bc6f58b2c4
- Implemented `2_exact_casimir_projectors.py` to construct exact SO(3) isotypic projectors using the Casimir operator, replacing Monte-Carlo methods. - Created `3_apply_exact_projectors.py` to apply the exact projectors to example states, calculating ||A_j||_* estimates with improved precision. - Developed `4_spherical_basis.py` to build a Condon-Shortley-consistent spherical basis for a single spin-1 leg using ladder operators. - Introduced `6_six_j_recoupling_proof.py` to provide a complete proof of the cut-recoupling formula for full collective SU(2) symmetry, verifying the relationship between reduced blocks A_j^(1) and A_p^(2). - Added a README file to document the execution order and purpose of each script in the symmetric states project.
56 lines
1.9 KiB
Python
56 lines
1.9 KiB
Python
"""
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Apply the EXACT Casimir-based isotypic projectors (projectors_exact.npz)
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to the two example states, at both cuts, replacing the earlier
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Monte-Carlo-based ||A_j||_* estimates with machine-precision values.
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Uses T1, T2, C12 (saved by recoupling_check.py) so that "Example 1"
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(pure psi1) and "Example 2" (Xi at alpha) are both obtained from the
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SAME three fixed tensors, no new quantum simulation.
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"""
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import numpy as np
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T1 = np.load("T1.npy")
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T2 = np.load("T2.npy")
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C12 = np.load("C12.npy")
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overlap = 0.25 # <psi1|psi2>, real (checked earlier)
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P = np.load("projectors_exact.npz")
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def state_tensor(alpha):
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N2 = 1 + np.sin(2*alpha)*overlap
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return (np.cos(alpha)**2*T1 + np.sin(alpha)**2*T2
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+ np.cos(alpha)*np.sin(alpha)*C12) / N2
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def report(label, alpha):
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T = state_tensor(alpha)
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M_c1 = T.reshape(27,27) # cut ABC|DEF
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M_c2 = T.reshape(9,81) # cut AB|CDEF
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print(f"\n=== {label} (alpha={alpha}) ===")
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print("-- cut ABC|DEF --")
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tot = 0.0
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for j in range(4):
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Pj = P[f"P3_{j}"]
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block = Pj @ M_c1 @ Pj
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nn = np.linalg.svd(block, compute_uv=False).sum()
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Aj = nn/(2*j+1)
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tot += (2*j+1)*Aj
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print(f" j={j}: ||A_j||_* = {Aj:.6f}")
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raw_nn = np.linalg.svd(M_c1, compute_uv=False).sum()
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print(f" sum_j (2j+1)||A_j||_* = {tot:.6f} vs. ||M||_* direct = {raw_nn:.6f}")
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print("-- cut AB|CDEF --")
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tot = 0.0
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for j in range(3):
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Pj_src = P[f"P2_{j}"]
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Pj_tgt = P[f"P4_{j}"]
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block = Pj_src @ M_c2 @ Pj_tgt
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nn = np.linalg.svd(block, compute_uv=False).sum()
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Aj = nn/(2*j+1)
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tot += (2*j+1)*Aj
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print(f" j={j}: ||A_j||_* = {Aj:.6f}")
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raw_nn = np.linalg.svd(M_c2, compute_uv=False).sum()
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print(f" sum_j (2j+1)||A_j||_* = {tot:.6f} vs. ||M||_* direct = {raw_nn:.6f}")
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report("Example 1 (pure psi1, three aligned singlets)", 0.0)
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report("Example 2 (superposition)", np.pi/5)
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