Chapter 5 Continuous systems

Lagrangians and Hamiltonians

Lagrangians \(\displaystyle L(q_{i},\dot{q}_{i})\)

Euler-Lagrange equation \(\displaystyle \frac{ \partial L }{ \partial q_{i} }-\frac{\mathrm{d}}{\mathrm{d}t}\frac{ \partial L }{ \partial \dot{q}_{i} }=0\)

Canonical momentum \(\displaystyle p_{i}=\frac{ \partial L }{ \partial \dot{q_{i}} }\implies \frac{\mathrm{d}}{\mathrm{d}t}(p_{i}\dot{q}_{i}-L)=0\)

Hamiltonian \(\displaystyle H=p_{i}\dot{q}_{i}-L\implies \frac{\mathrm{d}H}{\mathrm{d}t}=0\)

Hamilton's equation

\(\frac{\partial H}{\partial p_{i}} =\dot{q}_{i},\quad \frac{\partial H }{ \partial q_{i} } =- \dot{p}_{i}\)

Poisson bracket \(\{ A,B \}_{\mathrm{PB}}=\frac{ \partial A }{ \partial q_{i} } \frac{ \partial B }{ \partial p_{i} } -\frac{ \partial A }{ \partial p_{i} } \frac{ \partial B }{ \partial q_{i} }\)

For any function \(\displaystyle F\) \(\displaystyle \frac{\mathrm{d}F}{\mathrm{d}t}=\frac{ \partial F }{ \partial t }+\{ F,H \}_{\mathrm{PB}}\)

In quantum mechanics \(\displaystyle \{ A, B \}_{\mathrm{PB}}\to \frac{1}{\mathrm{i\hbar}}\langle [\hat{A},\hat{B}] \rangle\)

A charged particle in an electromagnetic field

Example

Lagrangians (Free particle): \(\displaystyle L=- \frac{mc^{2}}{\gamma}\) Electromagnetic field tensor

\[F_{\mu \nu}=\left(\begin{array}{cccc} 0 & E^{1} & E^{2} & E^{3} \\ -E^{1} & 0 & -B^{3} & B^{2} \\ -E^{2} & B^{3} & 0 & -B^{1} \\ -E^{3} & -B^{2} & B^{1} & 0 \end{array}\right)\qquad F^{\mu \nu}=\left(\begin{array}{cccc} 0 & -E^{1} & -E^{2} & -E^{3} \\ E^{1} & 0 & -B^{3} & B^{2} \\ E^{2} & B^{3} & 0 & -B^{1} \\ E^{3} & -B^{2} & B^{1} & 0 \end{array}\right)\quad F_{\mu \nu}F^{\mu \nu}=2(\boldsymbol{B}^{2}-\boldsymbol{E}^{2})\]

Electromagnetic lagrangian \(L=-\frac{1}{4} \int \mathrm{d}^{3} x F_{\mu \nu} F^{\mu \nu}\)

Lagrangian and Hamiltonian density

Example

\[H=\int \mathrm{d}^{3}x\mathcal{H}\qquad L=\int \mathrm{d}^{3}x\mathcal{L}\]

Conjugate momentum \(\displaystyle \pi (x)=\frac{ \partial \mathcal{L} }{ \partial \dot{\phi} }\implies \mathcal{H}=\pi \dot{\phi}-\mathcal{L}\) Four-vector version of Euler-Lagrange equation

\[ \frac{ \partial \mathcal{L} }{ \partial \phi } -\partial_\mu\left( \frac{ \partial \mathcal{L} }{ \partial (\partial_\mu \phi) } \right)=0 \]

Chapter 6 A first stab at relativistic quantum mechanics

Kelin-Gordon equation

Kelin-Gordon equation

\[ (\partial^{2}+m^{2})\phi(x)=0 \]

which has a negative solution. Think these as positive energy antiparticles.

Note

\[\phi (x)=\left[\begin{array}{c} \text { Incoming positive } \\ \text { energy particle } \\ \propto \mathrm{e}^{-\mathrm{i}(E t-\boldsymbol{p} \cdot \boldsymbol{x})} \end{array}\right]+\left[\begin{array}{c} \text { Outgoing positive } \\ \text { energy antiparticle } \\ \propto \mathrm{e}^{+\mathrm{i}(E t-\boldsymbol{p} \cdot \boldsymbol{x})} \end{array}\right]\]

Chapter 7 Examples of Lagrangians or how to write down a theory

A massless scalar field

Note

\[ \mathcal{L}=\frac{1}{2}\partial ^{\mu}\phi\partial_\mu \phi=\frac{1}{2}(\partial_\mu \phi)^{2}\]

Example

Using the E-L equation with

\[\frac{\partial \mathcal{L}}{\partial \phi}=0, \quad \frac{\partial \mathcal{L}}{\partial\left (\partial_{\mu} \phi\right)}=\partial^{\mu} \phi,\]

we have

\[\partial_{\mu} \partial^{\mu} \phi=0 .\]

This is the wave equation \(\partial^{2} \phi=0\) or

\[\frac{\partial^{2} \phi}{\partial t^{2}}-\nabla^{2} \phi=0,\]

and has wave-like solutions

\[\phi (x, t)=\sum_{\boldsymbol{p}} a_{\boldsymbol{p}} \mathrm{e}^{-\mathrm{i}\left (E_{\boldsymbol{p}} t-\boldsymbol{p} \cdot \boldsymbol{x}\right)},\]

with dispersion relation \(E_{\boldsymbol{p}}=c|\boldsymbol{p}|\) [though in our units \(c\)=1]

A massive scalar field

Lagrangian

\[ \mathcal{L}=\frac{1}{2}\left(\partial_{\mu} \phi\right)^{2}-\frac{1}{2} m^{2} \phi^{2} \]

Example

Using the Lagrangian

\[\frac{\partial \mathcal{L}}{\partial \phi}=-m^{2} \phi, \quad \frac{\partial \mathcal{L}}{\partial\left (\partial_{\mu} \phi\right)}=\partial^{\mu} \phi\]

and hence plugging into the Euler-Lagrange equation we have

\[\left (\partial_{\mu} \partial^{\mu}+m^{2}\right) \phi=0\]

The equation of motion for this field theory is the Klein-Gordon equation! The solution of these equations is again \(\phi (\boldsymbol{x}, t)=a \mathrm{e}^{-\mathrm{i}\left (E_{\boldsymbol{p}} t-\boldsymbol{p} \cdot \boldsymbol{x}\right)}\) , with dispersion \(E_{\boldsymbol{p}}^{2}= \boldsymbol{p}^{2}+m^{2}\).

An external source

Note

\[ \mathcal{L}=\frac{1}{2}[\partial_\mu \phi(x)]^{2}-\frac{1}{2}m^{2}[\phi(x)]^{2}+J(x)\phi(x) \]

Example

\[\left (\partial_{\mu} \partial^{\mu}+m^{2}\right) \phi (x)=J (x)\]

The \(\displaystyle \phi ^{4}\) theory