Download Electroweak Unification as Classical Field Theory

Electroweak Unification as Classical Field Theory
Sarah Hoffman
Oberlin College Physics Department
9 April 2010
Contents
Chapter 1.
Executive Summary
2
Chapter 2.
Introduction
3
Chapter 3. Electromagnetism
1. Maxwell’s Equations in terms of fields
2. Formulation in terms of potentials
2.1. Poisson’s Equation and distance dependence
3. Covariant formulation
4
4
4
5
5
Chapter 4. Lagrangian Formulation
1. Lagrangian Field Theory
2. Lagrangian Derivation of Maxwell’s Equations
7
7
7
Chapter 5. Electroweak Theory
1. The Weak Force
2. Field vs. Gauge Boson Formulation
3. A note on units
4. Weak Potentials, Currents and Fields
9
9
9
10
10
Chapter 6. Electroweak Maxwell’s Laws
1. Electroweak Lagrangian
2. Equations of Motion
11
11
12
Chapter 7. Interpreting the Equations
1. Vector notation for weak interactions
2. Potentials as sources of electromagnetic field
3. Gauge freedom
4. Where are there weak fields?
4.1. Chirality
5. Weak fields due to weak charge
6. Example: right-handed electrons
7. Example: left-handed electrons
15
15
15
16
17
17
18
18
18
Chapter 8. Concluding Remarks
1. Results
2. Further questions
3. Acknowledgment
21
21
21
21
Chapter 9. Appendix
1. Euler-Lagrange equations for a function and its complex conjugate
22
22
Bibliography
24
1
CHAPTER 1
Executive Summary
Electromagnetism – the interaction between electrically charged particles – is (along
with gravitation and the strong and weak interactions) one of the four fundamental forces
that account for all physical phenomena. Although the word brings to mind applications
like magnets and electrical circuits, electromagnetism is also responsible for much more of
the world we experience, including the very structure of matter: this is the force that holds
electrons to nuclei within atoms and binds atoms into molecules.
In classical physics, we think of the electromagnetic force as acting through the mechanism of fields. An electrical charge or current fills the surrounding space with an electric
or magnetic field, which then exerts a force on any other electrical charge or current that
happens to be present. We can represent these fields by vectors, or arrows, at every point
in space, and connect these arrows to form a picture of field lines. Maxwell’s Laws are a
set of simple and elegant equations that describe the behavior of electromagnetic fields.
Classical electromagnetism as summarized by Maxwell’s Laws is quite sufficient for almost all situations. However, according to our most complete understanding of physics,
Maxwell’s Laws do not tell the whole story. The electromagnetic interaction is fundamentally intertwined with another – the weak interaction. Weak interaction processes are much
less familiar to us than electromagnetic ones, and with good reason: the weak force acts only
over a miniscule range, far smaller than the width of an atom’s nucleus. (Its best-known
effect is a certain type of radioactive decay.) This interaction can usually be ignored because
its range is so small, but to be completely accurate, we cannot describe electromagnetism
without involving the weak interaction.
That unified description of the electromagnetic and weak interactions – electroweak or
GWS theory – was developed in the 1960s, primarily by Sheldon Glashow, Abdus Salam, and
Steven Weinberg, and recognized with the 1979 Nobel Prize in physics [1]. Its mathematical
predictions have been thoroughly confirmed. However, this theory is formulated in ways
that are much harder to picture or understand conceptually than the classical field theory
of electromagnetism.
This thesis works to bridge that conceptual gap. Our starting point is the electroweak
theory as presented in the quantum field theory text by F. Mandl and G. Shaw [2]. We
discuss some qualitative features of the weak interaction that are implied by the mathematical formulation. Next, and most importantly, we derive the electroweak-corrected version
of Maxwell’s Laws, a version with additional terms to describe the electromagnetic fields
produced by weak fields. Finally, we apply this corrected set of Maxwell’s Laws to a few
examples and begin to develop a classical field-lines picture of the electroweak interaction.
2
CHAPTER 2
Introduction
Almost every physicist is familiar with the fact of electroweak unification. Two of the
four fundamental forces, the electromagnetic and the weak, are inseparable; on a deep level
they are two manifestations of the same interaction. Few, though, could say what exactly
this unification means for the classical electromagnetism that we learned in our first physics
courses. We depicted the electromagnetic interaction with field lines, and we summarized
the behavior of the field lines with Maxwell’s Equations. How do those equations and that
field-line picture change when we include the weak force, which electroweak unification says
that, to be entirely accurate, we must?
This thesis addresses these questions by reformulating the electroweak theory – given
in Lagrangian form in Mandl and Shaw’s Quantum Field Theory – as a classical field
theory. We derive an electroweak version of Maxwell’s Equations and discuss qualitative
features of the electroweak fields. We are, in a sense, investigating the classical limit of
an interaction that is not actually manifested on a classical scale: the reason the weak
interaction can so often be ignored is that its range is orders of magnitude smaller than the
size of a nucleus. The investigation, however, is valuable to our conceptual understanding
of electromagnetism and of the weak interaction. There is also a potential application in
cosmology, as in the earliest moments after the Big Bang the symmetry between the two
interaction was unbroken and weak fields had the same unlimited range as electromagnetic
ones.
3
CHAPTER 3
Electromagnetism
1. Maxwell’s Equations in terms of fields
Classical electrodynamics is described by Maxwell’s Equations:
(3.1)
(3.2)
(3.3)
∇·E=ρ
1
∇×B=
c
∂E
j+
∂t
∇·B=0
1 ∂B
c ∂t
where E represents the electric field, B the magnetic field, ρ the charge density, and j the
current density. Here the units used are Lorentz-Heaviside, or rationalized cgs, units.
As well as providing a complete mathematical formulation of the electromagnetic interaction, these equations are relatively easy to interpret conceptually, allowing us to visualize
that interaction in terms of electric and magnetic field lines. Equation (3.1), Gauss’s Law,
tells us that electric charge creates electric field, which diverges from positive charge and
converges at negative charge. Equation (3.2), Ampere’s Law with Maxwell’s correction, tells
us that electric current and change in electric field both create magnetic field, which does
not diverge, but curls in loops. Equation (3.3) tells us that there is no magnetic equivalent
of charge on which magnetic field lines begin and end. Equation (3.4), Faraday’s Law, tells
us that change in magnetic field creates curling electric field.
This conceptual picture is familiar to anyone who has taken an introductory course in
electromagnetism. Our goal is to develop a similar qualitative insight into the electroweak
interaction.
(3.4)
∇×E=−
2. Formulation in terms of potentials
We can also describe the electromagnetic interaction in terms of a scalar potential and
a vector potential. The divergence of a field is zero if and only if that field is the curl of
some vector, so Eq. (3.3) implies the existence of a vector potential A such that
(3.5)
B = ∇ × A.
Combining this with Eq. (3.4), and with the fact that the curl of a field is zero if and only
if it is the gradient of some scalar, we can define a scalar potential φ such that
1 ∂A
.
c ∂t
Making these substitutions, and using the D’Alembertian
(3.6)
(3.7)
E = −∇φ −
=
1 ∂2
− ∇2 ,
c2 ∂t2
4
3. COVARIANT FORMULATION
5
Eq. (3.1) can be rewritten as
(3.8)
φ −
1∂
c ∂t
1 ∂φ
+∇·A
c ∂t
=ρ
and Eq. (3.2) as
(3.9)
A + ∇
1 ∂φ
+∇·A
c ∂t
1
= j.
c
In short: defining the potentials is equivalent to the no-magnetic-charges law and Faraday’s Law. Gauss’s Law and Ampere’s Law are then expressed in terms of the potentials to
complete the set of Maxwell’s Equations. This potential formulation contains all the same
information as the field formulation.
On a conceptual level, there are “hills” of scalar potential at positive charges and “valleys” at negative charges, and vector potential arrows point in the direction of current.
2.1. Poisson’s Equation and distance dependence. We have focused on the relationship between the fields or potentials and the charges. Note, however, that Maxwell’s
Equations also contain a description of the fields’ or potentials’ dependence on distance.
For example, in the static case, Eq. (3.8) reduces to Poisson’s equation,
∇2 V = −ρ.
(3.10)
For a localized volume charge distribution, this equation has the solution
Z
1
ρ(r0 )
(3.11)
V (r) =
dτ 0 ,
4π
(r − r0 )
indicating that V goes as the inverse of (r − r0 ), the distance from the source. If we apply
this formula to a point charge Q at the origin, we have
(3.12)
V (r) =
1 Q
,
4π r
which leads to Coulomb’s Law,
1 Q
r̂,
4π r2
the 1/r2 dependence of the electric field [8, 120–121]. Similarly, Eq. (3.9) in the static case
is
(3.13)
(3.14)
E(r) =
∇2 A = −j,
with the solution
(3.15)
1
A(r) =
4π
Z
j(r0 )
dτ 0 ,
(r − r0 )
so A falls off as 1/r, which leads to B falling off as 1/r2 [8, 258–259].
3. Covariant formulation
Electromagnetism can be expressed somewhat more elegantly in the covariant or relativistic formulation, using tensors and four-vectors rather than ordinary vectors and scalars.
We will need this notation when we come to the electroweak interaction, so we introduce it
for electromagnetism here.
We define the contravariant space-time four-vector components as
(3.16)
xµ = (ct, x, y, z) = (ct, x),
6
3. ELECTROMAGNETISM
and the covariant components as
xµ = (ct, −x, −y, −z) = (ct, −x),
(3.17)
using the index µ = (0, 1, 2, 3). We use Einstein summation notation, where
µ
(3.18)
a bµ =
3
X
aµ bµ .
µ=0
Two properties worth noting: we can change an index from subscript to superscript if we
make the opposite change to the same index within a product,
(3.19)
aµ bµ = aµ bµ ,
F µν aµ bν = Fµν aµ bν ,
etc.
and if an index is to be summed, we can freely replace it with a different index, as
aµ bµ = aν bν .
(3.20)
The x-, y-, and z-components of electric and magnetic field are combined in the antisymmetric field tensor


0
Ex
Ey
Ez
−Ex
0
Bz −By 
.
(3.21)
F µν = 
−Ey −Bz
0
Bx 
−Ez By −Bx
0
From the charge density and current density we can form the four-vector current
sµ = (cρ, j).
(3.22)
The scalar potential and vector potential form a four-vector potential
Aµ = (φ, A).
(3.23)
Equations (3.5) and (3.6), relating the fields to the potentials, now become
(3.24)
F µν = ∂ ν Aµ − ∂ µ Aν ,
where we use the notation
(3.25)
∂µ =
∂
∂xµ
and ∂µ =
∂
.
∂xµ
Just as the definitions of the potentials did in Section 2 above, Eq. (3.24) contains the
information of two of Maxwell’s Laws, Eqs. (3.3) and (3.4), which in this notation can be
written as the single equation
(3.26)
∂ λ F µν + ∂ µ F νλ + ∂ ν F λµ = 0.
The rest of Maxwell’s Laws, Eqs. (3.1) and (3.2), are expressed in this notation by
1
(3.27)
∂ν F µν = sµ .
c
CHAPTER 4
Lagrangian Formulation
1. Lagrangian Field Theory
Suppose a system is specified by a set of fields. These fields could be scalars, vectors,
four-vectors, tensors, etc. We label the components of those fields as φ1 , φ2 , ... , φn . So,
for example, φ4 might represent F 03 , the (µ = 0, ν = 3) component of the field tensor.
The system’s Lagrangian density is some function of the field components and their first
derivatives:
(4.1)
L = L(φ1 , ..., φn , ∂µ φ1 , ..., ∂µ φn ).
Although a Lagrangian density could potentially depend on second or higher derivatives of
the fields, we will not need that more general and complex case here.
By the principle of least action [3, 27–28], the Euler-Lagrange equations of motion for
the system are
∂L
∂L
= ∂µ
, with K = 1, 2, ..., n.
(4.2)
∂φK
∂(∂µ φK )
The meaning of such derivatives is not obvious – derivatives are usually taken with
respect to independent variables, not with respect to complicated functions. The answer is
that while carrying out the derivative we treat the function as a variable or mere placeholder,
ignoring what it represents. Afterwards, in the result, we replace that placeholder with the
function’s actual identity. For example, to find
∂L
,
∂(∂µ φK )
one could replace ∂µ φK with the symbol X every time it appears in the Lagrangian density,
∂L
calculate ∂X
, and then replace every X in the result with ∂µ φK .
2. Lagrangian Derivation of Maxwell’s Equations
The classical electromagnetic Lagrangian is
(4.3)
1
1
L = − Fµν F µν − sµ Aµ .
4
c
Applying the definition of the potential, (3.24), we have
(4.4)
1
1
L = − (∂ν Aµ − ∂µ Aν )(∂ ν Aµ − ∂ µ Aν ) − sµ Aµ
4
c
1
1
= − (∂ν Aµ ∂ ν Aµ − ∂µ Aν ∂ ν Aµ − ∂ν Aµ ∂ µ Aν + ∂µ Aν ∂ µ Aν ) − sµ Aµ
4
c
1
1 µ
ν µ
µ ν
= − (∂ν Aµ ∂ A − ∂ν Aµ ∂ A ) − s Aµ .
2
c
7
8
4. LAGRANGIAN FORMULATION
Thus our Lagrangian is dependent on a single four-vector field Aµ and its derivative
∂ν Aµ . The Euler-Lagrange equation of motion is
∂L
∂L
(4.5)
= ∂ν
.
∂Aµ
∂(∂ν Aµ )
We calculate
∂L
1
(4.6)
= − sµ
∂Aµ
c
and
∂L
1
(4.7)
= − [2(∂ν Aµ ) − 2(∂µ Aν )] = −(∂ ν Aµ − ∂ µ Aν ) = −F µν ,
∂(∂ν Aµ )
2
so the equation of motion is
1
(4.8)
∂ν F µν = sµ ,
c
which is the covariant form of Maxwell’s Equations (3.27). In other words, Maxwell’s
Equations are the equations of motion for the electromagnetic Lagrangian density.
CHAPTER 5
Electroweak Theory
1. The Weak Force
What is the weak force? The simplest response might be that it is the force responsible
for the decays of certain subatomic particles, including the neutron. A more sophisticated
answer adds that it is a force of extremely short range, on the order of 10−17 m; that it is
intrinsically weaker than the strong and electromagnetic forces but stronger than gravity;
that it violates parity; that it is carried by massive bosons, some of them electrically charged
and some of them neutral; that unlike the strong and electromagnetic forces it acts on all
of the elementary particles. Qualitative introductions to the weak force can be found in
Watkins [4, 16–17 and 46–48] and Griffiths [5, 56 and 65–72].
Our conceptual grasp of the weak force, on the whole, is not nearly as clear as our
understanding of electromagnetism. There is a good reason for this: the weak force works
only on subatomic scales. There are no macroscopic weak force phenomena for us to experience firsthand the way we can experience electromagnetic phenomena by handling magnets,
circuit components, and so on.
2. Field vs. Gauge Boson Formulation
Because it only becomes apparent on a subatomic scale, the theory of the weak interaction was a quantized theory from the beginning. Griffiths, [5, 44 and 56], gives some of the
history. In a quantum formulation, forces are considered to be transmitted not by fields,
but by the exchange of virtual particles called gauge bosons. Quantized theories like this
are generally formulated in terms of Lagrangians or Hamiltonians, or they are pictured with
Feynman diagrams, from which one calculates quantities such as lifetimes and scattering
cross sections.
Electromagnetism, of course, also has a quantized form, Quantum Electrodynamics
(QED), in which virtual photons replace the fields as carriers of the force. But the fields
formulation – the classical limit – is by no means obsolete and it is certainly easier to
understand. Perhaps a classical limit of the electroweak theory would similarly be more
transparent than the usual formulations.
One might question whether there is a classical limit. Does it even make sense to talk
about the classical picture of a force that only acts over quantum-scale distances? Consider,
though, that the weak interaction does not inherently need to be of such a miniscule range.
The force has a short range because the gauge bosons that carry it are massive, and the boson masses we plug into the theory are calculated from experimentally measured parameters
[2, 301–302]. We could take the same electroweak Lagrangian and plug in smaller, or zero,
boson masses; presumably it would then describe a weak force that did act on a macroscopic
scale. This, in a sense, is the realm we are investigating. In fact, such conditions once existed: the underlying symmetry between weak and the electromagnetic interactions, which
is broken in the world we see today, is unbroken above some very high critical temperature.
For a brief time after the Big Bang – a period called the electroweak epoch – the universe
9
10
5. ELECTROWEAK THEORY
exceeded this temperature, and the weak interaction produced long-range forces like the
Coulomb force [6].
3. A note on units
Mandl and Shaw give the theory of the electroweak interaction in natural units – that
is, c.g.s. units with ~ and c set to 1 [2, 96–97]. In this system, for example, the covariant
formulation of Maxwell’s Equations (3.27) becomes
∂ν F µν = sµ ,
(5.1)
with the factor of 1/c having been absorbed into the definition of sµ .
Natural units make the formulas easier to work with, but perhaps harder to interpret
or understand: since all quantities now have dimensions of mass raised to some power, and
many different quantities share the same dimensions, much of the insight that could be
gained from dimensional analysis is lost. For now, we follow Mandl and Shaw’s choice of
natural units.
4. Weak Potentials, Currents and Fields
In analogy to the electromagnetic potential field Aµ , we introduce the two “charged
weak” potential fields W µ and W †µ [2, 237] and the “neutral weak” potential field Z µ [2,
271]. W µ and W †µ describe interactions mediated by exchange of electrically charged W ±
bosons, while Z µ describes interactions mediated by exchange of electrically neutral Z 0
bosons. In classical terms, W †µ stands for the complex conjugate of W µ [2, 298].
To describe the way in which particles act as sources of electromagnetic field, we have
the electromagnetic current sµ . For each of our weak potentials, then, we introduce an
analogous quantity called a weak current: two charged weak currents J µ and J †µ [2, 236],
and the neutral weak current [2, 272] JZµ . These names are meant only to suggest which
current is associated with which potential – not the electrical charge, or lack thereof, of
the currents themselves. For example, charged particles such as electrons can participate
in processes mediated by the Z 0 , and can therefore make up a “neutral weak current.”
This simple notation hides some differences between the weak and the electromagnetic
interactions which would be revealed if we expressed the currents in terms of particle field
operators; these include the violation of parity and the fact that one particle species is
converted to another. For our purposes, however, is useful to treat the currents in this
parallel way.
For each of the weak potentials, we define a tensor [2, 242, 274]:
(5.2)
µν
FW
(5.3)
†µν
FW
(5.4)
FZµν
= ∂ν W µ − ∂µW ν
= ∂ ν W †µ − ∂ µ W †ν
= ∂ν Z µ − ∂µZ ν .
These tensors, simply by virtue of their definitions, have an important feature in common
with the electromagnetic field strength tensor F µν : they are antisymmetric, and so each
one has only six independent components. We could choose to label those six components
as the components of two vectors, as in Eq. (3.21). This is encouraging, because it means
that we can think about weak-interaction analogues to electric and magnetic field – what
we might call EW , BW , E†W , B†W , EZ , and BZ .
We can even say something more about those weak E and B fields: the definitions (5.2)
- (5.4) guarantee us, in each case, the equivalents of Eqs. (3.3) and (3.4), the second pair of
Maxwell’s Laws.
CHAPTER 6
Electroweak Maxwell’s Laws
1. Electroweak Lagrangian
Our Lagrangian density for the electroweak interaction is from Mandl and Shaw [2,
299-300], with some modifications: first, the Higgs field terms, which are outside the scope
of this project, are omitted, although particle mass terms are included, so what we have
here is the Glashow model [275] and is not gauge-invariant. Second, the source terms are
written using the currents introduced above rather than field operators, for example −sµ in
place of eψ̄l γ µ ψl , and free-lepton terms are left out because they will not contribute to the
equations of motion for the fields. Mandl and Shaw’s treatment covers only leptons, not
hadrons [2, 236].
(6.1)
L=
−
(6.2)
−
(6.3)
−
(6.4)
+
+(∂µ Wν − ∂ν Wµ )W †ν Z µ − (∂µ Wν† − ∂ν Wµ† )W ν Z µ ]
(6.5)
(6.6)
1
Fµν F µν
4
1 †
F µν + m2W Wµ† W µ
F
2 W µν W
1
1
FZµν FZµν + m2Z Zµ Z µ
4
2
†
ig cos θW [(Wµ Wν − Wν† Wµ )∂ µ Z ν
+
ie[(Wµ† Wν − Wν† Wµ )∂ µ Aν
+(∂µ Wν − ∂ν Wµ )W †ν Aµ − (∂µ Wν† − ∂ν Wµ† )W ν Aµ ]
(6.7)
(6.8)
+
g 2 cos2 θW [Wµ Wν† Z µ Z ν − Wν W †ν Zµ Z µ ]
(6.9)
+
e2 [Wµ Wν† Aµ Aν − Wν W †ν Aµ Aµ ]
(6.10)
+
(6.11)
+
(6.12)
−
(6.13)
−
(6.14)
−
eg cos θW [Wµ Wν† (Z µ Aν + Aµ Z ν ) − 2Wν W †ν Aµ Z µ ]
1 2 †
g Wµ Wν [W †µ W ν − W µ W †ν ]
2
s µ Aµ
g
√ [J µ† Wµ + J µ Wµ† ]
2 2
g
J µ Zµ
cos θW Z
Line (6.1) describes the free electromagnetic field, and line (6.12) describes that field’s
interaction with the electromagnetic current; these terms are just as in the classical Lagrangian (4.3). Lines (6.2) and (6.3) are for the free charged weak and neutral weak fields,
respectively. Lines (6.4) through (6.11) are for interactions between the various gauge
bosons, while lines (6.13) and (6.14) are for interaction with the charged weak and neutral
weak currents respectively.
Here mW and mZ are the gauge boson masses. The weak mixing angle or Weinberg
angle θW and the weak charge unit g are related to the electromagnetic charge unit e by
11
12
6. ELECTROWEAK MAXWELL’S LAWS
[2, 271]
(6.15)
g sin θW = e,
and experiments have determined [2, 272]
sin2 θW = 0.227 ± 0.014.
(6.16)
2. Equations of Motion
Our task now is to find the equations of motion for this electroweak Lagrangian. There
will be four equations, one for each of the potential fields Aµ , W µ , W †µ , and Z µ . We will
show in detail the derivation of the first of these, the equation
∂L
∂L
=
.
(6.17)
∂ν
∂(∂ν Aµ )
∂Aµ
On the right-hand side, only the terms of the Lagrangian that contain Aµ will contribute.
Those are lines (6.7), (6.9), (6.10), and (6.12). Note that lines (6.1) and (6.6) do not
contribute: they are functions of the derivative of Aµ , but not of Aµ itself.
∂L
∂[(6.7)] ∂[(6.9)] ∂[(6.10)] ∂[(6.12)]
=
+
+
+
.
∂Aµ
∂Aµ
∂Aµ
∂Aµ
∂Aµ
We can rewrite line (6.7), switching subscripts and superscripts by property (3.19) and then
using the definitions (5.2) and (5.3) to simplify, as
µν
†µν
Wµ† )Aµ ,
Wν − FW
ie[(∂ µ W ν − ∂ ν W µ )Wν† Aµ − (∂ µ W †ν − ∂ ν W †µ )Wν Aµ ] = ie(FW
so we find
∂[(6.7)]
µν
†µν
Wµ† ).
Wν − FW
= ie(FW
∂Aµ
The next term requires some product rules:
∂
(6.18)
(Aµ Aµ ) = 2Aµ
∂Aµ
(6.19)
∂
(W µ Wν† Aν Aµ ) = W µ Wν† Aν + W †µ Wν Aν .
∂Aµ
With these we calculate
∂[(6.9)]
= e2 (Wν† Aν W µ + Wν Aν W †µ − 2Wν† W ν Aµ ).
∂Aµ
In the same way we find
∂[(6.10)]
= eg cos θW (Wν† Z ν W µ + Wν Z ν W †µ − 2Wν† W ν Z µ ),
∂Aµ
and the last term is simply
∂[(6.12)]
= −sµ .
∂Aµ
Turning to the left hand side, the relevant terms are lines (6.1) and (6.6). From the
calculation in Section 2 we know
∂[(6.1)]
= −F µν ,
∂(∂ν Aµ )
and the other term is linear in ∂ν Aµ so it straightforwardly gives us
∂[(6.6)]
= ie(W †ν W µ − W †µ W ν ).
∂(∂ν Aµ )
2. EQUATIONS OF MOTION
13
Putting all these terms together, the equation of motion for Aµ is
(6.20)
∂ν [−F µν + ie(W †ν W µ − W †µ W ν )] =
†µν
µν
ie(FW
Wν − FW
Wµ† ) + e2 Wν† Aν W µ + Wν Aν W †µ − 2Wν† W ν Aµ ) +
eg cos θW (Wν† Z ν W µ + Wν Z ν W †µ − 2Wν† W ν Z µ ) − sµ .
Rearranging this to separate out the dependence on each of the potentials, we have
∂ν F µν =
(6.21)
sµ + (2e2 Wν† W ν )Aµ − (e2 Wν† Aν + eg cos θW Wν† Z ν )W µ
−(e2 Wν Aν + eg cos θW Wν Z ν )W †µ + (2eg cos θW Wν W †ν )Z µ
µν
†µν
+ie[(FW
Wν† − FW
Wν ) + ∂ν (W †ν W µ − W †µ W ν )].
The last line can be further rewritten using product rules, the definitions of the field strength
tensors (5.2) and (5.3), and the Lorentz condition [2, 242]
∂µ W µ = 0.
(6.22)
The result is
(6.23)
∂ν F µν =
sµ + (2e2 Wν† W ν )Aµ − (e2 Wν† Aν + eg cos θW Wν† Z ν )W µ
−(e2 Wν Aν + eg cos θW Wν Z ν )W †µ + (2eg cos θW Wν W †ν )Z µ
+ie[Wν† (2∂ ν W µ − ∂ µ W ν ) − Wν (2∂ ν W †µ − ∂ µ W †ν )].
This is the electroweak analogue of Maxwell’s Equations in covariant form (5.1). Note that
each term here contains a summation over the index ν, but not over µ, so (6.23) represents
four equations, one for each value of µ = 0, 1, 2, 3.
The calculations of the equations of motion for W µ , W †µ , and Z µ are similar, and we
will only present the results. These are given in the form of Eq. (6.21) rather than expanded
like Eq. (6.23). The equation for W µ ,
∂L
∂L
(6.24)
∂ν
=
,
∂(∂ν Wµ )
∂Wµ
turns out to describe not the fields of W µ , but those of W †µ :
†µν
(6.25) ∂ν FW
=
g †µ
√ J − (e2 Wν† Aν + eg cos θW Wν† Z ν )Aµ − (g 2 Wν† W †ν )W µ
2 2
+(g 2 cos2 θW Zν Z ν + e2 Aν Aν + 2eg cos θW Aν Z ν + g 2 Wν† W †ν − m2W )W †µ
−(g 2 cos2 θW Wν† Z ν + eg cos θW Wν† Aν )Z µ
†µν
+ig cos θW [Zν FW
− Wν† FZµν + ∂ν (W †µ Z ν − W †ν Z µ )]
†µν
+ie[Aν FW
− Wν† F µν + ∂ν (W †µ Aν − W †ν Aµ )]
and, in turn, the equation derived from
"
(6.26)
∂ν
∂L
∂(∂ν Wµ† )
#
=
∂L
∂Wµ†
14
6. ELECTROWEAK MAXWELL’S LAWS
describes the W µ fields:
µν
(6.27) ∂ν FW
=
g µ
√ J − (e2 Wν Aν + eg cos θW Wν Z ν )Aµ
2 2
+(g 2 cos2 θW Zν Z ν + e2 Aν Aν + 2eg cos θW Aν Z ν + g 2 Wν W ν − m2W )W µ
−(g 2 Wν W ν )W †µ − (g 2 cos2 θW Wν Z ν + eg cos θW Wν Aν )Z µ
rµν
+ig cos θW [Zν FW
− Wν† FZµν + ∂ν (W µ Z ν − W ν Z µ )]
µν
+ie[Aν FW
− Wν F µν + ∂ν (W µ Aν − W ν Aµ )].
The Appendix addresses a concern about how to treat W µ and W †µ when finding these
equations. Finally,
∂L
∂L
(6.28)
∂ν
=
∂(∂ν Zµ )
∂Zµ
gives us
(6.29)
∂ν FZµν =
g
JZµ + (2eg cos θW Wν W †ν )Aµ
cos θW
−(g 2 cos2 θW Wν† Z ν + eg cos θW Wν† Aν )W µ
−(g 2 cos2 θW Wν Z ν + eg cos θW Wν Aν )W †µ
+(2g 2 cos2 θW Wν W †ν − m2Z )Z µ
µν
†µν
+ig cos θW [FW
Wν† − FW
Wν + ∂(W †ν W µ − W †µ W ν )].
CHAPTER 7
Interpreting the Equations
1. Vector notation for weak interactions
In Section 4, we noted that we could define weak analogues to electromagnetic quantities
such as E and B. With that in mind, we introduce some notation to describe the weak
interaction using scalars and three-vectors instead of the harder-to-picture four-vectors.
We separate the weak four-potentials into weak scalar and vector potentials: just as Aµ =
(φ, Ax , Ay , Az ) = (φ, A), we define
W µ = (φW , Wx , Wy , Wz ) = (φW , W)
W †µ = (φ†W , Wx† , Wy† , Wz† ) = (φ†W , W† )
(7.1)
Z µ = (φZ , Zx , Zy , Zz ) = (φZ , Z).
Similarly we separate the weak four-currents into weak charge and current densities. In
EM we had, in natural units, sµ = (ρ, j), so now we say
J µ = (ρW , jW ),
(7.2)
J †µ = (ρ†W , j†W ),
JZµ = (ρZ , jZ ).
We also define EW = (EW x , EW y , EW z ) and BW = (BW x , BW y , BW z ) with components
given by

(7.3)
µν
FW

0
EW x
EW y
EW z
−EW x
0
BW z −BW y 
 = ∂ν W µ − ∂µW ν ,
=
−EW y −BW z
0
BW x 
−EW z BW y −BW x
0
or
(7.4)
BW = ∇ × W,
EW = −∇φW −
∂W
,
∂t
and the same for W †µ and Z µ .
2. Potentials as sources of electromagnetic field
Since we don’t have much intuition about weak fields, the most immediately interesting
of the equations of motion is Eq. (6.23), the one that describes the electromagnetic field.
What it tells us is that not only electromagnetic currents but also all four electroweak
potentials are sources of electromagnetic field.
15
16
7. INTERPRETING THE EQUATIONS
To see this more clearly, let us put Eq. (6.23) in terms of three-vectors. The µ = 0
equation is
(7.5)
∇·E=
ρ + 2e2 (φ†W φW − W† · W)φ
−[e2 (φ†W φ − W† · A) + eg cos θW (φ†W φZ − W† · Z)]φW
−[e2 (φW φ − W · A) + eg cos θW (φW φZ − W · Z)]φ†W
+2eg cos θW (φ†W φW − W† · W)φZ
+ie[W† · (2∇φW +
∂W
∂W†
) − W · (2∇φ†W +
)],
∂t
∂t
which simplifies to
(7.6)
∇·E=
ρ − 2e2 (W† · W)φ
+[e2 (W† · A) + eg cos θW (W† · Z)]φW
+[e2 (W · A) + eg cos θW (W · Z)]φ†W
−2eg cos θW (W† · W)φZ
+ie[W† · (2∇φW +
∂W†
∂W
) − W · (2∇φ†W +
)],
∂t
∂t
The µ = 1, 2, 3 equations together are
(7.7)
∂E
=
∂t
j + 2e2 (φ†W φW − W† · W)A
∇×B−
−[e2 (φ†W φ − W† · A) + eg cos θW (φ†W φZ − W† · Z)]W
−[e2 (φW φ − W · A) + eg cos θW (φW φZ − W · Z)]W†
+2eg cos θW (φ†W φW − W† · W)Z
∂W
∂W†
) − φW (∇φ†W + 2
)
∂t
∂t
+2(W† · ∇)W − 2(W · ∇)W† + W · ∇W† − W† · ∇W],
+ie[φ†W (∇φW + 2
In Eq. (7.5), electric field lines diverge not only from the electric charge ρ, but also from
the electromagnetic scalar potential φ and the three weak scalar potentials φW , φ†W , φZ . In
Eq. (7.7), magnetic field lines curl around not only the electric current j and the change
in electric field, but also the electromagnetic vector potential A and the three weak vector
potentials W, W† , Z.
3. Gauge freedom
This direct dependence of the fields on the potentials is a bit concerning. In classical
electromagnetism, we are free to alter the potential by a gauge transformation of the form
(7.8)
Aµ → Aµ0 = Aµ + ∂ µ f
without altering the fields F µν , and therefore without changing any measurable quantities.
How is that possible with Eq. (6.23), in which ∂ν F µν depends explicitly on Aµ ? Don’t the
fields now depend on our choice of gauge?
4. WHERE ARE THERE WEAK FIELDS?
17
The answer is that in electroweak theory we cannot freely change Aµ alone. A gauge
transformation affects the other potentials too. The simplest such transformation is
( µ
A → Aµ0 = Aµ − cos θW ∂ µ f
(7.9)
Z µ → Z µ0 = Z µ + sin θW ∂ µ f.
This gauge transformation, and another that affects all four potentials, can be derived from
[2, 269–270], which presents them in a different form. It is easy to check that Eq. (6.23) is
in fact invariant under this coupled transformation.
4. Where are there weak fields?
All of these new terms disappear if no charged weak field is present, that is if W µ =
W †µ = 0. Our next question, then, is under what conditions the weak potentials are
nonzero.
As has been mentioned, because the weak interaction is carried by massive bosons, it
has an extremely short range. So over most of space all three weak potentials are essentially
zero, and Eq. (6.23) reduces to the classical Mawell’s Equations (5.1), as we expect. Only
within about 10−17 m of their sources are the weak potentials significant.
What are their sources? We glossed over this point when we introduced the weak
currents. Which particles have the property of “weak charge,” in the same way that, say,
electrons have electromagnetic charge but neutrinos do not? There are in fact two different
kinds of weak charge – the property that interacts with W µ and W †µ (what I will term
“W-charge”) is different from the property that interacts with Z µ (“Z-charge”). [4, 54–
55] discusses weak charge qualitatively, and [2, 263–272] has a mathematical treatment.
All leptons (and antileptons) have Z-charge. Only left-handed leptons (and right-handed
antileptons) have W-charge. The W ± bosons also carry weak charge, but they are not part
of the currents; those only describe the coupling to leptons. Table 1 gives the values of the
charges.
Particle
Q/e W-charge
Z-charge
1
1
−
−
−
2
Left-handed e , µ , τ
-1
−2
2 − cos θW ≈ −0.27
2
−
−
−
Right-handed e , µ , τ
-1
0
sin θW
≈ 0.23
1
1
Left-handed νe , νµ , ντ
0
2
2
Right-handed νe , νµ , ντ
0
0
0
W ± boson
±1
±1
±(cos2 θW ) ≈ ±0.77
Table 1. Weak and Electromagnetic Charges. Q is the electromagnetic charge. W-charge is the weak isospin. Z-Charge is calculated as
(cos2 θW )Q/e − Y , where Y is the weak hypercharge [2, 267]. Right-handed
neutrinos are never found, and are included only as a theoretical matter, to
note that if they exist they do not participate in any electroweak interactions.
4.1. Chirality. For a massless particle, handedness, or chirality, is the same as helicity
– whether the particle’s spin is in the same direction as its velocity (right-handed) or in the
opposite direction (left-handed) [5, 124]. For a massive particle, however, helicity is not
Lorentz-invariant: since the particle’s speed is less than that of light, its direction of motion
depends on the reference frame. Its chirality, though, can be thought of as simply one of the
particle’s quantum numbers, and does not change with the reference frame. In the context
of weak interactions, when we say “handedness” we mean chirality, not helicity, so there is
no paradox of weak forces existing in one reference frame but not in another. Klauber [7]
18
7. INTERPRETING THE EQUATIONS
neatly distinguishes between the two concepts. Neutrinos are always left-handed, and antineutrinos always right-handed, but massive particles like the electron and positron come in
both chirality states.
5. Weak fields due to weak charge
Calculating W µ , W †µ , and Z µ exactly would be a daunting task: Eqs. (6.27), (6.25),
and (6.29) show that each of them, along with Aµ , depends in a complex way on all of the
others. We can, however, gain some insight into the weak fields by ignoring for the moment
those “potentials-as-sources” terms and considering just the term due to the current, the
first term in each of the equations:
g
µν
√ J µ + ...
(7.10)
∂ν FW
=
2 2
g
†µν
√ J †µ + ...
(7.11)
∂ν FW
=
2 2
g
(7.12)
∂ν FZµν =
J µ + ...
cos θW Z
This part of each equation has exactly the same form as the electromagnetic Eq. (5.1). So
the picture of fields due to charges and currents is the same for each of the weak interactions
as for the electromagnetic interaction.
∂EW
= jW + ..., etc.
∂t
EW diverges from ρW , BW curls around jW , and so on.
Note that, because the weak field equations are of the same form as the electromagnetic
equation, the argument sketched out in Ch. 3 Sect. 2.1 applies to all of them. In a classical
field-theory formulation, therefore, it seems that weak fields have the same distance dependence as electromagnetic field. The actual very short range of the weak force – or any force
whose gauge bosons have mass – is a quantum phenomenon.
(7.13)
∇ · EW = ρW + ...,
∇ × BW −
6. Example: right-handed electrons
We can now qualitatively describe the electroweak field lines for some elementary cases.
We begin with the simplest: a stationary electron in the right-handed state. It has negative electromagnetic charge and Z-charge, but no W-charge (see Table 1). Thus there
are no charged-weak fields, and all of the “potentials-as-sources” terms disappear from the
equations. We have the familiar electrostatic field, with E lines directed radially into the
electron, and no B lines. And per the argument above, we have a similar-looking neutralweak field: EZ lines directed radially inward, no BZ lines. The magnitudes of E and EZ
are different, but their behavior is the same.
Suppose instead of a stationary particle, we have a long line of right-handed electrons
moving at constant velocity. This constitutes a steady electromagnetic current and a steady
Z-current. There will be a magnetostatic field B looping around the direction of the electrons’ motion, and along with it a BZ , which is, again, identical except in its magnitude.
Since we have not matched the electrons with an equal number of positive charges, we also
have the cylindrically radial E field of a line of charge, and a similar EZ field.
7. Example: left-handed electrons
Now take an electron in the left-handed state. In our first approximation, considering
only the current terms as in Section 5, the electromagnetic and neutral-weak fields are just
7. EXAMPLE: LEFT-HANDED ELECTRONS
19
the same as in the right-handed case. The difference is that this electron has negative Wcharge as well. So it produces EW and E†W field lines radially inward, just like E and EZ
except in magnitude. A steady current of left-handed electrons produces loops of BW and
B†W just like B and BZ .
Note that we can also describe the situation in terms of scalar and vector potentials.
φW , φ†W , and φZ behave just like φ, becoming increasingly negative as we approach the
electron(s). Around the steady current, W, W† , and Z are just like A, vectors pointing in
the direction of current ĵ. To be specific, the scalar potential of a straight line of charge
density λ is of the form
φ=−
(7.14)
λ
ln r + [constant]
2π
and the vector potential of a straight current I is [8, 261]
I
(7.15)
A = − ln r + [constant] ĵ.
2π
The situation is more complicated that this, however, because the “potentials-as-sources”
terms are not necessarily zero. We refer to Eqs. (7.6) and (7.7), and consider a point P
at some distance r < 10−17 m from the electrons. At this point ρ = j = 0, so in classical
electromagnetism, E would have no divergence and B would have no curl. However, φ, φW ,
φ†W , and φZ are all nonzero (negative) at P.
In the case of the stationary electron, A, W, W† , and Z are – at least in the first
approximation – all zero, so the “potentials-as-sources” terms once again disappear in both
equations. In the case of the current, however, the vector potentials are not zero. Furthermore, they all have the same direction, ĵ, so we can simplify their dot products:
W† · W = W † W,
(7.16)
W† · A = W † A,
etc.
Applying this simplification, Eqs. (7.6) becomes
(7.17)
∇·E=
−2e2 (W † W )φ + [e2 (W † A) + eg cos θW (W † Z)]φW
+[e2 (W A) + eg cos θW (W Z)]φ†W − 2eg cos θW (W † W )φZ
+ie[W† · (2∇φW +
∂W†
∂W
) − W · (2∇φ†W +
)],
∂t
∂t
and Eq. (7.7) becomes
(7.18)
∂E
=
∂t
2e2 (φ†W φW − W † W )A
∇×B−
−[e2 (φ†W φ − W † A) + eg cos θW (φ†W φZ − W † Z)W
−[e2 (φW φ − W A) + eg cos θW (φW φZ − W Z)]W†
+2eg cos θW (φ†W φW − W † W )Z
∂W
∂W†
) − φW (∇φ†W + 2
)
∂t
∂t
+2(W† · ∇)W − 2(W · ∇)W† + W · ∇W†
+ie[φ†W (∇φW + 2
−W† · ∇W].
20
7. INTERPRETING THE EQUATIONS
Because the situation is static, we assume all the time derivatives are zero:
∂W
∂W†
∂E
=
=
= 0.
∂t
∂t
∂t
Because the vector potentials have the same direction, it does not matter which of them in
a product is a vector and which are magnitudes:
(7.19)
(7.20)
(W † W )A = (W † A)W = (W A)W† = (W † W A)ĵ,
etc.
Furthermore, because the electromagnetic charge, the W-charge, and the Z-charge are at
the same location, the four scalar potentials are the same except for the magnitude of their
respective charges, and the same is true of the four vector potentials:
φW
W
W-charge
(7.21)
=
=
, etc.
φ
A
Q/e
so we can say that
(7.22)
(7.23)
φW φ†W A = φW φW † = φ†W φW,
†
†
W W φ = W AφW
=
W Aφ†W ,
etc.
etc.
Then Eq. (7.17) simplifies to
(7.24)
∇ · E = 2ie[W† · (∇φW ) − W · (∇φ†W )],
and Eq. (7.18) to
(7.25)
∇×B=
ie[φ†W (∇φW ) − φW (∇φ†W ) + 2(W† · ∇)W
−2(W · ∇)W† + W · ∇W† − W† · ∇W].
Many of the terms of Eqs. (7.6) and (7.7) have disappeared. This is not a coincidence
or a peculiarity of the example. It happens because the currents j, jW , j†W , jZ are not
really independent. The electromagnetic and the weak charges belong to the same matter
particles, so the electromagnetic and weak currents will always be in the same place and
have the same (or opposite) direction, and much of this simplification will apply.
As for the remaining terms, if we write out the real and imaginary parts of the complex
φW , φ†W and W, W† , we find that the imaginary portions cancel – as they should, since
the left-hand side of each equation is real – but the real portions add. So we do in fact
have nonzero divergence of electric field and curl of magnetic field at a point containing no
electric charge or current.
CHAPTER 8
Concluding Remarks
1. Results
In this thesis, we analyzed electroweak unification – which is almost always treated in
quantum field theory – as a classical field theory.
By calculating the equation of motion of the electromagnetic four-potential Aµ from
the electroweak Lagrangian, we derived an electroweak version of Maxwell’s Laws. We also
calculated the equations of motion for the weak four-potentials.
Based on these equations and on parallels in the formulations of the electromagnetic and
weak interactions, we developed an electroweak field-lines picture and applied it to several
simple cases. Notably, we showed that electroweak theory implies that electromagnetic and
weak potentials act as sources of electromagnetic field.
2. Further questions
Our electroweak form of Maxwell’s Equations, Eqs. (7.6) and (7.7), is general, but we
discussed the field-lines picture only for static examples. What if the leptons accelerate?
In classical electromagnetism, J.J. Thomson analyzed the field lines of an accelerated point
charge and showed that they have transverse “kinks” representing electromagnetic radiation [9, 55–62]. One continuation of our thesis would be to extend Thomson’s analysis to
electroweak theory.
For that matter, we worked our examples on the assumption that the dominant contributions to the fields were still the original source terms and not the new “potentials as
sources” terms. This assumption allowed us to make progress, but it is not necessarily valid
in all cases. In electromagnetic radiation, for example, electric and magnetic fields produce
each other and propagate far from any charge or current source. Since their equations of
motion have a similar form, the fields we termed EW and BW could in theory do the same,
as could EZ and BZ . These “weak radiations” would not travel far in practice – they are
cut off by the W and Z bosons’ short range, which, as noted, our classical field theory fails to
account for. But might they have been prominent in the moments after the Big Bang when
the electroweak symmetry was unbroken? Would there have been combined “electroweak
radiation” in which electromagnetic and weak fields propagated each other? It would be
interesting to develop a picture of this radiation.
3. Acknowledgment
The author thanks Dan Styer, Schiffer Professor of Physics at Oberlin College, for
proposing the project and for invaluable guidance at every step.
21
CHAPTER 9
Appendix
1. Euler-Lagrange equations for a function and its complex conjugate
A particularly troubling case is that of the charged weak potentials W µ and W †µ . These
two functions are definitely not independent; they are a function and its complex conjugate:
1
W µ = √ (W1µ − iW2µ )
2
(9.1)
1
W †µ = √ (W1µ + iW2µ )
2
where W1µ and W2µ are real and independent. Can we really ignore this relationship when
finding the Euler-Lagrange equations for W µ and W †µ ? We can – a proof follows.
Let the Lagrangian density for our system be some function of a field φ, its complex
conjugate φ? , and their derivatives:
∂φ ? ∂φ?
,φ ,
(9.2)
L = L φ,
∂x
∂x
where
(9.3)
φ = R + iI
and φ? = R − iI.
R and I are real and independent. Therefore it is certainly valid to calculate the equations
of motion for R and I,
#
"
∂L
∂
∂L
(9.4)
=
∂R
∂x ∂( ∂R
∂x )
(9.5)
"
#
∂L
∂
∂L
=
.
∂I
∂I
∂x ∂( ∂x
)
We want to show that is is equally valid to calculate the equations of motion for φ and φ?
directly,
"
#
∂L
∂
∂L
=
(9.6)
∂φ
∂x ∂( ∂φ )
∂x
"
(9.7)
#
∂L
∂L
∂
=
.
∂φ?
∂x ∂( ∂φ? )
∂x
From Eqs. (9.3) and the chain rule,
?
∂L
∂L
∂φ
∂L
∂φ
∂L
∂L
=
+
=
+
?
∂R
∂φ
∂R
∂φ
∂R
∂φ ∂φ?
?
(9.8)
∂L
∂L
∂φ
∂L
∂φ
∂L
∂L
=
+
=i
−
.
∂I
∂φ
∂I
∂φ?
∂I
∂φ ∂φ?
22
1. EULER-LAGRANGE EQUATIONS FOR A FUNCTION AND ITS COMPLEX CONJUGATE
23
Also from Eqs. (9.3),
∂φ
∂R
∂I
=
+i
∂x
∂x
∂x
(9.9)
and
∂φ?
∂R
∂I
=
−i ,
∂x
∂x
∂x
so the chain rule gives
?
(9.10)
∂φ
∂φ
∂L
∂L ∂( ∂x )
∂L ∂( ∂x )
∂L
∂L
= ∂φ
+ ∂φ?
= ∂φ + ∂φ?
∂R
∂R
∂R
∂( ∂x )
∂( ∂x ) ∂( ∂x ) ∂( ∂x ) ∂( ∂x )
∂( ) ∂( ∂x )
" ∂x
#
∂φ
∂φ?
∂L ∂( ∂x )
∂L ∂( ∂x )
∂L
∂L
∂L
= ∂φ
+
=i
− ∂φ? .
?
∂I
∂I
∂I
∂( ∂x
)
) ∂( ∂φ
∂( ∂x
)
∂( ∂x ) ∂( ∂x
∂( ∂φ
∂( ∂x )
∂x )
∂x )
Substituting from Eqs. (9.8) and (9.10), Eq. (9.4) becomes
"
#
∂L
∂L
∂L
∂L
∂
(9.11)
+
+
=
∂φ ∂φ?
∂x ∂( ∂φ ) ∂( ∂φ? )
∂x
∂x
and Eq. (9.5), once we cancel a factor of i, becomes
#
"
∂L
∂L
∂L
∂
∂L
(9.12)
.
−
−
=
∂φ ∂φ?
∂x ∂( ∂φ ) ∂( ∂φ? )
∂x
∂x
Adding Eq. (9.12) and Eq. (9.11) yields Eq. (9.6), while subtracting Eq. (9.12) from Eq.
(9.11) yields Eq. (9.7). Therefore, Eqs. (9.4) and (9.5) are equivalent to Eqs. (9.6) and (9.7).
Bibliography
[1]
[2]
[3]
[4]
[5]
[6]
[7]
“Press Release: The 1979 Nobel Prize in Physics.” Nobel Web. 15 October 1979. http://nobelprize.org.
F. Mandl and G. Shaw, Quantum Field Theory, John Wiley and Sons Ltd., New York (1984).
D. Soper, Classical Field Theory, Dover Publications, Inc., New York (2008).
P. Watkins, Story of the W and Z, Cambridge University Press, Cambridge (1986).
D. Griffiths, Introduction to Elementary Particles, John Wiley and Sons, Inc., New York (1987).
S. Weinberg, “Gauge and global symmetries at high temperature,” Phys. Rev. D, v. 9, 3375 (1974).
R. Klauber, “Chirality vs. Helicity Chart,” Pedagogic Aids to Quantum Field Theory (2010),
http://www.quantumfieldtheory.info/.
[8] M. Nayfeh and M. Brussel, Electricity and Magnetism, John Wiley and Sons, Inc., New York (1985).
[9] J.J. Thomson, Electricity and Matter, Yale University Press, New Haven, Connecticut (1904).
24