TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/SEED-FIELD.txt
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000062 | 2. COMPLETE STATE, ALL FIELDS AND THE SAME TIME
000063 | 
000064 | For the actual collar length ell>0, beta=2ell, set
000065 | 
000066 |  C=(1+exp(2ell h))^-1,
000067 |  X=(2cosh(ell h))^-1,
000068 |  P_h=[[1-C,X],[X,C]].                                (3)
000069 | 
000070 | This is the full graph projection, including both crossings. If it is the
000071 | actual quantum covariance of the given Clifford source, CAR and positivity
000072 | fix its state and every word. Quasifreeness is then not an independent fit.
000073 | 
000074 | In the sea-relative excitation representation of THIS h let
000075 | 
000076 |  H=sum_j |lambda_j| n_j^exc,
000077 |  Z_beta=product_j(1+exp(-beta |lambda_j|)),
000078 |  rho=exp(-beta H)/Z_beta.                            (4)
000079 | 
000080 | Smooth bounded Phi on the circle gives Tr exp(-beta |h|)<infinity, so these
000081 | formulas define the normal free source for every beta>0. Finite zero modes
000082 | give finite multiplicity and occupation 1/2 in (3); they cause no divergent
000083 | sea subtraction. The infinite-dimensional H is defined by (4), not by a
000084 | difference of two divergent absolute energies.
000085 | 
000086 | The a in the following field map is the OLD signed charged field, expressed
000087 | in these particle/hole excitation coordinates. Explicitly, with spectral
000088 | Q_+=1_[0,infinity)(h), Q_-=1_(-infinity,0)(h),
000089 |  a_old(f)=b(Q_+ f)+d^*(overline(Q_- f)).
000090 | These Q signs are spectral signs of h, not the grading signs in E. Thus
000091 | occupation is C(h) and creators transform by exp(it h). Replacing a_old by
000092 | a bare excitation annihilator would incorrectly give C(|h|) and the wrong
000093 | time on negative h modes. The a below always denotes a_old.
000094 | 
000095 | On HS(K) put Omega=sqrt(rho), L_A Y=A Y, R_A Y=Y A and
000096 | Gamma Y=Pi Y Pi, with Pi the genuine excitation parity of the whole source.
000097 | The actual anticommuting field families are
000098 | 
000099 |  a_R=L_a, a_L=-Gamma R_a, a_L^*=Gamma R_(a^*).
000100 | 
000101 | The native complex-linear selfdual field map is exactly
000102 | 
000103 |  iota[c(T_faces(v0,v1,bar(w0),bar(w1)))]
000104 |    =a_R^*(v0)-a_L^*(v1)+a_R(w0)-a_L(w1).             (5)
000105 | 
000106 | Creators are linear in their tests; annihilators are antilinear. This map
000107 | has covariance (3) on all bounded CAR words. The GNS map from the graph
000108 | source to (5) is onto the existing thermal HS representation; it adds no
000109 | independent source copy or new physical species.
000110 | 
000111 | The given geometric group and this source group obey
000112 | 
000113 |  iota[c(exp(t Ical B)u)]=U_t iota[c(u)] U_t^*,
000114 |  U_t Y=exp(itH)Y exp(-itH),                            (6)
000115 | 
000116 | with the SAME t. In particular both creator faces transform by exp(it h)
000117 | and the conjugate slots by exp(-it bar(h)). The HS generator L_H-R_H is
000118 | distinct from positive H; its energy differences have both signs. Neither
000119 | operator is simply the classical Dirac D. Equation (6) is already proved
000120 | inside the actual graph source, not a new conditional time reconstruction.