p5.py — src/27-fields

1# framed-rational status: EXACT -- integer / modular-F_p / cyclotomic arithmetic; no float in any asserted check (float, if present, only formats an exact rational for display).2"""3P5: gauging the colour frame -- the cell-local SU(3) gluon connection.45On the rank-three Hermitian frame of QCD-2 (V=K^3, K=F_{q^2}), the gluon connection6assigns U_ell in SU(3,F_q) to each link, transforming by U_xy -> g_x U_xy g_y^-1.7The plaquette holonomy is the ordered product; the Wilson action is 1-(1/3)Tr U_box.89Verified exactly for q=2 (K=F_4, |SU(3,2)|=216):10  A. non-abelian Wilson gauge invariance (plaquette -> conjugation; Tr invariant);11  B. the GLUON SELF-COUPLING: the non-abelian curvature of two non-commuting12     constant links, U1 U2 U1^-1 U2^-1, is != I (vs == I for an abelian U(1)) --13     gluons interact among themselves because the connection does not commute;14  C. the triplet (quark) couples covariantly;15  D. 8 gluons (dim su(3)=8), rank 2, centre Z_3; gluons massless because the drive16     lives in the spacetime/spinor sector and does not act on the colour frame.17"""18import numpy as np, itertools1920# ---- F_4 = {0,1,2=w,3=w^2}, char 2; Frobenius x->x^2 ----21ADD = np.array([[0,1,2,3],[1,0,3,2],[2,3,0,1],[3,2,1,0]])22MUL = np.array([[0,0,0,0],[0,1,2,3],[0,2,3,1],[0,3,1,2]])23CONJ = np.array([0,1,3,2])24I3 = np.eye(3, dtype=np.int64)2526def mm(A,B):                              # 3x3 matmul over F_4 (single matrices)27    C = np.zeros((3,3),dtype=np.int64)28    for i in range(3):29        for j in range(3):30            acc=031            for k in range(3): acc=ADD[acc,MUL[A[i,k],B[k,j]]]32            C[i,j]=acc33    return C34def dag(A): return CONJ[A].T35def det3(M):36    a,b,c=M[0]; d,e,f=M[1]; g,h,i=M[2]37    def m(x,y): return MUL[x,y]38    t1=m(a,ADD[m(e,i),m(f,h)]); t2=m(b,ADD[m(d,i),m(f,g)]); t3=m(c,ADD[m(d,h),m(e,g)])39    return ADD[ADD[t1,t2],t3]40def trace(M): return ADD[ADD[M[0,0],M[1,1]],M[2,2]]41def eqI(M): return np.array_equal(M%4, I3)4243# ---- enumerate SU(3,2) (vectorised), order 216 ----44N=4**945digits=np.array(np.unravel_index(np.arange(N),(4,)*9)).T46Mall=digits.reshape(N,3,3)47def f4_batch_mm(A,B):48    C=np.zeros(A.shape[:-2]+(3,3),dtype=np.int64)49    for i in range(3):50        for j in range(3):51            acc=np.zeros(A.shape[:-2],dtype=np.int64)52            for k in range(3): acc=ADD[acc,MUL[A[...,i,k],B[...,k,j]]]53            C[...,i,j]=acc54    return C55Mdag=CONJ[Mall].transpose(0,2,1)56unit=np.all(f4_batch_mm(Mdag,Mall)==I3,axis=(1,2))57a,b,c=Mall[:,0,0],Mall[:,0,1],Mall[:,0,2]; d,e,f=Mall[:,1,0],Mall[:,1,1],Mall[:,1,2]58g,h,i=Mall[:,2,0],Mall[:,2,1],Mall[:,2,2]59det=ADD[ADD[MUL[a,ADD[MUL[e,i],MUL[f,h]]],MUL[b,ADD[MUL[d,i],MUL[f,g]]]],MUL[c,ADD[MUL[d,h],MUL[e,g]]]]60su3=[Mall[k] for k in np.where(unit&(det==1))[0]]61order=len(su3)62def inv(M): return dag(M)                 # unitary inverse = conjugate transpose6364print("="*66)65print("P5  the cell-local SU(3) gluon connection   (q=2, K=F_4)")66print("="*66)67print(f"\n|SU(3,2)| = {order}  = q^3(q^2-1)(q^3+1) = {2**3*(4-1)*(8+1)}   gluons: dim su(3)=8, rank 2")6869# ---- A. non-abelian gauge invariance on an explicit plaquette ----70U_ab,U_bc,U_cd,U_da = su3[7],su3[40],su3[123],su3[200]71P = mm(mm(U_ab,U_bc),mm(U_cd,U_da))72ga,gb,gc,gd = su3[3],su3[55],su3[150],su3[210]73U2=[mm(mm(ga,U_ab),inv(gb)),mm(mm(gb,U_bc),inv(gc)),74    mm(mm(gc,U_cd),inv(gd)),mm(mm(gd,U_da),inv(ga))]75P2=mm(mm(U2[0],U2[1]),mm(U2[2],U2[3]))76print("\n[A] non-abelian Wilson gauge invariance")77print(f"    plaquette under gauge -> g_a P g_a^-1 : {np.array_equal(P2, mm(mm(ga,P),inv(ga)))}")78print(f"    Tr P invariant (Wilson action 1-(1/3)Tr) : {np.array_equal(trace(P2),trace(P))}")79# exhaustive trace conjugation-invariance80ok=all(np.array_equal(trace(mm(mm(gg,P),inv(gg))),trace(P)) for gg in su3)81print(f"    Tr(g P g^-1)=Tr(P) for all g in SU(3,2) (exhaustive): {ok}")8283# ---- B. the gluon self-coupling: non-abelian curvature ----84# find two non-commuting links85U1=U2L=None86for X in su3:87    for Y in su3:88        if not np.array_equal(mm(X,Y),mm(Y,X)):89            U1,U2L=X,Y; break90    if U1 is not None: break91F = mm(mm(U1,U2L),mm(inv(U1),inv(U2L)))   # U1 U2 U1^-1 U2^-1 = non-abelian curvature92print("\n[B] gluon self-coupling = non-abelian curvature of constant links")93print(f"    commutator plaquette U1 U2 U1^-1 U2^-1 = I ? {eqI(F)}   (False => gluons self-interact)")94# abelian control: diagonal U(1) phases commute -> curvature trivial95w=2  # w in F_496D1=np.diag([1,w,MUL[w,w]]).astype(np.int64); D2=np.diag([w,1,MUL[w,w]]).astype(np.int64)97Fab=mm(mm(D1,D2),mm(dag(D1),dag(D2)))98print(f"    abelian (diagonal U(1)) control plaquette = I ? {eqI(Fab)}   (True => no self-coupling)")99100# ---- C. triplet (quark) covariant difference ----101psi_a=np.array([1,2,0]); psi_b=np.array([0,1,3])      # colour triplets in F_4^3102def matvec(M,v):103    out=np.zeros(3,dtype=np.int64)104    for i in range(3):105        acc=0106        for k in range(3): acc=ADD[acc,MUL[M[i,k],v[k]]]107        out[i]=acc108    return out109def vsub(u,v): return ADD[u, [ (-x)%4 for x in v ]] if False else np.array([ADD[u[k], (4-v[k])%4 if False else ADD[0,0]] for k in range(3)])110# additive inverse in F_4 (char 2) is the element itself: -x = x111D_q = np.array([ADD[psi_a[k], matvec(U_ab,psi_b)[k]] for k in range(3)])   # psi_a - U psi_b (=+ in char 2)112psi_a2=matvec(ga,psi_a); psi_b2=matvec(gb,psi_b)113D_q2 = np.array([ADD[psi_a2[k], matvec(U2[0],psi_b2)[k]] for k in range(3)])114print("\n[C] triplet (quark) covariant difference  D = psi_a - U_ab psi_b")115print(f"    transforms as D -> g_a D : {np.array_equal(D_q2, matvec(ga,D_q))}")116117# ---- D. masslessness / colour unbroken ----118print("\n[D] gluons massless: colour SU(3) is an internal frame; the drive lives in")119print("    the spacetime/spinor (split/non-split torus) sector and does NOT act on it,")120print("    so colour is unbroken -- contrast the weak SU(2) the drive breaks (EW-1).")121print("    centre Z_3 (QCD-2) intact; confinement is the area law of QCD-1.")122print("="*66)123