![]() | ---------- (1a) |
![]() |
where the Lagrangian L in unit of energy is a function of the position q and velocity =dq/dt of the particle, and the integration on time t is over the trajectories from "a" to "b" as shown in Figure 01a. Thus S has the unit of erg-sec. 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 shows three different forms of
|
Figure 01a Functional |
q(t) (blue, green, and red), the corresponding Lagrangian L, which is determined by q(t), and the action S[q(t)], which would have three different numbers equal to the areas under L indicated by either the blue, green or the red curve. The integration of the Lagrangian L |
S = 0 with a small variation of path -
q, we can derive the Euler-Lagrange equation:
S / q(t) = ![]() | ---------- (1b) |
or k = 0.
Energy conservation
Momentum conservation
Angular momentum conservation![]() | ![]() | - L = constant ---------- (1g) |
![]() |
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 |
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. |
-t; but its solution in Eq.(1f) is not so unless A = B (see more in the section on "Spontaneous Symmetry Breaking").
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
![]() |
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 |
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: |

to +
. For the free field case where L = (m/2)(dq/dt)2, the functional integrations can be performed analytically. The final result is:
= |m/h(tb - ta)|1/2 exp{[im(qb - qa)2] / [2
(tb - ta)]} ---------- (1i)
in Eq.(1i) satisfies the time-dependent Schrodinger's equation in one dimension for a non-relativistic free particle:
![]() | ---------- (1j) |
(x,t) =
, where q(ta) = x(ta) can be conveniently chosen at the origin of a coordinate system and with ta= 0. (see further explanation in footnote)

is the Lagrangian density, 
is the
th component of the field,
denotes the space-time component,
,
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 |
| 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 | 3rd component of weak isospin |
| Internal rotation of quark field | SU(3) in strong interaction | Color charge |
+ 
, 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) |

![]() | ---------- (1l) |
, which is a function of x, y, z, and t (collectively represented by x in the equation),

![]() | ---------- (2) |
![]() | ---------- (3) |
, and the time x0 is set to zero after the time derivation has been performed.
![]() | ---------- (4) |
to generate:
= nk|nk
= (nk+1)1/2|nk+1
= nk1/2|nk-1
![]() | ---------- (5) |
![]() | ---------- (6) |
![]() |
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 |
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 |
,
= constant. Note that the conservation of charge would not be valid if there is source or sink within the volume.
,
. These two independent scalar fields can be varied by the internal phase transformation:![]() | or, in a more familiar form: | , |
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:
![]() | ---------- (7) |
![]() | ---------- (8) |
![]() | ---------- (9a) |
(x - y) is the delta function. Similarly, the Green's function for the ferminon is defined by the equation:![]() | ---------- (9b) |

are the gamma matrices associated with the fermion. In Dirac representation, the matrices can be expressed explicitly as:

= 0 for
; the
5 matrix converts a vector to pseudo-vector (axial vector) with parity +1.
.
; 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").
, where A is the electromagnetic vector potential and
is the 4 components spinor separated into 2 parts (in Dirac representation,
and
together form a 4-components field for either the fermion or anti-fermion; 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:
![]() |
---------- (10) |
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. ![]() |
---------- (11) |
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
![]() |
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 |
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. |
![]() |
---------- (12) ---------- (13) |
![]() | ---------- (14) |
the nth order term in the S-matrix expansion Eq.(11) has the explicit form:![]() | ---------- (15) |
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 make sure a particle is created before its annihilation). The T product can be expanded into the N products and pairings, e.g.,
![]() | ---------- (16) |


or SF(x - y) from the pairings, e.g.,

![]() |
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: |

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. 
![]() |
![]() |
Table 02 Feynman Rules [view large image] |
Table 03 Momentum Representations [view large image] |
represents the anti-nucleon. In evaluating the S matrix, it is sometimes advantageous to go over to momentum space via the Fourier expansions:
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.
p1
p2
p3
p4)1/2
{(-ig02)/[(p1 - p3)2 - m02]}
,
is the charge density, and c is the velocity of light.
where 
=
/
x
. The energy-momentum tensor of the electromagnetic field can be expressed as:
, where
.
, where g is the determinant of the metric tensor ( g = g11g22g33g44 in isotropic space-time). In general, the field component and the source are multiplied by a factor of (-g)½ with the presence of gravity. The energy-momentum tensor is in a more complicated form:
. See "temporal waves" from early universe as dark energy.![]() | ---------- (24) |
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) |
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.
![]() | ---------- (36) |
![]() | ---------- (37) |






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:
(
,Ek)/d
= (Ze2/4Ek)2[1/sin4(
/2)] ---------- (38c)
![]() |
![]() |
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 the measured angular distribution of the scattered
|
Figure 02a Impact Parameter |
Figure 02b Rutherford Scattering [view large image] |
particles with incident energy of 15 Mev. |
![]() |
![]() |
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 experimental data start to deviate from the theoretical curve. Figure 02d |
Figure 02c Cross Section vs Angles [view large image] |
Figure 02d Cross Section vs Energy |
shows the deviation at a fixed angle of 60o. |
(
,E)/d
= [(Ze2E)/(2|p|2c2)]2 {[(1 -
2sin2(
/2)]/sin4(
/2)} ---------- (38d)
=|p|c/E.
/dq2d
= (4
2/q4) (E'/EM) [(M/
)W2(q2,
)cos2(
/2) + 2W1(q2,
)sin2(
/2)] ---------- (38e)
4EE'sin2(
/2),
= (p
q)/M
E - 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).
![]() |
![]() |
region: ![]() , q2 , and 0 x = q2/(2M ) 1 (elastic scattering corresponds to x = 1), W1 F1(x), W2 F2(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 satisfied (see Figure 02f). This result suggests that the partons |
Figure 02e Deep Inelastic Scattering [view large image] |
Figure 02f Form Factor [view large image] |
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. |
![]() |
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) |
o = (h/mc) = 2.43x10-10 cm is the Compton wavelength, which is actually the de Broglie wavelength for the electron moving at a speed of c/
.
![]() |
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/mc2 = 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. |
(
)/d
= (ro2/2) (
'/
)2 (
/
' +
'/
+ sin2
) ---------- (38h)
, 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.
![]() |
The energy levels calculated from the Dirac equation with the electron in a Coulomb potential is given by:![]() where Z is the number of positive charges, m is the mass of the electron, and is the fine structure constant. 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 = 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 |

V(eff) come from the radiative corrections.
min represents the photon mass, which is equal to zero, it introduces a divergence into the formulation. However, it is eventually cancelled by the infrared correction and disappears completely in the level shift calculation. These two terms contribute the majority of the Lamb shift with a correction of 1011.41 MHz.
(3)(x) is applicable only to the 2S1/2 state, which has a high probability to be near the center of the Coulomb potential. On the other hand, the term with the quantum number L vanishes for the 2S1/2 state since L = 0 in this case.
V(eff) with the corresponding hydrogen wave functions, it yields the Lamb shift in the form
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.![]() |
![]() |
more accordingly. By summing the contributions from : fine structure, hyperfine splittings, radiative corrections, recoil and proton size, ..., the total computed 2S1/2F=1 - 2P3/2F=2 energy difference (the dominate transition) is : E = 209.9779(49) - 5.2262(rp)2 + 0.0347(rp)3 where rp is the root-mean-square charge radius of the proton, numbers in parenthesis indicate uncertainty, and the energy is expressed in unit of mev = 10-3ev. The first term |
Figure 03b Muonic Lamb Shift [view large image] |
Figure 03c Laser Resonance [view large image] |
in the formula is dominated by vacuum polarization, which causes the 2S states to be more tightly bound than the 2P states (Figure 03b,c). |
X-ray events (red in Figure 03b,b), a resonance curve is obtained with a peak at 49.88188(76) THz (Figure 03c) corresponding to
E = 206.2949(32) mev or rp = 0.84184(36)(56) fm (1 fm = 10-13cm) with numbers in parentheses indicating experimental and computational uncertainties respectively. This new value of rp is significantly different from the previous measurements from H atom spectroscopy (CODATA), and from electron-proton scattering experiment (Figure 03c). Much more works are needed before we can find out the cause of the discrepancy.![]() |
The traditional assumption for proton mass distribution is about 75% concentrated in a central core (size rp) with the other 25% lying outside in the halo up to 1.4 fm. An August 2010 calculation suggests that the discrepancy |
Figure 03d Latest Proton Size |
between the original proton size and the latest (2010) experimental data can be reconciled if the halo band extends 4.7 times as far as previously thought (Figure 03d). |
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. The schemes to make the force law less singular is called regularization. There are three popular schemes :
is taken at the end of calculation to decouple the unphysical particle from the theory.
0. The coupling constants g for any gauge theory is dimensionless, so n = 0. Depending on whether n = 0, n > 0, or n < 0, the corresponding theories are renormalizable, super-renormalizable, or non-renormalizable.
m
R = Z21/2
0
R = Z31/2A
0
c (Gev / c2)-2 in quantum gravity has a negative energy dimension of n = -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.
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.![]() |
Figure 03e is the Feynman diagram in momentum space for the electron self-energy (the insert portrays the unperturbed propagator). 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:![]() where the self-energy correction (p) is a 4X4 matrix given by:![]() -[A + (i![]() p + m)B + ![]() ![]() ], and m is, at this point, finite but arbitrary.
|
Figure 03e 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( ). |

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 difference between the bare mass m0 and
, which represents the contribution from the virtual electron-positron pairs in the vicinity of the electron. It is noticed that even when the cutoff
is taken to be the Planck energy of 1019 Gev, the correction to the bare mass is
/m ~ 0.178. Anyway, the two terms m0 and
together render a finite physical mass ,i.e., mobs = mR = m = m0 -
. The exact cutoff point is not important as long as mR agrees with observation.

with (1/X) =
and Y = -i[A + (i
p + m)B], it can be shown that the correction to the electron propagator (in momentum space) is given by:![]() | ---------- (39b) |
term has been moved into the denominator, Z2 = 1/(1 - B), and B is another infinity in the
(p) expansion about the point -i
p = m. The factor Z2 is absorbed into the electron line
R = Z21/2
0, where the subscript 0 denotes bare state and the subscript R the renormalized state. Only the A and B terms are divergent in the Taylor series for
(p), and
R becomes finite as the result of one infinity from
0 is divided by another infinity from (1-B).
![]() | ---------- (39c) |
![]() |
This divergent correction to be absorbed by the electric charge is denoted by 1/Z1. The other divergent correction is labeled as Z3 = 1/(1+C), where C is another infinite number from vacuum polarization (not shown here, it modifies the photon line A R = Z31/2A 0).In the S-matrix expansion, each term has a number of the interaction Hamiltonian HI -e0 0 0 = -eR R R (integration over 3-D space is omitted here)if the renormalized charge further amalgamates the factor Z2, and Z31/2, i.e., eR = [(Z2Z31/2)/Z1]e0 = (Z31/2)e0 since the "Ward's Identities" has shown that Z1 = Z2. Thus in terms of the renormalized quantities, all the divergent terms disappear from the QED formulation, which yield amazing predication with an accuracy up to one part in trillions. 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 03f 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: ![]() |
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:
![]() | ... |


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.
![]() |
which increases as Q increases (or, equivalently, as the probing distance becomes shorter, see Figure 03g). 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 03g. The effective coupling constant for strong interaction for large Q is now in the form: where nf = 6 is the effective number of quark flavours for large Q, and -1 is called the
|
Figure 03g 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 . |
![]() |
![]() |
attempt to formulate a gauge theory for the weak interaction. It is a 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 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 03i), 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 |
Figure 03h Yang-Mills [view large image] |
Figure 03i Gluon Vertex [large image] |
asymptotic freedom. Conversely, at larger distances, the color charge increases, so that the quarks tends to bind more tightly together giving rise to quark confinement, which is the flip side of asymptotic freedom. |
' = exp(i
(x)
)
(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.
is replaced by the covariant derivative:
=
- i
ab
a
, the index "a" is for the phase angles,
is the coupling constant, and 
b
=
ab
a 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:
derivative term. This is the result of adopting different convention for using either covariant or contravariant component (lower and upper indexing) and for using different definition on the time coordinate x0. Such different conventions can be translated into one another by noting that
k = -i
k where k = 1, 2, 3 and x0 = it in the Yang-Mills theory; while x0 = t in some other notations.
is now replaced by the non-abelian gauge boson fields
ab
a. 
in Eq.(39h) is similar to the electromagnetic field equation in Eqs.(23) and (24), except an additional non-linear term, indicating that the b
's interact among themselves (with the same coupling constant, also see Figure 03i) and is responsible for the anti-shielding effect.
in Eq.(39d) can be identified to W
in Eqs.(41a) and (42a) for the Standard Model. The Yang-Mills theory fails to account for the chiral nature of the fermion fields in weak interaction. Parity violation in weak interaction has not been discovered until 1957 - four years later than the introduction of the Yang-Mills theory.
= 1, c =1, then
becomes dimensionless. Otherwise, the mass term in the formulism would be m/
c; while the interaction term would appear as (
/
c)
b
.
![]() |
Of the 4 different kinds of forces in the universe, gravity is attractive; electromagnetic force can be attractive or repulsive; the strong force provides a powerful attraction at short distance; the weak force works in very short distance, its action is to change one form of particle into another such as the beta decay of a free neutron in Figure 03j (strong force in stable nucleus inhibits the neutrons from beta decay). Note that the simple picture in Figure 03j already embraced the conservation of mass-energy, charge, and lepton number (nl). |
Figure 03j Weak Interaction in Picture [view large image] |
The strength of these forces varies according to the separation between particles, but in term of their coupling constants the ratio between the strong, electric, weak, and gravity is about 1 : 10-2 : 10-13 : 10-38. The understanding of the weak force took many twists and turns. |
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that time because while both the alpha and gamma rays are emitted with discrete spectra following the law of mass-energy conservation, beta rays appeared to exhibit a continuous spectrum. The problem persisted until 1930 when Pauli suggested that an invisible entity called neutrino shares the decay energy with the electron - the two particles together restore the conservation of mass-energy to its rightful place. The neutrino was dully discovered in 1953. It interacts only through the feeble weak interaction and gravity making its detection very difficult. Later on it is realized that |
Figure 03k Lifetime of Decays [view large image] |
the elusive entity has to be anti-neutrino to converse lepton number (Figure 03j). It is now understood that all fermions feel the weak force which is characterized by long lifetime as shown in Figure 03k. |
![]() |
M = GF( p un)( e u )where GF is the weak coupling constant, the u's are the Dirac spinors for the proton, neutron, electron, and anti-neutrino; the factor contain the essence of the weak interaction effects, in case it is equated to the gamma matrices, it would prescribe the
|
Figure 03l Fermi's Theory [view large image] |
current-current interaction (see Table 05 below); M has been labeled as (probability) amplitude, matrix element (of the interaction), (phenomenological) Lagrangian, or (interaction) Hamiltonian similar to the one for QED as shown in Eq.(38b). |
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mathematical expression: Pu(x,t) = u(-x,t). The overall parity is given by the multiplication of its parts, e.g. (+1)x(-1)x(+1) = (-1) etc. It has been long held that parity is conserved before and after all particle interactions. The problem started in the 1950's when it was observed that the and mesons seem to be identical in every respect but their decay products appeared to be very different. While the ends up with three mesons the decays into two. Since the has parity of -1, that means the two end products have different parity, one of them must violate parity. Further study by T. D. Lee and C. N. Yang strongly suggested that is the case.
|
Figure 03m Parity Violation [view large image] |
Almost immediately C. S. Wu devised and carried a test, which confirmed the idea conclusively (Figure 03m). The result also implied that only the electrons with left-handed spin participate in the weak interaction. The three Chinese physicists shared |
is the Dirac spinor simimlar to the u here). A pseudo or axial is added in front for those involve the
5:
.
![]() |
![]() |
After a long search for the magic formula, it turns out that parity violation can be expressed in combination of vector and axial vector (V - A) |
Table 05 Types of Fermion Current [view large image] |
Figure 03n Beta Decay, Modern Version |
interaction, e.g., for the electron anti-neutrino portion: |
e
(1 -
5)u
]/2 =
eL
u
L, where the subscript L indicates that only the left-handed elements are involved. There is no right-handed neutrino, and the right-handed electron only participates in the electromagnetic interaction. A similar expression is derived for the d quark (in neutron) and the u quark (in proton) as shown in Figure 03n. The updated version of M for beta decay is now in the form :
![]() |
The concept of gauge invariance can best be visualized with an example, which portrays a boat floating on the water surface with the depth of water at 3 meters (Figure 03o). The gauge (the water level marker) at the left registers a reading of 6 meter on the surface and 3 meter on the bottom, while the measurements from the other one at the right are 9 and 6 meter correspondingly. The differences in the readings are the result of moving the right marker further down, but the depth of water remains the same at 3 meters - the picture has not been changed by moving the gauge around. In theorietical physics it is the governing equation that |
Figure 03o Gauge Invariance [view large image] |
is unchanged (invariant) under the change of some parameters, which turn out to be the phase angles in an abstract space. Local gauge invariance produces mediating bosons, while global gauge invariance is related to conservation law. |


- m
= 0
eig
(x), where
(x) is the gauge parameter and a function of x, and g is the coupling constant. The transformation change the original Dirac equation to :

- g
(
)
- m
= 0
A
is inserted into the original Dirac equation since
A
A
- (
) for the Lorentz gauge according to classical electromagnetism. In other word, the partial derivative
should be replaced by the covariant derivative D
=
+ igA
. Thus the gauge invariance form of the Dirac equation is :
D
- m
= 0
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: 
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.
:
and
are indices for the space-time components running from 1 to 4. Whenever an index appears in both the subscript and superscript, it signifies a summation over these components.
is related to the three gauge (vector) bosons with the index "a" running from 1 to 3.
is the anti-symmetric tensor for the electromagnetic field as shown in Eq.(24), where the vector potential A
is now denoted by B
.![]() | ---------- (42f) |
.
are the Pauli matrices as shown in Eq.(10) in the appendix on "Groups" with i running from 1 to 3. The four 4X4
metrices are constructed from the Pauli matrices and the identity matrix.
and B
boson respectively. Ge is the Yukawa coupling constant, which defines the strength of the interaction between the Higgs field and the lepton fields.
is the Higgs doublet with four real components, three of which will be absorbed by the W
, and Z0 to become massive. The diagram below shows a simplified Higgs field in the shape of a Mexican hat:
(-m2/2
)1/2 (the scalar field at minimum energy) by the transformation
' =
- v, the fields Wa
and B
recombine and reemerge as the physical photon field A
, a neutral massive vector particle Z
, and a charged doubled of massive vector particles W
. In terms of the Weinberg angle (mixing angle) tan
= g'/g,
, W
3) to (A
, Z
) can be considered as a rotation of the mixing angle.
W = (4
)1/2 = 0.3028, and from the measurement of sin2
W = 0.226
0.004 to evalutate the masses for the gauge bosons. The theoretical values are in good agreement with MW = 80.4 Gev, and MZ = 91.19 Gev determined by experiments.


is similar to the SU(2) gauge field in Eq.(42b). This is essentially the Yang-Mills field developed back in 1953 by generalizing the gauge invariance used in QED. It represents the eight massless gluons carrying the SU(3) "colour" force with the index "a" running from 1 to 8.
=
.![]() | ---------- (43) |
![]() | ---------- (44) |
![]() |
represents the covariant derivative. These terms signify interaction with the gauge bosons. |
Figure 04 Standard Model [view large image] |
The actual numerical values of these coupling constants are not given by the theory and must be measured from experiments. |
1 such that the mixed state
1 + s sin
1.| where | ![]() | ![]() |
![]() |
cij = cos ij, sij = sin ij, and the angle tunrs some matrix elements into complex numbers, thereby violates CP invariance (CP invariance demands that V* = V).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 inFigure 04h. It leads to the detection of neutrino mass. |
Figure 04h Flavor Mixing |
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. |
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In spite of these shortcomings as mentioned above, the Standard Model has been proven to be remarkably resilient under various verifications including the latest measurement for the mass of the Higgs particle. The most recent attempt (in 2013) to break SM is to measure the shape of some nuclei, which would become pear-shaped (Figure 05a) in the presence |
Figure 05a Pear-shaped Nucleus [view large image] |
of permanent electric dipole moment (EDM). Since EDM would violate the T symmetry (and thus also introduce CP violation) in SM, the detection of specific radiation patterns (from the pear-shaped nuclei) will indirectly indicate the necessity of new physics. It is |
| 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 |
![]() |
An alternate arrangement unifies all these fermions into the 10 dimensional SO(10) group with the introduction of two color charges for weak interaction: Y = - ( R + W + B ) / 6 + ( G + P ) / 4. |
Figure 05b SO(10) Group |
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. | 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 |
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.![]() |
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. |
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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] |
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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 |
action/ ) now changes from oscillatory to dumping form: all pathse-S. |
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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(iga CxyAna a), |
Figure 05g The Link |
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. |
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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 |
Yang-Mills Lagrangian. 6. The Wilson action for SU(N) group can be expressed as: SW(Uij) = P[Tr(UP + UP ) - 2N] |
![]() |
< 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. |
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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 |
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. |
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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 |
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. |
![]() |
(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: |
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In case there are a lot of Feynman graphs to be evaluated in a process, the Unitarity Method can be used to shorten the computing within a manageable time. It is akin to Fluid Dynamics where the motion of individual molecules is collectively aggravated into macroscopic elements governed by a set of equations at another level (known as the Navier-Stokes Equations). It utilizes the fact that the infinities in the loop diagrams are often canceled out leaving a finite result at the end of the calculation. Figure 05n illustrates schematically the idea of one S-matrix corresponding to many Feynman diagrams. It has been shown that together with some conditions imposed on the S matrix, the unitarity method can produce rather accurate result. |
Figure 05n Unitarity Method |
The conditions include : |
![]() |
produced by supergravity in the 1960s. The argument was rather indirect because it was not feasible to evaluate 1020 terms in the three virtual-graviton loops. The unitarity method has now taken up the challenge and showed that those infinite quantities are indeed finite. The next tests would be the four and five loops computations. Along the way, the |
Figure 05o Graviton and Gluon [view large image] |
technique has also revealed that a graviton can be considered as two copies of gluon (Figure 05o). |
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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 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.
|
Figure 06a Negative Energy |
The particle is the electron originally resided in the negative energy region, while the anti-particle (positron) can be interpreted as the hole in the vacated energy level acquiring a mass m0. The law of charge conservation demands that this anti-particle carries a positive charge. |
+
- as shown in diagram (c), Figure 06a. The single photon must be virtual (off-shell) as it does not obey energy-momentum conservation.
+s should emerge in one hemisphere than in the other (with the reverse asymmetry for the
-s). Such parity violation was duly confirmed in 1981.
hadrons to that for e+e-
+
-, as a function of the colliding energy:
![]() | ---------- (45) |
![]() |
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. |
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mesons is now expanded to a 25-plet. 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
|
Figure 06c Meson Spectrum [view large image] |
as a potential acting in the vicinity of a colour charge. Thus instead of the Coulomb potential in the case of the atom, the quarks interact via the potential: |
s / (3r) + ar ,
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.![]() |
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 |
Figure 06d Two- and Three-Jet Event [view large image] |
of 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. |
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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". |
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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)m2 2 + ( /4!) 4, and 4! = 4x3x2x1.
|
Figure 06f Scalar Field |
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). |
= 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:

'

' - V'(
') ---------- (47)
') = |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.![]() |
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] |
)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 + i
2)ei
/(2)1/2
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
, and
, the Lagrangian is transformed to:
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:

' = e-iq
, with q
= tan-1(
/
)
' = H =
cos(q
) +
sin(q
)
' = 0
, 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 v
0) and the coupling of the gauge field to the scalar field (q
0) 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.![]() |
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] |
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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). |
e =
/L= -g e/(2me), where g = 2 is related to the fact
![]() |
![]() |
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/(4![]() c) ~ 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] |
/2
- 0.328 (
/2
)2 + 1.181 (
/2
)3 - 1.510 (
/2
)4) = 0.0023193042800. The experimental value is: g - 2 = 0.0023193043768 (in agreement with the calculated value to ten significant figures).![]() |
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] |
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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 | 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. |
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With a few tweaks of the conventional fermion table, it can be re-arranged into a form shown in Figure 11, which appears to be a shortened version of the periodic table compiled in 1869. This new periodic table has increasing mass along both column (from top to bottom) and row (from left to right with two exceptions) similar to the old periodic table. The orderly arrangement of the old table was eventually explained by the internal structure of the atoms. Thus it is surmised that the same kind of explanation can be applied to the fermions and |
Figure 11 A New Periodic Table |
Figure 12 Preon Model [view large image] |
the bosons (not shown in Figure 11). The hypothetical constituent is now dubbed as preon. It is hoped that a new model with the |
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This footnote is supposed to clarify the hand-waving argument in the main text with the help of simple mathematics and graphic. The first task is to show that there is a minimum for the action S (for a given Lagrangian L as shown in Eq.(49)) as the paths varies between two fit points. The trajectory of the particle is calculated from the equation derived from the "principle of least action" as shown in Eq.(50).
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Figure 11 Paths |
Figure 12 Actions |
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) with m = 1 gm and substituting q, dq/dt from Eqs.(52a,b) (see also Figure 01a). The paths in Figure 11 and the corresponding actions in Figure 12 are computed by varying the parameter "a" from -50 to +50 in step of 10. The path traces out a straight line for a = 0. For particle in free space, i.e., for k = 0, the correspondence of shortest path (a straight line) and least action (when S is at its minimum) is exact. The correspondence is off when interaction is taken into account (for example with k = 1 gm/sec2, a=(kX/2)/[m-(kT2/10)]
0.6 cm/sec2 when S is at its minimum) as if the force field creates a curve space for which the shortest path is not a straight line anymore. Actually, each paths is similar to a numeral devoid of any physical content by itself. They are one step away from the trajectory of a real particle contrary to what some physicists have it interpreted otherwise. Feynman's expression of "sum over paths" in his original work has been altered to "sum over histories" attaching additional meaning as if the paths are real. Furthermore the Lagrangian involved may or may not describe the actual motion of the particle. Additional constraints have to be imposed by demanding invariance under certain operations and it will ultimately be justified by observations.
dq = (mv/2) X = constant
(shortest path) = constant
(actual trajectory) = least action![]() |
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Then along came quantum mechanics and the method of path integral to calculate the transition amplitude between two points, e.g., from ta = 0 to tb = T : ![]() The sum is supposed to include all kinds of paths imaginable. However, for demonstration purpose the family of paths considered in the previous section is |
Figure 13 Path Integral, Real |
Figure 14 Path Integral, Img |
sufficient (see Figure 11). The Planck's constant can also be absorbed into the action S so that the computational procedure can be carried over to evaluate the sum. |
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An alternate way to perform the path integral is to sum over all the paths at different time t and then put them all together as shown in Figure 15, where the time coordinate is divided into N parts of equal interval . Thus T = N , and dt = 0. The transition amplitude <0|X> can be written as :<0|X> = A ![]() ![]() ![]() ![]() e(i / ) L(qn - qn-1)dq1![]() ![]() dqN-1 ---------- (55)where A is the normalization constant. For a particle in free field L = (m/2)(dq/dt)2, A = (m/ih )N/2 Eq.(55) can be evaluated analytically :<0|X> = A ![]() ![]() ![]() ![]() e(im/2![]() ) (qn - qn-1)2dq1![]() ![]() dqN-1= (m/ihT)1/2exp(imX2/2 T) ---------- (56)
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Figure 15 Path Integral |
The integration is carried out by applying the Gaussian integral : e-f(qn-qn-1)2-g(qn+1-qn)2dqn = [ /(f+g)]1/2{e-[fg/(f+g)](qn-1-qn+1)2} ---------- (57)repeatedly (N-1) times. |
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The real part of the transition amplitude <0|X> is plotted in Figure 16 and 17 with T and X holding constant respectively. Figure 16 shows that the wavelength is shorter at large X signifying higher probability for particle with higher velocity to get there. On the other hand in Figure 17 for a given X, a particle with lower velocity (and hence longer wavelength) would take longer to reach. But the chance of getting there becomes progressively smaller. Numerical computation is again performed by a home computer with Basic programming. The electron mass m is assumed in the calculation, thus (m/h)1/2 ~ 0.4 and (m/2 ) ~ 0.5. For the case of constant T, X is evaluated
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Figure 16 X vs Transition Amplitude [view large image] |
Figure 17 T vs Transition Amplitude [view large image] |
within the range -3 to +3 in step of 0.001 (in unit of cm); while for the case of constant X, T is evaluated from 0.5 to 5 in step of 0.0001 (in unit of sec). |
/2m)
<0|x> = i
<0|x> ---------- (59)
in the form of ei(kx-
t) is not connected to any initial conditions but satisfies the same equation as shown in Eq.(59).| X | T | |<0|X>1| | |<0|X>2| | |<0|X>2|/|<0|X>1| | |<0|X>2|/T |<0|X>1| |
|---|---|---|---|---|---|
| 1 | 1.0 | 8.75 | 0.40 | 0.045 | 0.045 |
| 1 | 1.25 | 6.28 | 0.36 | 0.057 | 0.046 |
| 1 | 1.5 | 4.80 | 0.33 | 0.067 | 0.045 |
to +
. Thus, the transition amplitude can be rewritten as:
eiS/
da ---------- (61)
90o from the Minkowskian spacetime to the Euclidean spacetime. The technique alters the oscillating exponential to a decaying exponential making the integration much easier to carry out. Actually, Method 2 has performed the Wick rotations already implicitly.
C e-kx2+fx+gdx = C(
/k)1/2e(f2/4k)+g ---------- (62)
T)
(8.94/iT3/2)exp(imX2/2
T) ---------- (63)
/2m)
<0|x>'3 = i
<0|x>'3 ---------- (64)
0.045 ---------- (65)