phy 2403f: problem set #1 due: wednesday, sept. 26,...
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PHY 2403F: PROBLEM SET #1 Due: Wednesday, Sept. 26, 2012
1. A Lorentz transformation xµ → x′µ = Λµνxν is such that it preserves the Minkowski metric ηµν ,
meaning that ηµνxµxν = ηµνx
′µx′ν for all x. Show that this implies that
ηµν = ηστΛσµΛτν .
Use this result to show that an infinitesimal transformation of the form
Λµν = δµν + ωµν
is a Lorentz transformation when ωµν is antisymmetric i.e. ωµν = −ωνµ. (Note that there an antisym-metric 4× 4 matrix has six parameters, as does a Lorentz transformation - 3 rotations and 3 boosts -so the counting works out).
Write down the matrix form for ωµν that corresponds to a rotation through an infinitesimal angle θabout the x3-axis. Do the same for a boost along the x1-axis by an infinitesimal velocity v.
2. In a theory of a single harmonic oscillator, define the coherent state |z〉 by
|z〉 = Neza†|0〉
where z is a complex number and N is a real positive constant, chosen such that 〈z|z〉 = 1. Coherentstates of the SHO are interesting because they smoothly interpolate between the classical and quantumworlds: for large z they become indistinguishable from classical oscillators. (Similarly, coherent statesof photons correspond to electromagnetic waves in the limit of large numbers of photons). They alsogive you good practice at manipulating creation and annihilation operators. As usual, H = ω(p2+q2)/2and the raising and lowering operators a and a† are defined as a = (q + ip)/
√2, a† = (q − ip)/
√2,
where the usual momentum P and position X are P =√µωp, X = q/
√µω.
(a) Find N .
(b) Compute 〈z|z′〉 and 〈z|H|z〉.(c) Show that |z〉 is an eigenstate of the annihilation operator a and find its eigenvalue. (Don’t be
disturbed by finding non-orthogonal eigenstates with complex eigenvalues; a is not a Hermitianoperator.)
(d) The statement that |z〉 is an eigenstate of a with well-known eigenvalue is, in the q-representation,a first-order differential equation for 〈q|z〉, the position-space wave-function of |z〉. Solve thisequation and find and sketch the wave-function. (Don’t bother with normalization factors here).
(e) Consider the time evolution of the system (work in the Heisenberg representation). Show that forreal z (this just sets the initial conditions) the expectation values of the position and momentumof the coherent state satisfy
〈z|X|z〉 =
√2
µωz cosωt
〈z|P |z〉 = −√
2µωz sinωt
By contrast, what are the expectation values of X and P for an oscillator in any state of definiteexcitation number n? Using a sketch, describe the behaviour of the wavepacket as a function oftime.
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3. A string of length a, mass per unit length σ and under tension T is fixed at each end. The Lagrangiangoverning the time evolution of the transverse displacement y(x, t) is
L =
∫ a
0
dx
[σ
2
(∂y
∂t
)2
− T
2
(∂y
∂x
)2]
where x identifies position along the string from one end point. By expressing the displacement as asine series Fourier expansion of the form
y(x, t) =
√2
a
∞∑n=1
sin(nπx
a
)qn(t)
show that the Lagrangian becomes
L =
∞∑n=1
[σ
2q2n −
T
2
(nπa
)2q2n
].
Derive the equations of motion. Hence, show that the string is equivalent to an infinite set of decoupledharmonic oscillators, and find their frequencies.
4. Even though we have set h = c = 1, we can still do dimensional analysis because we still have one unitleft, mass (or 1/length). In d space-time dimensions (1 time and d−1 space), what is the dimension inmass units of a canonical free scalar field, φ? (Work it out from the equal-time commutation relations.)Still in d dimensions, the Lagrange density for a scalar field with self-interactions might be of the form
L =1
2(∂µφ)
2 −∑n≥2
anφn.
What is the dimension (again in mass units) of the Lagrange density? The action? The coefficientsan? (as a check, whatever the value of d, a2 had better have the dimensions of mass2).
5. Show that replacing the lagrange density L = L(φa, ∂αφa) by
L′ = L+ ∂µΛµ(x),
where Λµ(x), µ = 0, . . . , 3, are arbitrary functions of the fields φa(x), does not alter the equationsof motion. Thus, when constructing the most general lagrange density for a field, we do not have toinclude terms which are total derivatives. This will simplify life.
6. (You can probably find this worked out in lots of places, but it’s good practice with working withfour-vectors, so I strongly encourage you to do it yourself!) Consider the Lagrangian for a real vectorfield Aµ:
L = − 12 [∂αAβ(x)]
[∂αAβ(x)
]+ 1
2 [∂αAα(x)]
[∂βA
β(x)]
+ µ2
2 Aα(x)Aα(x).
(a) Show that this leads to the field equations[gαβ(2 + µ2)− ∂α∂β
]Aβ(x) = 0,
and that the field Aα(x) satisfies the Lorentz condition
∂αAα(x) = 0.
(NB: If you are not careful with your indices and Einstein summation convention you will getyourself hopelessly messed up here.)
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(b) Consider the limiting case of a massless field, µ→ 0, and identify the field Aµ with the scalar and
vector potentials of electrodynamics: Aµ = (ϕ, ~A), where
~E = −∇ϕ− ∂ ~A
∂tand ~B = ∇× ~A.
Show that the field equations reproduce two of Maxwell’s equations, and that the other two holdas identities given the definitions of ~E and ~B in terms of ϕ and ~A.
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PHY 2403F: PROBLEM SET #2 Due: Wednesday, October 10, 2012
1. In a quantum theory, most observables do not have a definite value in the ground state of the theory.For a general observable, A, a reasonable measure of this quantum spread in the ground-state value ofA is given by the ground-state variance of A, defined by
varA ≡ 〈(A− 〈A〉2〉 = 〈A2〉 − 〈A〉2,
where the brackets indicate the ground-state expectation value. In the theory of a free scalar field ofmass µ, define the observable
A(a) ≡ π−3/2a−3∫d3~xφ(~x, 0)e−~x
2/a2 ,
where a is some length. Note that the Gaussian has been normalized so that its space integral is 1; thusthis is a smoothed-out version of the field averaged over a region of size a. Express the ground-state(vacuum) variance of A(a) as an integral over a single variable. You are not required to evaluate thisintegral except in the limiting cases of very small a and very large a. In both of these limits you shouldfind
varA(a) = αaβ + ...,
where α and β are constants that you are to find, and the triple dots denote terms negligible in thelimit compared to the term displayed. You should find that varA(a) goes to zero for large a, while itblows up for small a; speaking somewhat loosely, on large scales the average field is almost a classicalvariable, while on small scales quantum fluctuations are enormous.
2. In class we derived a general formula for the components of the linear momentum as an integral of theenergy-momentum tensor. Explicitly evaluate these for a free scalar field to find expressions for thelinear momentum as a function of creation and annihilation operators. Check your answer by seeingthat it agrees with the expression obtained earlier in class. (Normal-order freely).
3. A class of interesting theories is invariant under the scaling of all lengths by
xµ → x′µ = e−αxµ and φ(x)→ φ′(x) = eαdφ(eαx).
Here d is called the scaling dimension of the field. Consider the theory of a real scalar field defined bythe Lagrangian
L =1
2∂µφ∂
µφ− 1
2m2φ2 − gφp.
Find the scaling dimension d such the scaling is a symmetry of the derivative term ∂µφ∂µφ. For what
values of m and p is the scaling a symmetry of the theory? How do these conclusions change for ascalar field living in an (n+ 1)-dimensional spacetime instead of a 3+1-dimensional spacetime?
In 3+1 dimensions, use Noether’s theorem to construct the conserved current Jµ associate with scalinginvariance as a function of the fields and their derivatives.
4. Maxwell’s Lagrangian for the electromagnetic field is
L = −1
4FµνF
µν
where Fµν = ∂µAν − ∂νAµ, and Aµ is the 4-vector potential. (This is just the Lagrangian for a vectorfield from the first problem set with m = 0 and some slick new notation). Show that L is invariantunder the gauge transformation
Aµ → Aµ + ∂µξ
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where ξ = ξ(x) is a scalar field with arbitrary (differentiable) dependence on x. We will have muchmore to say about this later on in the course. Use Noether’s theorem and the spacetime translationalinvariance of the action to construct the energy-momentum tensor Tµν for the electromagnetic field.Show that the resulting object is neither symmetric nor gauge invariant. Consider a new tensor givenby
Θµν = Tµν − F ρµ∂ρAν .
Show that this object also defines four conserved currents. Moreover, show that it is symmetric, gaugeinvariant, and traceless. Comment: Tµν and Θµν are both equally good definitions of the energy-momentum tensor. However, Θµν clearly has nicer properties. Moreover, if you couple Maxwell’sLagrangian to general relativity, then it is Θµν which appears in Einstein’s equations.
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PHY 2403F: PROBLEM SET #3 Due: Friday, October 26, 2012
1. Explicitly verify Wick’s theorem for the case of three scalar fields:
T (φ(x1)φ(x2)φ(x3)) = : φ(x1)φ(x2)φ(x3) : +φ(x1)DF (x2 − x3)
+φ(x2)DF (x3 − x1) + φ(x3)DF (x1 − x2). (1)
2. Let us return to the problem of the creation of Klein-Gordon particles by a classical source. Recallthat this process can be described by the Hamiltonian
H = H0 +
∫d3x (−ρ(t, ~x)φ(x))
where H0 is the free Klein-Gordon Hamiltonian, φ(x) is the Klein-Gordon field, and ρ(x) is a c-number(i.e. classical number, not an operator) scalar function. We found that, if the system is in the vacuumstate before the source is turned on, the source will create a mean number of particles
〈N〉 =
∫d3p
(2π)31
2Ep|ρ(p)|2
where
ρ(p) =
∫d4y eip·yρ(y)
evaluated at 4-momenta p such that p2 = m2. In this problem we will solve this theory using theperturbation techniques we are developing, verifying the result above for the mean number of particles,and extracting more detailed information about the final state.
(a) Show that the probability that the source creates no particles is given by
P (0) =
∣∣∣∣〈0|T exp[i
∫d4xρ(x)φI(x)]
|0〉∣∣∣∣2 .
(b) Now let us do perturbation theory in powers of ρ. Evaluate the term in P (0) of order ρ2, andshow that P (0) = 1− λ+O(ρ4), where λ equals the expression given above for 〈N〉.
(c) Show that you can represent the term computed in part (b) as a Feynman diagram, where theFeynman rules for the theory include the usual propagator as well as the vacuum-to-one-particleinteraction below.
(Note that this is what you should expect by looking at the interaction Hamiltonian; it onlycontains a single field, which may either create or annihilate a single meson. The amplitude tocreate a meson of momentum k is proportional to the corresponding Fourier component of thesource).
(d) Now represent the whole perturbation series for P (0) in terms of Feynman diagrams. Show thatthis series exponentiates, so that it can be summed exactly:
P (0) = exp(−λ).
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(e) Compute the probability that the source creates one particle of momentum k. Perform thiscomputation first to O(ρ) and then to all orders, using the trick of part (d) to sum the series.
(f) Show that the probability of producing n particles is given by
P (n) = (1/n!)λn exp(−λ).
This is a Poisson distribution. (NOTE: The combinatorics get a bit nasty here. If you can’t dothe general case right off the bat, first try it for the first few values of n and look for the pattern.)
(g) Prove the following facts about a Poisson distribution:
∞∑n=0
P (n) = 1; 〈N〉 =
∞∑n=0
nP (n) = λ.
The first identity says that the P (n)’s are properly normalized probabilities, while the secondconfirms our previous result for 〈N〉. Compute the mean square fluctuation 〈(N − 〈N〉)2〉.
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PHY 2403F: PROBLEM SET #4 Due: Wednesday, November 7
1. In the theory of mesons and “nucleons”, compute, to lowest non-vanishing order in g, the centre-of-mass differential cross section dσ/d cos θ and the total cross section for “nucleon”-“antinucleon” elasticscattering as a function of s = (p1 + p2)2, µ2 and m2. For s, µ2 m2 (that is, for the nucleon massnegligible compared to the meson mass and beam energy) sketch a plot of the total cross section as afunction of s. (As we discussed in class, the singularity is called an s-channel pole - a pole in s due tothe intermediate meson going on shell. At higher orders in perturbation theory, the finite width of themeson smooths out the singularity.)
2. In the same theory, calculate the differential cross section dσ/d cos θ for “nucleon”-“antinucleon” an-nihilation into two mesons, again to lowest non-vanishing order in g. Do you find an s-channel pole?Why or why not?
3. Given the transformation laws for the spinors u+ and u− that we found in class,
u± → e−i~σ·eθ/2u±
for a rotation, andu± → e±~σ·eφ/2u±
for a boost, find how the components of V µ = (u†+u+, u†+~σu+) and Wµ = (u†−u−,−u
†−~σu−) transform
under (a) a rotation about the z axis, and (b) a boost along the x axis. Hence show that V µ and Wµ
do in fact transform as four-vectors.
4. (a) Using only the anticommutation relations
γµ, γν = 2gµν
prove the following γ matrix identities:
γλγλ = 4, γλγ
αγλ = −2γα
γλγαγβγλ = 4gαβ
γλγαγβγκγλ = −2γκγβγα.
(b) Find the corresponding identities in part (a) in d space-time dimensions. Assume d is a positiveinteger in your derivations, although by analytic continuation, the same formulas are valid forarbitrary (negative, rational, irrational, or even complex) d.
COMMENT: This is not as silly an exercise as it may seem. In PHY2404S you will learn thata particularly simple way to tame a class of divergences in quantum field theory is to do thecalculation in d dimensions (and hence finding the result as an analytic function of d), and onlytake the limit d → 4 at the end of the calculation. Intermediate steps will have poles at d = 4,and so by staying away from d = 4 intermediate stages of the calculation are well-defined. Thistechnique is called dimensional regularization.
(c)
Tr1 = 4
Tra/ = Trγ5γ5a/ = −Trγ5a/γ5 = 0
Tra/b/ = Tr(a/b/+ b/a/)/2 = 4a · b.
Carry on. Compute Tra/b/c/, Tra/b/c/d/ and the trace of up to four slashed vectors and one γ5. Thelast computation, Tra/b/c/d/γ5, will involve the ε symbol. To make sure we are all using the samesign conventions, choose ε0123 = +1.
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5. *** NEW: THIS PROBLEM IS POSTPONED TO THE NEXT (#5) PROBLEM SET ***
Consider adding scalar “nucleons” to the theory of mesons coupled to a dynamical source, considered inthe last problem set, and couple the “nucleons” in the usual way to the mesons. We can now calculatethe scattering of “nucleons” off a classical meson source ρ(t, ~x). This is analogous to Rutherfordscattering (scattering electrons off a heavy charged nucleus) in QED, since the charged nucleus is aphoton source, and photons couple to electrons analogously to our meson-“nucleon” coupling.
(a) Based on the Feynman rule for the interaction you derived in the previous question, draw theFeynman diagram which contributes to the matrix element
〈N(kf )|S − 1|N(ki)〉 (1)
at lowest nonvanishing order in g and use it to calculate the S matrix element for the scatteringof a “nucleon” off the external source at leading order.
(b) Let us now restrict ourselves to scattering off a static (time-independent) source; this would bethe case if it corresponded to a very heavy, static, nucleus. The Fourier transform of a time-independent source contains a delta function of energy (this just reflects energy conservation in atime-translation invariant system). It is then natural to define
〈kf |(S − 1)|ki〉 = iAfi(2π)δ(ωf − ωi). (2)
Show that the cross section for elastic scattering of a “nucleon” off the external source is
dσ
dΩ=|Afi|2
16π2. (3)
(You can derive this by putting the system in a box, as in lecture, and taking the limit T, V →∞at the end).
(c) Show that the differential cross section for scattering “nucleons” off a static point nucleus at theorigin (ρ(~x) = ρ0δ
(3)(~x)) is, at high energies where E m,µ
dσ
dΩ∝ ρ0
E4 sin4(θ/2)(4)
where E is the “nucleon” energy and θ is the scattering angle, and find the constant of proportion-ality. Show that if instead the charge has some finite extent, ρ(~x) = ρ0 f(~x), where
∫d3x f(~x) = 1,
the cross section may be written as
dσ
dΩ=
(dσ
dΩ
)point
|F (~q)|2, F (q) ≡∫d3x f(~x)ei~q·~x (5)
where ~q = ~kf −~ki. F (~q) is known as the form factor of the source. Note that as q → 0, F (~q)→ 1;this is because in the small q limit the virtual meson is soft, and with its large wavelengthcannot resolve the structure of the charge distribution. The greater the momentum transfer inthe scattering, the greater the detail with which the structure of the source may be resolved.
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PHY 2403F: PROBLEM SET #5 Due: Wednesday, November 21, 2012
1. Compute dσ/dΩ in the centre-of-mass frame, to lowest non-trivial order in perturbation theory, aver-aged over initial spins and summed over final spins, for meson-nucleon scattering in the scalar theorydiscussed in class,
LI = gψψφ.
2. The same as the previous problem, but for nucleon-antinucleon scattering in the pseudoscalar theory
LI = gψiγ5ψφ.
NB If you discover you have to take the trace of more than four Dirac matrices, you should think a bitharder about how you are implementing the trace formula.
3. (Postponed from the last problem set)
Consider adding scalar “nucleons” to the theory of mesons coupled to a dynamical source, considered inthe third problem set, and couple the “nucleons” in the usual way to the mesons. We can now calculatethe scattering of “nucleons” off a classical meson source ρ(t, ~x). This is analogous to Rutherfordscattering (scattering electrons off a heavy charged nucleus) in QED, since the charged nucleus is aphoton source, and photons couple to electrons analogously to our meson-“nucleon” coupling.
(a) Based on the Feynman rule for the interaction you derived in the previous question, draw theFeynman diagram which contributes to the matrix element
〈N(kf )|S − 1|N(ki)〉 (1)
at lowest nonvanishing order in g and use it to calculate the S matrix element for the scatteringof a “nucleon” off the external source at leading order.
(b) Let us now restrict ourselves to scattering off a static (time-independent) source; this would bethe case if it corresponded to a very heavy, static, nucleus. The Fourier transform of a time-independent source contains a delta function of energy (this just reflects energy conservation in atime-translation invariant system). It is then natural to define
〈kf |(S − 1)|ki〉 = iAfi(2π)δ(ωf − ωi). (2)
Show that the cross section for elastic scattering of a “nucleon” off the external source is
dσ
dΩ=|Afi|2
16π2. (3)
(You can derive this by putting the system in a box, as in lecture, and taking the limit T, V →∞at the end).
(c) Show that the differential cross section for scattering “nucleons” off a static point nucleus at theorigin (ρ(~x) = ρ0δ
(3)(~x)) is, at high energies where E m,µ
dσ
dΩ∝ ρ0
E4 sin4(θ/2)(4)
where E is the “nucleon” energy and θ is the scattering angle, and find the constant of proportion-ality. Show that if instead the charge has some finite extent, ρ(~x) = ρ0 f(~x), where
∫d3x f(~x) = 1,
the cross section may be written as
dσ
dΩ=
(dσ
dΩ
)point
|F (~q)|2, F (q) ≡∫d3x f(~x)ei~q·~x (5)
where ~q = ~kf −~ki. F (~q) is known as the form factor of the source. Note that as q → 0, F (~q)→ 1;this is because in the small q limit the virtual meson is soft, and with its large wavelengthcannot resolve the structure of the charge distribution. The greater the momentum transfer inthe scattering, the greater the detail with which the structure of the source may be resolved.
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PHY 2403F: PROBLEM SET #6 Due: Wednesday, December 5, 2012
1. A massive vector meson is minimally coupled to a charged Dirac particle. Compute, to lowest non-trivial order of perturbation theory, the amplitudes for fermion-fermion scattering and antifermion-fermion scattering. (You do not have to do spin sums or compute cross sections, just write downthe amplitudes). Explicitly verify that the contribution of the term in the vector-meson propagatorproportional to
kµkνµ2
vanishes.
2. In the same theory, we showed in class that for fermion-vector meson scattering, k(f)µ Mµν = 0 =
k(i)ν Mµν , where the amplitude for the process is given by ε
(f)∗µ ε
(i)ν Mµν . Here k(f) and k(i) are the
four momenta of the initial and final vector mesons respectively, and the ε’s are the correspondingpolarization vectors. Show that this is also true for charged scalar-vector boson scattering.
3. Compute the differential cross section dσ/d cos θ for e+e− → e+e− scattering. You may work in thelimit Ec.o.m. me, in which you can ignore the electron mass. Show that
dσ
d cos θ=πα2
s
[u2(
1
s+
1
t
)2
+
(t
s
)2
+(st
)2]
where the Mandelstam variables s, t and u are defined as
s ≡ (p+ p′)2, t ≡ (p− k)2, u = (p− k′)2
and p, p′, k and k′ are the 4-momenta of the incoming electron, incoming positron, outgoing electronand outgoing positron, respectively. Note that if we ignore the electron mass, s + t + u = 0. Rewritethis formula in terms of cos θ and graph it. What feature of the diagrams causes the differential crosssection to diverge as θ → 0?
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Quantum Field Theory: Example Sheet 1
Dr David Tong, October 2007
1. A string of length a, mass per unit length σ and under tension T is fixed at
each end. The Lagrangian governing the time evolution of the transverse displacement
y(x, t) is
L =
∫ a
0
dx
[
σ
2
(
∂y
∂t
)2
−T
2
(
∂y
∂x
)2]
(1)
where x identifies position along the string from one end point. By expressing the
displacement as a sine series Fourier expansion in the form
y(x, t) =
√
2
a
∞∑
n=1
sin(nπx
a
)
qn(t) (2)
show that the Lagrangian becomes
L =
∞∑
n=1
[
σ
2q2
n −T
2
(nπ
a
)2
q2
n
]
. (3)
Derive the equations of motion. Hence show that the string is equivalent to an infinite
set of decoupled harmonic oscillators with frequencies
ωn =
√
T
σ
(nπ
a
)
. (4)
2. Show directly that if φ(x) satisfies the Klein-Gordon equation, then φ(Λ−1x) also
satisfies this equation for any Lorentz transformation Λ.
3. The motion of a complex field ψ(x) is governed by the Lagrangian
L = ∂µψ∗∂µψ −m2ψ∗ψ −
λ
2(ψ∗ψ)2 . (5)
Write down the Euler-Lagrange field equations for this system. Verify that the La-
grangian is invariant under the infinitesimal transformation
δψ = iαψ , δψ∗ = −iαψ∗ (6)
Derive the Noether current associated with this transformation and verify explicitly
that it is conserved using the field equations satisfied by ψ.
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4. Verify that the Lagrangian density
L =1
2∂µφa∂
µφa −1
2m2φaφa (7)
for a triplet of real fields φa (a = 1, 2, 3) is invariant under the infinitesimal SO(3)
rotation by θ
φa → φa + θǫabcnbφc (8)
where na is a unit vector. Compute the Noether current jµ. Deduce that the three
quantities
Qa =
∫
d3x ǫabc φbφc (9)
are all conserved and verify this directly using the field equations satisfied by φa.
5. A Lorentz transformation xµ → x′µ = Λµνx
ν is such that it preserves the Minkowski
metric ηµν , meaning that ηµνxµxν = ηµνx
′µx′ν for all x. Show that this implies that
ηµν = ηστΛσµΛτ
ν . (10)
Use this result to show that an infinitesimal transformation of the form
Λµν = δµ
ν + ωµν (11)
is a Lorentz tranformation when ωµν is antisymmetric: i.e. ωµν = −ωνµ.
Write down the matrix form for ωµν that corresponds to a rotation through an in-
finitesimal angle θ about the x3-axis. Do the same for a boost along the x1-axis by an
infinitesimal velocity v.
6. Consider the infinitesimal form of the Lorentz transformation derived in the previous
question: xµ → xµ + ωµνx
ν . Show that a scalar field transforms as
φ(x) → φ′(x) = φ(x) − ωµν x
ν ∂µφ(x) (12)
and hence show that the variation of the Lagrangian density is a total derivative
δL = −∂µ(ωµνx
ν L) (13)
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Using Noether’s theorem deduce the existence of the conserved current
jµ = −ωρν [T µ
ρ xν ] (14)
The three conserved charges arising from spatial rotational invariance define the total
angular momentum of the field. Show that these charges are given by,
Qi = ǫijk
∫
d3x(
xjT 0k − xkT 0j)
(15)
Derive the conserved charges arising from invariance under Lorentz boosts. Show that
they imply
d
dt
∫
d3x (xi T 00) = constant (16)
and interpret this equation.
7. Maxwell’s Lagrangian for the electromagnetic field is
L = −1
4FµνF
µν (17)
where Fµν = ∂µAν − ∂νAµ and Aµ is the 4-vector potential. Show that L is invariant
under gauge transformations
Aµ → Aµ + ∂µξ (18)
where ξ = ξ(x) is a scalar field with arbitrary (differentiable) dependence on x.
Use Noether’s theorem, and the spacetime translational invariance of the action, to
construct the energy-momentum tensor T µν for the electromagnetic field. Show that
the resulting object is neither symmetric nor gauge invariant. Consider a new tensor
given by
Θµν = T µν − F ρµ ∂ρAν (19)
Show that this object also defines four conserved currents. Moreover, show that it is
symmetric, gauge invariant and traceless.
Comment: T µν and Θµν are both equally good definitions of the energy-momentum
tensor. However Θµν clearly has the nicer properties. Moreover, if you couple Maxwell’s
Lagrangian to general relativity then it is Θµν which appears in Einstein’s equations.
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8. The Lagrangian density for a massive vector field Cµ is given by
L = −1
4FµνF
µν +1
2m2CµC
µ (20)
where Fµν = ∂µCν − ∂νCµ. Derive the equations of motion and show that when m 6= 0
they imply
∂µCµ = 0 (21)
Further show that C0 can be eliminated completely in terms of the other fields by
∂i∂i C0 +m2C0 = ∂iCi (22)
Construct the canonical momenta Πi conjugate to Ci, i = 1, 2, 3 and show that the
canonical momentum conjugate to C0 is vanishing. Construct the Hamiltonian density
H in terms of C0, Ci and Πi. (Note: Do not be concerned that the canonical momen-
tum for C0 is vanishing. C0 is non-dynamical — it is determined entirely in terms of
the other fields using equation (22)).
9. A class of interesting theories are invariant under the scaling of all lengths by
xµ → (x′)µ = λ xµ and φ(x) → φ′(x) = λ−Dφ(λ−1x) (23)
Here D is called the scaling dimension of the field. Consider the action for a real scalar
field given by
S =
∫
d4x1
2∂µφ∂
µφ−1
2m2φ2 − gφ p (24)
Find the scaling dimension D such that the derivative terms remain invariant. For
what values of m and p is the scaling (23) a symmetry of the theory. How do these
conclusions change for a scalar field living in an (n+ 1)-dimensional spacetime instead
of a 3 + 1-dimensional spacetime?
In 3 + 1 dimensions, use Noether’s theorem to construct the conserved current Dµ
associated to scaling invariance.
4
Quantum Field Theory: Example Sheet 2
Dr David Tong, October 2007
1. A string has classical Hamiltonian given by
H =∞
∑
n=1
(
1
2p2
n + 1
2ω2
n q2
n
)
(1)
where ωn is the frequency of the nth mode. (Compare this Hamiltonian to the La-
grangian (3) in Example Sheet 1. We have set the mass per unit length in that ques-
tion to σ = 1 to simplify some of the formulae a little). After quantization, qn and pn
become operators satisfying
[qn, qm] = [pn, pm] = 0 and [qn, pm] = iδnm (2)
Introduce creation and annihilation operators an and a†n,
an =
√
ωn
2qn +
i√
2ωn
pn and a†n =
√
ωn
2qn −
i√
2ωn
pn (3)
Show that they satisfy the commutation relations
[an, am] = [a†n, a†
m] = 0 and [an, a†
m] = δnm (4)
Show that the Hamiltonian of the system can be written in the form
H =∞
∑
n=1
1
2ωn
(
ana†
n + a†nan
)
(5)
Given the existence of a ground state |0〉 such that an|0〉 = 0, explain how, after
removing the vacuum energy, the Hamiltonian can be expressed as
H =∞
∑
n=1
ωna†
nan (6)
Show further that [H, a†n] = ωn a†
n and hence calculate the energy of the state
|l1, l2, . . . , lN〉 =(
a†
1
)l1(
a†
2
)l2
. . .(
a†
N
)lN
|0〉 (7)
1
2. The Fourier decomposition of a real scalar field and its conjugate momentum in the
Schrodinger picture is given by
φ(~x) =
∫
d3p
(2π)3
1√
2E~p
[
a~p ei~p·~x + a
†
~p e−i~p·~x
]
(8)
π(~x) =
∫
d3p
(2π)3(−i)
√
E~p
2
[
a~p ei~p·~x − a
†
~p e−i~p·~x
]
(9)
Show that the commutation relations
[φ(~x), φ(~y)] = [π(~x), π(~y)] = 0 and [φ(~x), π(~y)] = iδ(3)(~x− ~y) (10)
imply that
[a~p, a~q] = [a†~p, a†
~q] = 0 and [a~p, a†
~q] = (2π)3δ(3)(~p− ~q) (11)
3. Consider a real scalar field with the Lagrangian
L = 1
2∂µφ ∂
µφ− 1
2m2φ2 (12)
Show that, after normal ordering, the conserved four-momentum P µ =∫
d3xT 0µ takes
the operator form
P µ =
∫
d3p
(2π)3pµ a
†
~p a~p (13)
where p0 = E~p in this expression. From this expression for P µ verify that if φ(x) is
now in the Heisenberg picture, then
[P µ, φ(x)] = −i∂µφ(x) (14)
4. Show that in the Heisenberg picture,
φ(x) = i[H, φ(x)] = π(x) and π(x) = i[H, π(x)] = ∇2φ(x) −m2φ(x) (15)
Hence show that the operator φ(x) satisfies the Klein-Gordon equation.
5. Let φ(x) be a real scalar field in the Heisenberg picture. Show that the rela-
tivistically normalized one-particle states |p〉 =√
2E~p a†
~p |0〉 satisfy
〈0|φ(x) |p〉 = e−ip·x (16)
2
6. In Example Sheet 1, you showed that the classical angular momentum of field is
given by
Qi = ǫijk
∫
d3x (xjT 0k − xkT 0j) (17)
Write down the explicit form of the angular momentum for a free real scalar field with
Lagrangian (12). Show that, after normal ordering, the quantum operator Qi can be
written as
Qi = −i ǫijk
∫
d3p
(2π)3a†
~p
(
pj ∂
∂pk
− pk ∂
∂pj
)
a~p (18)
Hence confirm that the quanta of the scalar field have spin zero (i.e. a stationary
one-particle state |~p = 0〉 has zero angular momentum).
7. The purpose of this question is to introduce you to non-relativistic quantum field
theory. This is the only place you will encounter such a thing in this course. Consider
the Lagrangian for a complex scalar field ψ given by
L = +iψ⋆∂0ψ −1
2m∇ψ⋆ · ∇ψ (19)
Determine the equation of motion, the energy-momentum tensor and the conserved
current arising from the symmetry ψ → eiαψ. Show that the momentum conjugate to
ψ is iψ⋆ and compute the classical Hamiltonian.
We now wish to quantize this theory. We will work in the Schrodinger picture.
Explain why the correct commutation relations are
[ψ(~x), ψ(~y)] = [ψ†(~x), ψ†(~y)] = 0 and [ψ(~x), ψ†(~y)] = δ(3)(~x− ~y) (20)
Expand the fields in a Fourier decomposition as
ψ(~x) =
∫
d3p
(2π)3a~p e
i~p·~x
ψ†(~x) =
∫
d3p
(2π)3a†
~p e−i~p·~x (21)
Determine the commutation relations obeyed by a~p and a†
~p. Why do we have only
a single set of creation and annihilation operators a~p, a†
~p even though ψ is complex?
What is the physical significance of this fact? Show that one particle states have the
energy appropriate to a free non-relativistic particle of mass m.
3
8. Show that the time ordered product T (φ(x1)φ(x2)) and the normal ordered product
: φ(x1)φ(x2) : are both symmetric under the interchange of x1 and x2. Deduce that
the Feynman propagator ∆F (x1 − x2) has the same symmetry property.
9. Verify Wick’s theorem for the case of three scalar fields:
T (φ(x1)φ(x2)φ(x3)) = : φ(x1)φ(x2)φ(x3) : +φ(x1)∆F (x2 − x3)
+φ(x2)∆F (x3 − x1) + φ(x3)∆F (x1 − x2) (22)
10. Consider the scalar Yukawa theory given by the Lagrangian
L = ∂µψ⋆∂µψ +
1
2∂µφ∂
µφ−M2ψ⋆ψ −1
2m2φ2 − gψ⋆ψφ (23)
Compute the amplitude for
• “Nucleon-Anti-Nucleon” annihilation ψ + ψ → φ at order g
• “Nucleon-Meson” scattering φ+ ψ → φ+ ψ at order g2
4
Quantum Field Theory: Example Sheet 3
Dr David Tong, October 2006
1. The Weyl representation of the Clifford algebra is given by,
γ0 =
(
0 1
1 0
)
, γi =
(
0 σi
−σi 0
)
(1)
Show that these indeed satisfy γµ, γν = 2ηµν , where 1 comes with an implicit 4 × 4
unit matrix. Find a unitary matrix U such that (γ′)µ = UγµU †, where (γ′)µ form the
Dirac representation of the Clifford algebra
(γ′)0 =
(
1 0
0 −1
)
, (γ′)i =
(
0 σi
−σi 0
)
(2)
2. Show that if γµ , γν = 2ηµν , then
[
γκγλ , γµγν]
= 2ηλµγκγν − 2ηκµγλγν + 2ηλνγµγκ − 2ηκνγµγλ . (3)
Show further that Sµν ≡ 1
4[γµ , γν ] = 1
2(γµγν − ηµν). Use this to confirm that the
matrices Sµν form a representation of the Lie algebra of the Lorentz group.
3. Using just the algebra γµ, γν = 2ηµν (i.e. without resorting to a particular
representation), and defining γ5 = iγ0γ1γ2γ3, /p = pµγµ and Sµν = 1
4[γµ, γν ], prove
the following results: (Some useful tricks include the cyclicity of the trace, and inserting
(γ5)2 = 1 into a trace).
i. Trγµ = 0
ii. Tr(γµγν) = 4ηµν
iii. Tr(γµγνγρ) = 0
iv. (γ5)2
= 1
v. Trγ5 = 0
vi. /p /q = 2p · q − /q /p = p · q + 2Sµνpµqν
vii. Tr( /p /q) = 4p · q
1
viii. Tr( /p1 . . . /pn) = 0 if n is odd
ix. Tr( /p1
/p2
/p3
/p4) = 4 [(p1 · p2)(p3 · p4) + (p1 · p4)(p2 · p3) − (p1 · p3)(p2 · p4)]
x. Tr(γ5 /p1
/p2) = 0
xi. γµ /p γµ = −2 /p
xii. γµ /p1/p2γ
µ = 4p1 · p2
xiii. γµ /p1
/p2
/p3γµ = −2 /p
3/p2
/p1
xiv. Tr(γ5 /p1
/p2
/p3
/p4) = 4i ǫµνρσ p
µ1pν
2p
ρ3p4
σ
4. The plane-wave solutions to the Dirac equation are
us(~p) =
( √p · σ ξs
√p · σ · ξs
)
and vs(~p) =
( √p · σ ξs
−√p · σξs
)
(4)
where σµ = (1, ~σ) and σµ = (1,−~σ) and ξs, with s = 1, 2, is a basis of orthonormal
two-component spinors, satisfying (ξr)† · ξs = δrs. Show that
ur(~p)† · us(~p) = 2p0δrs (5)
ur(~p) · us(~p) = 2mδrs
and similarly,
vr(~p)† · vs(~p) = 2p0δrs (6)
vr(~p) · vs(~p) = −2mδrs
Show also that the orthonality condition between u and v is
us(~p) · vr(~p) = 0 (7)
while taking the inner product using † requires an extra minus sign
ur(~p)† · vs(−~p) = 0 (8)
5. Using the same notation as Question 4 show that
2∑
s=1
us(~p)us(~p) = /p+m (9)
2∑
s=1
vs(~p)vs(~p) = /p−m (10)
2
where, rather than being contracted, the two spinors on the left-hand side are placed
back to back to form a 4 × 4 matrix.
6. The Fourier decomposition of the Dirac operator ψ(~x) and the conjugate field
ψ†(~x) is given by,
ψ(~x) =2∑
s=1
∫
d3p
(2π)3
1√
2E~p
[
bs~p us(~p)e+i~p·~x + c
s †
~p vs(~p)e−i~p·~x]
ψ†(~x) =2∑
s=1
∫
d3p
(2π)3
1√
2E~p
[
bs †
~p us(~p)†e−i~p·~x + cs~p vs(~p)†e+i~p·~x
]
(11)
The creation and annihilation operators are taken to satisfy
br~p, bs †
~q = (2π)3δrs δ(3)(~p− ~q)
cr~p, cs †
~q = (2π)3δrs δ(3)(~p− ~q) (12)
with all other anti-commutators vanishing,
br~p, bs~q = cr~p, c
s~q = br~p, c
s †
~q = br~p, cs~q = . . . = 0 (13)
Show that these imply that the field and it conjugate momenta satisfy the anti-
commutation relations,
ψα(~x), ψβ(~y) = ψ†
α(~x), ψ†
β(~y) = 0
ψα(~x), ψ†
β(~y) = δαβ δ(3)(~x− ~y) (14)
(Note: The calculation is very similar to that for the bosonic field, but at some point
you will need to make use of the identities (9) and (10)).
7. Using the results of Question 6, show that the quantum Hamiltonian
H =
∫
d3x ψ(−iγi∂i +m)ψ (15)
can be written, after normal ordering, as
H =
∫
d3p
(2π)3E~p
2∑
s=1
[
bs †
~p bs~p + c
s †
~p cs~p
]
(16)
(Note: Again, the calculation is very similar to that for the bosonic field. This time
you will need to make use of the identities derived in Questions 4 and 5).
3
9. The purpose of this question is to give you a glimpse into the spin-statistics theorem.
This theorem roughly says that if you try to quantize a field with the wrong statistics,
bad things will happen. Here we’ll see what goes wrong if you try to quantize a spin
1/2 field as a boson. We start with the usual decomposition (11). This time we choose
bosonic commutation relations for the annihilation and creation operators,
[br~p, bs †
~q ] = (2π)3δrs δ(3)(~p− ~q)
[cr~p, cs †
~q ] = −(2π)3δrs δ(3)(~p− ~q) (17)
with all other commutators vanishing. Note the strange minus sign for the c opera-
tors. Repeat the calculation of Question 6 to show that these are equivalent to the
commutation relations,
[ψα(~x), ψβ(~y)] = [ψ†
α(~x), ψ†
β(~y)] = 0
[ψα(~x), ψ†
β(~y)] = δαβ δ(3)(~x− ~y) (18)
Now repeat the calculation of Question 7, to show that, after normal ordering, the
Hamitonian is given by
H =
∫
d3p
(2π)3E~p
2∑
s=1
[
bs †
~p bs~p − c
s †
~p cs~p
]
(19)
This Hamiltonian is not bounded below: you can lower the energy indefinitely by
creating more and more c particles. This is the reason a theory of bosonic spin 1/2
particles is sick.
4
Quantum Field Theory: Example Sheet 4
Dr David Tong, November 2007
1. A real scalar field with φ4 interaction has the Lagrangian
L =1
2∂µφ∂
µφ−1
2m2φ2 −
λ
4!φ4 (1)
Use Dyson’s formula and Wick’s theorem to show that the leading order contribution
to 3-particle → 3-particle scattering includes the amplitude
p3
2p
p1
p1
2p
p3
//
/
= (−iλ)2i
(p1 + p2 + p3)2 −m2(2)
Check that this result is consistent with the Feynman rules for the theory. What other
diagrams also contribute to this process?
2. Examine 〈0|S|0〉 to order λ2 in φ4 theory. Identify the different diagrams with
the different contributions arising from an application of Wick’s theorem. Confirm
that to order λ2, the combinatoric factors work out so that the the vacuum to vacuum
amplitude is given by the exponential of the sum of distinct vacuum bubble types,
〈0|S |0〉 = exp ( + + + ...) (3)
3. Consider the Lagrangian for 3 scalar fields φi, i = 1, 2, 3, given by
L =3
∑
i=1
1
2(∂µφi)(∂
µφi) −1
2m2(
3∑
i=1
φ2
i ) −λ
8(
3∑
i=1
φ2
i )2 (4)
Show that the Feynman propagator for the free field theory (i.e. λ = 0) is of the form
〈0|Tφi(x)φj(y)|0〉 = δijDF (x− y) (5)
where DF (x− y) is the usual scalar propagator. Write down the Feynman rules of the
theory. Compute the amplitude for the scattering φiφj → φkφl to lowest order in λ.
1
4. The Lagrangian for Yukawa theory is given by
L = 1
2(∂φ)2 − 1
2µ2φ2 + ψ(i /∂ −m)ψ − λφψψ (6)
a) Consider ψψ → ψψ scattering, with the initial and final states given by,
|i〉 =√
4E~pE~q bs †
~p br †
~q |0〉
|f〉 =√
4E~p′E~q′ bs′ †
~p ′ br′ †
~q ′ |0〉 (7)
Show using Dyson’s formula and Wick’s theorem that the scattering amplitude at order
λ2 is given by,
A = (−iλ)2
(
[us′(~p ′) · us(~p)] [ur′(~q ′) · ur(~q)]
(p′ − p)2 − µ2−
[us′(~p ′) · ur(~q)] [ur′(~q ′) · us(~p)]
(q′ − p)2 − µ2
)
Draw the two Feynman diagrams that correspond to these two terms.
b) Consider now ψψ → ψψ scattering, with initial and final states given by
|i〉 =√
4E~pE~q bs †
~p cr †
~q |0〉
|f〉 =√
4E~p′E~q′ bs′ †
~p ′ cr′ †
~q ′ |0〉 (8)
Show that the amplitude is this time given by
A = −(−iλ)2
(
[us′(~p ′) · us(~p)] [vr(~q) · vr′(~q ′)]
(p− p′)2 − µ2−
[vr(~q) · us(~p)] [us′(~p ′) · vr′(~q ′)]
(p+ q)2 − µ2
)
(Be careful with minus signs!!). What are the Feynman diagrams that now contribute?
5. The Lagrangian for a pseudoscalar Yukawa interaction is given by
L = 1
2(∂φ)2 − 1
2µ2φ2 + ψ(i /∂ −m)ψ − λφψγ5ψ (9)
Write down the Feynman rules for this theory. Use this to write down the amplitude
at order λ2 for ψψ → ψψ scattering and ψψ → ψψ scattering.
6. Any vector function f(x) has a decomposition into a sum of transverse (zero diver-
gence) and longitudinal (zero curl) parts, namely
f = ∇× g + ∇h ≡ fT + fL (10)
where g and h are unique if one imposes the additional constraint ∇·g = 0 and certain
vanishing conditions at infinity. By taking the divergence and curl of equation (10),
determine g and h in terms of f . Show formally that
fT = f −∇(∇2)−1∇ · f (11)
2
Use this result to comment on the commutation relations of the quantized electromag-
netic gauge potential in Coulomb gauge.
7. Consider the Compton scattering process e−γ → e−γ in QED. Let the incom-
ing photon have polarization vector ǫµin, and the outgoing photon have polarization
ǫµout. Use the Feynman rules to derive the following amplitude associated to the lowest
order diagram,
outε /qε in q ()( )
u(p) u(p)/−
= i(−ie)2ur′(~p ′) /ǫout
( /p+ /q +m)
(p+ q)2 −m2/ǫinu
s(~p)
Compute also the contribution from the diagram
ε in q)( outε /q( )
u(p) u(p)/−
The complete amplitude at order e2 is the sum of these two contributions. Show that
the total amplitude vanishes if ǫin is replaced by the incoming photon momentum q
then the amplitude vanishes. Check that the same holds true if ǫout is replaced by q′ .
(Note that it will be helpful to recall the equation (/p − m)u(~p) = 0 satisfied by the
spinor).
8. Use the Feynman rules to show that the QED amplitude for e−e+ → µ−µ+ is
given at lowest order in e by,
µ−
µ+e+
e−
p,s
q,r
p,s
q,r
//
//
= (−ie)2[vr
e(~q)γµuse(~p)] [u
s′
m(~p ′)γµvr′
m(~q ′)]
(p+ q)2(12)
where the subscripts e and m denote whether the spinors satisfy the Dirac equation
for electrons or for muons.
3
9. Viki Weisskopf is one of the more charming characters from the history of quantum
field theory. This from his autobiography:
“Pauli asked me to calculate the amplitude for pair creation of scalar parti-
cles by photons. It was only a short time after Bethe and Heitler had solved
the same problem for electrons and positrons. I met Bethe in Copenhagen
at a conference and asked him to tell me how he did the calculations. I
also inquired how long it would take to perform this task; he answered, “It
would take me three days, but you will need about three weeks.” He was
right, as usual; furthermore, the published result was wrong by a factor of
two.”
Can you do better?
10. Now you understand the role played by fields in Nature, why do you think classical
physicists such as Faraday and Maxwell found it useful to introduce the concept of the
electric and magnetic field, but never fields for the electron or other particles?
4