lecture 17 april 1, 2010 physics 504, shapiro
TRANSCRIPT
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Lecture 17 April 1, 2010
Canonical Momentum Density
We have seen that in field theory the Lagrangian is anintegral of the Lagrangian density
L(φi, ∂φi/∂xν , xξ)
and the equations of motion come from the functionalderivatives of L with respect to the local values of thefields, but because the Lagrangian density is local, these
are given by∂L
∂φi(xν)and
∂L∂(∂ρφi(xν))
, which are
functions of xµ. The Euler-Lagrange equations involvednot a total momentum but a momentum density. For ascalar field φj this would be
Pµj (xρ) =∂L
∂∂µφj
∣∣∣∣xρ.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
For electromagnetism we have not a scalar but four fieldsAν , so we have four 4-vector fields
P µα :=
∂L
∂
(∂Aα(~x, t)∂xµ
) .Last time we saw that the lagrangian density for theelectromagnetic fields is
L = − 116π
FµνFµν −1cJµA
µ,
so the canonical momentum densities are
P µα :=
∂L∂(∂Aα(~x, t)/∂xµ)
= − 14πFµα,
because, as we saw last time, only the F 2 term dependson ∂Aα/∂xµ.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
The Stress (Energy-Momentum) Tensor
Discrete mechanics: H =∑
i Piqi − L, qi → PiField theory: Hamiltonian density
H(~x) :=∑i
Pi(~x)φi(~x)− L(~x) =∑i
∂L∂(∂φi/∂x0)
∂φi∂x0− L.
Time has been picked out. More generally, let
Tµν =∑i
∂L∂(∂φi/∂xµ)
∂φi∂xν− δµνL.
This object goes by the names energy-momentumtensor or stress-energy tensor or canonical stresstensor, and we see the hamiltonian density is the 00component of this tensor.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
T µν for electromagnetism
For electromagnetism, φi is replaced by Aλ,
Tµν =∂L
∂(∂Aλ/∂xµ)∂Aλ
∂xν− δµνL.
The first factor in the first term is
∂L∂(∂Aλ/∂xµ)
= − 14πFµλ,
so our first (tentative) expression for the energymomentum tensor is
Tµν = − 14π
(Fµλ
∂Aλ
∂xν− 1
4ηµνF
αβFαβ
).
This expression has good and bad properties!
We have seen T 00 is the Hamiltonian density.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
∫T 0i = P i
If T 00 is the energy density, is T 0j the density ofmomentum?
T 0i =1
4πF0λ∂iA
λ =1
4πEj∂iAj =
14π
(~E × ~B +
(~E · ~∇
)~A)i.
Poynting tells us the first term is the correct expression.The second term is unwanted, and also not gaugeinvariant.But we haven’t included charges in the momentum, and ifno charges, ~∇ · ~E = 0,
(~E · ~∇
)~Ai = ~∇ · (Ai ~E)−Ai~∇ · ~E is
a total derivative, and won’t affect the total momentum.
So we do have the good property∫d3xT 0µ = Pµ, the
total momentum. But we don’t have the right density.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Conservation of MomentumIf L has no explicit dependence on xµ, we can show∂µT
µν = 0, where ∂µ is the stream derivative. For
∂µTµν =
∑i
(∂µ
∂L∂(∂µφi)
)∂νφi+
∑i
∂L∂(∂µφi)
∂µ∂νφi−∂νL.
The derivative in the last term is given by the chain rule
−∂νL = −∑i
∂L∂φi
∂νφi −∑i
∂L∂ (∂µφi)
∂ν∂µφi
so
∂µTµν =
∑i
(∂µ
∂L∂(∂µφi)
− ∂L∂φi
)∂νφi
and the parenthesis vanishes by the equations of motion.Thus we have
∂µTµν = 0. (1)
This is a good thing.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Non-symmetry is a bad thingNote: this Tµν 6= T νµ. This is a problem. We expect theangular momentum density to be given by εijkxjT 0k, butthat requires Tµν to be symmetric.So our Tµν is good at giving the total momentum andbeing conserved, but bad in not being symmetric orgauge-invariant. Need modification keeping goodproperties but changing bad ones.If we have a tensor ψρµν = −ψµρν (antisymmetric in firsttwo indices) and we add ∂ρψ
ρµν to Tµν ,
∆ (∂µTµν) = ∂µ∂ρψρµν = 0,
so the new Tµν is also conserved. Furthermore,∫d3x∆T 0ν =
∫d3x∂ρψ
ρ0ν =∫d3x∂jψ
j0ν =∫Snjψ
j0ν → 0
for surface S →∞. So adding ∂ρψρµν keeps all the goodproperties.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Improving T µν
So consider ψρµν = AνFµρ/4π, and adding
14π∂ρ (AνFµρ) =
14π
(∂ρAν)Fµρ
because ∂ρFµρ = 0 in the absence of a source Jµ. But thisis just what we need to add to Tµν to make
Θµν = Tµν+1
4πFµρ∂ρA
ν = − 14π
(FµρF νρ −
14ηµνFαβFαβ
).
This expression has all the good properties and is alsogauge invariant and symmetric. Furthermore,
Θ0i = − 14πF 0jF ij =
14πEjεijkBk =
14π
( ~E × ~B)i,
the correct momentum density or energy flux, as given byPoynting.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Ambiguities in L− 1
16πFµνFµν is gauge invariant, but the interaction term
in the action, −1c
∫d4xJµ(xρ)Aµ(xρ) is not, so L is not a
unique function of the physical state ( ~E and ~B and Jµ).Is there an ambiguity in the action under a gaugetransformation Aµ → A′µ = Aµ + ∂µΛ? This adds a pieceto the action ∆A = −(1/c)
∫d4xJµ∂µΛ. But∫
d4xJµ∂µΛ =∫Snµ J
µ Λ︸ ︷︷ ︸−→S→∞
0
−∫d4xΛ ∂µJµ︸ ︷︷ ︸
0
,
so this will not affect the action.More generally, adding a total divergence to thelagrangian density in a field theory, like adding a totaltime derivative in a particle theory, does not affect theequations of motion, and is irrelevant to the physics.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Θµν with currentsThe energy-momentum tensor of the electromagnetic fieldis
ΘµνEM = − 1
4π
(FµρF νρ −
14ηµνFαβFαβ
),
and is conserved (∂µΘµνEM = 0 if there are no sources.
What if there are?
4π∂µΘµν = ∂µ
(FµρF ν
ρ +14ηµνFαβFαβ
)= (∂µFµρ)F ν
ρ + Fµρ∂µFν
ρ +12Fαβ∂νFαβ
=4πcJρF ν
ρ +12Fαβ
(∂αF
νβ − ∂βF
να + ∂νFαβ
)=
4πcJρF ν
ρ +12Fαβηνρ (∂αFβρ + ∂βFρα + ∂ρFαβ)︸ ︷︷ ︸
=0 as dF=ddA=0
=4πcJρF ν
ρ 6= 0.
Not conserved!
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
P νEM
is not conserved
Thus the total 4-momentum of the electromagnetic field
P νEM =1c
∫d3xΘ0ν(~x),
is not conserved, but rather
dP νEM
dt=
1c
d
dt
∫d3xΘ0ν(~x) =
∫d3x ∂0Θ0ν(~x)
=1c
∫d3xJρ(~x)F ν
ρ (~x)− 1c
∫d3x∂iΘiν
=1c
∫d3xJρ(~x)F ν
ρ (~x),
as the second term is the integral of a divergence.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Total Momentum is conserved
Consider a charged particle of mass mi, charge qi at point~xi(t). Its mechanical 4-momentum changes by
dP ν(i)
dt=
1γi
dP ν(i)
dτ=
1γi
qicF νρ(~xi)U
ρi .
This particle corresponds to a 4-current
Jρ = (cρ, ~J) = (cqiδ3(~x− ~xi), qiuiδ3(~x− ~xi)= qiγ
−1i Uρi δ
3(~x− ~xi).
Plugging this into our expression for the change in themomentum of the electromagnetic field, we have
dP νEM
dt=qic
∫d3xF ν
ρ (~x)γ−1i Uρi δ
3(~x−~xi) = − qicγi
F νρ(~xi)Uρi ,
and the total momentum, P νEM + P ν(i) is conserved.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Equations of Motion for Aµ
Euler-Lagrange tell us
∂σFσµ = ∂σ∂
σAµ − ∂µ∂σAσ =4πcJµ.
If we knew ∂σAσ = 0 (the Lorenz condition), we could
discard second term, and have
∂σ∂σAµ =
4πcJµ,
which has solutions given by1) a particular solution, given in terms of the Green’sfunction on J , and2) an arbitrary solution of the homogeneous waveequation ∂σ∂
σAµ = 0. The homogeneous solution is∑~k
(Aµ~k+
ei~k·~x−iω~kt +Aµ~k−
ei~k·~x+iω~kt
),
where ω = c|~k|, where ω = c|~k|.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
But we assumed the Lorenz condition, which constrainsthe coefficients ωA0
~k±∓ ~k · ~A~k± = 0. These are the
solutions for an electromagnetic wave in empty space.
Without imposing the Lorenz condition,
∂σ∂σAµ − ∂µ∂σAσ = 0,
which is inadequate to determine the evolution of Aµ(~x, t)in time. Fourier transform:kσk
σAµ(kν)− kµkσAσ(kν) = 0, which is not fourindependent equations, because dotting with kρ gives
kσkσkµA
µ(kν)− kµkµkσAσ(kν) = (k2 − k2)kρAρ(kν) = 0,
telling us nothing about Aρ(kν). Euler-Lagrange onlydetermine the components transverse to k.This is gauge invariance again. No physics constrains thegauge transformation Λ(~x, t) in the future, so Aµ isunderdetermined.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Solving the inhomogeneous equationBut we are free to impose the Lorenz condition. Let’s doso.
Now we turn to the inhomogeneous equation
Aµ = ∂β∂βAµ =
4πcJµ,
with the solution
Aµ(x) =4πc
∫d4x′D(x, x′)Jµ(x′),
where D(x, x′) is a Green’s function for D’Alembert’sequation
xD(x, x′) = δ4(x− x′).We are interested in solving this in all of spacetime. Noboundaries, translation invariance, soD(x, x′) = D(x− x′) = D(z). Solve by Fourier transform:
write D(z) =1
(2π)4
∫d4kµD(kµ)e−ikµz
µ.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Fourier transformed equationAs δ4(zµ) = 1
(2π)4
∫d4ke−ikµz
µ, we have k2D(kµ) = −1, so
the solution for the Green’s function is
D(kµ) = − 1k2, and D(zµ) = − 1
(2π)4
∫d4k
e−ikµzµ
k2.
As δ4(zµ) = 1(2π)4
∫d4ke−ikµz
µ, the solution for the
Green’s function is
D(kµ) = − 1k2, and D(zµ) = − 1
(2π)4
∫d4k
e−ikµzµ
k2.
Looks like what we did for Poisson, but here k2 = 0 ismore difficult, as it requires only k2
0 = ~k 2, not ~k = 0.For Poisson, trouble from ~k = 0 gives ambiguity ofψ = V0 + ~r · ~C, a uniform ~E and ambiguous constant in Φ.For wave equation: arbitrary waves satisfying free waveequation.Deform the integration path to resolve the singularities.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Disambiguate with Contour choice
Clarify ambiguity by specifying how to avoid thesingularities and writing
D(z) = − 1(2π)4
∫d3kei
~k·~z∫
Γdk0
e−ik0z0
k20 − |~k|2
.
Specifying contourΓ’s avoidance of thepoles at k0 = ±|~k|.Three such con-tours are shown.Integrand analyticexcept at poles sothe contours maybe deformed whileavoiding the poles.
r
aF
0k
The retarded (r), advanced(a), and Feynman (F ) con-tours for defining the Green’sfunction.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Retarded Green’s functionConsider D for contour r. If source at x′ = 0, and we lookfor A later, z0 > 0. Close contour in lower half plane,where
∣∣∣e−ik0z0∣∣∣ = e−|Im k0|z0 −→|k|→∞
0, so this semicircle
adds nothing. So D = −2πi times the sum of theresidues. minus because clockwise. The residues are
Resk0=|~k|
e−ik0z0
(k0 + |~k|)(k0 − |~k|)+ Resk0=−|~k|
e−ik0z0
(k0 + |~k|)(k0 − |~k|)
=e−i|
~k|z0
2|~k|+ei|~k|z0
−2|~k|= −isin(|~k|z0)
|~k|.
But if z0 < 0 close in upper half plane, no residues,D = 0, so all together
Dr(z) =Θ(z0)(2π)3
∫d3kei
~k·~z sin(|~k|z0)
|~k|.
D is rotationally invariant, so we may choose the Northpole along ~z using spherical coordinates.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
We get
Dr(z) =Θ(z0)(2π)2
∫ ∞0
k2 dk dθ sin θ eikR cos θ sin(kz0)k
=Θ(z0)2π2R
∫ ∞0
dk sin(kR) sin(kz0),
where R = |~z|.This is the retarded Green’s function aka causal, as theeffects on Aµ of the source are felt only after the sourceacts.
The contour a gives the advanced Green’s function usefulonly if you want to configure an incoming field whichwould magically be totally dissolved by a given source.
Finally the contour F gives the Feynman propagator,which is used in quantum field theory.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Simplifying DWe may simplify Dr by noting
sin(kR) sin(kz0) =12[cos(k(R− z0))− cos(k(R+ z0))
]=
14
[ei(z0−R)k − ei(z0+R)k + ei(z0−R)(−k)
+ei(z0−R)(−k)]
so Dr(z) =Θ(z0)8π2R
∫ ∞−∞
dk[ei(z0−R)k − ei(z0+R)k
]=
Θ(z0)4πR
[δ(z0 −R)− δ(z0 +R)]
=Θ(z0)4πR
δ(z0 −R),
where the second δ was dropped because both z0 and Rare positive. So the Green’s function only contributeswhen the source and effect are separated by a lightlikepath, with ∆z0 = |∆~z|.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Full solution for Aµ
So how do we describe the field when we know what thesources are throughout space-time? We can use any of theGreen’s functions to get the inhomogeneous contribution,and then allow for an arbitrary solution of thehomogeneous equation. Thus we can write
Aµ = Aµin(x) +4πc
∫d4x′Dr(x− x′)Jµ(x′)
= Aµout(x) +4πc
∫d4x′Da(x− x′)Jµ(x′).
If the sources are confined to some finite region ofspace-time, there will be no contribution from Dr at timesearlier than the first source, and Aµin(x) describes thefields before that time. Also after the last time that thesource influences things, the field will be given by Aµout(x)alone.
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Radiation Field.
Of course the source may be persistent, for example ifthere is a net charge, but we may often consider that theeffect of the source is confined to the change from Aµin(x)to Aµout(x). Then we define the radiation field to be
Aµrad(x) = Aµout(x)−Aµin(x) =4πc
∫d4x′D(x− x′)Jµ(x′),
where D(z) := Dr(z)−Da(z).
Physics 504,Spring 2010Electricity
andMagnetism
Shapiro
CanonicalMomentum,Tmunu
CanonicalMomentum forE&M
The Stress(Energy-Momentum)Tensor
Stress-Energyfor E&M
Ambiguities inLagrangiandensity
Θµν withcurrents
Equations ofMotion forAµ
Green’sfunction forwave equation
Better expression for Jµ from charges
The expression we wrote earlier for the current density ofa point charge,
Jρ = qiγ−1i Uρi δ
3(~x− ~xi)
can be written in this four-dimensional language as
Jρ(xµ) = qi
∫dtδ(t− x0/c)γ−1
i Uρi δ3(~x− ~xi(t))
= qic
∫dτδ4(xµ − xµi (τ))Uρi ,
where τ measures proper time along the path of theparticle.