• conservative force: work done independent of path taken in configuration space, $\vec{F} = -\vec{\nabla} V(\vec{r})$
• (A exerting force on B) = (A transferring its momentum to B)
• force on systems of particles are distinguished as external and internal
• weak, strong law of action and reaction come hand in hand with conservation of total momentum and total angular momentum, for a system of particles
• total kinetic energy decomposes into COM kinetic energy and kinetic energy of individual particles about COM

constraints

• holonomic constraints: $$f(\vec{r}_1,...,\vec{r}_n,t) = 0$$
• constraints introduce dependence between coordinates, decoupling of such facilitates the introduction of generalized coordinates.
• holonomic constraints are more tractable

d'alembert's principle

• the generalized coordinates form a manifold (configuration space), the motion of the particle traces a curve on the surface parameterized by affine parameter $t$
• virtual displacement: the displacement is consistent with current displacements and forces applied, yet the virtual displacement does not result in modification of applied force or constraint (as an actual indinitesmal displacement would)
• principle of virtual work: virtual work as of applied forces vanishes
• principle of virtual work + $\vec{F}_i - \dot{\vec{p}}_i$ = d'Alembert's principle: $$\sum_i [\vec{F}_i^{(a)} - \dot{\vec{p}}_i] \cdot \delta \vec{r}_i = 0$$
• transforming d'Alembert's principle into generalized coordinates gives Lagrange's equation.
• Lagrange's equation is not unique

velocity-dependent potentials

• lorentz force in EM is derivable from a velocity-dependent potential
• when not all forces acting on the system are derivable from a potential, Lagrange's equation can be written in a more generalized form

$$\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}_j}) - \frac{\partial L}{\partial q_j} = Q_j$$

for some generalized force $Q_j$ that can be put into

$$Q_j = - \frac{\partial U}{q_j} + \frac{d}{dt}(\frac{\partial U}{\partial \dot{q}_j})$$

Notice the velocity dependence.

• frictional force is derivable from Rayleigh's dissipation function, a 2nd order equation in velopcity

---

• The lagrangian is invariant under $$L \rightarrow L' = L + \frac{dF(q_1, ..., q_n, t)}{dt}$$
• the EM field is invariant under $$\vec{A}\rightarrow \vec{A} + \nabla \psi, \quad \phi \rightarrow \phi - \frac{1}{c}\frac{\partial \psi}{\partial t}$$

exercise 1.8x.pdf

minimum action principle

• hamilton's principle: $$I = \int_{t_1}^{t_2}Ldt = \int_{t_1}^{t_2}[T-V]dt$$ has vanishing 1st order differential with respect to all nearby paths for $L$ defined as a path
• holonomic constraint implies equivalence of hamilton's principle and lagrange's equation. This gives an alternative derivation of Lagr beside from d'alembert's principle of vanishing work from virtual displacements
• minimum action principle can be applied to solve
  • shortest path between 2 pts in space
  • minimum surface of revolution
  • brachistochrone

for nonholonomic constraints

• In deriving lagr, the constraints as holonomic ones were only used in the last step to move to generalized coordinates, this then gives independent equations of motion. Nonholonomic constraints clearly ruins this.
• semiholonomic: $f(q_1,..., q_n, \dot{q}_1, ..., \dot{q}_n) = 0$
• lagrange multipliers can treat nonholonomic constraints

lagr multiplier

A systematic approach for solving (sort of) semiholonomic constraints

• write the constraints in form $f_\alpha(q_i, ..., \dot{q}_i, ...) = 0$
• Then define lagrange multipliers, which are functions of $q_i,\dot{q}i, t$: $$\lambda\alpha = \lambda_\alpha(q_i, \dot{q}_i, t)$$
• It must be true that $\lambda_\alpha f_\alpha = 0$, summation implied. From our earlier derivation we found that hamilton's principle gives $$\delta \int_{t_0}^{t_1} L(q_i,\dot{q}_i,t) dt = 0$$

which implies

$$\delta \int_{t_0}^{t_1} \left[ L(q_i,\dot{q}i,t) + \lambda\alpha f_\alpha \right] dt = 0$$

Then apply the lagr eqm

$$\frac{d}{dt} \frac{\partial [ L+ \lambda_\alpha f_\alpha ] }{\partial \dot{q}i} - \frac{\partial [ L + \lambda\alpha f_\alpha ]}{\partial q_i} = 0$$

to obtain the desired eqm. KEEP IN MIND THE TOTAL TIME DERIVATIVE, PRODUCT RULE NEED TO BE USED, because $\lambda$ is time dependent.

Separation of COM and relative motion

• the relative energy of two particles about their COM is $$T' = \frac{1}{2}m_1 \vec{r}_1^2 + \frac{1}{2}m_2\vec{r}^2_2,$$ where $\vec{r}_1, \vec{r}_2$ are polar vectors to the COM. Goldstein shows $$T' = \frac{1}{2}\frac{m_1 m_2}{m_1 + m_2} \dot{\vec{r}}^2$$ where $\vec{r}$ is the separation between particle 1 and particle 2. Below I show an algebraic derivation.

algebraic derivation of separation of COM kinetic energy

$$T \propto m_1 \dot{q}_1^2 + m_2 \dot{q}_2^2$$

$$= \frac{1}{(m_1 + m_2)} \left[ m_1(m_1 + m_2) \dot{q}_1^2 + m_2(m_1 + m_2) \dot{q}_2^2 \right]$$

$$= \frac{1}{(m_1 + m_2)} \left[ m_1^2 \dot{q}_1^2 + m_2^2 \dot{q}_2^2 + m_1m_2 \dot{q}_1^2 + m_1m_2 \dot{q}_2^2 \right]$$

$$= \frac{1}{(m_1 + m_2)} \left[ (m_1 \dot{q}_1 + m_2 \dot{q}_2)^2 -2m_1m_2\dot{q}_1\dot{q}_2 + m_1m_2 \dot{q}_1^2 + m_1m_2 \dot{q}_2^2 \right]$$

$$= \frac{1}{(m_1 + m_2)} \left[ (m_1 \dot{q}_1 + m_2 \dot{q}_2)^2 + m_1m_2(\dot{q}_1 - \dot{q}_2)^2 \right]$$

$$=\frac{1}{m_1+m_2}(p_1^2 + p_2^2) + \frac{m_1 m_2}{m_1 + m_2}(\dot{q_1}-\dot{q_2})^2$$

This identifies a COM term and effective mass term.

goldstein 3.3

• for 2 bodies under central force, examining the magnitude of the radial vector gives a 1D problem. The fictitious centrifugal force is introduced:

$f \rightarrow f' = f + \frac{l^2}{mr^3} <=> V \rightarrow V' = V + \frac{1}{2}\frac{l^2}{2mr^2}$

goldstein 3.4

• by considering $G = \sum_i \vec{p}_i \cdot \vec{r}_i$, and taking time average, we arrive at $$ = -\frac{1}{2}<\sum_i \vec{F}_i \cdot \vec{r}_i>$$ this is the virial theorem
• goldstein claims that the virial theorem can be used to argue the ideal gas law

3.5 the differential equation for the orbit and integrable power-law potentials

• $l dt = mr^2 d\theta \Rightarrow \frac{d}{dt} = \frac{l}{mr^2}\frac{d}{d\theta}$
• [ ] make sense of the equation above, in terms of the differentials
• goldstein argues that if $\theta$ satisfies the eqm, $-\theta$ does as well
• not every power of $r$ in $V = ar^{n+1}$ can be solved by integration. Only for $n=1,-2,-3$ have trig solns. For $n=5,3,0,-4,-5,-7$ have elliptic soln.

3.7

• $$\theta = \theta' - \arccos \left[\frac{\frac{l^2 u}{mk}-1}{\sqrt{1+\frac{2El^2}{mk^2}}} \right]$$
• finds eqm for inverse square central force
• for elliptic orbits the major axis depends solely upon the energy, this is argued by solving a quadratic formula
• eccentricity (semiminor axis) governed by angular momentum
• [ ] eccentricity

intuition for semimajor and semiminor axis

• $$E = \frac{1}{2}m\dot{r}^2 + \frac{1}{2}\frac{l^2}{mr^2} - \frac{k}{r}$$ is the energy of the system, when $\dot{r} = 0$, this defines a purely algebraic, quadratic equation, the roots of which gives 2 exterme points of $r$, the bigger one is semimajor, smaller one is semiminor.
• angular velocity $\dot{\theta}$ attains its max value at perihelion, minimum at aphelion

3.8 motion in time in kepler problem

• it is customary to measure polar angle by setting $\theta=0$ at perihelion (point of closest approach, highest angular velocity)

3.10 scattering in central force fields

$$\sigma(\Omega) d\Omega = \frac{# \text{particles scattered into solid angle } d\Omega \text{ per unit time}}{\text{incident intensity}}$$ $d\Omega = 2\pi \sin \theta d \theta$

formulation of the problem

• system in equilibrium when potential is at extremum, if the extremum is minimum then the equilibrium is stable.
• we are interested in the motion of the system within the immediate neighborhood of stable equilibria, giving bound states
• by equilibrium condition and shifting the zero of potential, the potential can be written purely quadratically for small displacements in any system upto O(2).

simple derivation for soln of SHO

• assuming small displacements about stable equilibrium => generalized coordinates $\eta_i = q_i - q_{0i}$, the most general lagrangian is $$L = \frac{1}{2}[T_{ij} \dot{\eta}^i \dot{\eta}^j - V_{ij} \eta^i \eta^j]$$ giving EQM $T_{ij}\ddot{\eta}j + V{ij}\eta_j = 0$
• With ansatz $\eta_i = C a_i e^{-i\omega t}$, the EQM becomes $$[V_{ij}-\omega^2T_{ij}]a^{j} = 0$$. This demands the matrix $[V_{ij}-\omega^2T_{ij}]$ be singular, and we wish to solve for possible $\omega$ s that makes this matrix singular, the normal frequencies.

6.3

• the lengthy arguments in 6.2 demonstratest that the eqm of small displacements ($\eta_i$) can be solved by the ansats $$\eta_i = Ca_ie^{-i\omega t}$$ a complete solution is a superposition of allowed frequencies
• the soln. of the secular equations are called resonant frequencies of the system.

Obtaining the Hami

• in general, not any simpler. Can be simplified for special cases that overlap with physical interest $$H = \dot{q}_i p_i - L$$ $$p_i = \frac{\partial L}{\partial \dot{q}_i}$$

• cyclic coordinates $q_j$ does not appear in Lagr as a non-derivative term, they give conserved momenta:

$$p_j = \frac{\partial L }{\partial q_j}$$

$$\dot{p}_j = \frac{\partial L }{\partial q_j} = 0$$

• If $t$ does not appear in $L$, then $H$ will be a constant of motion.

exercise 8.15scratch.pdf

canonical trans

Apply cano. tran. defined by p->P, q->Q, H->K. Then Hamilton's principle takes form

$$\delta \int [P_i \dot{Q}_i - K(Q,P,t)]dt = 0$$

$$\delta \int [p_i \dot{q}_i - H(q,p,t)]dt = 0$$

The general solution takes the form

$$\lambda (p_i \dot{q}_i - H) = P_i \dot{Q}_i - K + \frac{d F}{dt}$$

for F function of two of p,q,P,Q, called generating function.

If $\lambda = 1$, the transform is called canonical transformation. If $\lambda \neq$, the transformation is called extended canonical transformation.

• any extended canonical transformation can be made up by a canonical one + scaled transformation.

Table 9.1 gi es trlations between the canonical trans and generating functions.

symplectic relation

reminder

Let $Q_i = Q_i(q,p)$, then

$$\dot{Q}_i = \frac{\partial Q_i}{\partial q_j} \dot{q}_j + \frac{\partial Q_i}{\partial p_j} \dot{p}_j$$

when combined with hami eqm, it is

$$\dot{Q}_i = \frac{\partial Q_i}{\partial q_j} \frac{\partial H}{\partial p_j} - \frac{\partial Q_i}{\partial p_j} \frac{\partial H}{\partial q_j}$$

This is written compactly in matrix form:

$$\dot{\vec{\eta}} = \vec{J} \frac{\partial H}{\partial \vec{\eta}}$$

where $\vec{J}$ is the 2nx2n matrix of

$$\left[\begin{array}{rr} 0 & I \ -I & 0 \end{array}\right]$$

and $\vec{\eta} = (q_1, q_2, ..., p_1, ..., p_n)$

As an illustration

$$ \left[\begin{array}{rr} \dot{q}_1 \ \dot{q}_2 \ \dot{p}_1 \ \dot{p}_2 \end{array}\right]

=

\left[\begin{array}{rr} 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \ -1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \end{array}\right]

\left[\begin{array}{rr} \partial_{q_1} H \ \partial_{q_2} H \ \partial_{p_1} H \ \partial_{p_2} H \end{array}\right] $$

symplectic structure

Now take restricted transformation $\vec{\zeta} = \vec{\zeta}(\vec{\eta})$. Then $\dot{\vec{eta}} = \vec{M} \dot{\vec{\eta}}$ where $\vec{M}$ is Jacobian. Together with Hami eqm, we have

$$\dot{\vec{\eta}} = \vec{M} \vec{J} \frac{\partial{H}}{{\partial \vec{\eta}}}$$

But $\frac{\partial H}{\partial \vec{\eta}} = \tilde{M} \frac{\partial H}{\partial \vec{\zeta}}$. The Hami eqm for $\zeta$ is

$$\dot{\zeta} = M J \tilde{M} \frac{\partial H}{\partial \zeta}$$

as oppose to $$\dot{\eta} = J \frac{\partial H}{\partial \eta}$$

But we also must have

$$\dot{\zeta} = J \frac{\partial H}{\partial \zeta}$$

This implies $$MJ\tilde{M} = J$$

note: for extedned canonical trans, it is $MJ\tilde{M} = J$

This is called symplectic condition, $M$ is called symplectic matrix.

Poisson Bracket

define

$$[u,v]_{q,p} = \frac{\partial u}{\partial q_i} \frac{\partial v}{\partial p_i} - \frac{\partial u}{\partial p_i}\frac{\partial v}{\partial q_i}$$

similarily, the definition can be written as

$$[u,v]_{\eta} = \left( \frac{\partial u}{\partial \vec{\eta}} \right)^T J \left( \frac{\partial v}{\partial \vec{\eta}} \right)$$

• $[q_j, q_k]{q,p} = 0 = [p_j, p_k]{q,p}$
• $[q_j, p_k]{q,p} = \delta{jk} = - [p_j, q_k]_{q,p}$

The above two relations are written neatly as $[\eta, \eta]_{\eta} = J$

• $[\zeta, \zeta]_\eta = M^T J M$
• This implies if $\eta$ -> $\zeta$ canonical, then $[\zeta, \zeta]_\eta = J$

The relation $[\zeta, \zeta]\eta = J = [\zeta, \zeta]\zeta$ is called fundamental poisson bracket. It is invariant under cano. trans.

some poisson bracket algebra

• [u,v] = 0
• [u,v] = - [v,u]
• (linearity) [a + bv, w] = a [u,w] + b[v,w]
• [uv,w] = [u,w]v + u[v,w]
• (nontrivial, Jacobi's identity) [u,[v,w]] + [v,[w,u]] + [w,[u,v]] = 0

some other cano invariants

• lagrange bracket ${u,v}$. It also satisfies $${u,u}[u,u] = - I$$
• the magnitude of volume element in phase space is canonically invariant

canonical trans. grp

Cano. trans. form a group. The subgrp that is analytic of continuous parameters form a lie grp.

• lie grp of parameter $\theta_i$ lie on flat vector space whose basis vectors constitute a lie algebra satisfying the poisson bracket $$[u_i, u_j] = \sum_k c_{ij}^k u_k$$ The elements of the lie grp are generated by $$Q(theta_i) = \exp \left[ \frac{i}{2} \sum \theta_i u_i \right]$$

• pauli matrices form a representation of rotation grp, the vectors add, in a sense.

Hamilton Jacobi eq

Demand that from the canonical transformation $K(P,Q) = 0$, together assuming $F_2$ type generating function, then we have

$$H(q_i; \frac{\partial F_2}{\partial q_i}; t) + \frac{\partial F_2}{\partial t} = 0$$

The equation is the Hamilton-Jacobi eq.

Write $S = S(qi; \alpha_i; t)$ as solution for $F_2$, where $\alpha$ are constants of integration. Then take $\alpha_i$ as momentum $P_i$, we have from generating function properties

$$p_i = \frac{\partial S}{\partial q_i}, \quad Q_i = \frac{\partial S}{\partial P_i} = \frac{\partial S}{\partial \alpha_i} \equiv \beta_i$$

Then we have parameterization

$$q_j = q_j(\alpha, \beta, t)$$

Hamilton's principal function is the generator of a canonical trans. to constant coordinates and momenta?

mean value theorem

In charge-free space, the value of the electrostatic potential at any point is the average of the potential over the surface of any sphere centered on that point. electrostatics = coulomb's law at various extents and contexts - $\rho \rightarrow$ potential $\rightarrow \vec{E} \rightarrow$ force

electrostatic units and SI

|ESU|SI| |--|--| |k=1|$k=(4\pi\epsilon_0)^{-1}$| |charge in units of statcoulomb|charge in coulombs| |field in statvolts/meter|field in volts/meter|

delta function in 8 equations

1. $\delta(x-a) = 0$ for $x\neq a$
2. $\int \delta(x-a) dx = 1$
3. $\int f(x)\delta(x-a)dx = f(a)$
4. $\int f(x) \delta'(x-a) dx = - f'(a)$, where $\delta' = \frac{\partial \delta(x-a)}{\partial x}$, $f' = \frac{\partial f(x)}{\partial x}$
5. $\delta[f(x)] = \sum_i |\frac{df}{dx}(x_i)|^{-1}\delta(x-x_i)$
6. $\delta(\vec{x}- \vec{X}) = \delta(x_1-X_1)\delta(x_2-X_2)\delta(x_3-X_3)$
7. $\int_{V}\delta(\vec{x}-\vec{X}) d^3x =$ 1 if $V$ contains $\vec{X}$ else 0
8. $\nabla^2 \frac{1}{|\vec{x} - \vec{x}'|} = -4\pi \delta (\vec{x}-\vec{x}')$

9. $\delta$ has units of inverse length

10. $\vec{E}$'s gauss law depends on 1. inverse square law, 2. central force nature, 3. linear superposition

11. differential form $\approx$ differential equation

12. a vector field can be specified almost completely if $(\nabla \cdot)$ and $(\nabla \times)$ are specified everywhere. This introducees potential
13. There is discontinuity of $\sigma/\epsilon_0$ in crossing surface for $\vec{E}$
14. Green's function is developed to handle boundary conditions

boundary conditions

• dirichlet's problem: potential specified on closed surface

• neumann's problem: normal derivatives specified

• green's function satisfy $$\vec{\nabla}^2 (\frac{1}{|\vec{x}- \vec{x}'|}) = - 4 \pi \delta(\vec{x}- \vec{x}')$$

jackson 1.8

• The formula $\Phi(\vec{x}) = \frac{1}{4\pi \epsilon_0} \int \frac{\rho(\vec{x}')}{|\vec{x}-\vec{x}'|} d^3 x'$ becomes inconvenient for certain problems
• green's theorem is developed to handle complicated boundary conditions.
• green's 1st identity: $$\int_V (\phi \nabla^2 \psi + \nabla \phi \cdot \nabla \psi)d^3x = \oint_S \phi \frac{\partial \psi}{\partial n} da$$
• green's 2nd identity $$\int_v (\phi \nabla^2 \psi - \psi \nabla^2 \phi)d^3x = \oint_S \left[ \phi \frac{\partial \psi}{\partial n} - \psi \frac{\partial \phi}{\partial n} \right] da$$
• letting $\psi = \frac{1}{R} \equiv \frac{1}{|\vec{x} - \vec{x}'|}$ gives a formulation for $\Phi$

jackson 1.9

• jackson proves uniqueness of $\Phi$ using green's identity for both dirichlet and neumann conditions

jackson 1.10

• green's function must satisfy $$\nabla'^2 G(\vec{x},\vec{x}') = - 4\pi \delta(\vec{x}-\vec{x}')$$ it has the most general form of $$G(\vec{x},\vec{x}') = \frac{1}{|\vec{x}-\vec{x}'|} + F(\vec{x},\vec{x}')$$ with $F$ satisfying the laplace equation inside volume $V$: $\nabla'^2 F(\vec{x},\vec{x}') = 0$
• the freedom in $F$ can be used to eliminate one of the surface integrals in (1.36), which one depends in the type of boundary condition (dirichlet/neumann)

method of images

• the total induced surface charge is the image charge, the negative of the real charge, or some combination of the two.

Lecture 6:Advanced Electrostatics III:

Green Functions for Poisson’s Equation Obtaining Green Functions from the Method of Images

• green operated by linear differential operators results in the delta function. see LN pg128.
• Green Function approach. The basic idea is to find the 'impulse' response function for the differential equation: the generalized potential one gets if one has a point-like source. Given the impulse response function, linearity, one can obtain the generalized potential for an arbitrary source function by convolving the impulse response function with that source function.
• The freedom F in G is crucial for satisfying nontrivial boundary conditions
• both F and G are symmetric in their arguments

• LN 3.47 discussion: pg 135 star

• the component F_D of the full Dirichlet Green Function G_D can be determined by the method of images in some cases.

jackson 1.11

• $W = \frac{1}{2}\int \rho(\vec{x}) \Phi(\vec{x}) d^3 \vec{x} = \frac{\epsilon}{2} \int |\vec{E}|^2 d^3 x$

• method of images is concerned with N point charges in presence of boundary surfaces, i.e. conductors at certain potentials
• the image charge must lie external to field of interest

2.2

• the charge density on a surface is the normal derivative of $\Phi$ on the surface
• force in mirror image configuration can be calculated by taking mirror charges into consideration directly.

GREEN FUNCTION

• a green function is uniquely specified once geometry and type of boundary condition is specified
• green function has a part that satisfies laplace and a part satisfy poisson

exercise 2.14bscratch.pdf

• finite element analysis (FEA) encompasses a variety of numerical approaches for the solution of a boundary value problem in physics and engineering
• jackson uses galerkin's method as illustrations, in 2D

LEGENDRE POLYNOMIALS!

I wish I knew these well before trying exercises in chapter 3 of Jackson! - $P_l$ is $l$-th order in $x$, - $P_l$ has only even/odd powers if $l$ is even/odd - $P_l(1) = 1, P_l(-1) = (-1)^l$ - $P_l(0)=[(−1)^n(2n−1)!!]/2^n n!$ for even $l$. $P_l(0)=0$ for odd $l$ - $\nabla^2 P_l(\cos \theta) = - \frac{l(l+1)}{r^2}P_l(\cos \theta)$

• legendre polynomials form a complete set of orthogonal functions on [-1,1]. Thus any function in this range can be expanded in legendre polynomials: $$f(x) = \sum_{l=0}^{\infty} A_l P_l(x), \quad A_l = \frac{2l+1}{2}\int_{-1}^{1} f(x) P_l(x) dx$$
• problems possessing azimuthal symmetry have solutions of form $$\Phi(r,\theta) = \sum_{l=0}^\infty [A_l r^l + B_l r^{-(l+1)}] P_l (\cos \theta)$$
• if the range of $\theta$ is restricted, then we might need a special set of legendre polynomials

3.5 associated legendre functions and spherical harmonics

• asimuthal given by $e^{im \phi}$, $m \neq 0$
• parity given by $(-1)^{l+m}$
• $ = \delta_{ll'}\delta_{mm'}$

$<.|.> = \int_0^{2\pi}d\phi \int^{\pi} \sin\theta [...]$ - to have finite solutions on [-1,1] the parameter $l$ must be nonzero integer, and integer $m$ takes on $2l+1$ values between $\pm l$. These legendre polynomial are called associated legendre functions $P^m_l (x)$,

$$P^m_l (x) = (-1)^m (1-x^2)^{m/2} \frac{d^m}{dx^m} P_l(x)$$ , in rodriguez form: $$P^m_l (x) = \frac{(-1)^m}{2^l l!} (1-x^2)^{m/2} \frac{d^{l+m}}{dx^{l+m}}[(x^2-1)^l]$$

• for fixed $m$, $P^m_l$ furnishes an orthogonal set in $l$ on $[-1,1]$: $$\int^1_{-1} P^m_{l'}(x) P^m_l(x) dx = \frac{2}{2l+1} \frac{(l+m)!}{(l-m)!} \delta_{l'l}$$
• together with the radial function, we can build a set of function that are orthonormal over $l,m$, on the surface of of a sphere $$Y_{lm}(\theta,\phi) = \sqrt{\frac{2l+1}{4\pi} \frac{(l-m)!}{(l+m)!}} P^m_l (\cos \theta) e^{im\phi}$$
• $Y_{l-m}(\theta,\phi) = (-1)^m Y^_{lm}(\theta, \phi)$ $$\int^{2\pi}_0 d\phi \int^\pi_0 (\sin \theta d\phi) Y^{l'm'} Y{lm} = \delta_{l'l} \delta_{m'm}$$ $$\sum_{l=0}^\infty \sum_{m=-l}^l Y^*{lm}(\theta',\phi') Y{lm}(\theta,\phi) = \delta(\phi-\phi')\delta(\cos \theta - \cos \theta')$$
• an arbitrary function over the surface of the sphere can be expanded in spherical harmonics $$g(\theta,\phi) \sum_{l=0}^\infty \sum_{m=-l}^l A_{lm} Y_{lm}(\theta,\phi)$$ $$A_{lm} = \int d \Omega Y^*_{lm}(\theta,\phi) g(\theta,\phi)$$
• the general solution for boundary-value problem in spherical coordinates can be written in terms of spherical harmonics and powers of r in generalization of (3.3) $$\Phi(r,\theta,\phi) = \sum_{l=0}^\infty \sum_{m=-l}^{l} [A_{lm}r^l + B_{lm} r ^{-(l+1)}]Y_{lm}(\theta,\phi)$$
• for $\theta = 0$, the expansion is simple: $$P^{m\neq0}l = 0$$ $$\Rightarrow Y{lm\neq 0}(\theta = 0, \phi) = Y_{lm\neq 0}(\theta = \pi, \phi) = 0$$ $$\Rightarrow g(\theta = 0, \phi) = \sum_{l=0}^{\infty} A_{l0}\sqrt{\frac{2l+1}{4\pi}}, \quad g(\theta = \pi, \phi) = \sum_{l=0}^{\infty} (-1)^l A_{l0}\sqrt{\frac{2l+1}{4\pi}}$$

3.6 additional theorems for spherical harmonics

the addition theorem expresses a legendre polynomial of order $l$ in the angle $\gamma$ in terms of products of the spherical harmonics of the angle $\theta,\phi$ and $\theta',\phi'$ $$P_l(\cos \gamma) = \frac{4\pi}{2l+1} \sum_{m=-l}^{l} Y^*{lm}(\theta',\phi') Y{lm}(\theta,\phi)$$ - gives expansion of $\frac{1}{|\vec{r} - \vec{r}'|}$

exercise 3.extra1(examples of legendre expansion).pdf

exercise 3.1scratch.pdf

exercise 3.3scratch.pdf

exercise 5.7bscratch.pdf

polarization

When complex notation is used, keep in mind that we are only talking about the real part.

• The wave equation $$\nabla^2 \vec{E} = c^{-2} \frac{d^2}{dt^2}\vec{E}$$ gives 3 wave equations in x,y,z components of E, they all travel at speed c. Yet these 3 components are not independent, Maxwell equation dictates the wave be transverse. So let $k$ = wave vector, $e_1,e_2$ two mutually orthogonal bases in transverse plane.
• The polarization vector gives the direction of E, which is a linear combination of $e_1, e_2$. If the direction of $\vec{E}$ is constant (see caution), we say the wave is linearly polarized, otherwise it's elliptically polarized.
• CAUTION: elliptical polarization is written by Jackson as $$\vec{E} (\vec{x},t) = (E_+ e_+ + E_- e_-)e^{i \vec{k} \cdot \vec{x} - i \omega t}$$ where $e_{\pm} = \frac{1}{\sqrt{2}}(e_1 \pm ie_2)$ represent circularly polarized waves of positive and negative helicity (check: fix $\vec{x} = 0$, Re($e_+e^{-i \omega t}$) $\propto \cos[\omega t] e_1 + \sin [\omega t] e_2$). The quantity $E_+ e_+ + E_- e_-$ is a constant, complex number. It does not mean the direction of $\vec{E}(\vec{x},t)$ is constant; for the real part of $(E_+ e_+ + E_- e_-)e^{i \vec{k} \cdot \vec{x} - i \omega t}$ may change.
• In the for of the elliptical polarized wave form $$\vec{E} (\vec{x},t) = (E_+ e_+ + E_- e_-)e^{i \vec{k} \cdot \vec{x} - i \omega t}$$ if the $E_+, E_-$ have same phase then the principal axes don't get a phase, if they differ by some phase $e^{i\alpha}$, this phase difference can be canceled out by the transformation $e_+ \rightarrow e^{i\alpha/2}e_+$, $e_- \rightarrow e^{-\alpha/2}$ which corresponds to 'rotating the ellipse'. See jackson's fig 7.4.

boundary value problems

• Use boundary conditions of maxwell equations to set up, the entire wave must be considered as a whole! e.g. you can not consider a wave propagating from A to B to C by successively applying result of A to B, then B to C, see exercise 7.2.

model for dispersive media

dispersive: whenever the speed of wave depends on its frequency, the medium is dispersive

• any small displacement near equilibrium is SHO <=> any function at local min resembles a quadratic polynomial

Microcanonical ensemble: macrostates defined by (N,V,E)

• The ensemble average of any physical quantity is identical to the value one expects to obtain on measurement

• The volume $\omega = \int d^{3N}p d^{3N}q$ integrated over allowed region is the volume of allowed region in phase space, give direct measure of multiplicity of states

=> we are led to define $\omega_0$ as volume of single state in phase space => $\omega_0 \equiv O(\hbar^N)$

Transition from quantum mechanical to classical

CM partition function:

$$Q_N = \frac{1}{N! h^{3N}} \int d\omega e^{\beta H(p,q)}$$

where $d\omega = d^{3N}p d^{3N}q$

QM partition function: $$Q_N = \int_0^\infty e^{\beta E} g(E) dE$$

• An ensemble of systems contain a copy for each microstate of the system
• Each system in the ensemble satisfies all external requirements placed on the original system and in this sense is 'just as good' as the actual system
• the fundamental postulate gives equal probability to each system in the ensemble

Microcanonical Ensemble: $\Omega = \Omega(N,V,E)$ Canonical Ensemble: $\Omega = \Omega(N,V,T)$

In canonical ensemble, $\textbf{Prob}(r) = \frac{e^{-\beta E_r}}{\sum_i e^{-\beta E_i}}$. This result can be obtained in two ways: 1. system in thermal equilibrium with large heat reservoir, 2. ensemble approach, then find most probable macrostate via lagrange multipliers.

density matrix

fundamental assumption: a closed system is equally likely to be in any of the quantum states accessible to it. By accessible, we mean 'compatible' with the physical description (temperature, volume, energy, pressure, etc.). This has nothing to do with which state the system will collapse to, but the number of states compatible (degeneracy) wich the state that the system collapses to. Usually this number is large.

• an ensemble is a collection of many systems, all constructed alike. Each system in the ensemble is a replica of the actual system in one of the quantum states accesible to the system. If there are g accessible states, then there will be g systems in the ensemble, one system for each state. The concept of ensemble facilitates turning the fundamental assumption into mathematical language.

• an ensemble of systems is composed of many systems, all constructed alike, each in one of the accessible states

• the probability distribution for state has an extremely sharp peak at the thermal equilibrium value suggests "most probable one" $\cong$ "the one"

helmholtz free energy

• $$F = U - \tau \sigma$$

- helmholtz free energy at constant temperature in thermal physics = potential energy of system in classical mechanics (with only a couple of particles)

• "mode" = specific vibration angular frequency modulo addition of $2n\pi$, in other words, a specific $\omega$ constitutes a "mode"
• "black" in a frequency = all radiations in such frequency range are absorbed

planck ditribution and stefan-boltzaman law

assuming lattice vibrational excitations: $\epsilon = s\hbar \omega$. Consider only mode $\omega$.

$$Z_\omega = \sum_s \exp(-s\hbar \omega / \tau) = \frac{1}{1- e^{-\hbar \omega/ \tau}}$$

$$\left< s \right>\omega = \frac{\left< \epsilon \right>\omega}{\hbar \omega} = \sum_s (s) \exp(-s\hbar \omega / \tau) = \frac{1}{e^{\hbar \omega / \tau} - 1}$$

for bosons, particle # need not be conserved, and $\left< s \right>_\omega$ can be interpreted as the thermal equilibrium expec number of $\omega$-energy photons/phonons in the system. This formalism is completely general to all bosons

In the realm of therst, $\left< \epsilon \right>\omega$ "IS" $\epsilon\omega$. So lifting our discussion to all $\omega$:

$$U = \sum_\omega D(\omega) \epsilon_\omega$$

Now transition to phase space with a standard treatment of degeneracy gives

• the stefan-boltzmann law for spatial energy density

$$\frac{U}{V} = \frac{\pi^2}{15 \hbar^3 c^3} \tau^4$$

• the the planck radiation law for spectral density

$$u_\omega = \frac{\hbar}{\pi^2 c^3} \frac{\omega^3}{ e^{\hbar \omega / \tau} - 1}$$

Kittel claims this is where all quantum theory began

debye phonons

• the distribution function for phonon/photon is the same:

$$\left< s(\omega) \right> = \frac{1}{\exp (\hbar \omega / \tau) - 1}$$

• the form $x = A e^{i\omega t}$ for hamonic oscillators leads us to assume amplitude and frequency are independent for phonons

• N coupled oscillators have 3N modes (reference: goldstein 6.4), so phonons have 3N modes in contrary to $\infty$ for photons. A bound is thus introduced for the integration in phase space, given by the debye frequency.

• Debye phonon is characterized by $C_V \propto \tau^3$ at low temperatures.

Heat flows from higher temperature to lower temperature. Particles flow from higher chemical potential to lower chemical potential.

ex: battery with electrons

diffusive contact: can exchange heat and particles

derivation of chemical potential written in helmholtz free energy

(reminder) Helmholdz free energy: $F = U - \tau \sigma$ will be a minimum for a system in thermal equilibrium

Then use consider two systems at thermal equilibrium

$$F = F_1 + F_2 = U_1 + U_2 - \tau (\sigma_1 + \sigma_2)$$

demand conservation of particle number $N = N_1 + N_2$, $dN_1 + dN_2 = 0$, and that helmholtz free energy at minimum $dF = 0$

The infinitesmal change of $F$ is

$$dF = \left(\frac{\partial F_1}{\partial N_1}\right)\tau dN_1 + \left(\frac{\partial F_2}{\partial N_2}\right)\tau dN_2 = 0$$

when combined with $dN_1 + dN_2 = 0$, it gives

$$\left(\frac{\partial F_1}{\partial N_1}\right)\tau = \left(\frac{\partial F_2}{\partial N_2}\right)\tau$$

which facilitates the definition $\mu(\tau, V, N) = \left( \frac{\partial F}{\partial N} \right)_{\tau, V}$

This facilitates the formulation diffusive equilibrium in chemical potential in analogy with thermal equilibrium in temperature.

THE CHEMICAL POTENTIAL IS EQUIVALENT TO A TRUE POTENTIAL ENERGY!

6. Curved Manifolds

Because Schutz does such a good job with clarity, the notes here only highlights key points and eqs.

Review of tensor algebra

1. a tensor field defines a tensor at every point
2. vectors and 1-forms are linear operators on each other, producing real numbers
3. tensors are linear operators on vectors and 1-forms, producing real numbers
4. If two tensors of the same type have equal components in a given basis, they have equal components in all bases and are said to be identical (or equal, or the same). Only tensors of the same type can be equal. In particular, if a tensor’s components are all zero in one basis, they are zero in all, and the tensor is said to be zero.
5. A number of manipulations of components of tensor fields are called ‘permissible tensor operations’ because they produce components of new tensors:
   1. Multiplication by a scalar field produces components of a new tensor of the same type.
   2. Addition of components of two tensors of the same type gives components of a new tensor of the same type. (In particular, only tensors of the same type can be added.)
   3. Multiplication of components of two tensors of arbitrary type gives components of a new tensor of the sum of the types, the outer product of the two tensors.
   4. Covariant differentiation (to be discussed later) of the components of a tensor of type $\begin{pmatrix}N\M\end{pmatrix}$ gives components of a tensor of type $\begin{pmatrix}N\M+1\end{pmatrix}$ .
   5. Contraction on a pair of indices of the components of a tensor of type $\begin{pmatrix}N\M\end{pmatrix}$ produces components of a tensor of type $\begin{pmatrix}N-1\M-1\end{pmatrix}$. (Contraction is only defined between an upper and lowe rindex)
6. Ifanequationisformedusingcomponentsoftensorscombinedonlybythepermissible tensor operations, and if the equation is true in one basis, then it is true in any other. This is a very useful result. It comes from the fact that the equation (from (5) above) is simply an equality between components of two tensors of the same type, which (from (4)) is then true in any system

Lengths and Volumes

$$l = \int_{\lambda_0}^{\lambda_1}|\vec{V} \cdot \vec{V}|^{1/2} d\lambda$$

Covariant differentiation

Christoffel Symbol (5.75)

$$\Gamma^\gamma_{\beta\mu} = \frac{1}{2}g^{\alpha \gamma}(g_{\alpha\beta,\mu}+g_{\alpha\mu,\beta}-g_{\beta\mu,\alpha})$$

$$V^\alpha_{,\beta} = \frac{\partial V^\alpha}{\partial x^\beta}$$

$$V^\alpha_{;\beta} = \frac{\partial V^\alpha}{\partial x^\beta} + V^\mu \Gamma^\alpha_{\mu \beta}$$

$$P_{\alpha;\beta} = P_{\alpha,\beta} - \Gamma^\mu_{\alpha \beta} P_\mu$$

$$T^{\alpha \beta}{;\gamma} = T^{\alpha\beta}{,\gamma} + \Gamma^\alpha_{\mu \gamma}T^{\mu \beta} + \Gamma^\beta_{\mu \gamma}T^{\alpha \mu}$$

metric

$$g_{\alpha \beta;\gamma} = 0 $$

in any basis

divergence

$$V^\alpha_{;\alpha} = V^\alpha_{,\alpha} + \Gamma^\alpha_{\mu\alpha} V^\mu$$

Parallel Transport

Let vector $V$ be taken by a curve parameterized by $\lambda$, parallel transport means

$$\frac{dV^\alpha}{d\lambda} = 0 $$

geodesic

a geodesic is parallel transport of parallel line, generalization of the idea of "staight line".

In curved space, parallel lines when extended do not remain parallel.

curvature tensor

riemann curvature tensor: quantifying the amount of change of parallel transport of a vector around a infinitesimally small rectangular loop.

$$R_{\alpha \beta \mu \nu} = -R_{\beta \alpha \mu \nu} = -R_{\alpha \beta \nu \mu} = R_{\mu \nu \alpha \beta}$$ $$R_{\alpha \beta \mu \nu} + R_{\alpha \nu \beta \mu} + R_{\alpha \mu \nu \beta} = 0$$

it has a relationship to commutator:

$$[\nabla_{\alpha}, \nabla_{\beta}] V^{\nu} = R^{\mu}_{\nu \alpha\beta} V^{\nu} $$

bianchi identities

$$R_{\alpha\beta\mu\nu;\lambda} + R_{\alpha\beta\lambda\mu;\nu} + R_{\alpha\beta\nu\lambda;\mu} = 0 $$

ricci tensor

$$R_{\alpha \beta} \equiv R^\mu_{\sigma \mu \beta} = R_{\beta\alpha}$$

einstein tensor

$$G^\alpha \beta \equiv R^{\alpha \beta} - \frac{1}{2}g^{\alpha \beta} R = G^{\beta\alpha}$$

$$G^{\alpha \beta}_{;\beta} = 0$$

quantum mechanics

• physical states are represented by rays (normalized vectors that differ only by a phase) in HIlbert space, a hilbert space is an infinite dimensional complex vector space with inner/scalar product defined
• observables are represented by hermitian operators.
• Important terms of Hilbert space operators/vectors include unitary, linearity, antiunitary, antilinearity, and adjoint
• For all Hilbert space vectors:

$$(\phi,\psi) = (\psi,\phi)^, \quad (\phi,a\psi_1+b\psi_2) = a(\phi,\psi_1) + b(\phi,\psi_2)$$ $$(a\phi_1+b\phi_2,\psi) = a^(\phi_1,\psi) + b^(\phi_2,\psi)$$ - Wigner proved that for any symmetry transformation, there is a corresponding operator $U$ in Hilbert space that is either unitary and linear or antiunitary and antilinear* - differential probability $\Leftrightarrow$ amplitude squared

symmetries

• a symmetry transformation is a change in POV that does not change the result of possible experiments
• wigner proved that symmetry transformations must be reflected in hilbert space by unitary and linear, or antiunitary and antilinear operators

---

unitary, antiunitary, linear, antilinear

Unitary and Linear means

$$(U\psi,U\phi) = (\psi,\phi) \quad U(a\psi_1 + b\psi_2) = aU\psi_1 + bU\psi_2$$

antiunitary and antilinear means

$$\braket{U\psi}{U\phi} = \braket{\psi}{\phi}^ \quad U(a\psi_1 + b\psi_2) = a^U\psi_1 + b^*U\psi_2$$

the adjoint for unitary is defined as

$$(\phi,A\psi) = (A^\dagger \phi,\psi)$$

while the adjoint for antiunitary operator is defined as

$$(\phi,A\psi) = (A^\dagger \phi,\psi)^*$$

• symmetry transformations that can be made continuously into the identity must be unitary and linear, because the identity is unitary and linear.
• observables and infinitesmal transformations: the generators of infinitesmal symmetry transformations must be hermitian (adjoint equals itself) and linear for such transformation to be unitary, which makes them candidates for observables. Indeed, most perhaps all of the physical observables, such as angular momentum, momentum, energy arise from symmetry transformation.
• the transformations in hilbert space that corresponds to a given symmetry transformation is said to furnish a representation of the symmetry transformation group

$$\textbf{Rep}(\Lambda_1)\textbf{Rep}(\Lambda_2)= N(\Lambda_1, \Lambda_2)\textbf{Rep}(\Lambda_1 \Lambda_2) \simeq \textbf{Rep}(\Lambda_1 \Lambda_2)$$

in other words, representations of symmetry transformations tell us how the states transform in response to this symmetry transformation. - [❤️] we are looking for Poincare-invariant theories thus we are interested in unitary representations of the Poincare group.

quantum lorentz trans

• the Poincare group is also known as the inhomogeneous lorentz group, while the group of boosts and rotations are known as the homogeneous poincare group, or the lorentz group.
• the subgroup of lorentz transformations with determinant 1 and $\Lambda^0_0\ge 1$ is known as the proper orthochronous lorentz group

  The proper orthochronous Lorentz group SO(1,3) consists of all Lorentz transformations that preserve the orientation and direction of time and are connected to the identity transformation.

• any lorentz transformation can be constructed with the proper orthocho. lorentz group together with a discrete transformation.

  any lorentz transformation is either proper and orthochronous, or may be written as the product of an element of the proper orthochronous lorentz group with a discrete transformation parity or time-reversal

• [❤️] the study of the whole lorentz group reduces to the study of its proper orthochronous subgroup $+$ parity and time-reversal

the poincare algebra

• much of the information about any Lie symmetry group is contained in properties near the identity, ie infinitesmal transformations
• the condition that $\Lambda g \Lambda = g$ requires that the infinitesmal transformation $\omega_{\mu \nu}$ obeys $$\omega_{\mu \nu} = -\omega_{\nu \mu}$$
• [❤️] we find the Lie algebra for the Poincare grp $$ i[J^{\mu\nu}, J^{\rho \sigma}] = \eta^{\nu \rho} J^{\mu \sigma} - \eta^{\mu \rho}J^{\nu \sigma} - \eta^{\sigma \mu} J^{\rho \nu} + \eta^{\sigma \nu} J ^{\rho \mu} \ i[P^\mu, J^{\rho \sigma}] = \eta^{\mu \rho} P^{\sigma} - \eta^{\mu \sigma} P^{\rho}\ [P^\mu, P^\rho] = 0 $$

or equivalently with $P={P^1,P^2,P^3}$, $J={J^{23},J^{31},J^{12}}$, $K={J^{01},J^{02},J^{03}}$, we have

$$ [J_i, J_j] = i \epsilon_{ijk} J_k, \quad [J_i, K_j] = i \epsilon_{ijk} K_k, \quad [K_i, K_j] = -i \epsilon_{ijk} J_k, \quad [J_i, P_j] = i \epsilon_{ijk} P_k$$

$$ [K_i, P_j] = - i H \delta_{ij}, \quad [K_i, H] = -i P_i, \quad [J_i, H] = [P_i, H] = 0 $$

1particle states

In this section, we consider the representation of the lorentz group, and classify the states by their little group, and we find that such classification corresponds to that of massive and massless particles. - We begin with a general statement of how representation of a lorentz transformation should act on a state (2.5.3) - no matter how we lorentz transform, we can never transform a time-like particle into a massive particle, so we can classify the particle by their momentum into 'cliques', where momenta from two different cliques are never related by a lorentz transformation. - we pick a standard momentum in each 'clique', we define a standard transformation that takes the standard momentum to any other momentum within the clique - The standard momentum and standard transformation above allows us to factor a general operator induced by a lorentz transformation into an operator induced by the standard transformation and some other operator reduced by a transformation that fixes the momentum, we call this 'other transformation' the 'little group' - On Lil grp: $p^\mu = L^{\mu}{\nu} p^\nu(p) k^{\mu}$ with $k^\nu$ being the standard momentum, along with $\psi{p,\sigma} \equiv N(p) U(L(p))\psi_{k \sigma}$ gives $U(\Lambda) \psi_{p,\sigma} = N(p) U(L^{-1}\Lambda L(p)) \psi_{k\sigma}$ Observe that $L^{-1}\Lambda L(p)$ belongs to the subgrp of the homogeneous lorentz group that fixes $k$, this is a more rigorous definition of the Wigner's little group. - Some simple further derivation leads quickly to (2.5.11), which turns the problem of finding the representation of lorentz group into finding representation of the little group. At a more fundamental level, what (2.5.11) says is that the representation of a general $\Lambda$ can be written as scaling by some number $\frac{N(p)}{N(\Lambda p)}$ together with a representation of the little group $D(W, p)$. - we fix the normalization factor in (2.5.11), shown in (2.5.18) - [❤️] the little group is SO(3) and ISO(2) for massive and massless particles}

parity and time reversal

• Parity and time-reversal are only conserved without the context of weak-nuclear force, the inclusion of which puts them at approximately conserved

Properties of P and T on the Poincare generators

$$\textbf{P} J \textbf{P}^{-1} = J \quad \textbf{P} K \textbf{P}^{-1} = -K \quad \textbf{P} P \textbf{P}^{-1} = -P$$

$$\textbf{T} J \textbf{T}^{-1} = -J \quad \textbf{T} K \textbf{T}^{-1} = K \quad \textbf{T} P \textbf{T}^{-1} = -P$$

massive 1particle states

• $P \Psi_{\vec{p}, \sigma} = \eta \Psi_{P\vec{p}, \sigma}$
• $T \Psi_{\vec{p}, \sigma} = \xi(-)^{j-\sigma} \Psi_{P\vec{p}, -\sigma}$

massless 1particle states

• $P \Psi_{\vec{p}, \sigma} = \eta_{\sigma} \exp{\mp i \pi \sigma} \Psi_{P\vec{p}, -\sigma}$ this happens because parity inverts momentum but not the direction of angular momentum, thus the helicity is inverted, the state also gains a phase factor dependent on the helicity. the sign in $\mp$ is dependent on whether the two-component of $\vec{p}$ is positive or negative.
• this also tells us that the existence of a symmetry under parity requires that any species of massless particle with non-zero helicity be accompanied with another of opposite helicity.
• We know that $T$ changes both the direction of angular momentum and momentum, so its logical that it doesnt change helicity, indeed it doesnt.

$$T \Psi_{\vec{p}, \sigma} = \xi_{\sigma} \exp{\pm i \pi \sigma} \Psi_{P\vec{p}, \sigma}$$

again the $\pm$ depends on whether the two component of $\vec{p}$ is positive or negative

exercise 2.4scratch2.pdf

exercise 2.4scratch.pdf

Scattering theory

IN and OUT states

formalism on actual physical states

• states that are energy eigenstates can not be localized in time, we must consider wave packets, superpositions of states.
• we assume that the state evolved like a free one long before the interaction took place
• we use $\Psi$ tp denote the actual physical state with interaction
• by convention, we let $\alpha$ denote momentum, spin and such, then we convene $$\braket{\Psi_\alpha}{\Psi_{\alpha'}}=\delta(\alpha,\alpha')$$ since $\alpha$ can contain discrete labels, we define $$\int d \alpha = \sum_{\text{discrete labels}} \int d \text{(continuous labels)}$$ the completeness relation reads $$\Psi = \int d\alpha \braket{\Psi}{\Psi_\alpha}\Psi_\alpha$$ the eigenvalue when acted on by the full hamiltonian is $$H \Psi_\alpha = E_\alpha \Psi_\alpha$$
• Weinberg uses $\Psi^+$ for in states and $\Psi^-$ for out states, they describe the states at time $t=-\infty$ and $t=\infty$ respectively
• energy eigenstates can not be localized in time, because the time evolution operator would yield an inconsequential phase factor
• we need to consider wave packet states, superpositions of states that are given by some amplitude $$\Psi p \int d\alpha g(\alpha) \Psi_\alpha$$

relation to free states to Lipp-Schwinger

• We decompose $H = H_0 + V$, and let $\Phi$ be the solution to $H_0$
• we convene $$H_0 \Phi_\alpha = E_\alpha \Phi_\alpha$$ $$\braket{\Phi_\alpha}{\Phi_{\alpha'}}=\delta(\alpha,\alpha')$$
• notice that we convened the same spectrum of $H$ on $\Psi$ as $H_0$ on $\Phi$, this requires that the masses appearing in $H_0$ be the physical masses that are actually measured, which are not necessarily the bare mass in $H$, if there is any difference it must be accounted for in $V$
• we then find $$\Psi_\alpha^{\pm} = \Omega(\mp \infty) \Phi_\alpha,$$ where $$\Omega(\tau) = \exp{iH\tau} \exp{-iH_0 \tau}$$ and additionally $$\braket{\Psi_\alpha^\pm}{\Psi_{\beta}^\pm} = \delta(\alpha - \beta)$$

• further derivations leads to the Lippmann-Schwinger formula $$\Psi_\alpha^{\pm} = \Phi_\alpha + \int d\beta \frac{T_{\beta \alpha}^{\pm} \Phi_\beta}{E_\alpha - E_\beta \pm i\epsilon}, \text{\quad \quad} T_{\beta \alpha}^{\pm} \equiv \braket{\Phi_\beta}{V \Psi_\alpha^{\pm}}$$

• as a convenient representation for the factor of $\frac{1}{E_\alpha - E_\beta \pm i \epsilon}$, we have $$\frac{1}{E \pm i \epsilon} = \frac{\mathscr{P}}{E} \mp i \pi \delta(E)$$ where $\mathscr{P} = \frac{E}{E^2 + \epsilon^2}$

S-matrix

• we define the S matrix by $$S_{\beta \alpha} = \braket{\Psi_\beta^-}{\Phi_\alpha^+}$$

• If there were no interaction then the in and out states would be the same, meaning $$S_{\beta \alpha} = \delta(\alpha-\beta),$$ thus the rate of reaction $\Psi_\alpha \rightarrow \Psi_\beta$ is $|S_{\beta \alpha} - \delta(\alpha - \beta)|^2$ \footnote{the in and out states are not differentiated, they are only labeled differently here, and they only differ in how they are created or their asymptotic time evolution, in other words $$\braket{\Psi_\beta^-}{\Phi_\alpha^+} = \braket{\Psi_\beta}{\Phi_\alpha}$$

• S matrix connects two complete sets of orthonormal states, it must be unitary $$\int d\beta S_{\beta \gamma}^* S_{\beta \alpha} = \braket{\Psi_\gamma}{\Psi_\alpha} = \delta(\gamma - \alpha), \quad S^\dag S = 1$$

• If we define the operator analog of the S matrix as $$\braket{\Phi_\beta}{S \Phi_\alpha} = S_{\beta \alpha}$$ we will find $$S = \Omega(\infty)^\dag \Omega(-\infty) = U(\infty, -\infty)$$ $$U(\tau, \tau_0) = \exp{iH_0 \tau }\exp{-iH(\tau - \tau_0)}\exp{-iH_0 \tau }$$

Lipp-Schwin analog of S

• We now try to derive a similar formula for $S^\alpha_\beta$ using the Lippmann-Schwinger formula
• the method is to instead of taking the $t \rightarrow -\infty$ limit for $\Psi^+$, we take the $+\infty$ limit, this means we have to close the contour of integration for $E_\alpha$ in the lower half plane now, and now we do pick up a contribution from the singular factor $(E_\alpha - E_\beta +i\epsilon)$, by the method of the residues and taking $t\rightarrow \infty$, we obtain $$\mathscr{I}\beta^+ \equiv \int d\alpha \frac{e^{-i E\alpha t} g(\alpha) T_{\beta \alpha}^{\pm}}{E_\alpha - E_\beta \pm i\epsilon} \quad \rightarrow -2i\pi e^{-iE_\beta t} \int d\alpha \delta(E_\alpha - E_\beta)g(\alpha) T_{\beta \alpha}^+$$
• we then find $$S_{\beta \alpha} \propto \delta(\beta - \alpha) - 2i\pi \delta(E_\alpha - E_\beta) \braket{\Phi_\beta}{V \Phi_\alpha}$$ which is known as the Born approximation, an approximation for the S-matrix at weak interactions.

Symmetries of the S-matrix

Lorentz-invariance

• We state the Lorentz-invariance condition as $$S_{\alpha \beta} = \braket{\Psi_\alpha}{\Psi_\beta}= \braket{U(\Lambda, a)\Psi_\alpha}{ U(\Lambda, a) \Psi_\beta}$$
• We do not get Lorentz-invariance of the $S$-matrix for free, that is, it is only for certain special choices of interaction Hamiltonian that will give the theory's S-matrix lorentz invariance, as defined above

• As another caveat, how do we know $U(\Lambda)$ acts the same on the in and out state? We dont, but we can apply the transformation rules in Hilbert space and we will find that the inner product will be invariant if the theory satisfies

  $$ [S, U(\Lambda)_0] = 0$$ , which when expressed in terms of the infinitesmal lorentz transformation, is

  $$[H_0,S]=[P,S]=[J_0, S]=[K_0,S]=0$$

• Weinberg also goes to show that an alternative formulation of the condition also makes Smat lorentz invariant $$[V,P_0] = [V, J_0]=0$$

Internal Symmetries

• We apply the same procedure as above but to internal symmetries, we find that the condition now becomes $$[U_0(T), H_0] = [U_0(T), V] = 0$$
• a classic example of conservation law from internal symmetry is electric charge
• More examples include baryon #(protons, neutrons, hyperons), lepton #(electrons, muons, $\tau$ particles, neutrinos), but these examples are believed to be only very good approximations.
• other examples of this type that are definitely approximation: conservation of strangeness
• isotopic spin invariance is only believed to be a good approximation, because such invariance is not respected by interactions such as EM\footnote{different members of the isospin multiplet have different electric charges and slightly different masses}

parity

• parity $P$ is not conserved in the following sense: $$ \begin{cases} [P, H_0] \approx 0 & \text{not conserved in weak interaction between left and right handed ferions} \ [P, V] \approx 0 & \text{conserved in EM but not weak interaction} \end{cases}$$the weak interaction is responsible for $\beta$-decay

• The naive definition of the parity operator is $$U(P) \Psi_{p, \sigma} = \eta \Psi_{P p, \sigma} $$where $\eta$ is the intrinsic parity (a phase factor that arises as an eigenvalue of the parity operation) of the state. In a subtle way, this gives us the freedom to redefine $P$ if $P$ were to be conservedm

  • Parity being conserved means $$[P, H_0] = [P, V] = 0$$ If we have a conserved internal symmetry $T$ means

    $$[U_0(T), H_0] = [U_0(T), V] = 0$$

    thus

    $$[P U_0(T), H_0] = [P U_0(T), V] = 0$$ $$P' = PU_0(T) \text{ for internal symmetry } T$$ - Must the intrinsic parity always take on $\pm 1$, ignoring normalization? - Yes, Suppose it doesn't, then we have $$P^2 \Psi = e^{i\theta} \Psi $$ if the theory we are considering has a continuous internal symmetry, then we can redefine $\textbf{P} \rightarrow \textbf{P} I$ for some internal symmetry $I$ that cancels the $e^{i\theta}$ exactly, and this new $\textbf{P}'$ will give intrinsic parity of $\pm 1$, we then redefine it to be the parity operator $\Box$ - No, if we don't have continuous internal symmetry. - $P^2 \Psi = \eta \Psi$ means $P^2$ acts like an internal symmetry

    time-reversal

    • we expect that time-reversal swaps the in and out states
    • As usual, applying the transformation rules of symmetry transformation to states in hilbert space we find the requirement for a thoery to be time-reversal invariant $$[T_0, H_0] = [T_0,V] = 0 $$ we will then find $$T \Omega(-\infty) \Psi_{\alpha} = \Omega(\infty) \Psi_{T \alpha } $$ which makes sens
    • time reverasl symmetry is much more difficult to experimentally verify than parity, because experimentally, it is extremely difficult to set up $\text{Ni}^{60} + e^- + \bar{nu} \rightarrow \text{Co}^{60}$

    PT

    • PT would have to be anti-unitary, because P is unitary and T is anti-unitary

  $$S_{\beta \alpha} \rightarrow -e^{2i(\delta_\alpha + \delta_\beta)} S^*_{\textbf{P} \textbf{T} \beta \textbf{P} \textbf{T} \alpha}$$

  $\textbf{P} \textbf{T}$ would preserve the momenta while swapping spin, so invariance of S-matrix under $\textbf{P} \textbf{T}$ implies there wouldnt be any preference for the electron in the decay $\text{Co}^{60} \rightarrow \text{Ni}^{60} + e^- + \bar{nu}$ to be emitted in the same or opposite direction to the $\text{Co}^{60}$ spin - the 1957 experiment did not rule out time-reversal symmetry immediately, but ruled out PT by demonstrating the above statement is false

  C

  It is understood today that C is not conserved in the weak interaction, just like how P is not conserved

CP

It is thought to be true that CP is ''more conserver" than C and P in weak force, but nevertheless CP is still not a conserved quantity

CPT

There is good reason to believe that CPT is conserved exactly, which would be really nice as it 1. gives a good interpretation of antiparticles 2. the fact that CPT commutes with the Hamiltonian tells us that the particle and antiparticle have the same mass

Rates and Cross-section

Weinberg: this section is more like a mnemonic, because it seems like no interesting open problems in physics hinge on getting the fine points right regarding these matters

Implications of Unitarity

Optical theorem

• By the requirement that the S-matrix being unitary, we can actually find that the imaginary part of $M_{\alpha \alpha}$ (Recall that the $M$ matrix is part of the S-matrix after we removed the part where nothing happens) is actually somewhat interesting. $$\Im M_{\alpha \alpha } = - \pi \int d\beta \delta^{4}(p_\beta - p_\alpha ) |M_{\beta \alpha }|^2 $$
• This is the optical theorem
• If we take the optical theorem to the high energy limit, we will find that unstable particles and their corresponding antiparticles have precisely the same lifetimes.

Boltzmannn H-theorem

• The optical theorem is obtained using $S^\dagger S =1$, using the other condition of $SS^\dagger = 1$, we find

$$\Im M_{\alpha \alpha } = - \pi \int d\beta \delta^{4}(p_\beta - p_\alpha ) |M_{\alpha \beta }|^2$$

which implies

$$\int d\beta \delta^{4}(p_\beta - p_\alpha ) |M_{\alpha \beta }|^2 = \int d\beta \delta^{4}(p_\beta - p_\alpha ) |M_{\beta \alpha }|^2$$

• further derivation shows $$-\frac{d}{dt} \int d\alpha P_\alpha \ln{P_\alpha/c_\alpha} \ge 0 $$which called the Boltzmann H-theorem, equivalent to entropy never decreases.

Cluster Decomposition Principle - We can express the Hamiltonian by giving its matrix elements between states with arbitrary number of particles. We will do so via a bunch of creation and annihilation operator, the benefit of this approach is that if a certain condition on the $\ap \ap^\dag $s in the Hamiltonian are satisfied, the S-matrix will satisfy the cluster decomposition principle, which will imply locality is obeyed.

local: events separated by spacelike vectors can not influence each other

• It can be argued that the cluster-decomposition principle plays a role in field theory inevitable, because past attempts to formulate a relativistic quantum theory without using field theory have either failed lorentz invariance or the CDP.

Structure of this chapter:

• discuss the basis of states containing arbitrary # of bosons and fermions
• define the creation and annihilation operators
• show how to construct the Hamiltonian that yields the S-matrix that satisfies the CDP

Bosons and Fermions

• the bose and fermi statics state that bosons are symmetric under exchange while fermions are antisymmetric $$\Phi_{p_1, p_2} = \pm \Phi_{p_2, p_1}$$
• we will define the bosons to be the ones that take the plus sign, and fermion as the ones that take the minus sign
• claim: all particles are either bosons or fermions
  • Suppose there exists another particle that is not fermion or boson, meaning its eigenvalue under exchange is $\alpha \neq \pm 1$, then consider an arbitrary state under two operations of exchange, find that the original state gets scale by a nontrivial value $\alpha^2$ $\Box$
  • the above argument depends on two operations of exchange get us back to the initial state, but this doesn't have to be the case, the state can take a tour that doesnt end back where it began. This would be a problem in two-dimensional space, but not in 3,4, and will come in section 9.7. This would be a problem if the $\alpha$ depends on more things than the species of the particles, and is assumed away in the cluster decomposition principle
• We now want to mathematically define a bunch of states that contain bosons and fermions, and is symmetric under exchagne of boson-boson, boson-fermions, but asymmetric under exchange of fermion-fermion. Such is accomplished with delta functions on all permutations, shown on page 172

Creation and Annihilation Operators

• The creation and annihilation operators in Weinberg specifically operate on the states that obey the exchange statistics of bosons and fermions defined by Weinberg
• The annihilation operators of both bosons and fermions annihilate the vacuum
• These operators obey the following commutation/anti-commutation relation: $$a(q')a^\dag(q) \mp a^\dag(q)a(q') = \delta(q' - q)$$ where the anticommutator is for fermion-fermion, commutator for everything else
  • Claim: Any operator maybe expressed as a sum of products of creation and annihilation operators
    • Begin with $\braket{\Psi_0}{\mathscr{O}\Psi_0} = C_00$, the rest follows pretty intuitively as on page 175
• the lorentz transformation rules for the creation operators are on page 177
• the transformation rules for the creation operators under the discrete transformations are beneath

Cluster Decomposition and Connected Amplitudes

• The cluster decomposition principle simply states that experiments that are sufficiently separated in space have unrelated results
• Consider a process where $A = {a_1, a_2, a_n} \rightarrow B = {b_1,b_2, ..., b_m}$, the S-matrix element would be $S_{AB}$ -Think about all the possible processes that could have led to this happening, we can have $a_1 \rightarrow b_1$, the rest doing the corresponding, or $a_1 \rightarrow b_2$, the rest doing the corresponding, or $a_1, a_2 \rightarrow b_1, b_2$, and the rest doing the corresponding...
  • without a doubt, there is a way to represent such multistate transition with combinatorics, and it will involve using $(-1)^?$ terms because some of the terms in the combinatoric monster will involve fermion-fermion exchange.
  • Indeed, we find that $S_{\beta \alpha}$ corresponds to a sum over all different ways of partitioning the particles in $A$ to clusters that goes to clusters in $B$
• the cluster decomposition principle is a statement that if $\alpha_1 \rightarrow \beta_1 $, $\alpha_2 \rightarrow \beta_2 $, ... $\alpha_n \rightarrow \beta_n $ are studied in $n$ labs that are very far from each other, the amplitude $S_{\beta \beta}$ factorizes as it would for Prob[AB]=Prob[A]Prob[B], in other words, we can treat the processes as independent events
• the recursively defined connected parts of the S-matrix gives us a way to rewrite $S_{\beta \alpha}$, thus another way to state the cluster decomposition principle, the recursiveness comes from the fact that the connected 1-particle component to define the connected 2-particle component, ...
• This gives a restatement of the cluster decomposition principle, into the statement that $S_{\beta \alpha}^C$ vanishes if any particles in the state $\beta$ or $\alpha$ are far from any others

Structure of the interaction

• How do we formulate a Hamiltonian that respects the Cluster decomposition principle? The answer is if the Hamiltonian satisfies (4.4.1), (4.4.2).
• "the trusting reader may prefer to skip the rest of this chapter, and move on to consider the implications in chapter 5", I trust you! So I ll skip!

Quantum Fields and Antiparticles

In this chapter, quantum fields will be introduced, during its construction, as a result of the union between special relaticity and quantum mechanics, we will encounter

• the connection between spin and statistics
• the existence of antiparticles
• relationships beteween particles and antiparticles, such as CPT

Free fields

• We would like to construct a Hamiltonian out of creation and annihilation operators, from (4.2.12) we know that under Lorentz transformations each of those operators is multiplied by a matrix that depends on the momentum of the operator, and the whole thing may not look as nice, because the Hamiltonian has different operators at different momentum the coefficients would be fked up, we would like to solve this issue.
• the solution is to construct the Hamiltonian out of creation and annihilation fields, $\Psi^-, \Psi^+$, respectively, (5.1.4, 5.1.5)
• the nice thing about this is under lorentz transformation, each field is multiplied by a position-independent matrix, furthermore, we find that these $D$-matrices furnish a representation of the homogeneous lorentz group.

  See (5.1.6, 5.1.7, the $D_{\bar{l} l}$) on the RHS is the matrix that weinberg is referring to

• (4.2.12) gives us the transformation rules of the $\text{a}{\text{p}}^\dag$, we take its adjoint and find the transformation rules of both $\text{a}{\text{p}}, \text{a}{\text{p}}^\dag$. Then putting the transformation rules of $\text{a}{\text{p}}, \text{a}{\text{p}}^\dag$ with that of $\Psi^\pm$, we find the transformation rules of coefficients $u,v$ of $\text{a}{\text{p}}, \text{a}_{\text{p}}^\dag$ in $\Psi^\pm$ under general Lorentz transformation (5.1.13, 5.1.14), these are the fundamental requirements that will allow us to calculate the $u,v$

• [❤️] We then take the rule above, and plug in pure translations, boosts, and rotations, and find

• translation: we get a general rule for arbitrary homogeneous lorentz transformation $\Lambda$ (5.1.19, 5.1.20)
• boosts: we get a relation between $u,v$ at $0$ momentum and some nonzero momentum (5.1.21, 5.1.22)
• rotations: we get a relation between $u,v$ of different spin $\sigma, \bar{\sigma}$ (5.1.23-5.1.26), notice that the summation on the LHS of these equations is over $\bar{\sigma}$, and that on the right is over $l$, which is an index in the coefficients $u,v$.

Causality

• We check the causality condition of the general free field constructed with coefficients in front of a continuous spectrum of creation and annihilation operators.
• The tator \footnote{tator means commutator/anticommutator}
  $[\psi^+(x),\psi^-(y)]$ in (5.1.30) does not vanish in general. This means we can not have fields that are created purely with creation operators, or fields that are purely annihilation operators. So the natural thing is to try linear combinations of the creation and annihilation fields: $$\kappa \psi^+ + \lambda \psi^-$$ We will see later that WLOG $\kappa = \lambda$ so we are only free to choose the scale of the field.
• we see that in order for this theory to conserve quantum numbers like electric charge, there must be a doubling of particle species carrying non-zero values of such quantum numbers: if a particular componenet of the annihilation field destrooys a particle of species $n$, then the same component of the creation field must create particles of species $\bar{n}$, this is the reason for antiparticles.

Conserved Quantum Number

• Lets say each particle carries some quantum number, and there is some sort of conservation law for that quantum number. Then creation of a particle would add such number and annihilation takes away such number.
• Formulating this in terms of mathematics: Suppose the Hermitian operator $Q$ corresponds to the observable of quantum number $q$ that the particle carries. Then we necessarily have $$[Q,\text{a}{\text{p}}] = Q\text{a}{\text{p}} - \text{a}{\text{p}} Q = -q \text{a}{\text{p}}$$ $$[Q, \text{a}{\text{p}}^\dag] = Q\text{a}{\text{p}}^\dag - \text{a}{\text{p}}^\dag Q = q \text{a}{\text{p}}^\dag$$ \footnote{because $\text{a}{\text{p}} Q$ first makes a measurement, then takes the particle away, and $Q \text{a}{\text{p}}$ makes the measurement after the particle is taken away, so the former measurement seems to observe an additional $q$. Similar idea for the latter tator relation.}
• Recall the form of our Hamiltonian $\mathscr{H}$ in terms of the creation and annihilation fields (5.1.9)

It would be fair to say $\mathscr{H}$ is a polynomial in $\psi^+$, $\psi^-$

$$\mathscr{H} = \sum_{NM} \sum_{l'1 ... l'_N} \sum{l_1 ... l_M} g_{l'1...l'_N,l_1...l_M} \psi^-{l'1}...\psi^-{l'N} \psi^+{l_1}...\psi^+_{l_M}$$

Weinberg claims that in order for the Hamiltonian to commute with $Q$, it is necessary that the Hamiltonian be formed out of fields that have simple commutation relations with $Q$, meaning $[Q, \psi_l] = -q_l \psi_l$\footnote{I don't understand why this HAS to be the case right now, I see why it would be nice, but I ll take his words. - But this is clearly not the case for $\psi = \text{a}{\text{p}} + \text{a}{\text{p}}^\dag$, unless $Q=0$. \footnote{for $[Q,\psi] = [Q, \text{a}{\text{p}} + \text{a}{\text{p}}^\dag] = -q(\text{a}_{\text{p}}- ap^\dag)$.} - A possible solution would be to introduce another particle that carries the opposite charge. This is the idea behind \textbf{antiparticles}.

causal: fields commute at spacelike separations.

Causal Scalar Fields

what 'scalar' means

• scalar representation means the field only takes on a scalar value at each point in spacetime, for that value to be Lorentz invariant it must be a Lorentz scalar.
• [👉] Recall the definition of the annihilation field in terms of its operator and coefficient

$$\Psi^+{l}(x) = \sum{\sigma n} \int d^3 u_{l} (x;\textbf{p}, \sigma, n) a(\textbf{p}, \sigma, n) $$

having scalar field means we can remove this $l$ index, again, keep in mind that the field only takes on a scalar value at each point in spacetime! - [👉] the field being just a scalar means the matrix $D_{\bar{l} l}(\Lambda)$ that we were talking about is simply a *scalar: $D(\Lambda)$***! - [👉] In his book weinberg considers the simplest representation where $D(\Lambda)=1$

spin 0 is necessary

\begin{itemize} - Recall that the $D^{(j)}$ representation of the rotation group is dim $2j + 1$, so for this representation to be a scalar representation we necessarily have spin = 0. A scalar field necessarily describes particle with 0 spin. - [👉] we can drop the $\bar{\sigma},\sigma$ labels because they only take on the value of 0, assuming we are working with one species of particles, we can drop the $n$ as well, so we write $$u = u(p), v = v(p)$$ - it is conventional to normalize the annihilation and creation fields so that $u(0) = v(0) = \sqrt{\frac{1}{2m}}$, so we have $$u(\textbf{p}) = \sqrt{\frac{1}{2p^0}}, \quad v(\textbf{p}) = \sqrt{\frac{1}{2p^0}}$$ \end{itemize}

implications of causality

• That's for the invariance under lorentz transformation, for causality, we need the fields to commute at spacelike separations. We find that while $[\phi^\pm , \phi^\pm]{\mp} =0$, $[\phi^\pm , \phi^\mp]{\mp}$ is not necessarily 0 for spacelike separations.
• To address this problem, we find that (5.2.6-5.2.9) the constants for both fields $\kappa, \lambda$ must be equal in magnitude, and we must be considering the boson, in other words, the commutator instead of anticommutator. This solution works by exploiting the evenness of $\Delta_+$.
• we can redefine the relative phase of the creation and annihilation operators so that $\kappa$ and $\lambda$ are equal exactly (5.2.10)

scalar fields need anti-particles for conserved charge

• The issue here is that $[Q, \phi^+] = -q\phi^+$, $[Q, \phi^{+\dag}] = q\phi^+$, we will have $$[Q, \phi^+ + \phi^{+\dag}] = q (\phi^{+\dag} - \phi^+)$$ this is not a commutation relation that allows for charge conservation.
• Indeed, if the particles that are destroyed and created by $\phi$ carry some conserved quantum number like electric charge, then $\mathscr{H}$ will conserve the quantum number if and only if each term in $\mathscr{H}$ contains equal number of $\text{a}{\text{p}}$ and $\text{a}{\text{p}}^\dag$, but this would be impossible if $\mathscr{H}$ is a polynomial in $\text{a}{\text{p}} + \text{a}{\text{p}}^\dag$.
• [❤️] In order for the scalar field to have conserved charge, we must imagine there is another field of the particle with opposite charge， let these two fields be $\phi^+, \phi^{+c}$, then we form the field by

$$ \phi = \phi^+ + \phi^{+c\dag}$$ $$ = \int \frac{d^3 p}{(2\pi)^{3/2} (2 p^0)^{1/2}} \left[ a_{\textbf{p}}e^{ip\cdot x} + a_{\textbf{p}}^{c\dag}e^{-ip \cdot x}\right] $$

MISC

• The commutator of the complex scalar field with its adjoint is $$[\phi(x), \phi^\dag(y)] = \Delta(x-y) = \Delta_+(x-y) - \Delta_+(y-x)$$
• Parity: The intrinsic parity of a state containing a spinless particle and its antiparticle is even.
• Charge Conju: The intrinsic charge conjugation parity of a state consisting of a spinless particle and antiparticle is even. If the particle is its own antiparticle, the charge conjugation must be real, $\pm 1$
• Merlin: even, real if its own antiparticle.

Causal Vector Fields

We first follow the same procedure as we did for the scalar field (5.3.1-5.3.5), we encounter a difference when it comes to spin of the particle: vector field can have nontrivial spin (5.3.6, 5.3.7). Thus we need to examine spin by looking at the rotation transforamtion properties of the vector field, by looking at the rotation generators $\mathscr{J}^\mu_\nu$

Rotation Generator

\begin{itemize} - The rotation generators are $$\left(\begin{array}{rrr} 0 & 0 & 0 \ 0 & 0 & -i \ 0 & i & 0 \end{array}\right), \left(\begin{array}{rrr} 0 & 0 & i \ 0 & 0 & 0 \ -i & 0 & 0 \end{array}\right), \left(\begin{array}{rrr} 0 & -i & 0 \ i & 0 & 0 \ 0 & 0 & 0 \end{array}\right)$$ squaring them, and adding them up yields $2\delta^i_j$. - We know that in the four vector representation of the rotation group, the time component is zero, pluggint this into (5.3.6, 5.3.7) we obtain (5.3.12, 5.3.14). Plugging in our our last result of $2\delta$ into (5.3.6, 5.3.7), we find (5.3.13, 5.3.15). \footnote{Recall that $(J^{(s)})^2 = s(s+1)$, so it may seem like for the $2\delta$ to be true, we must have $s =1$, but that would be the case if the rotation generators are the entire story of this representation. Which isnt the case because there is also another component of time. If the spatial components is nonzero and nontrivial, then we necessarily have $s=1$. However, there is another solution of setting the spatial components trivially to be 0, which would allow us to set $s=0$. } - Possibility 1: at $\textbf{p}=0$, only $u^0, v^0$ are nonzero and $s$ (or $j$) equals 0. - Possibility 2: at $\textbf{p} =0$, only $u^i, v^i$ are nonzero and $s = 1$.

Spin 0

• The conventional values for $u^0, v^0$ are $$u^0, v^0 = \pm i \sqrt{\frac{1}{2m}}$$
• as before, we drop the $\sigma$ label for they only take on a single value of $0$
• [❤️] We find that for general momenta, the coefficients are (5.3.16, 5.3.17), and we observe that the annihilation and creation fields are simply the \textbf{4-gradient of the scalar field}. Then weinberg be like thats enough i dont feel like exploring this option no more.

Spin 1

• For spin 1 the label $\sigma$ can take on between $-1, 0, 1$.
• By the analogy of the past normalization conventions, we find the 0-momenta, 0-spin coefficients (5.3.20)
• Using the raising and lowering operators $J_1^{(1)}\pm iJ_2^{(1)}$, we find the spin 1, -1 coefficients at $\textbf{p} = 0$ (5.3.21, 5.3.22)
• [❤️] the above two procedures give us the coefficients for $\sigma = -1, 0, 1$, now we can apply a boost to find the coefficients for any general momenta, and any of those three spins! \footnote{The $e^\mu(\textbf{p}, \sigma)$ vectors are a simpler way to state the the coefficients for arbitary momenta in terms of the three basis vectors $e(0,-1), e(0,0), e(0,1)$} and we find the fields $\phi^{+\mu}, \phi^{- \mu}$.

Spin 1 causality

• In computing the tators relations for spin 1 particle at spacelike separations, we find this $\Pi^{\mu \nu}$, which by some algebra is reduced to the familiar $\Delta_+$, which we know is even.
• [👉] we apply the same procedure for causality as we did for the scalar field, and we find that for causality it is necessary and sufficient that the spin 1 particle is \textbf{boson} and $|\kappa| = |\lambda|$. So we do the same procedure as scalar to adjust the pahse of the creation and annihilation operators to get them to be exactly equal.
• In addition, we find that this field is self-conjugate, so it is real which prohibits conservation of quantum number like electric charge.
• What if we want to conserve a charge? We do the same thing where we imageine there is another boson with exactly the same mass, spin, but opposite charge. This will give us a complex field.
• again, we can let the particle be neutral, and the boson would be its own antiparticle, and we simply conserve charge = 0.

spin 1 MISC

• A quick check indicates that the spin 1 field satisfies the KG equation.
• we can not get electrodynamics by simply taking the $m\rightarrow0$ limit of the massive spin 1 theory, because the spacelike commutator contains a $m^2$ at the bottom (5.3.30) which blows up.
• weinberg then discusses something related to photon vs spin1 massive boson, then talk about discrete symmetries.

Dirac Formalism

• The Dirac formalism is just one representation of the homogeneous lorentz group, but it has some special importance This formalism was introduced by Dirac, but weinberg wants to introduce it mathematically rather than historically
• from the point of view we are following here, the structure and properties of any quantum field are dictated by the representation of the homogeneous lorentz group under which it transforms.

Clifford Algebra

• By a representation of the homogeneous Lorentz group, we mean a set of matrices $D(\Lambda)$ satisfying $$D(\Lambda_1)D(\Lambda_2) = D(\Lambda_1 \Lambda_2)$$
• by looking at the infinitesmal transformations, we get the lie algebra of the lorentz group (5.4.4). \footnote{Any set of matrices satisfying the commutation relations in 5.4.4 acts as a set of generators for the some representation of the lorentz group.}
• the key here is that we find a set of matrices satisfying 5.4.4 via the \textbf{clifford algebra} (5.4.5 - 5.4.7).
• The properties of clifford algebra is explored by weinberg, the essential take away is that the clifford algebra allows us to obtain \begin{itemize}
• lorentz scalars (5.4.9),
• lorentz vectors (5.4.8),
• lorentz rank-2 tensors (5.4.10),
• lorentz rank-3 tensors (5.4.11),
• lorentz rank-4 tensors (5.4.12). \end{itemize} All these $\uparrow$ are representations of the Lorentz group, so we can form representations of the Lorentz group using clifford algebra!
• the clifford algebra automatically defines a parity operator (5.4.13-16)

Return to 4D spacetime dimensions

\begin{itemize} - In the 4D spacetime dimensions, weinberg considers an example set of $\gamma$ (5.4.17), the Pauli spinors. - recall that the pauli spinor matrices give the projection of the electron's spin along a direction. - using the relation between $\mathscr{J}$ and $\gamma$ given in (5.4.6), we find the lorentz group generators (5.4.19,20), and we find that these generators are block-diagonal, meaning we have found a reduciable representation that is simply a direct sum of two irreducible ones. \end{itemize}

Convenient way to write the clifford rank4tensor

• Weinberg considers how to write the rank4 tensor $\mathscr{P}^{\rho \sigma \tau \eta}$ in a more compact way>
• The answer relates to the $\gamma^5$ matrix defined by $\gamma^5 = -i\gamma^0 \gamma^1 \gamma^2 \gamma^3 $, with dis definition, we then find $$\mathscr{P}^{\rho \sigma \tau \eta} = 4!i\epsilon^{\rho \sigma \tau \eta} \gamma^5$$
• This matrix $\gamma^5$ is a pseudoscalar in the following sense $$[\mathscr{J}^{\rho \sigma}, \gamma_5] = 0, \quad \beta \gamma_5 \beta^{-1} = - \gamma_5.$$
• Moreover, we find $\mathscr{A}^{\rho \sigma \tau} = 3! i \epsilon^{\rho \sigma \tau \eta} \gamma_5 \gamma_\eta$.
• The 16 independent 4x4 matrices can therefore be taken as the components of the scalar 1 (D1), the vector $\gamma^{\rho}$ (D4), the antisymmetric tensor $\mathscr{J}^{\rho \sigma}$ (D 6), the axial vector $\gamma_5 \gamma_\eta$ (D4), and the pseudoscalar $\gamma_5$ (D1).
• recall from earlier that weinberg made a statement about how the clifford scalar, vector, tensors furnish representations of the lorentz group, in addition, they are all linearly independent, and we overall got 16 linearly independent componenet? What he has done here is writing the two tensors with $\gamma_5$ matrix, simplifying this representation.
• Weinberg then shows that the $\gamma_5$ matrix has unit square and anticommutes with the other four cliford matrices. This is particularly convenient, because this anticommutation relation gives us a D5 clifford algebra.

Causal Dirac Field

• Naturally, we now apply the Dirac representation This simply means plugging the $\mathscr{J}$ that we obtained in the last section using $\gamma$ into the formula

$$\sum_{\bar{\sigma}}u_{\bar{l}}(0, \bar{\sigma}) J^{(j)}{\bar{\sigma} \sigma} = \sum{l} \mathscr{J}_{\bar{l}l} u_l(0, \sigma)$$

we find the two equations above (5.5.3). - By a theorem from group theory, we find that the spin of the representation must be $\frac{1}{2}$, and we further find the basis vectors (equations above 5.5.6 )

The following are read at an extremely cursory pace

• Weinberg notes that the coefficients $\pm c, \pm b$ are determined by the parity operator. he illustrates this point to (5.5.17, 5.5.18).
• Then weinberg does what he did with all the other field: examining its causality. He finds that \textbf{the dirac formalism has to describe fermions}!
• then he explores the discrete transformation properties of the dirac field.

Massless Particle fields

• We begin by assuming that the massless particle field is also made out of the a field with creation and annihilation operators. Using the transformation relations (2.5.42), we arrive at (5.9.2, 3, 4)
• We find that the coefficients $u,v$ must satisfy (5.9.6,7), we simplify these equations by considering momentum-related transformations as not going from $p_1 \rightarrow p_2$, but $k \rightarrow p_2$. So we can express the general momentum coefficient $u(\textbf{p}),v(\textbf{p})$ in terms of the standard momentum $u(\textbf{k}),v(\textbf{k})$: (5.9.8,9).
• Now we zoom into the $\textbf{k}$ case, this transforms by the little group: (5.9.10,11).
• we find that no 4vector field can be constructed from the annihilation and creation operators for a particle of mass zero and helicity $\pm 1$.
• way out: use gauge invariance, and \textbf{tensor} field!
• we further find that causality implies the particle be its own charge conjugate, which would be the case for a photon.

The Feynman Rules

• There is obvious advantage in using a perturbation technique which keeps lorentz invariance and causality (cluster decomposition principle) manifest at all stages of the expansion.
• The old-fashion perturbation theory was not one of those advantageous techniques. The Feynman-Schwinger-Tomonaga is one of those advantageous technique above.
• This chapter outlines the diagrammatic calculational techniques first described by Feynman at Poconos Conference in 1948.

Feynman was led to these diagrammatic rules through his development of the path-integral approach, which will be the subject of section 9.

Derivation of the Rules via Dyson series

\begin{itemize} - The Dyson's series expresses the matrix element as (6.1.1)

Recall the expression of $S$ in terms of Dyson series (3.5.10), using our newest creation and annihilation operator formalism, we put the free fields on both sides of (3.5.10), and putting the corresponding creation and annihilation operators between the free fields and $S$ yields (6.1.1)

• The field of a specific particle that transforms under a particular representation of the homogeneous lorentz group, is given by (6.1.3)

This can be understood as the 'wave equation' of the particle, which contains everything we can possibly know about such particle. The integral over those coefficients imply that this is a localized particle, or 'wave packet'.

• weinberg here convenes a distinction between particles and antiparticle:
  • field operators that destroy particles and create antiparticles are '\textbf{fields}'
  • field operators that destroy antiparticles and create particles are '\textbf{field adjoints}'
• We now move the $\ap$s on the left of (6.1.1) to the right, using the commutator relations (6.1.4, 5, 6).
• In this way, the contributions to (6.1.1) are those arising from the delta function terms on the right-hand side of (6.1.4). The contribution to (6.1.1) of a given order in each of the terms $\Ham_i$ in the polynomial $\Ham(\psi(x),\psi^\dag(x))$ is given by a sum, over all the ways of pairing creation and annihilation operators, of the integrals of products of factors, in the 6 ways in (5.1.9-14)\footnote{These pairings are derived very trivially from 6.1.4 and 6.1.3}
• The S-matrix is obtained by multiplying these factors together, along with additional numerical factors to be discussed below, then integrating over $x_1,x_2,...,x_N$, then summing over all pairings, and then over the numbers of interactions of each type.
• It is probably worth nothing that Feynman diagrams are simply a diagrammatic mnemonics for keeping tracks of the terms that we encounter in moving the annihilation from left to right in (6.1.1) according to the rules between (6.1.9) and (6.1.14), which tells you how to handle $\ap$ with $\psi$, $\ap$ with $\ap^\dag$, and $\ap$ with time-ordering, respectively. Keep in mind that the lines represent a pairing that you can have. Figure 6.1 tells you more specifically what the pairings corresponds to.

The rules

The rules for calculating the S-matrix are conveniently summarized in temrs of Feynman diagrams. Each vertex in the diagrams represent one of the $\Ham_i$, and each of the lines represent a pairing described above.

• arrows always point \textbf{in} the direction that a \textbf{particle} is moving, and \textbf{opposite} to the direction an \textbf{antiparticle} is moving.
• To compute the contribution to the S-matrix for a given process, of a given order $N_i$ in each of the interaction terms $H_i$ in (6.1.2), we carry over the following steps:
  • draw all feynman diagrams consisting of $N_i$ vertices of each type $i$, and containing a line coming into the diagram from below for each particle/antiparticle in the initial state, and a line going upwards out of the diagram for eevery orticle in the final state, together with any number of internal lines running from one vertex to another, as required to give each vertex the proper number of attached lines. Each vertex is labeled with an interaction type $i$ and spacetime coordinate $x^\mu$. Each internal or external line is labeled at the field $\psi_l(x)$ or $\psi_l^\dag(x)$ that creates or destroys the orticle at that vertex, and each external line where it enters or leaves the diagram is labeled with the quantum numbers $\pp, \sigma, n$ or $\pp' \sigma', n'$ of the initial or final orticle.
  • For each vertex of type $i$, include a factor $-i$, obviously from $(-i)^N$, and a factor $g_i$, the coupling constant. For every final orticle, include a factor of (6.1.9,10), For every initial orticle, include a factor of (6.1.11,12). For internal lines include factors of (6.1.13,14)
  • Integrate the product of all these factors over the coordinates $x_1, x_2, ...$ of each vertex
  • Add up the results obtained in this way from each Feynman Diagram. The complete perturbation series for the S-matrix is obtained by adding up the contribution of each order in each interaction type, up to whatever order our strength permits.
• Some special rules apply for specific scenarios:
  • Suppose that an interaction $\Ham_i(x)$ contains $M$ factors of the same field. Then we get an additional factor of $M!$ in front of each diagram. The traditional method to combat this is to adjust the coupling factor to $\frac{g}{M!}$ in the $M$-th order expansion.
  • Remember how the permutations of the vertices cancel with the factor of $\frac{1}{N!}$, this is not always the casae because some permutations of the vertices do not necessarily produce a new diagram. This occurs when we have loop diagrams. In loop diagrams, we need to add an additional factor of $\frac{1}{N}$.
  • Whenever permutation puts the annihilation part $\psi^+(x)$ of a field in $\Ham$ just to the left of the creation part $\psi^{+\dag}(y)$ of a field adjoint in $\Ham(y)$, the permutation that puts the annihilation part $\psi^{-\dag}(y)$ of the field adjoint just to the left of the creation part $\psi^-(x)$ of the field involves one extra interchange of fermion operators, yielding the minus sign in the second term of (6.1.14) for fermions.
• The overall sign of the S-mat does not matter, but the relative signs do.
• general Feynman diagrams form either chains of lines that pass through the diagram with arbitrary number of interactions (traditional >--< diagrams), or else fermionic loops (loop ---->O< diagrams)

the canonical formalism

• all of the most familiar qft furnish canonical systems, and these can easily be put in a Lagrangian formalism
• The benefit of using Lagrangian formalism, a classical theory with a lorentz-invariant lagrangian density will when canonically quantized lead to a lorentz-invariant quantum theory.
• it would be hopeless to try to guess at a form of the Hamiltonian in non-abelian gauge theories without starting with a lorentz-invariant and gauge-invariant lagrangian density

canonical variables

Weinberg formalism to Lagr to Hamiltonian

• this section shows that various quantum theories that we have constructed so far satisfy the commutation rules and equations of motions of the Hamiltonian version of the canonical formalism, this means identifying the operators $q,p$ in theories we have considered so far (scalar, vector, tensor, etc) that satisfy the canonical commutator or anticommutation relations $$[q(\vec{x},t),p(\vec{y},t)]{\mp} = i\delta^3(\vec{x}-\vec{y})$$ $$[q(\vec{x},t),q(\vec{y},t)]{\mp} = 0$$ $$[p(\vec{x},t),p(\vec{y},t)]_{\mp} = 0$$
• Then weinberg uses the commutator relations between fields to find the right $p,q$ for each theory, specifically the real scalar field, complex scalar field, real vector field, dirac field of non-majoranna spin 1/2 particle (7.1.4-16).
• Then weinberg defines the quantum mechanical functional derivative (7.1.17,18) for a functional that takes in a system of operators that satisfy the canonical tator relations.
• Then he transitions to the Hamiltonian formalism by identifying that the functional derivative corresponds to the time-dependence of the operators $p,q$, and observing that the free Hamiltonian $H_0$ derived in weinberg formalism is equal to that of Hamiltonian formalism up to a constant.

what Lagr gives free Hami

• This question may be answered by Legendre transformation from Hamiltonian to Lagrangian (7.1.26)
• Lagr: lorentz-invariant, Hami: can be used to get S-mat elements, constraint, the Lagr must correspond to the right free field hami.

Can an interacting theory be formulated via canonical variable?

• weinberg introduces the canonical variables in heisenberg picture. He uses capital letters for those canonical variables.
• These canonical variables are quantized in the same way as the non-interacting ones
• weinberg notes that the relation between the canonical conjugates in interacting fields $P$ and the field variables $Q$, and the field variables $\dot{Q}$ is in general not the same as for the free theory operators.

the lagr formalism

How do we choose the hami?

• the easiest way to enforce lorentz invariance is via choosing a suitable lagr then derive the hami.
• The lagrangian is a functional of $\psi^l$ and $\dot{\psi}^l$ while the conjugate fields $\pi_l$ are defined to be the $\cdot{\psi}^l$ dependence of the lagr.
• pretty standard stuff, gets Euler lagrange in (7.2.9)
• The hamiltonian is obtained via legendre transformation, it would be a functional of the lagr, $\pi_l$ and $\cdot{\psi}$.
• after obtaining the equations of motion from the lagr and hami formalism, it would be tempting to identify the field variables $\psi$ and $\pi$ with canonical variables $q,p$. This works for the simple example of real scalar field, but \textbf{does not work in general}, because there are field variables, such as the time component of a vector field or the hermitian conjugate of a dirac field, that are not canonical variables, and do not have canonical conjugates, yet lorentz invariance requires them to be in the lagr.

getting the 'non-canonical' terms in the lagrangian

• the terms that do not behave like canonical variables are denoted $C^r$, so now we have $$P_n(\vec{x},t) = \frac{\delta L[Q(t),\dot{Q}(t),C(t)]}{\delta \dot{Q}^n(\vec{x},t)}, \quad 0 = \frac{\delta L[Q(t),\dot{Q}(t),C(t)]}{\delta C^r(\vec{x},t)}$$
• in a simple case, these equations can be solved to give $C$ in terms of $Q,P$.

global symmetries

the real point of lagr is that it provides a natural framework for the qm implementation of symmetry principles

constraints and dirac brackets

• the chief obstacle to deriving the hami from lagr is the constraints, there are two types
  • \textbf{primary} constraints are either imposed on the system, or arise from the structure of the lagr
  • \textbf{secondary} constraints arise from the requirement that the primary constraint be consistent with the equations of motion.
• weinberg says the the distinction between primary, secondary constraints is not important and he will treat them equally, but then why bother mentioning it?
• the constraints we have found for the massive vector field are of a type known as second class, which has a standard prescription for the tator relations.

the standard prescription for some constraints

• defines poisson bracket, notice that poisson brackets satisfy the algebraic properties of commutators and jacobi identity as well.
  • a constraint is known to be \textbf{first} class if its poisson bracket with all other constraints vanishes when we impose the constraints.
  • after the first class constraints are eliminated by a choice of gauge, the remaining constraint equaitons are such that no linear combination of the poission brackets of these constraints with each other vanishes, it follows that the matrix of the poisson brackets of the remaining constraints is non-singular, these are called second class constraints

    there must be an even number of second class constraints

• Dirac suggests that when all constraints are second class, the commutator relations will be given by the dirac bracket (7.6.19-20), which satisfies the commutator relations and jacobi identity as well (7.6.21-23)
• Maskawa and Nakajima: for any set of canonical variables, governed by second class constraints, it is always possible by a canonical transformation to construct two sets of variables $Q^n, \mathscr{Q}^r$ and their respective conjugates $P_n, \mathscr{P}_r$ such that the constraints read $\mathscr{Q}^r = \mathscr{P}_r = 0$.
• we then find that the dirac bracket is equal to the poisson bracket calculated in terms of the reduced set of unconstrained canonical variables.

qed

• infer the need for a principle of gauge invariance from the difficulty of formulating quantum theory of massless particles with spin
• deduce the features of electrodynamics from gauge invariance
• take gauge invariance as a starting point and deduce the existence of a vector potential for massless particles of unit spin

gauge invariance

• there is no way to construct a 4-vector as a linear combination of the creation and annihilation operators for helicity $\pm 1$
• Instead of banishing $A_\mu$ from the action, we shall require that the part of the action $I_M$ for matter and its interaction with radiation be invariant under a gauge transformation $A_\mu \rightarrow A_\mu +\partial_\mu \epsilon$
• we'd find that the lorentz invariance of $I_M$ requires $\partial_\mu \frac{\delta I_M}{\delta A_\mu (x)} = 0$ which implies $\frac{\delta I_M}{\delta A_\mu (x)}$ is a conserved current
• How to construct such current? We know that infinitesmal internal symmetries of the action implies conserved currents, in fact consider the transformation $$\delta \psi^l(x) = i \epsilon(x) q_l \psi^l (x)$$ that leaves the matter action invariant for a constant $\epsilon$, when the matter field satisfies the field equations this transformation defines a conserved current $J^\mu$.
• we can construct a lorentz-invariant theory by coupling the vector field $A_\mu$ to the conserved $J^\mu$ in the sense that $\delta I_M/\delta A_\mu(x)$ is taken to be proportional to $J^\mu$ from above: $$\frac{\delta I_M}{\delta A_\mu(x)} = J^\mu (x)$$

• the requirement above can be restated as a principle of invariance: the matter action is invariant under the adjoint transformations: $$\delta A_\mu(x) = \partial_\mu \epsilon(x), \quad \delta \psi_l(x) = o \epsilon(x) q_l \psi_l(x)$$

  • If $\epsilon$ is a constant, this is a global symmetry, or gauge invariance of the first kind
  • If $\epsilon(x)$ is arbitrary, then this is a local symmetry, or gauge invariance of the second kind

• Taking the radiation action to be that of (8.1.14), and imposing the equation of motion yields $0 = \frac{\delta}{\delta A_\nu} [I_\gamma + I_{M}] = \partial_\mu F^{\mu \nu} + J^\nu$, which is the inhomogeneous maxwell equations.

constraints and gauge conditions

• there are constraints that arise when we try to quantize the theory whose lagrangian we just studied above
• the first constraint arises from the fact that the lagrangian density is independent of the time-derivative of $A_0$, meaning $\Pi_0 = 0$. this is a primary constraint. this primary constraint leads immediately to the secondary constraint of $$\partial_i \Pi^i = - J^0$$
• because of the partial arbitrariness of $A_\mu$ due to gauge invariance, it is not possible to apply the canonical quantization procedure directly, rather, we need to first choose a gauge.

choosing coulomb gauge

choosing the coulomb gauge, and then applying dirac's method of removing second class constraints, we done.

path integral methods

• the awkwardness in obtaining simple results in the last chapter hinders canonical formalism from being applied to more complex theories (non-abelian gauge, general relativity, etc)
• (1960s) fadeev, popov and dewitt showed how to apply it to non-abelian gauge theories and general relativity, this revived path integral approach
• (1971) t'hooft used path-integral methods to derive the feynman rules for spontaneously broken gauge theories, including weak and electromagnetic interactions, in a gauge that made the high energy behavior of these theories transparent. this was a turning point for path integral
• in this chapter we derive the path integral formalism from canonical formalism, and see what additional sorts of vertices are needed to supplement the simplest version of the feynman path-integral method.

the general path integral formula

• weinberg begins with the general formalism for quantum mechanics (9.1.1-12)
• the key insight is that after considering general $$, weinberg considers the transition into a state with infinitesmal increment of time: $< q': \tau + d\tau | q: \tau>$
• this can be written with $d\tau$ taken to be $N$ tiny slices of $d\tau/N$, and we obtain a semi-main equation (9.1.26) in the limit of $N \rightarrow \infty$ this becomes a integral and we obtain a main equation (9.1.34) which is a constrained path integral. Physically this is the integral of all path that take $q(\tau)$ from $q$ at $\tau = t$ to $q'$ at $\tau = t'$.
• the great advantage of writing matrix elements in this way is that the path integrals are easy to calculate when expanded in powers of the coupling constants in $H$
• the path integral formalism allows us to not only compute transition probability between states, but also matrix elements between states with time-ordered products of general operators in between (9.1.35-38).

transition to the s-matrix

• if we want to convert the result of (9.1.34) to a notation appropriate to qft, we shall let the index $a$ run over all points $\vec{x}$

The idea being instead of all paths from point to point, we are integrating over all fields that take on the initial and final conditions

in space and over a spin and species index $m$, and replacing $Q_a(t)$ with $Q_a(\vec{x},t)$, same for $P$.

• Then to get the $S$ matrix element we just need to take the $t\rightarrow \pm \infty$ limits in the final and initial states.
• some derivations later we find an expression for the annihilation operators in the in and out states (9.2.7), together with (9.2.8), we have a differential equation to solve
• going with a gaussian ansatz, and the solution of vacuum expectation value of time ordered products can be expressed as (9.2.17). recall that in 6.4 we learned that the S-mat element is like the vacuum expectation value of t-ordered products, so this marks a intermediate stoppoint.

lagr version of path integral

• In (9.1.38), if $H$ is of the form (9.3.1), then the exponential part of (9.1.38) can be written as the corresponding lagrangian (9.3.1-10)
• however, there are some additional caveats with (9.3.10), being the determinant of $\mathscr{A}$.
  • recall $$M = \frac{ < VAC | T{...} | VAC> }{< VAC | VAC > }$$ so if this determinant amounts to some constant, then we have no problem (this would be the case for a set of scalar fields)
  • HMANASS, weinberg gives an example of a 'nonlinear $\sigma$ model' whose lagrangian density amounts to the determinant of $\mathscr{A}$ having exponential dependence on the trace of a term in the lagrangian. we can regard this determinant as providing a correction ($\Delta \mathscr{L}$) to the effective lagrangian density. This means we can still interpert the exponential part as an exponential of a lagrangian, but the lagrangian needs to be modified.
  • HMANASS, weinberg gives another example where we need to add a correction to the lagrangian density in order to keep the $e^{i\int \mathscr{L}}$ interpretation.
  • [❤️] In order to keep the exponentiated action interpretation, we add a $i\epsilon \times (\text{terms})$ to the lagrangian, giving (9.3.11).

path integral derivation of the feynman rules

• using the equation for $M$ (9.4.1), and our newly derived interpretation of $e^{i I[\psi]}$ as the weight function in the path integral (9.3.11), we obtain a formula for $M$: (9.4.2).
• dividing the lagrangian into a free and interaction part, we get the action in free and interaction parts as well (9.4.4-7)
• by adopting the actio weighting function interpretation, we have assumed that the lagrangian is quadratic in the fields, so we can always write the free action in the \textbf{generalized quadratic form} (9.4.8-9.4.11)
• To deal with interactions, we will expand the exponential in powers of $I_1$ and then $I_1$ in powers of the field (9.4.12)
• The general integral that we encounter both in the numerator and the denominator of (9.4.2) are of the form $\mathscr{J}$ in (9.4.13) > the $\psi$s can be taken to be 1 to get the denominator using some formula in the appendix, this can be written as (9.4.14), which amounts to the instructions of drawing feynman diagrams. what's left is to check that the factors of each vertex/line match the feynman rules we derived earlier
• the propagator in the path integral approach can be written as (9.4.15), some derivations later we find that its the same as the propagator we found earlier (9.4.16-18).
• weinberg then goes on to show that the same equivalence between path integral and feynman rules holds for another unperturbed lagrangian (the previous example was with a scalar field), and that the same holds for a lagrangian with derivative couplings.

non-abelian gauge theories

• qed uses U(1) as its gauge group, but the qft that have proved successful in describing the real world are all non-abelian gauge theories.
• it is natural that local gauge invariance should be extended to invariance under local non-abelian gauge transformations

gauge invariance

• in generalizing qed to non-abelian gauge symmetry, we begin with a general, non-abelian gauge symmetry group with nontrivial generators $$\delta \psi_l (x) = i\epsilon^\alpha(x)(t_\alpha)_l^m \psi_m(x)$$
• by assuming that these transformations can be made infintesmal, this non-abelian group has to be a lie group and we derive the commutator relations (structure constants) $[t_\alpha, t_\beta] = i C^\gamma_{\alpha \beta}t_\gamma$, and find its adjoint representation
• from the need to make the lagrangian gauge invariant with derivative couplings, we have to include a gauge field $A_\mu$, with gauge terms $\partial_\mu \epsilon$ in order to cancel non-gauge-invariant terms in the lagrangian. This naturally introduces $D_\mu \psi$ , which transforms just like $\psi$ under gauge transformations. The $\partial_\mu \epsilon$ terms in $A$ introduces $\partial_\nu \partial_\mu \epsilon$ terms in $\partial_\mu A_\mu$, which calls for an antisymmetric tensor $F = \partial_\mu A_\nu-\partial_\nu A_\mu$ as definition of the derivative of $A$. But this antisymmetric tensor has its own problem of having $\partial_\mu \epsilon-\partial_\nu \epsilon$ terms, which is resolved by finding its covariant version $$[D_\nu, D_\mu]\psi = -i(t_\gamma) F \psi, \quad F \cong \partial_\nu A - \partial_\mu A + C A\nu A\mu$$ we find that $F$ must transform like a matter field that happens to belong to the adjoint representation of the gauge grp.
• we find that by choosing lorentz transformation $\Lambda$ to act on the state in hilbert space, we can make the gauge field $A^\alpha_{\mu \Lambda(x)}$ vanish at any one point. Furthermore, $\Lambda$ can be chosen such that one component of $A$ vanishes completely, so we'll do that.
• a field is pure gauge if it can vanish purely from a choice of gauge

connections to gr

• connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. $$\textbf{gauge:} \quad \partial_\mu\psi \xrightarrow{A_\mu (x)} \text{cov. derivative of } \psi$$ $$\textbf{gr:} \quad ? \xrightarrow{\Gamma^\mu_{\nu \lambda}(x)} \text{cov. derivative of } T$$
• $F \sim R$ (Riemann-christofell curvature tensor)

gauge theory lagrangians and simple lie groups

• just as in electrodynamics, for any massless particle of unit spin, the lagr must contain a free-particle term quadratic in $\partial_\mu A^\alpha_\nu - \partial_\nu A^\alpha_\mu$

  for the 4-momentum reason

• it is not possible to introduce a kinematic term for the gauge field $A^\alpha_\mu(x)$ without also including self-interactions fo the field: $$\mathscr{L}'A = -\frac{1}{4}g{\alpha \beta}F^\alpha_{\mu \nu} F^{\beta \mu \nu}$$ this is another place where non-abelian gauge theories resemble general relativity, where the field self-interaction takes place via $\frac{-\sqrt{g}R}{8 \pi G}$ in the einstein-hilbert lagr density. This can be explained by observing the gauge field interacts with anything that transforms according to a nontrivial representation of the gauge group, including itself.
• [$\rightarrow$] the term $-\frac{1}{4}F^2$ in qed does not entail interactions, it's purely a kinematic term for the photon. when the gauge group is nontrivial, however, it is a self-interaction.
• we find that in order for things to work, the lie algebra has to be the direct sum of commuting compact simple and U(1) subalgebras.

field equations and conservation laws

• we determine the full lagrangian in ordinary derivatives (15.3.1), solves its eqm (15.3.2), and find a conservation law $\partial_\nu \mathscr{J}^\nu_\alpha$ where $\mathscr{J}^\nu_\alpha = -\partial_\mu F^{\mu\nu}_\alpha =$ for the current with ordinary derivatives rather than gauge covariant derivatives.
• the gauge invariance of the eqm is made manifest by replacing the ordinary derivatives in the eqm with gauge-covariant ones (15.3.5-8), this gives a gauge-covariant conservation law $D_\nu J^\nu_\alpha = 0$ with $\mathscr{J}^\nu_\alpha = -D_\mu F^{\mu\nu}_\alpha =$
• weinberg makes a connection to gr

quantization

• in quantizing gauge-invariant theory, we encounter the same problem as we did in electrodynamics where it's not clear that we can satisfy all eqms at once. A similar approach to before (choosing the right gauge) is warranted, and for the moment we choose the axial gauge.
• after identifying the canonical variables, we write the path integral for th amplitude, and with some criterion checking (hamiltonian being quadratic in $p$, $A_{\alpha0}$ being independent variable of integration) we find that the exponent in the path integral is just the action. $$ \propto \int_{\psi \text{ pth}} \int_{A \text{ pth}} (O...) e^{iI + \epsilon \text{ terms}}$$

dewitt-faddeev-papov method

• the axial gauge picked earlier hides the inherent lorentz and rotational invariance of the amplitude, in order to make that manifest, a change of gauge is warranted.
• something important stated in chapter 9: field-independent factors in the functional path integral affect only the vacuum fluctuation part of expectation values and S-matrix elements, and so are irrelevant to the calculation of the connected parts f the S-matrix.
• recognize that the functional integral in gauge qft is a special cause of a more general functional integral (15.5.1): $$\mathscr{J} = \int[\prod_{n,x}d\phi_n(x)]\mathscr{G}B[f[\phi]]\det \mathscr{F}[\phi]$$ $f[\phi]$ a non-gauge-invariant gauge-fixing functional, $B$ some functional.
• [$\rightarrow$] theorem: the integral (15.5.1) is actually independent of the gauge-fixing functional $f_\alpha[\phi;x]$ and depends on the choice of the functional $B[f]$ only through an irrelevant constant factor. and we get (15.2.21) as a powerful formula, whose effect on the lagr can be written as adding a term to the effective lagrangian $$\mathscr{L}{\text{EFF}} = \mathscr{L} - \frac{1}{2\zeta}f\alpha f_\alpha$$
• this is a powerful theorem that lets us derive feynman rules in a more convenient gauge. As our purpose is to make the amplitude manifestly lorentz invariant, the simplest choice is the lorenz gauge: $\partial_\mu A_\mu = 0$
• then the lorentz gauge is chosen to derive the feynman rules in this gauge invariant theory, which makes this theory manifestly lorentz invariant.

ghosts

• the determinant term in equation (15.5.21) may have non-trivial interpretations, in this context of non-abelian gauge theory.
• recall from section 9.5 that the determinant of any matrix $\mathscr{F}_{\alpha x, \beta y}$ can be written as a path integral over fermionic fields (15.6.1-2).
• the $\omega^*$ and $\omega$ in (15.6.2) are not necessarily complex conjugates of each other, because they are to be interpreted as separate path integrals.
• Thus the determinant of $\mathscr{F}$ term in (15.5.21) can be accounted for by path integrating over some effective action over $\omega, \omega^\star$ fields given by (15.6.2,4)
• the fields $\omega, \omega^\star$ are lorentz scalars so they are spin 0, but in order for the determinant $\rightarrow$ integral procedure to work they have to be grassman numbers (obeying fermi statics, or equivalently, anticommute), so it would appear that spin and statics is broken. yet we are fine because no particles described by these fields can appear in initial or final states. Thus $\omega, \omega^\star$ are called the ghost and antighost particles.
• the ghost action is $I_{GH} = \int d^4x d^4y \omega^\star \omega \mathscr{F}$, $\mathscr{F}$ is the matrix whose determinant we wish to evaluate. the modified action is $I_{MOD} = \int d^4x [\mathscr{L} - \frac{1}{2\zeta}f_\alpha f_\alpha] + I_{GH} = I_{EFF} + I_{GH}$
• Further inspection of (15.6.2) shows that the action respects the conservation of 'ghost number', equal to +1 for ghost, and -1 for antighost.
• to see what ghost vertices look like in feynman diagrams, we pick the simplest $\mathscr{F}$ (15.6.5), derive its interaction lagrangian density along with propagator term (15.6.5-13. we find that in this simplest theory the ghost vertex looks like a vector bosonic line with a ghost line and an antighost line.

brst

• when we made the amplitude lorentz invariant with dewitt-faddeev-papov method we had to pick a specific gauge, this would hide the gauge-invariance of the theory.
• this is a serious problem in proving renormalizability, as gauge invariance places restrictions on the counterterms available. so it's up to us to determine what kind of freedom we have in picking counterterms for canceling what kinds of infinities.
• we take the $I_{MOD} = I_{EFF} + I_{GHOST}$ and fourier transform the gauge fixing condition $B[f]$ in $\mathscr{L}{MOD}$ and to find ourselves having to integrate over some new fields $h\alpha$ (Nakanishi-Lautrup fields), weinberg creatively picks the name $I_{NEW}$ for the new action: $$I_{NEW} = \int d^4x(\mathscr{L} + \omega_\alpha^* \Delta_\alpha + h_\alpha f_\alpha + \frac{1}{2} \zeta h_\alpha h_\alpha)$$ $\omega$ is ghost field, $h$ is the nakanishi-lautrup field
• this action is not gauge invariant, yet it is invariant under brst: $$\delta_\theta \psi = i t_\alpha \theta \omega_\alpha \psi, \quad \delta_\theta A_{\alpha \mu} = \theta D_\mu \omega_\alpha, \quad \delta_\theta \omega_\alpha^* = - \theta h_\alpha,$$ $$\delta_\theta \omega_\alpha = - \frac{1}{2}\theta C_{\alpha \beta \gamma} \omega_\beta \omega_\gamma, \quad \delta_\theta h_\alpha = 0$$
• nilpotentcy in general is $A^N = 0$ (useless power), here means if $\delta_\theta F \cong \theta s F$, then $\delta_\theta(sF) = 0$
• weinberg shows in terms of nilpotency, $$I_{NEW} = \int d^4x \mathscr{L} + s\psi$$ and subsequently the brst symmetry of $I$. Notice that this equation indicates that the physical content of any gauge theory is in the kernel of the brst operator. The kernel modulo the image of any nilpotent transformation is said to form the cohomology of the transformation.
• btst symmetry will be useful in proving renormalizability as we will use the brst-invariance of the divergent terms in feynman diagrams
• brst transformation is not real $$<\alpha|Q = Q|\beta> = 0$$ physical states differ by $Q\ket{\beta}$ are indistinguishable physically, thus independent physical states correspond to states in the kernel of $Q$, modulo the image of $Q$, in other words, they correspond to the cohomology of $Q$.
• cohomology = reversing arrows

\subsection{external field methods}

\paragraph{effective action from external field}

\begin{itemize} \item $Z[J]$ is the vac-vac amplitude under external current $J$ coupled to $\phi$, it is given by $$Z[J] = \sum_{N=0}^{\infty}\frac{(iW[J])^N}{N!} = \exp{(iW[J])}$$ where $iW[J]$ is the sum of all connected vacvac amp, counting permutations as different diagrams. This means finding Z is finding W. \item define $\phi_J$ to be the vac expect value of the operator $\Phi(x)$ in the presence of the current $J$, or equivalently $$\phi_J = \frac{\delta}{\delta J(x)}W[J]$$ \item the quantum effective action is defined as $$\Gamma[\phi] \cong - \int d^4x \phi^r(x) J_{\phi r}(x) + W[J]$$ one finds $\frac{\delta \Gamma[\phi]}{\delta \phi^s(y)} = -J_{\phi s}(y)$ it is shown that $\Gamma[\phi]$ is the sum of all connected one-particle irreducible graphs in presence of $J_\phi$ \item $W_\Gamma [J,g]$ is for $W[J]$ when $I[\phi] \rightarrow g^{-1}\Gamma[\phi]$ \item[❤️] we find W via the effective action: $$iW[J] = \int_{\text{conn. tree}}\left[ \prod_{r,x}d\phi^r(x) \right] \exp{ \left[ i\Gamma[\phi] + i \int \phi^r(x)J_r(x) d^4x \right] }$$ \end{itemize}

\paragraph{application on scalar theory} \begin{itemize} \item take scalar action and add position-independent external field $\phi_0(x) = \phi_x$ assumed to be some functional over all space so we get a $\mathscr{V}_4 = \int d^4x$, and $\Gamma[\phi_0] = - \mathscr{V}_4 V(\phi_0)$ \item where $V(\phi_0)$ is known as the effective potential, if we wish to calculate it to one-loop order, we obtain (16.2.13,14) \item the divergence of the effective potential can be absorbed into appropriate constants (16.2.15) \end{itemize}

\paragraph{energy interpretation} \begin{itemize} \item \end{itemize}

• [ ] 5.8
• [ ] 5.10
• bound states are made significant by quark model
• perturbation theory was made to handle scattering and decaying, dont handle bound states well
• relativistic $\approx$ kinetic energy >> $mc^2$ or binding energy >> $mc^2$
• schro are treated as axioms in QM

hydrogen atom

• radial schro picks up an effective potential $$V(r) \rightarrow V(r) + \frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}$$ just like in CM
• relativistic correction together with spin-orbit coupling to form fine structure because they are both $O(\alpha^4)$
• lambshift: splits 2s, 2p, $O(\alpha^5)$
• hyperfine: spin-spin coupling via magnetic dipole, $O(\alpha^4)$
• quarkoniums are difficult to deal with because
  1. strong force is involved, whose potential we dont know
  2. light quarks (u,d,s) are intrinsically relativistic

light quark mesons

• within a multiplet, the particle with the higher charge is assigned the greater $I_3$
• $\pi^0$ is neither $u\bar{u}$ nor $d\bar{d}$, but the $I_3= 0$ state of the isospin triplet $$\text{isospin triplet} = \begin{cases} |11> = -u\bar{u}\ |10> = (u\bar{u} - d\bar{d})/\sqrt{2} \ |1-1> = d \bar{u} \end{cases}$$
• $\eta' = (u\bar{u} + d\bar{d} + s\bar{s})/\sqrt{3}$ is unaffected under SU(3) in the same sense that $\pi^0$ is an singlet under isospin SU(2)
• light quarks (u,d,s) belong to the fundamental rep. of SU(3), their antiquarks belong to the conjugate rep. of SU(3): $3 \otimes \bar{3} = 8 \oplus 1$
• the fact that quarks have unequal masses breaks mass symmetry in quark multilets

heavy quark baryons

• non-relativistic due to large mass
• 3 body systems, complicates things

• most similar to helium in theories we have already studied

• the 3-body difficulty is alleviated by considering $l=0$ states, where angular momentum of entire system comes from only coupling of the spins. This gives a j=3/2 quartet (totally symmetric), and two j=1/2 doublets (both antisymmetric among two particles): $2\otimes 2 \otimes 2 = 4 \oplus 2 \oplus 1$

• pauli's exclusion principle dictates which flavor states are allowed to couple with which color, spin, space state. The total spin of baryon being 3/2 or 1/2 implies the entire state must be totally antisymmetric.
• 3 light quarks in 3 flavors yields 1 decuplet, 2 octets, and a singlet: $3 \otimes 3 \otimes = 10 \oplus 8 \oplus 8 \oplus 1$
• every naturally occuring particle is a color singlet. Baryons have 3 particles with color, there is only 1 singlet, so that singlet must be the one baryons occupy: $$\psi(\text{color}) = (rgb-rbg + gbr - grb + brg - bgr)/\sqrt{6}$$
• Since a baryon is antisymmetric, its color being singlet means the rest of its wavefunction is symmetric
• natural occuring particles are color singlets because suppose it's a colorless non singlet then a SU(3) color transformation can give it a color

There are 3 experimental probes of elementary particle interactions: bound states, decays, scattering.

decaying

• We can not hope to calculate the lifetime of any particular particle, rather, we calculate the mean lifetime $\tau$ of a bunch of particles. The critical parameter, then, is the decay rate $\Gamma$, the probability per unit time that any muon will decay, for example.
• A short derivation finds the mean lifetime $\tau$ is $\tau = \frac{1}{\Gamma}$
• The branching ratio is the fraction of all particles of a given type that decay by each mode.
• For decays, the essential problem is to calculate the decay rate $\Gamma$, then it would be easy to compute everything else

Scattering

• The parameter of interest is the cross-sectional $\sigma$
• resonance: a special energy at which the particles involved 'like' to interact, forming a short-lived semibound state before breaking apart. resonance produces bumps in the graph of $\sigma$ v.s. $E$.

The golden rule

• the amplitude contains all the dynamical information, we calculate it by evaluating the feynman diagrams.
• the phase space factors contains only the kinematical information, it depends on the masses, energies, and momenta of the participant.
• The transition rate for a given process is determined by the amplitude and phase space according to Fermi's golden rule: $$\text{transition rate} = \frac{2\pi}{\hbar}|\mathscr{M}|^2\times (\text{phase space})$$

• [ ] renormalization

the dirac equation

• in nonrelativistic qm, particles are described by the schrodinger equation, in relativistic qm, particles of spin 0 are described by the KG equation, particles of spin 1/2 are described by the dirac equation, and particles of spin 1 are described by the proca equation.
• the derivation of dirac's equation involves first 'factor' the enerty momentum relation: $$(p^0)^2 - m^2c^2=(p^0+mc)(p^0-mc)=0$$
• the next brillance of dirac was the idea of making the $\gamma$s matrices instead of just a number, then we find the anticommutator relation (7.15), and some further derivation yields the dirac equation (7.20).
• Although the solution to Dirac equation has 4 components, it's not a 4-vector, rather it transforms like 2 spinors.

solutions to the dirac equation

• one solves the dirac equation in the rest frame ($\vec{p} = 0$) and finds 4 independent solutions (7.30).these shall be interpreted as a spin up, spin down electron, and a spin up, spin down antielectron. this is not a 4 vector!
• A quick examination of the solutions in (7.30) lets one find that these solutions only propagate in time, but not in space. The natural next step would be to extend the result at $\vec{p} = 0$ to general 4-momentum of a massive particle. Such planewaves furnish a representation of the corresponding little group.
• The above pursuit would yield the momentum space dirac equation, which then leads to the solution in (7.43), which after conventional normalization is written as (7.46)
• you might guess that $u^{(1)}$ describes an electron with spin up, and $u^{(2)}$ with spin down. but no. if we orient the $z$ axis so that it points along the direction of motion ($\vec{p}_x=\vec{p}_y = 0$), then $u^{(1)},u^{(2)},u^{(3)},u^{(4)}$ would correspond to the eigenstate of $S_z$ for electron spin up, electron spin down, antielectron spin up, antielectron spin down, respectively.

bilinear covariants

• griffiths makes an argument that the dirac spinor aint 4-vector (7.52-57)
• showed a way to construct scalar with dirac spinor (7.58-60), and asked is this a scalar or pseudoscalar? \footnote{the difference between scalar and pseudoscalar is how they transform under parity} It is a scalar (7.61-62). But is there another way to construct pseudoscalar out of dirac spinors? Yes, using $\gamma_5 = i \gamma^0\gamma^1\gamma^2\gamma^3$ (7.63-7.67)
• besides scalars and pseudoscalars, we can also construct vectors, pseudovectors, and antisymmetric tensor with dirac spinors (7.68-69)

the photon

• states the field forms of the maxwell equations (7.70), finds the index notation form of the inhomogeneous maxwell equations (7.70-74)
• for the homogeneous maxwell equations, introduce a vector potential $\vec{A}$, which gives us an index notation form of the homogeneous equations in terms of the potential (7.75-7.79), and notice that we can also write the inhomogeneous maxwell equations with the potential rather than field (7.80) -[❤️] we have two ways to write maxwell's equations, in EM fields, or in a single potential $A$. \href{https://en.wikipedia.org/wiki/Maxwell%27s_equations#Alternative_formulations}{wikipedia}
• in classical electrodynamics, the fields are the physical entities while the potentials are mathematical constructs that automatically satisfy the homogeneous equations. the subtlety in the potential formulations is that is only unique up to a gauge. There is no way to eliminate the gauge ambiguity, so we can either
  • live with the indeterminacy, meaning carrying along spurious degrees of freedom
  • impose an additional constraint, which spoils the manifest lorentz covariance of the theory
• griffiths choose the coulomb gauge $\nabla \cdot \vec{A} = 0$.
• in qed, the potential $A^\mu$ becomes the wave function of the photon, and we have a eqm for this photon: $\Box A^\mu = 0 $. solving this equation yields $A^\mu(x) = a e^{-(i/\hbar)p\cdot x}\epsilon^\mu(p)$, which can be interpreted as a wave propagation (exponential part) of some initial condition $\epsilon^\mu(p)$ at $x = 0$, this $\epsilon^\mu(p)$ is called the polarization vector.
• some further derivation indicates the polarization vector is orthogonal to the direction of the momentum vector, so we say a free photon is transversely polarized, for this reason the coulomb gauge is also called transverse gauge.

casimir's trick and the trace theorems

the feynman rules stated for qed works with spin, but we often dont care about the spins in actual experiments. In those cases the relevant cross section is an average over all initial spin configurations, and sum over all final spin configurations.

I. Crystal structure, symmetry and types of chemical bonds. (Chapter 1) • The crystal lattice • Point symmetry • The 32 crystal classes • Types of bonding (covalent, ionic, metallic bonding; hydrogen and van der Waals).

• II. Diffraction from periodic structures (Chapter 2) • Reciprocal lattice; Brillouin zones • Laue condition and Bragg law • Structure factor; defects • Methods of structure analysis • HRXRD. Experimental demonstration in the Physics Lab using Bruker D8 Discover XRD

• III. Lattice vibrations and thermal properties (Chapter 3) • Elastic properties of crystals; elastic waves • Models of lattice vibrations • Phonons • Theories of phonon specific heat; thermal conduction. • Anharmonicity; thermal expansion • Raman Scattering by phonons. Experimental demonstration in the Physics Lab using Ar-laser/SPEX 500M, CCD –based Raman Scattering setup

• IV. Electrons in metals (Chapters 4–5) • Free electron theory of metals • Fermi Statistics • Band theory of solids

• V. Semiconductors (Chapters 6–7) • Band structure. • Electron statistics; carrier concentration and transport; conductivity; mobility • Impurities and defects • Magnetic field effects: cyclotron resonance and Hall effect • Optical properties; absorption, photoconductivity and luminescence • Basic semiconductor devices • Photoluminescence. Experimental demonstration in the Physics Lab using Nd:YAG laser/SPEX –based Photoluminescence setup

• VI. Dielectric properties of solids (Chapters 8) • Dielectric constant and polarizability (susceptibility) • Dipolar polarizability, ionic and electronic polarizability • Piezoelectricity; pyro- and ferroelectricity • Light propagation in solids

• VII. Magnetism (Chapters 9) • Magnetic susceptibility • Classification of materials; diamagnetism, paramagnetism • Ferromagnetism and antiferromagnetism • Magnetic resonance • Multiferroic Materials • VIII. Superconductivity (Chapter 10)

• ordinary optical refraction happens at 500A in crystals

• bragg diffraction (crystal diffraction) can be used to select a special spectrum of beam

• bragg law: $2d \sin{\theta} = n \lambda$, this is the recurring theme of this chapter and will be echoed in the next chapter

reciprocal lattice and fourier

• in the fourer expansion of a function on the lattice, periodicity only allows terms with the same periodicity as the lattice in phase space. This defines the reciprocal lattice. Thus we can say the function is expanded on the reciprocal lattice in phase space: $$n(x) = \sum_p n_p \exp{i \frac{2 \pi p}{a} x}$$ where $\frac{2 \pi p}{a}$ is a point in the reciprocal lattice, with dimension of inverse distance.

• periodicity is the realm of fourier analysis

• If $\vec{a}_1, \vec{a}_2, \vec{a}_3$ are primitive vectors of the crystal lattice, the primitive vectors of the reciprocal lattice $\vec{b}_1, \vec{b}_2, \vec{b}_3$ are give by $$\vec{b}_i = 2 \pi \frac{\vec{a}_j \times \vec{a}_k }{\vec{a}_i \cdot \vec{a}_j \times \vec{a}_k}$$

• the diffraction pattern of the crystal is a map of the reciprocal lattice

• a reciprocal lattice vector $G=h b_1 + kb_2 + lb_3$ is normal to the plane (hkl) planes of direct lattice

brillouin zones

• a brillouin zone is defined as a wigner-seitz primitive cell in the reciprocal lattice

• only waves whose wavevector $\vec{k}$ drawn from the origin terminates on the surface of the brillouin zone can be diffracted by the crystal

• sc has reciprocal of sc

• bcc and fcc are reciprocals of each other

some fourier analysis

• by assuming refraction amplitude proportional to electron density, the total amplitude of the scattered wave in direction $k'$ from original wave in direction $k$ can be written as a function modulo the lattice with a function of the directions, such as below $$n(\vec{r})dV \times \exp{[i(\vec{k} - \vec{k}') \cdot \vec{r}]}$$ scattering amplitude for crystal with $N$ cells can be written as $$F_G = N \int_{cell} dV n(\vec{r}) \exp{-i\vec{G} \cdot \vec{r}} = N s_G$$ where $s_G$ is the structure factor.
• the structure factor contains single-cell information
• atomic form factor is integrated over all space

TODO:

• understand structure factor and form factor

• inert gas ~ noble gas ~ rare gas

inert bonds

• van der waal potential ~ london potential ~ induced dip-dip potental. It is the principal attractive interaction in inert gas. The othe major contribution to inert gas interactions is pauli exclusive principle, repulsive. These two forces add to give lennard jones.

• lennard jones potential: a combination of induced-dipole (or van der waal, or london) amd pauli exclusion principle $$U(R) = 4 \epsilon [(\frac{\sigma}{R})^{12} - (\frac{\sigma}{R})^6]$$

• if we neglect the kinetic energy of the inert gas atoms, the cohesive energy of an inert gas crystal is given by summing the lennard-jones potential over all pairs of atoms in the crystal.

ionic crystals

• the interaction of ionic crystals are largely coulomb, van der waal in ionic crystals only make contribute 1-2% in ionic crystals
• the name for electrostatic energy in ionic crystals is called madelung energy, because madelung was the first one to compute complicated sums over whole lattice of electrostatics
• the madelung constant is a sum of inverse of lattice separation over whole lattice, it decouples lattice structure in cohesive energy
• ionic bond examples: NaCl, CsCl

covalent bonds

• covalent bonds consists of two antisymmetric electrons

• in metals, bonds are formed due to lowering of valence electron energy as compared with free atoms without bonding

• hydrogen bonds: an atom of H is attracted to 2 other atoms. h-bond is important for th einteraction between H2O molecules and is responsible together with the electrostatic attraction of edip for water and ice

• the elastic properties of a crystal is viewed by consideration of it as a continuous homogeneous medium, rather than a lattice. This is valid for elastic waves with $\lambda$ longer than E-6 cm or E12 Hz.

• when a wave propagates in a crystal, we model it with entire planes of atoms moving in phase with displacements either parallel or perpendicular to wave vector (longitudinal/transverse)

• the author builds a model using hook's law assuming contributions from planes after the most two closest planes vanish, then solved as a coupled oscillator. The dispersion relation is then obtained

• only the elastic waves in the first brillouin zone are physically significant, for those outside such can be lattice-transformed into the first brillouin zone.

• at the boundaries of the first brillouin zone, the solution to the wave is not traveling, i.e. standing. Recall that for refraction the boundaries on the first brillouin zone give the only wavevectors that can refract.

• zero point energy $\approx$ GND state energy

• wave-like solutions do not exist for certain frequencies in polyatomic lattices. for diatomic, this is between $\sqrt{2C/M_1}$ and $\sqrt{2C/M_2}$. There is a frequency gap at the boundary $K_{max} = \pm \frac{\pi}{a}$ of first brillouin zone.

TODO: - [x] understand the connection between elastic waves, refraction, bragg law, first brillouin zone. When bragg condition is satisfied, traveling waves dont form, so we obtain standing waves that oscillate back and forth

• the free electron model have the valence electrons in metals move freely through the volume of the metal
• free electron model most useful when we focus on kinematic properties of conduction electrons
• [ ] 3s conduction band
• electrons in metals can have mean free path of $10^8$ interatomic spacing. This observation is incompatible with treating particles as solid balls, so wave-theory (QM) is needed. Further, infrequency of electron-electron scattering may be interpreted by pauli xclu principle in free electron fermi gas model.

energy levels in 1D

• orbital denotes solutions to the wave equation for a system of only 1 electron. The orbital formalism is relevant for when we treat N particle wave func as N x 1 particle wave func
• uses ISW model
• pauli xclu pcpl allows only one e- per orbital
• pauli xclu pcpl demand ISW orbitals are filled from bottom up in pairs
• fermi energy is the energy of topmost filled level, for N e- in GND state: $$\epsilon_F \frac{\hbar^2}{2m}(\frac{N \pi}{2L})^2$$

effect of T on the fermi-dirac dist

• fermi-dirac dist gives the probability that an orbital at energy $\epsilon$ will be occupied in an ideal electorn gas in thermal equilibrium: $$f(\epsilon)= \frac{1}{\exp{[(\epsilon-\mu(T))/k_B T]} + 1}$$
• By imposing $f(\epsilon)$ be binary at $T = 0$, we find $\mu(T=0) = \epsilon_F$
• The $\epsilon - \mu >> k_B T$ is the classical limit (giving Boltzmann Maxwell dist)

free electron gas in 3D

• 3D generalization of 1D is straightforward (see Kittel/Griffiths/Glazer&Wark/etc.)
• By transitioning to $k$ space, we obtain the density of state and find fermi energy $k_F = (\frac{3\pi^2N}{V})^{1/3}$, $\epsilon_F = \frac{\hbar^2}{2m}(\frac{3\pi^2 N }{V})^{2/3}$
• Density of state: $$D(\epsilon) = \frac{dN}{d\epsilon} = \frac{V}{2\pi^2}( \frac{2m}{\hbar^2} )^{3/2}\epsilon^{1/2}$$

heat capacity of the electron gas

• classical mechanics predicts $\frac{3}{2}k_B$, yet the actual heat capacity is 0.01 of this value
• this is due to electrons below fermi energy being 'locked'
• only a fraction $T/T_F$ can be excited at $T$, giving energy O(kT)
• At room temp, fermi temperature is approximately $5\times10^4$ K

experimental heat capacity of metals

• debye + free electron gas gives $$C = \gamma T + AT^3$$ this $\gamma$ is given the name of sommerfeld parameter
• thermal effective mass measures discrepancy between observed and actual values of $\gamma$. This discrepancy is due to other interactions besides pauli exclu, lattice inperfection, phonon scattering, etc.

electrical conductivity & Ohm

• use mv = $\hbar k$, a classical interperetation of electorn quantum momentum + lorentz force law
• in the absence of collisions, the fermi sphere accelerates, But the collisions and lattice imperfections introduce a 'frag'
• The collisions are measured with $\tau$, 'collision time'

• free electron model fails to disitinguish metals, semimetals, semiconductors, insulators; positive hall coeff. etc.
• energy bands/gaps = gaps in spectrum where no electron is permitted
• explanation of bands require periodicity potential to be taken into account

nearly free electron model

• treat periodic potential as perturbation
• bragg reflection occurs for waves in crystals, electorns are waves so can reflect braggly, giving gaps

bloch functions

• bloch functions: solutions of the schro for a periodic potential must be of form

$$\psi_{\vec{k}} (\vec{r}) = u_{\vec{k}} (\vec{r}) \exp[ i \vec{k} \cdot \vec{r}]$$

griffiths has a section on bloch's theorem in his qm book - bloch functions can be decomposed into a sum of traveling waves