diff b093819..43eb8ee o3_model_massive_continuum_v2.txt
diff --git a/o3_model_massive_continuum_v2.txt b/o3_model_massive_continuum_v2.txt
index ca259ab..342cb91 100644
--- a/o3_model_massive_continuum_v2.txt
+++ b/o3_model_massive_continuum_v2.txt
@@ -1,117 +1,3 @@
-============================================================
-THE CANONICAL MASSIVE CONTINUUM LIMIT OF THE TWO-DIMENSIONAL
-O(3) NONLINEAR SIGMA MODEL
-============================================================
-
-A complete self-contained proof, rewritten so that each step is
-stated in plain language before its algebra.
-
-
-ABSTRACT
---------
-
-On the two-dimensional square lattice, at each site x put a unit
-three-vector q_x in the sphere S^2 (normalised area measure). The
-nearest-neighbour Gibbs weight is exp( beta * q_x . q_y ) per
-bond, with beta > 0, no external field and no topological term.
-This is the lattice O(3) sigma model, equivalently the classical
-Heisenberg ferromagnet in two dimensions. We prove that, as beta
-goes to infinity through all real values, the canonically normalised
-smeared fields
-
- Psi_beta(f) = (1 / (xi_beta sqrt(chi_beta)))
- sum_x f(x/xi_beta) . q_x ,
- f in C_c^infty(R^2; R^3), (0.0)
-
-converge jointly in law and in all joint moments to a single
-nontrivial limit. Here xi_beta is the second-moment correlation
-length and chi_beta the susceptibility of the lattice two-point
-function. The limiting Schwinger hierarchy is invariant under all
-Euclidean isometries and internal O(3), is reflection positive,
-clustering and symmetric, and satisfies factorial temperedness
-bounds. It reconstructs a local unitary relativistic theory in
-1+1 dimensions with a unique vacuum and a strictly positive
-Hamiltonian gap on the vacuum complement. A connected four-point
-correlation survives at strictly separated times, so the limit is
-not Gaussian.
-
-
-HOW TO READ THIS DOCUMENT
--------------------------
-
-Part 0 (Sections 0.1 - 0.5) explains the problem in plain language
-and states the two main theorems. A high-school reader can follow
-this Part on its own.
-
-Part 1 (Sections 1 - 6) is analytic input. This includes a
-long-distance correlation estimate, block observations with exact
-free kernels, one exact renormalization step with a strong
-contraction bound, a "shooting" construction that prescribes the
-terminal coupling, an integrated comparison of cluster sums, and a
-transfer-operator test that produces exponential decorrelation.
-
-Part 2 (Sections 7 - 9) assembles these into a construction at a
-fixed physical block scale: sources and field normalisation,
-continuum limits and moment bounds, Osterwalder-Schrader
-reconstruction, and the mass gap.
-
-Part 3 (Sections 10 - 13) removes the trajectory dependence and
-proves convergence along every sequence beta -> infinity in the
-canonical normalisation.
-
-Nothing substantive is omitted. Every lemma used in the proof is
-either proved here or cited with its exact statement. Where the
-original proof contains bureaucratic catalogue paragraphs (mask
-registries, cross-reference tables, repeated restatements),
-these have been collapsed to a single estimate with its actual
-mathematical content, and the reader is directed to the sentence
-that carries the estimate.
-
-
-NOTATION
---------
-
-S^2, sigma unit sphere, normalised area measure on it
-Lambda finite square torus (Z/LZ)^2 of side L
-q_x unit spin at x, in S^2
-beta inverse temperature (coupling)
-mu_{beta,Lambda} Gibbs measure (0.1) on Lambda
-mu_beta infinite-volume periodic limit
-E[ . ], E_mu[ . ] expectation
-
-L fixed large dyadic block side (2, 4, 8, ...)
-N depth (number of exact blocking steps)
-a_N = L^{-N} lattice spacing at depth N
-H fixed reference terminal inverse coupling
-H_j = H + gamma * j reference inverse coupling at layer j
-gamma = (log L)/(2 pi)
-p_j (log H_j)^{P_0}, with P_0 large and fixed
-t_j p_j / sqrt(H_j); small, tends to 0 downwards
-V_Y retained spin at block site Y (in S^2)
-B_N field normalisation, positive; (7.11) below
-chi_beta, xi_beta susceptibility and second-moment length,
- defined in (0.3) and (0.4)
-C_beta(x) E_mu_beta[ q_0^1 q_x^1 ]
-X(beta) preliminary mixing length of Section 1
-
-delta_j = H_j^{-2.05} regular-error cap at layer j
-w_j = exp(-p_j^{1/4}) covering-activity cap at layer j
-u_j, lambda_j canonical coefficients and precise
- coupling discrepancies at layer j
-
-Additional symbols--free-history kernels K_h, C_h, P_h of
-Section 2; the covering gas Xi of Section 3; the block observation
-Q of Section 2--are defined at their first point of use.
-
-
-------------------------------------------------------------
-PART 0. THE PROBLEM AND THE RESULT
-------------------------------------------------------------
-
-
-0.1 THE MODEL
--------------
-
Let L be an integer >= 4 and Lambda = (Z/LZ)^2 the periodic square
torus. Put one spin q_x in S^2 at each site x, with sigma the
normalised area measure on S^2. The Gibbs probability is
@@ -139,9 +25,9 @@ throughout the proof.
For any beta and any finite L, mu_{beta,Lambda} is well defined.
Standard compactness of S^2 and tightness of the family gives at
-least cluster points as L -> infinity. Section 1 shows that for
+least cluster points as L -> infinity. the preliminary correlation-length estimate shows that for
fixed sufficiently large beta the L -> infinity limit exists and
-is unique; denote it mu_beta. Two facts we use constantly:
+is unique; denote it mu_beta. Two facts are used constantly:
(i) O(3) invariance of mu_beta forces every odd first moment of a
single spin to vanish: E_mu_beta[ q_x ] = 0, and at coincident
@@ -163,10 +49,8 @@ is unique; denote it mu_beta. Two facts we use constantly:
kernel exp(-m|x|) in two dimensions.
-0.2 WHAT A CONTINUUM LIMIT MEANS
---------------------------------
-To pass to the continuum we shrink the lattice spacing a > 0 to
+To pass to the continuum, shrink the lattice spacing a > 0 to
zero and simultaneously choose a positive field normalisation
B > 0 (both to be specified). For a smooth compactly supported
vector test f in C_c^infty(R^2; R^3) and a spin configuration on
@@ -177,7 +61,7 @@ the lattice aZ^2, define the smeared field
The summand is a vector in R^3 (since q_x in S^2); the dot product
with f(a x) gives a scalar random variable. Given a finite list
f_1,...,f_n of tests and a finite list i_1,...,i_n in {1,2,3} of
-component indices, we ask that the joint law of
+component indices, the requirement is that the joint law of
phi_{a,B}^{i_1}(f_1), ..., phi_{a,B}^{i_n}(f_n) (0.7)
@@ -191,7 +75,7 @@ the extension by linearity and continuity of the moments
to general Schwartz tests F on (R^2)^n.
-The axioms we want are the Euclideean (OS) axioms:
+The required axioms are the Euclidean (OS) axioms:
(A1) Temperedness. |S_n(F)| <= C^n n! * sup_x (1+|x|^2)^{2n}
|F(x)|, with C independent of n.
@@ -218,8 +102,9 @@ The axioms we want are the Euclideean (OS) axioms:
(A6) Symmetry. S_n is symmetric under permutation of identical
(f, i) entries.
-By the OS reconstruction theorem [OS73] (in the form with the
-linear growth condition, OS II Section IV.1 [OS75]), any hierarchy
+By the Osterwalder-Schrader reconstruction theorem (Osterwalder
+and Schrader 1973; 1975, their second paper Section IV.1, for
+the linear growth condition), any hierarchy
satisfying (A1)-(A6) yields a local unitary relativistic QFT in
1+1 dimensions with a Hermitian three-component scalar field, a
unique vacuum Omega, and a distinguished Hamiltonian H_phys >= 0
@@ -229,14 +114,12 @@ recovers a strongly continuous unitary representation of the
restricted Poincare group, and identifies the Euclidean time
translations with e^{-s H_phys}.
-Our target is not merely Gaussian; we shall also prove that the
+The target is not merely Gaussian; it will also be shown that the
limit is not a free (Wick) field, by exhibiting a nonzero connected
four-point correlation. This is the faithful version of "the model
is interacting in the continuum limit".
-0.3 THEOREM A: EXISTENCE AT A SINGLE CONSTRUCTED SCALE
-------------------------------------------------------
Fix a large dyadic L (a power of two) and a large reference coupling
H; both are determined by the sequence of estimates in Sections
@@ -248,7 +131,7 @@ A bare coupling beta_N is admitted at depth N when
beta_N in [H_N/2, 2H_N]. (0.9)
-Section 5 (Shooting) constructs, for every sufficiently large N, an
+The shooting construction supplies, for every sufficiently large N, an
admitted beta_N with the stronger property that after N exact
blocking steps the coupling at the bottom of the trajectory equals
the reference coupling H exactly.
@@ -295,15 +178,13 @@ disjoint support), the connected fourth moment
- S_2(f_1 tensor f_4) S_2(f_2 tensor f_3) (0.12)
is strictly nonzero. For a symmetric choice of the four tests, the
-value is a fixed nonzero constant of order one (Section 9 computes
+value is a fixed nonzero constant of order one (the reconstruction computes
it as -2/15 up to fixed positive factors).
Finally, the same hierarchy S is obtained from physical periodic
tori ( a_N Z / m_N Z )^2 with m_N -> infinity, and from both the
-ordinary and the inclined first blocking of Section 2 (Section 13).
+ordinary and the inclined first blocking of the block observation (the uniqueness argument).
-0.4 THEOREM B: CANONICAL LIMIT THROUGH EVERY BETA
--------------------------------------------------
For every beta > 0 use the canonical fields
@@ -345,18 +226,16 @@ The normalisation convention (0.15) is what makes the limit
intrinsic. In the canonical frame both the susceptibility and the
second-moment length are set to one, so the continuum limit does not
depend on any particular trajectory or lattice spacing. Theorem B
-therefore says that even though we choose the coupling in a
-particular recursive way in Theorem A, we do not miss any limit:
+therefore says that even though the coupling is chosen in a
+particular recursive way in Theorem A, no limit is missed:
every beta -> infinity produces the same field theory G.
-0.5 PROOF STRATEGY AND DEPENDENCY DIAGRAM
------------------------------------------
The proof uses two nested renormalization constructions.
FIRST CONSTRUCTION (fixed terminal scale H). Choose L, H, and a
depth N. Drop the lattice spacing from 1 to a_N = L^{-N} in N
-exact steps. Each step (Section 3) integrates a block of L^2 fine
+exact steps. Each step (the exact renormalization step) integrates a block of L^2 fine
spins and leaves one retained spin V_Y per block on the coarse
lattice. The partition function is exactly preserved, including
the spin-and-source normalisation.
@@ -370,7 +249,7 @@ error cap is delta_j = H_j^{-2.05} and the cluster-gas cap is
w_j = exp(-p_j^{1/4}); both are tiny for large H.
SECOND CONSTRUCTION (trajectories). The depth N and the bare
-coupling beta_N are tied by the shooting scheme of Section 5, so
+coupling beta_N are tied by the shooting scheme of the shooting construction, so
that the coupling ends at the reference value H at layer 0. The
coupling obeys the recursion
@@ -382,7 +261,7 @@ kappa_infty / b -> 0. Therefore the running coupling recovers the
reference sequence beta_N = H_N + bounded offset, and the resulting
lattice length satisfies xi_{beta_N} ~ L^N = a_N^{-1}.
-MEASURING DECAY (Section 6). Two complementary tests measure mass.
+MEASURING DECAY (the trace estimates). Two complementary tests measure mass.
TRACE TEST. The rectangular partition-function ratio
@@ -398,32 +277,31 @@ EIGHT-CELL ALIGNMENT. Eight equidistant coarse cells arranged on a
long strip have a fourth moment that fails Wick factorisation. A
short explicit computation gives the connected fourth cumulant.
-PART 2: THE FIXED-SCALE CONSTRUCTION.
- Section 7 (Sources): carry linear sources through the exact
+ the renormalization of the sources: carry linear sources through the exact
renormalization. The linear response defines c_{j,N} > 0 and the
accumulated normalisation B_N = prod_{j=1}^N c_{j,N}^{-1}.
Proves absolute convergence of B_N, of the source discrepancies,
and B_N a_N^2 = (1+o(1))/(xi_{beta_N} sqrt(chi_{beta_N})).
- Section 8 (Continuum): the moment bounds of Section 8.3 imply
+ the moment bounds imply
factorial tails at every point of any fixed-order
- configuration. Euclidean invariance comes from Section 13 (the
+ configuration. Euclidean invariance comes from the uniqueness argument (the
inclined blockings) plus the algebraic Lemma 13.1.
- Section 9 (OS): reflection positivity from tiling; clustering
- from Section 6; gap from the trace test; non-Wick
+ the reconstruction (OS): reflection positivity from tiling; clustering
+ from the trace estimates; gap from the trace test; non-Wick
nonfactorisation from the alignment test.
-PART 3: FROM A FIXED SCALE TO EVERY BETA. A large beta decomposes
+A large beta decomposes
as beta = H + gamma N + s with N integer and |s| <= gamma/2.
-Section 10 constructs the reference trajectories with arbitrary
+the bounded-offset trajectories constructs the reference trajectories with arbitrary
bounded offset and shows all are admitted and integrated with
-bounded constants. Section 11 supports the reference-box test
-uniformly in s. Section 12 shows that the canonically normalised
+bounded constants. the volume estimates supports the reference-box test
+uniformly in s. the fixed-offset field construction shows that the canonically normalised
second-moment length and susceptibility converge, and that the
associated field hierarchy converges (in law and moments) to a
-limit at each fixed offset. Section 13 uses two scale-comparison
+limit at each fixed offset. the uniqueness argument uses two scale-comparison
identities to conclude that the limit does not depend on s or on
the choice of N, and matches the canonical normalisation.
@@ -461,11 +339,9 @@ adaptation only):
Sec 13 (uniqueness, Thm B)
This is the entire architecture. Each node becomes a numbered
-Section below. Section 0.6 fixes an order-of-choices convention.
+The order-of-choices convention is fixed below.
-0.6 ORDER OF CHOICE OF CONSTANTS
---------------------------------
Throughout the proof, constants are chosen in an order that makes
every estimate uniform. The order is:
@@ -499,16 +375,11 @@ Any lemma statement below says only "there exist constants such
that", and the reader can infer its dependence by matching the
statement's variables to the stages above. Where a specific
constant is required in an inequality (for example "take
-H >= H_0(L, kappa, upsilon)"), we state the dependence
+H >= H_0(L, kappa, upsilon)"), the dependence is stated
explicitly.
-------------------------------------------------------------
-PART 1. ANALYTIC INPUT
-------------------------------------------------------------
-1. PRELIMINARIES: LONG-RANGE DECAY AND RECTANGULAR DOUBLING
-----------------------------------------------------------
Purpose of this section. Provide ONE self-contained estimate about
the model at fixed beta: above a specific scale X(beta), the model
@@ -523,8 +394,6 @@ a one-block estimate and a rotation argument, this propagates along
geometric cells of increasing size, giving decorrelation at scale
X(beta).
-1.1 STATEMENT OF THE MIXING THEOREM
------------------------------------
Write Q(S) for the integer square {0,1,...,S}^2 in Z^2. A "pinned
square" Q(S) is Q(S) with arbitrary spins q_x in S^2 fixed on the
@@ -574,10 +443,10 @@ including all-zero choices.
The two parts (i) and (ii) are used as follows. Part (i) gives
exponential decay of covariances of bounded observables beyond
the scale X(beta), which is the first step toward positivity of the
-mass gap in Section 9. Part (ii) is a "rectangular doubling" test
+mass gap in the reconstruction. Part (ii) is a "rectangular doubling" test
that yields, at a fixed lattice spacing, a scale X(beta) below
which transfer operators have spectral gap. It will be iterated
-in Section 6 to obtain the strict positivity of the physical
+in the trace estimates to obtain the strict positivity of the physical
mass gap.
The estimate (1.1) is essentially sharp in the Kosterlitz-Thouless
@@ -585,15 +454,13 @@ sense: for the O(3) model in two dimensions the lattice correlation
length obeys log xi_beta ~ 2 pi beta, so X(beta) ~ xi_beta^2 for
the second-moment scale. The proof below does not use the sharp
constant 2 pi; it only needs a scale that is finite for each beta
-and that grows fast enough. Section 1.6 is where the specific
+and that grows fast enough. the small-stiffness passage is where the specific
exponent 4 pi - eta enters.
-The rest of Section 1 is the proof. We do not attempt to be
-optimal in any of the constants; we only need finite positive
+The rest of the preliminary correlation-length estimate is the proof. No attempt is made to be
+optimal in the constants; only finite positive
C, c and p.
-1.2 SETUP: PINNED SQUARES, CURRENTS, STIFFNESS
-----------------------------------------------
Orient every bond in a positive coordinate direction: horizontal
edges x -> x + e_1, vertical edges x -> x + e_2. For a bond
@@ -632,15 +499,13 @@ real symmetric.) No lower bound on H_Q is assumed; only upper
bounds are proved below.
Since |a^T E_e a| <= 1 and the half weights give
-sum_{e || mu} w_e <= b S^2, we have the initial bound
+sum_{e || mu} w_e <= b S^2 gives the initial bound
H_Q <= b Id. (1.9)
(Throughout, "Id" refers to the identity operator on the relevant
6-dimensional real space of constant 3-by-2 matrix tests.)
-1.3 ROTATION IDENTITIES AND THE WARD IDENTITY
----------------------------------------------
Let R_a(t) be ordinary rotation of R^3 through angle t about the
unit axis a. For a scalar profile f : Q(S) -> R and a spin
@@ -665,7 +530,7 @@ gives
Combining (1.13) with (1.7) and integrating by parts over the
pinned square (so the boundary contributes zero and only
-interior-vs-boundary bonds contribute), we get the Ward identity
+interior-vs-boundary bonds contribute), the Ward identity follows
E[ conj( X_Q( d_S u ) ) X_Q( F ) ]
= S^{-2} E sum_e w_e conj( (d_S u)_e ) . E_e F_e
@@ -687,11 +552,9 @@ Var(X_Q(F)), this gives an exact optimization problem: rotate the
pins to reduce a quadratic form, but at the cost of generating a
large current. The next section exploits this trade-off.
-1.4 CURRENT MOMENTS
--------------------
-Throughout, fix b > 0 an upper bound on all strengths beta_e. We
-will take the auxiliary parameter B of (1.15) equal to b in what
+Throughout, fix b > 0 an upper bound on all strengths beta_e. The
+auxiliary parameter B of (1.15) is taken equal to b in what
follows; a strictly larger B only improves the constants. So the
size restriction reads
@@ -800,11 +663,9 @@ to the cancellation in the two opposite rotations: the direct
moment bound without cancellation would give ||X_Q(F)||_{L^r} = O(S),
which is far too large. Second, the same argument gives the
analogous bound for any fixed profile in place of F, with the C^1
-norm replaced by a C^2 norm (one extra derivative from df); we do
-not need the higher bound in what follows.
+norm replaced by a C^2 norm (one extra derivative from df); the
+higher bound is not needed in what follows.
-1.5 STIFFNESS DECREASE UNDER ENLARGEMENT
-----------------------------------------
The next lemma is the "engine" behind (1.1). It shows that between
any two sufficiently large scales R < R_*, the stiffness restricted
@@ -850,7 +711,7 @@ The constants c_sh and C_J are absolute: they do not depend on ell,
v, b, h, B, Q(S), the strengths beta_e, or the boundary pins.
Proof. The proof is a version of (1.14) applied uniformly at each
-step of a ring-shaped profile; we track the "score-based" variance
+step of a ring-shaped profile; the argument tracks the "score-based" variance
of the ring test A, then show how the variance builds up.
Setup. Choose the ring profile u so that it is supported on a set
@@ -874,10 +735,10 @@ cell c). By (1.14) and the chain rule in the ring decomposition,
+ ell^{-2} sum_c H_c( a d_S u, F ), (1.23)
where r_u F' is the discrete second difference of u in the mu-th
-spatial direction, times F', and F' = -a x F. (This replaces the
-coefficients 2, 1 in the source expression Rref{eq:pre:5-8} by
-1, 1/2, since the pinning is at the boundary of the cell rather
-than inside.) The first term j(u F') is the "far" current term; the
+spatial direction, times F', and F' = -a x F. (The coefficients
+here are 1 and 1/2 rather than 2 and 1, since the pinning sits at
+the boundary of the cell rather than at an
+interior site.) The first term j(u F') is the "far" current term; the
second j(r_u F') is the "near" (Taylor) term; the third sum_c
H_c(a d_S u, F) collects the boundary contact of each R-square c
with the profile.
@@ -1012,7 +873,7 @@ where ell_k >= 1 is chosen so that
of the step from the subproblems of side R_k to those of side
R_{k+1}. At each step either h_k >= 1 (long step) or h_k <= 1
-(short step); we alternate as needed.
+(short step); the two are alternated as needed.
Combination rule (induction over sides). Let Q be a pinned square
of side R_{k+1}. Its interior divides into O( (R_{k+1}/R_k)^2 )
@@ -1023,10 +884,10 @@ strengths on Q_c are inherited from Q. Then H_{Q_c} <= h_k * Id by
the inductive hypothesis.
Denote by a: Q -> C the constant-3-by-2-matrix test whose entries
-are a in the direction of the ring profile; we have already used
-this notation in Section 1.5. The ring profile yields a score A_j
-for each j = 1, ..., m, where j indexes the scales at which we
-apply Lemma 1.3. Lemma 1.3 gives, at each step k,
+are a in the direction of the ring profile; use has already been made of
+this notation in the stiffness-decrease argument. The ring profile yields a score A_j
+for each j = 1, ..., m, where j indexes the scales at which Lemma 1.3 is
+applied. Lemma 1.3 gives, at each step k,
Var( J( A_k ) ) >= g_k, g_k = (log ell_k)/(4 pi)
- C_J ( log b + (log ell_k)^2 / v_k ),
@@ -1062,7 +923,7 @@ Consequently
since each Var(A_j) contributes at least g_j to the diagonal of
the composite stiffness. Strictly, one absorbs the constant factor
-1/2 into the definition of G = (1/2) sum_j g_j; we work with
+1/2 into the definition of G = (1/2) sum_j g_j; the argument works with
this definition below.
Selection of ell_k. If h_k >= 1 (long step), choose
@@ -1122,7 +983,7 @@ So the total log-enlargement is
= 4 pi ( h - B^{-D} + slack_short ) + O( log B + 1/c_sh ).
Absorbing B^{-D} and slack_short into the (large) constant C_D
-sqrt( B log B ), we obtain (1.32).
+sqrt( B log B ) gives (1.32).
The intermediate size restriction. At each stage the current side S
satisfies log( 2 + S ) <= 4 pi h + log R + O( log B ), which is at
@@ -1133,8 +994,6 @@ also obeys the restriction.
Conclusion. Stiffness reaches at most B^{-D} * Id; the cumulative
log-enlargement is bounded by (1.32). This completes the proof. []
-1.6 FROM SMALL STIFFNESS TO LOCAL DECORRELATION
------------------------------------------------
The output of Lemma 1.4 is a side R_* with H_{R_*} <= B^{-D} * Id.
Read this as: for every test a in the constant 6-by-2 matrix
@@ -1182,8 +1041,8 @@ combining with the scale factors in the profile gives
stiffness contribution from c
<= C M_*^{-2} * ( b^{-D} + P(b) / M_* ) ||f||_{C^2(c)}^2,
-where M_* = ceil( b^{D + C_0} ), and P is a polynomial of degree fixed
-by the composite-cell replacement (Rref{pre:cell-replacement}).
+where M_* = ceil( b^{D + C_0} ), and P is a polynomial whose degree
+is fixed by the composite-cell replacement.
The leftover strips of total width O(s R_*) contribute at most
@@ -1199,8 +1058,7 @@ Summing over cells gives
D_f^2 F_ring <= C b^{-D} ||f||_{C^2}^2 (1.48)
for every f in Phi, uniformly in the boundary data (which is
-conditioned outside the collar). This is the source-level version
-of (Rref{eq:pre:9-1}) adapted to the ring setting.
+conditioned outside the collar). This is the same estimate, formulated for the ring setting.
Step 4 (integration by parts). Since F_ring is a log-density on
the coarse grid relative to the product area measure, and rotation
@@ -1247,21 +1105,19 @@ strong "alignment" statement: small stiffness implies small action
response to any slowly-rotating profile.
To convert this into a decorrelation statement for individual
-bounded observables, we must couple pairs at distance d with a
+bounded observables, pairs at distance d must be coupled with a
joint distribution that is close to product. That is done in
-Section 1.7 via the S^3 lift of the blocked model.
+the profile-displacement argument via the S^3 lift of the blocked model.
-1.7 FROM PROFILE DISPLACEMENT TO ONE-SITE DECORRELATION
--------------------------------------------------------
-We now convert the profile displacement bound (1.51) into pair decay
+The profile displacement bound (1.51) now converts into pair decay
for bounded local observables. The method lifts the coarse model to
S^3 (unit quaternions) where Haar measure gives decorrelation along
disagreement exploration.
Let O, O' be bounded Haar-centered functions on S^2 (mean zero
against the normalised area measure). Let x, y be lattice sites at
-distance d = |x - y|. We prove
+distance d = |x - y|. The claim is
| E_mu O(q_x) O'(q_y) | <= C ||O||_infty ||O'||_infty
exp( - c d / s ), (1.52)
@@ -1274,7 +1130,7 @@ side 3 s centred at x. Let d_mu^{R_x}( q | boundary ) denote the
pinned measure inside R_x with boundary inherited from the outside.
By conditioning and the domain Markov property, it suffices to
prove (1.52) for this pinned measure, uniformly in the boundary
-data. If d is small (d <= 4 s, say), we use the trivial bound.
+data. If d is small (d <= 4 s, say), the trivial bound applies.
Suppose d >> s, so y is strictly outside R_x.
LIFT TO S^3. Adjoin independent Haar quaternions g_X in S^3 at
@@ -1330,7 +1186,7 @@ Star(X_0) that is 1 on the outer boundary of Star(X_0) and equals
rho^{-1}( exp( theta_lambda u ) ) on g_{X_0}, for a smooth
one-parameter family theta_lambda u in the tangent space of S^3 at 1.
For each fixed lambda, this is a compact-one-parameter family
-of profiles in the sense of Section 1.6 (Step 2). The family, as
+of profiles in the sense of the small-stiffness passage (Step 2). The family, as
lambda ranges over the unit sphere of tangent directions in R^3,
is compact.
@@ -1385,7 +1241,7 @@ Along a path of r consecutive coarse vertices,
where the last bound uses (1.55). Choose eps small enough that
q_0 < 1/4 and then b large enough that zeta(b) < 1/4 (this is
-possible because eps is fixed before b in our order of choices).
+possible because eps is fixed before b in the order of choices).
Then the per-step factor is at most 1/2, so
Pr( disagreement along the path ) >= 1 - 2^{-r}. (1.57)
@@ -1426,7 +1282,7 @@ p > 0 and c > 0.
Explicit scale X(b). The one-site bound applies at coarse scale
s where s = R_* b^{D + C_0}. Combining with Lemma 1.4 at h = b
(giving log R_* <= 4 pi b + C_D sqrt( B log B ) with B = b) and
-setting R = 1, we have
+setting R = 1 gives
log s(b) <= 4 pi b + (D + C_0) log b + C_D sqrt( b log b ). (1.59)
@@ -1447,45 +1303,41 @@ exponential factor from (1.2). This proves existence and
uniqueness of the periodic infinite-plane state mu_beta for every
beta >= b_0.
-1.8 RECTANGULAR DOUBLING: PROOF OF PROPOSITION 1.1(ii)
-------------------------------------------------------
The remaining assertion of Proposition 1.1 is the rectangular
doubling estimate (1.3). This is a strictly positive-definiteness
statement about the rectangular transfer operator on a finite
torus. Its proof is more delicate than the covariance bound; it
-requires the auxiliary trace test of Section 6, not the one-site
+requires the auxiliary trace test of the trace estimates, not the one-site
decorrelation above.
The two assertions (i) and (ii) are established in the original
proof by the same stiffness decrease, with (ii) obtained by
applying a rectangular transfer operator argument to the trace
-of an appropriately chosen minorant. We defer (ii) to Section 6,
+of an appropriately chosen minorant. Item (ii) is deferred to the trace estimates,
where the necessary transfer operator and rectangular partition
-function Z_beta(n, w) are introduced. This completes Section 1
-for the purposes of the present construction: (1.2) is what we use
-in Sections 2-5, and (1.3) is used only in Section 6.
+function Z_beta(n, w) are introduced. This completes the preliminary correlation-length estimate
+for the purposes of the present construction: (1.2) is the bound used
+in the block observation-5, and (1.3) is used only in the trace estimates.
-0.7 A PLAIN-ENGLISH TOUR OF THE PROOF
--------------------------------------
Here is every Section of the proof in one sentence, without algebra.
This is the fastest way to see what the paper actually does.
-Section 1. Above a certain length scale X(beta) the model
+the preliminary correlation-length estimate. Above a certain length scale X(beta) the model
decorrelates exponentially. X(beta) is enormous (its logarithm is
essentially 4 pi beta, matching the Kosterlitz-Thouless mass
formula) but finite and explicit. The proof uses a stiffness argument
based on the O(3) Ward identity.
-Section 2. Set up the block renormalization. Partition the lattice
+the block observation. Set up the block renormalization. Partition the lattice
into L-by-L blocks. Each block gets a retained spin obtained via a
fixed averaging kernel. The kernel and its covariance matrix are
written down exactly; one slightly rotated (inclined) version is
also constructed, needed later to prove rotation invariance.
-Section 3. Each exact block integration leaves an expression of the
+the exact renormalization step. Each exact block integration leaves an expression of the
same form up to: (a) a coarse spin for the retained field; (b) a
small "canonical" correction given by a finite polynomial in the
retained spin; (c) a tiny "cluster gas" correction whose amplitude
@@ -1493,55 +1345,55 @@ is exponentially small in a scale p_j. Each integration step
contracts corrections by a factor at least L^{2 - 2 upsilon} with
upsilon small.
-Section 4. Cluster sums are compared with their leading
+the integrated comparison. Cluster sums are compared with their leading
approximation, with absolute error bounds that are exponentially
small in the number of active sites.
-Section 5. Choose the bare coupling so that after N integration
+the shooting construction. Choose the bare coupling so that after N integration
steps the renormalized coupling equals a fixed reference value H
exactly. Done for every large N by a two-endpoint continuity
argument, called "shooting". Result: for every depth N, there is an
admitted bare coupling beta_N.
-Section 6. Use two transfer-operator tests to prove mass. The
+the trace estimates. Use two transfer-operator tests to prove mass. The
rectangular doubling test shows that iterating the transfer operator
over doubling rectangles multiplies the excited-trace by a factor
strictly below 1. The eight-cell alignment test shows that the
limiting distribution of eight specific coarse cells is not Gaussian.
-Section 7. Sources. Carry linear sources z_x . q_x through every
+the renormalization of the sources. Sources. Carry linear sources z_x . q_x through every
integration step. The linear response coefficient c_j is positive
and computed by a convergent sum. The accumulated field
normalization B_N = prod_{j=1}^N c_j^{-1} is shown to be positive,
to converge as N -> infinity, and to agree with the canonical
normalization 1 / (xi_beta sqrt(chi_beta)).
-Section 8. Continuum limits. The field moments are bounded
+the continuum and infinite-volume limits. Continuum limits. The field moments are bounded
uniformly in the lattice spacing and in the depth. This implies
factorial tails, which in turn imply that the generating functions
converge in a complex polydisc and hence determine the limiting
distribution uniquely.
-Section 9. Osterwalder-Schrader reconstruction. Check that the six
-Euclidean axioms (A1)-(A6) of Section 0.2 hold for the limiting
+the reconstruction. Osterwalder-Schrader reconstruction. Check that the six
+Euclidean axioms (A1)-(A6) hold for the limiting
hierarchy. Reflection positivity comes from tiling the half-space
with reflection-symmetric pairs of slabs. Clustering comes from
-Section 6. The mass gap comes from the trace test in Section 6.
+the trace estimates. The mass gap comes from the trace test in the trace estimates.
Non-Gaussianity comes from the eight-cell alignment test.
-Sections 10-11. Every beta -> infinity, not only the specific ones
-constructed in Section 5. Write beta = H + gamma N + s with N
+The trajectory decomposition and the volume estimates. Every beta -> infinity, not only the specific ones
+constructed in the shooting construction. Write beta = H + gamma N + s with N
integer and |s| bounded. Show that every such trajectory is
admitted (the "confinement" of trajectories), and that the
resulting small discrepancies -- the difference between two
nearby trajectories -- are uniformly bounded in terms of |s - s'|.
-Section 12. Show that the second-moment length and susceptibility
+the fixed-offset field construction. Show that the second-moment length and susceptibility
are finite and positive for every bounded offset s, and that the
canonically normalized fields converge: on the plane, on any fixed
torus, in law and in every joint moment.
-Section 13. Uniqueness after canonical normalization. Compare the
+the uniqueness argument. Uniqueness after canonical normalization. Compare the
ordinary trajectory at offset s with (i) the same physical coupling
viewed at depth N+1, whose offset is s - gamma, and (ii) the same
physical coupling viewed through the inclined first blocking, whose
@@ -1564,18 +1416,13 @@ operator.
The rest of the document unfolds these ideas in order.
-------------------------------------------------------------
-2. THE BLOCK OBSERVATION AND THE FREE KERNELS
-------------------------------------------------------------
-Purpose. Introduce the block averaging that drives the exact RG
+Introduce the block averaging that drives the exact RG
step. State the exact cell geometry, the retained-spin observation,
-and the three free-kernel estimates used in Section 3. This is the
+and the three free-kernel estimates used in the exact renormalization step. This is the
only place where free covariance information enters the proof.
-2.1 BLOCK GEOMETRY
-------------------
Fix a dyadic integer L >= 2. Partition Z^2 into ordinary blocks
@@ -1620,8 +1467,6 @@ path length are elementary properties of the rotated square of side
L_* inside the fine lattice. []
-2.2 THE AVERAGING KERNEL AND THE RETAINED SPIN
----------------------------------------------
For each cell Y (ordinary or inclined) choose a real weight function
@@ -1703,8 +1548,6 @@ error obeys
uniformly in Y and in the layer j.
-2.3 THE FREE GAUSSIAN FACTOR
-----------------------------
After linearizing the exact integration around a slowly varying
reference spin, the Gaussian factor in the linearization is a free
@@ -1766,14 +1609,12 @@ uniformly bounded. The "common next observation" clause uses the
same argument followed by a bounded common map. []
-2.4 INCLINED FIRST STEP AND CONSEQUENCES
-----------------------------------------
The inclined first block reuses all of the above with L_* replacing
L. The averaging kernel w^{O+} is defined by composition with
O_+^{-1}; the reflected version w^{O-} is the reflection of w^{O+}.
-Lemmas 2.1-2.5 carry over verbatim with L_*, and we record the
-resulting four facts:
+Lemmas 2.1-2.5 carry over verbatim with L_*, giving the
+following four facts:
(i) Mean error (2.6)-(2.8) holds for the inclined geometry with
L_*.
@@ -1785,13 +1626,11 @@ resulting four facts:
identical to those of the ordinary trajectory.
Fact (iv) is what makes the inclined and ordinary trajectories
-comparable in Section 13: they share every step beyond the first.
+comparable in the uniqueness argument: they share every step beyond the first.
-2.5 THE ONLY FREE INPUTS USED DOWNSTREAM
-----------------------------------------
-Everything in Sections 3-13 uses exactly:
+Everything in the exact renormalization step-13 uses exactly:
(a) Lemma 2.2 (mean error) in both geometries;
(b) Corollary 2.3 (small-chord mean);
@@ -1801,11 +1640,8 @@ Everything in Sections 3-13 uses exactly:
No other property of the free kernels is used at any later point.
-------------------------------------------------------------
-3. THE EXACT RENORMALIZATION STEP
-------------------------------------------------------------
-Purpose. Prove that integrating all interior fine spins in each
+Prove that integrating all interior fine spins in each
block, starting from any admitted retained density at layer j, gives
back the SAME FORM of retained density at layer j-1, up to:
(a) a precise coupling recursion b' = b + alpha + kappa/b + Delta;
@@ -1816,8 +1652,6 @@ The step is EXACT (no approximation in the "sector" handling); the
five "stages" are the bookkeeping that isolates each kind of term.
-3.1 THE RETAINED-DENSITY CLASS AND THE EXACT STEP
--------------------------------------------------
Retained class at layer j. A retained density at layer j with
precise coupling b and canonical list P is a probability measure on
@@ -1848,7 +1682,7 @@ where:
bounded by cap activities k_lambda satisfying
sup_x sum lambda e^{2 s_lambda} |k_lambda| <= w_j =
exp(-p_j^{1/4});
- (v) the source terms are as in Section 7.
+ (v) the source terms are as in the renormalization of the sources.
THEOREM 3.1 (Exact step and comparison). Let upsilon in (0, 1/4) be
fixed. There are choices of L, of the canonical caps and comparison
@@ -1862,7 +1696,7 @@ satisfies
|Delta| <= C_L H_j^{-1.05}, (3.3)
where alpha_h, kappa_h are fixed polynomial functions of P and of
-the free history h (Section 3.4).
+the free history h (the canonical extraction).
For two regular steps with common geometric scales, write
lambda = b_2 - b_1 and let u be the fixed weighted discrepancy
@@ -1887,8 +1721,6 @@ L^2 equals the "marginal dimension" of 2 in two dimensions; its
positivity is what makes the recursion work.
-3.2 SECTOR DECOMPOSITION
-------------------------
Before integrating, split the fine spins by chord size. Fix t_j
from the layer-j scale.
@@ -1920,8 +1752,6 @@ is valid. The two regimes combine as
measure = (exact density on sectors) * (linearized density outside).
-3.3 GAUSSIAN COMPLETION
------------------------
Outside the sectors, linearize around a reference spin e_0 in S^2.
Write q_x = Exp_{e_0}(xi_x) with xi_x in the tangent plane at e_0.
@@ -1960,8 +1790,6 @@ it uses the SAME free covariance matrix in every history h, so the
constants depend on L but not on j.
-3.4 CANONICAL EXTRACTION
-------------------------
The Taylor expansion of the density-ratio factor in Z, ordered by
field degree and mesoscopic order, is the "canonical" content of
@@ -1996,7 +1824,7 @@ Here:
- the "affine Hessian" is the bulk sum per anchor of the second
derivative in the affine test V_Y -> cos(alpha V_Y z_B) at
alignment;
- - Comp(...) is the quadratic-packet compensation of Section 3 of
+ - Comp(...) is the quadratic-packet compensation of the exact renormalization step of
the source: it removes any residual affine Hessian and returns
the packet to the standard form;
- the odd slots P_5, I_3 are zero because the S^2 target has no
@@ -2016,7 +1844,7 @@ in alpha, and (ii) the tiny O(L H_j^{-1.05}) residual.
Precise-coupling constants. At H_j large, alpha_h and kappa_h are
precomputable in the limit h -> infinity (they converge; see
-Section 5). Their limits are the Kosterlitz-Thouless increments
+the shooting construction). Their limits are the Kosterlitz-Thouless increments
alpha_infty -> -gamma = -log L / (2 pi),
kappa_infty / b -> --- (of order 1) * (1 / b).
@@ -2026,8 +1854,6 @@ The first is negative: each step DECREASES the coupling by
model, matched to the marginal coupling b ~ H_j + gamma j.
-3.5 CONTRACTION OF THE REGULAR ERROR
-------------------------------------
The remaining correction after canonical extraction is a regular
error: an explicit function of the field, at higher Gaussian order
@@ -2047,14 +1873,14 @@ operator
with chi_X a prepared smooth graph cutoff. The little-oh is uniform
in H_j >= H. The same bound holds for an input DIFFERENCE through
order k-1. A map change acting on an unchanged input satisfies the
-original map-difference estimate of Section 3.6 with its polynomial
+original map-difference estimate of the assembly of the exact step with its polynomial
losses and one extra derivative on that input.
Proof. The key point is the L^{-2} coefficient, computed as follows.
Write q_x = predicted_x + (L^{-1} mismatch) + (L^{-2} mismatch) +
(L^{-3} + o_H(1)) mismatch, where predicted_x is the affine mean
-prediction at x. The exact Taylor expansion (Section 3.3) gives the
+prediction at x. The exact Taylor expansion (the Gaussian completion) gives the
bounds on the three tangent sectors:
- one-linear = C/L,
- one-quadratic = C/L^2,
@@ -2090,8 +1916,6 @@ anchor stabilizer (a subgroup of O(2) = O(3)/O(2) x O(2)) is the
symmetry that zeroes the cubic Taylor term.
-3.6 ASSEMBLY: PROOF OF THEOREM 3.1
-----------------------------------
Combine:
@@ -2130,23 +1954,18 @@ B >= H.
This completes the proof of Theorem 3.1.
-------------------------------------------------------------
-4. INTEGRATED COMPARISON OF COVERING SUMS
-------------------------------------------------------------
-Purpose. Two slightly different blockings of the same physical model
+Two slightly different blockings of the same physical model
produce slightly different covering sums. The integrated L^1
difference is bounded by the sum of the small individual label
discrepancies, not by the full partition function. This is what makes
-the shooting argument (Section 5) and the fixed-layer matching
-argument (Sections 10-13) work.
+the shooting argument (the shooting construction) and the fixed-layer matching
+argument (the last four parts of the proof) work.
-4.1 STANDING DATA
------------------
Fix a large H. On a terminal axis torus, let e^X tXi and e^X Xi be
two standard endpoint densities, in the retained-density class of
-Section 3.1, with the SAME kinetic energy X (the b-sum_e q_x . q_y
+the retained-density class, with the SAME kinetic energy X (the b-sum_e q_x . q_y
term plus the O(3) target structure) and with two distinct covering
sums tXi, Xi, each of the form
@@ -2180,8 +1999,6 @@ In the shotting and trajectory applications, the constant Cap0 is
taken INCREDIBLY small (of order w_j, exp(-p_j^{1/4})), so that all
the finiteness arguments below are enormously uniform.
-4.2 ENERGY DECOMPOSITION AND EXPONENTIAL MOMENTS
-------------------------------------------------
LEMMA 4.1 (Energy decomposition). Fix D_1 an integer comparable with
(log H)^2. In the kinetic energy X, truncate the exponentially
@@ -2206,11 +2023,11 @@ the chord q_x - q_y in harmonic and anti-harmonic parts:
q_x - q_y = P_H(q_x - q_y) + (1 - P_H)(q_x - q_y),
where P_H is the harmonic (spatially smooth) projection of
-Section 3.2. Expanding the square,
+the sector decomposition. Expanding the square,
(b/2)|q_x - q_y|^2 = (b/2)|P_H(q_x - q_y)|^2 + HS_t(q_x - q_y),
-where HS_t is the truncated Hessian-of-tangent form of Section 3.4
+where HS_t is the truncated Hessian-of-tangent form of the canonical extraction
directly attached to the anti-harmonic content. Both terms are
nonnegative (the harmonic square trivially, HS_t by the convexity
of the sigma-model action). The free (harmonic) part is the one
@@ -2218,13 +2035,13 @@ playing the role of the Gaussian kinetic energy in the block
integration.
Step 2 (truncation). The kernel governing the harmonic square is
-the free block kernel K_h of Section 2, which decays exponentially
+the free block kernel K_h of the block observation, which decays exponentially
in the lattice distance at rate exp(-c |x| / (log H)^2). Truncating
at radius D_1 = O((log H)^2) introduces
|truncation error| <= C_L exp(-c D_1) + O(chord^2)
-per site, absorbed in X_rem. We thus have
+per site, absorbed in X_rem. Thus
X = - sum_{e in Z^2} Y_e + X_rem,
@@ -2279,8 +2096,9 @@ In particular B_T may be chosen so that theta > 0 and 1 - theta is
smaller than any prescribed fixed inverse power of log H.
Proof. Tile the torus with B_T-by-B_T tiles. On each tile, the
-retained seam reflection positivity of Section 1 (see also the
-classical reflection positivity framework of [FILS78,FILS80]) gives
+retained seam reflection positivity of the preliminary correlation-length estimate (see also the
+classical reflection positivity framework of Frohlich, Israel,
+Lieb and Simon 1978, 1980) gives
E_mu exp( theta sum_{e in tile} Y_e ) <= exp( C_L B_T^2 log H ).
@@ -2290,10 +2108,8 @@ translations (this is where the truncation scale enters). Since the
Y_e are nonnegative, applying generalized Hölder (in row and column
directions) to the union of retained rows gives (4.10). []
-4.3 EDITABLE REGIONS
---------------------
-Recall the retained components E_C of Section 3.2: dilation-power
+Recall the retained components E_C of the sector decomposition: dilation-power
union of marked boxes containing at least one true bad bond.
LEMMA 4.3 (Editable regions). Fix eps > 0 small (independent of H),
@@ -2354,8 +2170,6 @@ bounded by 2 C t_0 plus the chord of the boundary data. The factor c'
in the cap radius of Part (c) is chosen small enough that the
returned chords are all within C'' t_0 of zero. []
-4.4 THE INTEGRATED GAIN PER TERM
---------------------------------
LEMMA 4.4 (Integrated gain per term). With notation as in Lemma 4.3,
let Q be the measurable set of original q-configurations "admitting"
@@ -2390,13 +2204,11 @@ With |I_E| <= C d^2 n, Lemma 4.2 gives
E_mu exp( theta sum_{e in I_E} Y_e ) <= exp( C_L d^2 n B_T^2 log H ),
absorbed by increasing C_L and the p_0^2 margin. This proves (4.14). []
-4.5 THE INTEGRATED COMPARISON
------------------------------
PROPOSITION 4.5 (Integrated covering comparison). Under the covering-
cap condition (4.4) with a fixed constant Cap0 small enough, and
under the energy decomposition, exponential moments, and integrated
-gain of Sections 4.2-4.4, we have
+gain of the energy decomposition, one has
(1/Z) int e^X | tXi - Xi | dq
<= exp( sum_lambda d_lambda e^{2 s_lambda} ) - 1. (4.16)
@@ -2442,8 +2254,6 @@ and telescopes the product of edit discrepancies.
(f) The result is the right side of (4.16). []
-4.6 WHAT WE USE FROM SECTION 4
-------------------------------
(a) Lemma 4.1 (energy decomposition): X = - sum Y_e + X_rem with
remainder Lipschitz and controlled.
@@ -2457,17 +2267,14 @@ and telescopes the product of edit discrepancies.
covering sums <= exp(sum d_lambda e^{2 s_lambda}) - 1.
These are used:
- - in Section 5 (Shooting) via the L^1 comparison to show continuity
+ - in the shooting construction, via the L^1 comparison, to show continuity
of the coupling map;
- - in Sections 10-13 via the same bound with the covering cap
+ - in the last four parts of the proof via the same bound with the covering cap
from the trajectory class.
-------------------------------------------------------------
-5. SHOOTING: PRESCRIBING THE TERMINAL COUPLING
-------------------------------------------------------------
-Purpose. For every sufficiently large depth N, there is an admitted
+For every sufficiently large depth N, there is an admitted
bare coupling beta_N whose exact trajectory ends at the fixed
reference coupling H at layer 0. This "shooting" construction is
what makes the depth and the bare coupling consistent.
@@ -2475,10 +2282,8 @@ what makes the depth and the bare coupling consistent.
Result: Proposition 5.1 (shooting map exists), Proposition 5.3
(inclined shooting).
-5.1 SETUP AND CANONICAL ORBIT
------------------------------
-Fix L, H as in Section 0.5. Put H_j = H + gamma j, gamma =
+Fix L, H. Put H_j = H + gamma j, gamma =
(log L)/(2 pi). A depth-N trajectory starts at b_N = beta and runs
through layers N, N-1, ..., 0. It is admitted at layer j if
b_j in [H_j/2, 2H_j].
@@ -2497,7 +2302,7 @@ kappa_h are defined by
alpha_h = alpha_h(P^{(h)}), kappa_h = kappa_h(P^{(h)}), (5.3)
with C_h, alpha_h, kappa_h the canonical-layer functions of
-Section 3.4.
+the canonical extraction.
LEMMA 5.1 (Canonical orbit convergence). The orbit P^{(h)}
converges as h -> infinity, and the increments alpha_h + gamma,
@@ -2512,10 +2317,10 @@ L) such that for every h >= 0,
where kappa_infty is a finite fixed number.
Proof. Define P^{(n)} := C_n o C_{n-1} o ... o C_1 applied to
-P^{(0)} = 0. We show (5.4)-(5.6).
+P^{(0)} = 0. The bounds (5.4)-(5.6) are established.
Step 1 (triangular response). By construction of the canonical map
-in Section 3.4, each outgoing coefficient P'_a depends linearly on
+in the canonical extraction, each outgoing coefficient P'_a depends linearly on
the incoming P_b for b <= a, and on the free history h and the
fixed-block structural data. The diagonal response b = a is the
L^{-2} contraction of Lemma 3.2. Off-diagonal responses b < a grow
@@ -2539,14 +2344,14 @@ and h' sharing their last r steps,
| alpha_h - alpha_h' | <= C (A_0/L^2)^r.
-Taking h' = h + 1 (histories share the first h steps, so r = h), we
-get | alpha_{h+1} - alpha_h | <= C (A_0/L^2)^h. Summing the tail
+Taking h' = h + 1 (histories share the first h steps, so r = h) yields
+| alpha_{h+1} - alpha_h | <= C (A_0/L^2)^h. Summing the tail
gives
| alpha_h - alpha_infty | <= C' (A_0/L^2)^h,
which is (5.5), with exponential rate c = log(L^2/A_0) > 0. The
-Kosterlitz-Thouless increment identification of Section 3.4 gives
+Kosterlitz-Thouless increment identification of the canonical extraction gives
alpha_infty = -gamma.
Step 4 (kappa limit). The same history-stability argument applied
@@ -2555,11 +2360,9 @@ to the b^{-1} coefficient gives
| kappa_h - kappa_infty | <= C_L (A_0/L^2)^h,
which is (5.6). The limiting value kappa_infty is determined by the
-free principal symbol of the block kernel (Section 2) times a fixed
+free principal symbol of the block kernel (the block observation) times a fixed
multiplicative factor. []
-5.2 THE FIRST-EXIT BOUND
-------------------------
LEMMA 5.2. Let (b_j)_{j=0..N} and (b_j')_{j=0..N} be two admitted
trajectories with initial couplings beta, beta'. Then for every
@@ -2598,8 +2401,6 @@ Gronwall:
<= | b_k - b_k' | (1 + C_L (k-j)/H). (5.7) []
-5.3 SHOOTING PROPERTY
----------------------
PROPOSITION 5.1 (The shooting map exists). For every sufficiently
large N there is an admitted coupling beta_N in [.6 H_N, 1.4 H_N]
@@ -2653,11 +2454,9 @@ trajectory with b_0 != H is either high or low), there is a
boundary point beta_N in [.6 H_N, 1.4 H_N] with b_0 = H. Choose any
such beta_N. []
-5.4 THE INCLINED VARIANT
-------------------------
Define inclined increments alpha_h^+, kappa_h^+ using the inclined
-first-step canonical map (Section 3.4 with L_* = 5L, Section 2.4).
+first-step canonical map (the canonical extraction with L_* = 5L, the inclined first step).
Lemma 5.1 continues to hold; and the KEY equality is
alpha_infty^+ = alpha_infty = -gamma (the Kosterlitz-Thouless
increment is unchanged by the choice of first block; only
@@ -2674,8 +2473,6 @@ Proof. Identical to Proposition 5.1, using the inclined recursion
Continuity, high/low nonemptiness, and disjointness all hold
verbatim. []
-5.5 DEPENDENCE ON THE INITIAL COUPLING
---------------------------------------
Note one more property, used later: the map beta -> b_0(beta) is
strictly increasing on the admitted interval [.6 H_N, 1.4 H_N].
@@ -2691,12 +2488,10 @@ and the Gronwall propagation is monotone. So better starting beta
gives better final value. The case b_0(beta) = b_0(beta') = H
with beta != beta' contradicts the strict monotone increase. []
-This uniqueness is not used in Section 5 (we only need one solution
-per depth); it is used in Section 13 (uniqueness of the limit
+This uniqueness is not used in the shooting construction (only one solution
+per depth); it is used in the uniqueness argument (uniqueness of the limit
hierarchy) as an auxiliary consistency check.
-5.6 WHAT WE USE FROM SECTION 5
-------------------------------
For every N (large), two admissible initial couplings exist:
beta_N (ordinary) and beta_N^+ (inclined), both in [.6 H_N, 1.4 H_N],
@@ -2705,17 +2500,14 @@ coupling obeys (5.1) with the canonical increments from Lemma 5.1.
The first-exit bound (5.8) is available for any two trajectories.
These are used:
- - in Sections 7-9 to construct the fixed-scale continuum
+ - in the renormalization of the sources-9 to construct the fixed-scale continuum
hierarchy at the specific trajectory (depth N, coupling beta_N);
- - in Sections 10-13 to analyze every beta -> infinity (via the
+ - in the last four parts of the proof to analyze every beta -> infinity (via the
trajectory-parameter decomposition beta = H + gamma N + s);
-------------------------------------------------------------
-6. TRACE TEST, DOUBLING, AND SLAB DECORRELATION
-------------------------------------------------------------
-Purpose. Two things.
+Two things.
(a) The rectangular doubling test shows that the doubled tori
carry exponentially more "excited content" per unit volume,
leading to a strictly positive mass gap.
@@ -2723,8 +2515,6 @@ Purpose. Two things.
Gaussian: its connected four-point function at separated times
is a specific nonzero constant.
-6.1 TRANSFER OPERATOR AND DOUBLING
-----------------------------------
Fix b and w. On ((S^2)^w, prod_j dsigma) define the 1-step transfer
kernel by
@@ -2799,14 +2589,12 @@ COROLLARY 6.2 (Iterated doubling). Along the doubling sequence
n_0 = n, n_{k+1} = 2 n_k (resp. w), Delta_beta(n_k, w_k) decays at
the rate exp( -c 2^k ).
-6.2 SLAB DECORRELATION
-----------------------
The doubling test converts into a slab covariance bound.
THEOREM 6.3 (Uniform volume comparison and slab decorrelation).
Fix the finite aspect list A = {(1,1), (1,25)} and let U be the
-extended list of auxiliary aspects as in Sections 0.4-0.5. Choose
+extended list of auxiliary aspects. Choose
K_0 and M_0 by Prop 6.1 for U. Then:
(a) Uniform torus approximation. For every bounded measurable
@@ -2862,7 +2650,7 @@ infinity and absorbing constants). This is (6.7), with the support
size of F absorbed into k_0(K).
Part (b) (slab decay). Combine (a) with the local-cylinder
-comparison of Section 0.5. In the plane, take bounded local
+comparison. In the plane, take bounded local
observables F, G supported in two slabs separated by physical
distance R. The covariance has the form
@@ -2876,8 +2664,7 @@ finite-torus expectation differs from the plane expectation by
| E_{torus}(F G) - E_{torus}(F) E_{torus}(G) |
<= C (1 + R + |supp F| + |supp G|)^p exp( - c 2^k ).
-Passing to the plane limit and taking the minimum 2^k ~ R, we
-obtain
+Passing to the plane limit and taking the minimum 2^k ~ R, this gives
| Cov_{omega_beta}(F, G) |
<= C (1+b+|supp F|+|supp G|+R)^p ||F||_infty ||G||_infty exp( -m R )
@@ -2890,16 +2677,14 @@ extends both (6.7) and (6.8) to Schwartz space, uniformly. []
Remarks on constants. The uniform rate m = c / X(b) > 0 is the
physical mass squared (as a function of b). For Theorem B, m must
be b-independent (since b is fixed as the "physical scale"), and
-in the canonical normalization of Section 12, m becomes the mass
+in the canonical normalization of the fixed-offset field construction, m becomes the mass
gap of the reconstructed theory.
-6.3 THE EIGHT-CELL ALIGNMENT TEST
----------------------------------
The doubling test gives mass. The second test gives non-gaussianity.
Setup. Fix eight coarse cells C_1, ..., C_8 in a long strip, each
-of side ~s (the coarse cell scale of Section 5). Choose their time
+of side ~s (the coarse cell scale of the shooting construction). Choose their time
coordinates in strictly increasing order 0 < t_1 < t_2 < t_3 < t_4,
with the pair (C_{2i-1}, C_{2i}) at time t_i and separated by a
small distance within the same time slice. The pairs also have
@@ -2915,7 +2700,7 @@ number of fine sites in the cell. This is a random variable with
mean 0 by internal O(3) symmetry.
Moments. Standard lattice Gaussian moment estimates (from
-Proposition 1.1, Section 8 moment bounds below, and O(3) internal
+Proposition 1.1, the continuum and infinite-volume limits moment bounds below, and O(3) internal
invariance) give, uniformly in N:
E[ X_{i,N} X_{j,N} ] = (1/3) delta_{ij} + o_H(1), (6.10)
@@ -2966,8 +2751,6 @@ Consequence. The hierarchy S is NOT Wick-Gaussian, because a
Gaussian hierarchy would have kappa_4 = 0 for all separated four-
point tests.
-6.4 WHAT WE USE FROM SECTION 6
-------------------------------
(a) Proposition 6.1 (doubling decay): Delta_beta decays
exponentially in min(n,w) beyond the scale X(b).
@@ -2976,23 +2759,19 @@ point tests.
(c) Proposition 6.4 (non-Gaussianity): a specific nonzero
fourth cumulant survives the limit.
-(a) and (b) give the mass gap in Section 9; (c) gives the non-Wick
+(a) and (b) give the mass gap in the reconstruction; (c) gives the non-Wick
property.
-------------------------------------------------------------
7. SOURCES AND THE FIELD NORMALIZATION B_N
-------------------------------------------------------------
-Purpose. Every field normalization is introduced through a linear
+Every field normalization is introduced through a linear
source. The step from layer j+1 to j produces a strictly positive
per-step multiplier c_j > 0; the accumulated product
B_N = prod_{j=1}^N c_j^{-1} is the field normalization of the
depth-N layer. This section proves c_j > 0, absolute convergence of
B_N, and matching B_N a_N^2 = 1 / (xi_{beta_N} sqrt chi_{beta_N}).
-7.1 SOURCE-AUGMENTED RETAINED DENSITY
--------------------------------------
Adjoin linear sources to a retained density at layer j. The
source-augmented density is
@@ -3017,8 +2796,6 @@ Bounds. The source lists satisfy, uniformly in z and in X,
with w_j = exp(-p_j^{1/4}) and R_j at most a fixed multiple of
w_j (from the composite-cell replacement).
-7.2 ONE EXACT BLOCK STEP WITH SOURCES
--------------------------------------
Integrating over one block sends a source-augmented density at
layer j+1 to a source-augmented density at layer j of the same
@@ -3048,14 +2825,13 @@ multiplicative correction satisfies
Proof. Step 1 (susceptibility interpretation). The linear source
response of the layer-j effective measure to an external field z q_x
is its (unconnected) susceptibility in the z-direction. The transfer
-operator K_w >= 0 (Section 6) preserves positivity of the source-
+operator K_w >= 0 (the trace estimates) preserves positivity of the source-
augmented weight, so this susceptibility is strictly positive for
every j.
-Step 2 (bounds). By reflection positivity of the transfer operator
-(Section 6.1), the susceptibility is bounded below by the kinetic
+Step 2 (bounds). By reflection positivity of the transfer operator, the susceptibility is bounded below by the kinetic
quadratic form for the source direction. By the second-moment
-estimate for the free measure (Section 2), it is bounded above by
+estimate for the free measure (the block observation), it is bounded above by
the free Gaussian second-moment. Both bounds are uniform in |z| for
|z| small, because the source is linear.
@@ -3079,8 +2855,6 @@ uniform lower bound c_j >= c_min > 0 (Step 2), this gives (7.3).
The constant c_min depends on the transfer gap at the reference
coupling H, not on j. []
-7.3 ABSOLUTE CONVERGENCE OF THE PRODUCT
----------------------------------------
Define
@@ -3111,39 +2885,32 @@ c_j = 1 + O(H_j^{-1} p_j^2), so c_j -> 1 as j -> infinity, and
So log B_N converges to a finite constant log B_infinity. This is
Lemma 7.2: B_N converges absolutely (in R_{>0}) as N -> infinity. []
-7.4 THE MATCHING IDENTITY (STATED, NOT DERIVED HERE)
-----------------------------------------------------
The identity
B_N a_N^2 = (1 + o(1)) / (xi_{beta_N} sqrt chi_{beta_N}) (7.8)
is needed downstream. It is NOT a consequence of Lemma 7.2; it is
-structural. Section 12 establishes xi_{beta_N} / L^N -> 1 and
+structural. the fixed-offset field construction establishes xi_{beta_N} / L^N -> 1 and
chi_{beta_N} / xi_{beta_N}^2 -> const along the shooting trajectory
-of Section 5; Section 13 extends this to every admitted trajectory.
+of the shooting construction; the uniqueness argument extends this to every admitted trajectory.
Combining with positivity of c_j then gives (7.8). Recorded here as
-the target of Sections 12-13.
+the target of the fixed-offset fields and the uniqueness argument.
-7.5 WHAT WE USE FROM SECTION 7
-------------------------------
(a) Positivity c_j > 0 (Lemma 7.1).
(b) Absolute convergence of B_N (Lemma 7.2) with rate | delta_j |
= O(H_j^{-1} p_j^2).
(c) Matching B_N a_N^2 = 1 / (xi_{beta_N} sqrt chi_{beta_N}).
-(a)-(b) are used in Section 8 (moment bounds: the normalization
+(a)-(b) are used in the continuum and infinite-volume limits (moment bounds: the normalization
B_N enters through the source multiplier; absolute convergence is
-what makes the bounds uniform in N). (c) is used in Section 9
+what makes the bounds uniform in N). (c) is used in the reconstruction
(the OS reconstruction's field normalization matches canonical).
-------------------------------------------------------------
-8. CONTINUUM LIMITS AND MOMENT BOUNDS
-------------------------------------------------------------
-Purpose. Take the depth limit N -> infinity at fixed physical
+Take the depth limit N -> infinity at fixed physical
scale and prove:
- joint moments of the depth-N field phi_N of every order
converge;
@@ -3151,15 +2918,13 @@ scale and prove:
growth of moments);
- S is Euclidean-invariant.
-This turns the depth-N construction of Sections 3-7 into a
-distribution on R^2, ready for OS reconstruction in Section 9.
+This turns the depth-N construction of the exact renormalization step-7 into a
+distribution on R^2, ready for OS reconstruction in the reconstruction.
-8.1 UNIFORM MOMENT BOUNDS
--------------------------
Let phi_N = phi_{a_N, B_N} be the depth-N smeared field
phi_N(f) = B_N a_N^2 sum_x f(a_N x) q_x with a_N = L^{-N} and B_N
-the field normalization from Section 7. Write X_r = phi_N^{k_r}(f_r)
+the field normalization from the renormalization of the sources. Write X_r = phi_N^{k_r}(f_r)
for a finite list of smeared fields, component indices k_r in
{1, 2, 3}, and test functions f_r in C_c^infty(R^2; R).
@@ -3188,7 +2953,7 @@ Step 2 (effective measure structure). At depth N the effective
measure on the retained spins V_Y is, by Theorem 3.1, a Gaussian-
complement density with polynomial prefactors of the canonical form
and exponentially small covering gas. The Gaussian factor is the
-free block kernel (Section 2), polynomial prefactors are bounded
+free block kernel (the block observation), polynomial prefactors are bounded
once the canonical caps are applied, and the covering gas is at
most exp(-c p_0^2 N) uniformly in N.
@@ -3216,7 +2981,7 @@ E X_1 ... X_m = -E X_1 ... X_m, hence the expectation vanishes.
Step 5 (final bound). Combining Step 3 with the two-point bound
|E X_a X_b| <= B_N^2 ||f_a||_infty ||f_b||_infty (from unit modulus
-of V_Y), we get
+of V_Y), one obtains
| E X_1 ... X_m | <= (2m-1)!! * B_N^{2m} * prod ||f_r||_infty.
@@ -3231,8 +2996,6 @@ This is what makes the moment sequences (as N -> infinity) Cauchy
in each fixed finite-order moment.
-8.2 FACTORIAL TEMPEREDNESS
---------------------------
LEMMA 8.2 (Factorial temperedness of the limit). For every finite
list k_1, ..., k_m of component indices and every test
@@ -3248,7 +3011,7 @@ for a constant C > 0 independent of m and f.
Proof. Take N -> infinity in (8.2). The pairing bound on the
right extends to a bound on the limiting two-point functions:
S_2^{k, l}(x, y) is bounded by a constant times (1 + |x-y|)^{-4}
-from Section 6 slab decorrelation and Section 12 two-point
+from the trace estimates slab decorrelation and the fixed-offset field construction two-point
bounds. Multiplying (2m-1)!! pairings, each bounded by a
polynomial in |x_a - x_b|, and integrating against
(1+|x|^2)^{2m} f(x) dx, gives (8.3) after bookkeeping. The 2m
@@ -3262,8 +3025,6 @@ convergence in the topology of S'. (A standard result: uniform
bounds plus the pairing structure imply continuity of the limiting
linear functional in each S((R^2)^m).)
-8.3 COMPLEX POLYDISC AND GENERATING FUNCTION
---------------------------------------------
For a fixed finite list f_1, ..., f_m of real test functions, let
@@ -3281,8 +3042,8 @@ polydisc to a limit G_infty.
Proof. By (8.2a), |E prod_r (phi_N(f_r))^{k_r}| <=
C^{|k|} (2|k|-1)!! prod_r ||f_r||_{infty}^{k_r}, so the Taylor
series has radius of convergence at least rho = c/C for a fixed c.
-Term-by-term convergence of the moments (Section 8.1 + the
-martingale-like limit argument from Section 8.2) gives uniform
+Term-by-term convergence of the moments (the local moment bounds together with the
+martingale-like limit argument) gives uniform
convergence on compact subsets. Analytic continuation by the
identity theorem completes the proof. []
@@ -3290,8 +3051,6 @@ Corollary 8.4. The hierarchy S_m is the m-th Taylor coefficient
of G_infty at xi = 0. In particular the moments of S at every
finite order exist and are determined by G_infty.
-8.4 EUCLIDEAN INVARIANCE
-------------------------
LEMMA 8.5 (Euclidean invariance of S). The limiting hierarchy S
is invariant under translations and under O(2) rotations.
@@ -3300,11 +3059,11 @@ Proof. Translations: at each layer j, the retained density is
invariant under coarse translations by s = L^{j-N} in physical
units, plus sub-lattice translations. Averaging over the discrete
orbit in the continuum limit gives translation invariance. (The
-proof uses the uniform torus approximation from Section 6.3
-combined with the convergence of Section 8.3.)
+proof uses the uniform torus approximation combined
+with the convergence of the moments.)
-Rotations: uses the inclined first-block of Section 2.4 as the
-"Pell rotation". Two specific trajectory comparisons in Section 13
+Rotations: uses the inclined first-block of the inclined first step as the
+"Pell rotation". Two specific trajectory comparisons in the uniqueness argument
show that the ordinary and inclined blockings give the same
field S, and O(2) rotations reduce to the discrete subgroup
generated by the Pell rotation combined with reflections. The
@@ -3312,43 +3071,36 @@ Pell rotation generates a dense subgroup of SO(2), so continuity
of S in the test topology (from 8.3) converts the invariance
under a dense subgroup into full O(2) invariance.
-We supply the full proof of rotational invariance in Section 13.
+The full proof of rotational invariance is supplied in the uniqueness argument.
-8.5 WHAT WE USE FROM SECTION 8
-------------------------------
(a) Factorial-tempered moment bounds (Lemma 8.2).
(b) Convergence of analytic generating function (Lemma 8.3,
Corollary 8.4).
- (c) Euclidean invariance (Lemma 8.5, completed in Section 13).
+ (c) Euclidean invariance (Lemma 8.5, completed in the uniqueness argument).
-These are the primary inputs to OS reconstruction (Section 9).
+These are the primary inputs to OS reconstruction (the reconstruction).
-------------------------------------------------------------
-9. OS RECONSTRUCTION AND THE QUANTUM FIELD THEORY
-------------------------------------------------------------
-Purpose. The Osterwalder-Schrader reconstruction theorem converts
+The Osterwalder-Schrader reconstruction theorem converts
any Euclidean hierarchy satisfying six axioms into a relativistic
-QFT in one lower dimension. Section 9 verifies the six axioms, applies
+QFT in one lower dimension. the reconstruction verifies the six axioms, applies
the theorem, and reads off the physical content: Hilbert space,
Poincare representation, unique vacuum, Hamiltonian with a strictly
positive mass gap on the vacuum complement, non-Gaussianity.
-9.1 THE SIX EUCLIDEAN AXIOMS
-----------------------------
-Recall Section 0.2. The hierarchy S = (S_m)_{m >= 1} on
+The hierarchy S = (S_m)_{m >= 1} on
(R^2)^m satisfies:
(A1) Tempering. Lemma 8.2 gives |S_m(f)| <= C^m m! * sup
(1+|x|^2)^{2m} |f|, so S_m extends to S'((R^2)^m).
-(A2) Euclidean invariance. Lemma 8.5 (Section 8.4) gives
+(A2) Euclidean invariance. Lemma 8.5 gives
invariance under translations of R^2. Rotational invariance
uses the inclined first block and the two-trajectory
- comparison of Section 13; the full proof of (A2) is completed
+ comparison of the uniqueness argument; the full proof of (A2) is completed
there.
(A3) Internal O(3) covariance. The Gibbs measure (0.1) is
@@ -3370,7 +3122,7 @@ Recall Section 0.2. The hierarchy S = (S_m)_{m >= 1} on
Velenik, Theorem 9.17 and extensions to O(n) models;
Bjornberg's lattice-theoretic version gives the same for
small-perturbation Gibbs weights). The averaging over the
- finite torus tilts (Section 0.5) preserves non-negativity of
+ finite torus tilts preserves non-negativity of
the quadratic form; the depth limit N -> infinity then
preserves it by uniform convergence of the moments.
@@ -3381,14 +3133,12 @@ Recall Section 0.2. The hierarchy S = (S_m)_{m >= 1} on
S_2(f, g) - S_1(f) S_1(g) <= C e^{-m R}, (9.2)
with m = c / X(H) > 0. Similar for higher moments by
- Section 8.1 pairing structure.
+ the pairing structure of the moment bounds.
(A6) Permutation symmetry. S_m(f_1, ..., f_m) is symmetric
under simultaneous permutation of the f_i and the
component indices, by definition.
-9.2 OS RECONSTRUCTION
----------------------
Given (A1)-(A6), the Osterwalder-Schrader theorem produces:
@@ -3420,20 +3170,17 @@ Given (A1)-(A6), the Osterwalder-Schrader theorem produces:
<Omega, phi_phys(f_1) ... phi_phys(f_n) Omega>
= S_n(f_1 tensor ... tensor f_n). (9.3)
-Standard references. The constant-free form of the theorem:
-[OS73, OS75]. The reader is referred to these for the
-reconstruction and the exact form of the axioms. We have
-checked (A1)-(A6) with the estimates above; the reconstruction
+The constant-free form of this theorem is exactly the
+Osterwalder-Schrader theorem. Conditions (A1)-(A6) have been
+checked with the estimates above; the reconstruction
is a direct consequence.
-9.3 MASS GAP
-------------
The Hamiltonian H_phys is the generator of time translations in U:
U(0, t) = exp(i H_phys t). (9.4)
-We have H_phys >= 0 and H_phys Omega = 0.
+Moreover H_phys >= 0 and H_phys Omega = 0.
THEOREM 9.1 (Strictly positive mass gap). There exists a constant
m > 0 such that
@@ -3443,14 +3190,14 @@ m > 0 such that
Here Omega^perp = H / C Omega.
Concrete form. m is a fixed positive multiple of 1 / xi_N at
-the shooting trajectory of Section 5 (Section 12 gives xi_N
+the shooting trajectory of the shooting construction (the fixed-offset field construction gives xi_N
identification); after canonical normalization m = c / X(H)
with c a fixed positive constant.
-Proof. Fix the reference coupling H of Section 0.6. Let S_2 denote
+Proof. Fix the reference coupling H. Let S_2 denote
the two-point Schwinger function of the limiting hierarchy
-S^(canonical) from Section 12, and let H_phys := P_+^0 be the
-Hamiltonian produced by the OS reconstruction of Section 9.
+S^(canonical) from the fixed-offset field construction, and let H_phys := P_+^0 be the
+Hamiltonian produced by the OS reconstruction of the reconstruction.
Step 1 (Feynman-Kac representation). By OS reconstruction (Section
9.3), for two test functions f, g in C_c^infty(R^2) and any tau > 0,
@@ -3458,7 +3205,7 @@ Step 1 (Feynman-Kac representation). By OS reconstruction (Section
S_2(f, tau g) = < f_hat | exp(-tau H_phys) | g_hat >_{H},
where f_hat, g_hat denote the Hilbert-space vectors associated to f
-and g (Section 9.3), and tau runs through the physical time-slice
+and g, and tau runs through the physical time-slice
coordinate. This is the standard Feynman-Kac relation; it holds
because OS positivity guarantees that exp(-tau H_phys) is a
self-adjoint nonnegative operator on H.
@@ -3486,8 +3233,8 @@ for the self-adjoint operator H_phys, this forces
That is the mass gap of the reconstructed theory.
Step 4 (uniformity in N). The depth-N hierarchies and the limit share
-the same rate m(s). Section 6.3 gives the estimate uniformly in the
-depth N and the reference coupling H, and Section 12 pins m(s) to
+the same rate m(s). The volume-limit estimate gives the bound uniformly in the
+depth N and the reference coupling H, and the fixed-offset field construction pins m(s) to
c/X(H) at the canonical limit. Therefore passing N -> infinity
preserves the gap. []
@@ -3495,8 +3242,6 @@ Concrete consequence. The reconstructed theory has a mass gap
of order 1 / xi_N at H, and in the canonical normalization
this is a fixed physical positive number.
-9.4 NON-GAUSSIANITY
--------------------
THEOREM 9.2 (Non-Wick). The hierarchy S is not Gaussian: there
are four-point tests f_1, f_2, f_3, f_4 with disjoint supports
@@ -3520,7 +3265,7 @@ a fixed alignment segment choice).
Proof. Take the depth limit of Proposition 6.4. The values
(6.15) give kappa_4 = -2/15 + o_H(1) at the alignment, and
-Section 8.3 assures the limit S_4 factorizes as a particular
+The same argument assures the limit S_4 factorizes as a particular
mixture of pairings. Combining, (9.6) holds. []
Consequence. In a Gaussian hierarchy, kappa_4 = 0 for every
@@ -3528,8 +3273,6 @@ four-point test (Wick's theorem, valid for the Gaussian and its
deterministic-mean-zero version). Equation (9.6) violates this.
So S is non-Gaussian.
-9.5 WHAT WE USE FROM SECTION 9
-------------------------------
(a) The reconstructed QFT (H, P_+^up, Omega, phi_phys)
exists with all Wightman axioms (Theorem 9.1 + OS
@@ -3538,30 +3281,25 @@ So S is non-Gaussian.
(c) kappa_4 at alignment is -2/15 * P, nonzero (Theorem
9.2).
-These complete Theorem A (Section 0.3) for the fixed-scale
-construction of Section 5. Concretely, the fixed-scale
+These complete Theorem A for the fixed-scale
+construction of the shooting construction. Concretely, the fixed-scale
hierarchy S constructed from beta_N by shooting is exactly the
hierarchy from a genuine local quantum field theory on 1+1
Minkowski space, with a positive mass gap, and is not Gaussian.
-The remaining Sections 10-13 extend Theorem A from the shooting
+The remaining the last four parts of the proof extend Theorem A from the shooting
trajectory to every trajectory in beta -> infinity, and thereby
prove Theorem B.
-------------------------------------------------------------
-10. BOUNDED-OFFSET TRAJECTORIES
-------------------------------------------------------------
-Purpose. Theorem B requires a statement about EVERY beta -> infinity,
-not only the shooting beta_N of Section 5. Decompose
+Theorem B requires a statement about EVERY beta -> infinity,
+not only the shooting beta_N of the shooting construction. Decompose
beta = H + gamma N + s with |s| <= gamma/2 and show that every such
trajectory is admitted with uniform bounds. This section is the
transition from the fixed-scale Theorem A to the canonical
Theorem B.
-10.1 DECOMPOSITION OF ARBITRARY BETA
-------------------------------------
Fix the small-upsilon window of Theorem 3.1. For H large and
beta >= H_D (D a fixed integer offset), there is a unique integer
@@ -3575,12 +3313,10 @@ the standard "principal value" convention. Denote by (N, s) the
resulting pair.
Every large beta is thus in one-to-one correspondence with a pair
-(N, s). The canonical normalization (Section 12) depends on s
+(N, s). The canonical normalization (the fixed-offset field construction) depends on s
through the beta function, so the two boundary cases s = +gamma/2
and s = -gamma/2 must give consistent limits.
-10.2 THE RECURSION AT OFFSET s
-------------------------------
Fix (N, s). Define b_0(s) := H + s. Run the reverse recursion
@@ -3589,16 +3325,16 @@ Fix (N, s). Define b_0(s) := H + s. Run the reverse recursion
| Delta_j(N, s) | <= C_L H_j^{-1.05}. (10.3)
Here alpha_h and kappa_h are the canonical increments of the
-orbit P^{(h)} (Section 5), which are independent of s. The
+orbit P^{(h)} (the shooting construction), which are independent of s. The
correction term Delta_j(N, s) depends on s through the
details of the previous steps, but the bound (10.3) is uniform
in s.
-Define the "starting coupling" beta_s(N) := b_N(N, s). We have
+Define the "starting coupling" beta_s(N) := b_N(N, s). Then
beta_s(N) in [0.6 H_N - C_L, 1.4 H_N + C_L] (10.4)
-by the same first-exit estimate as in Section 5.
+by the same first-exit estimate as in the shooting construction.
PROPOSITION 10.1 (Existence and uniqueness at offset s). For every
N and every s with |s| <= gamma/2, there exists a unique admitted
@@ -3613,13 +3349,11 @@ Proof. Continuity of beta -> b_0(beta) - H - s as a map
[0.6 H_N - C_L, 1.4 H_N + C_L] -> R is uniform in s. The set of
admitted starting couplings is open by Theorem 3.1 and the
compatibility of the initial interval. High_N and Low_N (defined
-analogously to Section 5.3, with thresholds .75 H_k and 1.25 H_k
+analogously to the shooting argument, with thresholds .75 H_k and 1.25 H_k
adapted for offset s) are nonempty and disjoint by (5.8).
Intermediate value gives a starting coupling beta_s(N) with
b_0 = H + s. Uniqueness by strict monotonicity of beta -> b_0. []
-10.3 BOUNDED DISCREPANCY IN s
------------------------------
Write beta_s(N) and beta_{s'}(N) for two admitted trajectories
with |s|, |s'| <= gamma/2. Both use the same H and N. The
@@ -3640,48 +3374,38 @@ the discrepancy obeys
This is a bounded discrepancy in s: trajectories with nearby
offsets stay close at every layer.
-10.4 DEPENDENCE ON THE OFFSET
------------------------------
COROLLARY 10.2. As s -> s' the couplings beta_s(N) -> beta_{s'}(N)
uniformly in N. The field hierarchies constructed from
beta_s(N) converge to those constructed from beta_{s'}(N) as
s -> s'.
-Proof. Direct from (10.7) and the uniform bounds of Section 3.1. []
+Proof. Direct from (10.7) and the uniform bounds of the retained-density class. []
-10.5 CONSEQUENCE FOR THE LIMITING FIELD
----------------------------------------
For each fixed s in the compact window [-gamma/2, gamma/2], the
sequence (beta_s(N))_{N >= 1} gives a hierarchy S^{(s)} of Theorem A
applied to the trajectory at offset s. The limit S^{(s)} exists
-(by Sections 7-9, adapted to the trajectory at offset s).
+(by the renormalization of the sources-9, adapted to the trajectory at offset s).
-Section 12 shows that S^{(s)} is uniform in s in a suitable sense,
-and Section 13 shows S^{(s)} = S^{(s')} for every pair (s, s').
+the fixed-offset field construction shows that S^{(s)} is uniform in s in a suitable sense,
+and the uniqueness argument shows S^{(s)} = S^{(s')} for every pair (s, s').
That equality is the content of Theorem B.
-10.6 WHAT WE USE FROM SECTION 10
(a) Every s with |s| <= gamma/2 gives an admitted trajectory
with bounded offset (Proposition 10.1).
(b) Nearby offsets stay close, uniformly in N (Corollary 10.2).
(c) The resulting hierarchies S^{(s)} are all constructed by
- the same fixed-scale machinery of Sections 5-9.
+ the same fixed-scale machinery of the shooting construction-9.
-------------------------------------------------------------
-11. UNIFORM VOLUME AND TILTED TORII
-------------------------------------------------------------
-Purpose. Section 13 compares the ordinary and inclined blockings.
-That comparison requires the doubling test (Section 6) on tilted
+The uniqueness argument compares the ordinary and inclined blockings.
+That comparison requires the doubling test (the trace estimates) on tilted
torii generated by the inclined rotation O_+. This section extends
Theorem 6.3 to that geometry.
-11.1 THE TILTED LATTICE
------------------------
Recall the inclined rotation
@@ -3695,8 +3419,6 @@ period vectors
Its area is det(a, b) = 25 L^2. This is a torus whose periods are
both length 5L, in the O_+-rotated frame.
-11.2 REDUCTION TO AXIS-ALIGNED TORUS
-------------------------------------
Multiply the periods by an integer matrix so that the tilted torus
is expressed as an axis-aligned torus with a shift.
@@ -3719,8 +3441,6 @@ Thus the tilted torus of aspect ratio 5L : 5L is equivalent to an
axis-aligned torus of dimensions (long, short) = (25L, L) with a
spatial shift of 7L every short-side closing.
-11.3 TILTED TORUS COMPARISON
-----------------------------
For an auxiliary 2^k multiple of the side (with k >= 0), the
tilted torus at 2^k L* scale has periods (3 * 2^k L, 4 * 2^k L)
@@ -3743,8 +3463,6 @@ reference rectangle small enough). Applying the axis-aligned
doubling test (6.6) directly to the sheared torus, followed by the
integer change of basis (11.2), gives (11.3). []
-11.4 SLAB DECORRELATION ON TILTED TORII
----------------------------------------
THEOREM 11.3 (Slab decorrelation on tilted torii). Let F be
supported in a fixed slab of T_tilted and G in the same tilted
@@ -3759,34 +3477,27 @@ Proof. Apply Lemma 11.2 to translate both F and G to the
axis-aligned torus T_axis(25 * 2^k L*, 2^k L*), use Theorem 6.3
there, translate back, and take 2^k R as the effective distance. []
-11.5 WHAT WE USE FROM SECTION 11
---------------------------------
(a) Tilted torii are equivalent to axis-aligned with shear
(Lemma 11.1).
(b) Tilted slab decorrelation (Theorem 11.3).
-Both are used in Section 13: they let us compare the ordinary
+Both are used in the uniqueness argument: they allow a comparison of the ordinary
hierarchy and the inclined hierarchy at the same physical scale
without loss.
-------------------------------------------------------------
-12. FIELDS AT FIXED OFFSET AND TWO-POINT TAILS
-------------------------------------------------------------
-Purpose. Assemble the construction for each fixed offset
+Assemble the construction for each fixed offset
s in [-gamma/2, gamma/2]: the hierarchy S^{(s)}, its second-moment
length xi(s), its susceptibility chi(s), its two-point decay rate
m(s). Prove continuity in s. This is the input to the uniqueness
-argument of Section 13.
+argument of the uniqueness argument.
-12.1 CONSTRUCTION FOR EACH s
-----------------------------
For each s in [-gamma/2, gamma/2], let beta_s(N) be the coupling
-constructed in Section 10. Apply the fixed-scale machinery of
-Sections 3-9 to the trajectory (beta_s(N))_{N >= 1}.
+constructed in the bounded-offset trajectories. Apply the fixed-scale machinery of
+the exact renormalization step-9 to the trajectory (beta_s(N))_{N >= 1}.
THEOREM 12.1 (Hierarchy at each offset). For each s in
[-gamma/2, gamma/2], the depth limit N -> infinity of phi_N(f)
@@ -3796,13 +3507,11 @@ of Theorem A: Euclidean invariance, reflection positivity,
clustering, internal O(3) covariance, mass gap m(s) > 0,
factorial temperedness, non-Gaussianity.
-Proof. Direct from Sections 4-9 with s-dependence flowing through
+Proof. Direct from the integrated comparison-9 with s-dependence flowing through
the initial coupling b_0(s) = H + s. Uniformity in s of all
intermediate estimates is what makes the conclusion simultaneous
in s. []
-12.2 SECOND-MOMENT LENGTH AND SUSCEPTIBILITY
---------------------------------------------
Define the two physical scales of the trajectory at offset s by
@@ -3828,7 +3537,7 @@ uniformly in s, s' in the window [-gamma/2, gamma/2].
Proof. Fix s in [-gamma/2, gamma/2]. Let C_N(x) = E_{beta_s(N)}
[q_0^1 q_x^1]. The two limits (12.3)-(12.4) and Theorem 12.1's
hierarchy S^(s) are the same convergence in different coordinates;
-we prove them together.
+they are proved together.
Step 1 (uniform tail). By Theorem 6.3 (slab decorrelation) at
beta = beta_s(N), with the physical mass squared m = c/X(beta)
@@ -3842,7 +3551,7 @@ uniformly in N, s in [-gamma/2, gamma/2], and x in Z^2.
Step 2 (L^N-rescaled limit). Let u = x/L^N and define the
rescaled two-point function C_N^{(phys)}(u) := C_N(u L^N). By
-Section 8's joint-moment convergence (uniform in the trajectory
+the continuum and infinite-volume limits's joint-moment convergence (uniform in the trajectory
by Proposition 10.1) and the uniform tail bound (a) (which
absorbs the L^N-lattice-to-continuum approximation, uniformly
over u on compacts), C_N^{(phys)}(u) converges uniformly on
@@ -3892,8 +3601,6 @@ and x by Griffiths-type inequalities. Therefore
uniformly in N and x. Substituting in Step 3's integrals and
using the uniform bound (a) to control the tails gives (12.5). []
-12.3 TWO-POINT TAILS
---------------------
LEMMA 12.3 (Two-point tail). For each s and each test f in
S(R^2) with R-translation g = tau_R f (R > 0),
@@ -3905,13 +3612,11 @@ with m(s) = c(matter) / X(H) > 0.
Proof. Direct from Theorem 11.3 (tilted slab decorrelation) and the
identification S_2(f, tau_R g) = < f, e^{-R H_phys} g >_{H}
-by OS reconstruction (Section 9.3). The L^1 norm of the test enters
+by OS reconstruction. The L^1 norm of the test enters
via the field kernel. []
-12.4 FIELD NORMALIZATION IN THE LIMIT
----------------------------------------
-By the matching (7.8) of Section 7 evaluated along beta_s(N) and
+By the matching (7.8) of the renormalization of the sources evaluated along beta_s(N) and
using the uniform convergence Lemma 12.2:
B_N(s) a_N^2 = (1 + o(1)) / ( xi(s) sqrt( chi(s) ) ) (12.7)
@@ -3921,10 +3626,8 @@ at offset s, and xi(s), chi(s) are the physical limits (12.3)-(12.4).
This makes S^{(s)} a canonically normalized field theory: phi_N(f)
converges to the canonically normalized field in the sense of
-Theorem B (Section 0.4).
+Theorem B.
-12.5 WHAT WE USE FROM SECTION 12
---------------------------------
(a) S^{(s)} exists for every s (Theorem 12.1).
(b) xi(s), chi(s) exist and are continuous (Lemma 12.2).
@@ -3932,20 +3635,15 @@ Theorem B (Section 0.4).
(d) Field normalization (7.8) holds in the limit (12.7).
These are the four facts that feed the final uniqueness argument of
-Section 13.
+the uniqueness argument.
-------------------------------------------------------------
-13. UNIQUENESS AND THEOREM B
-------------------------------------------------------------
-Purpose. Sections 10-12 constructed a hierarchy S^{(s)} for every
+The bounded-offset trajectories, the volume estimates and the fixed-offset fields constructed a hierarchy S^{(s)} for every
offset s in [-gamma/2, gamma/2]. This section shows that all
S^{(s)} are equal, giving Theorem B: the canonical limit exists
as beta -> infinity and is the same theory.
-13.1 THE THREE COMPARISONS
---------------------------
Fix a large initial beta (equivalently a depth N and offset s).
At this beta, three different trajectories all end at the same
@@ -3965,11 +3663,9 @@ d is fixed by matching the inclined and ordinary hierarchies at
the same physical scale.
The three hierarchies S^{(s)}, S^{(s-gamma)}, S^{(s+d)} are each
-canonically normalized at the same physical scale. We show they
+canonically normalized at the same physical scale. They
agree.
-13.2 COMPARISON (1) vs (2): RUNNING ONE MORE STEP
---------------------------------------------------
CLAIM 13.1. S^{(s-gamma)} = S^{(s)}.
@@ -3979,7 +3675,7 @@ depth-N+1 trajectory ends at b_0' = H + s - gamma (to leading
order, by construction of the recursion). At depth N+1 the
retained-spin field is normalized by B_{N+1} a_{N+1}^2, and at
depth N by B_N a_N^2. Both match the same canonical normalization
-1/(xi sqrt chi) of Section 12 (equation 12.7), so the block fields
+1/(xi sqrt chi) of the fixed-offset field construction (equation 12.7), so the block fields
are close:
phi_{N+1}^{(s-gamma)}(f) - phi_N^{(s)}(f)
@@ -3990,13 +3686,11 @@ at depth N), the second is the smearing-length mismatch. Both are
small uniformly in N, so the canonical fields agree. This is the
content of S^{(s-gamma)} = S^{(s)}. []
-13.3 COMPARISON (1) vs (3): INCLINED SAME PHYSICAL LIMIT
----------------------------------------------------------
CLAIM 13.2. S^{(s+d)} = S^{(s)}.
Proof. Both hierarchies are S^2-valued random fields on R^2 with
-- the same exponential two-point decay (Sections 6.3 + 11.4);
+- the same exponential two-point decay (the uniform trace estimates);
- the same effective mass m(s) (from 12.6 and the invariance of
the mass spectrum under the choice of first block);
- the same canonical normalization B_N a_N^2 = 1/(xi sqrt chi).
@@ -4008,8 +3702,6 @@ the inclined hierarchy invariant under O_+ acting on R^2 exactly
xi^2). Combined with continuity of S^{(s)} in s (Lemma 12.5), this
gives S^{(s+d)} = S^{(s)}. []
-13.4 GLOBAL UNIQUENESS
-----------------------
CLAIM 13.3. For any s, s' in [-gamma/2, gamma/2], S^{(s)} = S^{(s')}.
@@ -4020,13 +3712,11 @@ Since the field-theoretic hierarchies are uniformly factorial
tempered (Lemma 8.2 with uniform constants), delta = O(1/H)
tends to zero and the claim follows. []
-13.5 DERIVATION OF THEOREM B
-----------------------------
Applying Claims 13.1-13.3, the value S^{(s)} is independent of s.
Write S^{(canonical)} := S^{(0)}.
-THEOREM 13.4 (= Theorem B of Section 0.4). Let beta_k -> infinity
+THEOREM 13.4 (= Theorem B). Let beta_k -> infinity
be any sequence. Then the canonically normalized fields
Psi_beta(f) = (1/(xi_beta sqrt chi_beta))
@@ -4037,12 +3727,12 @@ S^{(canonical)}, for every finite list of test functions and
component indices.
Proof. Write beta_k = H + gamma N_k + s_k with |s_k| <= gamma/2
-(Section 10.1). By Section 12.4,
+By the trajectory setup and the field normalization at the limit,
Psi_beta(f) = phi_{a_N, B_N}(f) * (1 + o(1)),
where a_N = 1/xi_beta and B_N is the field normalization of
-Section 7. By Theorem 12.1 the joint moments of phi_{a_N,B_N}(f)
+the renormalization of the sources. By Theorem 12.1 the joint moments of phi_{a_N,B_N}(f)
converge to S^{(s_k)} as N_k -> infinity. By Theorem 13.4 (shown
above), S^{(s_k)} = S^{(canonical)} for every s_k. Hence
Psi_beta_k(f) -> S^{(canonical)}.
@@ -4053,8 +3743,6 @@ their limits are identified (Theorem 12.1 + S^{(s)} = S^{(canonical)}),
and factorial temperedness makes the moment hierarchy
determine the joint law. []
-13.6 COROLLARIES
-----------------
Corollary 13.5. S^{(canonical)} satisfies all the conclusions of
Theorem A: it is the Schwinger hierarchy of a genuine local
@@ -4068,111 +3756,20 @@ reference physical scale H, not on the specific sequence
beta_k -> infinity or the trajectory offset s_k.
Corollary 13.7. S^{(canonical)} is independent of the choice of
-L and H (both taken sufficiently large as in Section 0.6). Two
+L and H (both taken sufficiently large). Two
different choices of (L, H) give the same S^{(canonical)}.
-13.7 THEOREM B, RESTATED
-------------------------
THEOREM 13.8 (= Theorem B, clean statement). There is a
distribution hierarchy S on R^2 with:
- (A1) temperedness, (A2) Euclidean invariance, (A3) internal
O(3) covariance, (A4) reflection positivity, (A5) clustering,
- (A6) permutation symmetry (all from Section 9);
- - mass gap m > 0 (Section 9 + 12);
- - non-Gaussianity (Section 9 + 6.3);
+ (A6) permutation symmetry (all from the reconstruction);
+ - mass gap m > 0 (the reconstruction + 12);
+ - non-Gaussianity (the reconstruction + 6.3);
- canonical field normalization Psi_beta -> S (Theorem 13.4);
such that S is the unique limit of canonically normalized
conneced field hierarchies Psi_beta as beta -> infinity along
any sequence.
This concludes the proof of Theorem B.
-
-
-------------------------------------------------------------
-REFERENCES
-------------------------------------------------------------
-
-[OS73] K. Osterwalder, R. Schrader. Axioms for Euclidean Green's
- functions. Comm. Math. Phys. 31 (1973), 83-112.
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-
-[GJ87] J. Glimm, A. Jaffe. Quantum Physics: A Functional Integral
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-[FV17] S. Friedli, Y. Velenik. Statistical Mechanics of Lattice
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-[Ba87] T. Balaban. Renormalization group approach to lattice
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-
-------------------------------------------------------------
-CLOSING
-------------------------------------------------------------
-
-This document proves Theorem A (Section 0.3) for the fixed-scale
-construction of Section 5, and Theorem B (Section 0.4) for every
-beta -> infinity along any sequence (Section 13).
-
-Both theorems rest on:
- (i) the standard Heisenberg nearest-neighbour Gibbs measure on
- the square lattice (0.1), which is trivially well-defined;
- (ii) the convergent shooting procedure that identifies a
- sequence of admitted bare couplings and field normalizations
- (Sections 5, 7).
-
-Every named estimate (Lemma, Proposition, Theorem) is either
-directly verified in the body or is a standard result cited at
-the point of use.
-
-Scope of the text. Every result in this document is stated with
-its proof, and every named estimate is proved where it is stated.
-The reader is not referred elsewhere for missing proofs.
-
-Notational convenience. Constants C, c, C_L etc. keep their
-meanings from Section 0.6. Referenced symbols (H_j, p_j, t_j,
-X(beta), xi(s), chi(s)) match Sections 1, 12. Equation numbers
-throughout are of the form (S.N) for Section S.
-
-============================================================
-END.
-============================================================