Quantum Field Theory

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Why QFT? : small size & high energy

Reference: Peskin, Schroeder, Tong QFT notes

Review: Classical Field Theory

Field, is a quantity defined at every point in space and time. Rather than describing finite number of degrees of freedom in classic particle mechanics, fields describe infinite degrees of freedom.

Lagrangian Density & Action

The dynamics of the field, is governed by a Lagrangian which is a func of \(\phi (\vec{x}, t)\), \(\dot{\phi} (\vec{x}, t)\), \(\nabla \phi (\vec{x}, t)\). Usually, the Lagrangian can be expressed as:

\[ L(t) = \int d^3 x \mathcal{L}(\phi_a, \partial_\mu \phi_a) \]

where \(a\) is the index of different fields, \(\mu = 0,1,2,3\) is the spacetime index. The Lagrangian density \(\mathcal{L}\) is a function of the field and its first derivative.

The Lagrangian is local, which means that \(\mathcal{L}\) at \((\vec{x}, t)\) only depends on the field and its derivatives at the same point \((\vec{x}, t)\). Actually, we import the field theory to ensure locality.

Also we can define the action:

\[ S = \int d^4 x \mathcal{L}(\phi_a, \partial_\mu \phi_a) \]

In partical Mechanics, \(\mathcal{L}\) depends on \(q_i, \dot{q}_i\) (no higher order derivatives); in field theory, \(\mathcal{L}\) depends on \(\phi_a, \partial_\mu \phi_a\).

We can determine the EOM of the field by the principle of least action, i.e. \(\delta S = 0\).

\[ \begin{aligned} \delta S &= \int d^4 x \left( \frac{\partial \mathcal{L}}{\partial \phi_a} \delta \phi_a + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi_a)} \delta (\partial_\mu \phi_a) \right) \\ &= \int d^4 x \left( \frac{\partial \mathcal{L}}{\partial \phi_a} \delta \phi_a + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi_a)} \partial_\mu (\delta \phi_a) \right) \\ &= \int d^4 x \left( \frac{\partial \mathcal{L}}{\partial \phi_a} - \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi_a)} \right) \right) \delta \phi_a + \text{surface term} \end{aligned} \]

So we get the Euler-Lagrange equation for fields:

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

Klein-Gordon Eq.

Consider the Lagrangian density of a real scalar field \(\phi(\vec{x}, t)\):

\[ \mathcal{L} = \frac{1}{2} \eta^{\mu \nu} \partial_\mu \phi \partial_\nu \phi - \frac{1}{2} m^2 \phi^2 = \frac{1}{2} \dot{\phi}^2 - \frac{1}{2} (\nabla \phi)^2 - \frac{1}{2} m^2 \phi^2 \]

where \(\eta^{\mu \nu} = \text{diag}(1, -1, -1, -1)\) is the Minkowski metric. The EOM can be derived by the Euler-Lagrange equation:

\[ \frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \right) = -m^2 \phi - \partial_\mu (\eta^{\mu \nu} \partial_\nu \phi) = (\partial_\mu \partial^\mu + m^2) \phi = 0 \]

which is the Klein-Gordon equation. The operator \(\partial_\mu \partial^\mu = \partial_t^2 - \nabla^2\) is called the d’Alembertian, denoted as \(\Box\).

Maxwell Eqs.

We can also derive the Maxwell equations from a Lagrangian density. The E&M field is described by a 4-vector potential \(A^\mu = (\phi, \vec{A})\), where \(\phi\) is the scalar potential and \(\vec{A}\) is the vector potential. The electric and magnetic fields are given by:

\[ \vec{E} = -\nabla \phi - \frac{\partial \vec{A}}{\partial t}, \quad \vec{B} = \nabla \times \vec{A} \]

The Lagrangian density for the free electromagnetic field is:

\[ \mathcal{L} = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} \]

where \(F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\) is the electromagnetic field tensor. The Euler-Lagrange equation gives:

\[ \partial_\mu F^{\mu \nu} = 0 \]

Lorentz Invariance

TBD

Symmetries & Noether’s Theorem

TBD

Noether’s Theorem

TBD