TFPT scientific source excerpt Original repository path: experiments/theory-contracts/paper-source-reconstruction-20261025/GROUND-STATE-SOURCE-REDUCTION.txt Original file SHA-256: 908420944efb265dbb57f5fc53e2331a416170f76a626f75e14f3445299046cb Inclusive original lines: 1-200 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | GROUND STATE SOURCE REDUCTION 000002 | ============================= 000003 | 000004 | 25 October 2026. Direct CAR proof and application to the archived operational 000005 | seed. No Atlas, ledger, index or paper source was changed. 000006 | 000007 | Verdict 000008 | ------- 000009 | 000010 | PROVED AS A REDUCTION UNDER THE STATED FULL-CAR AND TIME-IDENTIFICATION 000011 | HYPOTHESES. 000012 | 000013 | Suppose the full graded CAR algebra of a one-particle space h is represented 000014 | on the same reconstructed seed Hilbert space as the original vacuum Omega. 000015 | Suppose the original Lorentzian time restricts to the Bogoliubov dynamics 000016 | 000017 | alpha_t(a(f)) = a(exp(i t K) f) (1) 000018 | 000019 | for one fixed self-adjoint K with a gap at zero, and is implemented by 000020 | 000021 | alpha_t(A)=exp(i t H) A exp(-i t H), 000022 | H >= 0, H Omega=0. (2) 000023 | 000024 | Then the restriction of the original vacuum state to the full CAR algebra is 000025 | not a further independent input. It is the unique pure gauge-invariant 000026 | quasifree ground state with covariance 000027 | 000028 | P_- = 1_(-infinity,0)(K). (3) 000029 | 000030 | In particular, one need not separately assume that omega restricted to CAR is 000031 | quasifree. What remains additional in the TFPT source bridge is the full 000032 | graded local CAR embedding and the identification of its induced time with 000033 | (1) for the same gapped K. The archived papers do not derive that embedding 000034 | from scalar P1. 000035 | 000036 | The statement is false on the even CAR algebra alone. The odd generators are 000037 | what turn positive energy into the two annihilation conditions below. 000038 | 000039 | 1. Convention and energy-transfer signs 000040 | --------------------------------------- 000041 | 000042 | Use the mathematical CAR convention in which the Hilbert-space scalar product 000043 | is anti-linear in its first entry and a(f) is anti-linear in f: 000044 | 000045 | {a(f),a(g)^*}=, {a(f),a(g)}=0. (4) 000046 | 000047 | If Kf=epsilon f, (1) gives 000048 | 000049 | alpha_t(a(f))=a(exp(i t epsilon)f) 000050 | =exp(-i t epsilon)a(f), (5) 000051 | 000052 | where anti-linearity supplies the minus sign. Thus a(f) has energy transfer 000053 | -epsilon. Taking adjoints gives 000054 | 000055 | alpha_t(a(f)^*)=exp(+i t epsilon)a(f)^*, (6) 000056 | 000057 | so a(f)^* has energy transfer +epsilon. 000058 | 000059 | Let 000060 | 000061 | P_+=1_(0,infinity)(K), P_-=1_(-infinity,0)(K). 000062 | 000063 | Because zero is outside the spectrum, these two projections sum to one. From 000064 | (2), no bounded operator with strictly negative energy transfer can create a 000065 | nonzero vector from Omega. Equations (5)--(6) therefore give 000066 | 000067 | a(f_+) Omega=0 for f_+ in P_+ h, (7) 000068 | a(f_-)^* Omega=0 for f_- in P_- h. (8) 000069 | 000070 | For pure point vectors this is immediate: a nonzero vector on the left would 000071 | be an H-eigenvector of energy -epsilon<0. For general K, smear (1) with a 000072 | Schwartz function whose Fourier support lies inside the positive or negative 000073 | spectral half-line. The resulting bounded CAR operator has Arveson spectrum 000074 | strictly below zero. Spectral approximation inside P_+h and P_-h then yields 000075 | (7)--(8). The gap prevents a zero-energy remainder. 000076 | 000077 | The requested signs are therefore exactly the stated ones: 000078 | 000079 | positive K mode: a(f_+) annihilates Omega; 000080 | negative K mode: a(f_-)^* annihilates Omega. 000081 | 000082 | 2. Covariance and Wick rule follow from the CAR 000083 | ------------------------------------------------ 000084 | 000085 | Define the ground-adapted annihilator 000086 | 000087 | b(f)=a(P_+ f)+a(P_- f)^*. (9) 000088 | 000089 | Equations (7)--(8) say b(f)Omega=0 for every f. The original a and a^* 000090 | operators are linear combinations of b and b^*. Repeatedly move every b to 000091 | the right by the CAR. Every term with a b at the far right vanishes, and every 000092 | commutation step produces only a two-point contraction. Induction on the word 000093 | length gives: 000094 | 000095 | * all odd moments vanish; 000096 | * every even moment is the signed Wick/Pfaffian sum of its two-point 000097 | contractions. 000098 | 000099 | The two basic contractions are, in the convention (4), 000100 | 000101 | omega(a(f)^* a(g))=, (10) 000102 | omega(a(f) a(g)^*)=. (11) 000103 | 000104 | The anomalous contractions omega(aa) and omega(a^*a^*) vanish. Hence omega is 000105 | the gauge-invariant quasifree state with one-particle density P_-. Since every 000106 | CAR polynomial has now been fixed and such polynomials are norm dense in the 000107 | CAR algebra, the state is unique. 000108 | 000109 | This is the zero-temperature ground-state form of the standard quasifree CAR 000110 | and Bogoliubov-automorphism framework. Primary source: 000111 | 000112 | Huzihiro Araki, 000113 | "On Quasifree States of CAR and Bogoliubov Automorphisms", 000114 | Publ. RIMS 6 (1970/71), 385--442, 000115 | DOI 10.2977/prims/1195193913. 000116 | 000117 | Araki supplies the state-symbol and Bogoliubov framework and characterizes 000118 | the corresponding quasifree equilibrium states. The short argument above is 000119 | included so that the present ground-state reduction does not rely on silently 000120 | replacing a finite-temperature KMS statement by its zero-temperature limit. 000121 | 000122 | 3. Why the full graded algebra is essential 000123 | ------------------------------------------- 000124 | 000125 | Take one CAR mode with {a,a^*}=1 and K=kappa>0. On the full algebra M_2, 000126 | 000127 | alpha_t(a)=exp(-i kappa t)a. 000128 | 000129 | Positive energy forces a Omega=0, so the full-CAR ground state is the empty 000130 | state. 000131 | 000132 | The even subalgebra is 000133 | 000134 | CAR_even=C 1 + C n, n=a^*a, (12) 000135 | 000136 | which is isomorphic to C direct-sum C. Its restricted dynamics is trivial, 000137 | because alpha_t(n)=n. Every state 000138 | 000139 | omega_lambda(n)=lambda, 0<=lambda<=1, (13) 000140 | 000141 | is therefore a ground state of the restricted even system. In particular the 000142 | empty state, the filled state and all mixtures have the same even positive 000143 | time. The missing odd operator a is exactly what detects that the filled state 000144 | would allow a negative-energy transition. 000145 | 000146 | Consequently an embedding of only the neutral current algebra, the even CAR 000147 | algebra or the F8 composite does not imply (3). The source bridge must contain 000148 | the odd local CAR fields and both adjoints. 000149 | 000150 | 4. What positive time the archived seed actually contains 000151 | --------------------------------------------------------- 000152 | 000153 | The bare operational seed is defined in 000154 | 000155 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000156 | sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex 000157 | lines 103--143. 000158 | 000159 | It lists tau_t in 000160 | 000161 | S=(A_loc,tau_t,Theta,omega,[u_Sigma],D_coll), 000162 | 000163 | requires a quasilocal Euclidean net with positive-time reflection and requires 000164 | omega to be reflection positive and clustering. The completion proof says 000165 | that reflection positivity and the positive-time net reconstruct the 000166 | one-sided OS Hilbert space. These lines do not themselves display a 000167 | Lorentzian H with (2). 000168 | 000169 | The later weakened ambient category makes the time premise clearer: 000170 | 000171 | documents/archive--tfpt-45--source_extracts--04_qft_source/sections/ 000172 | 006-section-internal-reduction-theorem-and-renormalized-obse.tex:13--29. 000173 | 000174 | Its U4 assumption says tau_t is a quasilocal dynamics with reflection 000175 | positivity and an admissible vacuum sector. This is still an ambient 000176 | admissibility assumption, not a derivation of the CAR time law (1). 000177 | 000178 | The required positive generator is supplied by the actual reconstruction 000179 | theorem only after the full OS hypotheses are met: 000180 | 000181 | documents/archive--tfpt-45--source_extracts--04_qft_source/sections/ 000182 | 008-section-relative-reflection-positivity-with-fermions.tex:215--254. 000183 | 000184 | Given the admissible Schwinger family, reflection positivity, temperedness, 000185 | graded Euclidean covariance, graded symmetry and clustering, the theorem 000186 | constructs a vacuum Omega and a strongly continuous semigroup 000187 | 000188 | T_s=exp(-s H_adm), H_adm>=0. (14) 000189 | 000190 | The class of the constant history is fixed by T_s, hence H_adm Omega=0. Its 000191 | analytic continuation supplies the reconstructed Wightman/CAR time. Thus the 000192 | positive-energy part of (2) is present on the reconstructed admissible theory, 000193 | conditional on the stated OS hypotheses. It is not an extra quasifree-state 000194 | postulate. 000195 | 000196 | For the proposed source reduction one must still verify that the embedded 000197 | full graded CAR subnet lies in this same reconstructed representation and that 000198 | the analytically continued original time restricts to (1). Once that is done, 000199 | (7)--(11) force its state. 000200 |