Quantum Field Theory

Contents

The Field Equation
Quantization
Green's Function and
Renormalization
Perturbation Theory and S Matrix
Feynman
Diagram
Quantum Electrodynamics (QED)
A Very Brief Overview of the
Standard Model

The Field Equation

Quantum Field Theory can be considered as "field + quantization".
Followings is a mathematical formulation (at 2nd year undergraduate level)
on the construction of quantum field and its application to elementary
particle physics.

The dynamic of the field is determined by the
field equation. The field equation for the neutral scalar meson field is a
very simple kind of Klein-Gordon Equation as shown below:

---------- (1)

where m is a parameter related to the mass, is the
field, which is a complex function (with real and imaginary parts) of x,
y, z, and t (simply represented by x in the equation),
,
and

are
the Laplacians in four dimensional space-time and three dimensional space
respectively. The wave equations and many systems in physics and
engineering are constructed with these Laplacian operators.

The
field can be expressed in a series expansion in terms of the harmonic
functions and the coefficients ck's, where k is a four
dimensional vector related to the momentum and energy of the
particle:

---------- (2)

which is just a Fourier Series where the coefficients are to be
determined by the field:

---------- (3)

Quantization

Quantization of the field is accomplished by demanding the coefficients
ck's to satisfy the following commutation rules:

---------- (4)

If a number operator Nk = ck*ck is
defined such that it operates on the state vector |nk> to
generate:
Nk|nk> =
nk|nk>
where nk is the number of
particles in the k state; it can be shown that

ck*|nk> =
(nk+1)1/2|nk+1>
ck|nk>
= nk1/2|nk-1>
Thus ck*
increases the number of particles in the k state by 1, while ck
reduces the number of particles in the k state by 1. They are called
creation and annihilation operator respectively. The complete set of
eigenvectors is given by:

---------- (5)

for all values of kl and nl. They
form an abstract space called the Hilbert space with all the eigenvectors
orthogonal (perpendicular) to each others and the norm (length) equal to
1.

In particular, the vacuum state is:

---------- (6)

which corresponds to no particle in any state - the vacuum.

Green's Function and
Renormalization

The above treatment is for the case of free field. The mathematics
becomes more complicated when there is interaction with the field. The
simplest case is to include the source of the field in the free field
equation. An additional term is inserted to the right of Eq.(1)
:

---------- (7)

where the label "o" designates quantities associated with the "bare
field".

Invoking the Green's function technique, the solution of
this equation is given by:

---------- (8)

where the first term is the free field solution and the Green's
function inside the integral is the solution of the equation with a point
source at point y in the general form:

---------- (9)

where the delta function on the right hand side equals to 1 for x = y,
and 0 otherwise.

It can be shown that the "bare field" can be
expressed in terms of ck's similar to the case of the free
field, but these coefficients are now modified by an additional term
related to the structure of the source. As a result the norm (length) of
the eigenvectors are no longer equal to 1. To recover this definition,
they have to be "renormalized" by the renormalization constant
Z1/2, which has the values in general;
it is equal to 1 for free field and 0 for a point source. The renormalized
field, mass, and energy are
and EnR = Z-1Eno
respectively. The physical mass mR is the experimentally
observed mass, mo is an unspecified parameter (called "bare
mass") which together with Z-1 determine a value in agreement
with experiment. In a realistic, fully quantized system, perturbation
theory must be used to obtain any numerical result. However, it is found
that perturbation theory calculations always lead to an infinity value for
Z so that the  ''bare" quantities such as mo would also be
infinity to yield a finite value for mR.

Perturbation Theory and S Matrix

It is not possible to obtain an analytical solution for the field
equation with the field itself in the interaction term. A perturbation
theory was developed to obtain approximate solutions step by step. The
interaction between a charged fermion and the photon is in the form ,
where A is the electromagnetic vector potential and is the
field for the fermion, which appears on both side of the Green's function
solution as shown in Eq.(8). An iteration procedure yields a sum of
integrals as shown in the formula below:

---------- (10)

where K is the Green's function. In this form the unknown field on the
left hand side is now expressed in terms of all known quantities on the
right hand side. The first term is the free field solution, and the
integration is over all the space-time x', x'',
x''', ... Note that each of the following term is multiplied by
the power of e, from e1, to e2, ... Since e=1/137
for the electromagnetic interaction, computation on a few terms would be
sufficient to obtain a result with acceptable accuracy.

Another
formulism is the S-matrix theory, which was very popular many years ago.
It is the transition probability expressed in an expansion as the result
of the iteration procedure on the transition operator, which transforms
the system from an initial state (at time negative infinity) to a final
state (at time positive infinity) as shown in the formula below.

---------- (11)

where HI involves the interaction fields (integrating over
all space and multiplied by the coupling strength) and
t1
> t2 > ... > tn-1. In this picture the
fields obey the free field equations, the interaction enters via
HI. It was thought that since we cannot measure the fields
directly, so we should not talk about it, while we do measure S-matrix
elements, so this is what we should be mindful about. It is now realized
that analyzing the S-matrix alone is not sufficient, information on the
quantum fields is also necessary. It can be shown that the various orders
in the S-matrix are related to the petrubation expansion in
Eq.(10).

Let us take the nucleon-pion system as an example of
S-matrix application:

---------- (12)
----------
(13)

where Eq.(12) is the free field equation for the pion, and Eq.(13)
is the free field equation for the nucleon (the Dirac equation).
Expressing in terms of the field itself, it can be shown that the
quantization rules in Eq.(4) become:

---------- (14)

where {a,b} = ab + ba is the anticommunition expression, and the
quantities on the right-hand side are the Green's functions for the pion
and nucleon respectively. Since the interaction is go
the nth order term in the S-matrix expansion Eq.(11) has the explicit
form:

---------- (15)

where the symbol N is the normal-order operator, which shifts all the
creation operators to the left (to avoid infinite vacuum energy), while T
is the time-order operator, which re-arranges the fields so that the one
associated with later time is on the left (to take care of the integration
limits in Eq.(11)).

Feynman Diagram

The first order term is:

---------- (16)

The mathematical entities inside the integral can be represented
graphically by the following conventions:

which
translates S(1) into a graph called the Feynman
diagram:

It
represents the process of the annihilation of a pair of nucleon and
anti-nucleon and the creation of a pion.
In the next higher order term
the normal operator will sometimes produces a Green's function such as
. This is usually referred to as propagator and graphically
represented by a wavy line running from x to y as show below:

which corresponds to the scattering of two nucleons by
exchanging a pion. Another graph such as the one below:

represents the process of virtual pair creation and
annihilation - the vacuum fluctuation.

The Feynman rules are
summarized in the table below:

where
the + and - superscripts refer to the positive frequency (eikx)
and negative frequency (e-ikx) expansion in Eq.(2), N
represents the nucleon and represents
the anti-nucleon.

Feynman diagrams can be divided into two types,
"trees" and "loops", on the basis of their topology. Tree diagrams have no
loops; that is, they only have branches. They describe process such as
scattering, which yields finite result and reproduces the classical value.
Loop diagrams, as their name suggests, have closed loops in them such as
the one for vacuum fluctuation. The process in "loops" diagrams involves
virtual particles and is usually divergent (becomes infinity at some
limits). The infinities are removed by the renormalization precedure
similar to the mass renormalization mentioned earlier. Virtual particles
can appear and disappear violating the rules of energy and momentum
conservation as long as the uncertainty principle is satisfied. They are
said to be "off mass-shell", because they do not satisfy the relationship
E2 = p2c2 + m2c4.

Quantum Electrodynamics (QED)

Quantum electrodynamics, or QED, is a quantum theory of the
interactions of charged particles with the electromagnetic field. It
describes mathematically not only all interactions of light with matter
but also those of charged particles with one another. QED is a
relativistic theory in that Albert Einstein's theory of special relativity
is built into each of its equations. That is, the equations are invariant
under a transformation of space-time. The QED theory was refined and fully
developed in the late 1940s by Richard P. Feynman, Julian S. Schwinger,
and Shin'ichiro Tomonaga, independently of one another. Because the
behavior of atoms and molecules is primarily electromagnetic in nature,
all of atomic physics can be considered a test laboratory for the theory.
Agreement of very high accuracy makes QED one of the most successful
physical theories so far devised.

The formulism for QED is very
similar to the nucleon-pion system in Eqs.(12) and (13). While Eq.(13) for
fermion is readily applicable (with appropriate value for ko,
which is proportional to the mass of the fermion); Eq.(12) is replaced by
the Maxwell equations:

where E, B are the electric and magnetic field
respectively, j is the current density, rho is the charge density,
and c is the velocity of light.

By virtue of the antisymmetry of Eq.(21), the continuity equation
for the charge-current density is automatically satisfied, i.e.,

The vector potential is introduced by:

---------- (24)

This definition is used for simplifying computations. It incorporates
Eqs.(19) and (20) into the formulism automatically. There is arbitrariness
when the Maxwell equations are written in this form. By imposing
particular conditions to the arbitrariness,

The vector potential A can always be decomposed into a
transverse component and a longitudinal component (with respect to the
direction of motion) as shown in Eq.(29) such that Eqs.(30) and (31) are
satisfied:

It can
be shown that the longitudinal component and Ao together
give rise to the instantaneous static Coulomb interactions between charged
particles, whereas the transverse component accounts for the
electromagnetic radiation of moving charged particles. The transverse
electromagnetic fields provides a simple and physically transparent
descriptions of a veaiety of processes in which real photons are emitted,
absorbed, or scattered. The three basic equations for the free-field case
are:

where A satisfies the transversality condition in
Eq.(30).

Eq.(33) is in a form very similar to the Klein-Gordon
Equation Eq.(1) or Eq.(12) except that the mass term vanishes (because the
photon has no rest mass) and the field is a vector (instead of scalar)
with two transverse components (polarization) perpendicular to each other.
Thus A can be expressed in Fourier series similar to Eq.(2):

The quantization rules for the electromagnetic field is very similar
to that in Eq.(4):

---------- (36)

where the ak's are related to the ck's by:

---------- (37)

Construction of the eigenvectors follows exactly the same way as in
Eqs.(5) and (6) with an additional index for
polarization.

Interaction between photon and fermion, e.g.,
electron can be introduced by demanding local gauge invariance for the
formulism. With this constraint on the quantum field theory, the ordinary
derivative in Eq.(13) becomes the covariant derivative:

and
the interaction takes the form:

where e
is the coupling constant. (See appendix on "Abelion/non-Abelion Groups and U(1), SU(2), SU(3)" for a
discussion about the concept of gauge or phase
transformation.)

The photon-electron processes are described by substituting Eq.(39)
to HI in the S matrix expansion Eq.(15):

The first order processes -

where the electron field has been decomposed to:

and the symbol X may stand for an interaction with
an external classical potential, the emission of a photon, or the
absorption of a photon. The realization of a particular process depends
entirely on the initial and final states.

The second order processes -

Two-photon annihilation - Pair of an electron and positron into
two gamma rays.

Compton scattering - It occurs when an electron and a photon
collide and scatter elastically.

Moller scattering - It is the scattering of two relativistic
electrons.

Bremsstrahlung - It is the process by which radiation is emitted
from an electron as it moves past a nucleus.

Higher order processes -

Vacuum polarization - It produces a correction to the coupling
constant (electric charge) and contributes to the Lamb shift, which is
a small splitting of hydrogen atom energy levels caused by the
interaction with the virtual pair (of electron and positron).

Vertex correction - A correction to the electron vertex function,
which contributes to the anomalous electron magnetic moment.

Self-energy - The interaction of a charged particle with its own
field and it usually gives rise to infinite self-energy and infinite
mass.

Mass renormalization - The observed mass is generated by combining
the bare mass and the (calculated) divergent mass.

In summary QED rests on the idea that charged particles (e.g.,
electrons and positrons) interact by emitting and absorbing photons, the
particles of light that transmit electromagnetic forces. These photons are
virtual; that is, they cannot be seen or detected in any way because their
existence violates the conservation of energy and momentum. The particle
exchange is manifested as the "force" of the interaction, because the
interacting particles change their speed and direction of travel as they
release or absorb the energy of a photon. Photons also can be emitted in a
free state, in which case they may be observed. The interaction of two
charged particles occurs in a series of processes of increasing
complexity. In the simplest, only one virtual photon is involved; in a
second-order process, there are two; and so forth. The processes
correspond to all the possible ways in which the particles can interact by
the exchange of virtual photons, and each of them can be represented
graphically by means of the Feynman diagrams. Besides furnishing an
intuitive picture of the process being considered, this type of diagram
prescribes precisely how to calculate the observable quantity involved.

A Very Brief Overview of the Standard
Model

It is beyond the scope of this webpage to present a comprehensive
review of the Standard Model. The following is a crude attempt to provide
a glance of the subject matter by introducing the Lagrangian density Standard Model for the Standard Model. The field
equations are derived by minimizing the "Action'' , which is related to the Lagrangian density.
Thus instead of writing down the field equations explicitly such as in
Eqs.(17) - (20) or Eqs.(13) and (38), the dynamics of the electro-weak
interaction can be expressed in term of the Lagrangian density:

This is known as the Weinberg-Salam Model. The
lepton Lagrangina density in
Eq.(40) consists of three parts. 1 is the gauge bosons part; 2 is the fermionic part; and 3 is the scalar Higgs sector, which generates mass for
the gauge bosons and the fermions.

Meaning of the symbols in the Lagrangina density :

The symbols and
are indices for the space-time components running from 1 to 4. Whenever
an index appears in both the subscript and superscript, it signifies a
summation over these components.

Wa is related to the three gauge (vector) bosons with
the index "a" running from 1 to 3.

F is the anti-symmetric tensor for the electromagnetic
field as shown in Eq.(24), where the vector potential
A is
now denoted by B.

fabc is the antisymmetric tensor such that
f123 = +1, f213 = -1,
f113 = 0, ... etc.

Since there is no right-handed neutrino in nature, the fermionic
field consists of a left-handed isodoublet and a right-handed isosinglet
as shown below -

---------- (42f)

This curious feature, that the electron is split into two parts is a
consequence of the fact that the weak interactions violate parity and
are mediated by V - A interactions, where V stands for vector and A
stands for axial vector (also known as pseudovector, which changes sign
under a parity transformation). The axial vector interaction is hidden
in the last term of Eq.(42d). The asymmetric forms for the fermion
fields in Eq.(42f) is a way to portrait the chiral  nature of the objects in weak interaction
- the left-handed version is different from the right-handed one. Note
that in Eq.(42c) the right-handed field R does not participate in the
weak interaction involving the vector bosons
Wa.

are
the Pauli matrices as shown in Eq.(10) in the appendix on  "Groups " with i running from 1 to 3. The four 4X4 metrices are constructed from the Pauli matrices and the
identity matrix.

g and g' are the coupling constants for the SU(2) and U(1)
interactions respectively.

The first three terms in Eq(41c) are responsible for generating the
mass of the gauge bosons, while the last term takes care of the fermion
mass.

is the
scalar Higgs field as depicted in the diagram below:

This form of field exhibits a drastic effect called
"spontaneous symmetry breaking". When the origin of the field is shifted
to v by the transformation ' = - v, the fields Wa and B
recombine and reemerge as the physical photon field
A, a neutral massive vector particle
Z, and a charged doubled of massive vector particles
W+/-:

The masses of the gauge bosons are obtained from the mass
sector:

Physically, the effect can be interpreted as an object moving from the
"false vacuum" (where = 0) to
the more stable "true vacuum" (where = v).
Gravitationally, it is similar to the more familiar case of moving from
the hilltop to the valley. In the case of Higgs field, the transformation
is accompanied with a "phase change", which endows mass to some of the
particles.

Experimentally, the predictions of the Weinber-Salam
model have been tested to about one part in 103 or
104. It has been one of the outstanding successes of the field
theory, gradually rivaling the predictive power of QED.

The above
formulism can be carried over to the electro-weak interactions between
quarks with the massless neutrino replaced by the up quark u (which has
mass), and the electron replaced by the down quark d. In order to get the
correct quantum numbers, such as the charge, the covariant derivatives are
different from the lepton as shown below:

where the coupling constants g and g' are also different
from the case of leptons.

The theory for the strong interaction is
called quantum chromodynamics (QCD), which has the Lagrangian
density:

Since the gauge bosons (the gluons) are massless, the
Lagrangian density appears to be in a much simpler form than the
electro-weak interactions in Eq.(40).

Meaning of the symbols in the QCD Lagrangina density :

Fa is similar to the SU(2) gauge field in Eq.(42a). This is
essentially the Yang-Mills field developed back in 1953 by generalizing
the gauge invariance used in QED. It represents the eight massless
gluons carrying the SU(3) "colour" force with the index "a" running from
1 to 8.

The index "i" is taken over the flavors, which labels the up, down,
strange, charm, top, and bottom quarks. The flavor index is not gauged;
it represents a global symmetry. However, the quarks also carry the
local colour SU(3) indices red, green, and blue (which is suppressed
here). In other words, quarks come in six flavors and three colours, but
only the colour index participates in the local gauge symmetry. It is
the colour force, which binds the quarks together.

=
.

mi denotes the mass for the various quarks.

The Lagrangian for the Standard Model then consists of three parts:

where WS stands for the Weinberg-Salam model, and lept. and
qurk. stand for the leptons and quarks that are inserted into the WS model
with the correct SU(2)XU(1) assignments. It is assumed that both the
leptons and quarks couple to the same Higgs field in the usual way. From
this form of the Standard Model, several important conclusions can be
drawn. First, the gluons from QCD only interact with the quarks, not the
leptons. Thus, symmetries like parity are conserved for the strong
interactions. Second, the chiral symmetry, which is respected by the QCD
action in the limit of vanishing quark masses, is violated by the weak
interactions. Third, quarks interact with the leptons via the exchange of
W and Z vector mesons.

Alıntı:  http://universe-review.ca/R15-12-QFT.htm

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