Investigate why two mathematically similar Fourier transform functions might have different causal properties despite being identical on the real line | Step-by-Step Solution

Hello! This is a fascinating problem that gets to the heart of why complex analysis is so powerful in physics. Let's explore this together!

1. What We're Solving:

We're investigating why two Green's functions G(ω) and H(ω) can be identical when we measure them on the real frequency axis, yet have completely different causal properties. This seeming paradox reveals deep connections between complex analysis and causality in physics.

2. The Approach:

Think of this like having two different movies that show identical scenes when viewed through a narrow window, but tell completely different stories when you see the full picture! We'll use the analytic properties of functions in the complex plane to understand causality, because:

• The real axis is just one "slice" of the complex plane
• Causal properties are encoded in how functions behave in the complex plane
• The Fourier transform connects time-domain causality to frequency-domain analyticity

3. Step-by-Step Solution:

Step 1: Set up our example functions Let's consider two specific Green's functions:

• G(ω) = 1/(ω - ω₀ + iγ) where γ > 0 (causal)
• H(ω) = 1/(ω - ω₀ - iγ) where γ > 0 (non-causal)

Step 2: Examine their behavior on the real axis When ω is real, both functions give:

• |G(ω)|² = |H(ω)|² = 1/[(ω - ω₀)² + γ²]
• They have identical magnitudes and similar phase relationships

Step 3: Analyze their poles in the complex plane Here's where the magic happens:

• G(ω) has a pole at ω = ω₀ - iγ (lower half-plane)
• H(ω) has a pole at ω = ω₀ + iγ (upper half-plane)

Step 4: Apply the causality criterion For a causal response (effect doesn't precede cause):

• Poles must be in the lower half of the complex plane
• This ensures the inverse Fourier transform gives zero for t < 0

Step 5: Compute the time-domain behavior Using contour integration and the residue theorem:

• G(t) ∝ e^(-iω₀t - γt) for t > 0, and 0 for t < 0 (causal!)
• H(t) ∝ e^(-iω₀t + γt) for t < 0, and 0 for t > 0 (non-causal!)

Step 6: Connect to physical meaning

• G(ω) represents a system that responds after being excited
• H(ω) represents a system that somehow "knows" about future excitations

4. The Key Insight:

Two functions can be identical on the real line but have completely different causal properties because causality is determined by the analytic structure in the complex plane, not just the real-axis behavior. The location of poles and branch cuts in the complex plane encodes the time-ordering information that we lose when we only look at real frequencies.

This is why:

• Causal functions must be analytic in the upper half-plane
• The real-axis values alone don't contain complete information
• We need the full complex structure to understand physical causality

5. Memory Tip:

Remember "Poles below, causality flows!" Causal Green's functions have their poles in the lower half of the complex plane, ensuring that cause precedes effect. Think of the poles as "gravitating" toward the past (negative imaginary axis) for causal systems.

This problem beautifully illustrates why complex analysis isn't just mathematical formalism—it's the language that nature uses to encode causality! Isn't it amazing how the universe "knows" about complex numbers?