Free Field Operator: Building Quantum Fields How Quantum Fields Evolve Without Interactions ๐ฏ Our Goal We aim to construct the free scalar field operator \( A(x,t) \), which describes a quantum field with no interactions—just free particles moving across space-time. ๐ง Starting Expression This is the mathematical formula for our field \( A(x,t) \): \[ A(x, t) = \frac{1}{(2\pi)^{3/2}} \int_{\mathbb{R}^3} \frac{1}{\sqrt{k_0}} \left[ e^{i(k \cdot x - k_0 t)} a(k) + e^{-i(k \cdot x - k_0 t)} a^\dagger(k) \right] \, dk \] x: Spatial position t: Time k: Momentum vector k₀ = √(k² + m²): Relativistic energy of the particle a(k): Operator that removes a particle (annihilation) a†(k): Operator that adds a particle (creation) ๐งฉ What Does This Mean? The field is made up of wave patterns (Fourier modes) linked to momentum \( k \). It behaves like a system that decides when and where ...
Understanding Delta Function Approximations: Gaussian Delta Sequence (Heat Kernel)
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Delta-Convergent Sequences — Refined Blog with SageMath Symbolics, Physics Insights, and Cleaner Code
In the previous blog, we understood the Lorentzian Delta Sequence (Cauchy Kernel). Let's take another step and explore the Gaussian Delta Sequence (Heat Kernel).
Why Study These Approximations?
Delta functions are central in many fields:
- Signal Processing: Ideal impulse, filter response
- Physics: Point charges/masses, Green's functions
- Spectral Theory: Lorentzian profiles in resonance
- Diffusion Models: Gaussians arise from the heat equation
- Numerics: Regularizing singular integrals
Each kernel has a story to tell.
Gaussian Delta Sequence (Heat Kernel)
Formula \[ f_t(x) = \frac{1}{2\sqrt{\pi t}} .e^{-\frac{x^2}{4t}} \]
- Smooth and fast-decaying
- Bell-shaped
- Natural from the heat equation
#Define the Function
var('x t')
f_gauss(x, t) = (1/(2*sqrt(pi*t))) * exp(-x^2 / (4*t))
f_gauss(x, t)
# Check Symbolic Integration
var('xi')
assume(t > 0) # Ensure t is positive
integral(f_gauss(xi, t), xi, -oo, oo).simplify_full()
#Limit Evaluation at x → 0
limit(f_gauss(0, t), t=0)
#Alternative Approach: Use Numerical Evaluation
#If the symbolic engine struggles, try evaluating the function numerically at progressively smaller values of ( t ):
t_values = [0.1, 0.01, 0.001, 0.0001]
[f_gauss(0, t).n() for t in t_values]
#Integral Test (Distributional Behavior)
var('a b')
assume(a < 0, b > 0) # Ensure a < 0 < b to match delta behavior
integral(f_gauss(xi, t), xi, a, b).simplify_full()
# Numerical Evaluation
# To see how the integral behaves for small ( t ):
import numpy as np
import matplotlib.pyplot as plt
import sage.all as sage
def gaussian_integral(t, a=-1, b=1):
from math import erf, sqrt, pi
return (1/2) * (erf(b / sqrt(4*t)) - erf(a / sqrt(4*t)))
# Test for different t values
t_values = np.logspace(-3, 0, 50) # Log-spaced values from 0.001 to 1
integral_values = [gaussian_integral(t) for t in t_values]
# Plotting
plt.figure(figsize=(8, 5))
plt.plot(t_values, integral_values, marker='o', linestyle='-', color='blue')
plt.axhline(y=1, color='r', linestyle='--', label="Expected Limit (1)")
plt.xscale("log")
plt.xlabel(r"$t$")
plt.ylabel(r"Integral Value")
plt.title("Numerical Verification: Gaussian Integral Convergence")
plt.legend()
plt.grid(True)
plt.show()
# Plot the Gaussian Sequence
p1 = plot(f_gauss(x, 0.5), (x, -5, 5), color='red', legend_label='t = 0.5') + \
plot(f_gauss(x, 0.2), (x, -5, 5), color='blue', legend_label='t = 0.2') + \
plot(f_gauss(x, 0.05), (x, -5, 5), color='green', legend_label='t = 0.05')
p1.show(title='Gaussian Approximation to ฮด(x)', ymin=0, ymax=3)
# Compute First and Second Derivatives
var('x t')
f_gauss(x, t) = (1/(2*sqrt(pi*t))) * exp(-x^2 / (4*t))
# First derivative (approximating ฮด'(x))
f_gauss_prime(x, t) = diff(f_gauss(x, t), x)
# Second derivative (approximating ฮด''(x))
f_gauss_double_prime(x, t) = diff(f_gauss_prime(x, t), x)
f_gauss_prime(x, t), f_gauss_double_prime(x, t)
#Plot the Derivatives
p1 = plot(f_gauss_prime(x, 0.5), (x, -5, 5), color='red', legend_label="t=0.5") + \
plot(f_gauss_prime(x, 0.2), (x, -5, 5), color='blue', legend_label="t=0.2") + \
plot(f_gauss_prime(x, 0.05), (x, -5, 5), color='green', legend_label="t=0.05")
p1.show(title="First Derivative of Gaussian Delta Approximation")
p2 = plot(f_gauss_double_prime(x, 0.5), (x, -5, 5), color='red', legend_label="t=0.5") + \
plot(f_gauss_double_prime(x, 0.2), (x, -5, 5), color='blue', legend_label="t=0.2") + \
plot(f_gauss_double_prime(x, 0.05), (x, -5, 5), color='green', legend_label="t=0.05")
p2.show(title="Second Derivative of Gaussian Delta Approximation")
#Plotting the Integrated Sequences
p1=plot(integral(f_gauss(x, 0.1), x, -5, 5), (x, -5, 5), color='green', legend_label="Gaussian")
p1.show(title="Integrated Delta Approximations")
๐ก Try It Yourself! Now You can copy and paste directly into here Run SageMath Code Here
Physics Note
Appears in diffusion, heat kernels, quantum mechanics (e.g., wavepacket spreading).
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