TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/ORIGINAL-STATE-MAP.txt
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000060 | review:1–46, together with the ORIGINAL stress/current map:33–115 and
000061 | source-quarter relation:166–180. The original current identity is
000062 |  Lcal_n^alpha=L_n^0+b J_n+(b^2/2)delta_n0, b=alpha-1/2.
000063 | The two quarter stress families are compared on the SAME CAR generators
000064 | by the actual state-preserving cover arrow; their difference reconstructs
000065 | the current. The other cover arrow crosses the sea and is not silently
000066 | declared vacuum preserving. The actual local support and unitary product
000067 | formula identify the local paired-stress algebra with A_J(I).
000068 | 
000069 | On the original neutral cyclic space
000070 |  H0=closure span{J_-n1 ... J_-nk Omega}, Q=0,
000071 | the existing circle excitation is H_alpha=L0^0. The same original Omega
000072 | is annihilated by L0^0,L1^0,L-1^0. Within the displayed affine improvement
000073 | class, these vacuum conditions force the subtraction b J_n+b^2 delta_n0/2;
000074 | they are not fitted to a target source.
000075 | 
000076 | The derived operators are
000077 |  P=L0^0-(L1^0+L-1^0)/2, D=(L1^0-L-1^0)/(2i).
000078 | The full normalized Laguerre basis takes its original one-current P to
000079 | multiplication x/2 in L2(R+,x exp(-x)dx). Completeness and the Carleman
000080 | core give its selfadjoint closure, spectrum[0,infinity), and zero kernel.
000081 | Second quantization on THIS already identified neutral current Fock space
000082 | gives P>=0 and ker P=C Omega. Its explicit unitary dilation is
000083 |  V_t f(x)=r exp((1-r)x/2)f(rx), r=exp(-t),
000084 | and V_t P V_t*=exp(-t)P. Direct differentiation and change of variables
000085 | verify the generator, unitarity and complete group law without a finite
000086 | matrix limit guess.
000087 | 
000088 | For M=A_J(0,infinity), N=A_J(a,infinity), a>0, the actual vacuum current
000089 | net supplies a standard half-sided inclusion. Its modular action is
000090 |  Delta_M^(it)=exp(i2pi tD),
000091 |  Delta_M^(it)exp(isP)Delta_M^(-it)=exp(i exp(-2pi t)sP).
000092 | The inclusion side is t<=0 for the source's N, and
000093 |  log Delta_N-log Delta_M=2pi aP on the actual common core/closure.
000094 | All signs agree with the explicit dilation. This is a genuine local net
000095 | and standard vector statement, not just a commutator of finite matrices.
000096 | 
000097 | The actual original quarter on Q0 is R=exp(-i pi L0^0/2). It carries the
000098 | four parabolic generators AND their moved cuts/algebras/modular groups.
000099 | It does not claim four half-sided preservers of one fixed cut. Opposite
000100 | parabolic averages recover the original neutral circle energy; P is not
000101 | set equal to that gapped circle energy.
000102 | 
000103 | PRIMARY APPLICATION CHECK: Tanimoto1807.11723v2 sections3.1–3.2 provides
000104 | the current net, essential selfadjointness and Sugawara representation.
000105 | GLW hep-th/9703129v2 section1, equations3–4 and Lemma1.3 supplies the
000106 | conformal local-net BW/standardity result. These primary passages were
000107 | opened in this review. Their hypotheses fit the specifically identified
000108 | Q0 current vacuum net, not an arbitrary interacting4D source.
000109 |  https://arxiv.org/pdf/1807.11723
000110 |  https://arxiv.org/pdf/hep-th/9703129
000111 |