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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

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

Axion

Preons

A footnote on Least Action Principle and Path Integral

---------- (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
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## 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 |

The "Action Principle" states that by minimizing the action

^{S / q(t) =} | ---------- (1b) |

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

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

m d

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

The potential for the Hooke's law is V(x) = (k/2)x

d

which is the Newton's law for the motion of harmonic oscillator. The general solution is:

x = A e

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) |

(m/2)(dx/dt)

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.
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## 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' -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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