Loading [MathJax]/jax/output/CommonHTML/jax.js







String Theory: Chapter 5 Problem Set Solutions for Barton Zwiebach's "A First Course in String Theory"


Warning: These are my own solutions for the given problems. If you are a student, use these at your own risk. They have not been subjected to grading in a university setting. Your professor may require more explicit statements, or other details worked out in the calculations.




Quick Calculations

Quick Calculation 5.1: Calculate the Variation δSnr for a Galilean Boost

Quick Calc 5.1: Solution

Quick Calculation 5.2: Show that you can Reparameterize dpμdτ=0 as dpμdτ=0 for τ(τ).

Quick Calc 5.2: Solution


Problems

Problem 5.1: Point Particle Equation of Motion and Reparameterizations

Problem 5.1: Solution

Problem 5.2: Particle Equation of Motion with Arbitrary Parameterization

Problem 5.2: Solution

Problem 5.3: Four-Current of a Charged Point Particle in D Dimensions

Problem 5.3 (a): Solution

Problem 5.3 (b): Solution

Problem 5.4: Hamiltonian for a Nonrelativistic Charged Particle ()

Problem 5.4 (a): Solution

Problem 5.4 (b): Solution

Problem 5.4 (c): Solution

Problem 5.5: Equations of Motion for a Charged Point Particle

Problem 5.5: Solution

Problem 5.6: Electromagnetic Field Dynamics with a Charged Particle (*) ()

Problem 5.6: Solution

Problem 5.7: Point Particle Action in Curved Space (*)

Problem 5.7: Solution

(*) Asterisk indicates solution may be fuzzy or lacking sufficient rigor.
(**) Double asterisk indicates solution may be significantly flawed.
() Dagger indicates the problem is marked in the textbook as referenced again in later chapters.
(Unmarked indicates solution should be correct other than minor quibbles.)


Quick Calculations

Quick Calculation 5.1:

Problem Statement: Calculate explicitly the variation of the action Snr under a boost.

Equation (5.2): Snr=Lnrdt=12mv2(t)dt, v2vv, v=dxdt, v=|v|

Solution:

In context with the textbook, the problem means the variation δ of the non-relativistic action Snr, under a Galilean boost vv+v0, where v0 is a constant:

δSnr=δLnrdt=12mv2new(t)dt
δSnr=δ12mvv0dt
δSnr=mvv0dt

The fraction 12 is cancelled by the symmetry of v,v0 varying by the same amount. We replace v2 with vv0, the magnitudes of each velocity vector modified by the difference in their directions. Since v0 is constant, it will not vary over the interval. Therefore that term can be pulled outside the integral:

δSnr=mv0vdt
δSnr=mv0dxdtdt
δSnr=mv0dx

Therefore, δSnr=0, because it is zero at the end points xi,xf. The variation of the action is thus zero under a Galilean boost. (Q.E.D.)


Quick Calculation 5.2:

Problem Statement: Show that dpμdτ=0 implies dpμdτ=0 for an arbitrary parameter τ(τ). What should be true about dτdτ for τ to be a good parameter when τ is?

Solution:

This is just a straight-forward consequence of the chain rule, which implicitly is a change of basis or coordinates:

dpμdτ=dpμdτdτdτ=0
dpμdτ=0

This holds for arbitrary τ(τ) provided dτdτ is continuous, such that it obeys the Inverse Function Theorem.



Problems

Problem 5.1

Problem Statement: Point particle equation of motion and reparameterizations.

If the path of a point particle is parameterized by proper time, the equation of motion (5.29). Consider now a new parameter τ=f(x). Find the most general function f which (5.29) implies: d2xμdτ2=0.

Solution:

The "most general" function f would be f(s), where f is a function of the spacetime interval s, rather than a parameter such as τ. This would follow from the textbook saying the action can be summed up by ds itself without using parameters. But by "general" the problem seems to mean for some arbitrary x:

τ=f(x)

This notation is awkward. Remember μ is an index, not an exponent, and this is not μ. Then by the chain rule:

dxμdx=dxμdτdτdx
dxμdx=dxμdτdfdx

Then we take the derivative with product rule:

d2xμdx2=ddx(dxμdτdfdx)
d2xμdx2=d2xμdτ2(dfdx)(dfdx)+dxμdτd2fdx2=0

The extra dfdx comes from having to do another chain rule to do the x derivative on dxμdτ. The problem gives us d2xμdτ2=0 as a condition:

0+dxμdτd2fdx2=0

dxμdτ0 in general. Therefore it follows that: d2fdx2=0, which integrates out to the function:

f=ax+b; a, b are arbitrary constants. (Q.E.D.)


Problem 5.2

Problem Statement: Particle equation of motion with arbitrary parameterization.

Vary the point particle action (5.15) to find a manifestly reparameterization invariant form of the free particle equation of motion.

Solution:

Equation (5.15): S=mctftiημνdxμdτdxνdτdτ

For the sake of context, "manifestly" reparameterization invariant in the textbook, Section 5.2, refers to the action S having the same form after changing the parameter τ to τ using the chain rule. In Section 5.3 the action is varied δS=mcPδ(ds), which ultimately leads to the general condition: d2xμds2=0. Therefore we can begin at this point.

You cannot generally replace ds with dτ in this equation, d2xμdτ20 for any arbitrary parameter τ. The equivalent expression must be expanded in terms of the parameter using the chain rule, and then manipulated into a form that shows it is manifestly reparameterization invariant:

dxλds=dxλdτdτds

d2xλds2=dds(dxλdτdτds)
d2xλds2=d2xλdτ2(dτds)(dτds)+dxλdτd2τds2=0
d2xλds2=d2xλdτ2(dτds)2+dxλdτd2τds2=0

Then we also know that ds is parameterized along the worldline with increments dτ:

ds=ημνdxμdτdxνdτdτ
(ds)2=ημνdxμdτdxνdτ(dτ)2

We can change the last term to get it all in terms of (dτds)2, making use of the chain rule on the last term:

d2xλdτ2(dτds)2+12dxλdτdds(dτds)2=0

We can then substitute in the definition of dτds from the above relations, but find it is inconveniently reciprocal:

d2xλdτ2(ημνdxμdτdxνdτ)1+12dxλdτdds(ημνdxμdτdxνdτ)1=0
d2xλdτ2(ημνdxμdτdxνdτ)1+12dxλdτdds(ημνdxμdτdxνdτ)1=0

This can be re-expressed in terms of positive exponents, by switching to the inverted (dτds)1=dsdτ, and getting all the derivatives in terms of the parameter τ:

d2xλdτ2((dτds)1)212dxλdτddτ((dτds)1)2=0
d2xλdτ2(ημνdxμdτdxνdτ)12dxλdτddτ(ημνdxμdτdxνdτ)=0

It is manifestly invariant with the same form under a reparameter change (dτdτ)2:

d2xλds2=d2xλdτ2(ημνdxμdτdxνdτ)12dxλdτddτ(ημνdxμdτdxνdτ)=0
d2xλds2=d2xλdτ2(dτdτ)2(ημνdxμdτdxνdτ(dτdτ)2)12dxλdτddτ(dτdτ)2(ημνdxμdτdxνdτ(dτdτ)2)=0
d2xλds2=d2xλdτ2(ημνdxμdτdxνdτ)12dxλdτddτ(ημνdxμdτdxνdτ)=0 (Q.E.D.)


Problem 5.3

Problem Statement: Current of a charged point particle.

Consider a point particle with charge q whose motion in a D = d + 1-dimensional spacetime is described by functions xμ(τ)={x0(τ),x} , where τ is a parameter. The moving particle generates an electromagnetic current jμ=(cρ,j).

(a) Use delta functions to write expressions for the current components j0(x,t) and ji(x,t).

(b) Show that your answer in (a) arise from the integral representation

jμ(t,x)=qcdτδD(xx(τ))dxμ(τ)dτ.

Here δD(x)δ(x0)δ(x1)...δ(xd).

Solution:

Problem 5.3 (a): Use delta functions to write expressions for the current components j0(x,t) and ji(x,t).

In non-relativistic classical electromagnetism, the charge moves along path x(t) relative to a coordinate system x, and the individual points along that path can be represented as a delta function. This is because the charge is concentrated at that position at that time t, such that the charge density is:

ρ=qδ(xx(t))

Which means the current density j=ρv is:

ρv=qvδ(xx(t))

In relativistic spacetime this path is instead the worldline, and t is replaced by the parameter τ for the proper time, where being stationary to the charges is the charge density and moving relative to the charges is the current density. The time component of the four-current jμ=(cρ,j) is then the charge density, and the spatial components j are the current density:

j0=cρ=cqδ(tct(τ))dτ
ji=j=qdxidτδi(xx(τ))dτ

The nature of delta functions is that, integrated over infinity, they are distributions concentrated at one point: +f(t)δ(tT)=f(T).

Problem 5.3 (b): Show that your answer in (a) arise from the integral representation jμ(t,x)=qcdτδD(xx(τ))dxμ(τ)dτ.

First notice that the δD term is composed of δxμ elements, including the time component:

δD(x)δ(x0)δ(x1)...δ(xd)

This is really a set of D integrals over the delta functions along the worldline, where the c is absorbed into the time component x0:

jμ(t,x)=q[dτδ(x0x0(τ))dx0(τ)dτ][dτδ(x1x1(τ))dx1(τ)dτ]...[dτδ(xdxd(τ))dxd(τ)dτ]

Integrating over the parameter τ gives the charge and current densities at each time point on the worldline. The first integral contains our expression for the charge density at a given time, and the remaining d integrals are our current density definition at each time point:

j0=q[dτδ(x0x0(τ))dx0(τ)dτdτ]=q[cδ(tct(τ))dτ]

ji=qdτδ(xixi(τ))dxi(τ)dτdτ=qdxidτδi(xx(τ))dτ

Thus, those terms arise from the D dimension integral. (Q.E.D.)


Problem 5.4

Problem Statement: Hamiltonian for a nonrelativistic charged particle.

The action for a nonrelativistic particle of mass m and charge q coupled to an electromagnetic field is obtained by replacing the first term in (5.34) by the nonrelativistic action for a free point particle:

S=12mv2dt+qcAμ(x)dxμdtdt

We have also chosen to use time to parameterize the second integral.

(a) Rewrite the action S in terms of the potentials (Φ,A) and the ordinary velocity v. What is the Lagrangian?

(b) Calculate the canonical momentum p conjugate to the position of the particle and show that it is given by p=mv+qcAv

(c) Show that the Hamiltonian for the charged particle is: H=12m(pqcA)2+qΦ.


Solution:

Equation (5.34): S=mcPds+qcPAμ(x)dxμ

Problem 5.4 (a): Rewrite the action S in terms of the potentials (Φ,A) and the ordinary velocity v. What is the Lagrangian?

Recall that: Aμ=(A0,A1,A2,...,An)=(Φ,A) and dxμdt=(dx0dt,dx1dt,dx2dt,dx3dt)=(c,v), where A0=Φ and x0=ct.

Then we can treat Aμ(x)dxμdtdt as an inner product, since the textbook in Section 5.4 says this is the dot product at each time point:

S=12mv2dt+qcAμ(x)dxμdtdt
S=12mv2dt+qc(Φc+Av)dt

Since S=Ldt it follows that the Lagrangian of the nonrelativistic charged particle is defined by the action as:

L=12mv2qΦ+qcAv (Q.E.D.)


Problem 5.4 (b): Calculate the canonical momentum p conjugate to the position of the particle and show that it is given by p=mv+qcA

The canonical momentum p=˙q conjugate to the position of the nonrelativistic charged particle is just the velocity derivative of the Lagrangian:

p=Lv
p=mv+qcvAv

However, this last term is an inner product depending on velocity (i.e. it is a scalar whose value changes linearly with v ), so it is a constant of the partial differentiation:

p=mv+qcAv (Q.E.D.)


Problem 5.4 (c): Show that the Hamiltonian for the charged particle is: H=12m(pqcA)2+qΦ

The Hamiltonian in classical mechanics is the Legendre transformation of the Lagrangian:

H=qipiL

Therefore we can simply write:

H=pvL
H=(mv+qcAv)v(12mv2qΦ+qcAv)

Since Av is a scalar, it is skipped over by the dot product:

H=(mv2+qcAv)(12mv2qΦ+qcAv)
H=12mv2+qΦ

Rather than try to derive H=12m(pqcA)2+qΦ, we will work backwards. At first glance, this is a problem:

(pqcA)2=(mv+qcAvqcA)2

Notice that (pqcA)2 if taken naively is the wrong dimensions. It means subtracting out the A dependence of p, the squaring process implicitly involves a dot product of v:

(pqcA)2=((m+qcAqcA)v)2=(m2vv0)=m2v2

Therefore it follows that this is an equivalent representation of the Hamiltonian H:

H=12m(pqcA)2+qΦ=12mv2+qΦ
H=12m(pqcA)2+qΦ (Q.E.D.)


Problem 5.5

Problem Statement: Equations of motion for a charged point particle.

Consider the variation of the action (5.34) under a variation δxμ(x) of the particle trajectory. The variation of the first term in the action was obtained in Section 5.3. Vary the second term (written more explicitly in (5.33)) and show that the equation of motion is (5.32). Begin your calculation by explaining why

δAμ(x(τ))=Aμxν(x(τ))δxν(x)

Solution:

Equation (5.32): dpμdτ=qcFμνdxνdτ

Equation (5.33): qcPAμ(x(τ))dxμdτ

Equation (5.34): S=mcPds+qcPAμ(x)dxμ


The second term in Equation (5.34) is just a more compact way of writing Equation (5.33):

S=mcPds+qcPAμ(x)dxμ
S=mcPds+qcPAμ(x(τ))dxμdτ

For the first term, which is Equation (5.5), recall the variation δS leads from Equation (5.18) to (5.25):

Equation 5.18: δS=mcδ(ds)
Equation 5.25: δS=tftidτddτ(δxμpμ)tftidτδxμ(τ)dpμdτ

The first term of Equation 5.25 vanishes when evaluated at the end points of the world-line. The second term gives the equation of motion when δS=0:

δS=tftidτδxμ(τ)dpμdτ

Therefore taking the variation of Equation (5.34) yields:

δS=mcPδ(ds)+qcPδ(Aμ(x)dxμ)
δS=tftidτδxμ(τ)dpμdτ+qctftidτ[(δAμ(x))dxμdτ+Aμ(x)(δ(dxμdτ))]

We notice from the problem statement that δAμ(x(τ))=Aμxν(x(τ))δxν(x), which makes sense because it expresses the variation of Aμ with respect to the variation of different coordinates xν. Then we can substitute this into the above equation for δS and suppress all the parameter parentheses for clarity:

δS=tftidτδxμdpμdτ+qctftidτ[(Aμxνδxν)dxμdτ+Aμ(δ(dxμdτ))]

Since we want to get the equation of motion, we have to use the standard trick of the derivative product rule to get the variations all in terms of δx:

Product Rule:
ddτ(Aμδxμ)=dAμdτδxμ+Aμδ(dxμdτ)
Aμδ(dxμdτ)=ddτ(Aμδxμ)dAμdτδxμ

Substituting this in (and recalling that expressions of the form tftiddτ(Aμδxμ)dτ=0 at the end points of δxμ ):

δS=tftidτδxμdpμdτ+qctftidτ[(Aμxνδxν)dxμdτ+ddτ(Aμδxμ)dAμdτδxμ]
δS=tftidτδxμdpμdτ+qctftidτ[(Aμxνδxν)dxμdτdAμdτδxμ]

This is in terms of variations δxν and δxμ. To express it in terms of Aν, we need another Aμxν term:

δS=tftidτδxμdpμdτ+qctftidτ[(Aμxνδxν)dxμdτAμxνdAνdτδxμ]

To get it in terms of only δxμ, we do a coordinate change on the left term: (Aμxνδxν)dxμdτ=Aνxμxμxνdxμdτδxν

δS=tftidτδxμdpμdτ+qctftidτ[(Aνxμδxμ)dxνdτAμxνdAνdτδxμ]

With the dAνdτ term, we just have A variation along the xν coordinates:

δS=tftidτδxμdpμdτ+qctftidτ[(Aνxμδxμ)dxνdτAμxνdxνdτδxμ]
δS=tftidτδxμdpμdτ+qctftidτ[(AνxμAμxν)dxνdτδxμ]

Therefore since Fμν=AνxμAμxν:

δS=tftidτδxμdpμdτ+qctftidτ[Fμνdxνdτδxμ]

δS=tftidτ(dpμdτ+qcFμνdxνdτ)δxμ

Therefore with δS=0 the Euler-Lagrange equation of motion is given by dpμdτ=qcFμνdxνdτ, which is Equation (5.32). (Q.E.D.)


Problem 5.6

Problem Statement: Electromagnetic field dynamics with a charged particle.

The action for the dynamics of both a charged point particle and the electromagnetic field is given by

S=mcPds+qcPAμ(x)dxμ14cdDxFμνFμν

Here dD=dx0dx1...dxd. Note that the action S' is a hybrid; the last term is an integral over spacetime, and the first two terms are integrals over the particle world-line. While included for completeness, the first term will play no role here. Obtain the equation of motion for the electromagnetic field in the presence of the charged particle by calculating the variation of S' under a variation δAμ of the gauge potential. The answer should be equation (3.34), where the current is the one calculated in Problem 5.3. [Hint: to vary Aμ(x) in the world-line action it is useful to rewrite this term as a full spacetime integral with the help of delta functions.]


Solution:

(Warning: This derivation may be essentially correct, but the last step eliminating the 14c term might be an improper handwave.)

Equation (3.34): Fμνxν=1cjμ

Problem 5.3: jμ(t,x)=qcdτδD(xx(τ))dxμ(τ)dτ; δD(x)δ(x0)δ(x1)...δ(xd)


We know from Problem 5.5 that the variation of the charged particle world-line by δAμ yields:

δS=tftidτδxμdpμdτ+qctftidτ[Fμνdxνdτδxμ]

The first term is irrelevant. The second term is the charged particle along its world-line, expressed in terms of Fμν instead of the gauge potential. If we want to convert this into an integral over spacetime in D dimensions, we can use delta functions like in Problem 5.3, which isolate the particle along its position at each time point:

δS=tftidτδxμdpμdτ+qctftidτ[FμνδD(xx(τ))dxνdτδxμ]

Recalling that the c term is swallowed by x0, we see that this second integral is really:

δS=tftidτδxμdpμdτ+tftidτ[Fμν1cjμδxμ]

Then we can replace this into the expression for S' and similarly eliminate c, with the factor of 4 coming from the symmetrical contributions of the μ,ν variations from FμνFμν: (*)

δS=tftidτδxμdpμdτ+tftidτ[Fμν1cjμδxμ]14cdDxFμνFμν
δS=tftidτδxμdpμdτ+tftidτ[Fμν1cjμδxμ]tftidτFμνFμνdxνdτδxμ
δS=tftidτδxμdpμdτ+tftidτ[Fμν(1cjμFμνxν)]δxμ

The problem is not concerned with the first term. The equation of motion for the electromagnetic field in the presence of the charged particle (when δSfield=0 ) is therefore Equation 3.34:

1cjμ=Fμνxν (Q.E.D.)


Problem 5.7

Problem Statement: Point particle action in curved space.

In Section 3.6 we considered the invariant interval ds2=gμν(x)dxμdxν in a curved space with metric gμν(x). The motion of a point particle of mass m on a curved space is studied using the action

S=mcds

Show that the equation of motion obtained by variation of the world-line is

dds[gμρdxμds]=12gμνxρdxμdsdxνds

This is called the geodesic equation. When the metric is constant we recover the equation of motion of a free point particle.


Solution:

(Warning: This derivation may be essentially correct overall, but the index manipulations may have been done too sloppily in reaching the equation of motion.)

Following the argument in Section 5.3 for Minkowski spacetime, we vary δS in terms of δ(ds):

δS=mcδ(ds)

It is more straight forward to instead vary ds2, and use proper time s itself instead of a parameter τ:

δ(ds2)=δ(gμν(x)dxμdxν)
2dsδ(ds)=δ(gμν(x)dxμds)dxνdsds2
2dsδ(ds)=(δ(gμν)dxμdsdxνds+gμνδ(dxμds)dxνds)ds2
2dsδ(ds)=(gμνxρdxμdsdxνdsδxρ+gμν(dδxμds)dxνds)ds2

The metric gμν is dynamical, so it cannot be skipped over when doing the variation. We replace the right term with a product rule:

Product Rule:
dds(δxμdxνds)=dδxμdsdxνds+δxμddsdxνds
dds(δxμdxνds)δxμddsdxνds=dδdxμdsdxνds

Inserting into the equation above gives the following, but the dds(δxμdxνds) vanishes at the end points of the world-line:

2dsδ(ds)=(gμνxρdxμdsdxνdsδxρ+gμν[dds(δxμdxνds)δxμddsdxνds])ds2
2dsδ(ds)=(gμνxρdxμdsdxνdsδxρ+gμνδxμddsdxνds)ds2

Replace these coordinate variations with a differential amount of the spacetime interval instead:

2dsδ(ds)=(gμνxρdxμdsdxνds+gμνd2xνds2)ds2δ(ds)

The second term is just the constancy of the momentum along the world-line, like in Equation 5.29 it vanishes, leaving the spacetime curvature term (the problem is specifically asking about the effect of the world-line variation):

2dsδ(ds)=(gμνxρdxμdsdxνds)ds2δ(ds)
dsδ(ds)=(12gμνxρdxμdsdxνds)ds2δ(ds)

This is consistent with the problem statement, which implies the form:

dds[gμρdxμds]=(ddsgμρ)dxμds+gμρdds(dxμds)
dds[gμρdxμds]=(ddsgμρ)dxμds+gμρ(d2xμds2)

Thus, when we consider the condition d2xμds2=0, leaving only the spacetime curvature term:

dds[gμρdxμds]=12gμνxρdxμdsdxνds

Therefore plugging that result into δS=0 simply gives this as the equation of motion. It is the geodesic because it is merely following the spacetime curvature, while if the metric is constant, this term vanishes and the action reduces to the free particle term for the equation of motion where d2xμds2=0. (Q.E.D.)



Up One Level: String Theory
Last updated: 12/25/2020