Quantum Field Theory Course

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Part I, Chapter 3 | Page 1 of 7

Real & Complex Scalar Fields

The simplest relativistic fields and their classical dynamics

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Lecture 3: Why Quantum Field Theory - MIT 8.323

Introduction to scalar fields and motivation for quantum field theory (MIT QFT Course)

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3.1 Real Klein-Gordon Field

The real scalar field Ο†(x) is the simplest relativistic field theory. The Lagrangian density is:

$$\mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{1}{2}m^2 \phi^2 - V(\phi)$$

For the free field, we set V(Ο†) = 0:

$$\mathcal{L}_{\text{free}} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{1}{2}m^2 \phi^2$$

Equation of Motion

The Euler-Lagrange equation yields the Klein-Gordon equation:

$$(\Box + m^2)\phi = (\partial_\mu \partial^\mu + m^2)\phi = 0$$

In explicit form:

$$\frac{\partial^2 \phi}{\partial t^2} - \nabla^2 \phi + m^2 \phi = 0$$

3.2 Plane Wave Solutions

Try a plane wave solution:

$$\phi(x) = A e^{-ip \cdot x} = A e^{-i(Et - \vec{p} \cdot \vec{x})}$$

where pΞΌ = (E, p) is the 4-momentum. Substituting into the Klein-Gordon equation:

$$(-p_\mu p^\mu + m^2)\phi = 0$$

Dispersion Relation

For non-trivial solutions, we need:

$$\boxed{p_\mu p^\mu = -E^2 + |\vec{p}|^2 = -m^2}$$

This gives the relativistic energy-momentum relation:

$$E^2 = |\vec{p}|^2 + m^2 \quad \Rightarrow \quad E = \pm\sqrt{|\vec{p}|^2 + m^2}$$

General Solution

The general solution is a superposition of plane waves:

$$\phi(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_{\vec{p}}}} \left[a(\vec{p}) e^{-ip \cdot x} + a^*(\vec{p}) e^{ip \cdot x}\right]$$

where we define the on-shell energy:

$$\omega_{\vec{p}} = \sqrt{|\vec{p}|^2 + m^2} > 0$$

and we choose only positive energy solutions (E > 0). The coefficients a(p) are Fourier modes.

3.3 Complex Scalar Field

A complex scalar field has two independent real components:

$$\phi(x) = \frac{1}{\sqrt{2}}(\phi_1(x) + i\phi_2(x))$$

where φ₁ and Ο†β‚‚ are real fields. The Lagrangian is:

$$\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^* \phi$$

Note: We use Ο†*βˆ‚ΞΌΟ† not (βˆ‚ΞΌΟ†*)(βˆ‚ΞΌΟ†) to get the correct sign.

Equations of Motion

Treating Ο† and Ο†* as independent fields, we get two Klein-Gordon equations:

$$(\Box + m^2)\phi = 0, \quad (\Box + m^2)\phi^* = 0$$

3.4 U(1) Global Symmetry

The complex scalar Lagrangian is invariant under global U(1) phase transformations:

$$\phi(x) \to e^{i\alpha} \phi(x), \quad \phi^*(x) \to e^{-i\alpha} \phi^*(x)$$

where Ξ± is a constant (same at all spacetime points). Check invariance:

$$\phi^* \phi \to e^{-i\alpha}\phi^* \cdot e^{i\alpha}\phi = \phi^* \phi \quad \checkmark$$
$$\partial_\mu \phi^* \partial^\mu \phi \to e^{-i\alpha}\partial_\mu \phi^* \cdot e^{i\alpha}\partial^\mu \phi = \partial_\mu \phi^* \partial^\mu \phi \quad \checkmark$$

Infinitesimal Transformation

For infinitesimal Ξ±:

$$\delta \phi = i\alpha \phi, \quad \delta \phi^* = -i\alpha \phi^*$$

Noether Current

By Noether's theorem, this U(1) symmetry leads to a conserved current:

$$j^\mu = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}\delta \phi + \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi^*)}\delta \phi^*$$

Computing:

$$\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} = \partial^\mu \phi^*, \quad \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi^*)} = \partial^\mu \phi$$

Therefore:

$$\boxed{j^\mu = i(\phi^* \partial^\mu \phi - \phi \partial^\mu \phi^*)}$$

This satisfies βˆ‚ΞΌjΞΌ = 0 (verify using the equations of motion).

Conserved Charge

The conserved charge is:

$$Q = \int d^3x \, j^0 = i\int d^3x \, (\phi^* \dot{\phi} - \dot{\phi}^* \phi)$$

After quantization, this will be identified with electric charge (or particle number). This is why U(1) symmetry is associated with charge conservation!

3.5 Energy and Momentum for Scalar Fields

From spacetime translation symmetries via Noether's theorem, we get:

Energy (Time Translation)

The Hamiltonian density (for real scalar field):

$$\mathcal{H} = \pi \dot{\phi} - \mathcal{L} = \frac{1}{2}\pi^2 + \frac{1}{2}(\nabla \phi)^2 + \frac{1}{2}m^2 \phi^2$$

Total energy:

$$E = \int d^3x \, \mathcal{H}$$

Momentum (Spatial Translation)

The momentum density:

$$\mathcal{P}^i = -\pi \partial^i \phi$$

Total momentum:

$$\vec{P} = -\int d^3x \, \pi \nabla \phi$$

These will be promoted to operators in the quantum theory, generating time and space translations.

Key Concepts (Page 1)

  • β€’ Real scalar: simplest relativistic field, Klein-Gordon equation
  • β€’ Dispersion relation: EΒ² = |p|Β² + mΒ²
  • β€’ Complex scalar: two real fields, U(1) phase symmetry
  • β€’ U(1) symmetry β†’ conserved current jΞΌ β†’ conserved charge Q
  • β€’ Energy and momentum from spacetime translation symmetries
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Chapter 3: Real & Complex Scalar Fields
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