|\left[\Phi (x),\Phi (y)\right]|0\rangle \Theta (x^{0}-y^{0}),}
where
[ Φ ( x ) , Φ ( y ) ] := Φ ( x ) Φ ( y ) − Φ ( y ) Φ ( x ) . {\displaystyle \left[\Phi (x),\Phi (y)\right]:=\Phi (x)\Phi (y)-\Phi (y)\Phi (x).}
Advanced propagator
A contour going anti-clockwise under both poles gives the causal advanced propagator. This is zero if x-y is spacelike or if y is to the past of x, so it is zero if x ⁰> y ⁰. This choice of contour is equivalent to calculating the limit
G adv ( x , y ) = lim ε → 0 1 ( 2 π ) 4 ∫ d 4 p e − i p ( x − y ) ( p 0 − i ε ) 2 − p → 2 − m 2 = − Θ ( y 0 − x 0 ) 2 π δ ( τ x y 2 ) + Θ ( y 0 − x 0 ) Θ ( τ x y 2 ) m J 1 ( m τ x y ) 4 π τ x y . {\displaystyle G_{\text{adv}}(x,y)=\lim _{\varepsilon \to 0}{\frac {1}{(2\pi )^{4}}}\int d^{4}p\,{\frac {e^{-ip(x-y)}}{(p_{0}-i\varepsilon )^{2}-{\vec {p}}^{2}-m^{2}}}=-{\frac {\Theta (y^{0}-x^{0})}{2\pi }}\delta (\tau _{xy}^{2})+\Theta (y^{0}-x^{0})\Theta (\tau _{xy}^{2}){\frac {mJ_{1}(m\tau _{xy})}{4\pi \tau _{xy}}}.}
This expression can also be expressed in terms of the vacuum expectation value of the commutator of the free scalar field. In this case,
G adv ( x , y ) = i ⟨ 0 | [ Φ ( x ) , Φ ( y ) ] | 0 ⟩ Θ ( y 0 − x 0 ) . {\displaystyle G_{\text{adv}}(x,y)=i\langle 0|\left[\Phi (x),\Phi (y)\right]|0\rangle \Theta (y^{0}-x^{0})~.}
Feynman propagator
A contour going under the left pole and over the right pole gives the Feynman propagator, introduced by Richard Feynman in 1948. This choice of contour is equivalent to calculating the limit
G F ( x , y ) = lim ε → 0 1 ( 2 π ) 4 ∫ d 4 p e − i p ( x − y ) p 2 − m 2 + i ε = { − 1 4 π δ ( τ x y 2 ) + m 8 π τ x y H 1 ( 1 ) ( m τ x y ) τ x y 2 ≥ 0 − i m 4 π 2 − τ x y 2 K 1 ( m − τ x y 2 ) τ x y 2 < 0. {\displaystyle G_{F}(x,y)=\lim _{\varepsilon \to 0}{\frac {1}{(2\pi )^{4}}}\int d^{4}p\,{\frac {e^{-ip(x-y)}}{p^{2}-m^{2}+i\varepsilon }}={\begin{cases}-{\frac {1}{4\pi }}\delta (\tau _{xy}^{2})+{\frac {m}{8\pi \tau _{xy}}}H_{1}^{(1)}(m\tau _{xy})&\tau _{xy}^{2}\geq 0\\-{\frac {im}{4\pi ^{2}{\sqrt {-\tau _{xy}^{2}}}}}K_{1}(m{\sqrt {-\tau _{xy}^{2}}})&\tau _{xy}^{2}<0.\end{cases}}}
Here, H1(1) is a Hankel function and K1 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,
G F ( x − y ) = − i ⟨ 0 | T ( Φ ( x ) Φ ( y ) ) | 0 ⟩ = − i ⟨ 0 | [ Θ ( x 0 − y 0 ) Φ ( x ) Φ ( y ) + Θ ( y 0 − x 0 ) Φ ( y ) Φ ( x ) ] | 0 ⟩ . {\displaystyle {\begin{aligned}G_{F}(x-y)&=-i\langle 0|T(\Phi (x)\Phi (y))|0\rangle \\[4pt]&=-i\left\langle 0|\left[\Theta (x^{0}-y^{0})\Phi (x)\Phi (y)+\Theta (y^{0}-x^{0})\Phi (y)\Phi (x)\right]|0\right\rangle .\end{aligned}}}
This expression is Lorentz invariant, as long as the field operators commute with one another when the points x and y are separated by a spacelike interval. The usual derivation is to insert a complete s