The World Particle content

Interactions

Schrodinger Wave Equation He started with the energy-momentum relation for a particle Expecting them to act on plane waves he made the quantum mechanical replacement: How about a relativistic particle？

The Quantum mechanical replacement can be made in a covariant form The Quantum mechanical replacement can be made in a covariant form. Just remember the plane wave can be written in a covariant form: As a wave equation, it does not work. It doesn’t have a conserved probability density. It has negative energy solutions.

The proper way to interpret KG equation is it is actually a field equation just like Maxwell’s Equations. Consider we try to solve this eq as a field equation with a source. We can solve it by Green Function. G is the solution for a point-like source at x’. By superposition, we can get a solution for source j.

Green Function for KG Equation: By translation invariance, G is only a function of coordinate difference: The Equation becomes algebraic after a Fourier transformation. This is the propagator!

Green function is the effect at x of a source at x’. KG Propagation Green function is the effect at x of a source at x’. That is exactly what is represented in this diagram. The tricky part is actually the boundary condition.

For those amplitude where time 1 is ahead of time 2, propagation is from 1 to 2. B B B B 2 1 1 C 2 C A A A A is actually the sum of the above two diagrams! To accomplish this,

Blaming the negative energy problem on the second time derivative of KG Eq., Dirac set out to find a first order differential equation. This Eq. still needs to give the proper energy momentum relation. So Dirac propose to factor the relation! For example, in the rest frame: Made the replacement First order diff. Eq.

Now put in 3-momenta: Suppose the momentum relation can be factored into linear combinations of p’s: Expand the right hand side: We get and we need:

It’s easier to see by writing out explicitly: Oops! What! or No numbers can accomplish this! Dirac propose it could be true for matrices.

2 by 2 Pauli Matrices come very close

Dirac find it’s possible for 4 by 4 matrices We need: that is He found a set of solutions: Dirac Matrices

Dirac find it’s possible for 4 by 4 matrices Pick the first order factor: Make the replacement and put in the wave function: If γ’s are 4 by 4 matrices, Ψ must be a 4 component column: It consists of 4 Equations.

The above could be done for 2 by 2 matrices if there is no mass. Massless fermion contains only half the degrees of freedom.

Now put in 3-momenta: Suppose the momentum relation can be factored into linear combinations of p’s: Expand the right hand side: We get and we need:

交叉項抵銷

Plane wave solutions for KG Eq. There are two solutions for each 3 momentum p (one for +E and one for –E )

Expansion of a solution by plane wave solutions for KG Eq. If Φ is a real function, the coefficients are related:

Plane wave solutions for Dirac Eq. Multiply on the left with There are two sets of solutions for each 3 momentum p (one for +E and one for –E )

How about u?

We need: You may think these are two conditions, but no. Multiply the first by So one of the above is not independent if

We need: or How many solutions for every p? Go to the rest frame!

uA is arbitrary Two solution (spin up spin down)

uB is arbitrary Two solution (spin down and spin up antiparticle)

There are four solutions for each 3 momentum p (two for particle and two for antiparticle) It’s not hard to find four independent solutions. - We got two positive and two negative energy solutions! Negative energy is still here! In fact, they are antiparticles.

Electron solutions:

Positron solutions:

Expansion of a solution by plane wave solutions for KG Eq. Expansion of a solution by plane wave solutions for Dirac Eq.

Bilinear Covariants Ψ transforms under Lorentz Transformation: Interaction vertices must be Lorentz invariant.

The weak vertices of leptons coupling with W μ νμ -ig

Bilinear Covariants Ψ transforms under Lorentz Transformation: Interaction vertices must be Lorentz invariant. How do we build invariants from two Ψ’s ? A first guess:

Maybe you need to change some of the signs: It turns out to be right! We can define a new adjoint spinor: is invariant!

In fact all bilinears can be classified according to their behavior under Lorentz Transformation:

Feynman Rules for external lines

How about internal lines? Find the Green Function of Dirac Eq. Now the Green Function G is a 4 ˣ 4 matrix

Using the Fourier Transformation Fermion Propagator

Photons: It’s easier using potentials: forms a four vector.

4-vector again Charge conservation

Now the deep part: E and B are observable, but A’s are not! A can be changed by a gauge transformation without changing E and B the observable: So we can use this freedom to choose a gauge, a condition for A:

For free photons: Almost like 4 KG Eq. Energy-Momentum Relation

Polarization needs to satisfy Lorentz Condition: Lorentz Condition does not kill all the freedom: We can further choose then Coulomb Guage The photon is transversely polarized. For p in the z direction: For every p that satisfy there are two solutions! Massless spin 1 particle has two degrees of freedom.

Feynman Rules for external photon lines

Gauge Invariance Classically, E and B are observable, but A’s are not! A can be changed by a gauge transformation without changing E and B the observable: Transformation parameter λ is a function of spacetime. But in Qunatum Mechanics, it is A that appear in wave equation: In a EM field, charged particle couple directly with A.

Classically it’s force that affects particles Classically it’s force that affects particles. EM force is written in E, B. But in Hamiltonian formalism, H is written in terms of A. Quantum Mechanics or wave equation is written by quantizing the Hamiltonian formalism: Is there still gauge invariance?

B does not exist outside.

Gauge invariance in Quantum Mechanics: In QM, there is an additional Phase factor invariance: It is quite a surprise this phase invariance is linked to EM gauge invariance when the phase is time dependent. This space-time dependent phase transformation is not an invariance of QM unless it’s coupled with EM gauge transformation!

Derivatives of wave function doesn’t transform like wave function itself. Wave Equation is not invariant! But if we put in A and link the two transformations: This “derivative” transforms like wave function.

In space and time components: The wave equation: can be written as It is invariant!

This combination will be called “Gauge Transformation” It’s a localized phase transformation. Write your theory with this “Covariant Derivative”. Your theory would be easily invariant.

There is a duality between E and B. Without charge, Maxwell is invariant under: Maybe there exist magnetic charges: monopole

Magnetic Monopole The curl of B is non-zero. The vector potential does not exist. If A exists, there can be no monopole. But quantum mechanics can not do without A. Maybe magnetic monopole is incompatible with QM.

But Dirac did find a Monopole solution:

Dirac Monopole It is singular at θ = π. Dirac String It can be thought of as an infinitely thin solenoid that confines magnetic field lines into the monopole.

But a monopole is rotationally symmetric. Dirac String doesn’t seem to observe the symmetry It has to! In fact we can also choose the string to go upwards (or any direction): They are related by a gauge transformation!

Since the position of the string is arbitrary, it’s unphysical. Using any charge particle, we can perform a Aharonov like interference around the string. The effects of the string to the phase is just like a thin solenoid: Since the string is unphysical. Charge Quantization

Finally…. Feynman Rules for QED

4-columns 4-rows 4 ˣ 4 matrices 4 ˣ 4 matrices

Dirac index flow, from left to right! 1 ˣ 1 in Dirac index

Numerator simplification using The first term vanish! In the Lab frame of e Photon polarization has no time component. The third term vanish!

Denominator simplification Assuming low energy limit: in low energy

2nd term:

Finally Amplitude: Amplitude squared

旋轉帶電粒子所產生之磁偶極 磁偶極矩與角動量成正比

帶電粒子自旋形成的磁偶極 Anomalous magnetic moment