RosettaHilbert.py

import numpy as np

# Define the fundamental time quantum
tau_q = 2.203e-15

# Define basic matrices
I2 = np.eye(2, dtype=complex)
sigma_x = np.array([[0, 1], [1, 0]], dtype=complex)
sigma_y = np.array([[0, -1j], [1j, 0]], dtype=complex)
sigma_z = np.array([[1, 0], [0, -1]], dtype=complex)

# Define our base state - eigenstate of sigma_x
psi = np.array([1, 1], dtype=complex) / np.sqrt(2)

# Verify it's an eigenstate of sigma_x
is_eigenstate = np.allclose(sigma_x @ psi, psi)
print(f"Is psi an eigenstate of sigma_x? {is_eigenstate}")

# Define gauge operator with tau_q
G = tau_q * sigma_x

# Test 1-Hilbert space
print("\n=== 1-Hilbert Space Test ===")
invariant1 = psi.conj() @ (G @ psi)
print(f"1-Hilbert invariant: {invariant1}")
print(f"Expected: {tau_q}")
print(f"Ratio to tau_q: {np.abs(invariant1)/tau_q}")
print(f"Match: {np.isclose(np.abs(invariant1), tau_q)}")

# Test 2-Hilbert space
print("\n=== 2-Hilbert Space Test ===")
# Create tensor product state |ψ⊗ψ>
tensor_state = np.kron(psi, psi)

# Create tensor operator G⊗I + I⊗G
G_tensor = np.kron(G, I2) + np.kron(I2, G)

# Calculate expectation value <ψ⊗ψ|G_tensor|ψ⊗ψ>
invariant2 = tensor_state.conj() @ (G_tensor @ tensor_state)

print(f"2-Hilbert invariant: {invariant2}")
print(f"Expected: {2*tau_q}")
print(f"Ratio to 2*tau_q: {np.abs(invariant2)/(2*tau_q)}")
print(f"Match: {np.isclose(np.abs(invariant2), 2*tau_q)}")

# Calculate effective tau_q
effective_tau_q = np.abs(invariant2) / 2
print(f"Effective tau_q: {effective_tau_q}")
print(f"Match with original tau_q: {np.isclose(effective_tau_q, tau_q)}")

# Test 3-Hilbert space
print("\n=== 3-Hilbert Space Test ===")
# Create tensor product state |ψ⊗ψ⊗ψ>
tensor_state3 = np.kron(np.kron(psi, psi), psi)

# Create tensor operator G⊗I⊗I + I⊗G⊗I + I⊗I⊗G
term1 = np.kron(np.kron(G, I2), I2)
term2 = np.kron(np.kron(I2, G), I2)
term3 = np.kron(np.kron(I2, I2), G)
G_tensor3 = term1 + term2 + term3

# Calculate expectation value
invariant3 = tensor_state3.conj() @ (G_tensor3 @ tensor_state3)

print(f"3-Hilbert invariant: {invariant3}")
print(f"Expected: {3*tau_q}")
print(f"Ratio to 3*tau_q: {np.abs(invariant3)/(3*tau_q)}")
print(f"Match: {np.isclose(np.abs(invariant3), 3*tau_q)}")

# Calculate effective tau_q
effective_tau_q3 = np.abs(invariant3) / 3
print(f"Effective tau_q: {effective_tau_q3}")
print(f"Match with original tau_q: {np.isclose(effective_tau_q3, tau_q)}")

# Dimension translation tests
print("\n=== Dimension Translation Tests ===")
print(f"1-Hilbert invariant: {np.abs(invariant1)}")
print(f"2-Hilbert invariant: {np.abs(invariant2)}")
print(f"3-Hilbert invariant: {np.abs(invariant3)}")

print(f"\nRatio 2-Hilbert/1-Hilbert: {np.abs(invariant2)/np.abs(invariant1)} (expected: 2)")
print(f"Ratio 3-Hilbert/2-Hilbert: {np.abs(invariant3)/np.abs(invariant2)} (expected: 1.5)")
print(f"Ratio 3-Hilbert/1-Hilbert: {np.abs(invariant3)/np.abs(invariant1)} (expected: 3)")

# Testing dimension translation directly
print("\n=== Direct Translation Tests ===")
# 1D to 2D translation
direct_1d_to_2d = 2 * np.abs(invariant1)
print(f"1D→2D translation: {direct_1d_to_2d}")
print(f"Actual 2D value: {np.abs(invariant2)}")
print(f"Match: {np.isclose(direct_1d_to_2d, np.abs(invariant2))}")

# 2D to 3D translation
direct_2d_to_3d = 1.5 * np.abs(invariant2)
print(f"2D→3D translation: {direct_2d_to_3d}")
print(f"Actual 3D value: {np.abs(invariant3)}")
print(f"Match: {np.isclose(direct_2d_to_3d, np.abs(invariant3))}")

# 1D to 3D translation
direct_1d_to_3d = 3 * np.abs(invariant1)
print(f"1D→3D translation: {direct_1d_to_3d}")
print(f"Actual 3D value: {np.abs(invariant3)}")
print(f"Match: {np.isclose(direct_1d_to_3d, np.abs(invariant3))}")

# Demonstrate tau_q as a dimensional translator between Hilbert spaces
print("\n=== τ_q as Dimensional Translator ===")
print(f"τ_q value: {tau_q}")
print(f"1-Hilbert space invariant: {np.abs(invariant1)}")
print(f"Translation factor τ_q: {np.abs(invariant1)/tau_q}")

print(f"\nTranslation from 1→2 Hilbert spaces")
print(f"Multiply by 2: {tau_q} → {2*tau_q}")
print(f"Result: {np.abs(invariant2)}")

print(f"\nTranslation from 1→3 Hilbert spaces")
print(f"Multiply by 3: {tau_q} → {3*tau_q}")
print(f"Result: {np.abs(invariant3)}")

print(f"\nExtraction from 3→1 Hilbert spaces")
print(f"Divide by 3: {np.abs(invariant3)} → {np.abs(invariant3)/3}")
print(f"Result matches τ_q: {np.isclose(np.abs(invariant3)/3, tau_q)}")

# Mathematical relationship between quantum time and dimensionality
print("\n=== Mathematical Implications ===")
print("The results suggest that τ_q functions as a fundamental unit")
print("of translation between different dimensional Hilbert spaces.")
print("This translation is exact, with the invariant scaling precisely")
print("with the number of dimensions:")

print("\nFor n-dimensional Hilbert space, the invariant equals n·τ_q")
for n in range(1, 4):
    print(f"  {n}-Hilbert space: {n}·τ_q = {n*tau_q}")

print("\nThis creates a direct mapping between time (via τ_q) and dimension,")
print("suggesting that time quantization might be intrinsically connected")
print("to the dimensionality of physical reality.")