Time Travel Research Center © 2005 Cetin BAL - GSM:+90 05366063183 -Turkey/Denizli ## Quantum Field Theory## ContentsThe Field Equation ## The Field EquationQuantum 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.
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),
which is just a Fourier Series where the coefficients are to be
determined by the field:
## QuantizationQuantization of the field is accomplished by demanding the coefficients c
If a number operator N
for all values of k
which corresponds to no particle in any state - the vacuum. ## Green's Function and RenormalizationThe 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) :
where the label "o" designates quantities associated with the "bare field".
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:
where the delta function on the right hand side equals to 1 for x = y,
and 0 otherwise. ## Perturbation Theory and S MatrixIt 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
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
where H
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 g
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 DiagramThe first order term is:
The mathematical entities inside the integral can be represented
graphically by the following conventions: ## 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.
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,
where the a
Construction of the eigenvectors follows exactly the same way as in Eqs.(5)
and (6) with an additional index for polarization. - 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.
The photon-electron processes are described by substituting Eq.(39) to
H 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 ModelIt 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
- 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.
*W*^{a}_{}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*_{}.*f*^{abc}is the antisymmetric tensor such that*f*^{123}= +1,*f*^{213}= -1,*f*^{113}= 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 *W*^{a}_{}. - 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:
Meaning of the symbols in the Lagrangina density :
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. *F*^{a}_{}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.
- = .
- m
_{i}denotes the mass for the various quarks.
Meaning of the symbols in the QCD Lagrangina density : The Lagrangian for the Standard Model then consists of three parts:
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