2.1. From Particles to Fields

In this chapter we discuss the Lagrangian formalism of particles and fields.

  1. Lagrangian and Euler-Lagrangian equation of single particle;
  2. Generalize single particle formalism to fields;
  3. Using the least action principle to derive some famous equation of motions: Klein-Gordon equation, Dirac equation, Maxwell equation, and equation for massive vector particles.
  4. Relation to statistical mechanics.

2.1.1. Classical Mechanics

For single particle, we define a Lagrangian \(L(q_i,\dot q_i, t)\) which is a function of generalized coordinate \(q_i\), its derivative, and time \(t\). To be precise, the generalized coordinates could be time dependent.

Action is defined as the integral of Lagrangian over time,

\[S = \int dt L(q_i,\dot q_i, t).\]

The principle that leads to the equation of motion that the particle obays is the one that extremize the action. Mathematically,

\[\delta S = 0,\]

which gives us the Euler-Lagrangian equation,

\[\partial_t \left( \frac{\partial L}{\partial \dot q_i} \right) - \frac{\partial L}{\partial q_i} = 0.\]

Derivation of Euler-Lagrangian Equation

  • [ ] Here goes the derivation

On the other hand, a Legendre transform of the Lagrangian is the Hamiltonian. For single particle

\[H = \dot q p - L,\]

where the generalized momentum is \(p = \partial L/\partial \dot q\). The equation of motion starting from Hamiltonian is

\[\begin{split}\dot q &= \frac{\partial H}{\partial p},\\ \dot p &= -\frac{\partial H}{\partial q}.\end{split}\]

2.1.2. Generalization to Fields

For single particle, the dynamics is described by where the particle is at a certain time. However, field spans over space and evolve over time. Thus we define a Lagrangian \(\mathcal L\) at each space point, which should be called Lagrangian density. Integrate over space we find the Lagrangian

\[L = \int d^3x \mathcal L.\]

To be more specific, Lagrangian density is a function of \(\phi_i(x^\mu)\), \(\phi_i(x^\mu)_{,\mu}\), where \(x^\mu\) are the space time coordinates. Then

\[L(t) = \int d^3x \mathcal L(\phi_i(x^\mu),\phi_i(x^\mu)_{,\mu}).\]

Lagrangian and Lagrangian Density

In quantum field theory, Lagrangian density is usually called Lagrangian.

Similar to single particle theory, action is defined as integral of Lagrangian over time,

\[S = \int dt L(t).\]

Apply the same principle as the single particle case, we can derive the Euler-Lagrangian for the fields

\[\partial_\mu \left( \frac{\partial \mathcal L}{\partial ( \phi_{i,\mu} )} \right) - \frac{\partial \mathcal L}{\partial \phi_i} = 0.\]

Derivation of Euler-Lagrangian Equation

  • [ ] Here goes the derivation

We can also define a momentum of field \(\phi_i\),

\[\Pi(x^\mu) = \frac{ \partial \mathcal L }{ \partial ( \partial_0 \phi_i ) }.\]

The Hamiltonian density follows

\[\mathcal H = \dot \phi_i \Pi - \mathcal L.\]

2.1.3. Examples of Lagrangian

2.1.3.1. Klein-Gordon Equation

2.1.3.2. Dirac Equation

2.1.3.3. Maxwell Equation

2.1.3.4. Equation for Massive Vector Particles

2.1.3.5. Shrodinger Equation

2.1.4. Refs and Notes

  1. Field Theory : A Modern Primer by Pierre Ramond. The first chapter of this book is “How to Build an Action Functional”, which is self-explanary.