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129A Lecture Notes
Quantum ElectroDynamics
1 Quantum ElectroDynamics
The true quantum mechanical and relatistic theory of electromagnetism is called Quantum ElectroDynamics, or QED for shot. It combines Dirac equa- tion to describe electron (and of course positron) and Maxwell equation for photon. The actual calculations of quantum mechanical amplitudes are based on perturbation theory, organized in terms of Feynman diagrams. Introduc- ing QED from the first principle takes a full semester (229A in the case of Berkeley graduate curriculum) and we cannot go through it in this course. Instead, I’d like to give you rough idea on how the theory works, without getting into hardcore formalism.
2 Photons
Early development of quantum mechanics was led by the fact that electro- magnetic radiation was quantized: photons. Here we show that the Maxwell field can be regarded as a collection of harmonic oscillators. Details of this discussion are not used later in this course. This section is for only those of you who are interested in seeing this explicitly.
2.1 Classical Maxwell Field
The vector potential A~ and the scalar potential φ are combined in the four- vector potential Aμ^ = (φ, A~). (1)
Throughout the lecture notes, we use the convention that the metrix gμν =
diag(+1, − 1 , − 1 , −1) and hence Aμ = gμν Aν^ = (φ, − A~). The four-vector coordinate is xμ^ = (ct, ~x), and correspondingly the four-vector derivative is ∂μ = (^1 c∂t∂ , ∇~). The field strength is defined as Fμν = ∂μAν − ∂ν Aμ, and
hence F 0 i = ∂ 0 Ai − ∂iA 0 = −
A/c − ∇~φ = E~, while Fij = ∂iAj − ∂j Ai = −∇~iAj^ + ∇~j Ai^ = −ijk B~k.
In the unit we have been using where the Coulomb potential is QQ′/ 4 πr without a factor of 1/ 0 , the action for the Maxwell field is
S = −
∫ dtd~x F μν^ Fμν − Aμjμ^ =
∫ dtd~x
( E~^2 − B~^2
)
. (2)
The gauge invariance of the Maxwell field is that the vector potential Aμ^ and Aμ^ + ∂μω (where ω is an arbitrary function of spacetime) give the same field strength and hence the same action. Using this invariance, one can always choose a particular gauge. For most purposes of non-relativistic systems encountered in atomic, molecular, condensed matter, nuclear and astrophysics, Coulomb gauge is the convenient choice, while for highly rela- tivistic systems such as in high-energy physics. We use Lorentz gauge in this lecture note:
∂μAμ^ =
c
∂t
φ + ∇ ·~ A~ = 0. (3)
Because of this condition, we can simplify the action. The original action is
Fμν F μν^ = (∂μAν − ∂ν Aμ)(∂μAν^ − ∂ν^ Aμ) = 2 ∂μAν (∂μAν^ − ∂ν^ Aμ) = 2 ∂μAν ∂μAν^ − 2(∂ν^ Aν )(∂μAμ) + surface terms. (4)
Using the Lorentz gauge condition, we can drop the second term in the last line. Therefore, the Maxwell action Eq. (2) becomes
S = −
∫ dtd~x ∂μAν ∂μAν^. (5)
2.2 Quantization
In order to quantize the Maxwell field, we first determine the “canonically conjugate momentum” for the vector potential Aμ. Following the definition pi^ = ∂L/∂ q˙i^ in particle mechanics, we define the canonically conjugate mo- mentum from the simplified action Eq. (5),
πμ =
∂L
∂ A˙μ^
c^2
A˙μ. (6)
Following the normal canonical commutation relation [qi, pj^ ] = i¯hδij^ , we set up the equal-time commutation relation
[Aμ(~x, t), πν (~y, t)] = [Aμ(~x, t), −
c^2
A˙ν (~y, t)] = i¯hδμν δ(~x − ~y). (7)
(“longitudinal”). But the longitudinal polarization vector does not satisfy the constraint Eq. (11), and this degree of freedom is removed. The scalar one is actually a gauge freedom, because under the gauge transformation Aμ^ → Aμ^ + ∂μω, aμ^ changes by an amount proportional to pμ^ and hence to μS. Therefore we can also omit μS because it is unphysical. In general, for the momentum vector pμ^ = p(1, sin θ cos φ, sin θ sin φ, cos θ), the circular polarization vectors are given by
μ ±(p) =
(±μ 1 + iμ 2 ) , (12)
where the linear polarization vectors are given by
μ 1 (p) = (0, cos θ cos φ, cos θ sin φ, − sin θ), (13) μ 2 (p) = (0, − sin φ, cos φ, 0). (14)
We can check that these two vectors are orthogonal:
μ λ∗ (p)μλ′^ (p) = δλ,λ′^. (15)
This property is satisfied for both the basis with linear polarizations i, j = 1, 2 and the helicity basis i, j = ±. Given the polarization vectors, we re-expand the vector potential in terms of the creation/annihilation operators
Aμ(~x) =
√ 2 π¯hc^2 L^3
∑
~p
ωp
∑ ±
(μ ±(p)a±(p)e−ipμx
μ/¯h
μ/¯h (16)).
With this expansion, the Lorentz gauge condition is automatically satisfied, while the creation/annihilation operators obey the commutation relations
[aλ(p), a† λ′ (q)] = δλ,λ′ δ~p,~q (17)
for λ, λ′^ = ±. We could also have used the linearly polarized photons
Aμ(~x) =
√ 2 π¯hc^2 L^3
∑
~p
ωp
∑^2
λ=
(μλ(p)aλ(p)e−ipμx μ/¯h
- μλ(p)a† λ(p)eipμx μ/¯h (18)).
with [aλ(p), a† λ′ (q)] = δλ,λ′ δ~p,~q (19)
for λ, λ′^ = 1, 2. Clearly, two sets of operators are related by
a 1 =
(a+ + a−), a 2 =
i √ 2
(a+ − a−). (20)
Once we have the mode expansion for the vector potential, one can work out the Hamiltonian
H =
∫ d~x
( E~^2 + B~^2
)
∑
~p
∑
±
¯hωp(a†±(p)a±(p) +
Because ¯hωp = c|~p|, the dispersion relation for the photon is the familiar one E = c|~p| for a massless relativistic particle. The “vacuum” which does not have any photons is simply defined in the similar way as in the harmonic oscillator,
aμ(p)| 0 〉 = 0. (22)
A single photon state is given by acting a creation operator on the vacuum,
|~p, λ〉 = a† λ(p)| 0 〉. (23)
Multi-photon states are obtained by acting the creation operator multiple times. This way, the system of photons is nothing but (an infinite collection of) harmonic oscillators.
3 Feynman Diagrams
We now have electrons and positrons described by Dirac equation, and also photons from quantized Maxwell theory. What we need next is their inter- action. The inteaction is described in terms of Feynman Diagrams, which represent systematic perturbative expansion in powers of fine-structure con- stant α = e^2 /¯hc = 1/137. The basic ingredient is the emission and absorption of a photon by an electron (or a positron). This is all there is to describe all electromagnetic interactions in the QED. Let us first look at a classic example of Rutherford scattering. You have a point positive charge situated at the origin, and you send in another charged particle (say, an electron). The point charge creates a Coulomb field around
only over the time interval ∆t given by the energy-time uncertainty relation ∆E∆t ∼ ¯h. The more energy is violated, the shorter the state survives. In the case of Coulomb scattering, a “kick” is due to the momentum of the virtual photon ~q^2 = −q^2 = 2p^2 (1 − cos θ) = 4p^2 sin^2 θ/2. This photon would normally have the energy c|~q|, but it’s energy vanishes because the initial and final energies of the electron are the same. This is the violation of energy conservation by ∆E = 2cp sin θ/2. This virtual photon is allowed to exist only for ∆t ∼ h/¯ ∆E = ¯h/(2cp sin θ/2), and hence can go only over the distance c∆t ∼ ¯h/(2p sin θ/2). This is commensurate with the classical idea that small kick (small θ) corresponds to a large impact parameter. If two electrons scatter, called Møller scattering, they exchange a virtual photon. But two electrons are identical, and you can draw another diagram where two final state electrons are interchanged. The quantum mechanical amplitude is given by the first diagram minus the second diagram, where the relative minus sign represents that of the Fermi statistics. Positrons are represented by lines going backward in time. Don’t try to attach any philosophical meaning to this statement. It just turns out to be a convenient book keeping of electrons and positrons in the Feynman diagrams. Let us take the arrow of time to be upwards in the diagram. When an electron and a positron scatter, they can exchange a virtual photon sideways, but also upwards. This new diagram represents the annihilation of the electron and the positron into a virtual photon, which materializes as a pair of an electron and a positron. In the center-of-momentum frame, the virtual photon is pure energy with no momentum. Again the vanishing momentum suggests vanishing energy for the photon, but it has finite energy and hence “virtual.” The latter diagram can be used to produce other types of matter other than electron/positron. For instance, the virtual photon can become a pair of a muon and an anti-muon. The muon is similar to the electron, with the same electric charge, spin, and statistics, but 200 times heavier and decays within 2μ sec. An electron can also be “virtual.” When you consider Compton scat- tering, it proceeds via two different Feynman diagram. In one diagram, the electron absorbs the photon, and becomes a “virtual” electron. This virtual electron emits a photon and becomes real again. In the other diagram, the electron emits a photon first and becomes virtual. Then the virtual elec- tron absorbs the photon and becomes real. These two diagrams are added together with positive sign, and allow you to work out the rate of Comp-
ton scattering, its angular distribution, etc. The resulting formula is called Klein–Nishina formula, and is one of the earliest quantitative success of the QED.
Made on 9-Sep-1993 13:24:10 by DREVERMANN with DALI_D1.
ALEPH^ DALI^ Run=15995^ Evt=
Figure 1: Bhabha scattering. The tracks are and electron and a positron because they shower in the electromagnetic calorimeter, represented in the inner red circle. The electron and positron beams are perpendicular to the event display and they collide at the center. The display is a projection on the plane transverse to the beams.
4 Casimir Effect
So far, we had not cared much about the zero-point energy of the photons in Eq. (21). It is often said that the zero-point energy just amounts to the baseline energy and all other energies are measured relative to it. In other words, pretending it doesn’t exist is enough. But the zero-point energy plays a role if a change in the system affects the zero-point energy itself. (See Milonni, P. W., and Shih, M. L., 1992, Contemp. Phys. 33 , 313. for a review.) To be concrete, put two conducting plates parallel to each other. The distance between the plates is d (at z = 0 and z = d), and the plates are very large of area L^2 where we take L → ∞. To simplify the discussions,
we also place thin plates at x = 0, x = L and y = 0, y = L to form a box. A conducting plate imposes boundary conditions on the radiation field E‖ = 0 and B⊥ = 0. Given these boundary conditions, the vector potential is expanded in modes as
A^ ~(~x) ∼ ~ sin(nx, ny, nz )
( (^) πn x L
x +
πny L
y +
πnz d
z
) , (26)
for nx, ny, and nz non-negative integers. For each wave vector ~k = (πnx/L, πny/L, πnz /d),
the polarization vector is transverse: ~k · ~ = 0. If one of the n’s vanishes, however, say ~k = (0, πny/L, πnz /d), the polarization vector ~ = (1, 0 , 0) is not allowed because that would give Ax 6 = 0 for any x (and generic y, z) and violates the boundary condition E‖ = 0. Therefore, we have only one polarization whenever one of n’s vanishes. The sum of zero point energies in this set up is therefore given by
U (d) =
¯hω × 2 =
∑ (^) ′ nx,ny ,nz
¯hc
[( πnx L
) 2
( πny L
) 2
( πnz d
) 2 ] 1 / 2
. (27)
The summation
∑ (^) ′ means that whenever one of n’s vanishes, we drop the multiplicity 2 for possible transverse polarizations. Because we regard the size of the plate to be large L → ∞, we can replace the sum over nx, ny in terms of integrals over kx,y = πnx,y/L,
U (d) =
( L
π
) (^2) ∑ ′ nz
∫ (^) ∞
0
dkx
∫ (^) ∞
0
dkyhc¯
[ k x^2 + k^2 y +
( (^) πn z d
) 2 ]^1 /^2
. (28)
Recall that we are interested in the difference in the zero-point energy when d is varied. Therefore, it is useful to compare it to the case when the plates don’t exist. We can compute the zero-point energy density in the infinite volume as usual, and then calcualte the energy U 0 (d) multiplied by the volume L^2 d:
U 0 (d) = Ld^2
∫ (^) d~k
(2π)^3
¯hc
[ k^2 x + k y^2 + k z^2
] 1 / 2
= Ld^2
∫ (^) ∞
0
dkx π
∫ (^) ∞
0
dky π
∫ (^) ∞
0
dkz π
hc¯
[ k^2 x + k^2 y + k z^2
] 1 / 2
. (29)
What is observable is the difference
U (d) − U 0 (d)
L^2
π^2
¯hc
∫ (^) ∞
0
dkxdky
∑ (^) ′ nz
[ k^2 x + k^2 y +
( πnz d
) 2 ] 1 / 2 −
d π
∫ (^) ∞
0
dkz
[ k^2 x + k y^2 + k z^2
] 1 / 2
.
(30)
Switching to the circular coordinates,
∫ (^) ∞
0
dkxdky =
∫ (^) ∞
0
k⊥dk⊥
∫ (^) π/ 2
0
dφ =
π 4
∫ (^) ∞
0
dk^2 ⊥,
we find
U (d) − U 0 (d)
=
L^2
π^2
¯hc
π 4
∫ (^) ∞
0
dk ⊥^2
∑ (^) ′ nz
[ k ⊥^2 +
( (^) πn z d
) 2 ] 1 / 2 −
d π
∫ (^) ∞
0
dkz
[ k ⊥^2 + k^2 z
] 1 / 2
.
(31)
This expression appears problematic because it looks badly divergent. The divergence appears when the wave vector is large, which corresponds to high frequency photons. The point is that the conducting plates are transparent to, say, gamma rays, or in general for photons whose wavelengths are shorter than interatomic separation a. Therefore, there is naturally a damping factor f (ω) = f (c(k^2 ⊥ + k^2 z )^1 /^2 ) with f (0) = 1 which smoothly cuts off the integral for ω ∼> c/a. In particular, f (∞) = 0. Then the expression is safe and allows us to use standard mathematical tricks. Changing to dimensionless variables u = (k⊥d/π)^2 and nz = kz d/π,
U (d) − U 0 (d) =
π^2 L^2 ¯hc 4 d^3
∫ (^) ∞
0
du
{ ∑ (^) ′ nz
√ u + n^2 z −
∫ (^) ∞
0
dnz
√ u + n^2 z
} f (ω).
(32) Interchanging the sum and integral, we define the integral
F (n) =
∫ (^) ∞
0
du
u + n^2 f (ω) =
∫ (^) ∞
n^2
du
uf (ω) (33)
with ω = πc
u/d in the last expression. Using this definition, we can write
U (d) − U 0 (d) =
π^2 L^2 ¯hc 4 d^3
{ 1 2
F (0) +
∑^ ∞
n=
F (n) −
∫ (^) ∞
0
dnF (n)
}
. (34)
By comparing both sides of Eq. (38), we now find
∑^ ∞ n=
F (n) =
∫ (^) ∞
0
F (x)dx + 1 2 F^ (0)^ −
∑^ ∞
k=
B 2 k 1 (2k)! F^
(2k−1)(0). (43)
Moving first two terms in the r.h.s. to the l.h.s, we obtain the Euler–McLaurin formula
1 2 F (0) +
∑^ ∞ n=
F (n) −
∫ (^) ∞
0
F (x)dx = −
∑^ ∞
k=
B 2 k 1 (2k)! F (2k−1)(0). (44)
Wondering why all odd Bk vanish except for B 1? It is easy to check that
x ex^ − 1
1 2 x = x(2 + ex^ − 1) 2(ex^ − 1) = x 2 coth x 2 (45)
which is manifestly an even function of x. Going back to the definition of our function F (n) Eq. (33), we find
F ′(n) = − 2 n^2 f (πcn/d). (46)
As we will see below, we have d of order micron in our mind. This distance is far larger than the interatomic spacing, and hence f (πcn/d) is constant f = 1 in the region of our interest. Therefore, we can ignore derivatives f (n)(0), and hence the only important term in the Euler–McLaurin formula Eq. (35) is F ′′′(0) = −4. We obtain
U (d) − U 0 (d) = −
π^2 L^2 ¯hc 4 d^3
F ′′′(0)
B 4 = −
π^2 L^2 ¯hc 720 d^3
In other words, there is an attractive force between two conducting plates
F = [U (d) − U 0 (d)]′^ =
π^2 L^2 ¯hc 240 d^4
which is numerically 0 .013dyn/cm^2 (d/μm)^3
per unit area. This is indeed a tiny force, but Sparnay has observed it for the first time in 1958. He placed chromium steel and aluminum plates at distances between 0.3–2μm, attached to a spring. The plates are also connected to a capacitor. By measuring the capacitance, he could determine the distance, while the known sping constant can convert it to the force.
5 Lamb Shift
We learned with Dirac equation that states of hydrogen atom with the same principal quantum number n and the total angular momentum j remain de- generate despite the corretions from spin-orbit coupling, relativistic correc- tions, and Darwin term. They are, however, split as a result of full quantum interactions between the electron and photons. This is what Willis Lamb found after he worked on war-time radar technology during the WWII and came back to his lab, applied his radar technology to the hydrogen atom. He found transition spectrum between 2s 1 / 2 and 2p 1 / 2 states of about 1 GHz. You heard about the Darwin term pushing the s-states up because the Zitterbewegung smears the electric field and gives rise to a delta-function potential at the origin. If there is an additional reason for the jitter of the electron, it would contribute more to the similar effect. The additional reason is the zero-point fluctuation of the radiation field. Each momentum mode of the photon has the zero-point fluctuation, and each of them jiggles the electron. That would make the electron jitter a little bit more in addition to the Zitterbewegung and pushes the s-state further up. Let us for simplicity treat the electron non-relativistically and see how much it gets jiggled by the zero-point motion of the electric field. The clas- sical equation of motion for the “jiggle” part of the electron position is
δ~x¨ =
e m
E.~ (50)
As we discussed in the case of the Darwin term generated by the Zitterbewe- gung, such a “jiggle” would smear the electric field and generate additional potential term
∆V =
〈δxiδxj^ 〉
∂^2 eφ ∂xi∂xj^
〈(δ~x)^2 〉∆(eφ). (51)
We are interested in the electric field caused by the zero-point motion. For each frequency mode of the photon, with frequency ω, we find then
δ~xω = −
e mω^2
E~ω. (52)
Therefore the size of the fluctuation in the electron position is
〈(δ~xω)^2 〉 =
e^2 m^2 ω^4
〈 E~ ω^2 〉. (53)
with α = e^2 /¯hc = 1/137. Following the calculation of the energy shift of s- states from the Darwin term, we find the additional potential term Eq. (51) to be
∆V =
〈(δ~x)^2 〉∆(eφ)
2¯h^2 α πm^2 c^2
log
Zα
4 πZe^2 δ(~x)
4¯h^3 Zα^2 3 m^2 c
log
Zα
δ(~x). (58)
The resulting energy shift for nS-states is
∆En '
4¯h^3 Zα^2 3 m^2 c
log
Zα
|ψn(0)|^2
[ 8(Zα)^4 α 3 π
log
Zα
] mc^2
2 n^3
Therefore, this contribution is suppressed relative to the fine-structure by α/π but is enhanced by a logarithm log α−^1 = 4.9. For n = 2, it gives about 1 GHz for the microwave resonance frequency between 2p and 2s, in rough agreement with data as we will see below. The standard calculation uses Feynman diagrams, where the electron emits a virtual photon before it interacts with the Coulomb potential, and after the interaction it reabsorbes the virtual photon. This diagram, called the vertex correction, is actually divergent both in the ultraviolet and the infrared; reminiscent of the discussion above. It turns out, however, that the ultraviolet divergence is of a different character. The piece that corresponds to the amount of “jiggling” of the electron, more correctly called the “charge radius” of the electron, is actually ultraviolet finite in the fully relativistic calculations, supporting the rough “cutoff” at ω ∼ mc^2 /¯h employed above. The ultraviolet divergence in this Feynman diagram is properly cancelled by another ultraviolet divergence called “wave function renormalization.” When you use the second-order perturbation theory, your state |φn〉 is modified to
|φn〉 → |χn〉 = |φn〉 +
∑
i 6 =n
|i〉〈i|V | 0 〉 E 0 − Ei
∑
i,j 6 =n
|j〉〈j|V |i〉〈i|V | 0 〉 (En − Ej )(En − Ei)
where | 0 〉 is the zeroth order state (not the vacuum) and |i〉 other states that mix with | 0 〉 due to the perturbation V. However, this state is not correctly
normalized, because
〈χn|χn〉 = 1 +
∑
i
|〈φn|V |i〉|^2 (En − Ei)^2
To correctly normalize the perturbed state |χn〉, we need to “renormalize” it as
|χn〉′^ = |χn〉
∑ i |〈φn|V^ |i〉|^2 /(En^ −^ Ei)^2
In the case of the QED, it corresponds to the Feynman diagram where the electron emits a virtual photon and reabsorbes it without any other interac- tions, which is also ultraviolet divergent. This change in the normalization of the state can be shown to precisely cancel the ultraviolet divergence in the vertex correction, and hence there is no problem with the apparent di- vergences. The present theoretical and experimental situation is reviewed, for exam- ple, in M.I. Eides, H. Grotch and V.A. Shelyuto, “Theory of light hydrogen- like atoms,” Phys. Rep., 342 , 63-261, (2001). The best experimental value of the 2s–2p splitting is 1 .057 845(3) GHz. (63)
The theoretical calculations depend on variety of other corrections in addition to the effect I had discussed, including the fact that the charge of the proton is not strictly point-like. The charge radius of the proton is not well determined experimentally, and limits the theoretical accuracy in calculating the level splitting. Using one particular measurement of the proton charge radius 0 .862(12) fm, the theory gives
1 .057 833(2)(4) GHz, (64)
which disagrees with data at more than 2 sigma level. But other measure- ments of the proton charge radius disagree with this measurement, and the discrepancy becomes larger. The inconsistency among the data makes it im- possible for us to draw any conclusions beyond a simple qualitative statement that theory and data agree for 6 digits.
6 Anomalous Magnetic Moment
We learned from the Dirac equation that the gyromagnetic ratio g = 2. This is certainly in a good agreement with data. But both experiments and
theoretical calculations had progressed greatly since old days. (See V.W. Hughes and T. Kinoshita, Rev. Mod. Phys. 71 , S133-S139 (1999). ) The crucial theoretical progress usually attributed to Schwinger is that there is a correction at O(α) to the g-factor. Because of the quantum fluc- tuations, an electron is sometimes not just an electron but is accompanied by a photon. More precisely, the process is the following. An electron inter- acts with the magnetic field with g = 2. But before the electron interacts with the magnetic field, it can emit a (virtual) photon, violating the energy conservation. The electron then interacts with the magnetic field, and only after that, it reabsorbes the photon. Schwinger’s calculation showed that the g-factor is not longer precisely 2 but is corrected as
g = 2
( 1 +
α 2 π
)
. (65)
More recently, Kinoshita has calculated the correction up to O(α^4 ) with 891 Feynman diagrams. It is customary to quote ae = (ge − 2)/2. He found
ae =
α 2 π
( (^) α
π
) 2
( (^) α
π
) 3
( α π
) 4
- 4.393(27) × 10 −^12. (66)
The last constant is a contribution from known heavier particles, such as μ, τ leptons, hadrons, and W , Z bosons. In order to obtain a numerical value, we need to know the fine structure constant α very precisely. The best measurement comes from the Quantum Hall Effect,
α−^1 = 137.036 003 7(33). (67)
Using this value as the input, the QED predicts the electron magnetic mo- ment to be ae = 1 159 652 153.5(1.2)(28.0) × 10 −^12. (68)
The first uncertainty comes from the estimated error in the numerical in- tegrations involved in the calculations, while the latter (and the dominant) error is in the input value of α. The experiment also had made a dramatic progress. The most precise measurement was done using Penning trap. If g were exactly two, an electron moving in a magnetic field would have precisely the same values for the
cylotron frequency and the spin precession frequency. The difference between them measures g − 2. The best values due to Van Dyck are
ae− = 1 159 652 188.4(4.3) × 10 −^12 , (69) ae+^ = 1 159 652 187.9(4.3) × 10 −^12. (70)
The agreement between experiment and theory is truly amazing. The exper- imental accuracy is 4 × 10 −^12 , which agrees with theory at 1.3 sigma level. Incredible! This is the most dramatic success of quantum physics, most likely of all physical sciences. The anomalous magnetic moment of muon is also interesting. Because muon is short-lived (lifetime is microsecond), the experimental measurement is more difficult. The trick is to actually use the decay product of the muon, μ−^ → e−^ ν¯eνμ, where you can detect the electron (but not neutrinos). Luckily, parity is violated in this decay (!),^1 and the direction of the decay electron is correlated with the muon spin. By measuring the direction of the de- cay electrons, we can measure the muon spin precession and hence gμ − 2. Theoretically, muon is heavier and the anomalous magnetic moment is more sensitive to heavier particles than that of electron. In fact, the contribution from hadrons (pions, various mesons, protons, etc) is quite important. You may even hope that it may detect the effect of yet-undiscovered particles. The theoretical prediction is (see, e.g., U. Chattopadhyay, A. Corsetti, P. Nath, http://arXiv.org/abs/hep-ph/0204251)
aμ = 11 659 176.8(6.7) × 10 −^10. (71)
Currently a new experiment is being conducted at Brookhaven National Laboratory and has measured the (anti-)muon magnetic moment. They re- ported (see H. N. Brown et al. [Muon g − 2 Collaboration], Phys. Rev. Lett. 86 , 2227 (2001), http://arXiv.org/abs/hep-ex/0102017. See also http://phyppro1.phy.bnl.gov/g2muon/index.shtml)
aμ = 11 659 202(14)(6) × 10 −^10. (72) (^1) According to Leon Lederman’s book “The God Particle,” he conceived the parity- violation experiment in muon decay when he heard of the rumor that C.S. Wu found “large” parity violation in nuclear β-decay. Together with his collaborators, he rushed to the laboratory where a graduate student was mounting his thesis experiment. Quickly they disassembled it (!), and mounted the parity violation experiment. They result is reported in Physical Review Letters right after Wu’s paper. You should feel lucky if your supervisor is not too keen on a timely success.
