Quantum Field Theory

Contents

Action Principle
Path Integral
Field Equation
Second Quantization
Noether's Theorem and Charge Conservation
Green's Function and Renormalization
Perturbation Theory and S Matrix
Feynman Diagram
Quantum Electrodynamics (QED)
From Coulomb Scattering to Deep Inelastic Scattering
Compton Scattering
Lamb Shift
Renormalizable Theories
The Divergence
Yang-Mills Theory
The Standard Model
The SO(10) Group and Unification
Lattice Theory
Quantum Vacuum
Spontaneous Symmetry Breaking
Gyromagnetic Ratio and Anomalous Magnetic Moment
Axion

Action Principle

Formulation of theoretical physics usually starts with the "Action Principle", which was originally used to derive the equation of motion for a particle in classical mechanics. The "action" for such case is:

---------- (1a)
Functional where the Lagrangian L is a function of the position q and velocity =dq/dt of the particle, and the integration is over the trajectories from "a" to "b" as shown in Figure 01b. Note that S is not just a simple function of t - rather it is a function of the entire set of points q(t). It is a function of the function q(t), or a "functional" of q(t). In other words, to say S is a functional of the function q(t) means that S is a number whose value depends on the form of the function q(t), where t is just a parameter used to specify the form of q(t). The action S is sometimes written as S[q(t)] to emphasize that it depends on the form of q(t).

Figure 01a Functional
[view large image]

 Figure 01a shows pictorially two different forms of q(t) (dark and red), the corresponding Lagrangian L, which is determined by q(t), and the action S[q(t)], which would have two different numbers equal to the areas under L indicated by either the dark or the red curve.
The integration of the Lagrangian L between the end points to yield a value for S[q(t)] is responsible for its dependence on the entire path of q(t) from one end point to the other.

The "Action Principle" states that by minimizing the action S, i.e, by taking S = 0 with a small variation of path - q, we can derive the Euler-Lagrange equation:

S / q(t)  = ---------- (1b)
from which the trajectory of the particle can be computed. It turns out that the path always follows the shortest route between the two fixed end points as indicated by the one at the trough in Figure 01c.

For a particle with mass m moving in a potential V (x), where x =q,

L = (m/2)(dx/dt)2 - V(x), ---------- (1c)

the Euler-Lagrange equation has the explict form:

m d2x/dt2 = -dV/dx = F, ---------- (1d)

which is the Newtonian equation of motion, where the negative gradient of the potential is the force F.

The potential for the Hook's law is V(x) = (k/2)x2, where k is the spring constant, the equation of motion then becomes:

d2x/dt2 + (k/m) x = 0, ---------- (1e)
which is the Newton's law for the motion of harmonic oscillator. The general solution is:

x = A ei(k/m)1/2t + B e-i(k/m)1/2t ---------- (1f).

Note that there would be no oscillation when m or k = 0.

The form of Lagrangian can be constrained by imposing various conditions. The most obvious one is to demand that it should be independent of the orientation of the coordinate system. Under this condition the Lagrangian has to be a scalar in three dimensional space (for classical mechanics) or four dimensional space-time (for relativistic mechanics). Other conditions can also be specified in constructing the Lagrangian. The Noether's Theorem states that each of this condition corresponding to a conservation law:

Time independence Energy conservation
Space independence Momentum conservation
Rotational independence Angular momentum conservation

In other word, the conservation laws are the consequence of Lagrangian symmetries under the transformations of time, space, and rotational angle respectively. Symmetry in physical systems carries a different meaning than simple geometrical invariance. Instead of checking whether an experimental arrangement looks identical when rotated (geometrical invariance), we want to know if the laws of physics are invariant, i.e., if objects behave in the same way when the system is rotated. For example, it can be shown that if the Lagrangian is time independence then
- L = constant ---------- (1g)
if L is given by Eq.(1c), then it becomes:

(m/2)(dx/dt)2 + V(x) = constant ---------- (1h)

Symmetry which is the conservation of total energy (kinetic energy + potential energy) for a particle moving in a potential V(x). The momentum is not conserved in this case. However the dependence on x is eliminated in the free field case where V(x) = constant. It follows from Eq.(1h) that dx/dt = constant. This is the Newton's first law, which states that the object would not experience acceleration if there is no external force acting on it. Thus both the momentum m(dx/dt) and energy (m/2)(dx/dt)2 are conserved; or in term of symmetry, the system is now independent of time and space. Figure 01b shows schematically the similarity and difference between geometrical and theoretical physics symmetries. By symmetry, they both mean something is unchanged after some sort of rearrangement.

Figure 01b Symmetry
[view large image]

 However, in the case of geometrical symmetry it is the configuration that remains the same after the operation. Whereas in theoretical physics the invariance is about the form of the equations before and after the transformation.
Although the laws of nature may seem to be simple and symmetrical, the real world isn't. It is messy and complicated. The reason is related to the fact that we do not observe the laws of nature, but instead its outcome or solution such as Eq.(1f), which is the solution of Eq.(1e). For example, the law in Eq.(1e) dictates that it is symmetrical (invariant) under time reversal:
t' -t; but its solution in Eq.(1f) is not so unless A = B (see more in the section on "Spontaneous Symmetry Breaking").

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Path Integral

In quantum theory all the paths in Figure 01c are possible, not just the one corresponding to S = 0. However, not all paths are equally likely; we must find an appropriate weighting for the paths. According to Feynman, each path is weighted by the factor eiS/. The classical path is still the one that makes S stationary under small changes of path. In its vicinity paths have a strong tendency to add up constructively, while far from the classical path (the ones that go high up on the sides of the trough) the phase factors will tend to produce cancellations. The amount the particle can stray from the classical path depends on the magnitude of the corresponding action relative to . The steeper the slope of the trough, the closer it is approaching the
Path Integral classical theory. The classical trajectory is recovered as 0. In general, some of the paths may indicate movement at greater than light speed, or in violation of energy-momentum conservation law. Those paths would be related to the virtual particles; they need only to obey the uncertainty principle. Their contribution to the transition amplitude is usually very small. Figure 01c shows the different paths in 3D perspective dimensions, each one of which contributes a value to the action S (defined by Eq.(1a)) as indicated by an arrow or the axis

Figure 01c Action Principle
[view large image]

 labelled [q(t)]. The quantum mechanical (probability) amplitude for going from q(ta) to q(tb), where "a" and "b" denoting the end points, is known as the Feynman Path Integral:


Eq.(1h) can be evaluated by breaking up every path into an infinite number of intermediate points: x1, x2, ..., xn, ...xN (we have equated q with x here), and then takes the sum of each point, e.g., xn in all the paths from - to +. For the free field case where L = (m/2)(dx/dt)2, the functional integrations can be performed analytically. The final result is:

= |m/h(tb - ta)|1/2 exp{[im(xb - xa)2] / [2(tb - ta)]} ---------- (1i)

If we vary one of the end points, e.g. xb, it can be shown that the transition amplitude in Eq.(1i) satisfies the time-dependent Schrodinger's equation in one dimension for a non-relativistic free particle:
---------- (1j)
The link between Eqs.(1i) and (1j) illustrates the equivalence of the path integral and canonical quantization in quantum theory. The wave function in the Schrodinger's equation can now be equated to the transition amplitude: (x,t) = , where x(ta) can be conveniently chosen at the origin of a coordinate system and with ta= 0.

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Field Equation

Quantum field theory can be developed by adopting the path integral formulism as mentioned earlier, or by combining the field equation with canonical quantization as shown in the next sub-topic. Anyway, the concept of Action principle can be generalized to derive the field equation with some modifications for the action. A similar procedure for minimizing the action can be used to derive the Euler-Lagrange equation, which is now called the field equation. Thus, the action of the field is now expressed in the form:



where is the Lagrangian density, is the th component of the field, denotes the space-time component,
,
and the integration is over a four-volume R. Similar to the case of particle Lagrangan in Eq.(1a) S has to be a scalar if is an invariant function of its arguments. The form of is further constrained by demanding that it is invariant under certain transformations such as space-time translation, Lorentz transformation, the gauge transformation, or the conformal transformation, etc.... This is to ensure that the equation of motion (field equation) is unchanged (symmetrical) under these operations. The corresponding conservation laws for these symmetries are summarized in Table 01 below.

Symmetry Group Conservation Law
Space-time translation Poincare (x x + a) Energy-momentum tensor
Space-time rotation Lorentz (x = x, = -) Angular momentum,
velocity of light
Interal rotation of complex field U(1) in electromagnetic interaction Electric charge
Internal rotation of lepton or quark field SU(2) in weak interaction Weak charge
Internal rotation of quark field SU(3) in strong interaction Color charge

Table 01 Conservation Laws

Note: The gauge symmetries in Table 01 are global (meaning everywhere), local gauge symmetries (at specific point) leads to the interactions with gauge bosons. See the sub-topic on Noether's Theorem for more details about charge conservation.

The "Action Principle" for field theories states that if we perform an arbitrary variation of the field, + , subject to the boundary condition = 0 for t = t1 and t = t2, then the solution of the variational problem S = 0 yields a set of the Euler-Lagrange equations from which the equations of motion, or field equations can be derived:
---------- (1k)
For the neutral scalar meson field

which yields the Klein-Gordon Equation according to Eq.(1k):
---------- (1l)
where m represents the mass of the one component scalar field , which is a function of x, y, z, and t (collectively represented by x in the equation),


are the d'Alembertian operator in 4-dimensional space-time and the Laplacian operator in 3-dimensional space respectively. The repeated dummy index in the equations is understood to be summed over the 4 space-time coordinates.

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)
where , and the time x0 is set to zero after the time derivation has been performed.

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Second 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 Fock 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.

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Noether's Theorm and Charge Conservation

Gauss Theorem The Noether's theorem in field theory takes the form of the conservation of current:


where the four-current J is defined as:
,
where is the index for the space-time, indicates the component of the field, and represents a small variation in the th parameter denoted as .

Figure 01d Gauss Theorem
[view large image]

 From this conserved current, we can also establish a conserved charge by integrating the equation over a volume in the three dimensional space. Applying the Gauss theorem (see Figure 01d), the term with spatial derivative can be converted into a surface integral, which
would vanish if the three-current diminishes sufficiently on the surface. Under this condition, we obtain:
,
which implies the charge = constant. Note that the conservation of charge would not be valid if there is source or sink within the volume.

The Noether's theorem can be illustrated in more details by using the two-component Klein-Gordon Equation as an example. The explicit form of the Lagrangian density in this case is:

,
where . These two independent scalar fields can be varied by the internal phase transformation:
    or,  in a more familiar form: ,
where is the parameter corresponding to ( is omitted for single parameter). This U(1) symmetry generates the four-current:. It follows that the conserved charge corresponding to this current is:

where N and N' represent the sums of number operators (in momentum space). In this form, N can be interpreted as the number of particles carrying charge -e (with various momentum), while N' is the number of anti-particles with charge +e. The sum of these numbers is a constant within a volume containing no source or sink.

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Green's Function and Renormalization

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.(1l) :
---------- (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 G(x - y) inside the integral is the solution of the equation with a point source at y in the form:
---------- (9a)
where (x - y) is the delta function. Similarly, the Green's function for the ferminon is defined by the equation:
---------- (9b)
where m0 is the mass of the fermion, and are the gamma matrices associated with the fermion. In Dirac representation, the matrices can be expressed explicitly as:

where g11 = g22 = g33 = 1, g44 = -1, g = 0 for ; the 5 matrix converts a vector to pseudo-vector (axial vector) with parity +1.

The chiral representation is in the form:
.
It splits the Dirac equation into 2 self-contained pieces (for the left-handed and right-handed leptons respectively) more suitable for formulating the Standard Model.

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 Z, which has the values ; it is equal to 1 for free field, 0 for a point source, and depends on the structure of the source in general. 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. Since Z-1 is infinite for a point source, mo has to be slightly more or slightly less infinite to yield a finite value for mR. This technique of replacing the ignorance in detailed structure of the source by measurement is the more general definition of renormalization, although it is now more often referred to as the method to cancel the infinities in quantum field theory. For example, it involves a many-body Schrodinger equation to compute the viscosity in the Navier-Stokes equations. Since it is impossible to obtain the solution for such a complicated system, its value is determined by experiment instead. A renormalizable theory is one in which the details of a deeper scale are not needed to describe the physics at the present scale, save for a few experimentally measurable parameters (see more in the section about "Renormalizable Theories").

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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. For example, the interaction between a charged fermion and the photon is in the form , where A is the electromagnetic vector potential and is the two component spinors (in Dirac representation, and represent the fermion and anti-fermion respectively; while in chiral representation they stand for the left-handed and right-handed fermions called Weyl spinors), which now appears on both sides in Eq.(8) (with (y) replaced by the interaction term, replacing , and G defined by Eq.(9b)). An iteration procedure yields a sum of integrals as shown in the formula below:
Perturbation Theory ---------- (10)
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 free field solution is denoted by 0, 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 e2=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 expansion. It is the transition amplitude expressed in a series as the result of the iteration procedure on the transition operator, which transforms the system from an initial state (at negative infinity time) to a final state (at positive infinity time) as shown in the formula below.
S Matrix ---------- (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 green's function and S-matrix formulations are equivalent.

The S-matrix elements of the expansion are taken between the initial state (i) at t = - and the final state (f) state at t = +, e.g., Sfi = f|S|i. Usually, one initial state can produce one or more final states as shown in Figure 01e, where three different initial states are taken as examples - namely, the electron positron scattering, the Compton scattering and the deep inelastic scattering. Each of this process produces many final states, but only a few have been shown just for illustration purpose. The Sfi is a complex number in general. It is called probability amplitude and is related to the probability of going from the i to f
S-matrix align= states. It has to satisfy the unitary condition, i.e., f S*fiSfi = 1, which guarantees that probability is conserved in the process. Such relationship indicates that the matrix (Sfi) has an inverse, which in turn implies that it is possible to return to the initial state from the final state at least in principle although the probability is almost zero in practice so that the

Figure 01e S-Matrix
[view large image]

 second law of thermodynamics is "almost" never violated. This property is also related to the conservation of information, which caused so much trouble for Stephen Hawking.

Note: For example in the electron positron scattering process, there are three possible finally states as illustrated in Figure 01e. The sum of the probabilities for each one of these has to be: S*11S11 + S*21S21 + S*31S31 = 1 to insure that the final state is in one of the three possibilities.

Now let us take the nucleon-pion system as an example of S-matrix application:
Field Eqs. ---------- (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 (see Eqs.(9a) and (9b)). Since the interaction HI = go the nth order term in the S-matrix expansion Eq.(11) has the explicit form:
---------- (15)
where the integration runs from - to + over all space-time, 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)). The T product can be expanded into the N products and pairings (see below), e.g.,


If the coupling constant g0 is a small number less than 1, the successive higher terms would be getting even smaller as the proportional constant is in the form of (g0)n. Therefore the perturbation series can be terminated up to certain term depending on the requirement of accuracy.

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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 generate Green's function such as or SF(x - y) from the pairings, e.g.,

The pairing is referred to as propagator or internal line and graphically represented by a line running from a vertex at x to another vertex at y as shown in the diagram below:


which represents the annihilation of a pair of nucleon and anti-nucleon through a virtual nucleon.

Nucleons Scattering Figure 01e corresponds to the scattering of two nucleons by exchanging a pion. The internal line represents the probability amplitude for a virtual particle to travel from one place to another (x y) in a given time with greater than light speed, or to travel with off mass-shell 4-momentum k, which could have arbitrary value in violation of energy-

Figure 01e Nucleon-Nucleon Scattering [view large image]

 momentum conservation but allowed by the uncertainty principle. Mathematically, it is expressed by the Green's function:

where is a small positive real constant - a mathematical device taking advantage of the technique of contour integral; + 0 will be taken after the integration. The subscript F refers to the Feynman prescription for integrating the Green's functions. They are in a form such that positive energy solutions are carried forward in time, while negative energy solutions are carried backward in time. The latter solution can be interpreted as the anti-particle with positive energy moving forward in time.

Another graph such as the one below:


this loop diagram represents the process of virtual pair creation and annihilation - the vacuum fluctuation (time runs horizontally in this graph). It is this kind of graphs, which give rise to divergent results.

The Feynman rules are summarized in the tables below:

Feynman Rules Feynman Rules, Momentum Space

Table 02 Feynman Rules [view large image]

Table 03 Momentum Representations [view large image]
where in Table 02, the + and - superscripts refer to the positive frequency (eikx) and negative frequency (e-ikx) terms as shown in the Fourier decompositions below (also see Eq.(2)) . In the tables, N represents the nucleon and represents the anti-nucleon. In evaluating the S matrix, it is sometimes advantageous to go over to momentum space via the Fourier expansions:

Then the Feynman rules (see Table 03) can be expressed in terms of the energy p and k = (mo2 + k2)1/2, and 4-momentum p and k, for the fermion and boson respectively, while v(p,s) and u(p,s) are Dirac spinors with spin s representing the nucleon and anti-nucleon respectively . The appearance of m0 in the formulas for both fermion and boson is rather confusing, it just indicates the rest mass for whichever particle in the process.

    As an example to demonstrate the power of Feynman diagram, the nucleon-nucleon scattering in Figure 01e will be used to evaluate the corresponding S matrix from the Feynman diagram (in momentum space, see Table 03):

  1. Collect the 4 factors from external nucleon lines (for both incoming and outgoing).
  2. Multiply 2 coupling constants and 2 delta functions from the vertex x and y respectively.
  3. Write down a propagator for the internal line.
  4. Integrate over internal momenta. In this case of tree diagram, the delta functions enforce the rule for energy-momentum conservation such that the internal momenta k = p1 - p3 = p4 - p2. Thus, the integration can be carried out trivially. Such is not the case for a loop diagram (see for example the self-energy diagram in Figure 03b).
  5. The integration of the k-space yields: -ig02/[(p1 - p3)2 - m02]
  6. The final result is: S(2) = (m/V)2/(p1p2p3p4)1/2 {(-ig02)/[(p1 - p3)2 - m02]} ,
    where m and m0 is the mass of the nucleon and pion respectively.
  7. The scattering cross section is proportional to the squared S matrix: |S(2)|2.
Evaluation of the S matrix with the Feynman rules in Table 02 yields the same result. For example, the external lines together with the contribution from the propagator would combine into a factor of exp[i(k+p3-p1)x], which gives a delta function with the same argument upon integrating over the x-space, and similarly for the y-space integration. Then the k-space integration can be performed exactly as outlined above. The rest is to collect all the factors associated with each of the steps. In fact, this is the procedure to derive the Feynman rules in momentum space.

It is worthwhile to repeat once again that Feynman diagrams can be divided into two types, "trees" and "loops", on the basis of their topology. Tree diagrams 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 loop diagrams involve "off mass-shell" virtual particles and is usually divergent (becomes infinity). Such 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.

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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, 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 3-vector potential A can always be decomposed into a transverse component and a longitudinal component (parallel to the direction of motion) as shown in Eq.(29) such that the transversality condition Eq.(30) (which dictates that the transverse components are perpendicular to the direction of motion) and the irrotational condition of the longitudinal component Eq.(31) are satisfied:


It can be shown that and A0 together give rise to the instantaneous static Coulomb interactions between charged particles with the total Hamiltonian in the form (the Hamiltonian H is related to the Lagrangian L by the formula H = p - L):
---------- (32)
where and A0 have been completely eliminated in favor of the instantaneous Coulomb interaction in the last term. It is instantaneous in the sense that its value at t is determined by the charge distribution at the same instant of time. This formalism was derived by E. Fermi in 1930. Since the separation into different terms is not relativistically covariant, nor is the transversality condition itself, the whole formalism appears noncovariant; each time we perform a Lorentz transformation, we must simultaneously make a gauge transformation to obtain a new set of A and A0. It had been shown that it is possible to develop manifestly covariant calculational techniques starting with this Hamiltonian. It is also possible to construct a formalism, which preserves relativistic covariance at every stage.

Note that the long range instantaneous Coulomb interaction does not imply a force travelling faster than the speed of light. Although the intereaction is instantaneous, it can be considered as the interaction between two overlapping Coulomb tails (clouds of virtual photons) of the two charged particles, so there is no need for the interaction to travel from one point to another in zero time.

QED concerns mainly with the transverse components, which account for the electromagnetic radiation of accelerating charged particles. The transverse electromagnetic fields provide a simple and physically transparent description of a variety 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.(33b) is in a form very similar to the Klein-Gordon Equation Eq.(1l) 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.)

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. Interaction also occurs by the exchange of virtual electron/positron. The exchange of virtual photon is manifested as the "force" in the electromagnetic 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 real particles occurs in a series of graphs of increasing complexity. In the first order graph, no virtual photon or virtual electron/positron is involved; in the second-order graph, there are either one virtual photon or one virtual electron/positron; and so forth. The graphs correspond to all the possible ways in which the particles can interact by the exchange of virtual photons and virtual electrons/positrons, 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.

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From Coulomb Scattering to Deep Inelastic Scattering

Coulomb scattering is the scattering of charged particles off one another as a result of the electrostatic force between them. Such process belongs to the first order graphs in the Feynman diagrams. Since there is no virtual particles involved, classical mechanics can be used to show that for a given kinetic energy Ek of the incident particle, the scattering angle is related inversely to the impact parameter b as depicted in Figure 02a. It also shows that for a given the incident particle can probe closer to the target (smaller b) with higher Ek. Since there is no way to measure b experimentally, the cross section = b2 is defined such that the incident particle is initially directed anywhere within such area. The Rutherford scattering formula is derived by identifying the alpha particles to be the incident beam:

d(,Ek)/d = (Ze2/4Ek)2[1/sin4(/2)] ---------- (38c)

where Z is the number of charges in the "nucleus", which contains all the positive charges and most of the mass in an atom.
Impact Parameter Rutherford Scattering This formula is related to the probability of an incident particle with kinetic energy Ek scattered into an solid angle d in the direction . The experimental apparatus to verify the Rutheford scattering is shown in Figure 02b. Figure 02c is

Figure 02a Impact Parameter [view large image]

Figure 02b Rutherford Scattering [view large image]

 the measured angular distribution of the scattered particles with incident energy of 15 Mev.
Rutherford Scattering Cross Section, Vary Angles Rutherford Scattering Cross Section, Varying Energy The agreement between theoretical prediction and experimental data overruled J. J. Thomson's "Plum Pudding Model", in which electrons are suspended in a pudding-like positively charged substance that contains most of atom's mass. The success of Rutherford's theory gives him the distinction of having discovered the atomic nucleus. However, there is a problem. When the incident energy is over about 27.5 Mev, the

Figure 02c Cross Section vs Angles [view large image]

Figure 02d Cross Section vs Energy [view large image]

 experimental data start to deviate from the theoretical curve. Figure 02d shows the deviation at a fixed angle of 60o.
The relativistic correction has been obtained by N. F. Mott within the framework of QED but without recourse to perturbation theory. The improved formula has the form:

d(,E)/d = [(Ze2E)/(2|p|2c2)]2 {[(1 - 2sin2(/2)]/sin4(/2)} ---------- (38d)

where p is the momentum of the incident particle, E = (mo2c4+p2c2)1/2, and =|p|c/E.

When the incident energy increases to over 1 Gev, the collision becomes inelastic. Under the impact of such high energy, the target nucleon is likely to disintegrate. High energy scatterings are used to probe the structure of nucleons. According to the parton model, at high enough energy the probe would see only the point-like parton inside the structure instead of seeing the whole nucleon as a coherent object as shown in Figure 02e. In general, the inelastic scattering cross section takes on the form:

d2/dq2d = (42/q4) (E'/EM) [(M/)W2(q2,)cos2(/2) + 2W1(q2,)sin2(/2)] ---------- (38e)

where in the limit when the mass of the incident particle is negligible, q2 = (k - k')24EE'sin2(/2), = (pq)/ME - E'. They are the 4-momentum transfer squared, and the energy transfer respectively representing such transfers from the electron to the "off mass-shell" photon with q2 0 (see Figure 02e for the definition of the various notations).

In Eq.(38e) the cross section is expressed in term of the simple Rutherford scattering formula. The W1 and W2 in the rest of the equation are dimensionless form factors. It is used to correct the point particle description. In general the precise structure of the nucleon is unknown, but the form of W1,2 is severely restricted by Lorentz invariance and electromagnetic current conservation. This is an extremely general and important tool in the absence of a complete theory. In the deep inelastic region:
Parton Model Form Factor , q2, and 0 x = q2/(2M) 1 (elastic scattering corresponds to x = 1), W1F1(x), W2F2(x). The F1(x) and F2(x) are finite and depend only on x. This relation is called Bjorken scaling. The usefulness of the parton model is that we can compare the scaling behavior of F1 and F2 against the various predictions for spin-0 and spin-1/2 partons, e.g., F1(x) = 0, and 2xF1(x) = F2(x) respectively for each case. Experimentally, the spin-1/2 model is reasonably

Figure 02e Deep Inelastic Scattering [view large image]

Figure 02f Form Factor [view large image]


 satisfied (see Figure 02f). This result suggests that the partons are, in fact, just the quarks. Thus we have progressed from the discovery of atomic nucleus to unravelling the structure of the nucleon in about 100 years.

[Top]

Compton Scattering

Compton Scattering The quantum theory of light postulates that photons behave like particles except for the absence of any rest mass. If this is true, then it should be possible for us to treat collisions between photons and, say, electrons in the same manner as billiard-ball collisions are treated in classical mechanics. Figure 02g shows how such a collision might be represented, with an X-ray photon striking an electron at rest and being scattered away from its original path while the electron receives an impulse and begins to move. Using just

Figure 02g Compton Scattering [view large image]

 kinematics the change in the wavelength of the X-ray can be expressed in the form:

' - = o (1 - cos) ---------- (38f)
where o = (h/mc) = 2.43x10-10 cm is the Compton wavelength, which is actually the de Broglie wavelength for the electron moving at the speed of light.
Compton Scattering Cross Section The Compton scattering belongs to the second order graphs in the Feynman diagrams. Since there is no loop diagram involved, it is still susceptible to classical treatment. The cross section has been computed by J. J. Thomson in the 1900's:

d()/d = (ro2/2) (1 + cos2) ---------- (38g)

where ro = e2/4mc2 = 2.82x10-13 cm stands for the classical radius of the electron.

Figure 02h Scattering Cross Section[view large image]


 The experimental scattering cross section at different incident energy are shown in Figure 02h. At the low energy range of about l Kev, the cross section in Eq.(38g) shows good agreement with minima at = 90o and 270o, and maxima at = 0o and 180o. Asymmetric patterns appear at higher energy with the maximum shifted to the forward direction.
The relativistic correction has been derived within the framework of QED. It is known as the Klein-Nishina formula:

d()/d = (ro2/2) ('/)2 (/' + '/ + sin2) ---------- (38h)

where , and ' is the frequency before and after the scattering as related in Eq.(38f). Already in the late 1920's, the angular distribution of -ray scattering had been measured to show that the observed deviation from the Thomson formula is precisely what one expects from Eq.(38h). The agreement achieved then is one of the earliest quantitative triumphs of the Dirac theory, which is an attempt to unify the theories of quantum mechanics and special relativity. It is the linearized version of the Schrodinger equation in consistence with special relativity, i.e., it is Lorentz invariant as shown in Eq.(13). The Dirac equation describes particles with spin-1/2. Note also that the deep inelastic scattering is just a special kind of Compton scattering with virtual photon hitting a necluon (such as the proton) instead of real photon colliding an electron as shown in Figure 02e.

[Top]

Lamb Shift

The energy levels calculated from the Dirac equation with the electron in a Coulomb potential is given by:

Lamb Shift where Z is the number of positive charges. The formula predicts that the energy levels of the hydrogen atom should depend only on the principal quantum number n and the total angular momentum j (J = L + S), so that the energy levels of the 2S1/2 (n=2, j=1/2) and 2P1/2 (n=2, j=1/2) states should be degenerate (the same). In 1947, Lamb and Retherford demonstrated that this is not the case. The 2P1/2 and 2S1/2 energy levels are separated by more than 1000 MHz (see Figure 03a).

Figure 03a Lamb Shift
[view large image]
This small shift in energy levels is caused by various high-order radiative corrections as mentioned earlier. The regular Coulomb potential is now supplemented by an effective potential V(eff) (up to the 2nd order):
By taking the average of V(eff) with the corresponding hydrogen wave functions, it yields the Lamb shift in the form

The total shift amounts to 1052.10 MHz. When other smaller contributions such as the 4th order radiative corrections and the mass corrections are included into the calculation, the final theoretical value is 1057.864 0.014 MHz. Comparing with the experimental value of 1057.862 0.020 MHz, it provides an excellent indicator of the basic correctness of QED. Actually, it is instrumental in restoring confidence to QED from the dubious methodology of renormalization.

The discoverer of Lamb Shift - Willis Eugene Lamb, Jr. passed away on May 15, 2008. Lamb was born in Los Angeles on 12 July 1913. He studied at the University of California, Berkeley, where he obtained his first degree (in chemistry) in 1934 and his PhD (in physics, under the supervision of Robert Oppenheimer) in 1938. He had taken up posts at various institutions, and became professor of physics and optical sciences at the University of Arizona since 1974. His effort on discovering and measuring the Lamb shift earned him the Nobel Prize of Physics in 1955.

[Top]

Renormalizable Theories

As mentioned earlier, it is very difficult to compute the viscosity from the Schrodinger equation, although the former must somehow follow from the latter. However, in relativistic theory of elementary particles, the form of the equations is highly restrictive by the requirement of relativistic invariance. For that reason it is actually not difficult to carry out the renormalization procedure in a relativistic theory, at least in the perturbation framework.

It is also mentioned that divergences (infinities) appear with certain loop diagrams in QED. The origin of these infinities can be traced back to the singular nature of the inverse-square law 1 / r2 as r 0. But there is no direct evidence that this law is still valid below a distance of some 10-16 cm. If QED is renormalizable, then the detail at the deeper scale is not important. The inverse-square law can be replace by a constant 1 / r02 for r r0 ~ 10-16 cm. It turns out that QED is indeed renormalizable.

If a S-matrix element diverges like lnn(r0) / r0D as r0 0, for some D 0, then its degree of divergence is said to be D. It is found that D can be determined for any Feynman diagram by the formula:

D = 4 - (3/2)F - B - kV ---------- (39a)

where F and B are respectively the number of external fermion and boson lines, V is the number of vertices in the Feynman diagram, and k is the energy dimension of the coupling constant. The coupling constants g for any gauge theory is dimensionless, so k = 0. Depending on whether k = 0, k > 0, or k < 0, the corresponding theories are renormalizable, super-renormalizable, or non-renormalizable.

According to the formula for D, only a small variety of graphs in a renormalizable theory are divergent. It is summerized in Table 04 below:

Graph Diagram F B D D' D'+1 Parameter(s)
Vacuum 0 0 4 4 5 (Ignored)
Photon Self-energy 0 2 2 0 1 Z3
Electron Self-energy 2 0 1 1 2 m and Z2
Vertex 2 1 0 0 1 Z1-1
Photon-Photon Scattering 0 4 0 0 > 0 > Nil (Convergent)
Electron-Electron Scattering 4 0 -2 -2 -1 Nil (Convergent)

Table 04 Degree of Divergence

For a super-renormalizable theory, k > 0, so there are even fewer divergent diagrams than the renormalizable theories. Moreover, if V is large enough, Eq.(39) indicates that all diagrams would be primitively convergent.

A non-renormalizable theory has k < 0, so D grows indefinitely with V. Since D + 1 divergent parameters are required in each case, a non-renormalizable theory contains an infinite number of divergent parameters. Even if we had an infinite number of bare parameters to adjust, there would be an infinite number of parameters in the theory. It would take a prohibited long time to measure all of them. The gravitational constant G = 6.7087x10-39 c (Gev / c2)-2 in quantum gravity has a negative energy dimension of k = -2 and hence it is non-renormalizable. Presumably some new physics will happen at the scale of 1019 Gev to turn quantum gravity into a renormalizable or a finite theory.

[Top]

The Divergence

This section will examine the origin and the forms of the divergence. The formulation will adopt the Feynman rules in momentum space as shown in Table 03 with a few minor changes (such as 5 ) to converse the system from nucleon-pion to electron-photon. Since the divergence arises from the virtual photon with off mass-shell momentum k in the internal line, the main focus will be directed to the integration of k in the S matrix.

Electron Self-energy Figure 03b is the Feynman diagram in momentum space for the electron self-energy. Using the Feynman rules, the S matrix can be written down for the transition of an electron from state (p, s) to state (p', s') in the form:
and m denotes the bare mass.

Figure 03b Electron Self-energy [view large image]



 Counting powers of the off mass-shell 4-momentum |k| = (k2 - E2)1/2 in the numerator and denominator of the integral suggests that it diverges linearly like d|k|. However, due to cancellations in the integrand, it diverges only logarithmically like d|k|/|k| log().
More detailed analysis shows that the k-integration modifies the electron propagator (in momentum space) as follows:

---------- (39b)
where the formally divergent self-energy can be expressed as Z2 is derived from vertex correction. The self-energy becomes infinite as the cut-off (in the upper limit of the k space integration) is made to go to infinity. This is referred to as ultraviolet divergence since it is caused by the infinite value of |k|. The physical mass of the electron is now the sum of the bare mass m and , which represents the contribution from the virtual electron-positron pairs in the vicinity of the electron. The two infinite terms m and together somehow render a finite physical mass.

The vertex correction in Figure 03c modifies the first order interaction of the electron with the external potential as shown in the small insert. According to the Feynman rules, the vertex correction has the following form:
---------- (39c)
Vertex Correction Denoting (Z1)-1 the correction factor to be applied to the electric charge due to the vertex correction, together with Z2 from the electron self-energy, and (Z3)1/2 from the vacuum polarization, the total correction to the electric charge is:

eR = Z1-1Z2Z31/2e0 ---------- (39d)

It is found that when the bare charge is replaced by renormalized electric charge in Eq.(39d), most of the divergent terms disappear from the formulation.

As one part of Eq.(39c) gives rise to Z2, which is divergent at |k| , the other part diverges logarithmically at low value of |k| as |k| 0. This is an example of what is known

Figure 03c Vertex Correction [view large image]



 as the "infrared divergence". A possible way to dispose of this difficulty is to temporarily assign a very small but finite mass to the photon; this can be accomplished by modifying the photon propagator in Eq.(39c) according to the following prescription:
where min is the "mass" of the photon. Under such scheme, virtual photons of very low energies (min) do not get emitted or absorbed. The net result is that, for small values of q = p' - p, that part of Eq.(39c) can be written as follows:
...
where the second term containing the generalized Pauli matrices can be identified as the second order correction to the electron magnetic moment. It can be shown that the infrared divergence is cancelled by the similar divergence arisen from the bremsstrahlung process to all orders in perturbation theory. Thus, QED is free of divergence seemingly out of a mathematical miracles.

It can be shown that for large momentum transfer Q = |p' - p|, the effective coupling constant in QED including vacuum polarization is modified to:

---------- (38m)
Effective Charge which increases as Q increases (or, equivalently, as the probing distance becomes shorter, see Figure 03d).

Similar vacuum polarization effects occur in QCD with similar Feynman graph but for gluon exchange between quarks and antiquarks. There is an additional contribution from the gluon self-coupling as shown in Figure 03d. The effective coupling constant for strong interaction for large Q is now in the form:

---------- (38n)

where nf = 6 is the effective number of quark flavours for large Q, and -1 is called the

Figure 03d Effective Coupling Constant [view large image]

 confinement length since s becomes infinite when Q = . The phenomenon of asymptotic freedom in QCD is associated with the fact that s 0 as Q .

[Top]

Yang-Mills Theory

The Yang-Mills (Figure 07e) theory is an early attempt to formulate a gauge theory for the weak interaction. It is a
Yang Mills Vertex for Gluon generalization of the U(1) gauge theory for QED with non-abelian group (for the internal space). It fails to account for the mass of the gauge bosons in the weak interaction. However, it provides a good framework for QCD in strong interaction as its mediating bosons are massless. Since non-abelian gauge bosons can be emitted and absorbed from the gauge bosons themselves (see Figure 03f), they create anti-shielding for the quark anti-quark pairs in the virtual particle cloud (the Yukawa cloud). The color charge becomes weaker at smaller probing distance, and leads to asymptotic freedom. Conversely, at larger distances, the color charge

Figure 03e Yang-Mills [view large image]

Figure 03f Gluon Vertex [large image]

 increases, so that the quarks tends to bind more tightly together giving rise to quark confinement, which is the flip side of asymptotic freedom.

The mathematical formulation starts with the local gauge transformation of the fermion field:

' = exp(i(x)) ---------- (39b)

where (x) is a function of x denoting the phase angles in the internal space, and represents three 2x2 non-commuting matrices (generators) for the SU(2) group; and eight 3x3 non-commuting matrices for the SU(3) group. In other word, this kind of transformation is non-abelian.

The problem with the construction is that the derivatives of the fermion field are not covariant under this transformation. It can be shown that the formulation is invariant only if the derivative is replaced by the covariant derivative:

D = - iaba ---------- (39c)

where the space-time component is labeled by the index , the index "a" is for the phase angles, is the coupling constant, and b = aba represents the new fields (the gauge bosons) introduced to render an invariant theory under the local gauge transformation. Then the Lagrangian density takes the form:



[Top]

The Standard Model

The Standard Model, based on the gauge group SU(3) X SU(2) X U(1), is one of the great successes of the gauge theory. It can describe all known fundamental forces except gravity. However, the Standard Model is certainly not the final theory of particle interactions. It was created by crudely splicing the electroweak theory and the theory of quantum chromodynamics (QCD). It involves 18 unknown parameters, cannot explain the origin of the quark masses or the various coupling constants. The theory is rather unwieldy and inelegant. Nevertheless, not only is it renormalizable, it can explain a vast number of results from all areas of particle physics. In fact, there is no piece of experimental data that violates 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 (38a), 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. 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).

The Lagrangian for the Standard Model then consists of three parts:
---------- (43)
where WS stands for the Weinberg-Salam model, 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.

The Standard Model can be written either in mathematical form as summarized in Eq.(43), or in pictorial form, using the Feynman diagrams as shown in Figure 04. It is convenient to rewrite Eq.(43) into a shorthand notation in describing the Feynman diagrams:
---------- (44)
where G stands for the gluon field, W for the vector mesons, F for the photon, H for the Higgs, and fi for the fermions.

It was mentioned earlier that there is a certain amount of mixing between the d quark and s quark from different generations. Although the Standard Model does not explain the origin of this mixing. The Cabbibbo angle, however, allows us to parametrize our ignorance. It is found that the most general form of mixing can be expressed by the mixing matrix M:

D' = M D,
where
Flavor Mixing cij = cosij, sij = sinij, and the angle tunrs some matrix elements into complex numbers, thereby violating CP invariance (CP invariance demands that M* = M).

Experimentally, the mixing angles ij are either smaller than or comparable to the Cabibbo angle 1 ~ 15o. Thus, the quark mixing is relatively unimportant. A similar mixing matrix exists in neutrino mixing between the flavor states (e, , ) and the mass (mixed) states (1, 2, 3). Neutrino mixing is large in comparison to the quark mixing as shown in
Figure 05a. It leads to the detection of neutrino mass.

Figure 05a Flavor Mixing
[view large image]


 The mixing angle must have the same value for every electroweak process. It is observed to have the same value everywhere, to an accuracy of about one percent. Other successful predictions include the existence of the W and Z bosons, the gluon, the charm and the top quarks. Z boson decays have been confimred by LEP in 20 million of such events.
However, the Standard Model contains 26 free parameters:

3 coupling constants + 2 Higgs parameters + 2 x [3 generations x (2 fermions masses) + 4 CKM parameters] + 1 instanton§

For massless neutrinos and no leptonic mixing angles, there are still 19 free parameters. With so much arbitrariness, the Standard Model should be considered only as the first approximation to the true theory of subatomic particles, i.e., it is an effective theory to be explained by more fundamental theory.

    Following is a list of subjects that the Standard Model fails to explain. Either it is not in the formulation or it is just plugged into the theory without explanation of its origin.

  1. The cosmological constant or vacuum energy.
  2. Dark energy.
  3. The inflaton in the first fraction of a second of the Big Bang.
  4. Matter-antimatter asymmetry.
  5. Cold dark matter.
  6. The form of the Higgs field.
  7. Hierarchy problem - huge Higgs boson mass implies huge mass for all elementary particles.
  8. Gravity.
  9. Masses of the quarks and leptons.
  10. Three generations of elementary particles.
Half of the list above from 1 to 5 is related to cosmology and astronomy. It emphasizes that understanding of the largest and the smallest phenomena must come together. Supersymmetry can address items 1, 3 - 7, while superstring theory may be able to explain items 8, 9, 10. Thus item 2 about dark energy remains to be the most enigmatic subject in physics and astronomy.

§ Instantons is the solution of the Euclidean version of Yang-Mills equations. The purpose is to probe the nonperturbative realm of gauge theories. It is called instanton because it creates an almost instantaneous blip (peak) in the Lagrangian. They are not particles and have no direct physical interpretation. Rather, it reveals that the vacuum of Yang-Mills theory actually consists of an infinite number of degenerate vacua, so the true vacuum must be a superposition of all of them.

[Top]

The SO(10) Group and Unification

The 16 fermions in the standard model are summarized in Table 05, where the superscript denotes electric charge, and the subscript denotes color charge. All the quarks in the SU(3) gauge group participate in strong interaction, all left-handed fermions undertake weak interaction, and those fermions carrying electric charge are eligible for electromagnetic interaction.

Gauge-Group /
Handedness		 SU(3) SU(2) U(1)
Left-handed ur2/3, uw2/3, ub2/3, dr-1/3, dw-1/3, db-1/3 0, e-1    
Right-handed ur2/3, uw2/3, ub2/3, dr-1/3, dw-1/3, db-1/3   e-1 0

Table 05 Fermions in Standard Model

SO10 Group An alternate arrangement unifies all these fermions into the 10 dimensional SO(10) group with the introduction of two color charges for weak interaction:
  • The 5 color charges include the red (R), white (W), blue (B) in strong interaction, and the green (G) and purple (P) in weak interaction. Each one is represented by a separate 2-dimensional plane (with '+' and '-' components) to fit into the 10 dimensions in SO(10).
  • Figure 05b lists all the fermions of the standard model in the first column. The left-handed species are not labeled, while the right-handed ones are represented by the anti-particles of the left-handed fermions and denoted with the superscript "c" since these two kinds are similar except for the opposite charges.
  • The electric charge is now generalized to the hypercharge
    Y = - ( R + W + B ) / 6 + ( G + P ) / 4.
  • The rules for assigning color charge to the fermions are such that complementary color charges, e.g., (+ +) or (- - -), ..., cancel each other. The particle carries the color charge corresponding to the left over '+' or '-' sign in the strong color charges (R, W, B) and to the single '+' sign in the weak color charges (G, P). Consequently, all the right-handed fermions do not participate in weak interaction (the cause of parity violation), and all the leptons do not take part in strong interaction.
  • The hypercharge Y is computed by substituting +1 or -1 into the formula according to the '+' or '-' entry in the table respectively. Its value is actually the average of the sub-grouping. For example, the left-handed neutrino and electron carries 0 and -1 electric charge respectively, thus the average is -1/2.

    • Figure 05b SO(10) Group
    [view large image]
  • The right-handed neutrino N is the odd-man/woman out in the standard model. It is there to provide a mechanism for the mass of the neutrinos. It does not involve in any of the interactions - that's why it has not been observed (smelling like dark matter ?).
  • The SO(10) group breaks into many schemes. One of them is the SU(3)SU(2)U(1) corresponding to the charges of (R, W, B), (G, P), and Y respectively. It is this particular grouping that fit all the elementary fermions snugly together without much room to maneuver. The breaking down to various subgroups occurred long time ago soon after the Big Bang.
  • In addition, the SO(10) has a 16-dimensional spinor representation, which fits exactly into the 16 fermions in Figure 05b.


    • To complete the list of all particles in the standard model, table 06 below displays all the force mediating bosons, where the single quote (') denotes anti-charge similar to the + charge of the positron in the electromagnetic interaction:

    Charge Gauge Field(s) Gauge Particle(s)
    r grr', grw', grb', gluon fields Gluons from red color charge
    w gww', gwr', gwb', gluon fields Gluons from white color charge
    b gbb', gbw', gbr', gluon fields Gluons from blue color charge
    g ggg', ggp', vector meson fields W vector mesons
    p gpp', gpg', vector meson fields Z0 vector meson
    e A, electromagnetic field Photon

    Table 06 Gauge Bosons

    There is actually only 8 independent bosons for SU(3) and 3 independent bosons for SU(2) since the trace (trace of a matrix A: TrA = ann) of their matrix representations is equal to zero - a condition to reduce the number of independent bosons by one. This is a requirement for the unitarity of the gauge group.

    Since the force mediating gluon itself carries color charges, the corresponding Feynman diagram is interpreted in slightly different way than those in QED. Figure 05c shows two quarks with blue and red colors moving toward each other initially.
    Strong Interaction They interact by gluon exchange such that the blue quark emits a blue-antired gluon thereby transforming itself into a red quark, while the red quark absorbs this gluon to become a blue quark. In other word, the gluon manages the creation and annihilation of the red-antired color charge pair, while fetches and donates the blue color charge from one quark to another

    Figure 05c Strong Interaction

    		resulting in the exchange of color charge on the quarks during the course of the interaction.


    Since all the color charges are SO(10) group members that can be turned into each others, it is expected that at high probing
    Unification by SUSY energy, which enables penetration through the shielding by the virtual particles, the coupling strength of the various kinds of charges would merge into one unifying point. As shown by the diagram on the left of Figure 05d, it almost works, but not quite (the experimental errors is indicated by the width of the lines). The expectation comes true by introducing supersymmetry (SUSY), it even brings the outcast gravity to close proximity of the merging point (see diagram on the right of Figure 05d). This additional symmetry

    Figure 05d Unification by SUSY [view large image]

    		 requires a new partner for all the particles. They have the same charges (of all kinds) as their known partner, but different masses (heavier) and spins (integer 1/2 integer). Supersymmetry in effect doubles the number of particles in the standard model.

    [Top]

    Lattice Theory

    The lattice theory uses brute force calculation by computer to obtain low-energy features of the hadron world in QCD. There are many problems associated with theoretical evaluation in this particular area of quantum field theory:

    Lattice Theory 1. There are three color charges in QCD (red, green and blue), whereas QED comes with only two (the positive and negative charges).
    2. The force carrier (the photon) in QED does not interact with each other, but the gluons in QCD interact among themselves. This is the reason why the photon fields spread out in QED, while the gluon fields confine to a flux tube in QCD (see Figure 05e).
    3. The QCD coupling constant at large distance (~ 10-14cm) approaches to 1 making it impractical to use perturbative method. The lattice theory resorts to the use of path integral by generating thousands of paths weighted according to their likelihood under the particular rule that governs the physical evolution of the system.

    Figure 05e Lattice Theory [view large image]
      On the other hand, there is a price to pay for putting QCD on lattice:

    1. Since the metric is Euclidean, it means that calculations with the theory are limited to the static properties of QCD such as confinement and perhaps the low-energy spectrum of states. It has difficulty calculating scattering amplitudes, which are defined in Minkowski space.
    2. Lattice theory explicitly breaks continuous rotational and translational invariance, since space-time is discretized. All that is left is symmetry under discrete rotations of the lattice.
    3. It is constrained by the available computational power. Many important effects (e.g., virtual pair creation etc.) are not included in the calculation back in the 20th century.
    Implementation of lattice theory (with some special features for QCD):

    3-D Lattice 1. The three dimensional space is discretized into a finite, periodic gird as shown in Figure 05f.
    2. The fermions are located at the nodes (vertices) of the grid with link (edge) "a" in between.
    3. Normal time is converted to Euclidean time, i.e., t it. This 4th dimension will be added to the grid in Figure 05f. It follows that the transition amplitude ~ all pathseiS (where S is the

    Figure 05f 3-D Lattice
    [view large image]

    		 action/) now changes from oscillatory to dumping form: all pathse-S.


    The Link 4. In a Feynman diagram, it would be the gauge field that goes between the fermions. However if we want to retain gauge invariant in lattice theory, the link should be identified to:
    U(Cxy, An) = P exp(igaCxyAnaa),

    Figure 05g The Link
    [view large image]

    		 where Cxy is a path between x and y (Figure 05g), P denotes path ordering, An represents the gauge field at point n, g is the coupling constant, and a are the SU(3) generators in the form of 3X3 matrices.
    Wilson Loop 5. One way to construct gauge-invariant quantities out of the links is the Wilson loop (Figure 05h), which goes around along the links in a closed path, and takes the trace (Tr) of the product from each link (trace of a matrix A: TrA = ann). It can be shown that in the continuum limit a 0, the Wilson loop reduces to a form proportional to the

    Figure 05h Wilson Loop
    [view large image]

    		 Yang-Mills Lagrangian.
    6. The Wilson action for SU(N) group can be expressed as:
    SW(Uij) = P[Tr(UP + UP) - 2N]
    7. For any gauge-invariant operator O, its average < O(U) > is proportional to:
    Gluon Fields < O(U) > ~ dUijO(Uij)exp[-SW(Uij)],
    where denotes multiplicaltion of objects with indices i, j. In particular, the average of the Wilson loop is:
    < W(C) > = < Tr(U1, U2, ... Un)C > ~ exp(-TA)
    at the strong coupling limit, where A is the area encircled by C, and T is called the string tension. This result implies a static potential V(r) ~ Tr, where r is the separation between the quarks. Since V(r) increases with the value of r, thus it validates the concept of confinement.

    Figure 05i Gluon Fields in Grid [view large image]

    		 8. Finally, the lattice theory has to be verified that it goes over to QCD at the continuum limit as the link a 0 and the Euclidean time it is reverted back to the normal time t.

    Figure 05i shows a typical pattern of activity in the gluon fields from a quark within the grid. In Figure 05j, all the gluon fields are concentrated in a "flux tube" as an anti-quark is added nearby to cancel the fluctuations shown in Figure 05i.

    Confinement For the simplest possible group with only two elements, if the lattice is 8x8x8x8 in size, then the sum contains 2212 ~ 101228 terms, which is clearly prohibitive even by running all these different paths with a super computer. The Monte Carlo technique (as inspired by winning the game), however, evades this problem by the statistical sampling method wherein it does some random sampling first in order to discover the algorithms that would yield the best fit. The first result from lattice calculation was published by K. G. Wilson in 1974, who set up the Wilson loop to perform the computation in the strong-coupling limit for quark confinement as shown in Figure 05j. The latest effort has derived a 'u' mass of 1.9 Mev and a 'd' mass of 4.4 Mev. Thus the uud quarks in a proton weigh 8.2 Mev, which is only about 1% of the proton's

    Figure 05j Confinement
    [view large image]

    		 measured mass (938 Mev). All the rest of the proton mass comes from the energy that binds the quarks together. It reveals that not only the atoms are mostly empty space; now we know the nuclei inside atoms are also mere puffballs with almost no solid substance.

    QCDOC Computers and QCD QCDOC (QCD On Chip), is a series of computers designed specifically for lattice QCD calculations and dedicated more or less exclusively to that task. The glass-fronted cabinets in Figure 05k, at Brookhaven National Laboratory, hold two QCDOC machines, each with 12288 processors; a third QCDOC is at the University of Edinburgh. A typical lattice has 32 nodes along each of the three spatial dimensions and 128 nodes along the time dimension. That's about

    Figure 05k QCDOC
    [view large image]

    Figure 05l Lattice


    		4 million nodes altogether, and 16 million links between nodes. Some lattice QCD simulations are run on "commodity clusters" - machines assembled out of hundreds or thousands of off-the-shelf computers. Figure 05l is an artist's imagination about the Lattice Theory - blending rows of computers with the nucleons and its quarks within.

    A paper published in November 2008 unveiled the masses of many light hadrons calculated with the lattice methodology
    Masses of Hadrons (see Figure 05m). The results (filled circles) are in remarkable agreement with the experimental values (horizontal lines). Each symbol (, K and so on) refers to a different type of hadron. The widths of the bands indicate the experimental decay widths, the invert of which are related to the finite lifetime of the particles. The vertical error bars denote the theoretical error estimates. Three of the hadrons (, K and ) have no error bars because they are used to fix the theory's parameters. Since there are (3x3x4) + (8x6) = 84 numbers at each node (quark fields have three flavours, three colours, and four components accounting for spin and antiparticles; gluon fields have eight internal

    Figure 05m Masses of Hadrons [view large image]

    		 directions in its symmetry group, and for each direction there are six fields: three electric and three magnetic) no computer can handle the astronomical number crunching in the lattice calculation, certain approximations are adopted to make the task manageable:
    1. Lattice Size - Because of asymptotic freedom, the short-wavelength fluctuations of the fields can be replaced by Gaussian random (free) fields. Thus there is a minimum lattice size, which helps to cut down the number of links. The effects of the missing fluctuations can be computed analytically and added back in.
    2. Lattice Volume - It is possible to control finite-volume errors by varying the simulated volume and making theoretically informed extrapolations.
    3. Small u and d quark masses - Solutions become hard to obtain with the very small physical values of these masses. They are handled by sophisticated, theoretically informed extrapolation from simulations using larger mass values.
    4. Finite computer power - Even after acceptable levels of discretization and restriction to finite volume, the space that should be surveyed in the lattice calculation is far too large for even the most powerful modern computer banks to handle. So in place of a complete survey, it is replaced by a statistical sample. This introduces errors that can be estimated by the standard techniques of statistics.
    It seems that such huge effort to calculate the masses of the hadrons is a waste of time and money since they are known experimentally already. Actually, it provides another proof for the correctness of QCD, and the developed technique can be used to compute other interesting quantities that are very difficult to measure experimentally.

    [Top]

    Quantum Vacuum

    In classical physics, empty space is called the vacuum. The classical vacuum is utterly featureless. However, in quantum theory, the vacuum is a much more complex entity. The uncertainty principle allows virtual particles (each corresponding to a quantum field) continually materialize out of the vacuum, propagate for a short time and then vanish. These zero-point vibrations mean that there is a zero-point energy associated with any quantum field. Since there are an infinite number of harmonic oscillators per unit volume, the total zero-point energy density is, in fact, infinite. The process of renormalization is usually implemented to yield a zero energy density for the standard quantum vacuum, which is defined as no excitation of field quanta, i.e., no real particles are present. In other word, the quantum vacuum is at a state of minimum energy - the ground state.

    Negative Energy States Another kind of vacuum structure is prescribed by the Dirac theory of spin-1/2 particles. It admits the existence of both positive- and negative-energy particles: E = (m02c4 + p2c2)1/2. The concept of negative-energy entities is wholly alien to our knowledge of the universe. All things of physical significance are associated with varying amounts of positive energy. To get around the problem, Dirac proposed an energy spectrum containing all electrons in the universe (see Figure 06a). In addition to the normal positive-energy spectrum, it also contains the negative-energy variety, which spans the spectrum from -m0c2 down to negative infinity. All the negative-energy levels are filled, thus the positive-energy particle is inhibited from transition into these lower energy states. Thus, there is no observable effect in the real world. Only when there

    Figure 06a Negative Energy [view large image]

    		 is enough energy available, e.g., E 2m0c2, a real particle and anti-particle pair with positive-energy can be created from this unseen sea of negative-energy particles.

    Reaction Ratio Closer examination of diagram (e), Figure 06a reveals that the intermediate product is a pair of quarks as shown in the insert within Figure 06b. The quark confinement process creates the observed hadrons in the final state. As the quarks are observed to be point-like (in deep inelastic scattering) and spin 1/2, the intermediate process e+e- q is very similar to the process e+e- +-, the only difference being that the charges on the quarks are only some fractions of that on the muons. This explains the constancy of the

    Figure 06b Reaction Ratio [view large image]



    		 ratio R mentioned earlier and displayed in Figure 06b. As for the pronounced spikes which punctuate the curve in Figure 06b. These shapes are formed at certain energies of the e+e- collision, when the q pair have just the correct mass to appear as a single-meson resonance. They are the SU(3) flavour symmetry mesons with mass ~ 1 Gev.
    As the collision energies increase, new resonance spikes occur at 3.096 Gev and 3.687 Gev. It was discovered subsequently that they are the products of a new type of quark called "charm", which together with the "strange" quark form the second generation of quark. With this discovery the number of mesons is expanded into the 16-plet of spin-0 mesons generated by SU(4) flavour symmetry. Similarly, the spikes at 10 Gev and 10.40 Gev in Figure 06b lead to the discovery of the bottom quark for the third generation. The SU(4) flavour symmetry has to be enlarged to SU(5) so that the basic multiplet of spin-0 mesons is now expanded to a 25-plet.
    Meson Spectrum It is further realized that the c mesons form a spectrum from the different values for the spin and the orbital angular momentum of the constituent quarks as shown in Figure 06c. In the same notation for the atomic spectra, S, P and D in the diagram refer respectively to orbital angular momentum 0, , and 2. The resemblance to the atomic spectrum is understandable because of asymptotic freedom as the c and bound themselves together loosely. The force between the quarks can be formulated as a potential acting in the vicinity

    Figure 06c Meson Spectrum [view large image]

    		 of a colour charge. Thus instead of the Coulomb potential in the case of the atom, the quarks interact via the potential:

    V = - 4s / (3r) + ar ,

    with which the Schrodinger equation in non-relativistic quantum mechanics yields a most satisfactory match to the observed meson mass level (see Figure 06c). This formula combines a Coulomb potential at short ranges with an attractive potential rising linearly at longer distance, giving rise to the ever-increasing forces of quark confinement. Thus, the masses of the c mesons provide direct support for the QCD picture of inter-quark forces containing both asymptotic freedom at short ranges and confining forces at longer distance.

    Jets As the beam energy is increased, the quark and antiquark are produced with very large momenta, moving in opposite directions known as two-jet event. The fragmentation into hadrons then takes place, preferentially along the direction of the motion the quark and antiquark, resulting in jets of hadrons which become more and more collimated as energy is increased (see left diagram of Figure 06d). The measurement of the angular distribution of

    Figure 06d Two- and Three-Jet Event [view large image]

    		 jet axes confirms that the spin of quarks is indeed 1/2. At even higher energy the quark or antiquark radiating a gluon, which forms a separate jet of its own. This three-jet event is shown in the right diagram of Fgiure 06d.


    The next topic is about the Higgs field, which prescribes yet another kind of structure to the quantum vacuum.

    [Top]

    Spontaneous Symmetry Breaking

    Symmetry Breaking The existence of asymmetric solutions to a symmetric theory is common to many branches of physics. The reason lies in the fact that the symmetric state is not the state of minimum energy, i.e., the ground state, and that in the process of evolving towards the ground state, the intrinsic symmetry of the system has been broken. Figure 06e shows that the initial position of the marble on top of the bump is symmetric but not in a state of minimum

    Figure 06e Symmetry Breaking [view large image]

    		 energy. A small perturbation will cause the rotational symmetry to be broken and the system to assume the ground state configuration. When the symmetry of a physical system is broken in this way, it is often referred to as "spontaneous symmetry breaking".


    Scalar Potential This idea can be applied to account for the mass of gauge boson in the electroweak interaction in the Standard model. Let us start by examine how such unstable symmetry can arise mathematically. Considering the Lagrangian for a scalar field with the potential V:

    L = (1/2) - V() ---------- (46)

    where V() = (1/2)m22 + (/4!)4, and 4! = 4x3x2x1.

    Figure 06f Scalar Field
    [view large image]

    		 If m2 > 0, the system has real mass, the potential exhibits a minimum at the origin, where = 0. This system is associated with a unique vacuum (see Figure 06f).

    When m2 < 0, the vacuum at = 0 is unstable; a particle would prefer to move down the potential to a lower-energy state at the bottom of one of the wells at = v = (-6m2/)1/2. The states at = 0, and = v are referred to as false vacuum, and true vacuum respectively (Fiigure 06f). Eq.(46) indicates that the system is symmetrical under -. If the origin at the false vacuum is shifted to v, i.e., ' = - v, then the Lagrangian in Eq.(46) becomes:

    L = (1/2)'' - V'(') ---------- (47)

    with V'(') = |m|2'2 + (1/6)v'3 + (/4!)'4. In this way, the scalar particle has acquired a positive mass squared given by 2|m|2, but the original symmetry between and - has been spontaneously broken because the field has been shifted (resulting in the occurrence of the '3 term) and there is a true vacuum at ' = 0. Note that the valley in diagram (b) has a depth of (3/2)(m4/). The zero point of the potential can be shifted up or down without any effect on subsequent calculation.

    Complex Scalar Field The formulation can be generalized to complex scalar field with two independent components corresponding to positively and negatively charged fields. In a slightly different notations, the Lagrangian for a complex scalar field has the form similar to Eq.(46):

    L = ()(*) - V(,*) ---------- (48)
    where V() = m2* + (/4)(*)2.

    Figure 06g Complex Scalar Field [view large image]
    In this case there is a circle of degenerate minima giving by (Re)2 + (Im)2 = v2, where v = (-4m2/)1/2 (see Figure 06g). Therefore, there are infinitely many possibilities for the stable configuration: any scalar field satisfying the condition for minimum energy will do. The Lagrangian in Eq.(48) is invariant under the global gauge transformation:

    = (1 + i2)ei/(2)1/2

    which is just a rotation in the 1-2 plane. In analogy to the previous example, the origin of one of the field components, e.g., 1 is shifted to a point at the circle of minimum energy:
    = 1 - v
    = 2
    In terms of , and , the Lagrangian is transformed to:

    Once again, the symmetry is spontaneously broken as before. The mass term can be easily located by looking for the one with quadratic fields. It reveals that the field acquires a (positive) mass squared of -2m2. The novel feature in this example is that the field remains massless. Such massless modes, which arise from the degeneracy of the ground state after spontaneous symmetry breaking, are called "Goldstone bosons". The appearance of Goldstone bosons seems to be in contrary to real world experience since no such massless, spin-0 particle exists. However, if the global gauge transformation in the formulation is replaced by local gauge transformation, it can be shown that the Goldstone boson is absorbed by the formerly massless gauge boson, which has now acquired a mass. The corresponding model is called "scalar electrodynamics", but when it it spontaneously broken it is then referred to as the "Higgs mechanism". Before the symmetry breaking the Lagrangian for the interacting scalar field and electromagnetic vector potential A has the form:

    If the vector potential is transformed as:

    where q is the coupling constant and v is the scalar field at the true vacuum as defined earlier. By a suitable choice of local gauge transformation such as:

    ' = e-iq, with q = tan-1(/)

    then the scalar field components become:

    ' = H = cos(q) + sin(q)
    ' = 0

    The Lagrangian re-emerges in the form:

    which shows that the symmetry is broken (by the odd power terms in H), and the Goldstone mode has been completely removed by the gauge-transformed boson A', which has acquired a mass qv. The remaining scalar field H has also acquired a mass (-2m2)1/2. The total number (four) of degrees of freedom is unaltered. Instead of a massless gauge boson, having two (transverse) modes, plus a complex field composed of two real components, we now have a massive vector field A' having three modes - two transverse and one longitudinal (a requirement for massive spin-1 boson), plus one real scalar field H. This is just a theoretical model to illustrate the effect of spontaneous symmetry breaking. Of course, the photon remains massless in the real world. It requires both spontaneous symmetry breaking (with v0) and the coupling of the gauge field to the scalar field (q0) to acquire a mass. A more complicated version of this model is applicable to the electroweak interaction in the Standard model. It has been shown subsequently by 't Hooft that the spontaneous symmetry breaking formulation remains renormalizable; the ultraviolet divergences encountered are no worse than those occurring in QED.

    In the Standard model the scalar field is identified as the Higgs field responsiable for the mass of fermions and gauge bosons. Supersymmetry increases the number of Higgs particles to five with masses ranging from about 100 - 400 Gev. The Higgs fields are supposed to permeated throughout the universe uniformly and isotropically since the Big Bang. The spontaneous symmetry breaking occurred at a temperature corresponding to about 250 Gev in the electroweak ear soon after the end of inflation. Above that temperature (the phase is known as the symmetric phase) all the particles become massless. The period after the transition is called the Higgs phase. In a way, the Higgs fields are similar to the hypothetical ether of the pre-relativity era, but with a crucial difference - that it is formulated as a relativistic invariant theory, which would not prescribe an absolute frame of reference.

    Higgs Field As each term in the Lagrangian of the Standard model represents a different process, Figure 06h shows the various Higgs interactions in the form of Feynman diagrams. Diagram (a) represents a fermion emitting or absorbing a Higgs particle. Diagram (b) shows the corresponding process for the gauge bosons. They can also interact simultaneously with two Higgs, as shown in (c), which also represents a gauge boson scattering a Higgs particle. The Higgs also interacts with itself, as shown in diagrams (d) and (e), which are related to the shape of the scalar potential (Figure 06g). Diagram (f) depicts an electron acquiring its mass.

    Figure 06h Higgs Field Interaction [view large image]

    [Top]

    Gyromagnetic Ratio and Anomalous Magnetic Moment

    The anomalous magnetic moment for the electron and muon illustrates the progress of our understanding in particle physics from classical mechanics to quantum theory, quantum field theory, the Standard Model, and beyond.

    Classical g-ratio A classical electron moving around a nucleus in a circular orbit has an orbital angular momentum, L=mevr, and a magnetic dipole moment, = -evr/2, where e, me, v, and r are the electron´s charge, mass, velocity, and radius, respectively. A classical electron of homogeneous mass and charge density rotating about a symmetry axis has an angular momentum, L=(3/5)meR2, and a magnetic dipole moment, = -(3/10)eR2, where R and are the electron´s classical radius and rotating frequency, respectively. The classical gyromagnetic ratio of an orbiting or a spinning electron is defined as the ratio of the

    Figure 07 Classical g-ratio [view large image]

    		 magnetic moment to the angular momentum. In both cases one finds cl = /L= -e/(2me). The minus sign indicates that is in the opposite direction to L (see Figure 07).

    In quantum theory, the interaction between an electron and a magnetic field can be portrayed by the Feynman diagram in Figure 08a, which shows a photon from the magnetic field is absorbed by the electron and thus altered its trajectory. The gyromagnetic ratio derived from the Dirac equation takes the form: e = /L= -g e/(2me), where g = 2 is related to the fact
    Magnetic Moment Interaction Vertex Correction that the spin of the electron is equal to /2. If the vertex correction as shown in Figure 08 is taken into account, then g = 2 ( 1 + /2), where = e2/(4c) ~ 1/137.036 is the fine structure constant giving g - 2 = 0.002322814. The extra term arises from the electron self-interaction, in which it emits and reabsorbs a virtual photon, making a loop in the Feynman diagram as shown in Figure 08b. The same process also applies to the muon.

    Figure 08a Quantum Description [view large image]

    Figure 08b Vertex Cor-rection [view large image]
    The more accurate calculation including higher loop diagrams up to the 4th order term yields the following expression:
    g - 2 = 2 ( /2 - 0.328 (/2)2 + 1.181 (/2)3 - 1.510 (/2)4 + ... + 4.393 X 10-12) = 0.0023193042800.
    The experimental value is: g - 2 = 0.0023193043768 (in agreement with the calculated value to ten significant figures).

    In calculating the effects of the cloud of virtual particles, we need to include not just the effects of virtual photons and virtual electron-positron pairs, but also virtual quarks, virtual Higgs particles, and, in fact, all the particles of the Standard Model. It turns out, though, that because of the larger muon mass, any such heavy particles would affects the muon magnetic moment more than the electron magnetic moment. The muon g - 2 has been calculated with the Standard Model to a precision of 0.6 ppm (parts per million). The calculated value with the combined effect is g - 2 = 0.0023318360. A remarkable fact is that the muon g - 2 factor not only can be predicted to high precision, but also measured to equally high precision. The measurements at Brookhaven National Laboratory in 2001/2004 (Figure 09) yields an average value of g - 2 = 0.0023318416. Thus, the comparison of measurement and theory provides a sensitive test of the Standard Model. If there is physics not included in the
    g-2 Experiment current theory, and such new physics is of a nature that will affect the muon's spin, then the measurement would differ from the theory. This is what appears to have been observed, although there are several interpretations of the result that must be considered. One of the missing pieces in the theoretical calculation is the exotic particles predicted by the theory of supersymmetry. Although these particles are rare and unstable their mere existence in the vacuum would modify observable quantities such as the muon magnetic moment.

    Figure 09 g-2 Experiment [view large image]

    [Top]

    Axion

    Axion Axion was postulated to explain why CP violation is not observed in strong interaction, although it should be according to the Standard Model. The CP violation in SM arises from a certain non-zero parameter related to the QCD vacuum (see instanton). It was shown in 1997 that the parameter could be driven to zero by a Higgs field. A side effect of the transformation is the introduction of a new particle - the axion. If axions do exist, they each would have a

    Figure 10 Axion
    [view large image]

    		mass of around 10-5 ev, but there could be so many of them in the Universe, that they contribute a large proportion of the overall mass in the form of dark matter.

    A report in March, 2006 claimed the detection of axions in laboratory. However, the results contradict other observations and do not fit with constraints deduced from astrophysics. Since axions interact only weakly with baryonic matter, they would be able to stream out from the central cores of stars into space virtually unobstructed, carrying energy away with them and cooling the stellar cores. This cooling is more effective the heavier the axions are, and if each axion had a mass greater than 0.01 ev this would affect the appearance of stars and the way supernovas exploded. Further experiments such as the one shown in Figure 10 will confirm (or refute) the discovery.

    Axion is also proposed to explain the 20 minutes cycles of X-ray flares coming from the center of the Milky Way. In the 1990s, computer simulations of clouds of dark matter made of axions showed that giant bubbles of these particles would burst out from the clouds. These axion bubbles would expand and contract with a period of 20 minutes - matching the period of infrared and X-ray flares from Sagittarius A*. The model relies on a controversial version of gravity, which proposes that gravity starts to repel as the gravitational field gets stronger. Confirmation by observation would prove the existence of axion and would also raise question about Einstein's General Relativity.

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    Finally, there is a website for anyone who aspires to a career in theoretical physics. The website provides links to teach people on the intricacy of particle theories. It was created by Gerard 't Hooft, who showed in 1971 that the gauge bosons in Standard Model could be made massive while preserving renormalizability. He is the recipient of the 1999 Nobel Prize.