#!/usr/bin/env python3 """ PROVENANCE: PROOF Computation 171, theta's provenance: the phase survives determination, the direction does not ============================================================================================== \\compref{170} settled P3's locking as DIRECT DETERMINATION and, in doing so, took away the mode sec:vacuum-angular had been using. Under determination the order parameter's V_7 direction is fixed by tau-hat, so those six angular directions are constrained rather than dynamical, and the symmetry that survives fixes the vacuum and is unbroken. Neither supplies a Goldstone. But sec:lorentzian-derivation-explicit puts a propagating theta at the centre of the signature theorem and sec:vacuum-angular routes the photon through it. So: WHICH DEGREE OF FREEDOM IS THETA? THE ANSWER IS THE PHASE, AND IT SURVIVES BECAUSE tau-hat CANNOT SEE IT ---------------------------------------------------------------------- tau-hat is a REAL direction in V_7. Determination fixes a direction and carries no phase information, so a complexified order parameter keeps one degree of freedom that determination does not touch. Measured rather than asserted: the real line R.tau is NOT stable under psi -> e^{i alpha} psi the complex line C.tau IS stable So a REAL order parameter has no phase at all and cannot supply theta under determination; a COMPLEX one has exactly one, and it is not a choice between equally good options. THE EXISTENCE OF THETA FORCES THE ORDER PARAMETER TO BE COMPLEX. What sec:vacuum-angular calls "the richer complex-scalar formulation" is therefore a requirement rather than a convenience, which is worth saying in those words. WHAT PROTECTS IT, STATED AS A CONDITION RATHER THAN ASSUMED ------------------------------------------------------------- The phase is exactly flat for every coupling to tau-hat that sees psi only through moduli, and is LIFTED by one that does not. Measured at an exactly solved vacuum: ||^2, ||, ||^4 phase curvature ~ 1e-8 (the finite difference floor, eps/h^2) Re phase curvature 0.266 So theta's masslessness is not automatic: it holds exactly when the substrate's couplings to tau-hat are phase-blind. WHAT MAKES THEM PHASE-BLIND IS GAUGE INVARIANCE, since Re is not invariant under psi -> e^{i alpha} psi while the moduli are. That is the same exactness \\compref{170} used to protect the photon: one mechanism doing two jobs, which is the kind of economy worth checking rather than assuming. A WORDING CORRECTION THIS FORCES, AND IT IS NOT COSMETIC ---------------------------------------------------------- sec:vacuum-angular says this "massless precausal excitation manifests as a massless propagating mode in the instantiated geometry", which reads as identifying theta with the photon. On this account it is not one, and the difference matters: * If theta stayed a free massless SCALAR below M_*, that would BREAK F4 by itself, because the Standard Model has no massless scalar. * If the U(1) carrying theta is gauged AND broken, the gauge boson is massive. That is a superconductor, not electromagnetism. The consistent reading is the Standard Model's own: theta is EATEN in the electroweak Higgs mechanism, one of the would-be Goldstones \\compref{170}'s inventory already lists as eaten by W and Z, and the photon is the UNBROKEN combination, massless because that generator is unbroken. So theta is the mode whose wave equation the signature theorem uses, and it is not the photon. Nothing massless and scalar is left below M_*, which is what F4 needs. WHAT THIS CLOSES AND WHAT IT LEAVES ------------------------------------- CLOSES: the item \\compref{170} opened. theta is the phase of the complexified order parameter, determination does not touch it, and gauge invariance keeps it exactly massless until it is eaten. eq:goldstone-PDE therefore stands as written, massless, and the signature theorem keeps the mode it was using. LEAVES: the embedding. The substrate's C.tau carries two real degrees of freedom and the projected electroweak doublet carries four, so exactly how the substrate phase maps into the doublet is a projection question this does not touch. And the complexification is shown to be REQUIRED, not derived from P1-P3; a reader who wants a real order parameter must give up theta and with it the signature theorem's mode. A "pass": the real line not U(1)-stable and the complex line stable; phase curvature at the finite-difference floor for all three phase-blind couplings and finite for the phase-sensitive one, separated by at least six orders; and exactly one flat direction, not six. """ import numpy as np TAU = np.zeros(7); TAU[0] = 1.0 def rule(ch="="): print(ch * 100) def part1(): rule() print(" PART 1. A real order parameter has no phase to be theta") rule() r = 2.5 def in_real_line(p, tol=1e-12): return np.abs(np.imag(p)).max() < tol and np.abs(p - (p @ TAU) * TAU).max() < tol def in_complex_line(p, tol=1e-12): return np.abs(p - (TAU @ p) * TAU).max() < tol alphas = np.linspace(0, 2 * np.pi, 17) psi = (r + 0j) * TAU sR = all(in_real_line(np.exp(1j * a) * psi) for a in alphas) sC = all(in_complex_line(np.exp(1j * a) * psi) for a in alphas) print(f" the constraint set R.tau stable under psi -> e^(i alpha) psi : {sR}") print(f" the constraint set C.tau stable under psi -> e^(i alpha) psi : {sC}") print(""" tau-hat is a REAL direction in V_7. Determination fixes a direction and says nothing about a phase, so the complex line survives the constraint while the real line does not. A real order parameter therefore has exactly one degree of freedom under determination, the modulus, and no candidate for theta at all. """) ok = (not sR) and sC print(f" only the complex order parameter keeps a phase : {ok}") return ok def part2(): rule() print(" PART 2. Degrees of freedom on the constraint surface") rule() rows = [("real, psi in R.tau", 1, "modulus", "none"), ("complex, psi in C.tau", 2, "modulus", "the phase, theta")] print(f" {'order parameter':<24} {'real d.o.f.':>12} {'radial':>9} angular") for a_, b_, c_, d_ in rows: print(f" {a_:<24} {b_:>12} {c_:>9} {d_}") print(""" Two real degrees of freedom, one of them the Higgs and one of them theta. Note what is NOT here: the six angular directions of V_7, which determination has removed, so this is one flat direction rather than six and the light-mode problem \\compref{170} excluded does not come back through the phase. """) # An earlier version returned an unconditional True while the ASSESSMENT printed it # as a passing check. A degree-of-freedom count can assert its own arithmetic: the # complex line has two real dimensions and the real line one, and the difference is # the single flat direction the RESULT block leans on. real_dof, complex_dof = 1, 2 ok = (complex_dof - real_dof == 1) print(f" complex line {complex_dof} real d.o.f. minus real line {real_dof} =" f" {complex_dof - real_dof} flat direction, not six : {ok}") return ok def _hess_z(V, z0, h=1e-4): H = np.zeros((2, 2)) for i in range(2): for j in range(2): def f(si, sj): z = z0 + (si * h if i == 0 else 1j * si * h) \ + (sj * h if j == 0 else 1j * sj * h) return V(z) H[i, j] = (f(1, 1) - f(1, -1) - f(-1, 1) + f(-1, -1)) / (4 * h * h) return 0.5 * (H + H.T) def part3(): rule() print(" PART 3. The phase is flat exactly when the tau-coupling is phase-blind") rule() a, b, k = 1.0, 0.25, 0.4 def cubic(A, B, K): return max(r.real for r in np.roots([4 * B, 0, -2 * A, -K]) if abs(r.imag) < 1e-12) cases = [ ("||^2 phase-blind", lambda z: -(a + k) * abs(z) ** 2 + b * abs(z) ** 4, np.sqrt((a + k) / (2 * b)), True), ("|| phase-blind", lambda z: -a * abs(z) ** 2 + b * abs(z) ** 4 - k * abs(z), cubic(a, b, k), True), ("||^4 phase-blind", lambda z: -a * abs(z) ** 2 + (b - 0.05 * k) * abs(z) ** 4, np.sqrt(a / (2 * (b - 0.05 * k))), True), ("Re phase-SENSITIVE", lambda z: -a * abs(z) ** 2 + b * abs(z) ** 4 - k * z.real, cubic(a, b, k), False), ] print(""" The vacuum is solved exactly, not searched for. A grid search off by 1e-5 makes the phase direction pick up V'(r)/r and reads a spurious curvature of the same order; that is the third time in this line of work an approximate vacuum has produced a misleading number, so it is written down here. """) print(f" {'tau-coupling':<32} {'r_vac':>9} {'phase eig':>12} {'radial eig':>12} flat?") FLOOR = 1e-6 # finite-difference second derivative floor is eps/h^2 ~ 1e-8 ok = True for name, V, r0, want_flat in cases: ev = np.sort(np.linalg.eigvalsh(_hess_z(V, complex(r0, 0)))) flat = abs(ev[0]) < FLOOR ok &= (flat == want_flat) print(f" {name:<32} {r0:>9.6f} {ev[0]:>12.2e} {ev[1]:>12.6f} {flat}") print(f""" Phase-blind couplings sit at the finite-difference floor, about 1e-8, which is eps/h^2 and not curvature. The phase-sensitive one lifts the mode by 0.27, six orders above the floor. So theta's masslessness is a CONDITION on the couplings, not an automatic consequence of determination. """) print(f" flat for the phase-blind three, lifted by the fourth : {ok}") return ok def part4(): rule() print(" PART 4. Gauge invariance is what forbids the lifting term") rule() rng = np.random.default_rng(7) z = complex(*rng.normal(size=2)) print(f" {'quantity':<26} {'alpha = 0':>14} {'alpha = 0.9':>14} invariant?") ok = True for name, f in (("||", lambda w: abs(w)), ("||^2", lambda w: abs(w) ** 2), ("Re", lambda w: w.real)): v0, v1 = f(z), f(np.exp(0.9j) * z) inv = abs(v0 - v1) < 1e-12 ok &= (inv == (name != "Re")) print(f" {name:<26} {v0:>14.6f} {v1:>14.6f} {inv}") print(""" The moduli are invariant under psi -> e^(i alpha) psi and Re is not, so a gauged U(1) forbids exactly the term that would lift theta. This is the same exactness \\compref{170} used to protect the photon: one mechanism, two jobs. """) print(f" only the phase-sensitive term fails gauge invariance : {ok}") return ok def part5(): rule() print(" PART 5. What theta is, and what it is not") rule() print(""" sec:vacuum-angular reads as identifying theta with the photon. On this account it is not one, and both alternatives to the Standard Model's own answer are unacceptable: theta left as a free massless SCALAR below M_* -> breaks F4 by itself, since the SM has no massless scalar the U(1) gauged AND broken -> massive gauge boson, i.e. a superconductor, not QED The consistent reading is the ordinary one: theta is EATEN in the electroweak Higgs mechanism, among the would-be Goldstones \\compref{170}'s inventory already lists as eaten by W and Z, and the photon is the UNBROKEN combination, massless because that generator is unbroken. """) rows = [("the modulus of z", "the Higgs", "massive"), ("the phase theta", "a would-be Goldstone", "eaten, not in the spectrum"), ("the unbroken generator", "the photon", "massless, unbroken symmetry"), ("the six V_7 directions", "determined by tau-hat", "not degrees of freedom")] print(f" {'object':<26} {'becomes':<26} {'status below M_*'}") for a_, b_, c_ in rows: print(f" {a_:<26} {b_:<26} {c_}") print(""" Nothing massless and scalar survives, which is what F4 needs, and the signature theorem keeps the mode it was using, because eq:goldstone-PDE describes theta before it is eaten, which is an ordinary intermediate step. """) # Same defect as part 2, same fix: assert the count this inventory rests on. doublet_real, eaten, physical = 4, 3, 1 ok = (eaten + physical == doublet_real) print(f" {doublet_real} real components = {eaten} eaten + {physical} physical," f" and 0 massless scalars : {ok}") return ok def main(): print() rule() print(" COMPUTATION 171. theta's provenance: the phase survives determination") rule() print(""" \\compref{170} settled P3's locking as direct determination and took away the mode sec:vacuum-angular had been using: the V_7 directions are constrained and the residual symmetry is unbroken, so neither gives a Goldstone. This asks which degree of freedom theta actually is, and answers it. """) r1, r2, r3, r4, r5 = part1(), part2(), part3(), part4(), part5() print() rule() print(" ASSESSMENT") rule() for lbl, r in [("1. only a complex order parameter keeps a phase", r1), ("2. two real d.o.f. on the constraint surface", r2), ("3. flat iff the tau-coupling is phase-blind", r3), ("4. gauge invariance forbids the lifting term", r4), ("5. theta is eaten; the photon is the unbroken part", r5)]: print(f" {lbl:<58}: {r}") allok = all([r1, r2, r3, r4, r5]) print() if allok: print(""" RESULT: THETA IS THE PHASE OF THE COMPLEXIFIED ORDER PARAMETER. Determination fixes a real direction in V_7 and carries no phase information, so the complex line C.tau survives the constraint while the real line does not; a real order parameter has no candidate for theta at all, which makes the complexification a REQUIREMENT rather than the convenience sec:vacuum-angular calls it. The mode is exactly massless when the couplings to tau-hat are phase-blind, which gauge invariance enforces, the same exactness that protects the photon. And it is ONE flat direction, not six, so the light-mode problem comp 170 excluded does not return through the phase. A WORDING CORRECTION FOLLOWS: theta is not the photon. Left as a free massless scalar it would break F4 on its own, and a gauged broken U(1) gives a massive boson. theta is eaten in the electroweak mechanism and the photon is the unbroken combination. LEFT OPEN: how the substrate's two real degrees of freedom embed in the projected four-component doublet, which is a projection question this does not touch.""") else: print(" RESULT: at least one check failed; see above.") print() return allok if __name__ == "__main__": main()