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🎨 QFT Course Features Demo

This page demonstrates all the interactive features added to the Quantum Field Theory course. These components enhance learning through visualization, practice problems, and connections to Quantum Mechanics.

🔗Course Connections

📚Prerequisites

Quantum Mechanics

Canonical Quantization

Understanding canonical commutation relations and operator formalism

→

Quantum Mechanics

Angular Momentum & Spin

Spin-1/2 fermions are essential for Dirac fields

→

Mathematics

Lie Groups & Symmetries

Gauge symmetries and Lorentz transformations

→

🔄Related Topics

Path Integrals

Harmonic Oscillator

1. Scalar Field Visualizer

Visualize real, complex, and Klein-Gordon scalar fields in 1D and 2D. See how different modes create field configurations and watch them evolve in time.

🌊Scalar Field Visualization

Mode Number: 1

Field Type: klein-gordon

• Klein-Gordon equation: (∂²/∂t² - ∇² + m²)φ = 0
• Relativistic wave equation for spin-0 particles
• Exhibits wave-like propagation with mass term
• Higher modes = higher energy states
• Animation shows time evolution of field configuration

Complex Scalar Field (2D)

🌊Scalar Field Visualization

Mode Number: 2

Field Type: complex

• Complex scalar field: φ(x,t) ∈ ℂ
• Describes charged spin-0 particles
• Conserved charge from U(1) symmetry
• Higher modes = higher energy states
• Animation shows time evolution of field configuration

2. Feynman Diagram Drawer

Explore fundamental QED processes through Feynman diagrams. Learn to read particle interactions, identify vertices, and understand perturbation theory.

📊Feynman Diagram Drawer

Select Diagram

Process

e⁻ → e⁻

Order

Tree level

Type

Tree

Legend:

• Blue line + arrow: Electron/fermion
• Pink line + arrow: Positron/anti-fermion
• Wavy orange line: Photon/gauge boson
• Yellow dot: Interaction vertex
• Arrows show particle flow direction (time flows left to right)

3. Physical Intuition

💡Why Do We Need QFT?

Quantum Mechanics works brilliantly for fixed numbers of non-relativistic particles. But what happens at high energies where E = mc²?

Problem 1: Special relativity allows particle-antiparticle pair creation. QM can't handle variable particle number!

Problem 2: The Schrödinger equation isn't Lorentz invariant. We need a relativistic framework that treats space and time on equal footing.

Solution: Quantum Field Theory treats particles as excitations of underlying fields. The field exists everywhere in spacetime, and particle number becomes a dynamical variable. This naturally incorporates both quantum mechanics and special relativity!

4. Common Mistakes

⚠️Common Mistakes to Avoid

❌

Mistake:

Thinking the quantum field φ(x,t) is just a wave function

🤔

Why it's wrong:

φ(x,t) is an operator-valued field, not a probability amplitude. It creates/annihilates particles at each spacetime point.

✅

Correct approach:

Treat φ̂(x,t) as a quantum operator that satisfies canonical commutation relations. The wave function describes states in Fock space, like |n,k⟩.

❌

Mistake:

Confusing real-space position with momentum-space modes

🤔

Why it's wrong:

In QFT, particles are excitations of momentum modes, not localized at definite positions.

✅

Correct approach:

Work in momentum space: particles have definite momentum k, not position x. Position is just a label for the field operator.

❌

Mistake:

Ignoring the iε prescription in propagators

🤔

Why it's wrong:

Without iε, the propagator is ambiguous at p² = m² (pole). The iε ensures correct time-ordering and causality.

✅

Correct approach:

Always include iε: D_F(p) = i/(p² - m² + iε). This shifts poles slightly off the real axis for proper contour integration.

5. QM vs. QFT Comparison

Quantum Mechanics vs. Quantum Field Theory

Understanding the fundamental differences between QM and QFT

Aspect	Quantum Mechanics	Quantum Field Theory
Fundamental Object	Wave function ψ(x,t)	Quantum field φ̂(x,t)
Number of Particles	Fixed (1, 2, ...N particles)	Variable (creation/annihilation)
Degrees of Freedom	Finite (3N for N particles)	Infinite (field at each point)
Relativity	Non-relativistic (Schrödinger)	Relativistic (Lorentz invariant)
Particle/Antiparticle	No natural antiparticles	Antiparticles emerge naturally
Spin-Statistics	Postulated	Derived (spin-statistics theorem)
Quantization	Canonical quantization of particles	Second quantization of fields

6. Computational Examples

Run this Python code directly in your browser to visualize scalar field modes and the relativistic dispersion relation \(E^2 = k^2 + m^2\) (in natural units where \(c = \hbar = 1\)).

Key Equations:

• Scalar field mode: \(\phi_k(x,t) = \frac{1}{\sqrt{2\omega_k}}\left(a_k e^{-i(\omega t - kx)} + a_k^\dagger e^{i(\omega t - kx)}\right)\)
• Dispersion relation: \(\omega_k = \sqrt{k^2 + m^2}\)
• Each mode \(k\) is a quantum harmonic oscillator with frequency \(\omega_k\)

Scalar Field Mode Expansion

Visualize quantum field modes and dispersion relations

import numpy as np
import matplotlib.pyplot as plt

# Scalar field mode expansion
def scalar_field_mode(x, t, k, m):
    """
    Calculate scalar field value for mode k at position x, time t
    φ(x,t) = (1/√(2ω)) [a_k e^{-i(ωt-kx)} + a_k† e^{i(ωt-kx)}]
    """
    omega = np.sqrt(k**2 + m**2)  # Dispersion relation
    # For visualization, set a_k = 1
    phi = (1/np.sqrt(2*omega)) * np.cos(omega*t - k*x)
    return phi

# Parameters
m = 0.5  # mass
k_values = [1, 2, 3]  # different modes
x = np.linspace(0, 4*np.pi, 200)
t = 0  # t=0 snapshot

# Plot different modes
plt.figure(figsize=(12, 6))
for k in k_values:
    phi = scalar_field_mode(x, t, k, m)
    plt.plot(x, phi, label=f'k={k}')

plt.xlabel('Position x')
plt.ylabel('φ(x,t=0)')
plt.title('Scalar Field Modes')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()

# Energy-momentum dispersion
k_range = np.linspace(0, 5, 100)
E = np.sqrt(k_range**2 + m**2)

plt.figure(figsize=(8, 6))
plt.plot(k_range, E, linewidth=2)
plt.xlabel('Momentum k')
plt.ylabel('Energy E')
plt.title(f'Dispersion Relation: E² = k² + m² (m={m})')
plt.grid(True, alpha=0.3)
plt.show()

Click Run to execute the Python code

First run will download Python environment (~15MB)

7. Practice Problems

📝 Practice Problems

Progress: 0/3 (0%)

⭐ Easy

Show that the Klein-Gordon equation (∂_μ∂^μ + m²)φ = 0 is Lorentz invariant.

⭐⭐ Medium

Calculate the canonical momentum π(x) conjugate to the scalar field φ(x) for the Klein-Gordon Lagrangian.

ℒ = ½(∂_μφ∂^μφ - m²φ²)

⭐⭐⭐ Hard

Derive the Feynman propagator for a scalar field in momentum space.

📊 Problem Set Statistics

Total Problems

Attempted

Completion

8. Equation Reference Database

Searchable database of all major QFT equations, organized by topic.

📐QFT Equation Reference

Showing 23 equations

Euler-Lagrange Equation

Classical Field Theory

Quantum Field Theory Course

🎨 QFT Course Features Demo

🔗Course Connections

📚Prerequisites

🔄Related Topics

1. Scalar Field Visualizer

🌊Scalar Field Visualization

Field Type: klein-gordon

Complex Scalar Field (2D)

🌊Scalar Field Visualization

Field Type: complex

2. Feynman Diagram Drawer

📊Feynman Diagram Drawer

Legend:

3. Physical Intuition

💡Why Do We Need QFT?

4. Common Mistakes

⚠️Common Mistakes to Avoid

Mistake:

Why it's wrong:

Correct approach:

Mistake:

Why it's wrong:

Correct approach:

Mistake:

Why it's wrong:

Correct approach:

5. QM vs. QFT Comparison

Quantum Mechanics vs. Quantum Field Theory

6. Computational Examples

Key Equations:

Scalar Field Mode Expansion

7. Practice Problems

📝 Practice Problems

📊 Problem Set Statistics

8. Equation Reference Database

📐QFT Equation Reference

Euler-Lagrange Equation

Klein-Gordon Equation

Dirac Equation

Noether's Theorem

Energy-Momentum Tensor

Canonical Commutation Relations

Field Mode Expansion

Ladder Operator Commutators

Feynman Propagator

Propagator (Momentum Space)

Path Integral Formula

Generating Functional

Wick's Theorem

S-Matrix

LSZ Reduction Formula

Differential Cross Section

Gauge Transformation

Covariant Derivative

Yang-Mills Field Strength

QCD Lagrangian

Beta Function

Callan-Symanzik Equation

QED Beta Function

📊 Database Statistics

🎯 What's Next?