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Quantum Field Theory


Action Principle and Symmetry
Path Integral
Field Equation
Second Quantization
Noether's Theorem and Charge Conservation
Conservation Rules
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
Weak Interaction
The Standard Model
The SO(10) Group and Unification
Lattice Theory
Unitarity Method
Quantum Vacuum
Spontaneous Symmetry Breaking
Gyromagnetic Ratio and Anomalous Magnetic Moment
A footnote on Least Action Principle and Path Integral

Action Principle and Symmetry

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 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 q(t) (blue, green, and red), the corresponding Lagrangian L, which is determined by q(t), and

Figure 01a Functional
[view large image]

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 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. It is equal to the area under the curve of L with respect to the L = 0 (horizontal) axis. (see further explanation in footnote)

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:

---------- (1b)
from which the trajectory of the particle can be computed.

For a particle with mass m moving in a potential V(q) :

L = (m/2)(dq/dt)2 - V(q), ---------- (1c)

the Euler-Lagrange equation has the explict form with the variable of the actual path labeled as x :

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 Hooke'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 is 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. However, in the case of geometrical symmetry it is the configuration that remains the same after the operation. Whereas in

Figure 01b Symmetry
[view large image]

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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